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--- author: - | Benjamin Kollenda, Philipp Koppe, Marc Fyrbiak\ Christian Kison, Christof Paar, Thorsten Holz bibliography: - 'bibliography.bib' title: | An Exploratory Analysis of Microcode\ as a Building Block for System Defenses --- <ccs2012> <concept> <concept\_id>10002978.10003006</concept\_id> <concept\_desc>Security and privacy Systems security</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10002978.10003022</concept\_id> <concept\_desc>Security and privacy Software and application security</concept\_desc> <concept\_significance>300</concept\_significance> </concept> </ccs2012>

--- abstract: 'We propose an unsupervised multi-conditional image generation pipeline: cFineGAN, that can generate an image conditioned on two input images such that the generated image preserves the texture of one and the shape of the other input. To achieve this goal, we extend upon the recently proposed work of FineGAN [@singh2018finegan] and make use of standard as well as shape-biased pre-trained ImageNet models. We demonstrate both qualitatively as well as quantitatively the benefit of using the shape-biased network. We present our image generation result across three benchmark datasets- CUB-200-2011[@welinder2010caltech], Stanford Dogs[@khosla2011novel] and UT Zappos50k[@finegrained].' author: - | Gunjan Aggarwal[^1]\ Adobe Inc, Noida, India\ `[email protected]`\ Abhishek Sinha$^{*}$[^2]\ Stanford University\ `[email protected]` bibliography: - 'egbib.bib' title: 'cFineGAN: Unsupervised multi-conditional fine-grained image generation' --- Introduction ============ Recent developments in deep learning and generative adversarial networks(GAN) have made it possible to generate realistic looking images of high resolution. The image generation techniques generally come in two forms : i) Unconditional image generation – starting from a noise vector, the generator generates an image [@goodfellow2014generative]. ii) Conditional image generation – given a condition, the aim is to generate an image adhering to some condition [@mirza2014conditional; @isola2017image; @zhu2017unpaired]. While a lot of work has been done in the domain of single-conditional image synthesis, the domain of unsupervised multi-conditional image synthesis is relatively new. We aim to generate an image conditioned on two inputs such that the generated image contains texture of the first and shape of the second conditioned image. Approach ======== Our work is based upon the recently released work FineGAN. The authors propose a GAN based framework that learns to disentangle the background, shape and texture of an image in an unsupervised manner. The network generates an image conditioned on input background, shape and texture codes. Our pipeline takes in two images $I_1$ and $I_2$ as input and generates an output image(O). The pipeline consists of three steps - i) Compute the texture code(T) that describes the first input image($I_1$), ii) Compute the shape code(S) that describes the second input image($I_2$), and iii) Feed the computed codes(T and S) as input to the pre-trained FineGAN network to get the desired output O. To compute the codes(T and S), we take a trained FineGAN and iterate over all the possible combinations of shape and texture codes for 10 different noise vectors(different noise vector lead to different orientations of the generated image). We denote as G the set of all such generated images. To compute the texture code(T), we compute the nearest neighbour of $I_1$ amongst G in the embedding space of ImageNet [@russakovsky2015imagenet] pre-trained ResNet50 model [@He_2016_CVPR]. The embedding space is defined by the Global Average Pooling layer output of the ResNet50 model. We repeat the same process for image $I_2$ to compute the shape code(S) with the exception of using a shape biased pre-trained ResNet50 network. The motivation for using a shape biased network stems from [@geirhos2018imagenet], where the authors show that the ImageNet trained models are biased towards texture details of image. The authors use stylized variants of the ImageNet dataset to train the network resulting in a shape-biased network. We hypothesize that the shape biasness of the network would allow it to better capture the shape details of the image, leading to correct identification of the shape code of an input image. We verify both quantitatively and qualitatively this design choice in the following section. Results and Discussions ======================= \[fig:results\] Our cFineGAN results over the three datasets - CUB-200-2011 [@welinder2010caltech], UT Zappos50k [^3] [@finegrained] and Stanford Dogs [@khosla2011novel] are shown in figure \[fig:results\]. Additional results can be found in the Appendix section. -- ------- 70.75 86.90 -- ------- \[tab:shape-acc\] \[fig:figure1\] \[fig:figure2\] To quantitatively evaluate the benefit of using a shape-biased pre-trained model for extracting the shape code, we compute the nearest neighbour in the embedding space for each generated image in G. We define accuracy as the fraction of times the query image and its nearest neighbour have the same shape code. As the shape code of all the generated images is known, we can compute this metric. Table \[tab:shape-acc\] shows that the accuracy achieved by the shape-biased model is much better than that of a standard model. Some qualitative results have been shown in figure \[fig:figure1\]. We baseline our method against the approach mentioned in [@singh2018finegan] where the authors train classifiers over the domain of generated images to predict the shape and texture codes given image as an input. Since the classifier is trained over the domain of generated images but is expected to predict the codes of natural images during evaluation time, the huge domain shift encountered between train and test settings lead to incorrect outputs. We show some qualitative comparisons against this baseline in figure \[fig:figure2\]. As can be seen cFineGAN better captures the shape and texture details of input images. Appendix ======== Additional Results ------------------ We show additional results over the three datasets in Figures \[fig:results\_add\_cub\], \[fig:results\_add\_dogs\] and \[fig:results\_adds\_shoes\]. ![Additional results over the CUB-200-2011 dataset.[]{data-label="fig:results_add_cub"}](blah_bird.png){width="13.5cm"} ![Additional results over the Stanford dogs dataset.[]{data-label="fig:results_add_dogs"}](new_additional_vert_dogs.jpg){width="13.5cm"} ![Additional results over the UT Zappos50k dataset.[]{data-label="fig:results_adds_shoes"}](blah_shoes.png){width="13.5cm"} [^1]: Authors contributed equally [^2]: work done while author was working at Adobe, India [^3]: We trained our own FineGAN model over this dataset

--- abstract: 'Wasserstein distance plays increasingly important roles in machine learning, stochastic programming and image processing. Major efforts have been under way to address its high computational complexity, some leading to approximate or regularized variations such as Sinkhorn distance. However, as we will demonstrate, regularized variations with large regularization parameter will degradate the performance in several important machine learning applications, and small regularization parameter will fail due to numerical stability issues with existing algorithms. We address this challenge by developing an Inexact Proximal point method for exact Optimal Transport problem (IPOT) with the proximal operator approximately evaluated at each iteration using projections to the probability simplex. The algorithm (a) converges to exact Wasserstein distance with theoretical guarantee and robust regularization parameter selection, (b) alleviates numerical stability issue, (c) has similar computational complexity to Sinkhorn, and (d) avoids the shrinking problem when apply to generative models. Furthermore, a new algorithm is proposed based on IPOT to obtain sharper Wasserstein barycenter.' author: - 'Yujia Xie, Xiangfeng Wang, Ruijia Wang, Hongyuan Zha [^1]' bibliography: - 'example\_paper.bib' title: A Fast Proximal Point Method for Computing Exact Wasserstein Distance --- INTRODUCTION {#section1} ============ Many practical tasks in machine learning rely on computing a Wasserstein distance between probability measures or between their sample points [@arjovsky2017wasserstein; @thorpe2017transportation; @ye2017fast; @srivastava2015wasp]. However, the high computational cost of Wasserstein distance has been a thorny issue and has limited its application to challenging machine learning problems. In this paper we focus on Wasserstein distance for discrete distributions the computation of which amounts to solving the following discrete *optimal transport* (OT) problem, $$\begin{aligned} \label{eq:target1} \begin{aligned} W(\bm{\mu},\bm{\nu}) = \sideset{}{_{\bm{\Gamma}\in\Sigma(\bm{\mu},\bm{\nu})}}\min \langle\bm{C},\bm{\Gamma}\rangle. \end{aligned}\end{aligned}$$ Here $\bm{\mu},\bm{\nu}$ are two probability vectors, $W(\bm{\mu},\bm{\nu})$ is the Wasserstein distance between $\bm{\mu}$ and $\bm{\nu}$. Matrix $\bm{C}=[c_{ij}]\in\mathbb{R}_{+}^{m\times n}$ is the *cost matrix*, whose element $c_{ij}$ represents the distance between the $i$-th support point of $\bm{\mu}$ and the $j$-th one of $\bm{\nu}$. The optimal solution $\bm{\Gamma}^*$ is referred as *optimal transport plan*. Notation $\langle\cdot,\cdot\rangle$ represents the Frobenius dot-product and $\Sigma(\bm{\mu},\bm{\nu})=\{\bm{\Gamma}\in\mathbb{R}_{+}^{m\times n}: \bm{\Gamma}\bm{1}_m=\bm{\mu}, \bm{\Gamma}^{\top}\bm{1}_n=\bm{\nu}\}$, where $\bm{1}_n$ represents $n$-dimensional vector of ones. This is a linear programming problem with typical super $O(n^3)$ complexity[^2]. An effort by Cuturi to reduce the complexity leads to a regularized variation of (\[eq:target1\]) giving rise the so-called Sinkhorn distance [@cuturi2013sinkhorn]. It aims to solve an entropy regularized optimal transport problem $$\begin{aligned} \label{eq:sinkhorn} \begin{aligned} W_{\epsilon}(\bm{\mu},\bm{\nu}) = \sideset{}{_{\bm{\Gamma}\in\Sigma(\bm{\mu},\bm{\nu})}}\min \langle\bm{C},\bm{\Gamma}\rangle+\epsilon h(\bm{\Gamma}). \end{aligned}\end{aligned}$$ The entropic regularizer $h(\bm{\Gamma})=\sum_{i,j}\Gamma_{ij}\ln\Gamma_{ij}$ results in an optimization problem (\[eq:sinkhorn\]) that can be solved efficiently by iterative Bregman projections [@benamou2015iterative], $$\bm{a}^{(l+1)} = \frac{\pmb{\mu}}{\bm{Gb}^{(l)}}, \quad \bm{b}^{(l+1)} = \frac{\pmb{\nu}}{\bm{G}^T \bm{a}^{(l+1)}}$$ starting from $b^{(0)} = \frac{1}{n}\pmb{1}_n$, where $\bm{G}=[G_{ij}]$ and $ G_{ij} = e^{-C_{ij}/\epsilon}$. The optimal solution $\bm{\Gamma}^*$ then takes the form $\Gamma^*_{ij} = a_i G_{ij} b_j$. The iteration is also referred as *Sinkhorn iteration*, and the method is referred as *Sinkhorn algorithm* which, recently, is proven to achieve a near-$O(n^2)$ complexity  [@altschuler2017near]. The choice of $\epsilon$ cannot be arbitrarily small. Firstly, $ G_{ij} = e^{-C_{ij}/\epsilon}$ tends to underflow if $\epsilon$ is very small. The methods in @benamou2015iterative [@chizat2016scaling; @mandad2017variance] try to address this numerical instability by performing the computation in log-space, but they require a significant amount of extra exponential and logarithmic operations, and thus, compromise the advantage of efficiency. More significantly, even with the benefits of log-space computation, the linear convergence rate of the Sinkhorn algorithm is determined by the *contraction ratio* $\kappa(\bm{G})$, which approaches $1$ as $\epsilon \to 0$ [@franklin1989scaling]. Consequently, we observe drastically increased number of iterations for Sinkhorn method when using small $\epsilon$. Can we just employ Sinkhorn distance with a moderately sized $\epsilon$ for machine learning problems so that we can get the benefits of the reduced complexity? Some applications show Sinkhorn distance can generate good results with a moderately sized $\epsilon$ [@genevay2017sinkhorn; @martinez2016relaxed]. However, we show that in several important problems such as generative model learning and Wasserstein barycenter computation, a moderately sized $\epsilon$ will significantly degrade the performance while the Sinkhorn algorithm with a very small $\epsilon$ becomes prohibitively expensive (also shown in @solomon2014wasserstein). In this paper, we propose a new framework, Inexact Proximal point method for Optimal Transport (IPOT) to compute the Wasserstein distance using generalized proximal point iterations based on Bregman divergence. To enhance efficiency, the proximal operator is inexactly evaluated at each iteration using projections to the probability simplex, leading to an **inexact** update at each iteration yet converging to the **exact** optimal transport solution. Regarding the theoretical analysis of IPOT, we provide conditions on the number of inner iterations that will guarantee the linear convergence of IPOT. In fact, empirically, IPOT behaves better than the analysis: the algorithm seems to be linearly convergent with just one inner iteration, demonstrating its efficiency. We also perform several other tests to show the excellent performance of IPOT. As we will discussed in Section \[section42\], the computation complexity is almost indistinguishable comparing to the Sinkhorn method. Yet again, IPOT avoids the lengthy and experience-based tuning of the $\epsilon$ and can converges to the true optimal transport solution robustly with respect to its own parameters. This is unquestionably important in applications where the exact sparse transport plan is preferred. In applications where only Wasserstein distance is needed, the bias caused by regularization might also be problematic. As an example, when applying Sinkhorn to generative model learning, it causes the shrinkage of the learned distribution towards the mean, and therefore cannot cover the whole support of the target distribution adequately. Furthermore, we develop another new algorithm based on the proposed IPOT to compute Wasserstein barycenter (see Section \[section5\]). Better performance is obtained with much sharper images. It turns out that the inexact evaluation of the proximal operator blends well with Sinkhorn-like barycenter iteration. PRELIMINARIES {#section2} ============= We then provide some background on optimal transport and proximal point method. Wasserstein Distance and Optimal Transport {#section21} ------------------------------------------ Wasserstein distance is a metric for two probability distributions. Given two distributions $\mu$ and $\nu$, the $p$-Wasserstein distance between them is defined as $$\begin{aligned} \label{eq:wd_c} \begin{aligned} W_p(\mu,\nu):=\Big\{\inf_{\gamma\in\Sigma(\mu,\nu)}\int_{\mathcal{M}\times\mathcal{M}}d^p(x,y)\text{d}\gamma(x,y)\Big\}^{\frac{1}{p}}, \end{aligned}\end{aligned}$$ where $\Sigma(\mu,\nu)$ is the set of joint distributions whose marginals are $\mu$ and $\nu$, respectively. The above optimization problem is also called the Monge-Kantorovitch problem or *optimal transport* problem [@kantorovich1942mass]. In the following, we focus on the $2$-Wasserstein distance, and for convenience we write $W(\cdot,\cdot)=W_2^2(\cdot,\cdot)$. When $\mu$ and $\nu$ both have finite supports, we can represent the distributions as vectors $\bm{\mu}\in\mathbb{R}_{+}^{m},\bm{\nu}\in\mathbb{R}_{+}^{n}$, where $\|\bm{\mu}\|_1=\|\bm{\nu}\|_1=1$. Then the Wasserstein distance between $\bm{\mu}$ and $\bm{\nu}$ is computed by (\[eq:target1\]). In other cases, given realizations $\{x_i\}_{i=1}^{m}$ and $\{y_j\}_{j=1}^{n}$ of $\mu$ and $\nu$, respectively, we can approximate them by empirical distributions $\hat{\mu}=\frac{1}{m}\sum_{x_i}\delta_{x_i}$ and $\hat{\nu}=\frac{1}{n}\sum_{y_j}\delta_{y_j}$. The supports of $\hat{\mu}$ and $\hat{\nu}$ are finite, so similarly we have $\bm{\mu}=\frac{1}{m}\bm{1}_{\{x_i\}}$, $\bm{\nu}=\frac{1}{n}\bm{1}_{\{y_j\}}$, and $\bm{C}=[c(x_i, y_j)]\in \mathbb{R}_{+}^{m\times n}$. The optimization problem (\[eq:target1\]) can be solved by linear programming (LP) methods. LP tends to provide a sparse solution, which is preferable in applications like histogram calibration or color transferring [@rabin2014adaptive]. However, the cost of LP scales at least $O(n^3 \log n)$ for general metric, where $n$ is the number of data points [@pele2009fast]. As aforementioned, an alternative optimization method is the Sinkhorn algorithm in [@cuturi2013sinkhorn]. Following the same strategy, many variants of the Sinkhorn algorithm have been proposed [@altschuler2017near; @dvurechensky2017adaptive; @thibault2017overrelaxed]. Unfortunately, all these methods only approximate original optimal transport by its regularized version and their performance both in terms of numerical stability and computational complexity is sensitive to the choice of $\epsilon$. {width="97.00000%"} Generalized Proximal Point Method --------------------------------- Proximal point methods are widely used in optimization [@afriat1971theory; @parikh2014proximal; @rockafellar1976augmented; @rockafellar1976monotone]. Here, we introduce its generalized form. Given a convex objective function $f$ defined on $\mathcal{X}$ with optimal solution set $\mathcal{X}^*\subset \mathcal{X}$, generalized proximal point algorithm aims to solve $$\begin{aligned} \label{eq:minimum} \begin{aligned} \sideset{}{_{x\in \mathcal{X}}}\argmin f(x). \end{aligned}\end{aligned}$$ In order to solve Problem (\[eq:minimum\]), the algorithm generates a sequence $\{x^{(t)}\}_{t=1,2,...}$ by the following generalized proximal point iterations: $$\begin{aligned} \label{eq:proximal} \begin{aligned} x^{(t+1)} = \sideset{}{_{x\in \mathcal{X}}}\argmin f(x) + \beta^{(t)} d(x, x^{(t)}), \end{aligned}\end{aligned}$$ where $d$ is a regularization term used to define the proximal operator, usually defined to be a closed proper convex function. For classical proximal point method, $d$ adopts the square of Euclidean distance, i.e., $d(x,y) =\|x-y\|_2^2$, in which case the sequence $\{x^{(t)}\}$ converges to an element in $\mathcal{X}^*$ almost surely. The generalized proximal point method has many advantages, e.g, it has a robust convergence behavior – a fairly mild condition on $\beta^{(t)}$ guarantee its convergence for some given $d$, and the specific choice of $\beta^{(t)}$ generally just affects its convergence rate. Moreover, even if the proximal operator defined in (\[eq:proximal\]) is not exactly evaluated in each iteration, giving rise to inexact proximal point methods, the global convergence of which with local linear rate is still guaranteed under certain conditions [@solodov2001unified; @schmidt2011convergence]. BREGMAN DIVERGENCE BASED PROXIMAL POINT METHOD ============================================== \[section3\] In this section we will develop the main algorithm IPOT. Specifically, we will use generalized proximal point method to solve the optimal transport problem . Recall the proximal point iteration , we take $f(\bm{\Gamma})=\langle \bm{C}, \bm{\Gamma} \rangle$, $\mathcal{X}=\Sigma(\bm{\mu},\bm{\nu})$, and $d(\cdot,\cdot)$ to be Bregman divergence $D_h$ based on entropy function $h(\bm{x})=\sum_i x_i \ln x_i$, i.e., $$\begin{aligned} \label{bregman_divergence} \begin{aligned} D_h (\bm{x},\bm{y}) = \sum_{i=1}^n x_i \log \frac{x_i}{y_i} - \sum_{i=1}^n x_i + \sum_{i=1}^n y_i. \end{aligned}\end{aligned}$$ As a result, the proximal point iteration for problem can be written as $$\begin{aligned} \label{proximal2} \bm{\Gamma}^{(t+1)} = \sideset{}{_{\bm{\Gamma}\in\Sigma(\bm{\mu},\bm{\nu})}}\argmin\langle \bm{C}, \bm{\Gamma} \rangle + \beta^{(t)} D_h(\bm{\Gamma},\bm{\Gamma}^{(t)}).\end{aligned}$$ However, it is still not trivial to solve the above optimization problem in each iteration, since optimization problems with such complicated constraints generally does not have a closed-form solution. Fortunately, with some reorganization, we can solve it with Sinkhorn algorithm. Substituting Bregman divergence into proximal point iteration (\[proximal2\]), with simplex constraints, we obtain $$\begin{aligned} \label{proximal3} \begin{aligned} \bm{\Gamma}^{(t+1)} = \sideset{}{_{\bm{\Gamma}\in\Sigma(\bm{\mu},\bm{\nu})}}\argmin \langle \bm{C}-\beta^{(t)} \log \bm{\Gamma}^{(t)}, \bm{\Gamma} \rangle + \beta^{(t)} h(\bm{\Gamma}). \end{aligned}\end{aligned}$$ Denote $\bm{C}'=\bm{C}-\beta^{(t)} \ln\bm{\Gamma}^{(t)}$. Note that for optimization problem , $\bm{\Gamma}^{(t)}$ is a fixed value that is not relevant to optimization variable $\bm{\Gamma}$. Therefore, $\bm{C}'$ can be viewed as a new cost matrix that is known, and problem is an entropy regularized optimal transport problem. Comparing to , problem can be solved by Sinkhorn iteration by replacing $G_{ij}$ by $G'_{ij} = e^{-C'_{ij}/\beta^{(t)}} = \Gamma_{ij}^{(t)} e^{-C_{ij}/\beta^{(t)}}$. As we will later shown in Section \[section32\], as $t\to \infty$, $\bm{\Gamma}^{(t)}$ will converge to an optimal transport plan. **Input:** Probabilities $\{\pmb{\mu},\pmb{\nu}\}$ on support points $\{x_i\}_{i=1}^m$, $\{y_j\}_{j=1}^n$, cost matrix $\bm{C}=[\|x_i-y_j\|]$ $\bm{b}\leftarrow \frac{1}{m}\pmb{1}_m$ $G_{ij}\leftarrow e^{-\frac{C_{ij}}{\beta}}$ $\bm{\Gamma}^{(1)} \leftarrow \pmb{11}^T$ $\bm{Q} \leftarrow \bm{G}\odot \bm{\Gamma}^{(t)}$ **for** [$l=1,2,3,...,L$]{} **do**$\quad$ // Usually set $L=1$ $\bm{a}\leftarrow \frac{\pmb{\mu}}{\bm{Qb}}$, $\bm{b}\leftarrow \frac{\pmb{\nu}}{\bm{Q}^T \bm{a}}$ **end for** $\bm{\Gamma}^{(t+1)} \leftarrow \text{diag}(\bm{a}) \bm{Q} \text{diag}(\bm{b})$ Figure \[fig:sketch\] illustrates how Sinkhorn and IPOT solutions approach optimal solution with respect to number of iterations in sense of Bregman divergence. First, let’s consider Sinkhorn algorithm. The objective function of Sinkhorn has regularization term $\epsilon h(\bm{\Gamma})$, which can be equivalently rewritten as constraint $D_h(\bm{\Gamma},\bm{11}^T)\leq \eta$ for some $\eta>0$. Therefore, Sinkhorn solution is feasible within the $D_h$ constraints and the closest to optimal solution set, as shown in Figure \[fig:sketch\] (a). Proximal point algorithms, on the other hand, solves optimization with $D_h$ constraints iteratively as shown in Figure \[fig:sketch\] (b)(c). Different from Sinkhorn algorithm, proximal point algorithms converge to the optimal solution with nested iterative loops. Exact proximal point method, i.e., solving (\[proximal3\]) exactly as shown in Figure \[fig:sketch\] (b), provides a feasible solution that is closest to the optimal solution set in each proximal iteration until the optimal solution reached. However, the disadvantage for exact proximal point method is that it’s not efficient. The proposed inexact proximal point method (IPOT) does not solve (\[proximal3\]) exactly. Instead, a very small amount of Sinkhorn iteration, e.g., only one iteration, is suggested. The reason for this is three-fold. First, the convergence of Sinkhorn algorithm in each proximal iteration is not required, since it is just intermediate step. Second, usually in numerical optimization, the first a few iterations achieve the most decreasing in the objective function. Performing only the first a few iterations has high cost performance. Last and perhaps the most important, it is observed that IPOT can still converge to an exact solution with small amount of inner iterations[^3]. The algorithm is shown in Algorithm \[algo1\]. For simplicity we use $\beta = \beta^{(t)}$. Denote $\text{diag}(\bm{a})$ the diagonal matrix with $a_i$ as its $i$-th diagonal elements. Denote $\odot$ as element-wise matrix multiplication and $\frac{(\cdot)}{(\cdot)}$ as element-wise division. We use warm start to improve the efficiency, i.e. in each proximal point iteration, we use the final value of $\bm{a}$ and $\bm{b}$ from last proximal point iteration as initialization instead of $\bm{b}^{(0)} = \pmb{1}_m$. Later we will show empirically IPOT will converge under a large range of $\beta$ with $L=1$, a single inner iteration will suffice. WASSERSTEIN BARYCENTER BY IPOT {#section5} ============================== We now extend IPOT method to a related problem – computing the Wasserstein barycenter. Wasserstein barycenter is widely used in machine learning and computer vision [@benamou2015iterative; @rabin2011wasserstein]. Given a set of distributions $\mathcal{P} = \{\bm{p}_1,\bm{p}_2,...,\bm{p}_K\}$, their Wasserstein barycenter is defined as $$\begin{aligned} \label{big_target} \begin{aligned} \bm{q}^*(\mathcal{P},\bm{\lambda}) = \sideset{}{_{\bm{q}\in\mathcal{Q}}}\argmin \sideset{}{_{k=1}^K}\sum \lambda_k W(\bm{q},\bm{p}_k) \end{aligned}\end{aligned}$$ where $\mathcal{Q}$ is in the space of probability distributions, $\sum_{k=1}^K \lambda_k = 1$, and $W(\bm{q},\bm{p}_k)$ is the Wasserstein distance between the barycenter $\bm{q}$ and distribution $\bm{p}_k$, which takes the form $$\label{eq:sampleW} W(\bm{q},\bm{p}_k) = \min_{\bm{\Gamma}} {\langle \bm{C}, \bm{\Gamma} \rangle },\quad \text{s.t.}\quad \bm{\Gamma}\textbf{1}=\bm{p}_k, \bm{\Gamma}^T\textbf{1}=\bm{q}.$$ The idea of IPOT method can also be used to compute Wasserstein barycenter. Substitute (\[eq:sampleW\]) into and reorganize, we have $$\begin{aligned} \bm{q}^*(\mathcal{P},\bm{\lambda}) = \sideset{}{_{\bm{q}\in\mathcal{Q}}}\argmin \sum_{k=1}^K \lambda_k {\langle \bm{C},\bm{\Gamma}_k \rangle }, \quad \text{s.t.} \quad \bm{\Gamma}_k\textbf{1}=\bm{p}_k,\text{ and } \exists \bm{q},\bm{\Gamma}_k^T\textbf{1}=\bm{q}.\end{aligned}$$ Analogous to IPOT, we take $$\begin{aligned} f(\{\bm{\Gamma}_k\})=\sum_{k=1}^K \lambda_k {\langle \bm{C}, \bm{\Gamma}_k \rangle },\end{aligned}$$ take $\mathcal{X}$ to be the corresponding constraints, and take $d(\{\bm{\Gamma}_k\},\{\bm{\Gamma}_k^{(t)}\})$ to be $\sum_{k=1}^K \lambda_k D_h(\bm{\Gamma}_k,\bm{\Gamma}_k^{(t)})$. The proximal point iteration for barycenter is $$\begin{aligned} \bm{\Gamma}_k^{(t+1)} = \argmin_{\bm{\Gamma}_k} \sum_{k=1}^K \lambda_k \left( \langle \bm{C},\bm{\Gamma}_k \rangle +\beta^{(t)} D_h(\bm{\Gamma}_k,\bm{\Gamma}_k^{(t)})\right) \quad \text{s.t.}\quad \bm{\Gamma}_k\textbf{1}=\bm{p}_k, \text{ and } \exists \bm{q},\bm{\Gamma}_k^T\textbf{1}=\bm{q}. \label{eq:bw_iter1}\end{aligned}$$ With further organization, we have $$\begin{aligned} \bm{\Gamma}_k^{(t+1)} = \argmin_{\bm{\Gamma}_k} \sum_{k=1}^K \lambda_k \left( \langle \bm{C}-\beta^{(t)} \log \bm{\Gamma}_k^{(t)},\bm{\Gamma}_k \rangle +\beta^{(t)} h(\bm{\Gamma}_k)\right) \quad \text{s.t.}\quad \bm{\Gamma}_k\textbf{1}=\bm{p}_k, \text{ and } \exists \bm{q},\bm{\Gamma}_k^T\textbf{1}=\bm{q}. \label{eq:bw_iter}\end{aligned}$$ On the other hand, analogous to Sinkhorn algorithm, @benamou2015iterative propose *Bregman iterative projection* that seeks to solve an entropy regularized barycenter, $$\begin{aligned} \label{eq:sink-wb} \begin{aligned} \bm{q}_{\epsilon}^*(\mathcal{P},\bm{\lambda}) = \sideset{}{_{\bm{q}\in\mathcal{Q}}}\argmin \sideset{}{_{k=1}^K}\sum \lambda_k W_{\epsilon}(\bm{q},\bm{p}_k). \end{aligned}\end{aligned}$$ Comparing and , the minimization in each proximal point iteration in can be solved by Bregman iterative projection [@benamou2015iterative] using the same change-of-variable technique in Section \[section3\]. \[alg:barycenter\] \[algo\_proximal\] **Input:** The probability vector set $\{\bm{p}_k\}$ on grid $\{y_i\}_{i=1}^n$ $\bm{b}_k\leftarrow \frac{1}{n}\textbf{1}_n,\forall k=1,2,...,K$ $C_{ij}\leftarrow c(y_i,y_j):=||y_i-y_j||^2_2$ $G_{ij}\leftarrow e^{-\frac{C_{ij}}{\beta}}$ $\bm{\Gamma}_k \leftarrow \mathbf{11}^T$ $\bm{H}_k \leftarrow \bm{G} \odot \bm{\Gamma}_k,\forall k=1,2,...,K$ $\bm{a}_k\leftarrow \frac{\bm{q}}{\bm{H}_k \bm{b}_k},\forall k=1,2,...,K$, $\bm{b}_k\leftarrow \frac{\bm{p}_k}{\bm{H}_k^T \bm{a}_k},\forall k=1,2,...,K$ $\bm{q} \leftarrow \prod_{k=1}^K (\bm{a}_k \odot (\bm{H}_k \bm{b}_k))^{\lambda_k}$ $\bm{\Gamma}_k \leftarrow \text{diag}(\bm{a}_k) \bm{H}_k \text{diag}(\bm{b}_k),\forall k=1,2,...,K$ **Return** $\bm{q}$ We provide the detailed algorithm in Algorithm \[algo\_proximal\], and name this algorithm *IPOT-WB*. Same as Algorithms \[algo1\], IPOT-WB algorithm can converge with $L=1$ and a large range of $\beta$. Since the sketch in Figure \[fig:sketch\] does not have restrictions on $f$ and $\mathcal{X}$, the sketch and the corresponding analysis for IPOT also applies to IPOT-WB, except the distance is in sense of convex combination of Bregman divergences instead of a single Bregman divergence. THEORETICAL ANALYSIS {#section32} ==================== Classical proximal point algorithm has sublinear convergence rate. However, after we replace the square of Euclidean distance in classical proximal point algorithm by Bregman distance, we can prove stronger convergence rate – a linear rate for both IPOT and IPOT-WB. First, we consider when the optimization problem (\[proximal3\]) is solved exactly, we have a linear convergence rate guaranteed by the following theorem. Let $\{x^{(t)}\}$ be a sequence generated by the proximal point algorithm $$\begin{aligned} \begin{aligned} x^{(t+1)} = &\sideset{}{_{x\in \mathcal{X}}}\argmin f(x) + \beta^{(t)} D_h(x,x^{(t)}), \end{aligned}\end{aligned}$$ where $f$ is continuous and convex. Assume $f^* = \min f(x) > -\infty$. Then, with $\sum_{t=0}^{\infty} \beta^{(t)}=\infty$, we have $$\label{mono-converge} f(x^{(t)})\downarrow f^*.$$ If we further assume $f$ is linear and $\mathcal{X}$ is bounded, the algorithm has linear convergence rate. More importantly, the following theorem gives us a guarantee of convergence when (\[proximal3\]) is solved inexactly. \[thom2\] Let $\{ x^{(t)} \}$ be the sequence generated by the Bregman distance based proximal point algorithm with inexact scheme (i.e., finite number of inner iterations are employed). Define an error sequence $\{ e^{(t)} \}$ where $$\begin{aligned} \begin{aligned} e^{(t+1)} \in \beta^{(t)} \left[ \nabla f(x^{(t+1)} ) + \partial \iota_{\mathcal{X}} (x^{(t+1)} ) \right] + \left[ \nabla h(x^{(t+1)} ) - \nabla h(x^{(t)} ) \right], \end{aligned}\end{aligned}$$ where $\iota_{\mathcal{X}}$ is the indicator function of set $\mathcal{X}$, and $\partial_{l_X} (\cdot)$ is the subdifferential of the indicator function $l_{X}$. If the sequence $\{ e^{k} \}$ satisfies $\sum_{k=1}^{\infty} \| e^{k} \| < \infty$ and $\sum_{k=1}^{\infty} \langle e^{k}, x^{(t)} \rangle$ exists and is finite, then $\{ x^{(t)} \}$ converges to $x^{\infty}$ with $f(x^{\infty}) = f^*$. If the sequence $\{ e^{(t)} \}$ satisfies that exist $\rho\in (0,1)$ such that$ \| e^{(t)} \| \le \rho^t$, $ \langle e^{(t)}, x^{(t)} \rangle \le \rho^t$ and with assumptions that $f$ is linear and $\mathcal{X}$ is bounded, then $\{ x^{(t)} \}$ converges linearly. The proofs of both theorems are given in the supplementary material. Theorem \[thom2\] guarantees the convergence of inexact proximal point method — as long as the inner iteration number $L$ satisfies the given conditions, IPOT and IPOT-WB algorithm would converge linearly. Note that although Theorem \[thom2\] manages to prove the linear convergence in inexact case, in practice the conditions is not trivial to verify. In practice we usually just adopt $L=1$. Now we know IPOT and IPOT-WB can converge to the exact Wasserstein distance and Wasserstein barycenter. What if an entropic regularization is wanted? Please refer to the supplementary material for how IPOT can achieve regularizations with early stopping. EMPIRICAL ANALYSIS {#section4} ================== In this section we will illustrate the convergence behavior with respect to inner iteration number $L$ and parameter $\beta$, the scalability of IPOT, and the issue with entropy regularization. We leverage the implementation of Sinkhorn iteration and LP solver based on Python package POT [@flamary2017pot], and use Pytorch to parallel some of the implementation. Convergence Rate {#section41} ---------------- A simple illustration task of calculating the Wasserstein distance of two 1D distribution is conducted as numerical validation of the convergence theorems proved in Section \[section32\]. The two input margins are mixtures of Gaussian distributions shown in the figure in the right lower of Figure \[fig:convergence\] (a): the red one is $0.4\phi(\cdot|60,8)+0.6\phi(\cdot|40,6)$, and the blue one is $0.5\phi(\cdot|35,9)+0.5\phi(\cdot|70,9)$, where $\phi(\cdot|\mu,\sigma^2)$ is the probability density function of 1 dimensional Gaussian distribution with mean $\mu$ and variance $\sigma^2$. Input vectors $\bm{\mu}$ and $\bm{\nu}$ is the two function values on the uniform discretization of interval $[1,100]$ with grid size 1. To be clear, the use of two 1D distribution is only for visualization purpose. We also did tests on empirical distribution of 64D Gaussian distributed data, and the result shows the same trend. We include more discussion in the supplementary material. Figure \[fig:convergence\] shows the convergence of IPOT under different $L$ and $\beta$. We also include the result of Sinkhorn method for comparison. IPOT algorithm has empirically linear convergence rate even under very small $L$. The convergence rate increases w.r.t. $\beta$ when $\beta$ is small, and decreases when $\beta$ is large. This is because the choice of $\beta$ is a trade-off between inner and outer convergence rates. On the one hand, a smaller $\beta$ usually lead to quicker convergence of proximal point iterations. On the other hand, the convergence of inner Sinkhorn iteration, is quicker when $\beta$ is large. Furthermore, the choice of $L$ also appears to be a trade-off. While a larger $L$ takes more resources in each step, it also achieves a better accuracy, so less proximal point iterations are needed to converge. So the choice of best $L$ is relevant to the choice of $\beta$. For large $\beta$, the inner Sinkhorn iteration can converge faster, so smaller $L$ should be used. For small $\beta$, larger $L$ should be used, which is not efficient, and also improve the risk of underflow for the inner Sinkhorn algorithm. So unless there are specific need for accuracy, we do not recommend using very small $\beta$ and large $L$. For simplicity, we use $L=1$ for later tests. [R]{}[0.5]{} {width="48.00000%"} We conduct the following scalability test to show the computation time of the proposed IPOT comparing to the state-of-art benchmarks. The optimal transport problem is conducted between the two empirical distributions of 16D uniformly distributed data (See Section \[section21\] for formulation). Besides proposed IPOT algorithm (see Algorithm \[algo1\] with $L=1$), the Sinkhorn algorithm follows @cuturi2013sinkhorn and the stabilized Sinkhorn algorithm follows @chizat2016scaling. The result of the scalability test is shown in Figure \[fig:scalability\]. The LP solver has a good performance under the current experiment settings. But LP solver is not guaranteed to have good scalability as shown here. Moreover, LP method is difficult to parallel. Readers who are interested please refer to experiments in @cuturi2013sinkhorn. Sinkhorn and IPOT can be paralleled conveniently, so we provide both CPU and GPU tests here. Under this setting, IPOT takes approximately the same resources as Sinkhorn at $\epsilon=0.01$. For smaller $\epsilon$, original Sinkhorn will underflow, and we need to use stabilized Sinkhorn. Stabilized Sinkhorn is much more expensive than IPOT, especially for large datasets and small $\epsilon$, as demonstrated by the experiment result of stabilized Sinkhorn at $\epsilon=10^{-2}$ and $10^{-4}$. Note that we also try to use the method proposed in @schmitzer2016stabilized for $\epsilon$ scaling, to help the convergence when $\epsilon \to 0$. However, although it is faster than Sinkhorn method when data size is smaller than $1024$, the time used at 1024 is already around $2\times 10^3$s. Therefore we didn’t include this method in the figure. Effect of Entropy Regularization -------------------------------- We have shown that IPOT can converge to exact Wasserstein distance with complexity comparable to Sinkhorn (see Figure \[fig:sketch\] and \[fig:scalability\]) and as we claimed in Section \[section1\] this is important in some of the learning problems. But in what cases is the exact Wasserstein distance truly needed? How will the entropy regularization term affect the result in different applications? In this section, we will discuss the exact transport plan with sparsity preference and the advantage of exact Wasserstein distance in learning the generative models. ### Sparsity of the Transport Plan {#section43} [R]{}[0.54]{} {width="50.00000%"} In applications such as histogram calibration and color transferring, an exact and sparse transport plan is wanted. In this section we conduct tests on the sparsity of the transport plan using the two distributions shown in Figure \[fig:convergence\] for both IPOT and Sinkhorn methods with different regularization coefficients. Figure \[fig:joint\_dist\] visualize the different transport plans. The red colormap is the result from Sinkhorn or IPOT method, where the black wire beneath is the result by simplex method as ground truth. To be clear, the different number of interaction of IPOT means the number of the outer iteration with still $L=1$ inner iteration. The proposed IPOT method can always converge to the sparse ground truth with enough iteration and it is very robust with respect to the parameter $\beta$, i.e., there is little visual difference with $\beta$ changing from $0.1$ to $0.001$. Furthermore, even with large $\beta=1$, the optimal plan is still sparse and acceptable. In addition, if some smoothness is wanted, IPOT method would also be able to work with early stopping. The degree of smoothness can be easily adjusted by adjusting the number of iterations if needed. On the other hand, the optimal plans obtained by Sinkhorn has two issues. If the $\epsilon$ is chosen to be large (i.e., $\epsilon=0.1$ or $0.01$), the optimal plan are blur i.e., neither exact nor sparse. In downstream applications, the non-sparse structure of transport plan make it difficult to extract the transport map from source distribution to target distribution. However if the $\epsilon$ is chosen to be small (i.e., $\epsilon=0.0001$), it needs more iterations to converge. For example, the Sinkhorn $\epsilon=0.0001$ case still cannot converge after 2000 iterations. So in Sinkhorn applications, $\epsilon$ needs to be selected carefully. This fine tuning issue can be avoid by the proposed IPOT method, since IPOT is robust to the parameter $\beta$. ![\[fig:generator2D\] The sequences of learning results using IPOT, Sinkhorn, and original WGAN. In each figure, the orange dots are samples of generated data, while the contour represents the ground truth distribution. ](generator2D_seq.png){width="95.00000%"} ### Shrinkage Problem in Generative Models {#section432} As shown in Equation (\[eq:sinkhorn\]), Sinkhorn method use entropy to penalize the optimization target and has biased evaluation of Wasserstein distance. The inaccuracy will affect the performance of the learning problem where Wasserstein metric is served as loss function. In order to better illustrate the affect of the inaccurate Wasserstein distance, we consider the task of learning generative models, specifically, Wasserstein GAN [@arjovsky2017wasserstein]. Similar to other GAN, WGAN seeks to learn a generated distrubution to approximate a target distribution, except using Wasserstein distance as the loss that measures the distance between the generated distribution and target distribution. It uses the Kantorovitch dual formulation to compute Wasserstein distance. In this section, we train a Wasserstein GAN with the dual formulation substituted by Sinkhorn and IPOT methods. Detailed derivation can be found in supplementary material. Meanwhile, the standard approach of using dual form proposed in @arjovsky2017wasserstein is also compared. Note that the purpose of this section is not to propose a new GAN but visualize how proposed IPOT can avoid the possible negative influence introduced by the inaccuracy of the entropy regularization in the Sinkhorn method. We claim that result of Sinkhorn method with moderate size $\epsilon$ tends to shrink towards the mean, so the learned distribution cannot cover all the support of target distribution. To demonstrate the reason of this trend, consider the extreme condition when $\epsilon \to \infty$, the loss function becomes $$\bm{\Gamma^*} = \argmin_{\bm{\Gamma}} h(\bm{\Gamma}) = \argmin_{\bm{\Gamma}} D_h(\bm{\Gamma},\bm{11}^T/mn).$$ So $\bm{\Gamma^*} = \bm{11}^T/mn$. If we view $\{x_i\}$ and $\{y_j:y_j=g_{\theta}(z_j)\}$ as the realizations of random variables $X$ and $Y$, the optimal Sinkhorn distance $W_{\epsilon}$ is expected to be $$\begin{aligned} \mathbb{E}_{X,Y}[W_{\epsilon}] & = \mathbb{E}_{X,Y}[\langle \bm{\Gamma^*}, \bm{C} \rangle ]\\ & = \mathbb{E}_{X,Y}[ \sideset{}{_{i,j}} \sum ||x_i-y_j||_2^2] \\ & = n^2(\text{Var}(X) + (\overbar{X}-\overbar{Y})^2 +\text{Var}(Y)) ,\end{aligned}$$ where $n$ is the data size, $\overbar{(\cdot)}$ is the mean of random variable, and $\text{Var}(\cdot)$ is the variance. At the minimum of the distance, the mean of generated data $\{y_j\}$ is the same as $\{x_i\}$, but the variance is zero. Therefore, the learned distribution would shrink asymptotically toward the data mean due to smoothing the effect of regularization. However, the proposed IPOT method is free from the above shrinking issue since the exact Wasserstein distance can be found with the approximately the same cost (see Section \[section41\] and Section \[section42\]). Now we illustrate the shrinkage problem by the following experiments. ***Experiments on 2D Synthetic Data*** First, we conduct a 2D toy example to demonstrate the affect of regularization. We use a 2D-2D NN as generator to learn a mapping from uniformly distributed noise to mixture of Gaussian distributed real data. Since as shown in Figure \[fig:convergence\], Sinkhorn may need more iteration to converge, in this experiment, IPOT uses 200 iterations and Sinkhorn uses 500 iterations. Figure \[fig:generator2D\] shows the results. As $\epsilon$ varies from $0.01$ to $0.1$, the learned distribution of Sinkhorn gradually shrinks to the mean of target distribution, again this is because the inaccurcy in calculating the Wasserstein distance. On the contrary, since IPOT can converge to the exact Wasserstein distance regardless of different $\beta$, the result robustly cover the whole support of target distribution. Furthermore, comparing to the dual form method used in the WGAN, the proposed IPOT method is better in small scale cases and can achieve similar performance in large scale cases [@genevay2017sinkhorn]. This is mainly because the discriminator neural network used in WGAN is susceptible to overfitting in low dimensional cases, and it exceeds the objective of this paper. ***Experiments on Higher Dimensional Data*** For higher dimensional data, we cannot visualize the final generated distribution as done in the 2D test. So in order to demonstrate IPOT has little shrinkage issue, we set the latent space to be 2D, and visualize it by plotting the images generated at dense grid points on the latent space. Due to the low dimensional latent space, We perform the experiment using MNIST dataset. Note that this is mainly for the convenience in visualization, the whole shrinkage-free property of IPOT method is also extendable to more complex learning problems. Associated with the MNIST dataset, we use a generator $g_{\theta}:\mathbb{R}^2 \mapsto \mathbb{R}^{784}$, noise data $\{z_j\}\sim \text{Unif}([0,1]^2)$ as input, and one fully connected hidden layer with 500 nodes. Figure \[fig:generative\_mnist\] shows an example of generated results. The Sinkhorn results look authentic, but we can only find some of the digits in it. This is exactly the consequence of shrinkage due to the inaccurate calculation of Wasserstein distance - in the domain where the density of learned distribution is nonzero, the density of target distribution is usually nonzero; but in some part of the domain where the density of target distribution is nonzero, the learned distribution is zero. In the example of Figure \[fig:generative\_mnist\] (a), the learned distribution cannot cover the support of digits 2,4,5,6 while when using IPOT to calculate the Wasserstein loss, all ten digits are can be recovered in \[fig:generative\_mnist\] (b), which shows the coverage of the whole domain of the target distribution. In supplementary material we provide more examples, e.g., if a larger $\epsilon$ is used, Sinkhorn generator would shrink to one point, and hence cannot learn anything, while the IPOT method is robust to its parameter $\beta$ and covers more digits. ![\[fig:sample\] The result of barycenter. For each digit, we randomly choose 8 of 50 scaled and shifted images to demonstrate the input data. From the top to the bottom, we show (top row) the demo of input data; (second row) the results based on @cuturi2014fast; (third row) the result based on @solomon2015convolutional; (fourth row) the result based on @benamou2015iterative; (bottom row) the results based on IPOT-WB. ](sample.png){width="70.00000%"} Computing Barycenter -------------------- We test our proximal point barycenter algorithm on MNIST dataset, borrowing the idea from @cuturi2014fast. Here, the images in MNIST dataset is randomly uniformly reshape to half to double of its original size, and the reshaped images have random bias towards corner. After that, the images are mapped into $50\times 50$ grid. For each digit we use 50 of the reshaped images with the same weights as the dataset to compute the barycenter. All results are computed using 50 iterations and under $\epsilon, \beta=0.001$. So for proximal point method, the regularization is approximately the same as $\epsilon=2\times 10^{-5}$, which is pretty small. We compare our method with state-of-art Sinkhorn based methods @cuturi2014fast, @solomon2015convolutional and @benamou2015iterative. Among the four methods, the convolutional method [@solomon2015convolutional] is different in terms of that it only handles structural input tested here and does not require $O(n^2)$ storage, unlike other three general purpose methods. We are also aware of that there are other literatures for Wassersetin barycenter, such as @staib2017parallel and @claici2018stochastic, but they are targeting a more complicated setting, and has a different convergence rate (i.e. sublinear rate) than the methods we provide here. The results (Figure \[fig:sample\]) from proximal point algorithm are clear, while the results of Sinkhorn based algorithms suffer blurry effect due to entropic regularization. While the time complexity of our method is in the same order of magnitude with Sinkhorn algorithm [@benamou2015iterative], the space complexity is $K$ times of it, because $K$ different transport maps need to be stored. This might cause pressure to memory for large $K$. Therefore, a sequential method is needed. We left this to future work. CONCLUSION ========== We proposed a proximal point method - IPOT - based on Bregman distance to solve optimal transport problem. Different from the Sinkhorn method, IPOT algorithm can converge to ground truth even if the inner optimization iteration only performs once. This nice property results in similar convergence and computation time comparing to Sinkhorn method. However, IPOT provides a robust and accurate computation of Wasserstein distance and associated transport plan, which leads to a better performance in image transformation and avoids the shrinkage in generative models. We also apply the IPOT idea to calculate the Wasserstein barycenter. The proposed method can generate much sharper results than state-of-art due to the exact computation of the Wasserstein distance. ACKNOWLEDGEMENT {#acknowledgement .unnumbered} =============== This work is partially supported by the grant NSF IIS 1717916, NSF CMMI 1745382, NSFC 61672231, NSFC U1609220 and 19ZR1414200. ------------------------------------------------------------------------ ------------------------------------------------------------------------ More Analysis on IPOT ===================== [r]{}[0.54]{} {width="50.00000%"} Convergence w.r.t. $L$ ---------------------- As mentioned in Section 6.1, we provide the test result of 64D Gaussian distributed data here. We choose the computed Wasserstein distance $\langle \bm{\Gamma}, \bm{C} \rangle$ as the indicator of convergence, because while the optimal transport plan might not be unique, the computed Wasserstein distance at convergence must be unique and minimized to ground truth. We use the empirical distribution as input distributions, i.e., $$\begin{aligned} \begin{aligned} &W(\{x_i\},\{g_{\theta}(z_j)\})= \sideset{}{_{\bm{\Gamma}}}\min \langle \bm{C}(\theta),\bm{\Gamma} \rangle \quad\\ &\text{s.t. } \bm{\Gamma} \pmb{1}_n = \frac{1}{n}\pmb{1}_n, \bm{\Gamma}^T \pmb{1}_n = \frac{1}{n} \pmb{1}_n. \end{aligned}\end{aligned}$$ As shown in Figure \[fig:convergence64\], the convergence rate is also linear. For comparison, we also provide the convergence path of Sinkhorn iteration. The result cannot converge to ground truth because the method is essentially regularized. [**Remark.**]{} When we are talking about amount of regularization, usually we are referring to the magnitude of $\epsilon$ for Sinkhorn, or the equivalent magnitude of $\epsilon$ computed from remark in Section 3 for IPOT method. However, the amount of regularization in a loss function should be quantified by $\epsilon/||\bm{C}||$, instead of $\epsilon$ alone. That is why in this paper, different magnitude of $\epsilon$ is used for different application. How IPOT Avoids Instability --------------------------- Heuristically, if Sinkhorn does not underflow, with enough iteration, the result of IPOT is approximately the same as Sinkhorn with $\epsilon^{(t)}=\beta/t$. The difference lies in IPOT is a principled way to avoid underflow and can converge to arbitrarily small regularization, while Sinkhorn always causes numerical difficulty when $\epsilon\to 0$, even with scheduled decreasing $\epsilon$ like @chizat2016scaling. More specifically, in IPOT, we can factor $\Gamma=\text{diag}(\bm{u}_1)\bm{G}^{t}\text{diag}(\bm{u}_2)$, where $(\cdot)^t$ is element-wise exponent operation, and $\bm{u}_1$ and $\bm{u}_2$ are two scaling vectors. So we have $\epsilon^{(t)}=\beta/t$. As $t$ goes infinity, all entries of $\bm{G}^{t}$ would underflow if we use Sinkhorn with $\epsilon^{(t)}=\beta/t$. But we know $\bm{\Gamma}^*$ is neither all zeros nor contains infinity. So instead of computing $\bm{G}^{t}$, $\bm{u}_1$ and $\bm{u}_2$ directly, we use $\bm{\Gamma}^{t}$ to record the multiplication of $\bm{G}^t$ with part of $\bm{u}_1$ and $\bm{u}_2$ in each step, so the entries of $\bm{\Gamma}^{t}$ will not over/underflow. The explicit computation of $\bm{G}^{t}$ is not needed. Therefore, by tuning $\beta$ and iteration number, we can achieve the result of arbitrary amount of regularization with IPOT. Learning Generative Models ========================== In this section, we show the derivation for the learning algorithm, and more tests result. For simplicity, we assume $|\{x_i\}|=|\{z_j\}|=n$. Given a dataset $\{x_i\}$ and some noise $\{z_j\}$ [@bassetti2006minimum; @goodfellow2014generative], our goal is to find a parameterized function $g_{\theta}(\cdot)$ that minimize $W(\{x_i\}, \{g_{\theta}(z_j)\})$, $$\begin{aligned} \label{lossG} \begin{aligned} W(\{x_i\},\{g_{\theta}(z_j)\}) & = \sideset{}{_{\bm{\Gamma}}}\min \langle \bm{C}(\theta),\bm{\Gamma} \rangle \quad \\ & \text{s.t. } \bm{\Gamma} \pmb{1}_n = \frac{1}{n}\pmb{1}_n, \bm{\Gamma}^T \pmb{1}_n = \frac{1}{n} \pmb{1}_n, \end{aligned}\end{aligned}$$ where $\bm{C}(\theta)=[c(x_i,g_{\theta}(z_j))]$.Usually, $g_{\theta}$ is parameterized by a neural network with parameter $\theta$, and the minimization over $\theta$ is done by stochastic gradient descent. In particular, given current estimation $\theta$, we can obtain optimum $\bm{\Gamma}^*$ by IPOT, and compute the Wasserstein distance by $\langle \bm{C}(\theta),\bm{\Gamma}^* \rangle$ accordingly. Then, we can further update $\theta$ by the gradient of current Wasserstein distance. There are two ways to solve the gradient: One is auto-diff based method such as @genevay2017sinkhorn, the other is based on the envelope theorem [@afriat1971theory]. Different from the auto-diff based methods, the back-propagation based on envelope theorem does not go into proximal point iterations because the derivative over $\bm{\Gamma}^*$ is not needed, which accelerates the learning process greatly. This also has significant implications numerically because the derivative of a computed quantity tends to amplify the error. Therefore, we adopt envelope based method. **Envelope theorem.** Let $ f(x,\theta )$ and $l(x)$ be real-valued continuously differentiable functions, where $x\in \mathbb {R} ^{n}$ are choice variables and $ \theta \in \mathbb {R} ^{m}$ are parameters. Denote $x^*$ to be the optimal solution of $f$ with constraint $l=0$ and fixed $\theta$, i.e. $$x^* = \argmin _{x}f(x,\theta ) \quad s.t. \quad l(x)= 0.$$ Then, assume that $V$ is continuously differentiable function defined as $V(\theta )\equiv f(x^{\ast }(\theta ),\theta)$, the derivative of $V$ over parameters is $$\frac {\partial V(\theta)}{\partial \theta}=\frac{\partial f}{\partial \theta}.$$ In our case, because $\bm{\Gamma}^*$ is the minimization of $\langle \bm{\Gamma},\bm{C}(\theta)\rangle$ with constraints, we have $$\begin{aligned} & \frac{\partial W(\{x_i\},\{g_{\theta}(z_j)\})}{\partial \theta} = \frac {\partial \langle \bm{\Gamma}^*,\bm{C}(\theta)\rangle}{\partial \theta} \\ & = \langle \bm{\Gamma}^*,{\frac {\partial \bm{C}(\theta)}{\partial \theta}}\rangle=\langle \bm{\Gamma}^*,2(g_{\theta}(z_j)-x_i){\frac {\partial g_{\theta}(z_i)}{\partial \theta}}\rangle,\end{aligned}$$ where we assume $C_{ij}(\theta)=\|x_i-g_{\theta}(z_j)\|_2^2$, but the algorithm can also adopt other metrics. The derivation is in supplementary materials. The flowchart is shown in Figure \[fig:arche\_more\], and the algorithm is shown in Algorithm \[algo\_gan\]. Note Sinkhorn distance is defined as $S(\{x_i\},\{g_{\theta}(z_j)\})= \langle \bm{C}(\theta),\bm{\Gamma}^* \rangle$, where $\bm{\Gamma}^*=\sideset{}{_{\bm{\Gamma}\in \Sigma( \pmb{1}/n,\pmb{1}/n)}}\argmin \langle \bm{C}(\theta),\bm{\Gamma} \rangle+\epsilon h(\bm{\Gamma})$. If Sinkhorn distance is used in learning generative models, envelope theorem cannot be used because the loss function for optimizing $\theta$ and $\bm{\Gamma}$ is not the same. In the tests, we observe the method in @genevay2017sinkhorn suffers from shrinkage problem, i.e. the generated distribution tends to shrink towards the target mean. The recovery of target distribution is sensitive to the weight of regularization term $\epsilon$. Only relatively small $\epsilon$ can lead to a reasonable generated distribution. **Input:** real data $\{x_i\}$, initialized generator $g_\theta$ Sample a batch of real data $\{x_i\}_{i=1}^n$ Sample a batch of noise data $\{z_j\}_{i=1}^n \sim q$ $C_{ij}:=c(x_i,g_{\theta}(z_j)):=||x_i-g_{\theta}(z_j)||^2_2$ $\bm{\Gamma}$ = IPOT($\frac{1}{n}\bm{1}_n,\frac{1}{n}\bm{1}_n,\bm{C}$) Update $\theta$ with $\langle \bm{\Gamma},[2(x_i-g_{\theta}(z_j)){\frac {\partial g_{\theta}(z_j)}{\partial \theta}}]\rangle$ [R]{}[0.4]{} {width="35.00000%"} Synthetic Test -------------- In section 5.1, we show the learning result of Sinkhorn and IPOT in 2D case. In Figure \[fig:generator1D\_comp\] we show sequences of results for a 1D-1D generator, respectively. The upper sequence is IPOT with $\beta=0.01,0.025,0.05,0.075,0.1$. The results barely change w.r.t. $\beta$. The lower sequence is the corresponding Sinkhorn results. The results shrink to the mean of target data, as expected. Also, we observe the learned distribution tends to have a tail that is not in the range of target data (also in 2D result, we do not include that part for a better view). It might be because the range of support that has a small probability has very small gradient when updated. Once the distribution is initialized to have a tail with small probability, it can hardly be updated. But this theory cannot explain why larger $\epsilon$ corresponds to longer tails. The tails can be on the left or right. We pick the ones on the left for easier comparison. {width="1.\textwidth"} MNIST Test ---------- The same shrinkage can be observed in MNIST data as well. See figure \[fig:generative\_mnist\_comp\]. While $\epsilon=0.1$ covers most shapes of the numbers, $\epsilon=1$ only covers a fraction, and $\epsilon=10$ seems to cover only the mean of images. Color Transferring ================== Optimal transport is directly applicable to many applications, such as color transferring and histogram calibration. We will show the result of color transferring and why accurate transport plan is superior to entropically regularized ones. The goal of color transferring is to transfer the tonality of a target image into a source image. This is usually done by imposing the histogram of the color palette of one image to another image. Since @reinhard2001color, many methods [@rabin2014adaptive; @xiao2006color] are developed to do so by learning the transformation between the two histograms. Experiments in @rubner1998metric have shown that transformation based on optimal transport map outperforms state-of-the-art techniques for challenging images. Same as other prime-form Wasserstein distance solvers [@pele2009fast; @cuturi2013sinkhorn], the proximal point method provide a transport plan. By definition, the plan is a transport from the source distribution to a target one with minimum cost. Therefore it provides a way to transform a histogram to another. One example is shown in figure \[fig:color\_transfer\]. We use three different maps to transform the RGB channels, respectively. For each channel, there are at most 256 bins. Therefore, using three channels separately is more efficient than treating the colors as 3D data. Figure \[fig:color\_transfer\] shows proximal point method can produce identical result as linear programming at convergence, while the results produced by Sinkhorn method differ w.r.t. $\epsilon$. General Bregman Proximal Point Algorithm ======================================== In the main body of the paper, we discussed the proximal point algorithm with specific Bregman distance, which is generated through the traditional entropy function. In this section, we generalize our results by proving the effectiveness of proximal point algorithm with general Bregman distance. Bregman distance is applied to measure the discrepancy between different matrices which turns out to be one of the key ideas in regularized optimal transport problems. Its special structure also give rise to proximal-type algorithms and projectors in solving optimization problems. Basic Algorithm Framework and Preliminaries ------------------------------------------- The fundamental iterative scheme of general Bregman proximal point algorithm can be denoted as $$\label{Bregman-iteration} x^{(t+1)} = \arg\min_{x\in X}\left\{ f(x) + \beta^{(t)} D_h (x , x^{(t)}) \right\},$$ where $t\in \mathbb{N}$ is the index of iteration, and $D_h(x,x^{(t)})$ denotes a general Bregman distance between $x$ and $x^{(t)}$ based on a Legendre function $h$ (The definition is presented in the following). In the main body of the paper, $h$ is specialized as the classical entropy function and as follows the related Bregman distance reduces to the generalized KL divergence. Furthermore, the Sinkhorn-Knopp projection can be introduced to compute each iterative subproblem. In the following, we present some fundamental definitions and lemmas. Legendre function: Let $h: X\rightarrow (-\infty, \infty]$ be a lsc proper convex function. It is called 1. [*[Essentially smooth]{}*]{}: if $h$ is differentiable on $\hbox{int}\,\hbox{dom}\, h$, with moreover $\| \nabla h(x^{(t)}) \|\rightarrow \infty$ for every sequence $\{ x^{(t)} \}\subset \hbox{int}\,\hbox{dom}\, h$ converging to a boundary point of $\hbox{dom}\, h$ as $t\rightarrow +\infty$; 2. [*[Legendre type]{}*]{}: if $h$ is essentially smooth and strictly convex on $\hbox{int}\,\hbox{dom}\, h$. Bregman distance: any given Legendre function $h$, $$\label{Bregman-distance} D_h (x,y) = h (x) - h (y) - \langle \nabla h(y), x - y \rangle,\quad \forall x\in \hbox{dom}\,h, \forall y\in \hbox{int}\,\hbox{dom}\, h,$$where $D_h$ is strictly convex with respect to its first argument. Moreover, $D_h (x,y)\ge 0$ for all $(x,y)\in \hbox{dom}\, h\times \hbox{int}\, \hbox{dom}\, h$, and it is equal to zero if and only if $x = y$. However, $D_h$ is in general asymmetric, i.e., $D_h(x,y)\ne D_h(y,x)$. Symmetry Coefficient: Given a Legendre function $h: X\rightarrow (-\infty,\infty]$, its symmetry coefficient is defined by $$\label{Symmetry-coefficient} \alpha (h) = \inf\left\{ \frac{D_h (x,y)}{D_h (y,x)}\ \big|\ (x,y)\in \hbox{int}\, \hbox{dom}\, h\times \hbox{int}\, \hbox{dom}\, h,\ x\ne y \right\}\in [0,1].$$ Given $h:X\rightarrow (-\infty,+\infty]$, $D_h$ is general Bregman distance, and $x$, $y$, $z\in X$ such that $h(x)$, $h(y)$, $h(z)$ are finite and $h$ is differentiable at $y$ and $z$, $$\label{three_point_lemma} D_h (x,z) - D_h (x,y) - D_h (y,z) = \left\langle \nabla h(y) - \nabla h (z), x - y \right\rangle$$ The proof is straightforward as one can easily verify it by simply subtracting $D_h(y,z)$ and $D_h(x,y)$ from $D_h(x,z)$. Theorem 5.1 and Theorem 5.2 --------------------------- In this section, we first establish the convergence of Bregman proximal point algorithm, [*[i.e.]{}*]{}, [**[Theorem 5.1]{}**]{}, while our analysis is based on @eckstein1993nonlinear [@eckstein1998approximate; @teboulle1992entropic]. Further, we establish the convergence of inexact version Bregman proximal point algorithm, [*[i.e.]{}*]{}, [**[Theorem 5.2]{}**]{}, in which the subproblem in each iteration is computed inexactly within finite number of sub-iterations. Note that here for simplicity we provide proof of $d(\bm{\Gamma},\bm{\Gamma}^{(t)})=D_h(\bm{\Gamma},\bm{\Gamma}^{(t)})$, i.e., the IPOT case. We can analogously prove it for $d(\{\bm{\Gamma}_k\},\{\bm{\Gamma}^{(t)}_k\})=\sum_{k=1}^K \lambda_k D_h(\bm{\Gamma}_k,\bm{\Gamma}^{(t)}_k)$, i.e., the IPOT-WB case, with very similar proof. This is because the latter is essentially just the weighted version of the former. Before proving both theorems, we propose several fundamental lemmas. The first Lemma is the fundamental descent lemma, which is popularly used to analysis the convergence result of first-order methods. (Descent Lemma) Consider a closed proper convex function $f:X\rightarrow (-\infty,\infty]$ and for any $x\in X$ and $\beta^{(t)} > 0$, we have: $$\label{Descent_Lemma} f(x^{(t+1)})\le f(x) + \beta^{(t)} \left[ D_h (x, x^{(t)}) - D_h (x,x^{(t+1)}) - D_h (x^{(t+1)},x^{(t)}) \right], \quad \forall x\in X.$$ The optimality condition of (\[Bregman-iteration\]) can be written as $$\left( x - x^{(t+1)} \right)^T \left[ \nabla f(x^{(t+1)}) + \beta^{(t)} \left( \nabla h (x^{(t+1)}) - \nabla h (x^{(t)}) \right) \right]\ge 0,\quad \forall x\in X.$$Then with the convexity of $f$, we obtain $$\label{Basic-inequality-1} f(x) - f(x^{(t+1)}) + \beta^{(t)} \left( x - x^{(t+1)} \right)^T \left( \nabla h (x^{(t+1)}) - \nabla h (x^{(t)}) \right) \ge 0.$$With (\[three\_point\_lemma\]) it follows that $$\left( x - x^{(t+1)} \right)^T \left( \nabla h (x^{(t+1)}) - \nabla h (x^{(t)}) \right) = D_h (x, x^{(t)}) - D_h (x,x^{(t+1)}) - D_h (x^{(t+1)},x^{(t)}).$$Substitute the above equation into (\[Basic-inequality-1\]), we have $$f(x^{(t+1)})\le f(x) + \beta^{(t)} \left[ D_h (x, x^{(t)}) - D_h (x,x^{(t+1)}) - D_h (x^{(t+1)},x^{(t)}) \right], \quad \forall x\in X.$$ Next, we prove the convergence result in [**[Theorem 5.1]{}**]{}. [**[Theorem 5.1]{}**]{} [*[ Let $\{x^{(t)}\}$ be the sequence generated by the general Bregman proximal point algorithm with iteration (\[Bregman-iteration\]) where $f$ is assumed to be continuous and convex. Further assume that $f^* = \min f(x) > -\infty$. Then we have that $\{f(x^{(t)})\}$ is non-increasing, and $f(x^{(t)})\rightarrow f^*$. Further assume there exists $\eta$, s.t. $$\label{assume1} f^* +\eta d(x) \leq f(x), \quad \forall x \in X,$$ The algorithm has linear convergence. ]{}*]{} 1. First, we prove the sufficient decrease property: $$\label{Sufficient-Decrease} f(x^{(t+1)}) \le f(x^{(t)}) - \beta^{(t)}(1 + \alpha(h)) D_h (x^{(t+1)},x^{(t)}).$$ Let $x=x^{(t)}$ in (\[Descent\_Lemma\]), we obtain $$\begin{aligned} f(x^{(t+1)})&\le& f(x^{(t)}) - \beta^{(t)} \left[ D_h(x^{(t)},x^{(t+1)}) + D_h (x^{(t+1)},x^{(t)}) \right]\nonumber\\ &\le& f(x^{(t)}) - \beta^{(t)}(1 + \alpha(h)) D_h (x^{(t+1)},x^{(t)}).\nonumber\end{aligned}$$ With the sufficient decrease property, it is obvious that $\{f(x^{(t)})\}$ is non-decreasing. 2. Summing (\[Sufficient-Decrease\]) from $i = 0$ to $i=t-1$ and for simplicity assuming $\beta^{(t)} = \beta$, we have $$\sum_{i=0}^{k-1} \left[ \frac{1}{\beta^{(t)}} \left( f(x^{(i+1)}) - f(x^{(i)}) \right) \right] \le - \left[ 1+\alpha(h) \right]\sum_{i=0}^{k-1} D_h (x^{(i+1)},x^{(i)})$$ $$\Rightarrow \qquad \sum_{i=0}^{\infty} D_h (x^{(i+1)},x^{(i)}) < \frac{1}{\beta(1+\alpha(h))} f(x^{(0)}) < \infty,$$which indicates that $D_h (x^{(i+1)},x^{(i)}) \rightarrow 0$. Then summing (\[Descent\_Lemma\]) from $i=0$ to $i=t-1$, we have $$k\left(f(x^{(t)}) - f(x)\right) \le \sum_{i=0}^{t-1} \left( f(x^{(i+1)}) - f(x) \right) \le \beta D_h (x,x^{(0)}) < \infty,\quad \forall x\in X.$$Let $t\rightarrow \infty$, we have $\lim_{t\rightarrow \infty} f(x^{(t)}) \le f(x)$ for every $x$, as a result we have $\lim_{k\rightarrow \infty} f(x^{(t)}) = f^*$. 3. Finally, we prove the convergence rate is linear. Assume $x^* = \argmin_{x} f(x)$ is the unique optimal solution. Denote $d(x)= D_h(x^*,x)$. Let also $\beta^{(t)} = \beta$, we will prove $$\label{rate} \frac{d(x^{(t+1)})}{d(x^{(t)})} \leq \frac{1}{1+\frac{\eta}{\beta}}$$ Replace $x$ with $x^*$ in inequality (\[Descent\_Lemma\]), we have $$\label{inqua1} f(x^{(t+1)})\le f^* + \beta \left[ d (x^{(t)}) - d(x^{(t+1)}) - D_h (x^{(t+1)},x^{(t)}) \right].$$ Using assumption \[assume1\], we have $$\label{inqua2} f^*+\eta d( x^{(t+1)})\leq f(x^{(t+1)})$$ Sum \[inqua1\] and \[inqua2\] up, we have $$\begin{aligned} \frac{\eta}{\beta} d( x^{(t+1)}) & \leq d (x^{(t)}) - d(x^{(t+1)}) - D_h (x^{(t+1)},x^{(t)}) \\ & \leq d (x^{(t)}) - d(x^{(t+1)})\end{aligned}$$ Therefore, $$\frac{d(x^{(t+1)})}{d(x^{(t)})} \leq \frac{1}{1+\frac{\eta}{\beta}}$$ Therefore, we have a linear convergence in Bregman distance sense. Assumption (\[assume1\]) does not always hold when $f$ is linear. In our specific case, $x$ is bounded in $[0,1]^{m\times n}$. More rigorously, we can prove the following lemma. Assume $\mathcal{X}$ is a bounded polyhedron, $x^*$ is unique, $d(x)$ is an arbitrary nonnegative convex function with $d(x*)=0$. If $f$ is linear, then there exist $\eta$, s.t. $f^* + \eta d(x) \leq f(x)$. Since $\mathcal{X}$ is a bounded polyhedron, any $x\in \mathcal{X}$ can be expressed as $x=\sum_{i=0}^n \lambda_i e_i$, where $e_i$ is the vertices of $\mathcal{X}$, $n$ is finite, and $\sum \lambda_i=1$. Also $f$ is linear, so $f(x)=\sum_{i=0}^n \lambda_i f(e_i)$ Since $f$ is linear, $\mathcal{X}$ is polyhedral and $x^*$ is unique, $x^*$ is a vertex of $X$. Denote $e_0=x^*$. Denote $\delta=\min_{i>0} f(e_i)-f^*$, then $\delta>0$, or else $x^*$ is not unique. Denote $d_{max}=\max_{i>0} d(e_i)$. Take $$\eta=\delta/d_{max},$$ we have $$\begin{aligned} f^*+\eta d(x)&= f^*+\eta d(\sum_{i=0}^n \lambda_i e_i)\\ \text{ (Jensen's Inequality)}& \leq f^*+\eta\sum_{i=0}^n \lambda_i d(e_i)\\ (d(e_0)=0)\quad& \leq f^*+(1-\lambda_0)\eta d_{max}\\ & = f^*+(1-\lambda_0)\delta\\ & = \sum_{i=1}^n \lambda_i (f^*+\delta)+\lambda_0 f^*\\ & \leq \sum_{i=0}^n \lambda_i f(e_i)\\ & = f(x).\end{aligned}$$ For more general cases, if $x^*$ is not unique, we can divide the vertices as “optimal vertices” and the rest vertices, instead of $e_0$ and the rest as above, the conclusion can be proved analogously. Furthermore, if $\mathcal{X}$ is not a polyhedron, as long as $\mathcal{X}$ is bounded, we can always prove the conclusion in a polyhedron $\mathcal{A}$ s.t. $\mathcal{X}\in \mathcal{A}$ and $x*$ is also the optimal solution of $\min_{x \in \mathcal{A}} f(x)$. Proof of more general cases can be found in @robinson1981some (This paper points out some fairly strong continuity properties that polyhedral multifunctions satisfy). Inequality (\[rate\]) shows how the convergence rate is linked to $\beta$. This is the reason we claim in Section 4.1 that a smaller $\beta$ would lead to quicker convergence in exact case. From above, we showed that the general Bregman proximal point algorithm with constant step size can guarantee convergence to the optimal solution $f^*$, and has linear convergence rate with some assumptions. Further, we prove the convergence result for the general Bregman proximal point algorithm with inexact scheme in [**[Theorem 5.2]{}**]{}. [**[Theorem 5.2]{}**]{} [*[Let $\{ x^{(t)} \}$ be the sequence generated by the general Bregman proximal point algorithm with inexact scheme (i.e., finite number of inner iterations are employed). Define an error sequence $\{ e^{(t)} \}$ where $$\label{Inexact-scheme} e^{(t+1)} \in \beta^{(t)} \left[ \nabla f(x^{(t+1)} ) + \partial \iota_{X} (x^{(t+1)} ) \right] + \left[ \nabla h(x^{(t+1)} ) - \nabla h(x^{(t)} ) \right],$$where $\iota_{X}$ is the indicator function of set $X$. If the sequence $\{ e^{(t)} \}$ satisfies $\sum_{k=1}^{\infty} \| e^{(t)} \| < \infty$ and $\sum_{k=1}^{\infty} \langle e^{(t)}, x^{(t)} \rangle$ exists and is finite, then $\{ x^{(t)} \}$ converges to $x^{\infty}$ with $f(x^{\infty}) = f^*$. If the sequence $\{ e^{(t)} \}$ satisfies that exist $\rho\in (0,1)$ such that$ \| e^{(t)} \| \le \rho^t$, $ \langle e^{(t)}, x^{(t)} \rangle \le \rho^t$ and with assumption (\[assume1\]), then $\{ x^{(t)} \}$ converges linearly.]{}*]{} [**[Remark]{}**]{}: If exact minimization is guaranteed in each iteration, the sequence $\{ x^{(t)} \}$ will satisfy that $$0 \in \beta^{(t)} \left[ \nabla f(x^{(t+1)} ) + \partial \iota_{X} (x^{(t+1)} )\right] + \frac{1}{\beta^{(t)}} \left[ \nabla h(x^{(t+1)} ) - \nabla h(x^{(t)} ) \right].$$As a result, with enough inner iteration, the guaranteed $e^{(t)}$ will goes to zero. This theorem is extended from [@eckstein1998approximate Theorem 1], and we propose a brief proof here. The proof contains the following four steps: 1. We have for all $k\ge 0$, through the three point lemma $$D_h (x,x^{(t+1)}) = D_h (x,x^{(t)}) - D_h (x^{(t+1)},x^{(t)}) - \langle \nabla h(x^{(t)}) - \nabla h(x^{(t+1)}), x^{(t+1)}-x \rangle,$$which indicates $$D_h (x,x^{(t+1)}) = D_h (x,x^{(t)}) - D_h (x^{(t+1)},x^{(t)}) - \langle \nabla h (x^{(t)}) - \nabla h (x^{(t+1)}) + e^{(t+1)}, x^{(t+1)} - x \rangle + \langle e^{(t+1)}, x^{(t+1)} - x \rangle.$$ Since $ \frac{1}{\beta^{(t)}} \left[ e^{(t+1)} + \nabla h(x^{(t)} ) - \nabla h(x^{(t+1)} ) \right] \in \nabla f(x^{(t+1)} ) + \partial \iota_{X} (x^{(t+1)} )$ and $0\in \nabla f(x^{*}) + \partial \iota_{X} (x^{*})$ if $x^*$ be the optimal solution, we have $$\begin{aligned} &&\langle \nabla h (x^{(t)}) - \nabla h (x^{(t+1)}) + e^{(t+1)}, x^{(t+1)} - x^{*} \rangle \nonumber\\ &=&\beta^{(t)} \langle \left[ \frac{1}{\beta^{(t)}} \left( \nabla h (x^{(t)}) - \nabla h (x^{(t+1)}) + e^{(t+1)} \right) \right] - 0, x^{(t+1)} - x^{*} \rangle \ge 0,\nonumber\end{aligned}$$because $\nabla f + \partial \iota_{X}$ is monotone ($f+ \iota_{X}$ is convex). Further we have $$D_h (x^*,x^{(t+1)}) \le D_h (x^*,x^{(t)}) - D_h (x^{(t+1)},x^{(t)}) + \langle e^{(t+1)}, x^{(t+1)} - x^* \rangle.$$ 2. Summing the above inequality from $i=0$ to $i=t-1$, we have $$D_h (x^*,x^{(t)}) \le D_h (x^*,x^{(0)}) - \sum_{i=0}^{t-1} D_h (x^{(i+1)},x^{(i)}) + \sum_{i=0}^{t-1} \langle e^{(i+1)}, x^{(i+1)} - x^* \rangle.$$Since $\sum_{t=1}^{\infty} \| e^{(t)} \| < \infty$ and $\sum_{t=1}^{\infty} \langle e^{(t)}, x^{(t)} \rangle$ exists and is finite, we guarantee that $$\bar{E}(x^*) = \sup_{t\ge 0} \left\{ \sum_{i=0}^{t-1} \langle e^{(i+1)}, x^{(i+1)} - x^* \rangle \right\} < \infty.$$Together with $D_h (x^{(i+1)},x^{(i)}) > 0$, we have $$D_h (x^*,x^{(t)}) \le D_h (x^*,x^{(0)}) + \bar{E}(x^*) < \infty,$$which indicates $$0 \le \sum_{i=0}^{\infty} D_h (x^{(i+1)},x^{(i)}) < D_h (x^*,x^{(0)}) + \bar{E}(x^*) < \infty,$$and hence $D_h (x^{(i+1)},x^{(i)}) \rightarrow 0$. 3. Based on the above two items, we know that the sequence $\{ x^{(t)} \}$ must be bounded and has at least one limit point $x^{\infty}$. The most delicate part of the proof is to establish that $0\in \nabla f(x^{\infty}) + \partial \iota_{X} (x^{\infty})$. Let $T = \nabla f + \partial \iota_{X}$, then $T$ denotes the subdifferential mapping of a closed proper convex function $f+ \iota_{X}$ ($f$ is a linear function and $X$ is a closed convex set). Let $\{ t_j \}$ be the sub-sequence such that $x^{t_j}\rightarrow x^{\infty}$. Because $x^{t_j}\in X$ and $X$ is a closed convex set, we know $x^{\infty}\in X$. We know that $D_h (x^*,x^{(t+1)}) \le D_h (x^*,x^{(t)}) + \langle e^{(t+1)},x^{(t+1)} - x^* \rangle$ and $\sum_{k=0}^{\infty} \langle e^{(t+1)},x^{(t+1)} - x^* \rangle$ exists and is finite. From [@polyak1987introduction Section 2.2], we guarantee that $\{ D_h (x^*,x^{(t)}) \}$ converges to $0 \le d(x^*) < \infty$. Define $y^{(t+1)} := \lambda_k \left( \nabla h (x^{(t)}) - \nabla h (x^{(t+1)}) + e^{(t+1)} \right) $, we have $$\lambda_k \langle y^{(t+1)}, x^{(t+1)} - x^* \rangle = D_h (x^*,x^{(t)}) - D_h(x^*,x^{(t+1)}) - D_h (x^{(t+1)},x^{(t)}) + \langle e^{(t+1)}, x^{(t+1)} - x^* \rangle.$$By taking the limit of both sides and $\lambda_k = \lambda > 0$, we obtain that $$\langle y^{(t+1)}, x^{(t+1)} - x^* \rangle \rightarrow 0.$$For the reason that $y^{k_j + 1}$ is a subgradient of $f+\iota_{X}$ at $x^{k_j + 1}$, we have $$f(x^*) \ge f(x^{k_j +1 }) + \langle y^{k_j +1}, x^* - x^{k_j +1} \rangle, \quad x^*\in X, x^{k_j +1}\in X.$$Further let $j\rightarrow \infty$ and using $f$ is lower semicontinuous, $\langle y^{(t+1)}, x^{(t+1)} - x^* \rangle \rightarrow 0$, we obtain $$f(x^{*})\ge f(x^{\infty}),\quad x^{\infty}\in X$$which implies that $0\in \nabla f(x^{\infty}) + \iota_{X} (x^{\infty})$. 4. Recall the inexact scheme (\[Inexact-scheme\]), we can equivalently guarantee that $$(x - x^{(t+1)})^T \left\{ \beta^{(t)} \nabla f(x^{(t+1)}) + \left[ \nabla h(x^{(t+1)}) - \nabla h(x^{(t)}) \right] - e^{(t+1)} \right\}\ge 0,\ \forall x\in X.$$Together the convexity of $f$ and the three point lemma, we obtain $$f(x^{(t+1)}) \le f(x) + \frac{1}{\beta^{(t)}} \left[ D_{h} (x,x^{(t)}) - D_{h} (x,x^{(t+1)}) - D_{h} (x^{(t+1)},x^{(t)}) - (x - x^{(t+1)})^T e^{(t+1)} \right].$$Let $x = x^*$ in the above inequality and recall the assumption (\[assume1\]), [*[i.e.]{}*]{}, $$f(x) - f(x^*) \ge \eta d(x),$$we have with $\beta^{(t)} = \beta$ $$\begin{aligned} \eta d(x^{(t+1)}) &\le& \frac{1}{\beta} \left[ d(x^{(t)}) - d(x^{(t+1)}) \right] + \frac{1}{\beta} \left( (x^{(t+1)} - x^*)^T e^{(t+1)} \right)\nonumber\\ &\le& \frac{1}{\beta} \left[ d(x^{(t)}) - d(x^{(t+1)}) \right] + \frac{1}{\beta} \left( C\|e^{(t+1)}\| + \langle x^{(t+1)},e^{(t+1)} \rangle \right),\nonumber\end{aligned}$$where $C:=\sup_{x\in X^*} \{ \|x\| \}$. The second inequality is obtained through triangle inequality. Then $$d^{(t+1)} \le \mu d^{(t)} + \mu \left( C\|e^{(t+1)}\| + \langle x^{(t+1)},e^{(t+1)} \rangle \right),$$where $\mu = \frac{1}{1 + \beta \eta} < 1$. With our assumptions and according to Theorem 2 and Corollary 2 in @So2017non, we guarantee the generated sequence converges linearly in the order of ${\cal{O}}\left( c^t \right)$, where $c = \sqrt{\frac{1+ \max\{ \mu,\rho \}}{2}}\in (0,1)$. Based on the above four items, we guarantee the convergence results in this theorem. [^1]: Yujia Xie, Ruijia Wang and Hongyuan Zha are affiliated with College of Computing at Georgia Institute of Technology. Xiangfeng Wang is affiliated with East China Normal University. Emails: [[email protected], [email protected], [email protected], [email protected]]{}. [^2]: Assume $O(n)=O(m)$. [^3]: Similar ideas are also used in accelerating the expectation-maximization algorithm with only one iteration used in the maximization step [@lange1995gradient].

--- abstract: 'We quickly review and make some comments on the concept of convexity in metric spaces due to Takahashi. Then we introduce a concept of convex structure based convexity to functions on these spaces and refer to it as $W-$convexity. $W-$convex functions generalize convex functions on linear spaces. We discuss illustrative examples of (strict) $ W-$convex functions and dedicate the major part of this paper to proving a variety of properties that make them fit in very well with the classical theory of convex analysis. Finally, we apply some of our results to the metric projection problem and fixed point theory.' author: - 'Ahmed A. Abdelhakim' title: A convexity of functions on convex metric spaces of Takahashi and applications --- Convex metric space,$W-$convex functions,metric projection 26A51, 52A01, 46N10, 47N10 Introduction and preliminaries ============================== There have been a few attempts to introduce the structure of convexity outside linear spaces. Kirk [@kirk1; @kirk2], Penot [@Penot] and Takahashi [@Takahashi], for example, presented notions of convexity for sets in metric spaces. Even in the more general setting of topological spaces there is the work of Liepinš [@Liepi] and Taskovič [@Taskovi]. Takahashi [@Takahashi] introduced a general concept of convexity that gave rise to what he referred to as convex metric spaces. \[def1\]([@Takahashi]) Let $(X,d)\,$ be a metric space and $I=[0,1].$ A continuous function $\emph{W}:X\times X\times I\,\rightarrow\, X$ is said to be a *convex structure* on $X$ if for each $x,y\in X$ and all $ t \in I$, $$\begin{aligned} \label{concon} d\left(u,\emph{W}(x,y; t )\right)&\leq& (1- t )\, d(u,x)+ t\, d(u,y)\end{aligned}$$ for all $u\in X$. A metric space $(X,d)$ with a convex structure ${W}$ is called a *convex metric space* and is denoted by $(X, W,d)$. A subset $C$ of $X$ is called *convex* if $\,W (x,y; t )\in C\,$ whenever $\,x,y \in C\,$ and $\, t \in I$. What makes Takahashi’s notion of convexity solid is the invariance under taking intersections and convexity of closed balls ([@Takahashi], Propositions 1 and 2). The convex structure $W$ in Definition \[def1\] has the following property which is stated in [@Takahashi] without proof. For the sake of completeness, we give a proof of it here. \[113\] For any $x,y$ in a convex metric space $(X,W,d)$ and any $\, t \in I$ we have $$\begin{aligned} d\left(x,\emph{W}(x,y; t )\right)\,=\, t \,d(x,y),\quad d\left(y,\emph{W}(x,y; t )\right)\,=\,(1- t )\, d(x,y).\end{aligned}$$ For simplicity, let $a$, $b$ and $c$ stand for $d\left(x,\emph{W}(x,y; t )\right)$, $d\left(y,\emph{W}(x,y; t )\right)$ and $d(x,y)$ respectively. Using (\[concon\]) we get $\,\,a\leq \, t \, c\,\,$ and $\,\,b\leq \,(1- t )\,c$. But $\,\,c\,\leq \,a+b\,\,$ by the triangle inequality. So $\,\,c\,\leq\, a+b \,\leq \,(1- t ) \,c + t \,c=c.\,$ This means $\,a+b=c\,$. If $\,a<t\,c\,$ then we would have $\,a+b<c\,$ which is a contradiction. Therefore, we must have $\,a= t \,c\,$ and consequently $\,b= (1- t )\, c.$ The necessity for the condition (\[concon\]) on $W$ to be a convex structure on a metric space $(X,d)$ is natural. To see this, assume that $ ( X, \parallel . \parallel_{X} ) $ is a normed linear space. Then the mapping $\,W:X\times X\times I\,\rightarrow\, X\;$ given by $$\begin{aligned} \label{cs} W(x,y; t )=(1- t )\,x+ t\,y, \quad x,y \in X,\;\; t \in I,\end{aligned}$$ defines a convex structure on $X$. Indeed, if $\rho$ is the metric induced by the norm $\,\parallel . \parallel_{X}\,$ then $$\begin{aligned} \rho \left(u,W(x,y; t )\right )&=& \parallel u-\left((1- t )x + t y \right) \parallel_{X} \\ &\leq& (1- t ) \parallel u-x\parallel_{X} + \, t \parallel u-y\parallel_{X}\\ &=& (1- t )\,\rho(u,x) + t \,\rho(u,y),\qquad \forall u\in X,\, t \in I.\end{aligned}$$ The picture gets clearer in the linear space ${\mathbb{R}}^2$ with the Euclidean metric and the convex structure given by (\[cs\]). In this case, given two points $\,x,y\in {\mathbb{R}}^2\,$ and a $\, t \in I$, $\;z=W(x,y; t )$ is a point that lies on the line segment joining $x$ and $y$. Moreover, Lemma \[113\] implies that if $\,\overline{xy}=L\,$ then $\,\overline{xz}= t \,L\,$ and $\,\overline{zy}=(1- t ) \,L\,$ and we arrive at an interesting exercise of elementary trigonometry to show that $\,\overline{uz}\,\leq\, (1- t )\,\overline{ux}+ t \,\overline{uy}\,$ for any point $u$ in the plane. (Hint: Apply the Pythagorean theorem to the triangles $\,\triangle uyv$, $\,\triangle uvz$ and $\,\triangle uvx$ in the figure below then use the fact that $\,\overline{xy}\,\leq\,\overline{xu}+\overline{uy}$). $$\begin{aligned} &\qquad\qquad \begin{tikzpicture} [scale=8] \draw (-0.25, 0.15) node[below left] {$\dfrac{\overline{xz}}{\overline{zy}}\,=\, \dfrac{ t }{1- t }$}; \draw (0.0, 0) -- (0.7, 0) node[below] {$y$}; \draw (0, 0) node[below left] {$x$}; \draw (0.5, 0.3) node[above right] {$u$}; \draw (0.5, 0) node[below] {$v$}; \draw[dotted] (0.5, 0) -- (0.5, 0.3); \draw (0.25, 0.0) node[below] {$z=W(x,y; t )$}; \draw (0.5, 0.3) -- (0.25, 0.0); \draw (0.5, 0.3) -- (0.0, 0.0); \draw (0.5, 0.3) -- (0.7, 0.0); \end{tikzpicture}\end{aligned}$$ Takahashi’s concept of convexity was used extensively in fixed point theory in metric spaces (cf. [@Tbk] and the references therein). One of its most important applications is probably iterative approximation of fixed points in metric spaces. There is quite huge literature on fixed point iterations (cf. [@berinde; @chidume]). Roughly speaking, the formation of most, if not all, known fixed point iterative procedures is based on that of the Mann iteration [@mann] and the Ishikawa iteration [@ishikawa] as its very first generalization. All of these sequences require linearity and convexity of the ambient topological space. Although Takahashi’s notion of convex metric spaces appeared in 1970, it was not until 1988 that Ding [@ding] exploited it to construct a fixed point iterative sequence and proved a convergence theorem in a convex metric space. To our best knowledge, this is the first time a fixed point iteration, other than the well-known Picard iteration, was introduced to metric spaces. Later, a lot of strong convergence results in convex metric spaces followed (see [@berinde]).\ In the light of Definition \[def1\], it is tempting to identify convex functions on convex metric spaces. Based on the idea of convex structures on metric spaces, we define and illustrate by examples what we call $W-$convex functions. In linear metric spaces with $W$ defined by (\[cs\]), $W-$convex functions coincide with traditional convex functions. We show throughout the paper that many of the main properties of convex functions on linear spaces are satisfied by $W-$convex functions. As expected some of these properties do not carry over automatically from linear spaces to convex metric spaces. In order to achieve such properties we had to require additional assumptions on the convex structure $W.$ For instance, while midpoint convex continuous functions on normed linear spaces are convex, midpoint $W-$convexity on its own seems insufficient to obtain an analogous result in convex metric spaces. Another example appears when we study the equivalence between local boundedness from above and local Lipschitz continuity of $W-$convex functions. To achieve this equivalence we required the convex metric space to satisfy a certain property that is naturally satisfied in any linear space. Other properties necessitated providing a suitable framework to prove. For example, to investigate the relation between $W-$convexity of functions and the convexity of their epigraphs, we had to design a convex structure on product metric spaces to be able to define convex product metric spaces and characterize their convex subsets. Finally, we apply some of our results on $W-$convexity to the metric projection problem and fixed point theory. For this purpose, we give a definition for strictly convex metric spaces that generalizes strict convexity in Banach spaces and relate it to a certain class of strictly $W-$convex functions. $W-$convex functions on convex metric spaces and their main properties ====================================================================== \[defnconv\] A realvalued function $f$ on a convex metric space $\left(X,\emph{W},d\right)$ is $W-$convex if for all $x,y\in X$ and $ t \in I,\,$ $f\left(\emph{W}\left(x,y; t \right)\right) \,\leq\,(1- t )f(x)+ t f(y).$ We call $f$ strictly $W-$convex if $\,f\left(\emph{W}\left(x,y; t \right)\right) \,<\,(1- t )f(x)+ t f(y)\,$ for all distinct points $x, y\in X$ and every $ t \in I^{\tiny o}=\,]0,1[.$ Consider the Euclidean space $\mathbb{R}^{3}$ with the Euclidean norm $\parallel.\parallel$. Let $\mathcal{B}$ be the subset of $\,\mathbb{R}^{3}\,$ that consists of all closed balls $B(\xi,r)$ with center $\xi \in \mathbb{R}^{3}$ and radius $r>0$. For any two balls $\,\displaystyle B(\xi_{1},r_{1}),\,B(\xi_{2},r_{2})\in \mathcal{B},\,$ define the distance function $\, d_{\mathcal{B}}\,\displaystyle \left(B(\xi_{1},r_{1}) \textbf{,}\,B(\xi_{2},r_{2})\right)\, \,=\,\parallel \xi_{1}-\xi_{2} \parallel+|r_{1}-r_{2}|$. It is easy to check that $(\mathcal{B},d_{\mathcal{B}})$ is a metric space. Let $\,\displaystyle W_{\mathcal{B}}:\mathcal{B}\times \mathcal{B}\times I\,\rightarrow\, \mathcal{B}\,$ be the continuous mapping given by $$\begin{aligned} \hspace{-1 cm}W_{\mathcal{B}} \left(B(\xi_{1},r_{1}),\,B(\xi_{2},r_{2}); \theta\right)= B\left((1-\theta)\xi_{1}+\theta \xi_{2},\, (1-\theta)\, r_{1}+\theta \,r_{2}\,\right), \quad \xi_{i} \in \mathbb{R}^{3},\, r_{i} >0,\, \theta \in I.\end{aligned}$$ Since for all $\,\theta \in I\,$ and any three balls $\,B(\xi_{i},r_{i})\in \mathcal{B},\,$ $i=1,2,3$, $$\begin{aligned} &&\hspace*{-2 cm} d_{\mathcal{B}}\, \left(W_{\mathcal{B}} \left(B(\xi_{1},r_{1})\textbf{,}\,B(\xi_{2},r_{2}); \theta\right)\textbf{,}\,B(\xi_{3},r_{3})\right)\\ &=& d_{\mathcal{B}}\, \left( B\left((1-\theta)\,\xi_{1}+\theta \,\xi_{2}\textbf{,}\, (1-\theta)\, r_{1}+\theta \,r_{2}\right)\textbf{,}\,B(\xi_{3},r_{3})\right)\\ &=&\parallel (1-\theta)\,\xi_{1}+\theta \,\xi_{2}-\xi_{3} \parallel+|(1-\theta)\, r_{1}+\theta \,r_{2}-r_{3}|\\ &\leq&(1-\theta)\,\left(\,\parallel \xi_{1}-\xi_{3}\parallel +| r_{1}-r_{3}|\,\right)+ \theta\,\left(\,\parallel \xi_{2}-\xi_{3}\parallel +| r_{2}-r_{3}|\,\right) \\ &=&(1-\theta)\,d_{\mathcal{B}} \left(B(\xi_{1},r_{1}) \textbf{,}\,B(\xi_{3},r_{3})\right)+ \theta\,d_{\mathcal{B}} \left(B(\xi_{2},r_{2}) \textbf{,}\,B(\xi_{3},r_{3})\right).\end{aligned}$$ Then $\left(\mathcal{B},W_{\mathcal{B}},d_{\mathcal{B}}\right)$ is a convex metric space. The function $ f:\mathcal{B}\rightarrow \mathbb{R}$ defined by $f\left(B(\xi,r)\right) := \,\parallel \xi \parallel+|r|$ is $W_{\mathcal{B}}-$convex. Let $\mathcal{I}$ be the family of closed intervals $ [a,b] $ such that $0\leq a\leq b\leq1$ and define the mapping $W_{\mathcal{I}}:\mathcal{I}\times \mathcal{I} \times I$ by $W_{\mathcal{I}}(I_{i},I_{j}; t ):= \left[\left(1- t \right)a_{i}+ t a_{j}, \left(1- t \right)b_{i}+ t b_{j}\right] $ for $I_{i}=[a_{i},b_{i}],\,I_{j}=[a_{j},b_{j}]\in \mathcal{I},$ $ t \in I.$ If $d_{\mathcal{I}}$ is the Hausdorff distance then $\left(\mathcal{I}, W_{\mathcal{I}},d_{\mathcal{I}}\right)$ is a convex metric space. This example of a convex metric space is given by Takahashi [@Takahashi].\ It is easy to verify that the Lebesgue measure defines a $W_{\mathcal{I}}-$convex function on $\left(\mathcal{I},W_{\mathcal{I}}, d_{\mathcal{I}}\right)$. \[excon3\] *(Composition with increasing convex functions).* Assume that $f$ is a $W_{X}-$convex function on the convex metric space $(X,W_{X},d_{X}).$ Let $g:f(X)\rightarrow \mathbb{R}$ be increasing and convex in the usual sense. Then $ g\circ f $ is $W_{X}-$convex on $X$. The composition $ g\circ f $ is strictly $W_{X}-$convex if $g$ is strictly convex or if $f$ is strictly $W_{X}-$convex and $g$ is strictly increasing. Given $x,y \in X$ and $ t \in I$, in the light of Definition \[defnconv\], it follows from the monotonicity of $g$ that $$\begin{aligned} g\left( f \left(W_{X}\left(x,y; t \right)\right)\right) \,\leq\,g\left((1- t )f(x)+ t f(y)\right) \,\leq\,(1- t )\, g \left(f(x)\right)+ t \, g \left(f(y)\right).\end{aligned}$$ \[excon4\] Let $(X,W_{X},d_{X})$ be a convex metric space and let $g:\mathbb{R} \rightarrow \mathbb{R} $ be increasing and (strictly) convex. Then the function $f:X \rightarrow \mathbb{R}$ defined by $ f(x):=g\left(d_{X}(x,x_{0}) \right) $ for some fixed $x_{0} \in X$ is (strictly) $W_{X}-$convex. Examples of the function $g$ include $g(x)=x,$ $g(x)=\chi_{[0,\infty[}(x)\,x^{2},$ $g(x)=\chi_{[0,\infty[}(x)\,|x|$ in the case of convexity and $g(x)=e^{x},$ $ g(x)=\chi_{[0,\infty[}(x)\,|x|^{\alpha} $ with $ \alpha>1 $ in the case of strict convexity. \[8tms\] Let $\left(X,W,d\right)$ be a convex metric space. Then 1. The restriction $g$ of a $W-$convex function $f$ on $X$ to a convex subset $C$ of $X$ is also $W-$convex. 2. \[2\] If $f$ is a $W-$convex function on $X$ and $\alpha \geq 0$ then $\alpha f$ is also a $W-$convex function on $X.$ 3. \[3\] The finite sum of $\,W-$convex functions on $X$ is $W-$convex. 4. \[4\] Conical combinations of $\,W-$convex functions is again $W-$convex. 5. The maximum of a finite number of $\,W-$convex functions is $W-$convex. 6. The pointwise limit of a sequence of $\,W-$convex functions is $W-$convex. 7. Suppose that $\left(Y_{n}\right)$ is a sequence of convex subsets of $X$ and that $f_{n}$ is a $W-$convex function on $Y_{n},$ $n\geq 1.$ Let $S=\cap_{n} Y_{n}$ and $M=\{x\in X: \sup_{n}f_{n}(x)< \infty\}.$ Then $M\cap S$ is convex and the upper limit of the family $\left(f_{n}\right)_{n\geq 1},$ the function $f=\sup_{n} f_{n},$ is $W-$convex on it. 8. If $ f:X\rightarrow \mathbb{R}$ is a nontrivial strictly $W-$convex function then $f$ has at most one global minimizer on $X$. <!-- --> 1. By the convexity of $C$, the restriction of $f$ to $C$ makes sense and the $W-$convexity of $g$ on $C$ follows from the $W-$convexity of $f$ on $X.$ 2. True since $\;\; \alpha f(W(x,y; t ))\leq \alpha\left( (1- t ) f(x)+ t f(y)\right)= (1- t ) \alpha f(x)+ t \alpha f(y).$ 3. Obvious from Definition \[defnconv\] and the linearity of the summation operator. 4. Follows from \[2\] together with \[3\]. 5. It suffices to show that $f=\max\{f_{1},f_{2}\}$ is $W-$convex on $X$ given the $W-$convexity of both $f_{1}$ and $f_{2}.$ For all $x,y \in X$ and $t \in I\,$ we have $$\begin{aligned} \;f_{i}(\emph{W}(x,y;t))\leq (1-t)\,f_{i}(x)+t\,f_{i}(y) \leq (1-t)\,f(x)+t\,f(y)\end{aligned}$$ which yields $\;f(\emph{W}(x,y;t)) \leq (1-t)\,f(x)+t\,f(y).$ 6. A consequence of the monotonicity of the limit. 7. Let $x,y \in M \cap S.$ Then $x,y \in Y_{n}$ for all $n\geq1,$ $\sup_{n}f_{n}(x)< \infty$ and $\sup_{n}f_{n}(y)< \infty.$ Fix $t\in I$ and $n\geq1.$ By the convexity of $Y_{n}$ we know that it contains $\emph{W}(x,y;t).$ Hence $\emph{W}(x,y;t) \in S.$ To prove the convexity of $ M \cap S $ it remains to show that $\emph{W}(x,y;t) \in M.$ This follows from the $W-$convexity of $f_{n}$ as $f_{n}\left(\emph{W}(x,y;t)\right) \leq (1-t)\,\sup_{n} f_{n}(x)+t \,\sup_{n} f_{n}(y)<\infty.$ Finally, invoking the completeness axiom for the reals, the latter inequality implies $\sup_{n} f_{n}\left(\emph{W}(x,y;t)\right) \leq (1-t)\,\sup_{n}f_{n}(x)+t\, \sup_{n}f_{n}(y)<\infty,$ which shows that $\sup_{n} f_{n}$ is $W-$convex on $M\cap S.$ 8. Assume there are two distinct points $x,y\in X$ such that $ f(x)=f(y)=\inf_{x\in X}{f(x)}$. By convexity of $X$ we have $W(x,y;\frac{1}{2})\in X$. Since $f$ is strictly $W-$convex then $\,f\left(W(x,y;\frac{1}{2})\right)< \frac{1}{2} f(x)+\frac{1}{2} f(y)= \inf_{x\in X}{f(x)} $ which is a contradiction. $W-$convexity and continuity ============================ We begin with proving Lipschitz continuity of $W-$convex functions on generalized segments in convex metric spaces. \[fpr\] Let $\left(X,W,d\right)$ be a convex metric space. Let $x$ and $y$ be two distinct points in $X$. Then a $W-$convex function $f$ on the set $\,\mathcal{L}(x,y)=\{W(x,y;\lambda): 0\leq \lambda \leq 1\}\,$ is Lipschitz continuous on it with a Lipschitz constant that depends only on $x$ and $y$. Moreover, if $\,|f(x)-f(y)|\,\leq\,\alpha\,d(x,y)\,$ for some $\alpha>0$ then $\,|f(z)-f(w)|\,\leq\,\alpha\,d(z,w)\,$ for all $z,w\in \mathcal{L}(x,y)$. Before proceeding with the proof of Proposition \[fpr\], we would like to make some remarks on the set $\mathcal{L}(x,y)$. If $X$ is a linear space and $W$ is defined by (\[cs\]) then $W(x,y;\lambda)$ is a unique vector in $X$ for each $\lambda\in I$ and the set $\mathcal{L}(x,y)$ is known ([@cb1]) as the line segment joining the two vectors $x$ and $y$. Clearly, the Euclidean geometry justifies this notion. In metric spaces the situation is different as, for $\lambda \in I^{\tiny o},$ $W(x,y;\lambda)$ is not necessarily a unique point. In fact the continuity of $W$ in $\lambda$ required by Definition \[def1\] is to be understood in the sense of continuity of multivalued functions. And if $\xi\in X$, the distance $d\left(\xi, W(x,y;\lambda) \right)$ should be thought of as a point-set distance, but this is just a technicality. Nevertheless Lemma \[113\] assures that every point in the set $W(x,y;\lambda)$ belongs to $S\left(x,(1-\lambda)\,d(x,y)\right) \cap S\left(y,\lambda\,d(x,y)\right)$ where $S(x_{0},r)$ is the usual sphere with center $x_{0}$ and radius $r>0$. Moreover, in the linear setting we have the symmetry $W(x,y;\lambda)=W(y,x;1-\lambda)$ which leads to the symmetry $\mathcal{L}(x,y)=\mathcal{L}(y,x)$. While, from Definition \[def1\] and Lemma \[113\] deduced from it, we have $$\begin{aligned} \hspace{-0.5 cm} d\big(W(x,y;\lambda),W(y,x;1-\lambda)\big) \leq&\; (1-\lambda)\, d\big(x,W(y,x;\lambda)\big)+ \lambda\, d\big(y,W(y,x;\lambda)\big)\\ =&\; \big((1-\lambda)^2+\lambda^2 \big)\,d(x,y).\end{aligned}$$ So, all that can be inferred in the convex metric space $\left(X,W,d\right)$ is $\,d\big(W(x,y;\lambda),W(y,x;1-\lambda)\big) < 2 d(x,y),$ $\lambda\in I^{\tiny o}.$ Consequently $\mathcal{L}(x,y)$ is not to be assumed symmetric in general. Finally, observe that $\mathcal{L}(x,y)$ is closed. Indeed, it follows from Lemma \[113\] that any $u\in \mathcal{L}(y,x)$ can be written as $\,u=W(x,y,{d(x,u)}/{d(x,y)}).$ So if $\left(z_{n}\right)$ is a sequence of elements of $ \mathcal{L}(x,y)$ then $\,z_{n}=W(x,y,{d(x,z_{n})}/{d(x,y)}),\,$ $n\geq 1$. If in addition $z_{n}\rightarrow z$ as $n \rightarrow \infty$ then, by the continuity of $d$ and $W,$ we formally get $\,z=\lim_{n\rightarrow \infty}\, W(x,y,{d(x,z_{n})}/{d(x,y)})= W(x,y,{d(x,z)}/{d(x,y)}).\,$ Since $\,d(x,z_{n})\leq d(x,y)\,$ then, passing to the limit, we also have $\,d(x,z)\leq d(x,y).$ This shows that $z\in \mathcal{L}(x,y).$ Now we prove Proposition \[fpr\]. Fix $x,y \in X$ so that $d(x,y)>0$. Let $z,w\in \mathcal{L}(y,x) $ be such that $z\neq w$. Then, by $W-$convexity of $f$, we have $$\begin{aligned} \label{z1} f\left(z\right)= f\left( W\left(x,y,\frac{d(x,z)}{d(x,y)}\right)\right) &\leq& \left( 1-\frac{d(x,z)}{d(x,y)}\right) f(x)+ \frac{d(x,z)}{d(x,y)} f(y).\end{aligned}$$ Similarly $$\begin{aligned} \label{w1} f\left(w\right)= f\left( W\left(x,y,\frac{d(x,w)}{d(x,y)}\right)\right) &\leq& \left( 1-\frac{d(x,w)}{d(x,y)}\right) f(x)+ \frac{d(x,w)}{d(x,y)} f(y).\end{aligned}$$ Considering (\[z1\]) and (\[w1\]), we have only two possibilities. Either $$\begin{aligned} f\left(z\right)-f\left(w\right) \nonumber \leq&\; \left( d(x,y) \right)^{-1} \, \left(d(x,w)-d(x,z)\right)\, \left( f(x)-f(y)\right)\\ \label{zw1}\leq&\; \left( d(x,y) \right)^{-1} \,\left| f(x)-f(y)\right| \, d(z,w).\end{aligned}$$ Or $$\begin{aligned} f\left(z\right)-f\left(w\right) \nonumber \leq&\; \left( d(x,y) \right)^{-1} \, \left(d(x,z)-d(x,w)\right)\, \left( f(x)-f(y)\right)\\ \label{zw2}\leq&\; \left( d(x,y) \right)^{-1} \,\left| f(x)-f(y)\right| \, d(z,w).\end{aligned}$$ Interchanging $z$ and $w$ in both sides of (\[zw1\]) or (\[zw2\]) we immediately get $$\begin{aligned} \label{zwf} |f\left(z\right)-f\left(w\right)| \,\leq\, \left( d(x,y) \right)^{-1} \,\left| f(x)-f(y)\right| \, d(z,w)\end{aligned}$$ which proves that $f$ is Lipschitz continuous on $\,\mathcal{L}(y,x)\,$ with the Lipschitz constant $\left( d(x,y) \right)^{-1} \,\left| f(x)-f(y)\right|$. The inequality (\[zwf\]) demonstrates the second assertion of the proposition as well. Let $\left(X,W,d\right)$ be a convex metric space. If a $W-$convex function $f$ on the set $\,\mathcal{L}(x,y)=\{W(x,y;\lambda): 0\leq \lambda \leq 1\}\,$ is such that $f(x)=f(y)$ then $f$ is constant on $\mathcal{L}(x,y)$. Continuous functions on convex metric spaces are $W-$convex provided that they are midpoint $W-$convex in a certain sense. We prove this in the following Proposition. Let $\left(X,W,d\right)$ be a convex metric space. Every continuous function $f:X\rightarrow \mathbb{R}$ such that $\,\displaystyle \,f\left( W \left( x,y;\frac{\mu+\nu}{2}\right)\right)\leq \frac{1}{2}\,f\left( W \left( x,y;\mu\right)\right)+ \frac{1}{2}\,f\left( W \left( x,y;\nu\right)\right),\,$ $x,y\in X,\,$ $\mu,\nu \in I,\,$ is $W-$convex. $$\begin{aligned} \begin{tikzpicture}[domain=0:1,xscale=8,yscale=2] %\draw[red] plot (\x, {0.25+\x/2+\x*\x/2}) node[right] {$v_1(x)$}; \draw plot (\x, {0.025+\x*\x}) node[right] {$y$}; \node [left] at (0,0) {$x$}; \node [below] at (0.13,0.03) {$W(x,y;\mu)$}; \node [above] at (0.33,0.2) {$W(x,y;\frac{\mu+\nu}{2})$}; \node [below right] at (0.56,0.3386) {$W(x,y;\nu)$}; \node at (0.1,0.035) {\textbf{\tiny $\times$}}; \node at (0.33,0.1339) {\textbf{\tiny$\times$}}; \node at (0.56,0.3386) {\textbf{\tiny$\times$}}; \node at (0.215,0.071225) {\textbf{\tiny$\parallel$}}; \node at (0.445,0.223025) {\textbf{\tiny$\parallel$}}; \node at (1.2,0.0) {$0\leq \nu \leq \mu \leq 1$}; \end{tikzpicture}\end{aligned}$$ Let $n$ be a nonnegative integer and let $\Lambda_{n}=\big\{{m}/{2^n}, m=0,1,...,2^{n}\big\}.$ By induction on $n$, we show that $$\begin{aligned} \label{conlmd} f\left( W(x,y;\lambda)\right) \leq (1-\lambda)\,f(x)+\lambda\,f(y), \quad \text{for every}\;x,y\in X,\;\lambda\in \Lambda_{n}.\end{aligned}$$ Since, by lemma \[113\], $ x=W\left(x,y;0\right) $ and $ y=W\left(x,y;1\right)$ then (\[conlmd\]) is valid when $n=0$ as $\Lambda_{0}=\{0,1\}$. Assume that (\[conlmd\]) is satisfied for any $\lambda \in \Lambda_{k}$ for some natural number $k$. Now let $x,y\in X$ and suppose that $\lambda \in \Lambda_{k+1}$. Obviously, there exist $s,t \in \Lambda_{k}$ such that $\lambda=(s+t)/2$. The induction hypothesis implies that $$\begin{aligned} \label{indhypu1} f\left( W(x,y;u)\right) \leq (1-u)\,f(x)+u\,f(y),\quad u\in\{s,t\}.\end{aligned}$$ By our assumption on $f$ we have $$\begin{aligned} \label{indhypu2} f\left( W(x,y;\lambda)\right) \leq \frac{1}{2} f\left( W(x,y;s)\right)+ \frac{1}{2} f\left( W(x,y;t)\right).\end{aligned}$$ Using (\[indhypu1\]) in (\[indhypu2\]) we obtain $$\begin{aligned} f\left( W(x,y;\lambda)\right) &\leq\, \frac{1}{2} \sum_{u\in\{s,t\}}\, \big( (1-u)\,f(x)+u\,f(y) \big) \\ &=\,\left(1-\frac{s+t}{2}\right) \,f(x)+\frac{s+t}{2}\,f(y) =\left(1-\lambda\right) \,f(x)+\lambda\,f(y).\end{aligned}$$ This proves (\[conlmd\]). Let $r\in I$ be arbitrary. Since the set $\Lambda=\cup_{n\geq 0} \Lambda_{n}$ is dense in $I$ then there exists a sequence $\left(r_{n}\right)\subset \Lambda$ that converges to $r$. Thus $$\begin{aligned} \label{dnscnt} f\left( W(x,y;r)\right) = f\left( W(x,y;\lim_{n\rightarrow \infty} r_{n})\right) = \lim_{n\rightarrow \infty} f\left( W(x,y; r_{n})\right)\end{aligned}$$ by the continuity of both the convex structure $W$ and the function $f$. Since $r_{n}\in \Lambda$ then there exists an integer $m\geq 0$ such that $r_{n}\in \Lambda_{m}$. By (\[conlmd\]), $\,f\left( W(x,y; r_{n})\right) \leq (1-r_{n})\,f(x)+r_{n}\,f(y)$. From the latter inequality, the monotonicity of the limit and (\[dnscnt\]) we obtain $$\begin{aligned} f\left( W(x,y;r)\right)\leq (1-\lim_{n\rightarrow \infty} r_{n})\,f(x)+ \lim_{n\rightarrow \infty} r_{n}\,f(y) = (1-r)\,f(x)+ r\,f(y).\end{aligned}$$ The next lemma paves the way to Proposition \[boundcon\] where we show that boundedness of $\,W-$convex functions on certain convex metric spaces is a necessary and sufficient condition for their continuity. In fact, our discussion in the rest of this section is confined to convex metric spaces $\left(X,W,d\right)$ that enjoy the property that for every two distinct points $x,y\in X$ and every $\,\lambda \in\, ]0,1[\, $ there exists $\xi\in X$ such that $x=W(y,\xi;\lambda)$ or there exists $\eta\in X$ such that $ \,y=W(x,\eta;\lambda)$. This property is naturally satisfied if $X$ is a linear space with $W$ defined as in (\[cs\]). In that case, $ \xi=\lambda^{-1}\left(x-y\right)+y$ and $\eta=\lambda^{-1}\left(y-x\right)+x$. \[lemxst\] Let $B(x_{0},r)$ be an open ball centered at $x_{0}$ with radius $r>0$ that is contained in $X$. If $f:X \rightarrow \mathbb{R}$ is $W-$ convex such that $|f(x)|\leq M$ on $B(x_{0},r)$ then $f$ is $\displaystyle \frac{2M}{\rho}-$Lipschitz on $B(x_{0},r-\rho),$ $\,0<\rho<r$. Let $x$ and $y$ be two distinct points in $B(x_{0},r).$ Then, by our assumption on $\left(X,W,d\right)$, there exists $\xi\in X$ such that $\displaystyle x=W\left(y,\xi;\frac{d(x,y)}{\rho+d(x,y)}\right)$ or there exists $\eta\in X$ such that $ \,\displaystyle y=W\left( x,\eta;\frac{\rho}{\rho+d(x,y)}\right)$. We shall deal with the first case and the second one can be treated analogously. First, since $f$ is $W-$convex then $$\begin{aligned} f(x)\leq \frac{\rho}{\rho+d(x,y)} f(y)+ \frac{d(x,y)}{\rho+d(x,y)} f(\xi).\end{aligned}$$ This implies $$\begin{aligned} \label{subsin} f(x)-f(y)\leq \frac{f(\xi)-f(x)}{\rho} d(x,y).\end{aligned}$$ Assume for the moment that $\xi\in B(x_{0},r)$. Using the boundedness of $f$ on $B(x_{0},r)$ the inequality (\[subsin\]) takes the form $$\begin{aligned} \label{subsin1} f(x)-f(y)\leq \frac{2M}{\rho} d(x,y).\end{aligned}$$ Interchanging the roles of $x$ and $y$ then exploiting the symmetry of the metric, we deduce from (\[subsin1\]) that $$\begin{aligned} |f(x)-f(y)|\leq \frac{2M}{\rho} d(x,y).\end{aligned}$$ To complete the proof, it remains to show that $\xi\in B(x_{0},r)$. From Lemma \[113\], we have $$\begin{aligned} d(\xi,x)= d(\xi,W\left(y,\xi;\frac{d(x,y)}{\rho+d(x,y)}\right)) =\frac{\rho d(\xi,y)}{\rho+d(x,y)} \leq \frac{\rho d(\xi,x)}{\rho+d(x,y)} +\frac{\rho d(x,y)}{\rho+d(x,y)}.\end{aligned}$$ Solving this inequality for $d(\xi,x)$ we find $d(\xi,x)\leq \rho$. Finally $$\begin{aligned} d(\xi,x_{0})\leq d(\xi,x)+d(x,x_{0})<\rho+r-\rho=r.\end{aligned}$$ \[boundcon\] A $W-$convex function $f$ on $X$ is locally bounded if and only if it is locally Lipschitz. Of course a locally Lipschitz function is continuous and therefore locally bounded. Let $f$ be locally bounded and let $x_{0}\in X$. Then there exists $r>0$ such that $f$ is bounded on $B(x_{0},r)$ and, by Lemma \[lemxst\], $f$ is Lipschitz on $B(x_{0},r/2)$. Since $x_{0}$ was arbitrary then $f$ is locally Lipschitz. \[bbove\] The local boundedness assumption on the $W-$convex function $f$ in Lemma \[lemxst\], and consequently in Proposition \[boundcon\], can be weakened to local boundedness from above. To prove this, assume that there exists $c>0$ such that $ f(\xi)\leq c $ for every $\xi\in B(x_{0},r)$ and let $x\in B(x_{0},r)$. Then there exists $y\in X$ such that $x_{0}=W(x,y,\frac{1}{2})$. Since, by Lemma \[113\], $ d(y,x_{0})=d(x,x_{0})<r $ then $y\in B(x_{0},r)$. Furthermore, by $W-$convexity of $f$, $\,2f(x_{0})\leq f(x)+f(y)$. So $$\begin{aligned} 2f(x_{0})-c \leq 2f(x_{0})-f(y) \leq f(x)\leq c\,\Longrightarrow\, |f(x)|\leq c+2|f(x_{0})|.\end{aligned}$$ Recall that a function $f:X\rightarrow \mathbb{R}$ is lower semicontinuous at $x_{0}$ if for every $t< f(x_{0})$ there is an open neighbourhood $\mathcal{N}_{x_{0}}$ of $x_{0}$ such that $f(x)>t$ for every $x\in \mathcal{N}_{x_{0}}$, and if $\,\forall\:t> f(x_{0})\;\exists\;\mathcal{N}_{x_{0}}:\; f(x)<t,\; \forall\:x\in \mathcal{N}_{x_{0}}$ then $f$ is upper semicontinuous at $x_{0}$. It follows from Proposition \[boundcon\] is that upper semicontinuous $W-$convex functions on open sets are continuous. The same applies to lower semicontinuous functions if and only if $X$ is complete. Furthermore, a family of continuous $W-$convex pointwise bounded functions on an open convex subset of a complete metric space is locally equi-bounded and locally equi-Lipschitz. The most important consequence of these facts is that pointwise convergence of sequences of continuous $W-$convex functions on open convex subsets of complete metric spaces is uniform on compact sets and preserves continuity. Since the proofs of these results (cf. [@cb1; @cb2; @cb3] ) is indifferent to the topology of the space and does not depend on linearity, we find it redundant to give them here. Epigraphs and sublevel sets of $W-$convex functions =================================================== The epigraph of a realvalued function $f$ on a set $C$ is the set $Epi (f)=\{(x,s)\in C\times \mathbb{R}: f(x)\leq s\}$ and the sublevel set of $f$ of height $h$ is is the set $ S_{h}(f)=\{x\in C: f(x)\leq h\}.$ In Proposition \[confunset\] below we show how $W-$convexity of functions is related to the convexity of their epigraphs and sublevel sets. First, let $\left(X, W_{X}, d_{X}\right)$ and $\left(Y, W_{Y}, d_{Y}\right)$ be two convex metric spaces. The mapping $\,d_{p}: \left( X\times Y \right)^{2} \rightarrow [0,\infty[$, $$\begin{aligned} d_{p}\left((x_{1},y_{1}),(x_{2},y_{2})\right)= \left\{ \begin{array}{ll} \big( \left(d_{X}(x_{1},x_{2})\right)^{p}+ \left(d_{Y}(y_{1},y_{2})\right)^{p}\big)^{\frac{1}{p}}, & \hbox{$1\leq p<\infty$;} \\\\ \max\left\{d_{X}(x_{1},x_{2}), d_{Y}(y_{1},y_{2})\right\}, & \hbox{ $p=\infty$,} \end{array} \right.\end{aligned}$$ is a metric on the cartesian product $X\times Y$ and $\left(X\times Y,d_{p} \right)$ is called a product metric space. Now, let $\left(X,W_{X},d_{X}\right)$ and $\left(Y,W_{Y},d_{Y}\right)$ be two convex metric spaces. We note the following: \[lemconprod\] The mapping $\,W_{X\times Y}: \left(X\times Y\right)^{2}\times I\rightarrow X\times Y\,$ given by $\,W_{X\times Y}\left( \left(x_{1},y_{1}\right), \left(x_{2},y_{2}\right);t\right)= \left( W_{X}(x_{1},x_{2};t), W_{Y}(y_{1},y_{2};t)\right)$ is continuous and defines a convex structure on the product metric space $\left(X\times Y,d_{1}\right).$ The continuity of $W_{X\times Y}$ follows from the continuity of the convex structures $W_{X}$ and $W_{Y}.$ Let $ t\in I $ and $ (x_{i},y_{i})\in X \times Y,$ $i=1,2,3.$ By the definition of $W_{X\times Y},$ it remains to prove that $$\begin{aligned} \nonumber & d_{1}\left( \left(x_{3},y_{3}\right), \left(W_{X} \left(x_{1},x_{2};t\right), W_{Y} \left(y_{1},y_{2};t \right)\right) \right)\\ &\label{inefin} \qquad\qquad\qquad\leq\, (1-t)\,d_{1}\left( \left(x_{1},y_{1}\right), \left(x_{3},y_{3}\right)\right)+ t \,d_{1}\left( \left(x_{2},y_{2}\right), \left(x_{3},y_{3}\right)\right).\end{aligned}$$ However, we shall pretend that we need to prove the inequality (\[inefin\]) for the metric $d_{p}$ with $ 1\leq p<\infty.$ This enables us to demonstrate the difficulty in the proof for the case $p>1$ and explain why the assertion of Lemma \[lemconprod\] is limited to the case $p=1.$ The metric $d_{\infty}$ is excluded for the same reason. Of course we could simply construct counterexamples for those cases but that would take us outside the scope of this paper. Now, for $1\leq p <\infty,$ we exploit the following facts: (i)$W_{X}$ and $W_{Y}$ are convex structures on $X$ and $Y$ respectively. (ii)The map $x\mapsto x^{p}$ is monotonically increasing on $ [0,\infty[.$ (iii)$(\mu+\nu)^p\leq 2^{p-1}\big(\mu^p+\nu^p\big),\,$ for all $\mu,\nu \geq 0.$ We then see that $$\begin{aligned} \nonumber &\;d^{p}_{p}\left( \left(x_{3},y_{3}\right), \left(W_{X} \left(x_{1},x_{2};t\right), W_{Y} \left(y_{1},y_{2};t \right)\right) \right)\\ \nonumber =&\;\big( d_{X}\left(x_{3},W_{X} \left(x_{1},x_{2};t\right)\right)\big)^{p}+ \big( d_{Y}\left(y_{3},W_{Y} \left(y_{1},y_{2};t\right)\right)\big)^{p}\\ \nonumber \leq &\; \big( (1-t)d_{X}(x_{1},x_{3})+t d_{X}(x_{2},x_{3}) \big)^{p} + \big( (1-t) d_{Y}(y_{1},y_{3})+t d_{Y}(y_{2},y_{3}) \big)^{p} \\ \nonumber \leq &\; 2^{p-1} (1-t)^{p} \big[ \big(d_{X}(x_{1},x_{3})\big)^{p}+ \big(d_{Y}(y_{1},y_{3})\big)^{p}\big]+ 2^{p-1} t^{p} \big[ \big(d_{X}(x_{2},x_{3})\big)^{p}+ \big( d_{Y}(y_{2},y_{3})\big)^{p} \big]\\ = &\; 2^{p-1} \,\big[ (1-t)^{p} d^{p}_{p}\big( \left(x_{1},y_{1}\right), \left(x_{3},y_{3}\right)\big)+ t^{p} \,d^{p}_{p}\big( \left(x_{2},y_{2}\right), \left(x_{3},y_{3}\right)\big) \big]\end{aligned}$$ which gives the desired inequality (\[inefin\]) when $p=1.$ Using Lemma \[lemconprod\], we can describe convex subsets of convex product metric spaces. \[zzcon\] A subset $Z$ of the convex product metric space $\left(X\times Y, W_{X\times Y}, d_{1}\right)$ is convex if $\,W_{X\times Y}\left(\left(x_{1},y_{1}\right), \left(x_{2},y_{2}\right);t\right) \in Z$ for all points $(x_{1},y_{1}),(x_{2},y_{2})\in Z$ and all $t\in I.$ In the light of Definition \[zzcon\] one can easily verify Lemma \[intersec\] below. \[intersec\] The intersection of any collection of convex subsets of the convex product metric space $\left(X\times Y, W_{X\times Y}, d_{1}\right)$ is convex. \[confunset\] Let $f$ be a realvalued function on a convex metric space $\left(X,W_{X},d_{X}\right).$ Then 1. The function $f$ is $W_{X}-$convex if and only if $Epi (f)$ is a convex subset of the convex product metric space $\left(X\times R, W_{X\times \mathbb{R}}, d_{X}+d_{\mathbb{R}}\right),$ where $W_{\mathbb{R}}$ and $d_{\mathbb{R}}$ are the usual convex structure and metric on $\mathbb{R}$ respectively. 2. If $f$ is $W_{X}-$convex then the sublevel set $S_{h}(f)$ is a convex subset of $X$ for every $h\in \mathbb{R}.$ <!-- --> 1. Suppose that $f$ is $W_{X}-$convex on $X$ and let $(x,s), (y,t)\in Epi(f).$ Then $$\begin{aligned} f(W_{X}(x,y; \lambda ))\leq (1- \lambda ) f(x)+ \lambda f(y) \leq (1- \lambda ) s+ \lambda t\end{aligned}$$ for all $\lambda \in I.$ Therefore $\,\left(W_{X}(x,y; \lambda ), (1- \lambda ) s+ \lambda t\right) \in Epi(f).$ That is $$\begin{aligned} W_{X\times\mathbb{R}} \left(\left(x,s\right), \left(y,t\right);\lambda\right)= \left(W_{X}(x,y;\lambda ), W_{\mathbb{R}}(s,t;\lambda )\right)\in Epi(f), \quad \lambda \in I.\end{aligned}$$ Hence $Epi (f)$ is a convex subset of $X\times R$. Conversely, suppose $Epi (f)$ is convex. Fix $x,y\in X$ and $t\in I.$ Since $(x,f(x)),(y,f(y)) \in Epi (f),$ then $$\begin{aligned} \left(W_{X} \left(x,y;t\right), W_{\mathbb{R}}\left(f(x),f(y);t\right)\right)= W_{X\times \mathbb{R}}\left(\left(x,f(x)\right), \left(y,f(y)\right);t\right)\in Epi(f).\end{aligned}$$ Thus $\;f\left(W_{X} \left(x,y;t\right)\right)\leq W_{\mathbb{R}}\left(f(x),f(y);t\right) = (1-t)\,f(x)+t\,f(y),\;$ which is to say that $f$ is $W_{X}-$convex. 2. Let $t \in I$ and let $x,y\in S_{h}(f)$ so that $f(x)\leq h$ and $f(y)\leq h.$ Since $f$ is $W_{X}-$convex then $\,f\left(W_{X} \left(x,y;t\right)\right)\leq (1-t)\,f(x)+t\,f(y) \leq h.$ Therefore $W_{X} \left(x,y;t\right) \in S_{h}(f)$ and $S_{h}(f)$ is convex. The following theorem is an application of Lemma \[intersec\] and Proposition \[confunset\]. The pointwise supremum of an arbitrary collection of $W-$convex functions is $W-$convex. Let $\left(X,W,d\right)$ be a convex metric space. Let $J$ be some index set and assume that $\{f_{i}\}_{i\in J}$ is a collection of $W-$convex functions on $X$. Then, by Proposition \[confunset\], $Epi (f_{i})$ is a convex subset of the convex product metric space $\,\left(X\times R, W_{X\times \mathbb{R}}, d_{X}+d_{\mathbb{R}}\right)\,$ for every $i\in J$. If $\,f:X\rightarrow \mathbb{R}\,$ is such that $\,f(x)=\sup_{i\in J}{f_{i}(x)},\;x\in X,\,$ then $\,Epi(f)=\cap_{i \in J} Epi \left(f_{i} \right).\,$ By Lemma \[intersec\], $\,Epi(f)\,$ is a convex subset of $\,\left(X\times R, W_{X\times \mathbb{R}}, d_{X}+d_{\mathbb{R}}\right)\,$ and, using Proposition \[confunset\], it follows that $f$ is $W-$convex on $X$. Applications to the projection problem and fixed point theory ============================================================= Let $Y$ be a nonempty subset of a convex metric space $\left(X,W,d\right)$. The distance map (cf. [@Papadopoulos]) $d_{Y}:X\rightarrow [0,\infty[$ is defined by $\,d_{Y}(x)=\inf_{y\in Y}{d(x,y)}.$ The distance map $d_{Y}$ is $W-$convex. Indeed, if $x_{1},x_{2}\in X,$ $y\in Y$ and $t\in I$ then, by the definition of $d_{Y}$, we have $$\begin{aligned} d_{Y}(W\left( x_{1},x_{2}; t\right))\leq \;d\left(W\left( x_{1},x_{2}; t\right),y\right) \leq (1-t)\,d(x_{1},y)+t\,d(x_{2},y)\end{aligned}$$ for every $y\in Y$. Hence, by positive homogeneity and subadditivity of the infimum, $$\begin{aligned} d_{Y}(W\left( x_{1},x_{2}; t\right)) \leq &\;\inf_{y\in Y}\big ((1-t)\,d(x_{1},y)+t\,d(x_{2},y)\big)\\ \leq &\;(1-t)\, \inf_{y\in Y}d(x_{1},y)+ t\, \inf_{y\in Y}d(x_{2},y)\\ = &\;(1-t)\, d_{Y}(x_{1})+t\, d_{Y}(x_{2}).\end{aligned}$$ If $Y$ is convex, then the metric projection operator (also called the nearest point mapping) (cf. [@Li]) $P_{Y}:X\rightarrow 2^{Y}$ is given by $\;P_{Y}(x):=\big\{y\in Y: d(x,y)=d_{Y}(x) \big\}$. If $P_{Y}(x)\neq \emptyset\,$ for every $x\in X$ then $Y$ is called proximal. $P_{Y}(x)$ is convex ([@beg], Lemma 3.2) and if $Y$ is closed then it is proximal. The proof of the proximality of $Y$ in this case is standard and given, in the setting of normed spaces, in many books (cf. [@Papadopoulos; @cb1]). We briefly sketch it here. There exists a minimizing sequence $ (y_{n})\subset Y $ such that $d(x,y_{n})\rightarrow d_{Y}(x),\;x\in X, $ as $n\rightarrow \infty$. So the sequence $ (y_{n}) $ is bounded and, up to replacing it by a subsequence, it converges to $y$, say. Consequently, $d(x,y_{n})\rightarrow d(x,y)$ as $n\rightarrow \infty$. Hence $d(x,y)=d_{Y}(x)$. Since $Y$ is closed then $y\in P_{Y}(x)$.\ The set of metric projections $P_{Y}(x)$, if nonempty, is not necessarily a singleton. If $P_{Y}(x)$ is a singleton for each $x\in X$ then the convex set $Y$ is called a Chebyshev set. It is well-known (cf. [@cb3]) that every closed convex subset of a strictly convex and reflexive Banach space is a Chebyshev set.\ We would like to describe sufficient conditions for a point $x\in X$ to have a unique projection in $Y$. We begin with introducing a definition for strict convexity in convex metric spaces. \[dd11\] A convex metric space $(X,W,d)$ is strictly convex if for each $x_{0}\in X$ and any two distinct points $x,y\in S\left(x_{0},\rho\right)\;$ with $\rho>0,\,$ we have $\,W(x,y;t)\in B\left(x_{0},\rho\right),\;\; \forall\;t \in I^{\tiny o}.$ If $X$ is a linear space endowed with a norm that induces the metric $d$ and $W$ is given by (\[cs\]) then Definition \[dd11\], after normalizing and translating to the origin, coincides with the known definition of strictly convex normed spaces [@cb3]. \[dd22\]( (Strict) $W-$convexity w.r.t spheres). Let $(X,W,d)$ be a convex metric space. Fix $x_{0}\in X,$ $\rho>0$ and $ \sigma\in\,]0,\rho[.\,$ We call a realvalued function $f$ on $ \overline{B\left(x_{0},\rho \right)}$ $W-$convex w.r.t the sphere $\,S\left(x_{0}, \sigma\right)\,$ if $$\begin{aligned} f\left(W(x,y;t)\right)\leq (1-t)\,f(x)+t\,f(y),\;\; \forall\; x,y\in S\left(x_{0}, \sigma\right), \, t\in I,\end{aligned}$$ and we call it strictly $W-$convex w.r.t the sphere $\,S\left(x_{0}, \sigma\right)\,$ if $$\begin{aligned} f\left(W(x,y;t)\right)< (1-t)\,f(x)+t\,f(y),\;\; \forall\; x,y\in S\left(x_{0}, \sigma\right)\;\text{with}\; x\neq y,\;\forall\; t\in I^{\tiny o}.\end{aligned}$$ The following proposition is a direct consequence of Definitions \[dd11\] and \[dd22\]. Let $(X,W,d)$ be a convex metric space. If for each $x_{0}\in X$ and $\rho>0,$ the function $f:X\rightarrow [0,\infty[\,$ defined by $\,f(x):=d(x,x_{0})\,$ is strictly $W-$convex w.r.t the sphere $S\left(x_{0}, \rho\right)$ then the space $X$ is strictly convex. The following theorem asserts that closed convex subsets of strictly convex metric spaces are Chebyshev sets. \[thm4\] Assume that $Y$ is a closed convex subset of a strictly convex metric space $(X,W,d)$. Then every $x\in X$ has a unique projection on $Y$. Since $Y$ is closed then $\,P_{Y}(x)\neq \emptyset,\;\forall\:x \in X\,$ by the discussion above. If $x\in Y$ then $P_{Y}=\{x\}.$ Let $x\in X-Y $ have two distinct projections $ y_{1},y_{2} \in Y$. Then $\,d(x,y_{1})=d(x,y_{2})=d_{Y}(x)$. Let $t\in I^{\tiny 0}.$ Since $Y$ is convex then $W(y_{1},y_{2};t) \in Y$, and since $X$ is strictly convex then $$\begin{aligned} d\left(W(y_{1},y_{2};t),x\right) <(1-t)\,d(y_{1},x)+t\,d(y_{2},x)= d_{Y}(x),\end{aligned}$$ which is a contradiction. Let $Y$ be a compact convex subset of a strictly convex complete metric space. If $\,f: Y\rightarrow Y\,$ is continuous then it has a fixed point in $Y$. Since $Y$ is compact then it is closed and, by Theorem \[thm4\] above, $Y$ is a Chebyshev set. The rest of the proof follows from Theorem 3.4 and Corollary 3.5 in [@beg]. \[thm555\] Let $(X,W,d)$ be a convex metric space and let $\,T:X \rightarrow X\,$ is a nonexpansive mapping. Assume that the function $\,f:X\rightarrow [0,\infty[\,$ defined by $\,f(x):=d\left(x,Tx\right)$ is strictly $W-$convex with a local minimum at $\,\xi \in X$. Then $\xi$ is a fixed point of $ T$. By proposition \[8tms\], the point $\xi$ is the unique global minimizer of $f$. Suppose that $\,T \xi \neq \xi.$ Since $X$ is convex then $W\left(\xi,T \xi; t\right) \in X\;\forall\;t\in I,\,$ and since $f$ is strictly $W-$convex on $X$ then, for all $\,t \in I^{\tiny o},\,$ we have $$\begin{aligned} \nonumber f\left(W\left(\xi,T \xi; t\right)\right) &<& (1-t)\,f(\xi)+t\,f(T \xi) \\ \nonumber &=& (1-t)\,d(\xi, T \xi)+t\,d(T \xi, T^{2} \xi)\\ \label{strff} &\leq& (1-t)\,d(\xi, T \xi)+t\,d( \xi, T \xi) \,=\,d(\xi, T \xi)\,=\,f(\xi),\end{aligned}$$ where we used nonexpansiveness of $f$ in estimating $\,d(T \xi, T^{2} \xi)\leq d(\xi, T \xi).$ The strict inequality (\[strff\]) contradicts the fact that $\,f(\xi)=\min_{x\in X} f(x)$. Therefore we must have $\,T \xi= \xi.$ The function $f$ is continuous by the continuity of $\,T$. Hence, if $X$ is compact then there does exist a point $\xi \in X$ such that $\,f(\xi)=\min_{x\in X} f(x),\,$ and we do not need to make such an assumption on $f$. 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--- abstract: 'We study the full counting statistics for the transmission of two identical particles with positive or negative symmetry under exchange for the situation where the scattering depends on energy. We find that, besides the expected sensitivity of the noise and higher cumulants, the exchange symmetry has a huge effect on the average transmitted charge; for equal-spin exchange-correlated electrons, the average transmitted charge can be orders of magnitude larger than the corresponding value for independent electrons. A similar, although smaller, effect is found in a four-lead geometry even for energy-independent scattering.' author: - 'F. Hassler' - 'G.B. Lesovik' - 'G. Blatter' title: Effects of Exchange Symmetry on Full Counting Statistics --- Introduction ============ Full counting statistics provides the ultimate information on the statistics of charge transport in mesoscopic systems. The first study of this type [@levitov:93] already provided a surprising result, a (quantum [@levitov:92; @binomial_s]) binomial distribution of charges, telling that the flux of incoming electrons is regular (a consequence of exchange correlations) and the scattering events are independent. This result applies to the situation where the scatterer is characterized by an energy-independent transmission and for a constant applied voltage. The modification due to adiabatic pumping [@muzykantskii:03] leaves the binomial result largely unchanged. Recently, much interest has concentrated on single particle sources feeding devices with individual electrons;[@lee:95; @levitov:96; @ivanov:97; @lebedev:05; @keeling:06] such single particle excitations are generated by (unit-flux) voltage pulses. Again, it turns out that the full counting analysis remains binomial when the system is driven by such voltage pulses.[@ivanov:97] These findings make explicit that exchange correlations are absent in the (energy-independent) scattering process. In this letter, we demonstrate that exchange effects manifest themselves when single particle sources inject electrons into devices with energy dependent scattering. ![Contour plots for the two-particle wave functions $|\Psi_{\text{in},\pm}(k_1,k_2)|^2$ with vanishing separation $\delta x$ between the particles. The three processes with 0, 1, and 2 particles transmitted draw their weight $P_n$, $n=0,1,2$, from the (shaded) regions labelled with 0, 1, and 2 (shown is the case for the quantum point contact, with a Lorentzian wave function Eq. (\[eq:wf\_lorentz\]).)[]{data-label="fig:redist"}](redist) The analysis of a setup with energy-dependent scattering and driven with a *constant* voltage has unveiled sensitivity to exchange in the noise but not in the average current.[@buttiker:92] Furthermore, exchange effects in the noise were found in the statistics of two particles escaping from a dot at short times [@lesovik:habil] and for two electrons traversing a reflectionless symmetric beam splitter [@burkard:00]. Here, we show that applying voltage *pulses* in the incoming lead and injecting individual particles produces non-linear transport due to exchange effects; i.e., a twice larger voltage pulse does not double the transmitted charge and hence the average current is sensitive to the exchange symmetry. Note that voltage pulses applied *across* the device are not expected to generate single particle excitations. The discussion of full counting statistics in the presence of an energy-dependent scattering is difficult to conduct in a second-quantized formalism. On the other hand, the recent insight regarding the correspondence between fidelity and full counting statistics [@lesovik:06:c] has enabled a first-quantized approach based on a wave-packet formalism. Here, we make use of this new approach in our investigation of exchange effects on the full counting statistics of charge transport and find interesting new results: while it is expected that the noise as well as higher-order correlators will be sensitive to the exchange symmetry, we find the astounding result that exchange can hugely enhance (or suppress) the *average charge* already. The effect is a consequence of the symmetry-induced reshuffling of weight in the momentum distribution of the two-particle wave function, see Fig. \[fig:redist\], combined with the energy dependence of the scattering matrix: the anti-symmetry under exchange moves weight away from the (Pauli-blocked) diagonal, which typically leads to an enhancement in the scattering channel transmitting one particle, at the expense of the channel where both particles are reflected. In a multi-channel/multi-lead setup the effect appears already for energy-independent transmission amplitudes. Single Lead =========== First Quantized Picture ----------------------- ![(a) Single channel quantum wire driven inductively ($L$) with a nearby current $I_V(t)$. Two Lorentzian voltage pulses $V(t) = L \dot{I}_V/c $ drive the wave packets $\psi_1$ and $\psi_2$ with overlap $S$. The scatterer at $x=x_\text{s}$ gives rise to a momentum-dependent transmission probability $T_k$. (b) Three lead fork geometry with incoming wave packets in leads 1 and 2 and the counter (black box) in lead 3. (c) Reflectionless four-lead beam splitter; the incoming wave packets in leads 1 and 2 undergo a reflectionless transmission into leads 3 and 4.[]{data-label="fig:setup"}](setup) We start with the discussion of a quantum wire with one conducting channel, cf. Fig. \[fig:setup\](a). The task is to find the full counting statistics of the charge transport through a scatterer with an energy-dependent transmission amplitude located at $x=x_\text{s}$. Using the wave-packet formalism of Ref. , we calculate the generating function for the full counting statistics for two incoming particles, the simplest case exhibiting the effect of exchange symmetry. The incoming particles are described by two wave packets of the form $$\label{eq:wave_packet} \psi_{\text{in},m}(x;t) = \int_0^\infty \frac{dk}{2\pi} f_m(k) e^{i k (x - v_\text{F} t)}$$ with the Fourier components $f_1(k)$ and $f_2(k)$. The normalization of the wave packets in momentum space reads $\int (dk/2\pi) | f_m(k) |^2 = 1$. At low temperatures the interesting physics takes place near the Fermi points and we can use a linearized spectrum $\epsilon = v_{\rm\scriptscriptstyle F} |k|$ with the Fermi velocity $v_{\rm\scriptscriptstyle F}$, the momentum $\hbar k$ and the energy $\hbar \epsilon$. Traversing the scatterer, the wave packet, Eq. (\[eq:wave\_packet\]), acquires a reflected and transmitted term $$\begin{gathered} \label{eq:wf_out} \psi^\sigma_{\text{out},m} (x,t) = \int_0^\infty \frac{dk}{2\pi} f_m(k) e^{- i k v_\text{F} t} \Bigl[ r_k e^{-ikx} \Theta(x_\text{s}-x) \\ + e^{i \sigma \lambda/2} \tau_k e^{ikx} \Theta(x-x_\text{s}) \Bigr],\end{gathered}$$ where $\tau_k$ ($r_k$) is the transmission (reflection) amplitude and we have introduced a counting field $\exp(i\sigma \lambda/2)$ in the transmitted part; the sign $\sigma = \pm 1$ differentiates between the forward- and back-propagating wavefunctions required in the wave-packet formalism of full counting statistics, cf. Ref. . The properly (anti)symmetrized two-particle wave functions assume the form $\Psi_{{\rm x},\pm}(x_1,x_2;t)\propto\psi_{{\rm x},1}(x_1;t)\psi_{{\rm x},2}(x_2;t)\pm (x_1\leftrightarrow x_2)$ with ${\rm x} = {\rm in},~{\rm out}$ (see, Fig. \[fig:setup\](c); throughout the paper, the subscript ‘$\pm$’ refers to the exchange symmetry, e.g., for two electrons this corresponds to the triplet or singlet states). The transport statistics is described by the generating function $\chi_\pm (\lambda) = \int dx_1 dx_2\, \Psi_{{\rm out},\pm}^{-1\,*} \Psi_{{\rm out},\pm}^{+1}$ and we obtain the result $$\begin{gathered} \label{eq:fcs_2wp} \chi_\pm (\lambda) = \frac{ \bigl[ 1 + (e^{i \lambda} - 1 ) \langle 1 | T | 1 \rangle \bigr] \bigl[1 + (e^{i \lambda} - 1 ) \langle 2 | T | 2 \rangle \bigr] } {1 \pm |S|^2} \\ \pm \frac{ \bigl[ S + (e^{i \lambda} - 1 ) \langle 1 | T | 2 \rangle \bigr] \bigl[ S^* + (e^{i \lambda} - 1 ) \langle 2 | T | 1 \rangle\bigr] } {1 \pm |S|^2}\end{gathered}$$ with the matrix elements $\langle n | T | m \rangle = \int (dk/2\pi) f_n^*(k) T_k$ $f_m(k)$ involving the transmission probability $T_k = |\tau_k|^2$ and the overlap integral $S = \int (dk/2\pi) f_1^*(k) f_2(k)$.[@lesovik:06:c] The Fourier transform $P_n = \int (d\lambda/2\pi) \chi(\lambda) e^{-i \lambda n}$ yields the probability $P_n$ for transmitting $n$ particles. ![Normalized average charge $q= Q/2e\langle T \rangle$ transmitted through a transmission step (left) and a transmission resonance (right). The step/resonance is placed at a distance $k_0$ away from the Fermi momentum $k_{\rm\scriptscriptstyle F}$, $\xi$ is the width of the wave packets in real space, and $\delta x$ denotes the separation between the wave packets. The top (bottom) figures show the average transmitted charge for particles with negative (positive) exchange symmetry. Note the huge enhancement in the average transmitted charge in the limit $\delta x \to 0$ and $k_0\xi \geq 1$ for the case of negative exchange.[]{data-label="fig:charge"}](mean_charge) For *energy-independent* transmission amplitudes $\tau_k \equiv \tau$, the exchange term in Eq. (\[eq:fcs\_2wp\]) cancels with the denominator and the counting statistics does not dependent on the exchange symmetry of the particles. This property is particular to the one-dimensional wire; in a multi-lead setup, this cancellation does not occur and interference terms can be observed even for energy-independent scattering amplitudes, see below. On the contrary, for an *energy-dependent* transmission, the exchange term has a dramatic effect on the statistical properties of the transferred charge. To keep the discussion simple, we consider the case of two incident wave packets with the same momentum distribution separated in position. Given the initial separation $\delta x$, the Fourier components of the wave packets satisfy $f_2(k) = f_1(k) e^{- i k \delta x}$ and thus, $\langle 1 | T | 1 \rangle = \langle 2 | T | 2 \rangle \equiv \langle T \rangle$ $= \int (dk/2\pi) T_k |f_1(k)|^2$. The overlap integral $S=\int (dk/2\pi) |f_1(k)|^2 \exp (i k \delta x)$ is the Fourier transform of the distribution in momentum space. The transmission probabilities $P_{n,\pm}$ are given by ($\sum_n P_{n,\pm}=1$) $$\begin{aligned} \label{eq:prob} P_{0,\pm} &= \frac{ ( 1 - \langle T \rangle )^2 \pm | S - \langle 1 | T | 2 \rangle |^2 } {1 \pm |S|^2}, \nonumber\\ P_{1,\pm} &= 2 \frac{\langle T \rangle ( 1 - \langle T \rangle) \pm \bigr[\text{Re} ( \langle 1 | T | 2 \rangle S^*) - | \langle 1 | T | 2 \rangle |^2 \bigr]} {1 \pm |S|^2}, \nonumber\\ P_{2,\pm} &= \frac{ \langle T \rangle ^2 \pm | \langle 1 | T | 2 \rangle |^2 } {1 \pm |S|^2}; \end{aligned}$$ they depend on the exchange symmetry provided that $\langle 1 | T | 2 \rangle \ne S \langle T \rangle$. These transmission probabilities are easily converted into cumulants of transmitted charge $Q = \int dt I(t)$. [@gustavsson:06] For two particles incident on the scatterer, the first two cumulants read ($e$ denotes the charge of the particles) $$\begin{aligned} \label{eq:cum} \langle Q/e \rangle_\pm &= P_{1,\pm} + 2 P_{2,\pm} \nonumber \\ \langle\langle (Q/e)^2 \rangle\rangle_\pm &= P_{1,\pm} ( 1- P_{1,\pm} ) + 4 P_{2,\pm} P_{0,\pm} \end{aligned}$$ and we find both depending on the exchange symmetry. While interference terms in the noise $\langle\langle Q^2 \rangle\rangle$ due to the exchange symmetry of electrons are expected and have been found before, [@buttiker:92; @lesovik:habil; @burkard:00] here, we find that already the transmitted charge $\langle Q \rangle$ is sensitive to the exchange symmetry. For a quantitative analysis of the effect, we need to specify the shape $f_1$ of the wave packets as well as the energy dependence $T_k$ of the transmission probability. Rather than postulating trivial Gaussian wave packets, here, we consider a more realistic situation where the wave packets are deliberately created by voltage pulses. The creation of such single-particle excitations requires voltage pulses of unit flux[@levitov:96] $\Phi_0 = hc/e$ and with Lorentzian shape,[@keeling:06] $V_{t_1}(t)=-(2v_{\rm\scriptscriptstyle F} \xi \Phi_0/c)/[v_{\rm\scriptscriptstyle F}^2( t-t_1)^2 + \xi^2]$, where we conveniently express the pulse duration $\xi/v_{\rm\scriptscriptstyle F}$ through the length parameter $\xi$. Such a voltage pulse generates a wave packet with amplitude ($x_1 = v_{\rm\scriptscriptstyle F}t_1$) $$\label{eq:lorentz} f_1(k) = \sqrt{4\pi\xi} e^{-\xi (k-k_{\rm\scriptscriptstyle F})-i k x_1} \Theta(k-k_{\rm\scriptscriptstyle F}),$$ and a Lorentzian shape in real space, $$\label{eq:wf_lorentz} |\psi_1|^2 = \frac{\xi/\pi}{(x-x_1-v_{\rm\scriptscriptstyle F} t)^2+\xi^2},$$ cf. Appendix \[sec:equiv\]. For two wave packets separated by $\delta x$ we obtain an overlap integral $S = e^{-i k_{\rm\scriptscriptstyle F} \delta x}/ (1 + i \delta x/2 \xi)$. Next, we concentrate on the scatterer where we consider two generic types: (i) for a sharp *transmission resonance*, e.g., due to a double barrier, the energy dependent transmission probability assumes the form $T_k^\text{res} = \alpha/[1 + \beta^2 (k - k_{\rm\scriptscriptstyle F} - k_0)^2]$ where $\alpha \leq 1$ is the amplitude of the resonance and $k_0>0$ is its position relative to the Fermi wave vector $k_{\rm\scriptscriptstyle F}$. The width $\beta^{-1}$ of the resonance has to be much smaller than the width $\xi^{-1}$ of the wave packet in $k$-space, $\beta^{-1} \ll \xi^{-1}$. The transmission probability $\langle T^\text{res} \rangle$ for a single wave packet with amplitude $f_1(k)$ assumes the form $\langle T^\text{res} \rangle \approx (2 \pi \alpha \xi /\beta) e^{-2 \xi k_0}$ for $\beta^{-1} \ll k_0$, i.e., a resonance away from the Fermi level. While a small parameter $k_0\xi$ produces a large overall signal, it does suppress the effect of exchange symmetry since the transmission is already saturated, hence, below we will be interested in intermediate and large values of $k_0\xi$. (ii) For a (sharp, $\beta^{-1} \ll \xi^{-1}$) *transmission step*, e.g., due to a quantum point contact, we use the Kemble function [@landau:3] $T_k^\text{qpc}= \alpha/[1+e^{-\beta (k-k_{\rm \scriptscriptstyle F} -k_0)}]$ producing the averaged transmission probability $\langle T^\text{qpc}\rangle\approx \alpha e^{-2\xi k_0}$; note that the small prefactor $\xi/\beta$ is absent in this case. For a sharp resonance, the exchange term assumes the simple form $\langle 1 | T^\text{res} | 2 \rangle\approx e^{-i (k_{\rm\scriptscriptstyle F}+k_0) \delta x} \langle T^\text{res} \rangle$ and its product with the overlap integral $S^*$ results in an expression $\propto \exp (-i k_0\delta x)$ independent of $k_{\rm\scriptscriptstyle F}$. The average transmitted charge, cf.Fig. \[fig:charge\], then exhibits oscillations in the separation $\delta x$, $$\begin{aligned} \label{eq:res_q} &\langle Q/e \rangle^\text{res}_\pm = 2 \langle T^\text{res} \rangle \\ &\quad \times \frac{1+ (\delta x / 2 \xi)^2 \pm [ \cos (k_0 \delta x) + (\delta x / 2 \xi) \sin(k_0 \delta x) ] } { 1+ (\delta x / 2 \xi)^2 \pm 1}. \nonumber\end{aligned}$$ For wave packets with a large separation $\delta x \gg \xi$, the exchange term decays as $(\delta x)^{-2}$ and the transmitted charge is given by $\langle Q/e \rangle^\text{res} = 2 \langle T^\text{res} \rangle$, independent on the sign of the exchange term. On the other hand, strongly overlapping wave packets with $\delta x \to 0$ reproduce the result for independent particles $\langle Q/e \rangle^\text{res}_+ = 2 \langle T^\text{res} \rangle$ for the symmetric exchange. For the anti-symmetric case the transmitted charge $$\label{eq:qminus_res} \langle Q/e \rangle^\text{res}_- = 2 \langle T^\text{res} \rangle (1 - 2 \xi k_0 + 2 \xi^2 k_0^2)$$ is reduced for narrow wave packets, $\xi k_0 < 1$ and enhanced for wave packets with $\xi k_0>1$; the reduction can be up to $50\%$ for $\xi k_0 = 1/2$, while the increase is unlimited for $\xi k_0>1$.[@note] Note that this increase is due to a large $P_{1,-}$ as $P_{2,-}$ vanishes. The quantum point contact yields almost identical results: The off-diagonal matrix element assumes the form $\langle 1 | T^\text{qpc} | 2 \rangle \approx e^{-i k_0 \delta x} S \langle T^\text{qpc} \rangle$ and the *interference term* in $P_2$ vanishes. The transmitted charge $$\label{eq:qpc_q} \langle Q/e \rangle^\text{qpc}_\pm = 2 \langle T^\text{qpc} \rangle \frac{1+ (\delta x / 2 \xi)^2 \pm \cos (k_0 \delta x) } { 1+ (\delta x/ 2 \xi)^2 \pm 1}$$ again exhibits oscillations with $k_0 \delta x$, cf. Fig. \[fig:charge\]. The various limits discussed above are reproduced as well, except for the case of anti-symmetric exchange and strongly overlapping wave packets, cf.(\[eq:qminus\_res\]), where the average transmitted charge now is given by $$\label{eq:qminus_qpc} \langle Q/e \rangle^\text{qpc}_- = 2 \langle T^\text{qpc} \rangle (1 + 2 \xi^2 k_0^2).$$ Eqs. (\[eq:qminus\_res\]) and (\[eq:qminus\_qpc\]) are the most striking results of our study: for a large parameter $\xi k_0$ the mean transmitted charge can be hugely increased as compared to the value expected for two independent wave packets. Note that the above results are valid in a regime where all relevant lengths remain below the phase breaking length $L_\varphi$. Finally, we provide the expressions for the generating function of the full counting statistics, $$\begin{aligned} \label{eq:2fcs} \chi^\text{res}_\pm =& 1 + \langle Q/e \rangle^\text{res}_\pm (e^{i\lambda} -1 )\\ &+ \langle T^\text{res} \rangle^2 \frac{(1 \pm 1) [ (\delta x /2 \xi)^2 + 1]} { (\delta x /2 \xi)^2 + 1 \pm 1 } (e^{i \lambda} -1)^2, \nonumber\\ \chi^\text{qpc}_\pm =& 1 + \langle Q/e \rangle^\text{qpc}_\pm (e^{i\lambda} -1 ) + \langle T^\text{qpc} \rangle^2 (e^{i\lambda} -1 )^2.\end{aligned}$$ The above enhancements in $\langle Q \rangle_-$ are entirely due to $P_1$; there are two crucial elements in the game, Pauli exclusion and dispersive scattering. Dispersive scattering generates broader wave functions and combined with Pauli exclusion we have a reduction in $P_0$ and $P_2$, and hence an increase in $P_1$. However, different settings may be thought of where $P_2$ is enhanced. E.g., a large $P_{2,-}$ (see (\[eq:fcs\_2wp\])) is obtained for wave packets with shifted amplitudes $f_2(k) = f_1(k+\delta k)$ in $k$-space and a large overlap integral $S$ combined with a transmission amplitude suppressing $k$-values in the overlap region. A large enhancement in the tunneling probability of two identical particles due to an enhanced $P_2$ was also found by Suslov and Lebedev.[@suslov] Second Quantized Picture ------------------------ In a second quantized picture, electronic excitations in the quantum wire are produced by a voltage pulse $V(t)$ applied at $x=0$.[@voltage] If the time dependence of the voltage is slow compared to the transition time of an electron through the loop, the potential can be considered as quasistatic. It can then be incorporated in the phase factor $\exp [i \phi(t - x/v_{\rm\scriptscriptstyle F}) \Theta(x)]$, where $v_{\rm\scriptscriptstyle F}$ is the Fermi velocity and the phase, $$\label{eq:phase} \phi(t) = \frac{e\Phi(t)}{\hbar c} = - \frac{e}{\hbar} \int_{-\infty}^t \!\! dt' V(t'),$$ is proportional to the flux $\Phi(t)$. The solution $\psi_{\text{L},k} (x;t) = \exp [i k (x- v_{\rm\scriptscriptstyle F} t) + i \phi(t- x/ v_{\rm\scriptscriptstyle F}) \Theta(x) ]$ of the time dependent Schrödinger equation[@lesovik:94] is a plane wave with well-defined energy $\hbar v_{\rm\scriptscriptstyle F} k$, left of the position of the voltage pulse, $x<x_\text{s}$. The wave function $\psi_{\text{L},k}(x;t)$ for $x>0$ can be decomposed into energy eigenmodes $\exp[ik(x-v_{\rm\scriptscriptstyle F} t) ]$ of the free Hamiltonian $$\label{eq:fourier} \psi_{\text{L},k} ( x_\text{s} > x >0;t) = \! \int \!\! \frac{dk'}{2\pi} U(k'-k) e^{i k' (x - v_{\rm\scriptscriptstyle F} t)}$$ where the transformation kernel $$\label{eq:u} U(q) = v_{\rm\scriptscriptstyle F} \int dt e^{i \phi(t) + i q v_{\rm\scriptscriptstyle F} t}$$ is the Fourier transform of the phase factor $\exp [i \phi(t)]$. In the present work, we are interested in applying integer flux pulses such that $\phi(t \rightarrow \infty) \in 2\pi \mathbb{N}$ as noninteger flux pulses do not produce clean single-particle excitations and lead to logarithmic divergences in the noise of the transmitted charge.[@lee:93; @levitov:96] In the case of integer flux pulses, the behavior of $\exp [i \phi(t)]\rightarrow 1$ for large times, $t \rightarrow \pm\infty$, leads to a Dirac delta function $2\pi \delta (q)$. The remaining part $U^\text{reg} (q) = U(q) - 2\pi \delta(q)$ is finite for a localized voltage pulse; $$\label{eq:ureg} U^\text{reg} (q) = v_{\rm\scriptscriptstyle F} \int dt \bigl[ e^{i \phi(t)} -1 \bigr] e^{i q v_{\rm\scriptscriptstyle F} t}.$$ The transformation $U(k'-k)$ describes the scattering amplitude for the transition from a momentum state $k$ (for $x<0$) to the state $k'$ (for $x>0$) due to the application of the voltage pulse. The statement that the wave function is in the momentum state $k'$ holds in the asymptotic region, i.e., for observation points with $x_\text{obs} \gg \xi$. The scattering matrix approach at the scatterer assign the momentum component $k'$ a transmission amplitude $\tau_{k'}$ for traversing the scatterer. Therefore, momentum eigenstates $k$ incoming from the left assume the form $$\label{eq:left} \psi_{\text{L},k} (x> x_\text{s} , t) = \int \frac{dk'}{2\pi} U(k'-k) e^{i k' (x - v_{\rm\scriptscriptstyle F} t)} \tau_{k'},$$ right of the scattering region. The states originating from the right do not enter the region where the voltage pulse is applied and consists of the incoming and the reflected part $$\label{eq:right} \psi_{\text{R},k} (x> x_\text{s} , t) = (e^{- i k x} + r_k e^{i k x}) e^{- i k v_{\rm\scriptscriptstyle F} t}$$ without any shift in energy. The time dependent field operator is given by $$\label{eq:field} \Psi (x>x_\text{s}, t) = \int \frac{dk}{2\pi} \bigl[ \psi_{\text{L},k} (x;t) a_k + \psi_{\text{R},k} (x;t) b_k \bigr]$$ in the region $x>x_\text{s}$ behind the scatterer; here, $a_k$ and $b_k$ denote fermionic annihilation operators for states coming from the left, right reservoir, respectively. Averaging the current operator, defined as $I (x > x_\text{s};t) = e \hbar\bigl(\Psi^\dagger(x;t) \partial_x \Psi(x;t) - [\partial_x \Psi^\dagger(x;t)] \Psi(x;t)\bigr)/2 i m$, over the Fermi reservoirs assuming the same Fermi distribution $n(k)$ at the far right and left of the interaction region and integrating the ensemble averaged current over time, the average transmitted charge $$\label{eq:mean_charge} \langle Q/e \rangle = \int \frac{dk}{2\pi} n(k) \int \frac{dk'}{2\pi} K(k'-k) T_{k'}$$ is obtained. It depends through the kernel $K(q) = |U^\text{reg} (q)|^2 + 4\pi \delta(q) \text{Re} \bigl[ U^\text{reg} (0) \bigr]$ on different transmission probabilities $T_{k+q}$ than incoming wave vector $k$; note that for $U^\text{reg} (q)$ whose real part is not continuous near $q=0$, $\text{Re} [U^\text{reg} (0)]$ has to be replaced by the symmetric limit $\text{Re} [U^\text{reg} (0 + i\varepsilon) + U^\text{reg} (0 - i\varepsilon)]/2$ ($\varepsilon \rightarrow 0)$. Equivalently, the kernel $K(q)$ can be defined as $$\label{eq:k} K(q) = v_{\rm\scriptscriptstyle F}^2 \int dt dt' \bigl[ e^{i \phi(t) - i\phi(t')} -1 \bigr] e^{i q v_{\rm\scriptscriptstyle F} (t-t')}.$$ The first quantized result, $\langle Q/e \rangle = \int (dk/2\pi) |f(k)|^2 T_k$, is the same as the second quantized, Eq. (\[eq:mean\_charge\]), provided that $\int (dk'/2\pi) n(k') K(k-k') = |f(k)|^2$, where $f(k)$ denotes the wave function in the first quantized picture. In a first step, we discuss a unit flux voltage pulse of Lorentzian shape $V_{t_1}(t)= - (2 v_{\rm\scriptscriptstyle F} \xi \hbar/e) /[v_{\rm\scriptscriptstyle F}^2( t-t_1)^2 + \xi^2]$, $v_{\rm\scriptscriptstyle F} \xi \gg 1$ applied at time $t_1$. The regular part of the transformation kernel is given by $$\label{eq:ureg1} U^\text{reg}_{x_1}(q) = - 2\pi (2\xi) e^{-\xi q - i q x_1} \Theta(q),$$ where $x_1 = v_{\rm\scriptscriptstyle F} t_1$. The $\Theta$-function reflects the fact that the Lorentzian pulse only increases the energy of the individual Fourier components. As we will discuss in Appendix \[sec:equiv\], the pulse produces a clean single particle excitation on top of the Fermi sea. Calculating $$\label{eq:k1} K_{x_1}(q) = (2\pi)^2 2\xi \Bigl[2 \xi e^{-2 \xi q} \Theta(q) - \delta(q) \Bigr]$$ and inserting the result in Eq. (\[eq:mean\_charge\]) yields the average charge transmitted, (for zero temperature with $n(k) = \Theta(k_{\rm\scriptscriptstyle F} -k)$, i.e., temperatures $\vartheta \ll \hbar\xi/v_{\rm\scriptscriptstyle F}$) $$\begin{aligned} \label{eq:mean_charge1} \langle Q/e \rangle_{x_1} &= \int_0^{k_{\rm\scriptscriptstyle F}} \!\!\! dk \int dk' 2 \xi \Bigl[ 2\xi e^{-2 \xi (k'-k)} \Theta(k'-k) \nonumber\\ & \quad - \delta(k' - k) \Bigr] T_{k'} = \int \frac{dk}{2\pi} |f_1(k)|^2 T_k.\end{aligned}$$ In the last step, we performed the integration over $k$ and renamed the remaining variable of integration. Furthermore, we dropped terms with $k \approx 0$. These terms correspond to the fact that the left states are shifted up in energy and therefore, currents from filled states on the right for small $k$ values are not canceled by corresponding currents on the left. This is an unphysical artifact of the linear spectrum approximation which is only valid close to the Fermi point, $k\approx k_{\rm\scriptscriptstyle F}$. Formally, one can cure the problem by setting the lower integration bound for the $k$ integration to $-\infty$ whenever one integrates over the Fermi sea. The empty states are then shifted to $k \rightarrow - \infty$ and do not appear in the final result. Alternatively, the transmission probability $T_k$ can be set to 0 for small values of $k$. Eq. (\[eq:mean\_charge1\]) is exactly the first quantized result for a wave packet of Lorentzian form, Eq. (\[eq:lorentz\]). More interesting than the single voltage pulse are two pulses separated by $\delta x = v_{\rm\scriptscriptstyle F} \delta t = x_2-x_1$ as they produce two wave packets and we expect interference terms to be found as in the first quantized formalism. For two pulses, the phase factor is a product of the phase factors of the individual pulses. Therefore, the Fourier transformed function $U_{x_1, x_2}(q)$ is the convolution of the transformation functions for single pulses $U_{x_1, x_2}(q) = U_{x_1} \! \ast \! U_{x_2} (q)$, $$\begin{aligned} \label{eq:ureg2} U^\text{reg}_{x_1, x_2} &(q) - U^\text{reg}_{x_1} (q)- U^\text{reg}_{x_2} (q) = U^\text{reg}_{x_1} \! \ast \! U^\text{reg}_{x_2} (q) \\ &= \frac{2 (2\pi) (2\xi)^2}{\delta x } e^{-\xi q - i q(x_1 + x_2)/2} \sin(q \delta x/2) \Theta(q). \nonumber\end{aligned}$$ In the expression for the transmitted charge, we need to know the kernel $K$ given by $$\begin{gathered} \label{eq:k2} K_{x_1, x_2}(q) = K_{x_1} (q) + K_{x_2} (q) + 2 (2\pi)^2 (2\xi)^2 e^{-2 \xi q } \Theta(q) \\ \times\Biggl( 2\frac{\bigl[\delta x \cos (q \delta x/2) - 2 \xi \sin(q \delta x/2) \bigr]^2}{(\delta x)^2} - 1 \Biggr).\end{gathered}$$ The first two terms correspond to the direct contribution, the last term describes the interference. At zero temperature the average transmitted charge $$\begin{aligned} \label{eq:mean_charge2} \langle & Q/e \rangle_{x_1, x_2} = \langle Q/e \rangle_{x_1} + \langle Q/e \rangle_{x_2} \\ & + 2 \int_0^\infty \!\!\!\! dq \,2 \xi e^{-2 \xi q} T_{k_{\rm\scriptscriptstyle F} + q} \frac{1 - \cos(q \delta x) - \frac{\delta x}{2\xi} \sin( q \delta x) }{(\delta x /2 \xi)^2} \nonumber\\ & = P_{1,-} + 2 P_{2,-}, \nonumber\end{aligned}$$ for two Lorentzian voltage pulses separated by $\delta x$ is obtained. When comparing Eq. (\[eq:mean\_charge2\]) to the first quantized result (\[eq:prob\]), it can be seen that the result in second quantization is the same as expected from the analysis in the first quantized picture for an antisymmetric wave function. For a single voltage pulse with double flux, i.e. $\delta x=0$, the increase in the transmitted charge as seen in the first quantized formalism is confirmed. Moreover, applying a voltage pulse carrying even more flux quantum, leads to a further increase in the transmitted charge with respect to the independent particle picture, cf.Appendix \[sec:many\_flux\]. So far, we considered only spinless particles. The electronic spin $1/2$ is included easily in the second quantized formalism. The summation over the spin degree of freedom provides a factor of two for each cumulant; i.e., each voltage pulse excites two particles in the same spatial state. In the first quantized formalism, a single voltage pulse which excites two particles in the state $\Psi_1 (x;t)$ can be described as a singlet $$\label{eq:spin1wf} \Psi (x_1, x_2, t) = \Psi_1(x_1 , t) \Psi_1 (x_2, t) \frac{| {\uparrow \downarrow} \rangle - | {\downarrow \uparrow} \rangle } {\sqrt{2}}.$$ Each of the two particles carries the same current and they are independent of each other which leads to a factor two as in the second quantized formalism. Accordingly, two unit flux voltage pulses excite a four particle state involving the wave functions $\Psi_{1,2} (x;t)$. The second quantized formalism leads a factor of two due to the spin degeneracy and an averaging over the energies of the antisymmetrized wave function $\Psi_- (x_1, x_2;t) = \bigl[ \Psi_1 (x_1;t) \Psi_2(x_2, t) - (x_1 \leftrightarrow x_2) ] \bigr] / \sqrt{2(1 - |S|^2) }$. In first quantization, the four-particle wave function with spin 0 is given by $$\begin{aligned} \label{eq:spin2wf} \Psi (x_1, x_2&, x_3, x_4, t) = \\ \frac{1}{\smash[b]{\sqrt{6}}} \Bigl[ &\Psi_- (x_1, x_2, t) \Psi_- (x_3, x_4, t) \bigl( | {\uparrow\uparrow\downarrow\downarrow}\rangle + | {\downarrow\downarrow\uparrow\uparrow}\rangle \bigr) \nonumber\\ + &\Psi_- (x_1, x_4, t) \Psi_- (x_2, x_3, t)\bigl( | {\uparrow\downarrow\downarrow\uparrow}\rangle + | {\downarrow\uparrow\uparrow\downarrow}\rangle \bigr) \nonumber\\ - &\Psi_- (x_1, x_3, t) \Psi_- (x_2, x_4, t)\bigl( | {\uparrow\downarrow\uparrow\downarrow}\rangle + | {\downarrow\uparrow\downarrow\uparrow}\rangle \bigr) \Bigr]. \nonumber\end{aligned}$$ As the spin is unimportant, we may integrate over the spin degrees of freedom and obtain a density matrix consisting of an equal mixture of the wave function $\Psi_- (x_1, x_2;t) \Psi_- (x_3, x_4;t)$ and wave functions obtained by coordinate relabeling. Relabeling does not change the physics of independent particles. We can choose any one of the wave functions in the first quantized picture. Both the antisymmetry ($\Psi_-$) and the trivial factor of two (product of two wave functions) are now also visible in the first quantized language. The generating function $\chi_\pm^{(4)} = \chi_\pm{}^2$ of the full counting statistics for the four spin degenerate particles is the square of the spinless generating functions $\chi_\pm$ of Eq. (\[eq:2fcs\]). The procedure outline can also be carried out for different forms of the voltage pulse. Note though that for pulses other than a Lorentzian voltage pulse there is no simple first quantized correspondence to the second quantized formalism. A unit flux voltage pulse is given by $$\label{eq:2nd_pulse} V(t) = - \frac{2 \hbar \dot{f}(t)/e}{1 + f(t)^2}$$ with the requirement that $f(t\rightarrow \pm \infty ) \rightarrow \pm \infty$; the Lorentzian voltage pulse, considered so far, is a special case obtained with $f(t) = v_{\rm\scriptscriptstyle F} t/ \xi$. The phase factor assumes the form $$\label{eq:2nd_phase} e^{i \phi(t)} = \frac{f(t) - i}{f(t) + i}.$$ As an example we consider the voltage pulse given by $f(t) = (v_{\rm\scriptscriptstyle F} t / \xi)^3$ which has a simple transformation function $U(q) = 2\pi \delta(q) + U^\text{reg} (q)$ with $$\begin{aligned} \label{eq:2nd_ureg} U^\text{reg}(q) &=- 2\pi \frac{2 \xi}{3} \bigl[ e^{\xi q} \Theta(-q) \\ &+ e^{-\xi q/2} [ \cos (\sqrt{3} \xi q/2) + \sqrt{3} \sin(\sqrt{3} \xi q/ 2) ] \Theta(q) \bigr] \nonumber\end{aligned}$$ The part with positive momentum transfer, $q>0$, corresponds to an excited electron as we have seen for the simple Lorentzian pulse. The negative momentum indicates holes accompanying the electron. The kernel $K$ assumes the form $$\begin{aligned} \label{eq:2nd_kernel} K(q)&= (2 \pi )^2 \frac{(2\xi)^2}{9} \bigl[ e^{2\xi q} \Theta(-q) \\ & + e^{-\xi q} [2 - \cos( \sqrt{3} \xi q ) + \sqrt{3} \sin( \sqrt{3} \xi q)] \Theta(q) \bigr] \nonumber \\ & - 2 (2 \pi)^2 (2 \xi) \delta(q)/ 3. \nonumber\end{aligned}$$ Plugging it in Eq. (\[eq:mean\_charge\]), yields the average transmitted charge $$\begin{aligned} \label{eq:2nd_current} \langle Q/e \rangle =& \frac{2 \xi}{9} \int dk \, T_k \Bigl[ - e^{- 2\xi(k_{\rm\scriptscriptstyle F} - k)} \Theta( k_{\rm\scriptscriptstyle F} - k ) \nonumber\\ &+ e^{-\xi (k - k_{\rm\scriptscriptstyle F})} \Bigl( 4 + \cos \bigl[\sqrt{3} \xi ( k - k_{\rm\scriptscriptstyle F})\bigr] \nonumber\\ &+ \sqrt{3} \sin \bigl[\sqrt{3} \xi (k - k_{\rm\scriptscriptstyle F}) \bigr] \Bigr)\Theta(k-k_{\rm\scriptscriptstyle F}) \Bigr]\end{aligned}$$ at zero temperature. The part with the negative momentum transfer carries a negative charge, hole, and the charge of the other part is positive and so the expected nature of excitations is confirmed. Note that in the case of a fully transparent wire ($T_k \equiv 1$), the average transmitted charge is exactly one electron $e$. Multi-Lead ========== In a multi-lead setup, exchange effects can be found for an energy-independent scattering already. Here, we discuss the three-lead fork geometry and the four-lead reflectionless beam splitter for two incoming particles in different leads 1 and 2 and hence vanishing initial overlap. The wave packets are assumed to have a Lorentzian shape (\[eq:lorentz\]) and to impinge on the scatterer with a delay $\delta t = \delta x/v_{\rm\scriptscriptstyle F}$. In the three-lead geometry, cf. Fig. \[fig:setup\](b), the generating function reads $$\label{eq:gen3} \chi^{{\hbox{\multilead A}}}_\pm = 1 + T_3 (e^{i \lambda} -1) + (1 \pm |S|^2) T_{13} T_{23} (e^{i \lambda} -1)^2,$$ where $T_{13}$ and $T_{23}$ denote the transmission probabilities for particles incident from leads $1$ and $2$ and propagating into lead $3$ (containing the counter) and $T_{3}=T_{13}+T_{23}$. The probabilities $$\begin{aligned} \label{eq:prob3} P^{{\hbox{\multilead A}}}_{0,\pm} &= (1 - T_{13}) (1-T_{23}) \pm |S|^2 T_{13} T_{23} \nonumber\\ P^{{\hbox{\multilead A}}}_{1,\pm} &= T_{13} ( 1 - T_{23}) + (1-T_{13}) T_{23} \mp 2 |S|^2 T_{13} T_{23} \nonumber\\ P^{{\hbox{\multilead A}}}_{2,\pm} &= (1 \pm |S|^2 ) T_{13} T_{23}\end{aligned}$$ depend on the exchange symmetry as long as $T_{13} T_{23} \ne 0$. For $\delta x=0$ the overlap integral $S = e^{-i k_{\rm \scriptscriptstyle F} \delta x}/(1+i\delta x/2\xi)$ becomes unity and hence $P_{2,-}=0$. Finally, the charge cumulants assume the form $$\begin{aligned} \label{eq:cum3} \langle Q/e \rangle^{{\hbox{\multilead A}}}_\pm \!&=\! T_3, \\ \langle\langle (Q/e)^2 \rangle\rangle^{{\hbox{\multilead A}}}_\pm \! &= \! T_{13}(1\!-\!T_{13}) + T_{23} (1\!-\!T_{23}) \pm \! 2 |S|^2 T_{13} T_{23}; \nonumber\end{aligned}$$ effects due to exchange symmetry are limited to the charge noise which is enhanced (reduced) for the (anti-)symmetric case. For the reflectionless four-lead beam splitter with the counter in lead 3, Fig. \[fig:setup\](c), we obtain the generating function, probabilities, and moments in the form $$\begin{aligned} \label{eq:gen4} \chi^{{\hbox{\multilead B}}}_\pm &= 1 + [ T_3 \pm |S|^2 ( T_3 -1 ) ] (e^{i \lambda} -1) \nonumber \\ & \quad + (1 \pm |S|^2) T_{13} T_{23} (e^{i \lambda} -1 )^2 \nonumber\\ P^{{\hbox{\multilead B}}}_{0,\pm} &= (1 \pm |S|^2) (1 - T_{13}) (1 - T_{23}) \nonumber\\ P^{{\hbox{\multilead B}}}_{1,\pm} &= (1 \pm |S|^2) [ (1 - T_{13}) T_{23} + T_{13} (1- T_{23}) ] \nonumber\\ & \quad\mp |S|^2 \nonumber\\ P^{{\hbox{\multilead B}}}_{2,\pm} &= (1 \pm |S|^2) T_{13} T_{23} \nonumber\\ \langle Q/e \rangle^{{\hbox{\multilead B}}}_\pm &= ( 1 \pm |S|^2) T_3 \mp |S|^2 \nonumber\\ \langle\langle (Q/e)^2 \rangle\rangle^{{\hbox{\multilead B}}}_\pm & = (1 \pm |S|^2) \bigl[ (1 - T_{13}) T_{13} + (1 - T_{23}) T_{23} \nonumber\\ & \quad \mp |S|^2(T_3 -1 )^2 \bigr].\end{aligned}$$ Particles with anti-symmetric exchange and $\delta x=0$ exhibit no partitioning and the noise vanishes,[@burkard:00] as $P^{{\hbox{\multilead B}}}_{0,-} = P^{{\hbox{\multilead B}}}_{2,-} = 0$ and $P^{{\hbox{\multilead B}}}_{1,-} = 1$, i.e., each of the incoming particles is transmitted in a different outgoing lead.[@burkard:00] In the nonsymmetric case $T_3\ne 1$ (note that $T_{13}+T_{14}=1$ and $T_{14}=T_{23}$ in the symmetric situation), an exchange effect shows up already in the average transmitted charge $\langle Q/e \rangle^{{\hbox{\multilead B}}}_\pm=(1\pm |S|^2)T_3 \mp |S|^2$. Experimental Verification ========================= In order to observe exchange effects in an experiment, we propose to test for the predicted non-linearity in the transport: comparing the average transmitted charge $\langle Q^{\scriptscriptstyle (1)}\rangle$ for a single flux pulse with the one ($\langle Q^{\scriptscriptstyle (2)}\rangle$) injected with doubled voltage, our analysis predicts the non-trivial result $\langle Q^{\scriptscriptstyle (2)}\rangle/2\langle Q^{\scriptscriptstyle (1)}\rangle = 1+2\xi^2 k_0^2$, cf., Eqs. (\[eq:qminus\_qpc\]); for a transmission resonance, $\langle Q^{\scriptscriptstyle (2)}\rangle/2\langle Q^{\scriptscriptstyle (1)}\rangle = 1-2\xi k_0+2\xi^2 k_0^2$, cf., Eqs.(\[eq:qminus\_res\]). Given a pulse length of duration $\sim 10^{-10}~{\rm s}$, such an experiment requires a device with a phase coherence length beyond $10~\mu{\rm m}$ and is preferably carried out on a quantum point contact where Coulomb effects are less prominent. In this case, we have to compare the spread in energy $\hbar v_{\rm \scriptscriptstyle F}/\xi$ of the wave packet with the charging energy $e^2/C$, where $C \sim \varepsilon L$ denotes the junction capacitance ($L$ the length of the point contact). For a sharp transmission step, we have $\xi/L<1$ small and the relation $\xi/L < \varepsilon (\hbar c/e^2)(v_{\rm \scriptscriptstyle F}/c)$ can be satisfied. Conclusion ========== In summary, we have studied effects of the exchange symmetry on the transport statistics in mesoscopic systems. We find a strong dependence (and possibly huge enhancement) of the average charge transmission on the exchange symmetry, provided that the wave function sufficiently overlap. For antisymmetric exchange such overlap pushes weight of the two-particle wave function to higher energies and combined with the energy dependent scattering this leads to the observed enhancement. For multi-channel/multi-lead setups, exchange effects can already be observed for energy-independent scattering probabilites. There the origin of the effect is rooted in simple Pauli-blocking. Furthermore, we have proposed a experiment to test for the most striking effect of exchange; the non-linearity in transport. We thank M. Indergand, A.V. Lebedev, M.V. Lebedev, and M.V. Suslov for discussions and acknowledge financial support from the CTS-ETHZ and the Russian Foundation for Basic Research (06-02-17086-a). Equivalence Between the First and Second Quantized Picture {#sec:equiv} ========================================================== Finally, we motivate the form (\[eq:lorentz\]) for the $k$-space amplitude $f_1(k)$ of the pulse-generated wave packet. In the adiabatic limit, a voltage pulse $V(t)$ applied at $x=0$ adds a phase $\phi(t)=-(e/\hbar) \int_{-\infty}^t dt' V(t')$, $e>0$, to the wave function across the origin [@lesovik:94]. We derive the scattering matrix $U_{k',k}$ describing the transitions from $k$ states at $x<0$ to $k'$ states at $x>0$. This is conveniently done in real space, where a particle described by the time-retarded state $|\tilde{x} \rangle$ with $\langle x,t|\tilde{x} \rangle = \delta(x- v_\text{F}t)$ for $x<0$ evolves to $\exp [i \phi(-\tilde{x}/ v_\text{F})]|\tilde{x}\rangle$ at $x>0$ under the action of the voltage pulse ($\tilde{x}=x-v_{\rm \scriptscriptstyle F} t$ is the retarded variable). An arbitrary incoming state at $x<0$ then is transformed into the corresponding outgoing state at $x>0$ via the unitary scattering operator $\hat U=\exp[i\int\! d \tilde{x} \phi(-\tilde{x}/v_{\rm \scriptscriptstyle F}) \Psi^\dagger (\tilde{x}) \Psi(\tilde{x})]$ with the time-retarded field operator $\Psi(\tilde x)$. We introduce the fermionic annihilation operators $a_k$, generating basis states $\exp (i k \tilde{x})$, and transform to $k$-space via $\Psi(\tilde x) = \int (dk/2\pi)\exp (i k \tilde x) a_k$ to find the matrix elements $U_{k',k} = U (k'-k) = v_{\rm \scriptscriptstyle F}\!\int\!dt \exp[i\phi(t)+i(k'-k) v_{\rm \scriptscriptstyle F} t]$. The scattering amplitude then relates to the amplitude $f_1(k)$ in (\[eq:lorentz\]) via $U_{x_1}(q)\!=\!-\sqrt{4\pi \xi}f_1(k_{\rm \scriptscriptstyle F} + q)$ $e^{ik_{\rm \scriptscriptstyle F}v_{\rm \scriptscriptstyle F} t_1}+2\pi\delta(q)$, cf. Eq. (\[eq:ureg1\]). The voltage pulse transforms the Fermi sea $|{\Phi}_{\rm \scriptscriptstyle F}\rangle$ to $|\bar{\Phi}_{\rm \scriptscriptstyle F}\rangle = \hat{U}_{x_1} |{\Phi}_{\rm \scriptscriptstyle F}\rangle$; the calculation of the density matrix at zero temperature $$\begin{aligned} \label{eq:matrix1} \langle \bar{\Phi}_{\rm\scriptscriptstyle F}| a^\dagger_{k'} & a^{\vphantom{\dagger}}_k | \bar{\Phi}_{\rm\scriptscriptstyle F} \rangle \! =\!\!\! \int \!\! \frac{dk''}{2\pi} U_{x_1}^*\! (k' - k'') U_{x_1} \!(k - k'') \Theta(k_{\rm\scriptscriptstyle F}-k'')\nonumber\\ =&\langle {\Phi}_{\rm\scriptscriptstyle F} | a^\dagger_{k'} a^{\vphantom{\dagger}}_k |{\Phi}_{\rm\scriptscriptstyle F}\rangle + f_1^*(k') f^{\vphantom{*}}_1(k),\end{aligned}$$ involves the transformed annihilation operators $$\label{eq:trans} \hat U^\dagger a_k \hat U = a_k + \int \frac{dk'}{2\pi} U^\text{reg} (k - k') a_{k'}$$ with $U^\text{reg} (q)$ defined in Eq. (\[eq:ureg\]). It tells us that the voltage pulse leaves back a filled Fermi sea above which an excitation is created with an amplitude $f_1(k)$ given by Eq. (\[eq:lorentz\]). Furthermore, we find that the two-particle correlator $\langle \bar{\Phi}_{\rm \scriptscriptstyle F}| a^\dagger_{k} a^\dagger_{k'} a^{\vphantom{\dagger}}_{k'} a^{\vphantom{\dagger}}_k |\bar{\Phi}_{\rm \scriptscriptstyle F} \rangle$ vanishes for $k,k' > k_{\rm\scriptscriptstyle F}$ and hence pair-excitations are absent. To observe the effects of the exchange symmetry, the overlap integrals are important. To create two particles, two pulses need to be applied in sequence, $\hat U= \hat U_{x_2} \hat U_{x_1}$. A straightforward calculation shows that $$\label{eq:matrix2} \langle \Phi_{\rm\scriptscriptstyle F} | \hat U_{x_1}{\!}^\dagger \hat U_{x_2}{\!}^\dagger a^\dagger_{k'} a^{\vphantom{\dagger}}_k \hat U_{x_2} \hat U_{x_1} | \Phi_{\rm\scriptscriptstyle F} \rangle = \langle \Phi_{\rm\scriptscriptstyle F} | a^\dagger_{k'} a^{\vphantom{\dagger}}_k | \Phi_{\rm\scriptscriptstyle F}\rangle + \int \frac{dk''}{2\pi} \frac{[f^*_1(k') f^*_2(k'') - f^*_2(k') f^*_1(k'')] [f_1(k) f_2(k'') - f_2(k) f_1(k'')]}{1 - |S|^2}, \nonumber$$ i.e., the resulting state is the unperturbed Fermi system with an antisymmetrized two particle wave packet on top of it. Therefore, the first quantized picture can be used even for two voltage pulses resulting in a two particle wave function in the first quantized picture and Eq. (\[eq:fcs\_2wp\]) gives the full counting statistics of the two transmitted electrons. Many Flux Lorentzian Voltage Pulse {#sec:many_flux} ================================== Applying a Lorentzian shaped $n$ flux voltage pulse $V_n (t) = - (2 n v_{\rm\scriptscriptstyle F} \xi \hbar/e)/[(v_{\rm\scriptscriptstyle F} t)^2 + \xi^2]$, an even more drastic nonlinear effect is expected than found for the two particle case. The phase factor $\exp [i \phi(t)]$ is the $n$-th power of the single pulse phase factor. The Fourier transformation of the phase factor can be carried out using Cauchy’s formula. We obtain $$\label{eq:u_n} U^\text{reg}_n (q) = 2\pi (2\xi) e^{-\xi q} \Theta(q) \sum_{l=1}^n \frac{n! (-1)^l (2 \xi q)^{l-1} }{(l-1)! l! (n-l)!}.$$ The kernel is then given by $$\begin{gathered} \label{eq:k_n} K_n (q) = - (2\pi)^2 n (2\xi) \delta(q) + (2\pi)^2 (2\xi)^2 e^{-2 \xi q} \Theta(q) \\ \times\sum_{l,m=1}^n \frac{ (n!)^2 (-1)^{l+m} (2\xi q)^{l+m-2}}{(l-1)! l! (n-l)! (m-1)! m! (n-m)!}\end{gathered}$$ To calculate the average transmitted charge, Eq. (\[eq:k\_n\]) need to be integrated over the Fermi sea, c.f. Eq. (\[eq:mean\_charge\]). The averaging over the zero temperature Fermi sea leads to $$\begin{gathered} \label{eq:kav_n} |f(k)|_n^2 = \int \frac{dk'}{2\pi} n(k') K(k-k') = 2\pi (2\xi) \Theta(k-k_{\rm\scriptscriptstyle F}) \\ \times\sum_{l,m=1}^n \frac{ (n!)^2 (-1)^{l+m} \Gamma(l+m-1, 2 \xi (k- k_{\rm\scriptscriptstyle F}) )} {(l-1)! l! (n-l)! (m-1)! m! (n-m)!}\end{gathered}$$ where $\Gamma(n,x_0) =\int_{x_0}^\infty dx\, x^{n-1} e^{-x}$ denotes the incomplete gamma function. The average charge transmitted is given by $\langle Q/e \rangle_n = \int (dk/2\pi) |f(k)|_n^2 T_k$. The result for the scattering resonance can then be calculated, $$\begin{aligned} \label{eq:res_n} \langle Q/e \rangle_2 &= 2 \langle T^\text{res}\rangle (1 - 2 \xi k_0 + 2 \xi^2 k_0^2) \nonumber\\ \langle Q/e \rangle_3 &= 3 \langle T^\text{res}\rangle (1 - 4 \xi k_0 + 8 \xi^2 k_0^2 - 16 \xi^3 k_0^3/3 \nonumber\\ &\quad+ 4 \xi^4 k_0^4/3) \nonumber\\ \langle Q/e \rangle_4 &= 4 \langle T^\text{res}\rangle (1 -6 \xi k_0 + 18 \xi^2 k_0^2 -68 \xi^3 k_0^3/2 \nonumber \\ &\quad+ 14 \xi^4 k_0^4 - 4 \xi^5 k_0^5 + 4 \xi^6 k_0^6/9) ,\end{aligned}$$ using the fact that the incomplete gamma function for integer $n$ is expressible through a polynomial times an exponential. For the quantum point contact, the equivalent expressions read $$\begin{aligned} \label{eq:qpc_n} \langle Q/e \rangle_2 &= 2 \langle T^\text{qpc}\rangle (1 + 2 \xi^2 k_0^2) \nonumber\\ \langle Q/e \rangle_3 &= 3 \langle T^\text{qpc}\rangle (1 + 4 \xi^2 k_0^2 - 8 \xi^3 k_0^3/3 + 4 \xi^4 k_0^4/3) \nonumber\\ \langle Q/e \rangle_4 &= 4 \langle T^\text{qpc}\rangle (1 + 6 \xi^2 k_0^2 - 8 \xi^3 k_0^3 + 22 \xi^4 k_0^4/3 \nonumber \\ & \quad - 8 \xi^5 k_0^5/3 + 4 \xi^6 k_0^6/9).\end{aligned}$$ Increasing the voltage, $V_n \propto n$, which corresponds to sending more electrons through the scattering region, there is a huge nonlinear increase in the transmitted charge if $\xi k_0 \gg 1$. [10]{} L.S. Levitov and G.B. Lesovik, JETP Lett. [**58**]{}, 230 (1993). L.S. Levitov and G.B. Lesovik, JETP Lett. [**55**]{}, 555 (1992). For constant voltage, binomial statistics is correct if one measures the particle flux away from the scattering region at distances $L\gg v_{\rm\scriptscriptstyle F} \hbar /e V$, see G.B. Lesovik and N.M. Chtchelkatchev, JETP Lett. [**77**]{}, 393 (2003). B.A. Muzykantskii and Y. Adamov, Phys. Rev. B [**68**]{}, 155304 (2003). H. Lee and L.S. Levitov, cond-mat/9507011 (1995). L.S. Levitov, H.W. Lee, and G.B. Lesovik, J. Math. Phys. [**37**]{}, 4845 (1996). D.A. Ivanov, H. Lee, and L.S. Levitov, Phys. Rev. B [**56**]{}, 6839 (1997). A.V. Lebedev, G.B. Lesovik, and G. Blatter, Phys. Rev. B [**72**]{}, 245314 (2005). J. Keeling, I. Klich, and L.S. Levitov, Phys. Rev. Lett. [**97**]{}, 116403 (2006). M. Büttiker, Phys. Rev. B [**46**]{}, 12485 (1992). G.B. Lesovik, habilitation thesis, ISSP RAS, 1997. G. Burkard, D. Loss, and E.V. Sukhorukov, Phys. Rev. B [**61**]{}, R16303 (2000). G.B. Lesovik, F. Hassler, and G. Blatter, Phys. Rev. Lett. [**96**]{}, 106801 (2006), cond-mat/0507200. S. Gustavsson, R. Leturcq, B. Simovic, R. Schleser, T. Ihn, P. Studerus, K. Ensslin, D.C. Driscoll, and A.C. Gossard, Phys. Rev. Lett. [**96**]{}, 076605 (2006). L.D. Landau and E.M. Lifshitz, [*Quantum Mechanics*]{}, vol. 3 of [*Course of Theoretical Physics*]{} (Pergamon Press, London, 1958). The exponential decrease $\propto \exp(-2k_0\xi)$ in $\langle T^\text{res}\rangle$ guarantees that $\langle Q/e \rangle^\text{res}\ll 2$ for $\xi k_0 >1$. M.V. Suslov and M.V. Lebedev, unpublished. Note that in the experiment [@kozhevnikov:00] on photon-assisted noise [@lesovik:94] instead of a voltage pulse localized at a specific position a bias voltage over the whole sample was used. This procedure might not work in our case as our effect is more sensitive and depends on the details of the voltage drop across the sample. G.B. Lesovik and L.S. Levitov, Phys. Rev. Lett. [**72**]{}, 538 (1994). H.W. Lee and L.S. Levitov, cond-mat/9312013 (1993). 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--- abstract: | We study the network structure of Wikipedia (restricted to its mathematical portion), MathWorld, and DLMF. We approach these three online mathematical libraries from the perspective of several global and local network-theoretic features, providing for each one the appropriate value or distribution, along with comparisons that, if possible, also include the whole of the Wikipedia or the Web. We identify some distinguishing characteristics of all three libraries, most of them supposedly traceable to the libraries’ shared nature of relating to a very specialized domain. Among these characteristics are the presence of a very large strongly connected component in each of the corresponding directed graphs, the complete absence of any clear power laws describing the distribution of local features, and the rise to prominence of some local features (e.g., stress centrality) that can be used to effectively search for keywords in the libraries. **Keywords:** Online mathematical libraries, Wikipedia, MathWorld, DLMF, complex networks, text search. author: - | Flavio B. Gonzaga$^{1,2}$\ Valmir C. Barbosa$^{1,}$[^1]$\mbox{\ }$\ Geraldo B. Xexéo$^1$\ \ $^1$Programa de Engenharia de Sistemas e Computação, COPPE\ Universidade Federal do Rio de Janeiro\ Caixa Postal 68511\ 21941-972 Rio de Janeiro - RJ, Brazil\ \ $^2$Núcleo de Ciência da Computação\ Universidade Federal de Alfenas\ 37130-000 Alfenas - MG, Brazil bibliography: - 'mlibs.bib' title: 'The Network Structure of Mathematical Knowledge According to the Wikipedia, MathWorld, and DLMF Online Libraries' --- Introduction {#intr} ============ Until a few decades ago, before it became commonplace to search the Web for information and knowledge, people desiring quick access to some mathematical concept or formula used to resort to printed encyclopedias or handbooks, such as the compilation by Abramowitz and Stegun [@as65], the more specialized tables put together by Gradshteyn and Ryzhik [@gr00], or still others [@i93; @bsmm04]. Of such volumes, the undisputed citations champion seems to be Abramowitz and Stegun’s [@bclo11], whose work has since been methodically expanded [@l03] into a NIST-sponsored publication [@olbc10]. Lately, though, the situation has become, if anything, more complex. For, while those printed works continue to be used and cited widely and their ranks continue to be enlarged by the addition of new works of a similar genre [@gbl08], the premier source, at least for a first approach, has undoubtedly become the Web. In fact, it seems safe to state that most mathematics-related queries on Google return Wikipedia[^2] or Wolfram MathWorld[^3] pages as prominently ranked. As mentioned, however, printed and online material still coexist and, curiously, movement has taken place in both directions: while in one direction MathWorld material has found its way into Weisstein’s encyclopedia [@w02], in the other the NIST volume has been turned into the Digital Library of Mathematical Functions, DLMF.[^4] Here we aim to explore the structure of mathematical knowledge as reflected in these three online libraries. By “structure” we do not mean the organization of material into the many mathematical areas and subareas. Nor do we mean the coalescence of all deduction chains that is behind all of mathematics and inherently amounts to an acyclic directed graph [@c02], i.e., one with no directed cycles. We mean, rather, the no longer acyclic directed graphs that reflect all the cross-referencing that took place as those libraries were created by several collaborators (and still takes place as the libraries evolve). Exploring their graph structures from the perspective of such hypertextual interconnections amounts to applying some of the complex-network notions and metrics developed during the past fifteen years or so, much as has been done so successfully to various other fields [@bs03; @nbw06; @bkm09]. It also amounts to a chance to globally view all the material compiled into each library and inquire, from a network-theoretic perspective, what traces remain, if any, as telltale signs of the essentially very distinct methods of construction employed to build them, all of a collaborative nature but supposedly more and more controlled as we move from Wikipedia to MathWorld and then to DLMF. In our analyses we use several frequency data, of both a network-wide nature as well as node-related, aiming not only to describe the libraries’ properties as such data reveal them, but also to discover how these properties relate to the libraries’ robustness in the face of accidental or intentional loss of material and to their ease of search in response to text queries. What has turned up is a collection of results that both sets the three mathematical libraries apart from the wider English-language Wikipedia and from the much wider Web, and at the same time groups the three libraries together insofar as they share important properties. Some of the most significant results include the characteristic that a very large fraction of each library’s pages are packed together in the sense of mutual reachability; the presence of clear signs that all three libraries result from decisions regarding the deployment of links that are leveraged by technical knowledge (rather than, say, some nontechnical measure of a page’s relevance, such as popularity); and the discovery of successful criteria for guiding text search within the libraries’ pages that differ significantly from those most commonly used (e.g., by Google). We proceed in the following manner. First, in Section \[graphs\], we introduce the five directed graphs that we use in all analyses (two for Wikipedia, two for MathWorld, one for DLMF) and also some basic notation. We then move, respectively in Sections \[global\] and \[local\], to a study of these graphs’ global and local network-theoretic features. Section \[local-scc\] is dedicated to an analysis of the five graphs’ robustness when nodes are lost either as a result of some random process or as a deterministic function of the graphs’ local features. We continue with Section \[local-search\], where we investigate the effect of such features in the ranking of nodes when responding to text queries. We conclude in Section \[concl\]. Five directed graphs {#graphs} ==================== In all three libraries it is possible to reach the technical-content pages by navigating through a hierarchy of specialized subdivisions from the main portal (the so-called category pages). Once the content pages are reached, further navigation is possible through the links that lead from one such page to another. Each of the directed graphs with which we work has a node for each content page and directed edges that reflect inter-page links. In all cases, links leading from a page to itself are ignored when building the graph, so no self-loops exist. Similarly, should multiple links exist from a page to another, only one edge is created in the graph between the corresponding nodes. In the case of Wikipedia and MathWorld, links can be categorized into those appearing in a page’s main text and those that are given in the page’s “See also” section when it exists. We perceive these two link types as playing entirely different roles. While in-text links are generally meant to clarify some of the terms used in the page, being therefore meant for quick side lookups before continuing on the main text, See-also links are used to point to pages where related material is to be found. For this reason, we use two different graphs for each of Wikipedia and MathWorld. They both have the same node set, but their edge sets differ, one reflecting in-text as well as See-also links, the other reflecting See-also links only. The case of DLMF requires no such special treatment. Although its pages, too, contain special, “Referenced by” links, such links are simply antiparallel versions of the library’s non-Referenced-by links. That is, page $a$ contains a non-Referenced-by link to page $b$ if page $b$ contains a Referenced-by link to page $a$. Referenced-by links in DLMF are therefore redundant as far as building its directed graph is concerned. They are for this reason ignored. These observations amount to five different graphs with which to work, as summarized in Table \[table-graphs\]. In the table, for each of the libraries and, when applicable, taking See-also links into account, we give the time frame within which the content pages were downloaded and the notation we use to refer to the corresponding graph. Some additional basic notation to be used throughout is the following. Given the graph under consideration, we let $n$ stand for its number of nodes and $m$ for its number of edges. For node $i$, $I_i$ is its set of in-neighbors (nodes from which edges are directed toward $i$) and $O_i$ its set of out-neighbors (nodes toward which edges are directed from $i$). Its in-degree is $\delta_i^+=\vert I_i\vert$, its out-degree is $\delta_i^-=\vert O_i\vert$, and its number of neighbors when edge directions are disregarded (henceforth referred to simply as its degree) is $\delta_i=\vert I_i\cup O_i\vert\le\delta_i^++\delta_i^-$. Clearly, it holds that $\max\{\delta_i^+,\delta_i^-\}\le\delta_i$. For any two nodes $i$ and $j$, $d_{ij}$ is the distance from $i$ to $j$, that is, the number of edges on a shortest directed path leading from $i$ to $j$. If none exists, then $d_{ij}=\infty$. We let $R_i$ be the set of nodes $j$ such that $0<d_{ij}<\infty$. Note that $R_i=\emptyset$ if and only if node $i$ is a sink, i.e., $O_i=\emptyset$. Global features {#global} =============== We give six global features for each graph. The first two are straightforward and provide simple relationships between the graph’s number of nodes, $n$, and its number of edges, $m$. The first one is simply the graph’s mean in-degree, denoted by $\delta^+$ and given by $$\delta^+ =\frac{1}{n}\sum_i\delta_i^+ =\frac{m}{n}$$ (necessarily equal to the graph’s mean out-degree). The second feature is the graph’s mean degree. Denoting it by $\delta$, we have $$\delta^+ \le\delta =\frac{1}{n}\sum_i\delta_i \le\frac{1}{n}\sum_i(\delta_i^++\delta_i^-) =2\delta^+.$$ Both $\delta^+$ and $\delta$ work as indicators of the graph’s edge density relative to its number of nodes. The value of $\delta$, in particular, may swing toward either of its bounds, $\delta^+$ and $2\delta^+$, indicating in the former case that every edge’s antiparallel counterpart is also present in the graph and in the latter case that none is. On average, then, the fraction of $\delta$ corresponding to antiparallel edge pairs is given by $(2\delta^+-\delta)/\delta=2\delta^+/\delta-1$. Our next global feature is the fraction $S$ of $n$ that corresponds to the nodes inside the graph’s largest strongly connected component (GSCC henceforth, where $G$ is for “giant”). A strongly connected component is either a singleton whose only member, say node $i$, is such that $i\notin R_j$ for every node $j\in R_i$ (no directed path exists back from any node that can be reached from $i$ through a directed path), or a larger set that is maximal with respect to the property that $j\in R_i$ for any two of its members $i$ and $j$ such that $j\neq i$. In the latter case, then, a directed path exists between any two distinct nodes inside the strongly connected component. Informally, the value of $S$ can be regarded as an indication of the network’s “degree of acyclicity.” If the graph is acyclic, then all its strongly connected components are singletons and $S=1/n$. The other extreme corresponds to the case in which all nodes are in the GSCC, so $S=1$. The fourth and fifth global features are both related to classifying a graph vis-à-vis the so-called small-world criteria [@ws98; @ajb99], namely small distances and large transitivity. We address the first criterion by computing the average distance between any two distinct nodes, so long as only finite distances are considered. We denote this average by $\ell$, which is then such that $$\ell=\frac{1}{N}\sum_i\sum_{j\in R_i}d_{ij},$$ where $N$ is the number of $i,j$ pairs contributing to the double summation. As for the second criterion, that of transitivity, we follow the usual trend of disregarding edge directions and computing the resulting graph’s clustering coefficient in its most common formulation [@n10]. If $C$ is the clustering coefficient, then this formulation lets $C=3t/T$, where both $t$ and $T$ refer to node triples in the graph, e.g., $i,j,k$. The value of $t$ is meant to reflect the number of triangles in the graph, that is, those triples in which an edge connects $i$ and $j$, another connects $j$ and $k$, and yet another connects $i$ and $k$. The value of $T$, on the other hand, counts the triples that are arranged as three-node (two-edge) paths. The factor $3$ in the numerator of the ratio defining $C$ reflects the fact that there are three triples of the latter type for each triangle in the graph. It follows that $0\le C\le 1$ (no transitivity through full transitivity). In our analysis of each graph’s clustering coefficient $C$, we present it side-by-side with the value it would have if every node $i$ continued to have the same degree $\delta_i$ but the connections were made at random [@n10]. This value, denoted by $C'$, is given by $$C'=\frac{(\delta^{(2)}-\delta)^2}{n\delta^3},$$ where $\delta^{(2)}=(1/n)\sum_i\delta_i^2$. Our last global feature is in fact a series of four assortativity coefficients. Each one is the Pearson correlation coefficient of two length-$m$ sequences of numbers. If $\alpha_1,\alpha_2,\ldots,\alpha_m$ and $\beta_1,\beta_2,\ldots,\beta_m$ are the sequences, $\mu_\alpha$ and $\mu_\beta$ are the corresponding means, and $\sigma_\alpha$ and $\sigma_\beta$ are the corresponding standard deviations, this coefficient is $$r_{\alpha,\beta}=\frac {(1/m)\sum_e\alpha_e\beta_e-\mu_\alpha\mu_\beta} {\sigma_\alpha\sigma_\beta}.$$ The original assortativity coefficient is obtained by letting $\alpha_e=\delta_i^-$ and $\beta_e=\delta_j^+$ for $e$ the edge directed from $i$ to $j$ [@n02; @n03]. That is, it measures how correlated the out-degrees of the edges’ tail nodes are with the in-degrees of the edges’ head nodes. A shorthand for this formulation is to use out,in in place of $\alpha,\beta$. We get the other three variations by selecting the other possible combinations (in,out; out,out; in,in) [@ffgp10; @ppz11]. The global features of the graphs in Table \[table-graphs\] are shown in Tables \[table-global1\] and \[table-global2\], which include an additional row for the directed graph, denoted by $W^+$, that corresponds to the entire English-language Wikipedia of a relatively recent past [@cscbdlc06; @zbsd06]. Table \[table-global1\], moreover, contains one further row for the whole Web, now based on data from an older past [@bkmrrstw00].[^5] The corresponding directed graph is denoted by $W^*$. Not all global features are available for $W^+$ or $W^*$, as indicated by blank entries in the tables. Graphs are arranged in Tables \[table-global1\] and \[table-global2\] in nonincreasing order of $n$, then in decreasing order of $m$. The data shown in Table \[table-global1\] indicate that edge density relative to the number of nodes, as given by $\delta^+$, has the same order of magnitude for most graphs, the exception being $W'$, the Wikipedia graph based exclusively on See-also links, whose $\delta^+$ value is one order of magnitude lower. Wikipedia contributors to the mathematical pages, therefore, seem to deploy See-also links considerably less methodically than those who contribute to MathWorld. It is also worth noting that the five mathematics-related graphs have fairly different values for the ratio $2\delta^+/\delta-1$, pointing at $W'$ as the graph with the fewest antiparallel edge pairs contributing to degrees on average, and to $M'$, the MathWorld graph based on See-also links, as having the most. Once again, then, MathWorld contributors appear more meticulous at providing cross-referencing information of the See-also form. One of the most striking contrasts in Table \[table-global1\] concerns the value of $S$, the size of the graph’s GSCC relative to $n$. While for the Web graph $W^*$ our current best estimate places about $28\%$ of the nodes inside the GSCC, for the Wikipedia graph $W^+$ and most of the mathematical-library graphs we have been considering the GSCC encompasses substantially more nodes (between $62$ and $82\%$). The exception, once again, occurs on account of graph $W'$, whose GSCC is sized at a mere $2\%$ of the nodes, and which as we have noted is only very sparsely interconnected by the See-also links. The remaining data in Table \[table-global1\] refer to $\ell$ and to $C$, a graph’s average path length (in the directed sense) and clustering coefficient (in the undirected sense), respectively. We first note that, for six of the seven graphs, $\ell$ is proportional to $\ln n$ by a constant of the order of $10^{-1}$, the exception being $W'$, for which the proportionality constant is roughly $1.45$ (this comes from the substantially larger distances in comparison to $W$, as expected from the substantially lower $\delta^+$ value). In all cases, however, distances are on average very small given the value of $n$, so all seven graphs qualify as small-world structures. Moreover, as is usually but not always the case [@n10], in all five mathematics-related graphs the value of $C$ is noticeably larger than that of $C'$. In fact, except for the DLMF graph $D$, $C$ surpasses $C'$ by a factor of at least two orders of magnitude. The construction of $D$, which has $C\approx 5.64C'$, seems to have been guided by forces that prevent the formation of triangles more than they do in the other four cases. One possible explanation is that, in comparison to Wikipedia or MathWorld, each DLMF page contains substantially more material, which in fact is reflected in the low number of nodes of graph $D$. Table \[table-global2\] contains all four assortativity coefficients for all five mathematics-related graphs and $W^+$, the unrestricted Wikipedia graph. The vast majority of all values is of the order of $10^{-2}$ at most, being therefore sufficiently near zero for the sequences involved to be taken as uncorrelated. In general this is indicative either of a random pattern of connections (which is not the case) or that criteria for edge deployment are at work that make no reference whatsoever to in- or out-degrees (which is more plausibly the case). Curiously, though, the same holds also for the only two exceptions, $W^+$ and $D$, for which the moderately negative but nonnegligible value of $r_\mathrm{out,in}$ is suggestive that in these two graphs connections are effected in such a way that promotes a small but noticeable degree of disassortative mixing of tail nodes’ out-degrees with head nodes’ in-degrees. That is, there is a slight tendency of nodes with larger (smaller) out-degrees to connect out to nodes with smaller (larger) in-degrees. This tendency is quantified very similarly by $r_\mathrm{out,in}$ for both $W^+$ and $D$ ($-0.150$ in the former case, $-0.169$ in the latter). Perhaps the aforementioned fact that the typical DLMF page contains more material than the other libraries’ pages somehow makes $D$ resemble $W^+$ in this one aspect. Local features {#local} ============== Presenting a graph’s local features requires that we value each feature of interest for each node and then provide some probability distribution of that feature over the entire graph. In this section we work with the feature’s complementary cumulative distribution (CCD henceforth), denoted by $F(z)$ for an admissible feature value $z$, which is the probability that a randomly chosen node has a feature value that surpasses $z$. We compute $F(z)$ as the fraction of $n$ representing the nodes for which the feature is valued beyond $z$. Clearly, if for a graph the feature in question is never valued beyond $Z$, then $F(z)=0$ for all $z\ge Z$. The most widely studied local features are a node’s in-degree, out-degree, and degree. Not only have they been measured in a variety of domains, but knowledge of how they are distributed can be used in the study of many other network properties [@nsw01]. These features are the first three we study, as characterizations of in- and out-degrees have over the years led to important discoveries regarding the Web and the unrestricted Wikipedia. Specifically, we know from at least two independent sources operating on different data that in-degrees in the Web graph (our $W^*$ graph in one case, a different version in the other) are distributed according to a power law [@ajb99; @bkmrrstw00; @dllm04]. That is, the probability that a randomly chosen node has in-degree $k>0$ is proportional to $k^{-\alpha}$ (so the corresponding CCD is approximately proportional to $k^{1-\alpha}$) for $\alpha\approx 2.1$. Similar power laws have also been reported for the graph’s out-degrees, but in this case there seems to be some disagreement [@dllm04]. As for the Wikipedia graph, $W^+$, its in-degree, out-degree, and degree distributions have all been found to follow power laws, of exponents $-2.21$, between $-2.65$ and $-2$, and $-2.37$, respectively [@cscbdlc06; @zbsd06]. Power laws are inherently scale-free [@n05], and their appearance in graphs such as $W^*$ and $W^+$ has been explained particularly well by the mechanism of edge deployment known as preferential attachment [@p76; @ba99; @baj00; @cscbdlc06]. The additional local features that we consider are the ones given in Table \[table-local\]. Four of them ($B_i$, $S_i$, $C_i$, and $G_i$) are measures of node $i$’s centrality in the graph, being therefore related to shortest directed paths in which $i$ participates in some way. The remaining three are related to search mechanisms on the Web. They are measures of how node $i$ qualifies as a hub ($y_i$) or an authority ($x_i$) in the HITS (Hyperlink-Induced Topic Search) mechanism, and the node’s page rank ($\rho_i$), which underlies Google searches. The centrality features can be computed through variations of a well-known algorithm [@ub01], and similarly the other three, though requiring iterative updates for convergence. In the case of the HITS-related features, first every $x_i$ and $y_i$ is initialized to $1$. Then the $x_i$’s and $y_i$’s are alternately updated via the rules given in Table \[table-local\]. The updating of the $x_i$’s is followed by a normalization of the resulting values so that $\sum_ix_i^2=1$, which is achieved by dividing each $x_i$ by the Euclidean norm of the vector of components $x_1,x_2,\ldots,x_n$. The updating of the $y_i$’s proceeds similarly. After convergence, all features are normalized so that $\sum_ix_i=\sum_iy_i=1$. As for the page-rank feature, once again every $\rho_i$ is initialized to $1$ and the update rule given in Table \[table-local\] is iteratively applied until convergence, at which time all $\rho_i$’s are normalized so that $\sum_i\rho_i=1$. For the two HITS-related features and page rank, our criterion for convergence has been that, for all nodes, the two latest feature values differ from each other by some quantity in the interval $[-10^{-16},10^{-16}]$. CCD plots for the local features are given in Figure \[fig-degrees\] (in-degree, out-degree, and degree), Figure \[fig-centralities\] (centralities), and Figure \[fig-searches\] (hub, authority, and page rank). One striking characteristic they all share is that no feature of any library seems to be expressible as a clear power law for any significant number of orders of magnitude. For example, although we have found the in-degree CCD for DLMF to be given approximately by a power-law of $\alpha=2.47$, this seems reasonable only for one order of magnitude (roughly between $10$ and $100$). In the case of Figure \[fig-degrees\], in particular, this widespread absence of a power law works to confirm the expectation that, in such a specialized domain as the five libraries’, it is expertise, rather than some popularity-based criterion such as preferential attachment, that guides the establishment of connections. In Figure \[fig-centralities\], the CCD plots for the $C_i$ and $G_i$ values share the peculiar property that all nodes are concentrated inside three relatively narrow centrality intervals. For each of the five libraries, first are the sink nodes, those for which $R_i=O_i=\emptyset$, having $C_i=G_i=0$. Then comes what in almost all cases is the most densely populated interval. Nodes in this interval have the relatively small centrality values typically associated with relatively large distances to the nodes in $R_i$. They are members of the graph’s largest so-called in-component, which encompasses the GSCC and all nodes from which at least one directed path leads to the GSCC. This explains the single exception, which once again concerns the small-GSCC graph of the Wikipedia library with only See-also links ($W'$). The third centrality interval contains the remaining nodes and is characterized by centrality values that in almost all cases bespeak relatively small distances to the nodes in $R_i$. These nodes lie outside the graph’s largest in-component and, once again, the single exception is relative to $W'$. Local features and GSCC disruption {#local-scc} ================================== In the graphs we have been studying, as in all graphs reflecting real-world networks, the existence of the GSCC is merely a matter of observation: we simply look for the graph’s strongly connected components and select the largest one. In a more abstract sense, however, random-graph models of networks have been studied for the existence of such components under a growth regime from relative sparseness to relative denseness (that is, as the graph’s number of nodes and/or edges is changed so that it becomes denser). Such studies were initiated with the Erdős-Rényi (ER) random graphs [@er59], which are undirected and characterized by a Poisson distribution of node degrees. Since edges do not have directions in the ER model, one looks for weakly (rather than strongly) connected components, or simply connected components, and for the GCC (rather than the GSCC). It turns out that, increasing $\delta$ (the mean degree) past $1$ as the graph becomes denser gives sudden rise to the GCC as a connected component that for the first time is set apart from the others by virtue of its size [@er60]. A similar phenomenon also occurs in many other random-graph models, including their directed variations with regard to the rise of the GSCC [@k90; @mr95; @mr98; @dms01; @nsw01]. Another similar phenomenon, often called site percolation, is the breakdown of the GCC or GSCC when nodes are continually isolated from the rest of the graph by the removal of all edges incident to them. In the case of ER graphs, for example, the GCC breakdown is expected to happen after a fraction $1-1/\delta$ of the nodes has been randomly isolated [@b01], provided $\delta>1$ to begin with (i.e., provided there really is a GCC initially). Results of this sort have been obtained also for undirected graphs with degrees obeying a scale-free distribution. However, unlike the ER graphs, with their degrees closely clustered about the mean, now there may exist high-degree nodes, so it makes sense to look at targeted as well as random node-isolation processes. As it turns out, for $\alpha=2.5$ (which is thought to describe the Internet graph) the GCC is only expected to disappear after at least $99\%$ of the nodes have been randomly isolated, although for relatively small graphs this can be as low as about $80\%$ [@cebh00]. Targeting highest-degree nodes first, though, implies that isolating fewer than $20\%$ of the nodes is expected to suffice [@ajb00]. We know of no similar studies for directed random-graph models regarding the impact of node isolation on the graph’s GSCC. So, despite the figures given above, we are essentially left without any meaningful clue as to what to expect when we conduct node isolation in our five mathematical libraries. The results we present in this section describe the evolution of $S$, the fraction of $n$ inside the GSCC, as nodes are isolated either randomly or targeting first the non-isolated node for which a specific local feature is highest. In the former case we provide the average value obtained from ten independent trials. As for the local feature in question, we report on all ten discussed in Section \[local\]. In all cases, node isolation is performed until no strongly connected component has more than one node. When isolation stops, then, all remaining nodes are either isolated (no in- or out-neighbors) or part of an acyclic portion of the current graph. Our results appear in Figure \[fig-break\], where the breakdown fractions for random isolation are seen to be in the $[0.4,0.6]$ interval for the non-See-also graphs, along with roughly $0.3$ for $M'$ and less than $0.01$ for $W'$. If we once again except $W'$, with its frail GSCC, and maybe $M'$ as well, we are left with figures that indicate what seem to be quite resilient GSCCs in $M$, $W$, and $D$. As we turn to the isolation of nodes following one of the local features, the data in Figure \[fig-break\] reveal that the specific feature in question is practically irrelevant, with the exception of graph centrality and closeness centrality in all cases but that of $W'$. These two features are, respectively, the second and third least effective means we have found to break the GSCC (following the random method, which is the least effective). Except for graph and closeness centrality, the data also reveal a breakdown fraction of about $0.2$ for $W$ and $D$, then a little above $0.1$ for $M$, then a little below $0.1$ for $M'$ and, finally, less than $0.01$ for $W'$. Targeting nodes based on any one of these local features, then, reveals a dividing line between the See-also and non-See-also graphs as well, with $W'$ once more at the lowest end and $M'$ in between $W'$ and the non-See-also group. Local features and text search {#local-search} ============================== Google’s search engine grew out of the notion of page rank, one of the ten local features examined in Section \[local\]. Page rank, however, is no more of a node descriptor than any of the other local features, so in principle it is at least conceivable that any of the others might be used instead. We explore such possibility in this section for each of the five directed graphs $W$, $W'$, $M$, $M'$, and $D$, regarding the search, in their nodes’ texts, for a number of the top keywords in mathematics as reported at the Microsoft Academic Search (MAS) site[^6] as of November 1, 2012. We follow the standard method outlined in [@br11]. For each graph and local feature, and for a given query (one of the aforementioned keywords), this method begins by identifying a list $A$ of answer nodes (sorted by nonincreasing feature value) as well as a set $R$ of relevant nodes. It then proceeds to calculating the well-known Precision and Recall metrics for each $k=1,2,\ldots,\vert A\vert$. These are given by the fraction of $k$ corresponding to the nodes in the size-$k$ prefix of list $A$ that are also in $R$ (Precision) and the fraction of $\vert R\vert$ that corresponds to these shared nodes (Recall). Note that, the more relevant nodes are ranked first in $A$, the higher Precision values are obtained for a larger stretch of Recall values. The elements of $A$ are simply those nodes whose texts contain the keyword in question. As for $R$, normally it would be identified by a group of experts. In the absence of one, however, we identify it by resorting to all ten local features, not just the feature that is being analyzed and was used to sort $A$, and letting each one “vote” for or against each potential candidate for inclusion in $R$. Set $R$, therefore, is as much a function of the feature used to sort $A$ as it is of the others. The following steps summarize the construction of set $R$: 1. Let $X$ be the set of nodes in whose texts the desired keyword appears. If $\vert X\vert\le 10$, go to Step 5. 2. Create ten sorted lists of the nodes in $X$, each by nonincreasing order of one of the ten local features. 3. Let $Y$ be the set of nodes that appear amid the top ten nodes in a strict majority (i.e., at least six) of the ten lists. 4. Let $R:=Y$ and stop. 5. Let $R:=\emptyset$ and stop. Note that requiring $\vert X\vert>10$ for termination to occur in Step 4 is necessary to avoid the trivial case of $R=X$, which allows for no discrimination of the local features vis-à-vis one another. When the requirement is not met and termination occurs in Step 5, the query in question is dropped. Our results, given next, refer to those MAS keywords, out of the top $300$, for which the procedure above terminated in Step 4 in our experiments. Whenever such keywords numbered more than $100$, we considered only the top $100$. As it turns out, we obtained the desired $100$ keywords for all graphs but $D$, which ended up with only $14$ keywords (i.e., only $14$ of the $300$ keywords were found in more than ten nodes). Figure \[fig-pr\] contains the resulting Precision-Recall plots. They are given as Precision averages relative to eleven Recall intervals, viz. $[0,0.1),[0.1,0.2),\ldots,[1,1]$, plotted respectively at the abscissae $0,0.1,\ldots,1$. According to the data in Figure \[fig-pr\], in order to search the mathematical portion of Wikipedia through the use of local features based on graph $W$ it is best to use page rank, followed very closely by either the hub or authority feature. Should the search be based on graph $W'$, however, then one should use the hub criterion as the absolute champion. Notice, notwithstanding this, that the use of $W'$ incurs a loss of Precision of about $10\%$ relative to the use of $W$ and cannot be recommended on any grounds. Still regarding Wikipedia, our data also indicate that, if one is willing to examine the list $A$ of answer nodes past the point at which about $70\%$ of the set $R$ of relevant nodes have been covered, then the nodes’ degrees and stress centralities turn out to be the local features to be preferred for sorting $A$. Turning to MathWorld we find a wholly different picture in the data, since now the best local feature to sort $A$ is the nodes’ stress-centrality values as given by graph $M$, regardless of how much of $R$ one is willing to examine. If one is willing to examine no more than about $50\%$ of $R$, though, then the nodes’ betweenness-centrality values are equally effective. As for using graph $M'$, and unlike the case of Wikipedia, only a small loss of Precision is incurred in comparison with $M$ (about $1$ or $2\%$), but now the local feature of preference to sort $A$ is betweenness centrality, followed very closely by the nodes’ degrees (for examining up to about $60\%$ of $R$). As for DLMF, the local feature of choice is once again betweenness centrality (for up to about $40\%$ of $R$), though the nodes’ authority values are equally effective (up to about $20\%$ of $R$), and so are the nodes’ stress-centrality values and in-degrees (up to about $10\%$ of $R$). Should one be willing to examine about $50\%$ of set $R$ or more, then stress centrality becomes the local feature to be preferred. Conclusions {#concl} =========== We have studied three online mathematical libraries, viz. the mathematical portion of Wikipedia, MathWorld, and DLMF, from the perspective of network theory. To this end, we considered directed graphs whose nodes are library pages and whose edges reflect the directed pairing of pages through the links that point from one to another. In the case of Wikipedia and MathWorld, these links come in two clearly identifiable categories (those that are in-text and those in the pages’ See-also sections), so we considered two separate graphs for each of these libraries. We focused on both global and local network-theoretic properties of these graphs, aiming at characterizing them, studying their resiliency to the accidental or intentional loss of material, and also assessing how best to perform text search in the pages that their nodes stand for. Among our key finds are the presence of GSCCs that in most cases encompass node fractions substantially larger than that of the Web, indications of small-world phenomena, practically no signs of relevant assortativity in the linking patterns, and the absence of any clear power laws describing the distributions of local features. We also found that most graphs are quite resilient to the accidental loss of material, though naturally less so when we consider the intentional destruction of pages. As for searching the libraries for the occurrence of specific keywords, only for Wikipedia do the customary criteria of page rank and the HITS-related features perform best. For the smaller MathWorld and DLMF, primacy is taken by local features that hitherto do not appear to have been considered for this purpose, notably stress centrality, betweenness centrality, and the nodes’ degrees. We believe that many of these finds can be attributed to one key distinguishing property of all three libraries. Unlike what happens in several other domains, where such intangibles as affinity or popularity dictate the establishment of connections, in building these libraries what matters is how knowledgeable each contributor is on the core material being treated and on how it relates to the other topics. That this key distinction should surface in the form of measurable effects such as the networks’ structural properties and their consequences, and that this should happen despite the typically large number of often independent contributors involved, is quite remarkable. We finalize with a note on some related work on MathWorld that precedes our own analysis [@jg09]. Such work is based on a December 2008 version of the library, so it predates the one we use by some eight months (cf. Table \[table-graphs\]). Despite this relatively short span of intervening time, our graph has $25\%$ more nodes (about $3\,000$ nodes beyond that work’s $12\,000$), so we conjecture that some intermittent failure during the download process may have caused the loss of material. In [@jg09] the authors give the distributions of in- and out-degrees and of betweenness centrality. Despite the considerable difference between the two graphs, our results agree with theirs in that neither in-degrees nor out-degrees are distributed as power laws. Their betweenness-centrality distribution also appears consistent with ours, though they seem to have missed the page for “Triangle,” which we find to be one of the top ten for this local feature but they do not. They also discuss clustering, average distance, and assortativity, but the definitions they use for these quantities are not the most commonly used and are incompatible with ours. Acknowledgments {#acknowledgments .unnumbered} --------------- The authors acknowledge partial support from CNPq, CAPES, and a FAPERJ BBP grant. [^1]: Corresponding author ([email protected]). [^2]: `http://en.wikipedia.org/wiki/Portal:Mathematics`. [^3]: `http://mathworld.wolfram.com`. [^4]: `http://dlmf.nist.gov`. [^5]: Slightly more recent data seem to indicate an $S$ value of roughly $0.33$ for a similarly sized Web [@dllm04], but no estimate is given for $\ell$. [^6]: `http://academic.research.microsoft.com/RankList?entitytype=8&topDomainID=15&subDomainID=0`.

--- abstract: 'We study the birational geometry of $\overline{M}_{3,1}$ and $\overline{M}_{4,1}$. In particular, we pose a pointed analogue of the Slope Conjecture and prove it in these low-genus cases. Using variation of GIT, we construct birational contractions of these spaces in which certain divisors of interest – the pointed Brill-Noether divisors – are contracted. As a consequence, we see that these pointed Brill-Noether divisors generate extremal rays of the effective cones for these spaces.' author: - David Jensen bibliography: - 'ref.bib' title: 'Birational Contractions of $\overline{M}_{3,1}$ and $\overline{M}_{4,1}$' --- Introduction ============ The moduli spaces of curves are some of the most studied objects in algebraic geometry. In recent years, a great deal of progress has been made on understanding the birational geometry of these spaces. Examples include the work of Hassett and Hyeon on the minimal model program for $\overline{M}_g$ [@HH1] [@HH2] and the discovery by Farkas of previously unknown effective divisors on $\overline{M}_g$ [@Farkas]. Nevertheless, many fundamental questions remain open. Many of these questions can be stated in terms of the cone of effective divisors $\overline{NE}^1 ( \overline{M}_g )$. Among the first to study this cone were Eisenbud, Harris and Mumford in a series of papers proving that $\overline{M}_g$ is of general type for $g \geq 24$ [@HMum] [@HE]. A key element of these proofs is the computation of the class of certain divisors on $\overline{M}_g$. The original paper of Harris and Mumford focused on the $k$-gonal divisor in $\overline{M}_{2k-1}$, a specific case of the more general class of Brill-Noether divisors. In their argument, they use this calculation to show that the canonical class can be written as an effective sum of a Brill-Noether divisor, boundary divisors, and an ample divisor, and hence lies in the interior of $\overline{NE}^1 ( \overline{M}_g )$. The search for effective divisors with this property eventually led to the Harris-Morrison Slope Conjecture. In their work, Harris and Eisenbud discovered that all of the Brill-Noether divisors lie on a single ray in $\overline{NE}^1 ( \overline{M}_g )$. One consequence of the Slope Conjecture would be that this ray is extremal. The Slope Conjecture has recently been proven false in [@FP] and subsequently in [@Farkas], but the statement is known to hold for certain small values of $g$. In several of these cases, the statement can be proved by use of the Contraction Theorem, which states that the set of exceptional divisors of a birational contraction $X \dashrightarrow Y$ span a simplicial face of $\overline{NE}^1 (X)$ (see [@Rulla]). In other words, the Slope Conjecture has been shown to hold for small values of $g$ by constructing explicit birational models for the moduli space in which the Brill-Noether divisor is contracted. Moreover, these models arise naturally as geometric invariant theory quotients. The purpose of this paper is to carry out a pointed analogue of the discussion above in some low genus cases. In [@Logan], Logan introduced the notion of **pointed Brill-Noether divisors**. Let $Z = (a_0 , \ldots , a_r )$ be an increasing sequence of nonnegative integers with $\alpha = \sum_{i=0}^r (a_i - i)$. Let $BN^r_{d,Z}$ be the closure of the locus of pointed curves $(p,C) \in M_{g,1}$ possessing a $g^r_d$ on $C$ with vanishing sequence $Z$ at $p$. When $g+1 = (r+1)(g-d+r)+ \alpha$, this is a divisor in $\overline{M}_{g,1}$, called a **pointed Brill-Noether divisor**. Logan’s original motivation was to prove a pointed version of the Harris-Mumford general type result. In this setting, it is natural to consider an analogue of the Slope Conjecture: Is there an extremal ray of $\overline{NE}^1 ( \overline{M}_{g,1} )$ generated by a pointed Brill-Noether divisor? We consider this question in certain low-genus cases. When $g=2$, this question was answered in the affirmative by Rulla [@Rulla]. He shows that the Weierstrass divisor $BN^1_{2,(0,2)}$ generates an extremal ray of $\overline{NE}^1 ( \overline{M}_{2,1} )$ by explicitly constructing a birational contraction of $\overline{M}_{2,1}$. Our main result is an extension of this to higher genera: There is a birational contraction of $\overline{M}_{3,1}$ contracting the Weierstrass divisor $BN^1_{3,(0,3)}$. Similarly, there is a birational contraction of $\overline{M}_{4,1}$ contracting the pointed Brill-Noether divisor $BN^1_{3,(0,2)}$. As a consequence, we identify an extremal ray of the effective cone. For $g = 3,4$, there is an extremal ray of $\overline{NE}^1 ( \overline{M}_{g,1} )$ generated by a pointed Brill-Noether divisor. The proof uses variation of GIT. In particular, we consider the following GIT problem: let $Y$ be a surface and fix a linear equivalence class $\vert D \vert$ of curves on $Y$. Now, let $$X = \{ (p,C) \in Y \times \vert D \vert \text{ } \vert p \in C \}$$ be the universal family over this space of curves. In the case where $(Y, \vert D \vert )$ is $( {\mathbb{P}\,}^2 , \vert {\mathcal{O}\,}(4) \vert )$ or $( {\mathbb{P}\,}^1 \times {\mathbb{P}\,}^1 , \vert {\mathcal{O}\,}(3,3) \vert )$, the quotient of $X//Aut(Y)$ is a birational model for $\overline{M}_{3,1}$ or $\overline{M}_{4,1}$, respectively. By varying the choice of linearization, we obtain a birational model in which the specified divisor is contracted. The outline of the paper is as follows. In section 2 we provide some background on variation of GIT. In section 3, we develop a tool for studying GIT quotients of families of curves on surfaces. In particular, we construct a large class of divisors on these spaces that are invariant under the automorphism group of the surface, called Hessians. In sections 4 and 5 we then examine separately curves on ${\mathbb{P}\,}^2$ and on ${\mathbb{P}\,}^1 \times {\mathbb{P}\,}^1$, yielding our result in the cases of $g = 3$ and 4. We plan on discussing similar results for genus 5 and 6 in a later paper. **Acknowledgements** This work was prepared as part of my doctoral dissertation under the direction of Sean Keel. I would like to thank him for his abundance of help and suggestions. I would also like to thank Brendan Hassett for his ideas. Variation of GIT ================ The birational contractions that we construct arise naturally as GIT quotients. This section contains a brief summary of results of Dolgachev-Hu [@DH] and Thaddeus [@Thaddeus] on variation of GIT. Given a group $G$ acting on a variety $X$, the GIT quotient $X//G$ is not unique – it depends on the choice of a $G$-ample line bundle. In particular, if ${\mathcal{L}\,}\in Pic^G (X)$, we have $$X //_{ {\mathcal{L}\,}} G = Proj \bigoplus_{n \geq 0} H^0 (X, {\mathcal{L}\,}^{\otimes n} )^G .$$ Following Dolgachev and Hu, we will call the set of all $G$-ample line bundles the **$G$-ample cone**. A study of how the quotient varies with the choice of the $G$-ample line bundle was carried out independently by Dolgachev-Hu [@DH] and Thaddeus [@Thaddeus]. The following theorem is a summary of some of the results of those papers: [@DH] [@Thaddeus] The $G$-ample cone is divided into a finite number of convex cones, called **chambers**. Two line bundles ${\mathcal{L}\,}$ and ${\mathcal{L}\,}'$ lie in the same chamber if $X^s ({\mathcal{L}\,}) = X^{ss} ({\mathcal{L}\,}) = X^{ss} ({\mathcal{L}\,}') = X^s ({\mathcal{L}\,}')$. The chambers are bounded by a finite number of **walls**. A line bundle ${\mathcal{L}\,}$ lies on a wall if $X^{ss}({\mathcal{L}\,}) \neq X^s ({\mathcal{L}\,})$. If ${\mathcal{L}\,}$ lies on a wall and ${\mathcal{L}\,}'$ lies is an adjacent chamber, then there is a morphism $X//_{{\mathcal{L}\,}'}G \to X//_{{\mathcal{L}\,}}G$. This map is an isomorphism over the stable locus. Both Thaddeus and Dolgachev-Hu examine the maps between quotients at a wall in the $G$-ample cone. Specifically, let ${\mathcal{L}\,}_+$, ${\mathcal{L}\,}_-$ be $G$-ample line bundles in adjacent chambers of the $G$-ample cone, and define ${\mathcal{L}\,}(t) = {\mathcal{L}\,}_+^t \otimes {\mathcal{L}\,}_-^{1-t}$. Suppose that the line between them crosses a wall precisely at ${\mathcal{L}\,}(t_0 )$. Following Thaddeus, define $$X^{\pm} = X^{ss} ({\mathcal{L}\,}_{t_0}) \backslash X^{ss} ({\mathcal{L}\,}_{\mp} )$$ $$X^0 = X^{ss} ({\mathcal{L}\,}_{t_0}) \backslash (X^{ss} ({\mathcal{L}\,}_+) \cup X^{ss} ({\mathcal{L}\,}_-))$$ \[Thaddeus\] [@Thaddeus] Let $x \in X^0$ be a smooth point of $X$. Suppose that $G \cdot x$ is closed in $X^{ss}({\mathcal{L}\,}_{t_0})$ and that $G_x \cong \mathbb{C}^*$. Then the natural map $X //_{{\mathcal{L}\,}_{ \pm }} G \to X //_{{\mathcal{L}\,}_{t_0}} G$ is an isomorphism outside of $X^{\pm}//_{{\mathcal{L}\,}_{\pm}} G$. Over a neighborhood of $x$ in $X^0 //_{t_0} G$, $X^{\pm} //_{{\mathcal{L}\,}_{\pm}} G $ are fibrations whose fibers are weighted projective spaces. In order to determine whether a point is (semi)stable, we will make frequent use of Mumford’s numerical criterion. Given a $G$-ample line bundle ${\mathcal{L}\,}$ and a one-parameter subgroup $\lambda : \mathbb{C}^* \to G$, it is standard to choose coordinates so that $\lambda$ acts diagonally on $H^0 (X, {\mathcal{L}\,})^*$. In other words, it is given by $diag(t^{a_1}, t^{a_2}, \ldots , t^{a_n} )$. We will refer to the $a_i$’s as the weights of the $\mathbb{C}^*$ action. For a point $x \in X$, Mumford defines $$\mu_{\lambda} (x) = min(a_i \vert x_i \neq 0).$$ Then $x$ is stable (semistable) if and only if $\mu_{\lambda} (x) < 0$ (resp. $\mu_{\lambda} (x) \leq 0$) for every nontrivial 1-parameter subgroup $\lambda$ of $G$ (see Theorem 2.1 in [@Mumford]). Hessians ======== Here we set up the GIT problem that appears in sections 4 and 5. We also identify a collection of $G$-invariant divisors that will be useful for analyzing this problem. Let $Y$ be a smooth projective surface over $\mathbb{C}$, ${\mathcal{L}\,}'$ an effective line bundle on $Y$, and $Z = {\mathbb{P}\,}H^0(Y, {\mathcal{L}\,}')$. Let $$X = \{ (p,C) \in Y \times Z \vert p \in C \}.$$ We denote the various maps as in the following diagram: $$\xymatrix{ X \ar[d]^f \ar[r]^i & Y \times Z \ar[d]^{\pi_2} \ar[r]^{\pi_1} & Y \\ Z \ar[r]^{id} &Z}$$ If ${\mathcal{L}\,}'$ is base-point free, then $X$ is a projective space bundle over $Y$, so it is smooth and $PicX \cong PicY \times {\mathbb{Z}\,}$. We will later study the GIT quotients of $X$ by the natural action of $Aut(Y)$. If $C$ is a curve on $Y$ and ${\mathcal{L}\,}$ is another line bundle on $Y$, then for every point $p \in C$ there are $n+1 = h^0(C, {\mathcal{L}\,}\vert_C )$ different orders of vanishing of sections $s \in H^0(C, {\mathcal{L}\,}\vert_C )$. When written in increasing order, $$a_0^{{\mathcal{L}\,}}(p) < \cdots < a_n^{{\mathcal{L}\,}}(p)$$ the orders of vanishing are called the **vanishing sequence** of ${\mathcal{L}\,}$ at $p$. The **weight** of ${\mathcal{L}\,}$ at $p$ is defined to be $w^{{\mathcal{L}\,}}(p) = \sum_{i=0}^n a_i^{{\mathcal{L}\,}}(p) - i$. A point is said to be an ${\mathcal{L}\,}$-**flex** if the weight of ${\mathcal{L}\,}$ at the point is nonzero. In other words, $p$ is an ${\mathcal{L}\,}$-flex if the vanishing sequence of ${\mathcal{L}\,}$ at $p$ is anything other than $0 < 1 < \cdots < n$. The **divisor of ${\mathcal{L}\,}$-flexes** is $\sum_{p \in C}w^{{\mathcal{L}\,}}(p)p$. It corresponds to a section $W_{{\mathcal{L}\,}}$ of a certain line bundle called the **Wronskian** of ${\mathcal{L}\,}$. We say that a curve $H$ on $Y$ is an **${\mathcal{L}\,}$-Hessian** if the restriction of $H$ to $C$ is precisely the divisor of ${\mathcal{L}\,}$-flexes. Returning to our family of curves $f : X \to Z$ above, suppose that ${\mathcal{L}\,}$ is a line bundle on $Y$ such that the pushforward $f_* ( \pi_1 \circ i )^* {\mathcal{L}\,}$ is locally free of rank $n+1$. We define a relative ${\mathcal{L}\,}$-Hessian to be a divisor $H \subseteq X$ whose restriction to each fiber is the divisor of $f_* ( \pi_1 \circ i )^* {\mathcal{L}\,}$-flexes. Relative ${\mathcal{L}\,}$-Hessians were studied by Cukierman [@Cukierman], who shows: \[Cukierman\] [@Cukierman] The class of the relative ${\mathcal{L}\,}$-Hessian is $$(n+1)c_1 ( \pi_1 \circ i)^* {\mathcal{L}\,}+ {{n+1}\choose{2}} c_1 \Omega^1_{X/Z} - c_1 f^* f_* ( \pi_1 \circ i )^* {\mathcal{L}\,}.$$ In our particular case, we can determine this class more explicitly. For $X,Y,$ and $Z$ as above, the class of the relative ${\mathcal{L}\,}$-Hessian is $$(n+1)c_1 ( \pi_1 \circ i)^* {\mathcal{L}\,}+ {{n+1}\choose{2}} ( c_1 \pi_1^* \Omega^1_Y \vert_X + c_1 ( \pi_1 \circ i)^* {\mathcal{L}\,}' + c_1 f^* {\mathcal{O}\,}_Z (1))$$ $$- h^0 (Y , {\mathcal{L}\,}\otimes {\mathcal{L}\,}'^* ) ( c_1 f^* {\mathcal{O}\,}_Z (1) ).$$ We follow the proof in [@Cukierman]. If $I$ is the ideal sheaf of $X$ in $Z \times Y$, then we have the exact sequence $$0 \to I/I^2 \to \pi_1^* \Omega^1_Y \vert_X \to \Omega^1_{X/Z} \to 0$$ so we have $$c_1 \Omega^1_{X/Z} = c_1 \pi_1^* \Omega^1_Y \vert_X - c_1 I/I^2.$$ Also, $X$ is the scheme of zeros of a section of the line bundle $E = ( \pi_1 \circ i)^* {\mathcal{L}\,}' \otimes f^* {\mathcal{O}\,}_Z (1)$ on $Y \times Z$. Note that $I/I^2 \cong E^* \otimes {\mathcal{O}\,}_X = E^* \vert_X$. It follows that $$c_1 \Omega^1_{X/Z} = c_1 ( \pi_1 \circ i)^* \Omega^1_Y \vert_X + c_1 E$$ $$= c_1 ( \pi_1 \circ i)^* \Omega^1_Y \vert_X + c_1 ( \pi_1 \circ i)^* {\mathcal{L}\,}' + c_1 f^* {\mathcal{O}\,}_Z (1).$$ Now, consider the exact sequence on $Y \times Z$ $$0 \to \pi_1^* L \otimes E^* \to \pi_1^* L \to \pi_1^* L \vert_X \to 0$$ From the projection formula, we see that $$\pi_{2*}( \pi_1^* {\mathcal{L}\,}\otimes E^* ) = H^0 (Y , {\mathcal{L}\,}\otimes {\mathcal{L}\,}'^* ) \otimes {\mathcal{O}\,}_Z (-1)$$ and $R^1 \pi_{2*} (\pi_1^* L \otimes E^* ) = 0$. This gives us the exact sequence on $Z$ $$0 \to \pi_{2*} ( \pi_1^* L \otimes E^* ) \to \pi_{2*} \pi_1^* L \to \pi_{2*} ( \pi_1^* L \vert_X ) \to 0$$ Since the middle term is a trivial bundle, the result follows from Proposition \[Cukierman\]. For the remainder of this section, we identify specific examples that will appear in the arguments to follow. In section 4 we consider the case that $Y = {\mathbb{P}\,}^2$ and ${\mathcal{L}\,}' = {\mathcal{O}\,}_Y (d)$ for some $d \geq 3$. By the above, we see that for every $m$ and $d$, a relative ${\mathcal{O}\,}_Y (m)$-Hessian $H_m$ exists. Since $c_1 \pi_1^* \Omega^1_Y \vert_X = {\mathcal{O}\,}_X (-3,0)$, if $m<d$, $H_m$ is cut out by a $G$-invariant section $W_m$ of $${\mathcal{O}\,}_X ((n+1)m + {{n+1}\choose{2}} (d-3), {{n+1}\choose{2}} ),$$ where $n+1 = h^0 (Y, {\mathcal{L}\,}) = {{m+2}\choose{2}}$. In particular, $H_1$ is cut out by a section $W_1 \in H^0( {\mathcal{O}\,}_X(3(d-2),3))$. $W_1$ vanishes at $(p,C)$ if $C$ is smooth at $p$ and the tangent line to $C$ at $p$ intersects $C$ with multiplicity at least 3, or if $p$ is a singular point of $C$. Similarly, $H_2$ is defined by a section of $W_2 \in H^0( {\mathcal{O}\,}_X(15d-33, 15))$. $W_2$ vanishes at $(p,C)$ if $C$ is smooth at $p$ and the osculating conic to $C$ at $p$ intersects $C$ with multiplicity at least 6, or if $p$ is a singular point of $C$. It is known that $H_2 = H_1 \cup H_2'$ is reducible ( see Proposition 6.6 in [@CF]). Indeed, if a line meets $C$ with multiplicity 3 at $p$, then the double line meets $C$ with multiplicity 6 at $p$. The points of $H_2' \cap C$ are classically known as the **sextatic points** of $C$, and $H_2 '$ is cut out by a $G$-invariant section $W_2 '$ of ${\mathcal{O}\,}_X (12(d-\frac{9}{4}),12)$. A simple calculation shows that $H_2 ' \cap C$ also contains those points of $C$ where $w^{{\mathcal{O}\,}_C (1)} (p) > 1$. These include singular points and points where the tangent line to $C$ is a **hyperflex** (a line that intersects $C$ at $p$ with multiplicity $\geq 4$). Similarly, in section 5 we consider the case that $Y = {\mathbb{P}\,}^1 \times {\mathbb{P}\,}^1$, and ${\mathcal{L}\,}' = {\mathcal{O}\,}_Y (d,d)$. Note that, for every $(m_1 , m_2 , d)$ with $m_i < d$, a relative ${\mathcal{O}\,}_Y (m_1 , m_2 )$-Hessian $H'_{m_1 ,m_2}$ exists. In this case, our formulas show that the rank of $f_* ( \pi_1 \circ i )^* {\mathcal{O}\,}_Y (m_1 , m_2 )$ is $$n+1 = h^0 ({\mathcal{O}\,}_Y (m_1 , m_2 )) = (m_1 + 1)(m_2 + 1).$$ Also, since $c_1 \pi_1^* \Omega^1_Y \vert_X = {\mathcal{O}\,}_X (-2,-2,0)$, we see that $H'_{m_1 ,m_2}$ is cut out by a section $W'_{m_1 ,m_2} \in H^0 ( {\mathcal{O}\,}_X (a_1 , a_2 , b))$ for $$a_i = (n+1)m_i + {{n+1}\choose{2}}(d-2)$$ $$b = {{n+1}\choose{2}}.$$ Since ${\mathbb{P}\,}^1 \times {\mathbb{P}\,}^1$ has a natural involution, we know that $W'_{m_1 , m_2}$ cannot be $G$-invariant if $m_1 \neq m_2$. Notice, however, that $W'_{m_1 , m_2} \otimes W'_{m_2 , m_1}$ is a $G$-invariant section of ${\mathcal{O}\,}_X (a, a, b)$ for $$n+1 = (m_1 + 1)(m_2 + 1)$$ $$a = (n+1)(m_1 + m_2 ) + 2{{n+1}\choose{2}}(d-2)$$ $$b = 2{{n+1}\choose{2}}.$$ We will use $W_{m_1 , m_2}$ to denote the $G$-invariant section described here, and $H_{m_1 ,m_2}$ to denote its zero locus. In particular, $W_{0,1} \in H^0 ({\mathcal{O}\,}_X (2(d-1),2(d-1),2))$. It vanishes at a point $(p,C)$ if $C$ intersects one of the two lines through $p$ with multiplicity at least 2 (or, equivalently, if the osculating $(1,1)$ curve is a pair of lines). Similarly, $W_{1,1} \in H^0 ({\mathcal{O}\,}_X (2(3d-4),2(3d-4),6))$. It vanishes at a point $(p,C)$ if there is a curve of bidegree $(1,1)$ that intersects $C$ with multiplicity 4 or more at $p$. Contraction of $\overline{M}_{3,1}$ =================================== In this section, we prove our main result in the genus 3 case: There is a birational contraction of $\overline{M}_{3,1}$ contracting the Weierstrass divisor $BN^1_{3,(0,3)}$. In order to construct a birational model for $\overline{M}_{3,1}$, we consider GIT quotients of the universal family over the space of plane quartics. The image of the Weierstrass divisor in this model is precisely the Hessian $H_1$, and we exhibit a GIT quotient in which this locus is contracted. For most of this section we will consider, more generally, plane curves of any degree $d \geq 3$. Specifically, following the set-up of the previous section, we let $$X = \{ (p,C) \in {\mathbb{P}\,}^2 \times \vert {\mathcal{O}\,}(d) \vert \text{ } \vert p \in C \}.$$ Then $\pi_2 : X \to \vert {\mathcal{O}\,}(d) \vert$ is the family of all plane curves of degree $d$. Our goal is to study the GIT quotients of $X$ by the action of $G = PSL(3,\mathbb{C})$. By the above, we know that $PicX \cong {\mathbb{Z}\,}\times {\mathbb{Z}\,}$, so the quotient $X //_{{\mathcal{L}\,}} G$ depends on a single parameter $t$ which we call the **slope** of ${\mathcal{L}\,}$. We say a line bundle ${\mathcal{L}\,}$ has **slope** $t$ if ${\mathcal{L}\,}= \pi_1^* {\mathcal{O}\,}(a) \otimes \pi_2^* {\mathcal{O}\,}(b)$ with $t = \frac{a}{b}$. We write $X^s (t)$ and $X^{ss} (t)$ for the sets of stable and semistable points, and $X //_t G $ for the corresponding GIT quotient. Here we describe the numerical criterion for points in $X$. Let $p = (x_0 , x_1 , x_2 )$ and $$C = \sum_{i+j+k=d} a_{i,j,k} x_0^i x_1^j x_2^k.$$ Then a basis for $H^0 ( {\mathcal{O}\,}_X (a,b))$ consists of monomials of the form $$\prod_{\alpha = 1}^a x_{l_{ \alpha }} \prod_{\beta = 1}^b a_{i_{\beta},j_{\beta},k_{\beta}}.$$ The one-parameter subgroup with weights $(r_0 , r_1 , r_2 )$ acts on the monomial above with weight $$\sum_{\alpha =1}^a r_{l_{\alpha}} - \sum_{\beta = 1}^b ( i_{\beta} r_0 + j_{\beta} r_1 + k_{\beta} r_2 ).$$ In our case, we will only be interested in maximizing or minimizing this weight, so it suffices to consider monomials of the form $x_l^a a_{i,j,k}^b$. In this case, the one-parameter subgroup acts with weight $ar_l - b(ir_0 + jr_1 + kr_2 )$, which is proportional to $$\mu_{\lambda} (x_l , a_{i,j,k} ) := tr_l - (ir_0 + jr_1 + kr_2 ).$$ The $G$-ample cone of $X$ has two edges, one of which occurs when $t=0$. In the case where $d=4$, we obtain the well-known moduli space of plane quartics. Descriptions of $X^s (0)$ and $X^{ss} (0)$ appear in [@Mumford], and the quotient $X //_0 G$ plays an important role in the birational geometry of $\overline{M}_3$. For example, Hyeon and Lee show that this quotient is a log canonical model for $\overline{M}_3$ [@HL], and the space also appears in work on moduli of K3 surfaces [@Artebani] and cubic threefolds [@CML]. We will see that, when $t$ is large, stability conditions reflect the inflectionary behavior of linear series at the marked point. Thus, as $t$ increases, the curve is allowed to have more complicated singularities, but vanishing sequences at the marked point become more well-behaved. Our first result is to identify the other edge of the $G$-ample cone. It is determined by the Wronksian $W_1$. An edge of the $G$-ample cone occurs at $t = d-2$. It suffices to show that $X^{ss}(d-2) \neq X^s (d-2) = \emptyset$. It is clear that $X^{ss}(d-2) \neq \emptyset$, since $W_1$ is a $G$-invariant section of ${\mathcal{O}\,}_{X} (3(d-2),3)$. To show that $X^s (d-2) = \emptyset$, we invoke the numerical criterion. Let $(p,C) \in X$. By change of coordinates, we may assume that $p = (0,0,1)$ and the tangent line to $C$ at $p$ is $x_0 = 0$. So in the coordinates described above, we have $a_{0,0,d} = a_{0,1,d-1} = 0$. Now consider the 1-parameter subgroup with weights $(-1,0,1)$. We have $$\mu_{\lambda} (x_2 , a_{i,j,k}) = d-2 + i - k$$ which is negative whenever $i-k-2 < -d = -i-j-k$, or $2i + j < 2$. This only occurs when both $i = 0$ and $j < 2$, in other words, when either $a_{0,0,d}$ or $a_{0,1,d-1}$ is nonzero. By assumption, however, this is not the case, so $(p,C) \notin X^s (d-2)$. Since $(p,C)$ was arbitrary, it follows that $X^s (d-2) = \emptyset$. Next, we identify the adjacent chamber in the $G$-ample cone. It lies between the slopes corresponding to the Wronskians $W_1$ and $W_2$. In what follows, we let $S$ denote the set of all pointed curves $(p,C)$ admitting the following description: $C$ consists of a smooth conic together with $d-2$ copies of the tangent line through a point $q \neq p$ on $C$. Notice that $S \subset H_2 '$. For any $t \in (d- \frac{9}{4} , d-2)$, $X^s (t) = X^{ss} (t) = X \backslash (H_1 \cup S)$. We first show that $X^{ss} (t) \subseteq X \backslash H_1$. Suppose that $(p,C) \in H_1$. As before, by change of coordinates, we may assume that $p = (0,0,1)$ and the tangent line to $C$ at $p$ is $x_0 = 0$. Since $(p,C) \in H_1$, either $p$ is a singular point of $C$ or this tangent line intersects $C$ at $p$ with multiplicity at least 3. Thus we have $a_{0,0,d} = a_{0,1,d-1} = 0$, and either $a_{1,0,d-1} = 0$ (if $p$ is singular) or $a_{0,2,d-2} = 0$ (if $p$ is a flex). We first examine the case where $p$ is a flex. In this case, consider the 1-parameter subgroup with weights $(-5,1,4)$. Then $$\mu_{\lambda} (x_2 , a_{i,j,k}) = 4t + 5i - j - 4k > 4d - 9 + 5i - j - 4k = 9i + 3j - 9$$ which is non-negative when $3i + j \geq 3$. Since, by assumption, $C$ has no non-zero terms with both $i = 0$ and $j < 3$, we see that $(p,C) \notin X^{ss} (t)$. Next we look at the case where $p$ is a singular point. Consider the 1-parameter subgroup with weights $(-1,-1,2)$. Then we have $$\mu_{\lambda} (x_2 , a_{i,j,k}) = 2t + i + j - 2k > 2d- \frac{9}{2} + i + j - 2k = 3i + 3j - \frac{9}{2}$$ which is non-negative when $i + j \leq \frac{3}{2}$. By assumption, $C$ has no non-zero terms where one of $i,j$ is 0 and the other is at most 1, so $(p,C) \notin X^{ss} (t)$. It follows that $X^{ss} (t) \subseteq X \backslash H_1$. Next we show that $X^{ss} (t) \subseteq X \backslash S$. Suppose that $(p,C) \in S$. Without loss of generality, we may assume that $C$ is of the form $$C = x_0^{d-2} ( a_{d,0,0} x_0^2 + a_{d-1,1,0} x_0 x_1 + a_{d-2,2,0} x_1^2 + a_{d-1,0,1} x_0 x_2 ).$$ Now, consider the 1-parameter subgroup with weights $(-1,0,1)$. Then $$\mu_{\lambda} ( x_l , a_{i,j,k} ) \geq -t + i - j > 2-d + i - j$$ which is non-negative when $i - j \geq d-2$. It follows that $(p,C) \notin X^{ss} (t)$. Now we show that $X \backslash (H_1 \cup S) \subseteq X^s (t)$. Suppose that $(p,C) \notin X^s (t)$. Then there is a nontrivial 1-parameter subgroup that acts on $(p,C)$ with strictly positive weight. By change of basis, we may assume that this subgroup acts with weights $(r_0 , r_1 , r_2 )$, with $r_0 \leq r_1 \leq r_2$. Since this is a nontrivial subgroup of $PSL(3, \mathbb{C})$, we know that $r_0 < 0 < r_2$ and $r_0 + r_1 + r_2 = 0$. We then have $$\mu_{\lambda} (x_l , a_{i,j,k} ) = tr_l - (r_0 i + r_1 j + r_2 k) > 0$$ We divide this into three cases, depending on $p$. **Case 1 – $p = (0,0,1)$:** In this case, $r_l = r_2$. If $r_1 \geq 0$, then $t r_2 < (d - 2)r_2 \leq 2r_1 + (d-2)r_2$. On the other hand, if $r_1 < 0$&lt; then $t r_2 < (d - 2)r_2 < r_0 + (d-1)r_2$. Since the subgroup acts with strictly positive wieght, it follows that $a_{0,0,d} = a_{0,1,d-1} = 0$, and either $a_{1,0,d-1} = 0$ or $a_{0,2,d-2} = 0$. Hence, $(p,C) \in H_1$. **Case 2 – $p$ lies on the line $x_0 = 0$, but not on the line $x_1 = 0$:** In this case, $r_l = r_1$. If $r_1 > 0$, then since $r_1 \leq r_2$, we have $t r_1 < dr_1 \leq r_1 j + r_2 (d-j)$, so we see that $a_{0,0,d} = a_{0,1,d-1} = \cdots = a_{0,d,0} = 0$. This means that $p$ lies on a linear component of $C$, and therefore $(p,C) \in H_1$. On the other hand, if $r_1 \leq 0$, then since $r_2 \geq -2 r_1$, we see that $t r_1 \leq (d-3)r_1 \leq (d-1)r_1 + r_2 \leq r_1 j + (d-j) r_2 + r_2$ for $j \leq d-1$, so $a_{0,0,d} = a_{0,1,d-1} = \cdots = a_{0,d-1,1} = 0$. This means that either $p$ lies on a linear component of $C$ or the only point of $C$ lying on the line $x_0 = 0$ also lies on the line $x_1 =0$. Again, we see that $(p,C) \in H_1$. **Case 3 – $p$ does not lie on the line $x_0 = 0$:** In this case, $r_l = r_0$. Since $r_0 < 0$ and $r_1 < r_0 < r_2$, we see that $tr_0 < (d-3)r_0 = (d-2)r_0 + r_1 + r_2 < r_0 i + r_1 j + r_2 k$ for $i \leq d-2, k \neq 0$. Now, if $r_0 \geq 4r_1$, then we have $tr_0 < (d - \frac{9}{4} ) r_0 = (d- \frac{5}{4} )r_0 + r_1 + r_2 \leq (d-1)r_0 + r_2$. It follows that $C$ is of the form $$C = \sum_{i+j=d} a_{i,j,0} x_0^i x_1^j.$$ In other words, $C$ is a union of $d$ lines. In this case, the tangent line to every point of $C$ is a component of $C$ itself, so $(p,C) \in H_1$. On the other hand, if $r_0 < 4r_1$, then $tr_0 < (d- \frac{9}{4} ) r_0 = (d-3)r_0 + \frac{3}{4} r_0 < (d-3)r_0 + 3 r_1$. It follows that $C$ is of the form $$C = x_0^{d-2} ( a_{d,0,0} x_0^2 + a_{d-1,1,0} x_0 x_1 + a_{d-2,2,0} x_1^2 + a_{d-1,0,1} x_0 x_2 )$$ hence $C \in S$. We now consider the wall in the $G$-ample cone determined by the Wronskian $W_2$. A wall of the $G$-ample cone occurs at $t = d- \frac{9}{4}$. More specifically, $X^{ss} (t) = X \backslash ((H_1 \cap H_2 ') \cup S)$, and $X^s (t) \subseteq X \backslash (H_1 \cup S)$. First, notice that if $(p,C) \notin H_2 '$, then $(p,C) \in X^{ss} (t)$, since $W_2 '$ is a $G$-invariant section of ${\mathcal{O}\,}_{X} (12(d- \frac{9}{4}), 12)$ that does not vanish at $(p,C)$. Moreover, by general variation of GIT we know that, when passing from a chamber to a wall, we have $$X^{ss} (t + \epsilon ) \subseteq X^{ss} (t)$$ $$X^s (t) \subseteq X^s (t + \epsilon )$$ Thus, $X^s (t) \subseteq X \backslash (H_1 \cup S)$ and $X \backslash ((H_1 \cap H_2 ') \cup S) \subseteq X^{ss} (t)$. Now, suppose that $(p,C) \in S$. Using the same argument as above with the same 1-parameter subgroup, we see that $(p,C) \notin X^{ss} (t)$. Next, suppose that $(p,C) \in H_1$. If $p$ is a singular point of $C$, then we see that $(p,C) \notin X^{ss} (t)$ by the same argument as before, using the subgroup with weights $(-1,-1,2)$. The only other possibility is that $p$ is a flex. In this case, we again consider the 1-parameter subgroup with weights $(-5,1,4)$. As before, we have $$\mu_{\lambda} (x_2 , a_{i,j,k}) = 4d - 9 + 5i - j - 4k = 9i + 3j - 9$$ which is non-negative when $3i + j \geq 3$. As before, we see that $(p,C) \notin X^s (t)$. Notice furthermore that if the tangent line to $C$ at $p$ is a hyperflex, then $a_{0,3,d-3} = 0$ as well, and so the expression $3i + j - 3$ above is strictly positive (rather than simply nonnegative), and thus $(p,C) \notin X^{ss} (t)$. If $(p,C) \in H_1 \cap H_2 '$, then either $p$ is a singular point of $C$, or $C$ is smooth at $p$ and the tangent line to $C$ at $p$ is a hyperflex. From our observations above, we may therefore conclude that $X^{ss} (t) \subseteq X \backslash ((H_1 \cap H_2 ') \cup S)$. We are left to consider the behavior of our quotient at the wall crossing defined by $t_0 = d-\frac{9}{4}$. As in Theorem \[Thaddeus\], we let $$X^{\pm} = X^{ss} (t_0) \backslash X^{ss} (t_0 \mp \epsilon )$$ $$X^0 = X^{ss} (t_0 ) \backslash (X^{ss} (t_0 + \epsilon ) \cup X^{ss} (t_0 - \epsilon ))$$ Our first task is to determine $X^-$ and $X^0$ in this situation. With the set-up above, $X^- = H_1 \backslash H_2 '$. $X^0$ is the set of all pointed curves $(p,C)$ consisting of a cuspidal cubic plus $d-3$ copies of the projectivized tangent cone at the cusp. The point $p$ is the unique smooth flex point of the cuspidal cubic. We have already seen that $X^{ss} (t_0 ) = X \backslash ((H_1 \cap H_2 ') \cup S)$ and $X^{ss} (t_0 + \epsilon ) = X \backslash (H_1 \cup S)$. Thus, $X^- = H_1 \backslash H_2 '$. To prove the statement about $X^0$, let $(p,C) \in X^0$. Notice that, since $X^0 \subseteq X^-$, $p$ is a smooth point of $C$ and the tangent line to $C$ at $p$ intersects $C$ with multiplicity exactly 3. Since $(p,C) \notin X^{ss} (t_0 - \epsilon )$, there must be a nontrivial 1-parameter subgroup that acts on $(p,C)$ with strictly positive weight. Again we assume that this subgroup acts with weights $(r_0 , r_1 , r_2 )$, with $r_0 \leq r_1 \leq r_2$. As before, we know that $r_0 < 0 < r_2$ and $r_0 + r_1 + r_2 = 0$. Again we have $$\mu_{\lambda} (x_l , a_{i,j,k} ) = tr_l - (r_0 i + r_1 j + r_2 k) > 0$$ We divide this into three cases, depending on $p$. **Case 1 – $p = (0,0,1)$:** In this case, $r_l = r_2$. Now, if $t r_2 \geq r_0 + (d-1)r_2$, then $(d - \frac{9}{4})r_2 > r_0 + (d-1)r_2$, so $r_1 > \frac{1}{4} r_2$. This means that $t r_2 < (d - \frac{9}{4})r_2 < 3r_1 + (d-3)r_2$. It follows that $a_{0,0,d} = a_{0,1,d-1} = 0$, and either $a_{1,0,d-1} = 0$ or $a_{0,2,d-2} = a_{0,3,d-3} = 0$. But we know that $p$ is a smooth point of $C$ and the tangent line to $C$ at $p$ intersects $C$ with multiplicity exactly 3, so neither of these is a possibility. **Case 2 – $p$ lies on the line $x_0 = 0$, but not on the line $x_1 = 0$:** Using the same argument as before, we see that $p$ lies on a linear component of $C$, which is impossible. **Case 3 – $p$ does not lie on the line $x_0 = 0$:** In this case, $r_l = r_0$. Again, since $r_0 < 0$ and $r_1 < r_0 < r_2$, we see that $tr_0 < (d-3)r_0 = (d-2)r_0 + r_1 + r_2 < r_0 i + r_1 j + r_2 k$ for $i \leq d-2, k \neq 0$. Notice that, if $tr_0 < (d-1)r_0 + r_2$, then as before we see that $C$ is the union of $d$ lines, which is impossible. We therefore see that $(d - \frac{12}{5})r_0 > t r_0 \geq (d-1)r_0 + r_2$. But then $ \frac{7}{5}r_0 < -r_2 = r_0 + r_1$, so $r_0 < \frac{5}{2} r_1$. It follows that $t r_0 < ( d- \frac{12}{5} )r_0 < (d-4)r_0 + 4r_1 \leq r_0 i + r_1 j$ for $j \geq 4$. We see that $C$ is of the form $$C = x_0^{d-3} (a_{d,0,0}x_0^3 + a_{d-1,1,0}x_0^2 x_1 + a_{d-2,2,0}x_0 x_1^2 + a_{d-3,3,0}x_1^3 + a_{d-1,0,1}x_0^2 x_2).$$ Thus, $C$ consists of a cuspidal cubic together with $d-3$ copies of the projectivized tangent cone to the cusp. The point $p$ is the unique flex point of the cuspidal cubic. It is clear that this $(p,C) \in X^{-}$, since the tangent line to $C$ at $p$ intersects $C$ with multiplicity exactly 3. To see that $(p,C) \notin X^{ss}(t_0 - \epsilon )$, consider again the 1-parameter subgroup with weights $(5,-1,-4)$. The statement then follows from the fact that all cuspidal plane cubics are projectively equivalent. The map $X //_{t_0 - \epsilon} G \to X //_{t_0} G$ contracts the locus $H_1 \backslash H_2 '$ to a point. Outside of this locus, the map is an isomorphism. Let $(p,C) \in X^0$. Since all cuspidal plane cubics are projectively equivalent, $G \cdot (p,C) = X^0$, so $G \cdot (p,C)$ is closed in $X^{ss}(t_0 )$ and $X^0 // G$ is a point. An automorphism of ${\mathbb{P}\,}^1$ extends to $(p,C)$ if and only if it fixes the point $p$ and the cusp, and thus the stabilizer of $(p,C)$ is isomorphic to $\mathbb{C}^*$. The conclusion follows from Theorem \[Thaddeus\]. We are particularly interested in the case where $d=4$, because in this case $X //_{t_0 - \epsilon} G$ is a birational model for $\overline{M}_{3,1}$. In particular, we have the following: There is a birational contraction $\beta : \overline{M}_{3,1} \dashrightarrow X //_{t_0 - \epsilon} G$. It suffices to exhibit a morphism $\beta^{-1}: V \to \overline{M}_{3,1}$, where $V \subseteq X //_{t_0 - \epsilon} G$ is open with complement of codimension $\geq 2$ and $\beta^{-1}$ is an isomorphism onto its image. To see this, let $U \subseteq X^{ss}(t_0 - \epsilon )$ be the set of all moduli stable pointed curves $(p,C) \in X^{ss}(t_0 - \epsilon )$. Notice that the complement of $U$ is strictly contained in the discriminant locus $\Delta$, which is an irreducible hypersurface in $X$. Note furthermore that there are stable points contained in both $X \backslash \Delta$ and $\Delta \cap U$. Thus, the containments $(X \backslash U) //_{t_0 - \epsilon} G \subset \Delta //_{t_0 - \epsilon} G$ and $\Delta //_{t_0 - \epsilon} G \subset X //_{t_0 - \epsilon} G$ are strict. It follows that the complement of $U // G$ in the quotient has codimension $\geq 2$. By the universal property of the moduli space, since $U \to Z$ is a family of moduli stable curves, it admits a unique map $U \to Z \to \overline{M}_{3,1}$. This map is certainly $G$-equivariant, so it factors uniquely through a map $U //_{t_0 - \epsilon} G \to \overline{M}_{3,1}$. Since every degree 4 plane curve is canonical, two such curves are isomorphic if and only if they differ by an automorphism of ${\mathbb{P}\,}^2$. It follows that this map is an isomorphism onto its image. There is a birational contraction of $\overline{M}_{3,1}$ contracting the Weierstrass divisor $BN^1_{3,(0,3)}$. Furthermore, the divisors $BN^1_{3,(0,3)}$, $BN^1_2$, $\Delta_1$ and $\Delta_2$ span a simplicial face of $\overline{NE}^1 ( \overline{M}_{3,1} )$. The composition $\overline{M}_{3,1} \dashrightarrow X //_{t_0 - \epsilon} G \to X //_{t_0} G$ is a birational contraction. By the above, the Weierstrass divisor is contracted by this map, so it suffices to show that $BN^1_2$ and the $\Delta_i$’s are contracted as well. We first note that every smooth curve in $X$ is canonically embedded and hence non-hyperelliptic, so the closure of the image of each of these divisors must be contained in the singular locus $\Delta$, which is an irreducible hypersurface. Since the generic point of $\Delta$ is an irreducible nodal curve, we see that the closure of the image of $\Delta_i$ is of codimension 2 or greater for all $i \geq1$. Moreover, since the set of singular hyperelliptic curves has codimension 2 in $\overline{M}_{3,1}$, and we have constructed an embedding of an open subset of $\Delta$ into $\overline{M}_{3,1}$, we may conclude that $BN^1_2$ is contracted as well. Contraction of $\overline{M}_{4,1}$ =================================== We now turn to the case of genus 4 curves. Our main result will be the following: There is a birational contraction of $\overline{M}_{4,1}$ contracting the pointed Brill-Noether divisor $BN^1_{3,(0,2)}$. In a similar way to the previous section, we will construct a birational model for $\overline{M}_{4,1}$ by considering GIT quotients of the universal family over the space of curves in ${\mathbb{P}\,}^1 \times {\mathbb{P}\,}^1$. Here, the Hessian $H_{0,1}$ is again the image of a pointed Brill-Noether divisor. As above, our goal is to find a GIT quotient in which this locus is contracted. Let $Y = {\mathbb{P}\,}^1 \times {\mathbb{P}\,}^1$ and $$X = \{ (p,C) \in Y \times \vert {\mathcal{O}\,}(d,d) \vert \text{ } \vert p \in C \}.$$ Then $\pi_2 : X \to \vert {\mathcal{O}\,}(d,d) \vert$ is the family of all curves of bidegree $(d,d)$. Our goal, as before, is to study the GIT quotients of $X$ by the action of $G = PSO(4, \mathbb{C} )$. By the above, we know that $PicX \cong {\mathbb{Z}\,}^3$, but we are only interested in those line bundles of the form ${\mathcal{O}\,}_X (a,a,b)$. We can therefore define the slope of a line bundle ${\mathcal{L}\,}\in PicX$ as above. We say a line bundle ${\mathcal{L}\,}$ has **slope** $t$ if ${\mathcal{L}\,}= \pi_1^* {\mathcal{O}\,}(a,a) \otimes \pi_2^* {\mathcal{O}\,}(b)$ with $t = \frac{a}{b}$. We write $X^s (t)$ and $X^{ss} (t)$ for the sets of stable and semistable points, and $X //_t G $ for the corresponding GIT quotient. Here we describe the numerical criterion for points in $X$. Let $p = (x_0 , x_1 : y_0 , y_1 )$ and $$C = \sum_{i_0 + i_1 = j_0 + j_1 = d} a_{i_0 ,i_1 ,j_0 ,j_1} x_0^{i_0} x_1^{i_1} y_0^{j_0} y_1^{j_1}.$$ Then a basis for $H^0 ( {\mathcal{O}\,}_X (a,a,b))$ consists of monomials of the form $$\prod_{\alpha_0 = 1}^a x_{l_{ \alpha_0 }} y_{m_{ \alpha_1 }} \prod_{\beta = 1}^b a_{i_{0 \beta},i_{1 \beta} , j_{0 \beta}, j_{1 \beta}}.$$ The one-parameter subgroup with weights $(-r_0 , r_0, -r_1 , r_1 )$ acts on the monomial above with weight $$\sum_{\beta = 1}^b ( r_0 (i_{0 \beta } - i_{1 \beta }) + r_1 (j_{0 \beta } - j_{1 \beta }) ) - \sum_{\alpha_0 =1}^a ((-1)^{l_{\alpha_0}} r_0 + (-1)^{m_{\alpha_1}} r_1 ) .$$ In our case, we will only be interested in maximizing or minimizing this weight, so it suffices to consider monomials of the form $x_l^a y_m^a a_{i_0 , i_1 , j_0 , j_1}^b$. In this case, the one-parameter subgroup acts with weight $b(r_0 (i_0 - i_1 ) + r_1 (j_0 - j_1 )) - a((-1)^l r_0 + (-1)^m r_1)$, which is proportional to $$\mu_{\lambda} (x_l , y_m , a_{i_0 , i_1 , j_0 , j_1} ) := r_0 (i_0 - i_1 ) + r_1 (j_0 - j_1 ) - t((-1)^l r_0 + (-1)^m r_1).$$ As in the previous section, when $t = 0$, we obtain a moduli space of curves of bidegree $(d,d)$. In particular, the case $d=3$ is notable for being a birational model for $\overline{M}_4$. We will see that as $t$ increases, stable curves are allowed to have more complicated singularities, but the vanishing sequences of linear series at the marked point become more well-controlled. We begin by identifying an edge of the $G$-ample cone corresponding to the Wronskian $W_{0,1}$. An edge of the $G$-ample cone occurs at $t=d-1$. It suffices to show that $X^{ss}(d-1 ) \neq X^s (d-1) = \emptyset$. It is clear that $X^{ss}(d-1) \neq \emptyset$, since $W_{0,1}$ is a $G$-invariant section of ${\mathcal{O}\,}_X (2(d-1), 2(d-1), 2)$. To show that $X^s (d-1) = \emptyset$, we invoke the numerical criterion. Let $(p,C) \in X$. By change of coordinates, we may assume that $p = (0,1:0,1)$. So, in the coordinates described above, we have $a_{0,d,0,d} = 0$. Now consider the 1-parameter subgroup with weights $(-1,1,-1,1)$. We have $$\mu_{\lambda} ( x_1 , y_1 , a_{i_0 , i_1 , j_0 , j_1} ) = 2(d-1) + (i_0 - i_1 ) + (j_0 - j_1 )$$ which is negative whenever $(i_0 - i_1 ) + (j_0 - j_1 ) < -2(d-1) = -i_0 - i_1 - j_0 - j_1 + 2$, or $i_0 + j_0 < 1$. This only occurs when $i_0 = j_0 = 0$, in other words, when $a_{0,d,0,d}$ is nonzero. By assumption, however, this is not the case, so $(p,C) \notin X^s (d-1)$. Since $(p,C)$ was arbitrary, it follows that $X^s (d-1) = \emptyset$. As above, we identify the adjacent chamber in the $G$-ample cone. It lies between the slopes corresponding to the Wronskians $W_{0,1}$ and $W_{1,1}$. In what follows, we let $S$ denote the set of all pointed curves $(p,C)$ admitting the following description: $C$ consists of a smooth curve of bidegree $(1,1)$ together with $d-1$ copies of the two lines through a point $q \neq p$ on $C$. Notice that $S \subset H_{1,1}$. For any $t \in (d- \frac{4}{3} , d-1)$, $X^s (t) = X^{ss} (t) = X \backslash ( H_{0,1} \cup S)$. We first show that $X^{ss} (t) \subseteq X \backslash H_{0,1}$. Suppose that $(p,C) \in H_{0,1}$. As before, by change of coordinates, we may assume that $p = (0,1:0,1)$. Since $(p,C) \in H_{0,1}$, $C$ intersects one of the two lines through $p$ with multiplicity at least 2. Without loss of generality, we may assume this line to be $x_0 = 0$. Thus, if we write $C$ as above, then $a_{0,d,0,d} = a_{0,d,1,d-1} = 0$. Now, consider the 1-parameter subgroup with weights $(-1,1,-2,2)$. Then $$\mu_{\lambda} ( x_1 , y_1 , a_{i_0 , i_1 , j_0 , j_1} ) = 3t + i_0 - i_1 + 2j_0 - 2j_1 > 3d-4 + i_0 - i_1 + 2j_0 - 2j_1$$ $$= 2(i_0 + 2j_0 - 2)$$ which is non-negative when $i_0 + 2j_0 \geq 2$. Since, by assumption, $C$ has no non-zero terms with both $i_0 \leq 1$ and $j = 0$, we see that $(p,C) \notin X^{ss} (t)$. Next we show that $X^{ss} (t) \subseteq X \backslash S$. Suppose that $(p,C) \in S$. Without loss of generality, we may assume that $C$ is of the form $$C = x_0^{d-1} y_0^{d-1} ( a_{d,0,d,0} x_0 y_0 + a_{d-1,1,d,0} x_1 y_0 + a_{d,0,d-1,1} x_0 y_1 ).$$ Now, consider the 1-parameter subgroup with weights $(1,-1,1,-1)$. Then $$\mu_{\lambda} ( x_l , y_m , a_{i_0 , i_1 , j_0 , j_1} ) \geq -2t - i_0 + i_1 - j_0 + j_1 > -2d+2 - i_0 + i_1 - j_0 + j_1$$ $$= -2(i_1 + j_1 - 1)$$ which is non-negative when $i_1 + j_1 \leq 1$. It follows that $(p,C) \notin X^{ss} (t)$. Now we show that $X \backslash (H_{0,1} \cup S) \subseteq X^s (t)$. Suppose that $(p,C) \notin X^s (t)$. Then there is a nontrivial 1-parameter subgroup that acts on $(p,C)$ with strictly positive weight. By change of basis, we may assume that this subgroup acts with weights $(-r_0 , r_0 , -r_1, r_1 )$, with $0 \leq r_0 \leq r_1$ and $r_1 > 0$. We then have $$\mu_{\lambda} (x_l , y_m , a_{i_0 , i_1 , j_0 , j_1} ) = r_0 (i_0 - i_1 ) + r_1 (j_0 - j_1 ) - t((-1)^l r_0 + (-1)^m r_1) > 0$$ We divide this into four cases, depending on $p$. **Case 1 – $p = (0,1:0,1)$:** In this case, $l = m = 1$. We have $t(-r_0 - r_1 ) > (d-1)(-r_0 - r_1 ) \geq -(d-2)r_0 - dr_1$. It follows that $a_{0,d,0,d} = a_{1,d-1,0,d} = 0$, so $(p,C) \in H_{0,1}$. **Case 2 – $p$ lies on the line $y_0 = 0$, but not the line $x_0 = 0$:** In this case, $l = 1$ and $m = 0$. Here, $t(-r_0 + r_1 ) > (d-2)(-r_0 + r_1 ) \geq -dr_0 + kr_1$ for all $k \leq d-2$. We therefore see that $a_{0,d,k,d-k} = 0$ for all $k \leq d-2$. If $a_{0,d,d,0} \neq 0$, then every point of $C$ that lies on the line $x_0 = 0$ also lies on the line $y_0 = 0$, a contradiction. We therefore see that $a_{0,d,d,0} = 0$ as well, but this means that $p$ lies on a linear component of $C$, and therefore $(p,C) \in H_{0,1}$. **Case 3 – $p$ lies on the line $x_0 = 0$, but not on the line $y_0 = 0$:** In this case, $l = 0$ and $m = 1$. Note that $t(r_0 - r_1 ) > d(-r_0 + r_1 ) \geq -dr_0 + kr_1$ for all $k \leq d$. It follows that $a_{k,d-k,0,d} = 0$ for all values of $k$, which means that $y_0 = 0$ is a linear component of $C$. Thus $(p,C) \in H_{0,1}$. **Case 4 – $p$ does not lie on either of the lines $x_0 = 0$ or $y_0 = 0$:** In this case, $l = m = 0$. Now note that $t(r_0 + r_1 ) > (d-2)(r_0 + r_1 )$, so $a_{k_0 ,d-k_0 ,k_1 ,d-k_1} = 0$ if $k_0$ and $k_1$ are both less than $d$. Furthermore, since $r_0 \leq r_1$, $(d-2)(r_0 + r_1 ) \geq dr_0 + (d-4)r_1$, so $a_{d,0,k,d-k} = 0$ for $k \leq d-2$. Now, if $(d- \frac{4}{3} ) (r_0 + r_1 ) \leq (d-4)r_0 + dr_1$, then $2r_0 \leq r_1$, so $t(r_0 + r_1 ) > (d- \frac{4}{3} )(r_0 + r_1 ) \geq dr_0 + (d-2)r_1$. It follows that either $a_{d,0,d-1,1} = 0$, in which case $C$ is a union of $2d$ lines and hence $(p,C) \in H_{0,1}$, or $a_{k,d-k,d,0} = 0$ for all $k \leq d-2$, in which case $C \in S$. We now consider the wall in the $G$-ample cone determined by the Wronskian $W_{1,1}$. A wall of the $G$-ample cone occurs at $t = d- \frac{4}{3}$. More specifically, $X^{ss} (t) = X \backslash ((H_{0,1} \cap H_{1,1}) \cup S)$, and $X^s (t) \subseteq X \backslash (H_{0,1} \cup S)$. First, notice that if $(p,C) \notin H_{1,1}$, then $(p,C) \in X^{ss} (t)$, since $W_{1,1}$ is a $G$-invariant section of ${\mathcal{O}\,}_X (6(d- \frac{4}{3} ), 6(d- \frac{4}{3} ), 6)$ that does not vanish at $(p,C)$. Moreover, by general variation of GIT we know that, when passing from a chamber to a wall, we have $$X^{ss} (t + \epsilon ) \subseteq X^{ss} (t)$$ $$X^s (t) \subseteq X^s (t + \epsilon )$$ Thus, $X^s (t) \subseteq X \backslash (H_{0,1} \cup S)$ and $X \backslash ((H_{0,1} \cap H_{1,1}) \cup S) = X^{ss} (t)$. Now, suppose that $(p,C) \in S$. Using the same argument as before with the same 1-parameter subgroup, we see that $(p,C) \notin X^{ss} (t)$. Next, suppose that $(p,C) \in H_{0,1}$. In this case, we again consider the 1-parameter subgroup with weights $(-1,1,-2,2)$. As before, we have $$\mu_{\lambda} ( x_1 , y_1 , a_{i_0 , i_1 , j_0 , j_1} ) = 3d-4 + i_0 - i_1 + 2j_0 - 2j_1 = 2(i_0 + 2j_0 - 2)$$ which is non-negative when $i_0 + 2j_0 \geq 2$. Since, by assumption, $C$ has no non-zero terms with both $i_0 \leq 1$ and $j = 0$, we see that $(p,C) \notin X^s (t)$. Notice furthermore that if $(p,C) \in H_{0,1} \cap H_{1,1}$, this means that the osculating $(1,1)$ curve to $C$ at $p$ is the pair of lines through that point, and this curve intersects $C$ with multiplicity at least 4. This means that either $a_{0,d,1,d-1} = 0$ or $a_{2,d-2,0,d} = 0$, which implies that the expression $i_0 + 2j_0 - 2$ above is zero for at most one term, and strictly positive for all of the others. Now consider the 1-parameter subgroup with weights $(-1 - \epsilon , 1 + \epsilon , -2,2)$. For $\epsilon > 0$, we see that any curve with $a_{0,d,1,d-1} = 0$ is unstable. Conversely, if $\epsilon < 0$, we see that any curve with $a_{2,d-2,0,d} = 0$ is unstable. It follows that $(p,C) \notin X^{ss} (t)$, and thus $X^{ss} (t) = X \backslash ((H_{0,1} \cap H_{1,1}) \cup S)$. Again, we want to use Theorem \[Thaddeus\] to study the wall crossing at $t_0 = d- \frac{4}{3}$. Again, we let $$X^{\pm} = X^{ss} (t_0) \backslash X^{ss} (t_0 \mp \epsilon )$$ $$X^0 = X^{ss} (t_0 ) \backslash (X^{ss} (t_0 + \epsilon ) \cup X^{ss} (t_0 - \epsilon ))$$ and determine $X^-$ and $X^0$. With the set-up above, $X^- = H_{0,1} \backslash H_{1,1}$. $X^0$ is the set of all pointed curves $(p,C)$ admitting the following description: $C$ consists of a smooth curve of bidegree $(1,2)$ (or $(2,1)$), together with $d-1$ copies of the tangent line to this curve through a point that has a tangent line, and $d-2$ copies of the other line through this same point. The marked point $p$ is the unique other point on the smooth $(1,2)$ curve that has a tangent line. We have already seen that $X^{ss} (t_0) = X \backslash ((H_{0,1} \cap H_{1,1}) \cup S)$ and $X^{ss} (t_0 + \epsilon ) = X \backslash (H_{0,1} \cup S)$. Thus, $X^- = H_{0,1} \backslash H_{1,1}$. To prove the statement about $X^0$, let $(p,C) \in X^0$. Notice that, since $X^0 \subseteq X^-$, exactly one of the two lines through $p$ intersects $C$ with multiplicity exactly 2. Since $(p,C) \notin X^{ss} (t_0 - \epsilon )$, there must be a nontrivial 1-parameter subgroup that acts on $(p,C)$ with strictly positive weight. Again we assume that this subgroup acts with weights $(-r_0 , r_0 , -r_1, r_1 )$, with $0 \leq r_0 \leq r_1$ and $r_1 > 0$. Again we have $$\mu_{\lambda} (x_l , y_m , a_{i_0 , i_1 , j_0 , j_1} ) = r_0 (i_0 - i_1 ) + r_1 (j_0 - j_1 ) - t((-1)^l r_0 + (-1)^m r_1) > 0$$ We divide this into four cases, depending on $p$. **Case 1 – $p = (0,1:0,1)$:** In this case, $l = m = 1$. Again we have $t(-r_0 - r_1 ) > (d-1)(-r_0 - r_1 ) \geq -(d-2)r_0 - dr_1$. Now, if $t(-r_0 - r_1 ) \leq -dr_0 - (d-2)r_1$, then $(d - \frac{4}{3})(-r_0 - r_1 ) < -dr_0 - (d-2)r_1$, so $r_1 > 2r_0$. This means that $t(-r_0 - r_1 ) < (d - \frac{4}{3})(-r_0 - r_1 ) < -(d-4)r_0 - dr_1$. It follows that $a_{0,d,0,d} = a_{1,d-1,0,d} = 0$, and either $a_{0,d,1,d-1} = 0$ or $a_{2,d-2,0,d} = 0$. But we know that exactly one of the two lines through $p$ intersects $C$ with multiplicity exactly 2, so neither of these is a possibility. **Case 2 – $p$ lies on the line $y_0 = 0$, but not the line $x_0 = 0$:** Following the same argument as above we see that either $p$ lies on a linear component of $C$, or every point of $C$ that lies on the line $x_0 = 0$ also lies on the line $y_0 = 0$. It follows that $(p,C) \notin X^-$, a contradiction. **Case 3 – $p$ lies on the line $x_0 = 0$, but not on the line $y_0 = 0$:** Again, following the same argument as above we see that $p$ lies on a linear component of $C$. This implies that $(p,C) \notin X^-$, which is impossible. **Case 4 – $p$ does not lie on either of the lines $x_0 = 0$ or $y_0 = 0$:** In this case, $l = m = 0$. As above, we see that $a_{k_0 ,d-k_0 ,k_1 ,d-k_1} = 0$ if $k_0$ and $k_1$ are both less than $d$, and $a_{d,0,k,d-k} = 0$ for $k < d-1$. Now, if $(d - \frac{3}{2})(r_0 + r_1 ) \leq (d-6)r_0 + dr_1$, then $3r_0 \leq r_1$, so $t(r_0 + r_1 ) > (d - \frac{3}{2})(r_0 + r_1 ) \geq dr_0 + (d-2)r_0$. It follows that either $a_{d,0,d-1,1} = 0$, in which case $C$ is a union of $2d$ lines, which is impossible, or $a_{k,d-k,d,0} = 0$ for all $k < d-2$. We therefore see that $C$ is of the form $$C = x_0^{d-2}y_0^{d-1} (a_{d,0,d,0}x_0^2 y_0 + a_{d,0,d-1,1}x_0^2 y_1 + a_{d-1,1,d,0}x_0 x_1 y_0 + a_{d-2,2,d,0}x_1^2 y_0 ).$$ Thus, $C$ consists of three components. One is a curve of bidegree $(2,1)$. The other two components consist of multiple lines through one of the points on this curve that has a tangent line. The point $p$ is forced to be the unique other such point. It is clear that this $(p,C) \in X^{-}$, since by definition, one of the lines through $p$ intersects $C$ with multiplicity greater than 1, and it is impossible for it to intersect a smooth curve of bidegree $(2,1)$ with higher multiplicity than 2, or for the other line through $p$ to intersect the curve with multiplicity at all. To see that $(p,C) \notin X^{ss}(t_0 - \epsilon )$, consider the 1-parameter subgroup with weights $(-1,1,-2,2)$. Finally, notice that all such curves are in the same orbit of the action of $G$, so $X^0$ must be the set of *all* such curves. To see this, note that if we fix the two points that have tangent lines to be $(1,0:1,0)$ and $(0,1:0,1)$, then the curve is determined uniquely by the third point of intersection of the curve with the diagonal. Since $PSL(2, \mathbb{C} )$ acts 3-transitively on points of ${\mathbb{P}\,}^1$, we obtain the desired result. The map $X //_{t_0 - \epsilon} G \to X //_{t_0} G$ contracts the locus $H_{0,1} \backslash H_{1,1}$ to a point. Outside of this locus, the map is an isomorphism. Let $C = x_1^{d-2}y_1^{d-1}(x_0^2 y_1 + x_1^2 y_0 )$, and $p = (0,1:0,1)$. Then $(p,C) \in X^0$. As we have seen, $X^0$ is the orbit of $(p,C)$, so $G \cdot (p,C)$ is closed in $X^{ss}(t_0)$ and $X^0 //_{t_0} G$ is a point. Notice that the stabilizer of $(p,C)$ must fix $p = (0,1:0,1)$, and the other ramification point, which is $(1,0:1,0)$. Thus, the stabilizer of $(p,C)$ must consist solely of pairs of diagonal matrices. A quick check shows that the stabilizer of $(p,C)$ is the one-parameter subgroup with weights $(-1,1,-2,2)$, which is isomorphic to $\mathbb{C}^*$. Again, the conclusion follows from Theorem \[Thaddeus\]. Our main interest is the case where $d=3$. As above, this is because in this case $X //_{t_0 - \epsilon} G$ is a birational model for $\overline{M}_{4,1}$. In particular, we have the following: There is a birational contraction $\beta : \overline{M}_{4,1} \dashrightarrow X //_{t_0 - \epsilon} G$. As above, it suffices to exhibit a morphism $\beta^{-1}: V \to \overline{M}_{4,1}$, where $V \subseteq X //_{t_0 - \epsilon} G$ is open with complement of codimension $\geq 2$ and $\beta^{-1}$ is an isomorphism onto its image. Again, we let $U \subseteq X^{ss}(t_0 - \epsilon)$ be the set of all moduli stable pointed curves $(p,C) \in X^{ss}(t_0 - \epsilon)$. The proof in this case is exactly like that in the case of ${\mathbb{P}\,}^2$, as the discriminant locus $\Delta \subseteq X$ is again an irreducible $G$-invariant hypersurface. By the universal property of the moduli space, since $U \to Z$ is a family of moduli stable curves, it admits a unique map $U \to Z \to \overline{M}_{4,1}$. This map is certainly $G$-equivariant, so it factors uniquely through a map $U //_{t_0 - \epsilon} G \to \overline{M}_{4,1}$. Since every curve of bidegree $(3,3)$ on ${\mathbb{P}\,}^1 \times {\mathbb{P}\,}^1$ is canonical, two such curves are isomorphic if and only if they differ by an automorphism of ${\mathbb{P}\,}^1 \times {\mathbb{P}\,}^1$. It follows that this map is an isomorphism onto its image. There is a birational contraction of $\overline{M}_{4,1}$ contracting the pointed Brill-Noether divisor $BN^1_{3,(0,2)}$. Moreover, if $P$ is the Petri divisor, then the divisors $BN^1_{3,(0,2)}$, $P$, $\Delta_1$, $\Delta_2$, and $\Delta_3$ span a simplicial face of $\overline{NE}^1 ( \overline{M}_{4,1} )$. The composition $\overline{M}_{4,1} \dashrightarrow X //_{t_0 - \epsilon} G \to X //_{t_0} G$ is a birational contraction. By the above, the given pointed Brill-Noether divisor is contracted by this map, so it suffices to show that $P$ and the $\Delta_i$’s are contracted as well. The proof is again the same as the ${\mathbb{P}\,}^2$ case. We note that every smooth curve in $X$ is Petri general, so the closure of the image of each of these divisors must be contained in the singular locus $\Delta$, which is an irreducible hypersurface. Since the generic point of $\Delta$ is an irreducible nodal curve, we see that the closure of the image of $\Delta_i$ is of codimension 2 or greater for all $i \geq1$. Moreover, since the set of singular curves with semi-canonical pencils has codimension at least 2 in $\overline{M}_{4,1}$, and we have constructed an embedding of an open subset of $\Delta$ into $\overline{M}_{4,1}$, we may conclude that $P$ is contracted as well.

--- abstract: 'Galaxy starlight at 3.6$\mu$m is an excellent tracer of stellar mass. Here we use the latest 3.6$\mu$m imaging from the Spitzer Space Telescope to measure the total stellar mass and effective radii in a homogeneous way for a sample of galaxies from the SLUGGS survey. These galaxies are representative of nearby early-type galaxies in the stellar mass range of 10 $<$ log M$_{\ast}$/M$_{\odot}$ $<$ 11.7, and our methodology can be applied to other samples of early-type galaxies. We model each galaxy in 2D and estimate its total asymptotic magnitude from a 1D curve-of-growth. Magnitudes are converted into stellar masses using a 3.6$\mu$m mass-to-light ratio from the latest stellar population models of Röck et al., assuming a Kroupa IMF. We apply a ratio based on each galaxy’s mean mass-weighted stellar age within one effective radius (the mass-to-light ratio is insensitive to galaxy metallicity for the generally old stellar ages and high metallicities found in massive early-type galaxies). Our 3.6$\mu$m stellar masses agree well with masses derived from 2.2$\mu$m data. From the 1D surface brightness profile we fit a single Sersic law, excluding the very central regions. We measure the effective radius, Sersic n parameter and effective surface brightness for each galaxy. We find that galaxy sizes derived from shallow optical imaging and the 2MASS survey tend to underestimate the true size of the largest, most massive galaxies in our sample. We adopt the 3.6$\mu$m stellar masses and effective radii for the SLUGGS survey galaxies.' author: - | Duncan A. Forbes$^{1}$[^1], Luciana Sinpetru$^{2}$, Giulia Savorgnan$^1$, Aaron J. Romanowsky$^{3,4}$, Christopher Usher$^{5}$ and Jean Brodie$^{4}$\ $^{1}$Centre for Astrophysics & Supercomputing, Swinburne University, Hawthorn VIC 3122, Australia\ $^{2}$Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ, UK\ $^{3}$Department of Physics and Astronomy, San José State University, One Washington Square, San Jose, CA 95192, USA\ $^{4}$University of California Observatories, 1156 High Street, Santa Cruz, CA 95064, USA\ $^{5}$Astrophysics Research Institute, Liverpool John Moores University, 146 Brownlow Hill, Liverpool L3 5RF, UK\ title: 'The SLUGGS Survey: stellar masses and effective radii of early-type galaxies from Spitzer Space Telescope 3.6$\mu$m imaging' --- \[firstpage\] galaxies: masses – galaxies: evolution – galaxies: individual Introduction ============ The total stellar mass is a fundamental parameter for any galaxy. Not only do many other galaxy properties vary with stellar mass, but an accurate measure of stellar mass is required to probe the dark matter content (i.e. the total mass minus the stellar mass) in a galaxy. However, measuring the total stellar mass is problematic, even once the total luminosity has been accurately measured. For example, a common approach is to measure the total luminosity of a galaxy at near-IR wavelengths for which the light mostly comes from old stars that dominate the mass, and the effects of dust are much reduced compared to optical wavelengths. A typical approach is to use the full-sky ground-based near-IR imaging of the 2MASS survey (Jarrett et al. 2003). However it has been reported that the 2MASS reduction pipeline systematically underestimates the total luminosity and size of large, nearby galaxies due to a truncation of their surface brightness profiles (Schombert & Smith 2012; Scott et al. 2013). An alternative approach is to use the 3.6$\mu$m band of the Spitzer Space Telescope (Werner et al. 2004) or the 3.4$\mu$m band of the WISE space telescope (Wright et al. 2010). Such wavelengths are particularly well suited to measure the stellar masses of galaxies. For example, Norris et al. (2014) concluded that photometry from WISE can [*“...provide extremely simple, yet robust stellar mass tracers for dust free older stellar populations..."*]{}. This is because the 3.4–3.6$\mu$m light from galaxies is dominated by the light from old stars, and it is less effected by variations in the star formation history than shorter wavelengths. Although intermediate-aged stars, hot dust and polycyclic aromatic hydrocarbons may contribute to the emission at 3.6$\mu$m, these sources are negligible for most early-type galaxies which are dominated by old stellar populations (Meidt et al. 2012; Querejeta et al. 2015). Here we use 3.6$\mu$m imaging from the Spitzer Space Telescope. The 3.6$\mu$m mass-to-light ratio (M/L$_{3.6}$) has virtually no dependence on metallicity, and only a very small dependence on age for old stellar ages. We use the latest single-burst stellar population models (Röck et al. 2015) which are based on empirical mid-infrared stellar spectra (Cushing et al. 2005; Rayner et al. 2009). These models cover a range of metallicity, ages and IMF slopes. They are shown to reproduce well the mid-infrared colours of early-type galaxies. For a Kroupa IMF, these models give M/L$_{3.6}$ $\sim$ 0.8 for a stellar population mean age of 9 Gyr, with a variation between different isochrones, i.e. from BaSTI and Padova, of $\sim$0.05. For a metallicity range of \[Fe/H\] = –0.4 to solar (i.e. typical of the mean values for massive early-type galaxies), the variation is insignificant at $\sim$0.02. We note that the Flexible Stellar Population Synthesis (FSPS; Conroy & Gunn 2010) models with AGB circumstellar dust included (Villaume et al. 2015) also give M/L$_{3.6}$ $\sim$0.8 for a 9 Gyr old, moderately metal-rich population. Meidt et al. (2014) adopted a constant value of M/L$_{3.6}$ = 0.6 for their S4G sample, although their sample was dominated by late-type galaxies with younger mean ages on average. Stellar mass-to-light ratios have a strong dependence on the Initial Mass Function (IMF). The M/L$_{3.6}$ values quoted above refer to a Kropua IMF. Salpeter and other IMFs tend to have higher M/L$_{3.6}$ values by a factor of $\sim$1.5-3 (Röck et al. 2015), which would lead to larger stellar masses for a given 3.6$\mu$m luminosity. Recent work indicates that the IMF for early-type galaxies is skewed to low mass stars (see e.g. Ferre-Mateu et al. 2013; Martin-Navarro et al. 2015; McConnell et al. 2016). Currently, it is not yet clear what is causing the IMF variations nor whether these variations are confined to galaxy central regions, high metallicity regions or spheroids. Here we adopt a Kroupa IMF for our global M/L$_{3.6}$ but caution that the stellar masses for massive elliptical galaxies may need revising upwards. Since M/L$_{3.6}$ varies with stellar age, an age-appropriate ratio should be employed. Here we adopt an age dependent mass-to-light ratio from the Röck et al. (2015) models using mean stellar ages from the literature. We assume a Kroupa IMF. The SLUGGS survey targets 25 nearby massive early-type galaxies in different environments and 3 so-called bonus galaxies (Brodie et al. 2014). We study the kinematics and metallicity of both the galaxy itself, and its system of globular clusters, to large galactocentric radii. The sample galaxies are chosen to cover a range of key parameters including stellar mass and physical size. Until now, the approach in the SLUGGS survey to measure stellar mass has been to obtain the extinction-corrected K-band (2.2$\mu$m) magnitude from the 2MASS extended galaxy catalog and apply the correction of Scott et al. (2013) for missing light. We then applied a constant M/L$_{2.2}$ = 1 irrespective of stellar metallicity or age. A value of unity is simplistic, but a reasonable approximation for a very old stellar population with a Kroupa IMF (Bruzual & Charlot 2003). The effective radii (R$_e$) of the SLUGGS galaxies are listed in Brodie et al. (2014, B14), which are based on Cappellari et al. (2011). Cappellari et al. used sizes from both optical and near-IR imaging. They noted that the near-IR sizes from the 2MASS survey (Jarrett et al. 2003) for the largest, most massive galaxies appear to be systematically underestimated and they scaled-up their near-IR sizes to match the optical sizes on average. A recent study by van den Bosch (2016) also found the 2MASS survey to underestimate the sizes (and total fluxes) of nearby galaxies. Accurate galaxy effective radii are important in order to compare galaxies on a similar relative scale. For example the SLUGGS survey and other integral field spectroscopy studies, derive kinematic profiles as a function of effective radii and measure properties such as specific angular momentum within 1 R$_e$ (Arnold et al. 2014; Alabi et al. 2015; Foster et al. 2016). Thus the photometric effective radii need to be accurate in order to correctly compare kinematic properties between different galaxies. Effective radii, combined with accurate stellar masses, are needed to probe the dark matter fraction within a given multiple of R$_e$. The Spitzer Space Telescope imaging presented here offers an opportunity to revisit the sizes and masses of the SLUGGS early-type galaxies. In the next sections we present the 3.6$\mu$m data from the Spitzer Space Telescope and our methodology for deriving total magnitudes, stellar masses, and effective radii for 27 SLUGGS early-type galaxies. These new measurements are compared with literature values. In an Appendix we list the measurements for six additional nearby early-type galaxies, which we include for the interested reader. Spitzer Data ============ Here we use images from the IRAC instrument of the Spitzer Space Telescope, which has a pixel scale of 1.22 arcsec over a 5.2 $\times$ 5.2 arcmin$^{-2}$ field-of-view. We have downloaded the latest (July 2016) available 3.6$\mu$m basic calibrated data frames from the Spitzer Heritage Archive. These Astronomical Observation Requests (AORs) are detailed in Appendix A (for SLUGGS galaxies) and C (for non-SLUGGS galaxies). These data have been corrected for scattered light, dark current, flat-fielded and flux calibrated. The MOPEX package is used to assemble the long-exposure ($>$1 s) frames into an image mosaic showing the field of view around each target galaxy. An example of the final mosaic for NGC 1407 is shown is Fig. 1. We follow a similar reduction procedure to that of Savorgnan & Graham (2016, SG16). Thus when multiple pointings are available, an overlap correction is applied to create a uniform background level. Using IRAF we determine the sky background level and rms at multiple points on the outskirts of each mosaiced image. The sky values are averaged to give a final value for the sky background of each image, which is subtracted from the image. Finally, bright stars and other unwanted objects are masked out of the mosaic. For further details see SG16. Unfortunately, the 3.6$\mu$m data for the low-mass SLUGGS galaxy NGC 4474 are not useful for measuring the total light (and hence mass) or galaxy size. In this case, the galaxy is only partially visible as it is near the edge of the available Spitzer pointing. We therefore adopt its stellar mass (log M$_{\ast}$ = 10.23) from its 2MASS K-band magnitude and its effective radius (R$_e$ = 1.5 kpc) from Brodie et al. (2014). Measuring total magnitudes ========================== To measure the total light from each galaxy we model the galaxy in 2D using the IRAF task [*ellipse*]{} and obtain the total magnitude of the galaxy model. The galaxy centre was initially allowed to vary, but if it varied by more than 1 pixel, we fixed it to the average central value. For a few galaxies (i.e. NGC 4486, 4594, 5866 and 7457) we could not obtain a good galaxy model with a radially varying position angle (PA) and so in these cases we fixed the PA to a representative value based on the radial trend. The model extends in galactocentric radius until the integrated magnitude at that radius is less than 0.02 mag different from the previous (penultimate) radius. Thus we effectively adopt the asymptotic total magnitude of the model galaxy. We estimate our combined photometric and systematic uncertainty to be $\pm$ 0.05 mag. We do not correct our 3.6$\mu$m magnitudes for Galactic extinction (which are less than 0.01 mag.). The total 3.6$\mu$m magnitudes that we measure in the Vega system, and other basic properties of the SLUGGS galaxies, are given in Table 1. ------------------ --------- ------ --------- --------- ----------- ----------------- ------------ -------------------- ------ Galaxy Type Core Dist. Age m$_{3.6}$ log M$_{\ast}$ R$_e$ $\mu_e$ n $\rm [NGC]$ \[Mpc\] \[Gyr\] \[mag\] \[M$_{\odot}$\] \[arcsec\] \[mag/arcsec$^2$\] (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) 720 E5 1 26.9 7.8 6.92 11.27 29.1 17.54 3.8 821 E6 3 23.4 12.9 7.57 11.00 43.2 19.03 6.0 1023 S0 3 11.1 13.5 6.01 10.99 48.0 17.61 4.2 1400 E1/S0 1 26.8 13.8 7.44 11.08 25.6 17.87 5.0 1407 E0 1 26.8 12.0 6.16 11.60 93.4 19.19 4.9 2768 E6/S0 2 21.8 13.3 6.68 11.21 60.3 18.70 3.8 2974 E4/S0 3 20.9 11.8 7.47 10.93 30.2 17.99 4.3 3115 S0 3 9.4 9.0 5.58 10.93 36.5 16.75 4.7 3377 E5-6 3 10.9 11.3 7.09 10.50 45.4 18.81 5.9 3607$^\dagger$ S0 1 22.2 13.5 6.51 11.39 48.2 18.33 5.3 3608 E1-2 1 22.3 13.0 7.41 11.03 42.9 19.00 5.3 4111 S0 – 14.6 6.0 7.24 10.52 10.1 15.71 3.0 4278 E1-2 1 15.6 13.7 6.84 10.95 28.3 17.51 6.2 4365 E3 1 23.1 13.4 6.31 11.51 77.8 18.96 4.9 4374 E1 1 18.5 13.7 5.81 11.51 139.0 19.71 8.0 4459 S0 3 16.0 11.9 6.76 10.98 48.3 18.52 5.4 4473 E5 1 15.2 13.0 6.74 10.96 30.2 17.67 5.0 4474 S0 3 15.5 11.1 – 10.23 17.0 – – 4486 E0/cD 1 16.7 12.7 5.30 11.62 86.6 18.24 5.1 4494 E1-2 3 16.6 11.0 6.68 11.02 52.5 18.53 4.5 4526 S0 – 16.4 13.6 6.17 11.26 32.4 17.05 3.6 4564 E6 3 15.9 13.3 7.78 10.58 14.8 16.93 3.2 4594$^\dagger$ Sa 1 9.5 12.5 4.56 11.41 72.0 17.06 3.2 4649 E2/S0 1 16.5 13.2 5.33 11.60 79.2 18.06 4.6 4697 E6 3 12.5 13.4 5.85 11.15 95.8 19.08 5.3 5846 E0-1/S0 1 24.2 12.7 6.50 11.46 89.8 19.38 5.2 5866$^{\dagger}$ S0 – 14.9 5.9 6.50 10.83 23.4 16.59 2.8 7457 S0 3 12.9 6.1 7.94 10.13 34.1 18.67 2.6 ------------------ --------- ------ --------- --------- ----------- ----------------- ------------ -------------------- ------ Notes: columns are (1) galaxy name, $\dagger$ = bonus galaxy, (2) Hubble type, (3) 1= core, 2 = intermediate, 3 = cusp central light profile, (4) distance from B14 (typical uncertainty is $\pm$0.05 dex), (5) mean stellar age from McDermid et al. (2015), see text for exceptions), (6) 3.6 micron apparent magnitude in the Vega system (typical uncertainty is $\pm$0.05), (7) stellar mass (typical uncertainty is $\pm$0.1 dex), (8) effective radius (typical uncertainty is +0.18 and –0.13 dex), (9) $\mu_e$ (typical uncertainty is +0.52 and –1.11 mag.), (10) Sersic n (typical uncertainty is +0.13 and –0.11 dex). Spitzer 3.6$\mu$m imaging is not available for NGC 4474: M$_{\ast}$ and R$_e$ are from 2MASS 2.2$\mu$m imaging. Total 3.6$\mu$m magnitudes for several SLUGGS galaxies from Spitzer data are available in the S4G study of Munoz-Mateos et al. (2015, S4G) and Savorgnan & Graham (2016, SG16). The former study measured total magnitudes using a curve-of-growth method and adopted the asymptotic magnitude. The uncertainty on the S4G total magnitudes for the SLUGGS galaxies is given as $\pm$ 0.001–0.002 mag. in the S4G online database. This is likely to be the formal uncertainty on fitting their asymptotic magnitude and does not include other sources of uncertainty. The latter study fit multiple components to the 1D surface brightness profile of each galaxy after 2D modelling. The combination of the different components was used to calculate the total magnitude, with an estimated uncertainty of $\pm$0.25 mag. Fig. 2 shows a comparison, for the SLUGGS galaxies, of our measured magnitude against 3.6$\mu$m total magnitudes from S4G (converted to the Vega system) and SG16. Our magnitudes generally lie in between these two studies. We are systematically brighter than S4G by $\sim$0.18 mag. on average. Our measurements agree fairly well with SG16 with the exception of four galaxies that are more than 0.3 mag brighter than us (and have even larger discrepancies with S4G magnitudes when in common). Three of the galaxies, NGC 5846, NGC 4374, and NGC 4486, feature a partially depleted core at their centre (Lauer et al. 2007; Krajnovic et al. 2013). The total magnitudes of SG16 do not take this into account, i.e they masked out the core and fit a Sersic profile (rather than a core-Sersic profile) to the galaxy light. This effectively overestimates the total magnitude of each galaxy by the amount of light ‘missing’ due to the depleted core, i.e. $\sim$0.2–0.3 mag. For NGC 4697 we suspect that the best-fit profiles of SG16 overestimated the spheroid effective radius and consequently its luminosity. Their 1D surface brightness profile for this galaxy is less extended than the best-fit effective radius itself. Calculating total stellar masses ================================ To calculate the total 3.6$\mu$m luminosity in solar units in the Vega system for each galaxy we convert the 3.6$\mu$m apparent magnitude into a total luminosity assuming that the absolute magnitude of the Sun to be M$_{3.6}$ = 3.24 (Oh et al. 2008), and taking its distance from Table 1. We note that the distances, usually based on surface brightness fluctuations, contribute an uncertainty of $\sim$0.1 dex to the luminosity. We multiply the luminosity by the 3.6$\mu$m mass-to-light ratio from the single stellar population models of Röck et al. (2015). In particular, we use M/L$_{3.6}$ appropriate for each galaxy’s mean mass-weighted stellar age within 1 R$_e$ taken from McDermid et al. (2015), supplemented by values from Rembold et al. (2005) for NGC 720, Norris et al. (2006) for NGC 3115, Spolaor et al. (2008) for NGC 1400 and 1407, and Sanchez-Blazquez et al. (2006) for NGC 4594. These ages are given in Table 1. The 3.6$\mu$m mass-to-light ratio that we apply varies from $\sim$0.60 to 1.0. We use the Padova isochrones for solar metallicity (which is a reasonable value for our early-type galaxies within 1 R$_e$; McDermid et al. 2015). We note that the equivalent BaSTI isochrones differ by only 0.01 for our typical age. We estimate that the uncertainty in M/L$_{3.6}$ due to differences in isochrone tracks ($\pm$0.03), mean age uncertainty ($\pm$0.05) and metallicity variations ($\pm$0.02), combined with our measurement and distance uncertainties, give a final uncertainty of about 0.1 dex in log stellar mass. We also assume a Kroupa IMF. As noted in the Introduction, although there is evidence for an IMF skewed to low mass stars, the effect seems largely limited to the central regions of the most massive galaxies. Nevertheless this gives rise to a systematic underestimate of the stellar masses of the most massive galaxies. Each galaxy in this study has a total stellar mass determined from the total K-band (2.2$\mu$m) magnitude from the 2MASS survey. The 2MASS 2.2$\mu$m magnitude is corrected for missing flux according to Scott et al. (2013) and we take the absolute magnitude of the Sun to be M$_{2.2}$ = 3.28 (table 2.1 from Binney & Merrifield 1998). The stellar mass has been calculated in previous SLUGGS papers assuming a fixed M/L$_{2.2}$ = 1.0 irrespective of stellar age (e.g. Alabi et al. 2016). A value of unity is reasonably representative of an old, metal-rich stellar population with a Kroupa-like IMF. In Fig. 3 we show a comparison of our new 3.6$\mu$m-based stellar masses versus the stellar mass obtained from the K-band. We find an excellent overall correspondence with the stellar masses derived using the K-band. Galaxies with lower 3.6$\mu$m masses relative to the previous 2.2$\mu$m masses (i.e. that lie above the unity line) tend to be those with young ($\sim$6 Gyr) mean stellar ages. The overall excellent agreement indicates that both 3.6$\mu$m and 2.2$\mu$m total magnitudes give reliable stellar masses (under the same assumption of a Kroupa IMF). Given the small difference in 2.2 vs 3.6 $\mu$m stellar masses, we will continue to adopt the K-band stellar mass of log M$_{\ast}$ = 10.23 for NGC 4474 for which we are unable to measure a 3.6$\mu$m magnitude. Measuring galaxy sizes ====================== Each galaxy surface brightness profile is fit with a single Sersic law (Graham & Driver 2005). We exclude the inner 2 pixels (2.44 arcsec), i.e. we only fit radii that are larger than the effective FWHM resolution of the Spitzer Space Telescope. This also means that the presence of any nuclear star cluster or AGN does not affect the fits. Most of our galaxies reveal central surface brightness profiles that can be well described as either a core or a cusp (Lauer et al. 2007; Krajnovic et al. 2013; Dullo & Graham 2013), as listed in Table 1. In the case of cusps, they are generally well fit by a single Sersic profile. On the other hand for core profiles we exclude the so-called depleted core region from the fits. Thus the fitting range for each galaxy is either $>$2 pixels for the cusp, intermediate and unknown galaxies and greater than the depleted core region for the core galaxies. The fits to the 3.6$\mu$m surface brightness profile for each SLUGGS galaxy are shown in Appendix B. The central region excluded from each fit is indicated by open circles, and can be most clearly seen in the Sersic profile minus data residual profiles. The code used for the fitting process is the same as SG16. For most galaxies SG16 fit multiple components to each galaxy. However, they did fit a single Sersic profile to three large SLUGGS galaxies. In these cases our effective radii agree very well with their value, i.e. NGC 4374: 139.0 vs 129.8 arcsec, NGC 4486: 86.6 vs 87.1 arcsec and NGC 5846: 89.8 vs 83.4 arcsec. We list the (equivalent circular) effective radii from the single Sersic fits to our 3.6$\mu$m surface brightness profiles in Table 1. A typical uncertainty associated with the effective radius of the SLUGGS galaxies is calculated based on the average of the uncertainties of the 14 early-type galaxies effective radii in the sample of SG16 with single Sersic fits compared to those from other studies (see SG16 for details). The 1$\sigma$ uncertainty of +0.18 dex and –0.13 dex thus takes into account both random and systematic errors. If we only considered random measurement errors a smaller uncertainty would result. Table 1 also lists the other fitting parameters, i.e. the Sersic n value and the surface brightness at the effective radius $\mu_e$. Again, we adopt the typical uncertainties found for the 14 early-type galaxies, i.e. +0.13 and –0.11 dex on n, and +0.52 and –1.11 mag on $\mu_e$. We note that Sersic parameters are strongly correlated with each other. In Fig. 4 we compare our new effective radii from the Spitzer images with those listed by Brodie et al. (2014), i.e. from Cappellari et al. (2011), based on optical data. We find good correspondence for small-sized galaxies and generally we measure greater effective radii for the large-sized galaxies. This implies that the sizes of the most massive galaxies are underestimated, as was suggested by Cappellari et al. (2011). Given the reasonable agreement in sizes for the smaller galaxies, the uncertainties from the work of SG16 may be an overestimate. Fig. 4 also shows the effective radii from the 2MASS Large Galaxy Atlas (Jarrett et al. 2003). In particular, we take the K-band effective radius along the semi-major axis ($k\_r\_eff$) and multiply it by an ellipticity correction ($\sqrt(k\_ba)$ to obtain an equivalent circular R$_e$ value. Cappellari et al. (2011) found that the 2MASS effective radii were on average 1.7$\times$ smaller than the optical radii from the RC3. The dotted line in this plot indicates sizes reduced by this factor and indeed the 2MASS R$_e$ values scatter about this line. The smaller sizes may be due to an oversubtraction of the sky background which truncates the 2MASS K-band surface brightness profile (Schombert & Smith 2012). ------------- ------------ ------------ ------------ ------------ ------------ ----------- Galaxy 3.6$\mu$m B14 Kor09 Chen10 Vika13 Other $\rm [NGC]$ \[arcsec\] \[arcsec\] \[arcsec\] \[arcsec\] \[arcsec\] (1) (2) (3) (4) (5) (6) (7) 1407 93 63 – – – 100 (P15) 4365 78 53 $\sim$154 97 78 126 (B12) 4374 139 53 $\sim$135 131 96 – 4486 87 81 $\sim$630 105 82 – 4649 79 66 $\sim$118 105 68 – 5846 90 59 – – – 81 (N14) ------------- ------------ ------------ ------------ ------------ ------------ ----------- Notes: columns are (1) galaxy name, effective radii in arcsec from (2) 3.6$\mu$m imaging (this work), (3) Brodie et al. (2014), (4) Kormendy et al. (2009), (5) Chen et al. (2010), (6) Vika et al. (2013), and (7) other published SLUGGS works (i.e. Pota et al. 2015; Blom et al. 2012 and Napolitano et al. 2014). We find that both the 2MASS and B14 effective radii underestimate the true size of the most massive, largest galaxies. In Table 2 we list R$_e$ values in arcsec from single Sersic fits to the surface brightness profiles of the six most massive SLUGGS galaxies from the literature, along with the B14 values and those measured in this work. We include the single Sersic fits to SDSS imaging from Chen et al. (2010) and Vika et al. (2013). In the case of Vika et al. we quote the z band R$_e$ value. The final column in Table 2 gives R$_e$ values derived by previous SLUGGS studies of Blom et al. (2012) and Pota et al. (2015) from deep optical imaging, and that adopted by Napolitano et al. (2014). We also include effective radii from Kormendy et al. (2009) who presented very deep optical imaging from multiple sources of Virgo cluster galaxies, and fit single Sersic profiles excluding the central core region (as we have done). We have converted their semi-major axis effective radii to equivalent circular ones using each galaxy’s average ellipticity. We also have four low mass Virgo galaxies in common with Kormendy et al. and we find good agreement for three (NGC 4473, 4564 and 4551), For NGC 4459 we find a somewhat larger size than Kormendy et al. (2009) but this galaxy is difficult to model correctly given the bright nearby foreground star (despite our efforts to mask it from the 2D modelling process). For the massive Virgo galaxies, we have excellent agreement for NGC 4374 (M84) with Kormendy et al. and Chen et al. For NGC 4365, 4486 and 4649 (M60), we have good agreement with Vika et al., but we find systematically smaller sizes than Chen et al., and significantly smaller sizes than Kormendy et al. (and Blom et al. 2012 for NGC 4365). We suspect that this is because these galaxies have elongated isophotes in their outer regions, i.e light that goes beyond the Spitzer mosaic (whereas NGC 4374 has very circular outer isophotes and is well contained within our mosaic). For NGC 4486 (M87) Kormendy et al. derive a size that is 6–8$\times$ that of other studies (corresponding to $\sim$50 kpc). Their single Sersic fit includes the extended light of the cD envelope. Our measured 3.6$\mu$m effective radii for NGC 1407 and NGC 5846 are similar to those used by Pota et al. (2015) and Napolitano et al. (2014) but significantly larger than B14. We compare our 3.6$\mu$m sizes and stellar masses with the Virgo cluster galaxies of Chen et al. (2010) in Fig. 5. Chen et al. fit single Sersic profiles to multi-filter SDSS imaging of $\sim$ 100 Virgo cluster early-type galaxies. We convert their measurements into physical properties assuming a Virgo distance of 16.5 Mpc, and log M/L$_g$ = 0.7 from Bell et al. (2003) for red galaxies. Fig. 5 shows that the distribution of sizes and stellar masses from our 3.6$\mu$m measurements for the SLUGGS galaxies are consistent with that of the Virgo early-type galaxies from Chen et al. (2010). Thus we expect our new stellar masses and effective radii for the SLUGGS galaxies to be representative of nearby early-type galaxies in general. It is clear from Table 2 that a wide variety of galaxy effective radii can be found in the literature for massive galaxies (even when restricted to a single Sersic fit). We recognise that our effective radii derived from Spitzer 3.6$\mu$m imaging may be updated by deeper imaging studies (perhaps leading to even larger R$_e$ values if a single Sersic continues to be adopted), however our Spitzer imaging provides a homogenous set of improved effective radii for the SLUGGS survey which we now adopt. Conclusions =========== Using archival Spitzer Space Telescope imaging of the nearby early-type galaxies from the SLUGGS survey, we have created 3.6$\mu$m mosaic images of each galaxy (excluding NGC 4474 for which Spitzer imaging was not available). After masking out foreground stars and other unwanted features, we modelled each galaxy and derived total asymptotic magnitudes. Our total magnitudes generally lie in between those of 3.6$\mu$m literature studies for the galaxies in common. These total magnitudes are converted into stellar masses using the single stellar population models of Röck et al. (2015). The 3.6$\mu$m flux from early-type galaxies is an ideal tracer of stellar mass as the mass-to-light ratio at this wavelength is very insensitive to metallicity and only mildly sensitive to age for old stellar populations. Here we use M/L$_{3.6}$ from Padova isochrones that is adjusted for each galaxy’s mean stellar age while assuming a Kroupa IMF. We estimate the final uncertainty in log stellar mass to be $\pm$0.1 dex. We find that our 3.6$\mu$m stellar masses have a strong linear correlation with stellar masses derived at 2.2$\mu$m, after correcting the 2MASS K-band fluxes for missing light. From our 2D galaxy modelling, we fit a single Sersic law to the 3.6$\mu$m surface brightness profile in 1D. We exclude the central 2 pixels in all galaxies (corresponding to the FWHM of the Spitzer Space Telescope) and in addition exclude the central core of the massive galaxies that contain so-called depleted cores. As well as new effective radii (R$_e$), we derive the surface brightness at 1 R$_e$ and the Sersic n parameter. Our 3.6$\mu$m sizes show good agreement with literature values from optical imaging as used by Brodie et al. (2014), and show that sizes from the 2MASS (K-band) survey systematically underestimate the true size (as found previously by Cappellari et al. 2011). For the larger, more massive galaxies we find that the optical sizes used by Brodie et al. (2014) are also systematically underestimated relative to the sizes from our 3.6$\mu$m imaging and deep optical imaging in the literature. Our new sizes and stellar masses show good agreement with those of Virgo cluster early-type galaxies measured from SDSS imaging. We now adopt the sizes and stellar masses, measured in a homogeneous way from our 3.6$\mu$m imaging, for galaxies in the SLUGGS survey. Our methodology can be adopted by other studies requiring more accurate effective radii and stellar masses for nearby early-type galaxies. Acknowledgements ================ We thank A. Ferre-Mateu and A. Alabi for useful comments. We thank the referee for several suggestions that have improved the paper. DAF thanks the ARC for financial support via DP130100388. JB and AR acknowledge NSF grants AST-1211995, AST-1616598 and AST-1616710. This work is based on observations made with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA. 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------------- ----------------------------------------------------------------------------------------------------------------------------------------------- Galaxy Astronomical Observation Requests $\rm [NGC]$ 720 r49345024, r49345280 821 r14569216, r49418752, r49419008 1023 r4432640, r50631168, r50631680, r50631936, r52778496, r52778752, r52779008, r52779264, r52779520, r52779776, r52780032 1400 r49436416 1407 r49348096, r49348352 2768 r18031872 2974 r18032384, r49613056 3115 r4441088 3377 r4444928, r49411328, r50545664, r50545920, r50546176, r52910336, r52910592, r52910848, r52911104, r52911360, r52911616, r52911872 3607 r4449536, r49389312, r49614592, r49614848 3608 r18033408, r49460736, r49614848 4111 r30984192, r31015424, r42249216, r42249472, r50528000, r50528256, r50528512, r52912128, r52912384, r52912640, r52912896, r52913152, r52913408 4278 r4461568, r49616128 4365 r11115264, r49358336, r49358592, r50576640, r50577152, r50577664, r52976640, r52976896, r52977152, r52977408, r52977664, r52977920 4374 r4463872, r50608128, r50608640, r50609152, r52971264, r52971520, r52971776, r52972032, r52972288, r52972544 4459 r11378944, r49501696, r49501952 4473 r11377920, r49339904, r49340160, r50554368, r50554880, r50555392, r53005312, r53005568, r53005824, r53006080, r53006336, r53006592 4474 – 4486 r12673792, r49337856, r49338112, r50576384, r50576896, r50577408, r52962304, r52962560, r52962816, r52963072, r52963328, r52963584 4494 r18035200 4526 r4472064, r49341440, r49341696, r49595904, r50644736, r50644992, r50645248, r52992768, r52993024, r52993280, r52993792, r52994048 4564 r14572032, r49510912, r49511168, r50647040, r50647296, r50647552, r52942592, r52942848, r52943104, r52943360, r52943616, r52943872 4594 r5517824, r5518080, r50595328, r50595840, r50596864, r52765696, r52765952, r52766208, r52766464, r52766720, r52766976, r52767232 4649 r4476672, r49337344, r49337600, r50590208, r50590976, r50591488, r52967680, r52967936, r52968192, r52968448, r52968704, r52968960 4697 r10896896, r49359872, r49360128, r50622464, r50622720, r50622976, r52809472, r52809728, r52809984, r52810240, r52810496, r52810752, r52811008 5846 r4491264, r16310272, r49363968, r49364224 5866 r5526016, r5526272 7457 r18037504, r50547200, r50547456, r50547712, r52946176, r52946432, r52946688, r52946944, r52947200, r52947456 ------------- ----------------------------------------------------------------------------------------------------------------------------------------------- Appendix B ========== Sersic fits to 3.6$\mu$m surface brightness profiles of SLUGGS galaxies are shown in Figures 6 to 14. ![\[fig:corrGC2\] Sersic fit to 3.6$\mu$m surface brightness profile and residuals as a function of circular equivalent radius. The [*upper*]{} panel shows the data points (with excluded data points shown by open circles) and the best fit Sersic profile in red. Parameters for the Sersic fit are given in the top right. The dashed line shows 3$\times$ the rms of the sky background level. $\Delta$ gives the rms of the residuals in mag arcsec$^{-2}$. The [*lower*]{} panel shows the residuals of the Sersic model fit minus the surface brightness data. ](montage1x3_A1.jpg){width="40.00000%"} ![\[fig:corrGC2\] Sersic fit to 3.6$\mu$m surface brightness profile and residuals as a function of circular equivalent radius. The [*upper*]{} panel shows the data points (with excluded data points shown by open circles) and the best fit Sersic profile in red. Parameters for the Sersic fit are given in the top right. The dashed line shows 3$\times$ the rms of the sky background level. $\Delta$ gives the rms of the residuals in mag arcsec$^{-2}$. The [*lower*]{} panel shows the residuals of the Sersic model fit minus the surface brightness data. ](montage1x3_A2.jpg){width="40.00000%"} ![\[fig:corrGC2\] Sersic fit to 3.6$\mu$m surface brightness profile and residuals as a function of circular equivalent radius. The [*upper*]{} panel shows the data points (with excluded data points shown by open circles) and the best fit Sersic profile in red. Parameters for the Sersic fit are given in the top right. The dashed line shows 3$\times$ the rms of the sky background level. $\Delta$ gives the rms of the residuals in mag arcsec$^{-2}$. The [*lower*]{} panel shows the residuals of the Sersic model fit minus the surface brightness data. ](montage1x3_A3.jpg){width="40.00000%"} ![\[fig:corrGC2\] Sersic fit to 3.6$\mu$m surface brightness profile and residuals as a function of circular equivalent radius. The [*upper*]{} panel shows the data points (with excluded data points shown by open circles) and the best fit Sersic profile in red. Parameters for the Sersic fit are given in the top right. The dashed line shows 3$\times$ the rms of the sky background level. $\Delta$ gives the rms of the residuals in mag arcsec$^{-2}$. The [*lower*]{} panel shows the residuals of the Sersic model fit minus the surface brightness data. ](montage1x3_A4.jpg){width="40.00000%"} ![\[fig:corrGC2\] Sersic fit to 3.6$\mu$m surface brightness profile and residuals as a function of circular equivalent radius. The [*upper*]{} panel shows the data points (with excluded data points shown by open circles) and the best fit Sersic profile in red. Parameters for the Sersic fit are given in the top right. The dashed line shows 3$\times$ the rms of the sky background level. $\Delta$ gives the rms of the residuals in mag arcsec$^{-2}$. The [*lower*]{} panel shows the residuals of the Sersic model fit minus the surface brightness data. ](montage1x3_A5.jpg){width="40.00000%"} ![\[fig:corrGC2\] Sersic fit to 3.6$\mu$m surface brightness profile and residuals as a function of circular equivalent radius. The [*upper*]{} panel shows the data points (with excluded data points shown by open circles) and the best fit Sersic profile in red. Parameters for the Sersic fit are given in the top right. The dashed line shows 3$\times$ the rms of the sky background level. $\Delta$ gives the rms of the residuals in mag arcsec$^{-2}$. The [*lower*]{} panel shows the residuals of the Sersic model fit minus the surface brightness data. ](montage1x3_A6.jpg){width="40.00000%"} ![\[fig:corrGC2\] Sersic fit to 3.6$\mu$m surface brightness profile and residuals as a function of circular equivalent radius. The [*upper*]{} panel shows the data points (with excluded data points shown by open circles) and the best fit Sersic profile in red. Parameters for the Sersic fit are given in the top right. The dashed line shows 3$\times$ the rms of the sky background level. $\Delta$ gives the rms of the residuals in mag arcsec$^{-2}$. The [*lower*]{} panel shows the residuals of the Sersic model fit minus the surface brightness data. ](montage1x3_A7.jpg){width="40.00000%"} ![\[fig:corrGC2\] Sersic fit to 3.6$\mu$m surface brightness profile and residuals as a function of circular equivalent radius. The [*upper*]{} panel shows the data points (with excluded data points shown by open circles) and the best fit Sersic profile in red. Parameters for the Sersic fit are given in the top right. The dashed line shows 3$\times$ the rms of the sky background level. $\Delta$ gives the rms of the residuals in mag arcsec$^{-2}$. The [*lower*]{} panel shows the residuals of the Sersic model fit minus the surface brightness data. ](montage1x3_A8.jpg){width="40.00000%"} ![\[fig:corrGC2\] Sersic fit to 3.6$\mu$m surface brightness profile and residuals as a function of circular equivalent radius. The [*upper*]{} panel shows the data points (with excluded data points shown by open circles) and the best fit Sersic profile in red. Parameters for the Sersic fit are given in the top right. The dashed line shows 3$\times$ the rms of the sky background level. $\Delta$ gives the rms of the residuals in mag arcsec$^{-2}$. The [*lower*]{} panel shows the residuals of the Sersic model fit minus the surface brightness data. ](montage1x3_A9.jpg){width="40.00000%"} Appendix C ========== Six additional (non-SLUGGS) galaxies have been measured in this study using the procedure described above. In Table 4 we list the Astronomical Observation Requests and in Table 5 the measured 3.6$\mu$m properties for these additional galaxies. Figures 15 and 16 show Sersic fits to their 3.6$\mu$m surface brightness profiles. ------------- ---------------------------------------------------------------------------------------------------------------------------------------------- Galaxy Astronomical Observation Requests $\rm [NGC]$ 1052 r11516672, r49600512, r49612288 2549 r26602240, r49447424, r49447680, r49619712 3379 r4445696, r49411584, r49411840, r50629376, r50629632, r50629888, r52832512, r52832768, r52833024, r52833280, r52833536, r52833792, r52834048 3665 r49465344, r49465600 3998 r4452608, r42242560, r42242816, r49622784, r50586368, r50586624, r50586880, r52892416, r52892672, r52892928, r52893184, r52893440, r52893696 4551 r49510400, r49510656 ------------- ---------------------------------------------------------------------------------------------------------------------------------------------- [ ]{} [@cccccccccc]{} Galaxy & Type & Core & Dist. & Age & m$_{3.6}$ & M$_{\ast}$ & R$_e$ & $\mu_e$ & n\ $\rm [NGC]$ & & & \[Mpc\] &\[Gyr\] & \[mag\] & \[M$_{\odot}$\] & \[arcsec\] & \[mag/arcsec$^2$\] &\ (1) & (2) & (3) & (4) & (5) & (6) & (7) & (8) & (9) & (10)\ \ 1052 & E3-4/S0 & 1 & 19.4 & 13.0 & 7.12 & 11.02 & 21.9 & 17.16 & 3.4\ 2549 & S0 & 3 & 12.3 & 8.9 & 7.75 & 10.28 & 14.7 & 16.88 & 3.1\ 3379 & E0-1 & 1 & 10.3 & 13.7 & 5.92 & 10.96 & 54.9 & 18.02 & 5.7\ 3665 & S0 & – & 33.1 & 13.2 & 7.12 & 11.48 & 50.5 & 18.95 & 5.4\ 3998 & S0 & 2 & 13.7 & 13.7 & 7.04 & 10.76 & 19.1 & 16.92 & 4.0\ 4551 & E & 3 & 16.1 & 13.2 & 8.67 & 10.24 & 13.8 & 17.45 & 2.1\ Notes: columns are (1) galaxy name, (2) Hubble type, (3) 1= core, 2 = intermediate, 3 = cusp central light profile, (4) distance, (5) mean stellar age from McDermid et al. (2015), except for NGC 1052 from Milone et al. (2007), (6) 3.6 micron apparent mag., (7) stellar mass, (8) effective radius, (9) $\mu_e$, (10) Sersic n. ![\[fig:corrGC2\] Sersic fit to 3.6$\mu$m surface brightness profile and residuals as a function of circular equivalent radius for non-SLUGGS galaxies. The [*upper*]{} panel shows the data points (with excluded data points shown by open circles) and the best fit Sersic profile in red. Parameters for the Sersic fit are given in the top right. The dashed line shows 3$\times$ the rms of the sky background level. $\Delta$ gives the rms of the residuals in mag arcsec$^{-2}$.The [*lower*]{} panel shows the residuals of the Sersic model fit minus the surface brightness data. ](montage1x3_B1.jpg){width="40.00000%"} ![\[fig:corrGC2\] Sersic fit to 3.6$\mu$m surface brightness profile and residuals as a function of circular equivalent radius for non-SLUGGS galaxies. The [*upper*]{} panel shows the data points (with excluded data points shown by open circles) and the best fit Sersic profile in red. Parameters for the Sersic fit are given in the top right. The dashed line shows 3$\times$ the rms of the sky background level. $\Delta$ gives the rms of the residuals in mag arcsec$^{-2}$.The [*lower*]{} panel shows the residuals of the Sersic model fit minus the surface brightness data. ](montage1x3_B2.jpg){width="40.00000%"} [^1]: E-mail: [email protected]

--- abstract: 'We suggest that order-of-magnitude reduction of the longitudinal-acoustic phonon scattering rate, the dominant decoherence mechanism in quantum dots, can be achieved in coupled structures by the application of an external electric or magnetic field. Modulation of the scattering rate is traced to the relation between the wavelength of the emitted phonon and the length scale of delocalized electron wavefunctions. Explicit calculations for realistic devices, performed with a Fermi golden rule approach and a fully three-dimensional description of the electronic quantum states, show that the lifetime of specific states can achieve tens of $\mu$s. Our findings extend the feasibility basis of many proposals for quantum gates based on coupled quantum dots.' author: - Andrea Bertoni - Massimo Rontani - Guido Goldoni - Filippo Troiani - Elisa Molinari title: 'Field-controlled suppression of phonon-induced transitions in coupled quantum dots' --- The loss of quantum coherence, brought about by the coupling between charge degrees of freedom and lattice vibrations, constitute a main limitation for single and coupled quantum dot (SQD and CQD) implementations of quantum logic gates, as well as for a wide class of novel few- and single-electron devices. In quantum dots (QDs) [@dotsrev] the discrete nature of the energy spectrum strongly reduces this interaction with respect to other semiconductor systems. In particular, the coupling with the dispersionless optical phonons, the main source of decoherence in structures of higher dimensionality, is either completely suppressed [@bockelmann90] or, in specific conditions, leads to localized polaronic states [@inoshita97hameau99]. Still, however, emission of longitudinal acoustic (LA) phonons [@jacak03] limits the charge coherence times to tens of ns in typical devices, since intra-band electronic transitions merge into the LA phonon continuum. Tailoring the coupling between electronic quantum states and LA phonons would be a fundamental step toward the design of quantum devices with optimal operation conditions. In this Letter we show that LA-phonon scattering rate of specific transitions can be suppressed in CQDs by proper structure design (as already discussed in Ref.  for quantum dot arrays) and by use of external magnetic and electric fields. This is accomplished by exploiting the interplay between the wavevector of the emitted or absorbed phonon and the length scale of the quantum states. While the above effect is small in typical SQDs, which are characterized by a single length scale, we show that order-of-magnitude modulation of the scattering rate can be achieved in coherently coupled QDs, where two characteristic length scales come into play, namely the dimension of each dot and the dimension of the CQD system as a whole. Decay times of a single electron in a SQD have been measured recently by transport spectroscopy [@fujisawa02]. These experiments, which specifically probe the decay of carriers injected in the first excited state to the ground state, showed a satisfactory agreement with a Fermi golden rule estimate of the electron relaxation times. Below we investigate the decay time of specific low energy transitions which are, therefore, directly accessible with similar techniques. As a prototypical system, we consider a GaAs/AlGaAs CQD structure formed by two identical QDs with cylindrical shape, coupled along the growth direction [@tarucha96rontani04]. We describe the electronic quantum states within the familiar envelope function approximation and consider a parabolic confinement in the $xy$ plane of the cylinders, and a symmetric double-well in the $z$ direction: $V({\bf r})= V_z(z) + \frac{1}{2}m\omega_0^2(x^2+y^2)$ where $V_z(z)= V_l$ if $L_b/2 \leq |z| \leq (L_b/2 + L_d)$ and $V_z(z)= V_h$ otherwise. Here $L_d$ is the thickness of the GaAs layers and $L_b$ is the thickness of the inter-dot layer. $(V_h-V_l)$ is the band-offset of GaAs/AlGaAs. Furthermore, we consider either a magnetic ($B$) or an electric ($E$) field applied in the $z$ direction. For this cylindrically symmetric configuration, the eigenfunctions can be given the separable form $\psi_{nmg}({\bf r})= \phi_{nm}(x,y) \chi_{g}(z)$, with $n=0,1,\dots$ the radial and $m=0,\pm 1,\dots$ the angular quantum number of the Fock-Darwin state $\phi_{nm}(x,y)$; $g=0$ ($1$) indicates the ground (first excited) eigenstate for the biased double-well potential along the $z$ direction. We use the standard deformation-potential model for the electron-LA phonon interaction, with the interaction Hamiltonian ${\bf H}_{e-p}=\sum_{\bf q} F(q) ( b_{\bf q} e^{i{\bf q r}} + b_{\bf q} ^{\dagger} e^{-i{\bf q r} }) $, where $b_{\bf q}$ and $b_{\bf q}^{\dagger}$ are the annihilation and creation operators for a LA-phonon with wavevector ${\bf q}$ and $F(q)= q \sqrt{{\hbar D^2}/{(2\rho\omega_qV)}}$. $D$ is the deformation potential, $\rho$ is the crystal density and $V$ the volume of the system. A linear dispersion approximation $\omega_q= v q$ is used for the LA branch, with $v$ longitudinal sound speed [@footnote1]. Furthermore, we assume that the device operates in a single-electron regime: electron-electron scattering is neglected accordingly. Below we shall concentrate on the scattering rate between specific excited (initial) states $\psi_i$, with energy $E_i$, and the ground (final) state $\psi_f$, with energy $E_f$ [@footnote2]. The scattering rate at zero temperature, given by the Fermi golden rule, is $\, \tau^{-1}=\frac{2\pi}{\hbar}\sum_{\bf q} |F(q)M_{fi}({\bf q})|^2 \delta(E_f-E_i+\hbar\omega_{\bf q}) \, , $ where $M_{fi}({\bf q})=\langle \psi_f | e^{-i{\bf q r} } | \psi_i\rangle$. It is convenient to write the phonon wave vector ${\bf q}$ in spherical coordinates and separate the scattering matrix $M_{fi}({\bf q})$ into a $z$ component $M_{fi}^{(z)}(q_z=q \cos\theta)$, to be evaluated numerically, and an in-plane component $M_{fi}^{(xy)}(q_x=q \cos\varphi\sin\theta,q_y=q \sin\varphi\sin\theta)$ which can be exactly derived [@bockelmann94]. Performing the integration over $q$, one finally obtains: $$\begin{aligned} \label{fermigoldenrule2} \tau^{-1}=\frac{q_0^2} {v(2\pi\hbar)^2} F(q_0) \int_0^{2\pi}\!d\varphi \int_0^{\pi/2}\! d\theta\sin\theta \nonumber \\ \times |M_{fi}^{(z)}(q_0 \cos\theta) M_{fi}^{(xy)} (q_0 \cos\varphi\sin\theta,q_0 \sin\varphi\sin\theta)|^2 ,\end{aligned}$$ with $q_0=(E_i-E_f)/(\hbar v)$. We emphasize that the decoupling of the in-plane and vertical degrees of freedom in Eq. (\[fermigoldenrule2\]) is approximately valid as long as the in-plane and vertical confinement have different length scales, as is achieved by many growth techniques. Now we focus on the $(n,m,g) = (0,-1,0) \rightarrow (0,0,0)$ transition, which only involves in-plane degrees of freedom. We choose a CQD system (see Fig. \[fig1\], caption) such that the tunneling energy is larger than $\hbar\omega_0$ and $(0,-1,0)$ is the lowest excited state. Figure \[fig1\] shows the computed scattering rate at $B=0$ T and zero temperature as a function of the in-plane confining energy $\hbar \omega_0$. The striking oscillations, which cover several orders of magnitude, have their origin in the phase relation between the phonon plane wave corresponding to the considered electronic transition and the electron wavefunction delocalized between the two QDs: two limiting conditions, one with maximum overlap, the other with minimum overlap, due to electron-phonon antiphase, occur repeatedly as the confinement energy (and, therefore, the energy of the matching phonon) is varied; the two cases correspond to a maximum and a minimum of the matrix element $M_{fi}^{(z)}$, respectively. In the inset of Fig. \[fig1\] we compare the scattering rate of two CQD structures, with different barrier widths, with the one of a SQD. As one may expect, the curves have a similar limiting behavior: when the confining potential and, as a consequence, the energy of the emitted phonon is very low, $\tau^{-1}$ decreases due to (a) the $q^2$ prefactor ensuing from the low-energy LA-phonon density of states and the linear $q$ dependence of $F(q)$ in Eq. (\[fermigoldenrule2\]), and (b) the decreasing value of $M^{(xy)}_{fi}({\bf q})$ due to orthogonality of the states (at vanishing $q$ the electron-phonon Hamiltonian becomes constant). In the opposite limit, for $\hbar\omega_0>$ 2 meV, the phonon oscillation length becomes much smaller than the typical length scale of the electron wave function; as a consequence $M_{fi}({\bf q})$ vanishes since the initial and final electron states do not posses the proper Fourier component able to trigger the phonon oscillation. In the [*intermediate regime*]{}, while the SQD curve shows a single well-defined maximum, the CQD curve oscillates. The oscillation pattern depends on the geometry of the system [@zanardi98]. If the distance between the dots is increased, the energy difference between the maxima decreases (see inset in Fig. \[fig1\]), with the CQD curve enveloped by the SQD curve. The modulation shown in Fig. \[fig1\] is obtained by changing the lateral confining potential, i.e., a structure parameter that could be hardly tuned in an experimental setup. We show in the following that the strong suppression of acoustic-phonon induced scattering can also be driven by a magnetic field B parallel to the growth direction. Indeed, B induces (a) a non-uniform shift in the energies $E_i$ and $E_f$ and, consequently, a shift in the emitted phonon wavevector $q_0$; (b) a modification of the in-plane scattering matrix $M_{fi}^{(xy)}$ originating from the stronger localization of the wavefunctions. Figure \[fig2\] shows that, for a CQD with a $2$ meV confining energy, the acoustic-phonon scattering rate is reduced by four orders of magnitude at $B = 1$ T. In order to test the robustness of this effect in realistic experimental conditions, and particularly to compare with possible pump-and-probe experiments [@fujisawa02], in which the electron energy could vary between the cycles leading to a finite energy uncertainty, we introduced a Gaussian smearing in the QD confining energy. The resulting scattering rate (Fig. \[fig2\], inset) still shows oscillations of about one order of magnitude when an uncertainty of 0.2 meV is assumed. Next, we analyze the transition rate between two different pseudo-spin levels, $(n,m,g)= (0,0,1)\rightarrow (0,0,0)$, i.e., with the $z$ component of the electron wave function decaying from the first excited to the ground state. We consider a sample with $\hbar\omega_0$ substantially larger than the tunneling energy, so that the above transition is the lowest (see Fig. \[fig3\], caption). In this case the wavelength of the emitted phonon is controlled by the tunneling energy, that can be tuned by means of an electric field applied in the $z$ direction. As in the previous cases, order-of-magnitude modulation of the lowest transition can be achieved at specific fields (Fig. \[fig3\], solid line); furthermore, scattering rates tend to be smaller at higher fields as a result of increasing the wavevector of the emitted phonon, whose energy is also shown. Note that the effect of the fields on the scattering rates generally depends on the specific transition, therefore making it possible to tailor the scattering rates and to make only specific states robust to phonon-induced decoherence. This should allow a selective population of excited states. As an example, in Fig. \[fig3\] we report the scattering rates of the $2^{\mbox{\scriptsize nd}}$ excited state to the lower ones (see inset): at $E \sim 200$ kV/m (0,1,0) decays preferentially to (0,0,1) which, in turn, is long lived. Finally, we remark that the observation of the field-dependent oscillation of the transition rate would serve as a signature for the coherent delocalization of electron states in CQDs. In summary we have shown that a strong suppression of LA-scattering, the main source of electron decoherence in quasi zero dimensional systems, can be obtained in CQDs with energy-level separation around 3 meV. Significantly, the relaxation time can be controlled by external magnetic or electric field and increased up to tens of $\mu$s; this should be compared to SQDs with similar level spacing, where scattering times are of the order of ns and not strongly affected by external fields. This work has been partially supported by projects MIUR-FIRB n.RBAU01ZEML, MIUR-COFIN n.2003020984, INFM Calcolo Parallelo 2004, and MAE, Dir.Gen. Promozione Cooperazione Culturale. [11]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{} , , , ** (, ); , , , , , , ****, (); , ****, (). , ****, (). , ****, (); , , , , , , , , ****, (). , , , , ****, (). , ****, (). , , , , , ****, (). , , , , , ****, (); , , , , , , , ****, (). In the numerical simulations we consider a GaAs/Al$_{0.3}$Ga$_{0.7}$As heterostructure and use the following parameters: $D=8.6$ eV, $\rho=5300$ kg/m$^3$, $v=3700$ m/s, $V_h-V_l= 243$ meV. The subscripts $i$ and $f$ indicate, in compact notation, the three quantum numbers $(n_i,m_i,g_i)$ and $(n_f,m_f,g_f)$ of the initial and final state, respectively. , ****, (). At negative B, $(0,-1,0)$ is higher than $(0,1,0)$; the $(0,1,0)\rightarrow (0,0,0)$ scattering rate can be inferred from that of $(0,-1,0)\rightarrow (0,0,0)$, reported in Fig. \[fig2\], by changing the sign of the field B. FIGURE CAPTIONS FIG. 1. Scattering rate in a CQD ($L_d=10$ nm, $L_b=3$ nm) for the transition $(n,m,g)=(0,-1,0)\rightarrow(0,0,0)$ as a function of the in-plane parabolic confinement energy $\hbar\omega_0$. The dotted line shows the value of $\hbar\omega_0$ used for the simulations with an applied magnetic field. Inset: the scattering rate for the same sample is compared with that for a CQD with a larger barrier ($L_b=8$ nm, dashed curve) and for a SQD ($L_d=10$ nm, dotted curve), in a linear scale. FIG. 2. Scattering rate in a CQD ($L_d=10$ nm, $L_b=3$ nm, $\hbar\omega_0=2$ meV) for the transition $(n,m,g)=(0,-1,0)\rightarrow (0,0,0)$ [@footnote4] as a function of the applied vertical magnetic field $B$ (see text) at three temperatures, $T=0$ K (solid), $100$ K (dashed), $300$ K (dotted). The proper Bose statistics for phonon modes has been used in the latter two simulations. Inset: two different energy-uncertainty conditions are compared at $T=0$K: the data are convoluted with a Gaussian uncertainty of about 10% (FWHM=$0.2$ meV, solid line) and 20% (FWHM=$0.4$ meV, dashed line) on the dot confining energy. FIG. 3. Scattering rate (solid line), for the lowest pseudo-spin transition $(n,m,g)= (0,0,1)\rightarrow (0,0,0)$ in a CQD with $L_d=12$ nm, $L_b=4$ nm, and $\hbar\omega_0=5$ meV as a function of an electric field applied along $z$. The scattering rate for $(0,1,0)\rightarrow (0,0,1)$ and $(0,1,0)\rightarrow (0,0,0)$ are shown with dotted and dashed lines respectively. The energy of the emitted LA phonon is reported for reference (thick dash-dotted curve, right axis). ![\[fig1\]](fig1){width=".98\linewidth"} ![\[fig2\] ](fig2){width=".98\linewidth"} ![\[fig3\] ](fig3){width=".98\linewidth"}

--- abstract: 'Magnetic excitations in [Sr$_2$RuO$_4$]{}  have been studied by inelastic neutron scattering. The magnetic fluctuations are dominated by incommensurate peaks related to the Fermi surface nesting of the quasi-one-dimensional $d_{xz}$- and $d_{yz}$-bands. The shape of the incommensurate signal agrees well with RPA calculations. At the incommensurate [**Q**]{}-positions the energy spectrum considerably softens upon cooling pointing to a close magnetic instability : [Sr$_2$RuO$_4$]{} does not exhibit quantum criticality but is very close to it. $\omega / T$-scaling may be fitted to the data for temperatures above 30 K. Below the superconducting transition, the magnetic response at the nesting signal is not found to change in the energy range down to 0.4meV.' address: | $^a$ Forschungszentrum Karlsruhe, IFP, Postfach 3640, D-76021 Karlsruhe, Germany\ $^b$Laboratoire Léon Brillouin, C.E.A./C.N.R.S., F-91191-Gif-sur-Yvette CEDEX, France\ $^c$ II. Phys. Inst., Univ. zu Köln, Zülpicher Str. 77, D-50937 Köln, Germany\ $^d$ Institut Laue-Langevin, Boite Postale 156, 38042 Grenoble Cedex 9, France\ $^e$ Department of Physics, Kyoto University, Kyoto 606-8502, Japan\ $^f$ International Innovation Center (IIC), Kyoto 606-8501, Japan author: - 'M.  Braden$^{a,b,c,*}$,Y.  Sidis$^b$, , P.  Bourges$^b$, P. Pfeuty$^b$, J. Kulda$^d$, Z. Mao$^e$, Y.  Maeno$^{e,f}$' title: 'Inelastic neutron scattering study of magnetic excitations in Sr$_2$RuO$_4$.' --- INTRODUCTION ============ [Sr$_2$RuO$_4$]{}  is still the only superconducting layered perovskite besides the cuprates[@maeno]; however, in contrast to the cuprate high temperature superconductors (HTSC), superconductivity in [Sr$_2$RuO$_4$]{}  develops in a well defined Fermi-liquid state [@maeno; @2]. Nevertheless the superconducting state and the pairing mechanism in [Sr$_2$RuO$_4$]{}  are highly unconventional. The present interest in this compound goes far beyond the simple comparison with the cuprate high temperature superconductors. The extreme sensitivity of the superconducting transition temperature on non-magnetic impurities suggests a non-phonon mechanism [@3]. It is further established that superconductivity in [Sr$_2$RuO$_4$]{}  is made of spin-triplet Cooper pairs and breaks time-reversal symmetry [@2; @4]. The strongest experimental argument for that can be found in the uniform susceptibility measured either by the NMR-Knight-shift or polarized neutrons experiments [@5; @hayden] and in the appearance of spontaneous fields in the superconducting state reported by $\mu$SR [@6]. A spin-triplet p-wave order parameter had been proposed before these experiments [@7], in the idea that superconductivity arrises from a dominant interaction with ferromagnetic fluctuations in analogy to superfluid $^3$He. Rice and Sigrist stressed the comparison with the perovskites SrRuO$_3$ and CaRuO$_3$ which order ferromagnetically or are nearly ferromagnetic respectively [@srruo3]. Evidence for ferromagnetic fluctuations in [Sr$_2$RuO$_4$]{}  was inferred from NMR-experiments : Imai et al. found that $1 \over {T_1T}$ of the O- and of the Ru-NMR exhibit a similar temperature dependence and interpreted that this could be only due to ferromagnetic fluctuations [@imai]. The macroscopic susceptibility in [Sr$_2$RuO$_4$]{}  is enhanced when compared with the band structure calculation but only weakly, in particular its temperature dependence remains flat [@2; @9]. There exist also layered ruthenates which are very close to ferromagnetic order at low temperatures : the purest [Sr$_3$Ru$_2$O$_7$]{}  samples show meta-magnetism and samples with somehow less quality even order ferromagnetically [@10]. A highly enhanced susceptibility pointing to a ferromagnetic instability is also observed in the phase diagram of [Ca$_{2-x}$Sr$_{x}$RuO$_4$]{}  [@11; @12] but for a rather high Ca concentration, [Ca$_{1.5}$Sr$_{0.5}$RuO$_4$]{} [@13]. In these nearly ordered layered ruthenates, the susceptibility is about two orders of magnitude higher than that in [Sr$_2$RuO$_4$]{}  and strongly temperature dependent. Some doubt about the unique role of ferromagnetism in [Sr$_2$RuO$_4$]{}  arose from the strong moment antiferromagnetic order observed in the Ca analog [@14], which inspired Mazin and Singh to perform a calculation of the generalized susceptibility based on the electronic band structure [@15]. Surprisingly they found that the dominating part is neither ferro- nor antiferromagnetic but incommensurate. The Fermi-surface in [Sr$_2$RuO$_4$]{}  is well studied both by experiment [@16] and by theory [@17] with satisfactory agreement. Three bands are contributing to the Fermi-surface which may be roughly attributed to the three t$_{2g}$-levels, the d$_{xy}$-, d$_{yz}$- and d$_{xz}$-orbitals, occupied by the four 4d-electrons of Ru$^{4+}$. The d$_{xy}$-orbitals hybridize in the xy-plane and, therefore, form a band with two-dimensional character, [$\gamma$-band]{}. In contrast, the d$_{xz}$ and the d$_{yz}$ orbitals may hybridize only along the x and the y-directions, respectively, and form quasi-one-dimensional bands, [$\alpha$-band]{}  and [$\beta$-band]{}, with flat sheets in the Fermi-surface, [$\alpha$-sheet]{}  and [$\beta$-sheet]{}. The latter give rise to strong nesting and enhanced susceptibility for ${\bf q}=(0.33,q_y,0)$ or ${\bf q}=(q_x ,0.33,0)$ [@15]. Along the diagonal both effects strengthen each other yielding a pronounced peak in the susceptibility at ${\bf q_{i-cal}}=(0.33,0.33,0)$. Using inelastic neutron scattering (INS) we have perfectly confirmed this nesting scenario [@sidis]. The dynamic susceptibility at moderate energies is indeed dominated by the incommensurate fluctuations occurring very close to the calculated position at ${\bf q_i}=(0.30,0.30,0)$. Furthermore, we found a pronounced temperature dependence which can explain the temperature dependence of both Ru- and O-$1 \over T_1T$-NMR [@imai]. In the meanwhile, several groups have worked on theories where superconductivity is related to the incommensurate fluctuations [@18; @19; @20]. These theories, however, do not yet give an explanation for the fine structure of the order parameter. Specific heat data on the highest quality samples clearly indicates the presence of line nodes in the superconducting state [@21]. Ultrasonic [@22] and thermal conductivity [@23] results would disagree with vertical line nodes which leave horizontal line nodes as the only possible ones. Zhitomirsky and Rice [@24] assume that superconductivity is transferred from the [$\gamma$-band]{}  to the [$\alpha$-band]{}  and [$\beta$-band]{}  by a proximity effect and get a conclusive explanation for the horizontal line nodes. In this model superconductivity should be mainly related to excitations associated with the active [$\gamma$-band]{}, which so far have not been characterized. Therefore, it appears still interesting to further deepen the study of the magnetic fluctuations in [Sr$_2$RuO$_4$]{} . In this paper we report on additional INS studies in [Sr$_2$RuO$_4$]{}  in the normal as well as in the superconducting phase. We present a more quantitative analysis of the incommensurate fluctuations related to the [$\alpha$-sheet]{}  and [$\beta$-sheet]{}  and discuss the possible contributions due to the two-dimensional [$\gamma$-band]{}. Experimental ============ Experimental setup ------------------ Single crystals of [Sr$_2$RuO$_4$]{}  were grown by a floating zone method in an image furnace; they exhibit the superconducting transition at T$_c$=0.7, 1.35 and 1.43K and have volumes of about 450mm$^3$ each. Since most of the measurements were performed in the normal phase, where the differences in T$_c$ should not affect the magnetic excitation spectrum, we aligned the three crystals together in order to gain counting statistics in the INS experiments. Count rates in the ruthenates experiments are relatively small already due to the steeper decrease of the Ru magnetic form-factor with increasing scattering vector. For the measurements below and across the superconducting transition we mounted only the two crystals with relatively high T$_c$ together. The mounting of the two or three crystals was achieved with individual goniometers yielding a total mosaic spread below 0.5 degrees. We used the thermal triple axis spectrometer 1T installed at the Orphée-reactor (Saclay, France) in order to further characterize the scattering in the normal state. The instrument was operated with double focusing pyrolithic graphite (PG) monochromator and analyzer, in addition PG filters in front of the analyzer were used to suppress higher order contamination, the final neutron energy was fixed at 14.7meV. All diaphragms determining the beam paths were opened more widely than usually in order to relax ${\bf Q}$-resolution, since the magnetic signals are not sharp in ${\bf Q}$-space. In most experiments, the scattering plane was defined by (1,0,0) and (0,1,0) directions in order to span any directions within the (ab) plane. An additional experiment has been made with the (1,1,0) and (0,0,1) directions with the plane to follow the spin fluctuations along the c$^*$ axis. Studies aiming at the response in the magnetic excitation spectrum on the opening of the superconducting gap require an energy resolution better than the expected value for twice the superconducting gap. Therefore, such experiment is better placed on a cold triple-axis spectrometer even though this implies a sensitive reduction in the flux. We have made preliminary studies using the cold spectrometers 4F at the Orphée reactor and a more extensive investigation on the spectrometer IN14 at the ILL. These instruments possess PG monochromators (double at 4F and focusing at IN14) and focusing analyzers. The final neutron energy was fixed at $E_f$= 5 meV on both cold source spectrometers where a Be filter has been employed to cut down higher-order wavelength neutrons. Cooling was achieved by use of a dilution- and a He$^3$-insert at 4F and IN14 respectively. Theoretical background for magnetic neutron scattering ------------------------------------------------------ The magnetic neutron cross section per formula unit is written in terms of the Fourier transform of the spin correlation function, $\rm S_{\alpha\beta}({\bf Q}, \omega)$ (labels $\alpha,\beta$ correspond to x,y,z) as [@Lovesey]; $$\frac{d^2 \sigma}{d \Omega d \omega}= \frac{k_f}{k_i} r_0^2 F^2({\bf Q}) \sum_{\alpha,\beta} (\delta_{\alpha,\beta}-\frac{Q_{\alpha}Q_{\beta}}{|{\bf Q}|^2}) S_{\alpha\beta}({\bf Q}, \omega) \; \label{eq:INS}$$ where k$_i$ and k$_f$ are the incident and final neutron wave vectors, $r_0^2$=0.292 barn, $F({\bf Q})$ is the magnetic form factor. The scattering-vector ${\bf Q}$ can be split into ${\bf Q}$ =${\bf q}$+${\bf G}$, where ${\bf q}$ lies in the first Brillouin-zone and ${\bf G}$ is a zone-center. All reciprocal space coordinates $(Q_x,Q_y,Q_z)$ are given in reduced lattice units of 2$\pi$/a or 2$\pi$/c. According to the fluctuation-dissipation theorem, the spin correlation function is related to the imaginary part of the dynamical magnetic susceptibility times the by one enhanced Bose-factor: $$S_{\alpha\beta}({\bf Q}, \omega)=\frac{1}{\pi (g \mu_B)^2} \frac{\chi_{\alpha\beta}"({\bf Q},\omega)} {1-\exp(-\hbar \omega / k_BT)} \; \label{eq:INSchi}$$ In case of weak anisotropy, which is usually observed in a paramagnetic state, $\rm \chi_{\alpha\beta}"({\bf Q},\omega)$ reduces to $\rm \chi"({\bf Q},\omega) \delta_{\alpha,\beta}$. Note that for itinerant magnets, anisotropy of the susceptibility can occur due to spin-orbit coupling. $\rm F({\bf Q})$ can be described by the Ru$^+$ magnetic form factor in first approximation. Once determined, the magnetic response is converted to the dynamical susceptibility and calibrated in absolute units by comparison with acoustic phonons, using a standard procedure depicted in [@sidis]. Results and discussion ====================== RPA analysis of the magnetic excitations ------------------------------------------ At low temperature [Sr$_2$RuO$_4$]{}  exhibits well defined Fermi-liquid properties; therefore, it seems appropriate to assess its magnetic excitations within a RPA approach basing on the band structure. Density functional calculations in LDA were performed by several groups and yield good agreement with the Fermi-surface determined in de-Haas van Alphen experiments. The bare non-interacting susceptibility, $\chi ^0({\bf q})$, can be obtained by summing the matrix elements for an electron hole excitation [@Lovesey]: $$\chi^0 ({\bf q}, \omega) = - \sum_{{\bf k},i,j} M_{{\bf k}i,{\bf k+q}j} \frac{f(\varepsilon_{{\bf k+q},j})-f(\varepsilon_{{\bf k},i})} {\varepsilon_{ {\bf k+q},j}-\varepsilon_{{\bf k},i}-\hbar \omega + i \epsilon} \; \label{eq:1}$$ where $\epsilon \rightarrow$0, $f$ is the Fermi distribution function and $\varepsilon_k$ the quasiparticle dispersion relation. This was first calculated by Mazin and Singh [@15] under the assumption that only excitations within the same orbital character are relevant (the matrix-elements $M_{{\bf k}i,{\bf k+q}j}$ are equal to one or zero). Mazin and Singh predicted the existence of peaks in the real part of the bare susceptibility at [$\omega$]{}=0 due to the pronounced nesting of the [$\alpha$-band]{}s and [$\beta$-band]{}s. These peaks were calculated at (0.33,0.33,0) and experimentally confirmed very close to this position at [$\mathbf{q_i}$]{}=(0.3,0.3,0), see Fig. 1. In addition to the peaks at [$\mathbf{q_i}$]{}, this study finds ridges of high susceptibility at (0.3, $q_y$,0) for $0.3<q_y<0.5$ and some shoulder for $0<q_y<0.3$. The susceptibility gets enhanced through the Stoner-like interaction which is treated in RPA by : $$\chi ({\bf q} ) = { \chi ^0 ({\bf q}) \over 1-I({\bf q})\chi^0({\bf q})} \; \label{eq:2}$$ with the ${\bf q}$-dependent interaction $I({\bf q})$. For the nesting positions Mazin and Singh get $I({\bf q})\chi^0({\bf q})$=1.02, which already implies a diverging susceptibility and a magnetic instability. In Fig. 1 we show a scheme of the (hk0)-plane in reciprocal space. Due to the body centering in space-group I4/mmm any (hkl)-Bragg-points have to fulfill the condition (h+k+l) even; (100) for instance is not a zone-center but a Z-point. The boundaries of the Brillouin-zones are included in the figure. Large filled circles indicate the position of the incommensurate peaks as predicted by the band-structure and as observed in our previous work. The thick lines connecting four of them correspond to the ridges of enhanced susceptibility also suggested in reference [@15]. In the meanwhile several groups have performed similar calculations which all agree on the dominant incommensurate fluctuations [@20; @24b; @25; @26; @27]. However there are serious discrepancies concerning the detailed structure of the spin susceptibility away from [$\mathbf{q_i}$]{}. These discrepancies mainly rely on the parameters used to describe the electronic band structure, on the choice of $\rm I({\bf q})$ and on the inclusion of more subtle effects such as spin-orbit or Hund couplings. In order to compare directly to the INS experiments, see equations (1) and (2), it is necessary to perform the RPA analysis by taking into account both real and imaginary part of the susceptibility. Morr et al. [@27] report such calculations obtained by fitting the band structure to the ARPES results [@28]. They find in addition to the peak at [$\mathbf{q_i}$]{}  quite strong intensity near ${\bf P_i}$=(0.3,0.5,0), (the middle of the walls, see Fig. 1), which is even comparable to that at [$\mathbf{q_i}$]{}  in the bare susceptibility. Similar results were obtained in references [@20; @24b]. Ng and Sigrist [@25] find much less spectral weight in the ridges at (0.3, $q_y$) for $0.3<q_y<0.5$ but stronger shoulders $0<q_y<0.3$. In addition, they calculate the separate contribution of the [$\gamma$-band]{}  which does not show a particular enhancement in the ferromagnetic ${\bf q}$-range but is little structured. Eremin et al. [@26] calculate the susceptibility taking into account strong hybridization and obtain results somehow differnt from the other groups. They find a strong signal at ${\bf P_i}$; in addition there is some enhancement of the susceptibility related to the van-Hove singularity of the [$\gamma$-band]{}. This contribution occurs quite close to the zone center at ${\bf q_{vH}}$=(0.15,0,0). We have performed the full RPA analysis basing on the LDA band structure reported in reference [@17; @15] in order to accompany our experimental investigations. We first calculate the bare electron hole susceptibility $\chi^0$ from the usual expression, equation (3). For the band energies $\epsilon(k_x,k_y,k_z)$ we use the expressions of Mazin and Singh [@17] for the three mutually non hybridizing tight-binding bands in the vicinity of the Fermi level: $$\begin{aligned} \epsilon_{xy}(k_x,k_y,k_z) & = & 400mev(-1+2(cos(k_x)+cos(k_y))\nonumber \\ & & -1.2cos(k_x)cos(k_y))\\ \epsilon_{xz}(k_x,k_y,k_z) & = & 400mev(-.75+1.25cos(k_x)-\nonumber\\ & & .5cos(k_x/2)cos(k_y/2)cos(k_z/2))\\ \epsilon_{yz}(k_x,k_y,k_z) & = & 400mev(-.75+1.25cos(k_y)-\nonumber\\ & & .5cos(k_x/2)cos(k_y/2)cos(k_z/2)) \end{aligned}$$ We also used the crude approximation that matrix-elements for transitions between bands of the same character are equal to one and others zero. The [**q**]{}-dependent “ferromagnetic” interaction (Stoner factor) I([**q**]{}) is taken to be equal to (following reference [@15]): $I({\bf q})=\frac{320mev}{1+0.08 {\bf q}^2}$. With this choice of I([**q**]{}) the calculated static susceptibility $\chi({\bf q}=0,\omega=0)$ is slightly lower than the measured one. If I([**q**]{}) is chosen larger an instability appears at the incommensurate wave vector. Our results for the imaginary part of the generalized susceptibility at an energy transfer of 6meV are given in the lower part of figure 1. Besides the dominating nesting peak near [$\mathbf{q_i}$]{}  there is a further contribution near (0.15,0.15,0) which is related to the [$\gamma$-sheet]{}. In contrast we find a small susceptibility near ${\bf P_i}$ and for ($q_x$,0,0) with small values of $q_x$. Since the ${\bf q}$-position of the magnetic excitations were found not to depend on energy, it is easiest to observe the signal in INS by scanning at constant energy. The scan paths are included in Fig. 1, they are purely transverse or rocking-like, [**r**]{}-scan, along a \[100\]-direction, [**x**]{}-scans, and in diagonal direction, [**d**]{}-scans. The observed signal is rather broad and, therefore, the scans performed are extremely wide covering complete cuts through the Brillouin-zone. This further implies that the background (BG) may be non-constant at least sloping. Also, the signals are relatively weak compared to typical triple axis spectrometry problems; this implies that sample independent BG-contributions which usually are negligible play a role. Fig. 2 presents the results of [**d**]{}-scans at different temperatures clearly demonstrating the gain in statistics compared to the previous work [@sidis]. The magnetic intensity shown in Fig. 2 disappears upon heating but this effect gets partially compensated by the gain through the Bose-factor. Shape of the incommensurate signal ----------------------------------- The fact that the incommensurate signal around a zone-center and around a Z-point, (001), are equivalent already indicates that the coupling between RuO$_2$-planes is negligible, i.e. that in-phase and out-of-phase coupling are indistinguishable. The 2D-character has been directly documented by Servant et al. [@29] who found no q$_l$-dependence at (0.3,0.3,$q_l$) for q$_l$ between -0.5 and 0.5. We find the same result by varying Q$_l$ in a broader range between 2 and 5 in (0.3,0.3,Q$_l$), see Fig. 3. This 2D character is actually surprising since the dispersion relation of the quasi-particles involved in the computation of $\rm \chi"({\bf q}, \omega)$ is not purely 2D [@15]. One may therefore expect weak spin correlation along the $c^*$-direction. Recently, it has been shown that these fluctuations freeze out into a spin density wave (SDW) ordering by a minor replacement of Ru by Ti [@srruti]. In this ordering a very short correlation length along the c-direction nicely reflects the 2D-character of the incommensurate inelastic signal. The SDW propagation vector finally favored in the static ordering corresponds to the out-of-phase coupling between neighboring layers. This static interlayer-coupling might be even due to the CDW always coupled to a SDW, which, however, has not yet been discovered so far. One may add that the stripe ordering in La$_{1.67}$Sr$_{0.33}$NiO$_4$, which differs from the [Sr$_2$Ru$_{1-x}$Ti$_x$O$_4$]{}-case in many aspects but nevertheless may be considered as a mixed SDW-CDW-ordering, occurs at the same propagation vector [@lsno]. The wider Q$_l$-dependence of the incommensurate signal shown in Fig. 3 can give information about the anisotropy of the excitation, since INS measures only the spin component perpendicular to Q. We consider the diagonal susceptibility $\chi"_{\alpha \beta}$ with tetragonal symmetry $\chi_{\pm}:=\chi"_{xx}=\chi"_{yy}\not= \chi"_{zz}$. The measured intensity is then given by : $$\frac{d^2 \sigma}{d \Omega d \omega} \propto F^2({\bf Q}) \lbrack(1-\frac{Q_l^2}{|{\bf Q}|^2}) \chi"_{zz} + (1+\frac{Q_l^2}{|{\bf Q}|^2}) \chi"_{\pm} \rbrack \; \label{eq:INSanisotropy}$$ which implies that the observations at high Q$_l$ favor the in-plane component of the susceptibility. For the detailed analysis, one has to compare with the form factor. In the figure we show the Q$_l$-dependence assuming that spin density is localized at the Ru-site and may be modeled by the form factor of Ru$^{1+}$ [@formfac]. The Ru-form-factor dependence underestimates the signal at higher Q$_l$-values; however, the Ru$^{1+}$-form factor is certainly a too crude approximation. Measurements of the spin-density distribution induced by an external field have not been very precise due to the small magnetic susceptibility and the resulting small moment in [Sr$_2$RuO$_4$]{} [@32]. However, in [Ca$_{1.5}$Sr$_{0.5}$RuO$_4$]{}  which exhibits ferromagnetic ordering below 1K and whose low temperature susceptibility is about two orders of magnitude higher than that in [Sr$_2$RuO$_4$]{} [@11; @12; @13], it has been possible to study the field induced spin-density distribution. These experiments revealed an extremely high amount of spin-density at the oxygen position, about one third of the total moment [@34] and an orbital contribution at the Ru-site. By use of the [Ca$_{1.5}$Sr$_{0.5}$RuO$_4$]{} spin-density distribution one obtains a good description of the $Q_l$-dependence given in Fig. 3. However, the form-factor in [Sr$_2$RuO$_4$]{}  should be even more complex. Since the main contribution originates from the flat $d_{xy}$-orbitals [@26], there must be an anisotropy in the effective form factor which indeed was observed in [Ca$_{1.5}$Sr$_{0.5}$RuO$_4$]{}[@34] . Qualitatively, the anisotropy in the form-factor has to be compensated by some weak anisotropy in the spin susceptibility, i.e. by an enhanced out-of-plane component. Such susceptibility anisotropy corresponds to the orientation of the spins in the SDW ordering phase in [Sr$_2$Ru$_{1-x}$Ti$_x$O$_4$]{}  [@srruti] where the spins are aligned parallel to the c-direction and also to the conclusion deduced by Ng and Sigrist from the influence of spin-orbit coupling in [Sr$_2$RuO$_4$]{}[@25]. [*– Shoulder of the incommensurate peak –*]{} In order to examine the shape of the incommensurate peak and to verify the existence of the ridges of additional nesting intensity or the additional peak ${\bf P_i}$, reported in different band structure analyzes [@15; @24b; @25; @26; @27], we have scanned across [$\mathbf{q_i}$]{}  along the \[100\] or \[010\]-directions, [**y**]{}-scans see Fig. 1. The results are shown in Fig. 4. One can recognize that the incommensurate peak is not symmetric but exhibits always a shoulder to the lower $q_x$-side in absolute units. The shoulder is seen in many scans in reversed focusing conditions of the spectrometer configuration excluding an experimental artefact. Our full RPA calculation nicely agrees with such shoulder; the thin line in Fig. 4 shows the calculated imaginary part of the susceptibility at the energy transfer of the scan and describes the observed signal perfectly besides a minor offset in the position of the incommensurate signal. In contrast, neither the experiment nor our full RPA analysis yield significant intensity in the ridges, i.e. the range ($q_x$,0.3,0) with $0.3<q_x <0.5$, see Fig. 1 and 4. In particular an intensity at $P_i$ only three times weaker than that at [$\mathbf{q_i}$]{}  would have been easily detected experimentally. The nesting peak appears to be isolated with two shoulders along \[100\] and \[010\] to the lower absolute $q_{x,y}$-sides. These shoulders are connected to the [$\gamma$-sheet]{}. Possible additional magnetic excitations ------------------------------------------ In none of the band-structure calculations there is evidence for a strong and sharp enhancement of any susceptibility exactly at the zone-center pointing to a ferromagnetic instability [@24b; @25; @26; @27; @15]. However, several of these calculations find some large susceptibility near the zone-center which can be associated with the van Hove-singularity of the [$\gamma$-band]{}  closely above the Fermi-level. Eremin et al. [@26] report this signal at ${\bf q}$=(0.1-0.2,0,0), other groups find a small peak along the diagonal (q,q,0) [@20; @25; @24b],. Our own analysis also yields such a signal which is found to be strongest at (0.15,0.15,0) and which levels out along the \[100\] and \[010\] directions, see Fig. 1. In order to address this problem we have mapped out the intensity for ($Q_x$,$Q_y$,0) with $-0.5<Q_x<-0.0$ and $0.6<Q_y <1.5$; these scans are shown in Fig. 5 after subtraction of the scattering angle dependent background. It is obvious that the incommensurate peak is by far the strongest signal. At low temperature one may roughly estimate any additional signal to be at least a factor 6 smaller than the nesting peak. The analysis of such weak contributions is quite delicate and demands a reliable subtraction of the background. Nevertheless the scans in the Fig. 5 indicate some scattering closer to the zone center. However, this contribution is not sharply peaked at the $\Gamma$-point but forms a broad square or a circle. The signal roughly agrees with the prediction that the [$\gamma$-band]{}  yields magnetic excitations near the zone-center. Since the sloping background is a major obstacle to analyze the additional contributions we tried to compensate these effects by scanning from ${\bf Q}$=(0 1 0) along the four diagonals, which are illustrated in Fig. 1, and by summing the four scans. The results are given in Fig. 6. In the four single scans, Fig. 6a), one recognizes the nesting signal with its intensity being determined by the form-factor and the sloping background. The summed scans, see Fig. 6c), should have constant background and the low temperature summed scan once more documents that the nesting signal is by far the strongest one. However, upon heating additional scattering contributions seem to become enhanced in intensity in particular compared to the nesting signal which decreases; note that the Bose-factor will already strengthen any signal by a factor three in the data in Fig. 6c). Also, at higher energy the additional scattering seems to be stronger as seen in Fig. 6d) (the background is strongly sloping even in the sum due to a smaller scattering angle). The energy and temperature dependence of the additional contribution corresponds to that predicted by the full RPA analysis for the [$\gamma$-band]{}  magnetic contribution shown in Fig. 6b). The spectral weight at ${\bf Q}$=(0.15,0.15,0) relative to that of the nesting feature increases upon increasing temperature or energy as it is expected for a signal directly related to the van Hove singularity. Due to the agreement between the INS results and the RPA calculations, we suggest to interpret the additional broad scattering as being magnetic in origin; however, a polarized neutron study would be highly desirable. This scattering further might be relevant for a quantitative explanation of the NMR-data [@imai; @sidis], in particular its temperature independent part. [Sr$_2$RuO$_4$]{}  is not close to ferromagnetic order but substitution of Sr through Ca yields such order for the concentration [Ca$_{1.5}$Sr$_{0.5}$RuO$_4$]{}[@13]. This doping effect was explained in a band structure calculation [@38] as arising from a down shift in energy of the [$\gamma$-band]{}  pushing the van Hove-singularity closer to the Fermi-level. In such samples the [$\gamma$-band]{}  magnetic scattering should therefore become strongly enhanced. Indeed first INS studies on these compounds reveal broad signals similar to the additional scattering described above, but much stronger [@friedt]. This strongly supports a magnetic interpretation of the scattering in figures 5 and 6. Combined temperature and energy dependence of the incommensurate signal ------------------------------------------------------------------------ The energy and temperature dependence has been studied in more detail by performing [**r**]{}-scans across the incommensurate position, since in this mode the background is almost flat. However, due to phonon contributions we could not extend the measurements to energies higher than 12meV, which is slightly below the lowest phonon frequency observed at [$\mathbf{q_i}$]{}   [@35]. In particular the large scans required to cover the broad magnetic signal prevent any analysis within the phonon band frequency range on a non-polarized thermal triple axis spectrometer. The results of the scans are given in Fig. 7. At low temperature we find an energy spectrum in good agreement to that published earlier [@sidis]. In the range up to 12meV we observe at all temperatures an energy independent peak width, which, however, increases upon increase of temperature. For temperatures much higher than 160K the background considerably increases, see Fig. 2, and prevents a detailed analysis within reasonable beam-time. In Fig. 7b) we show the temperature dependence of the peak width averaged over the different energies which agrees well to the results obtained from the single scans with less statistics reported in reference [@sidis]. Even at the lowest temperature the width of the signal remains finite. The spectral functions have been fitted by a single relaxor behavior [@36]: $$\chi "({\bf q_i}, \omega) = \chi '({\bf q_i}, 0) \frac{\Gamma \omega}{\omega ^2 + \Gamma ^2} \; \label{eq:3}$$ where $\Gamma$ is the characteristic energy and $\chi '({\bf q_i},0)$ the amplitude which corresponds to the real part of the generalized susceptibility at [$\omega$]{}=0 according to the Kramers-Kronig relation. The ${\bf Q}$-dependence of the signal may be described by a Lorentzian distribution with half width at half maximum $\kappa$, but due to experimental broadening the constant energy scans are equally well described by Gaussians. At low temperature the spectrum clearly exhibits a characteristic energy as seen in the maximum of the energy dependence, but this maximum is shifted to higher values upon heating. Fig. 7c) and 7d) report the results of the least squares fits with the single relaxor. Although the statistics is still limited, the tendencies can be obtained unambiguously. The height of the spectral functions, $\chi '({\bf q_i},0)$, rapidly decreases upon heating, which overcompensates the broadening of the signal in ${\bf Q}$-space. Our finding that the characteristic energy in the range 6-9meV is well defined only at low temperatures can be related with the far-infrared c-axis reflectance study by Hildebrand et al. [@hildebrand], since the optical spectrum shows a resonance in this energy range at low temperatures. In reference [@sidis] we have compared the temperature dependence of the incommensurate signal with that of the spin-lattice relaxation rate $T_1$ measured by both $^{17}$O and $^{101}$Ru NMR experiments[@imai]. These NMR-techniques probe the low energy spin fluctuations ($\omega \rightarrow 0$ with respect to INS measurements); furthermore, they integrate the fluctuations in ${\bf q}$-space. $(1/T_1T)$ is related to the generalized susceptibility and the INS results by [@nmr]: $$(1/T_1T) \simeq \frac{k_B\gamma_n^2}{(g\mu_B)^2} { {\sum_{q} { | A({\bf q}) |^2 \left.\frac{\chi "({\bf q},\omega)} { \omega} \right\vert_{\omega \rightarrow 0}}}} \; \label{eq:1/T1}$$ with $| A({\bf q}) |$ the hyper fine fields. $(1/T_1T)$ corresponds hence to the slope of the spectral function in Fig. 7a) times the extension of the signal in ${\bf Q}$-space. The new data perfectly agrees to the former result, the loss of the incommensurate signal upon heating may explain almost entirely the temperature dependent contribution to $(1/T_1T)$ [@imai]. The temperature dependence of the magnetic excitation spectrum at the incommensurate position may be analyzed within the results of the self consistent renormalization theory described in reference [@36]. In an antiferromagnetic metal the transition is governed by a single parameter related to the Stoner-enhancement at the ordering wave-vector, $\delta=1-I({\bf q_i})\chi^0({\bf q_i})$. The characteristic entities of the magnetic excitations are then given by: $$\kappa ^2 \propto \delta \, \label{SCR1}$$ $$\Gamma \propto \delta \, \label{SCR2}$$ $${1 \over \chi'({\bf q_i},0)} \propto \delta \ . \label{SCR3}$$ When the system approaches the phase transition, the unique parameter $\delta$ diminishes, which behavior should be observable in all three parameters. Equations (11-13) imply a sharpening of the magnetic response in ${\bf q}$-space as well as in energy and a divergence of the susceptibility at the ordering vector. Fig. 7 qualitatively confirms this picture. All the relevant parameters, see the right scales in Fig.7, decrease towards low temperature. Therefore, one may conclude that [Sr$_2$RuO$_4$]{}  is approaching the SDW transition related to the nesting effects upon cooling. However, all these parameters do not vanish completely but remain finite even at the lowest temperatures in agreement with the well known fact that [Sr$_2$RuO$_4$]{}  does not exhibit magnetic ordering. In particular the magnetic scattering remains rather broad in ${\bf q}$-space implying a short correlation length of just 3-4 lattice spacings. The temperature dependence of the magnetic excitations corroborate our recent finding that only a small amount of Ti is sufficient to induce SDW magnetic ordering [@srruti]. Since [Sr$_2$RuO$_4$]{}  is close to a quantum critical point it is tempting to analyze whether the excitation spectrum is governed by some ${\omega \over T}$-scaling, as it has been claimed for the high temperature superconductor [La$_{2-x}$Sr$_{x}$CuO$_4$]{} [@aeppli; @keimer; @sachdev] and for CeCu$_{5.9}$Au$_{0.1}$ [@schroeder]. One would expect that the susceptibility is given by : $$\chi"({\bf q_i},\omega,T)\propto T^{-\alpha}g({\omega\over T})\ .$$ In Fig. 8 we plot the $\omega^{\alpha}\chi"({\bf q_i},\omega)$-data of Fig. 7 and that obtained previously as a function of temperature [@sidis] against ${\omega \over T}$ for $\alpha$=0.75 and 1.0. Only the data at higher temperatures agree with the scaling concept, demonstrating that [Sr$_2$RuO$_4$]{}  is not a quantum critical point. The schematic inset may illustrate the phase diagram, where the magnetic transition is determined by some parameter $r$ (external pressure or composition). At the critical transition one would observe quantum criticality in the entire temperature range, whereas for r-values where the transition is suppressed quantum criticality is observed only at higher temperatures. One then may expect a cross-over temperature T$^*$ where the system transforms from an unconventional metal at high temperatures towards a Fermi-liquid at low temperatures [@sachdev2]. Only in the temperature range above T$^*$ the magnetic excitations should exhibit the related ${\omega \over T}$-scaling. Our data clearly shows that such scaling can be fitted to the data only for the three higher temperatures studied. The description with the scaling concept seems to be slightly better for the exponent $\alpha$=0.75. The temperature dependent data suggests a crossover near 30K. This cross-over agrees very well to that seen in electronic transport properties, where well defined Fermi-liquid behavior is only observed below about 25K [@maeno; @2]. Magnetic scattering in the superconducting phase -------------------------------------------------- As emphasized by Joynt and Rice [@joynt], the wave-vector- and energy-dependent spin susceptibility in superconductors reflects directly the vector structure of the superconducting (SC) gap function, allowing a complete identification of the SC order parameter symmetry. Inelastic neutron scattering experiments have the potential, in principle, to determine the superconducting order parameter. In high-$T_c$ superconductors, the spin-singlet $d$-wave symmetry SC gap induces a striking modification of the spin susceptibility in the superconducting state. As a consequence, the so-called “magnetic resonance peak” has been observed in the superconducting state of various copper oxide superconductors by INS[@rossat91; @tl2201]. Therefore, a similar experiment in [Sr$_2$RuO$_4$]{}would certainly be instructive about the SC gap symmetry. The enhanced spin susceptibility has been calculated in [@27; @39; @39bis], considering a spin-triplet p-wave superconductiviting state with [**d**]{}([**k**]{})=[**z**]{}($\rm k_x \pm i k_y$). Note that in such a case, the superconducting gap is isotropic due to the particular shape of the Fermi-surface in [Sr$_2$RuO$_4$]{}. For the wavevector [$\mathbf{q_i}$]{}, Kee et al. [@39] and Morr et al. [@27] predicted that below $T_c$ spectral weight is shifted from below twice the superconducting gap, $\Delta$, into a resonance like feature close to 2$\Delta$. Morr et al. find the resonance in the $zz$-channel yielding an enhancement of the magnetic excitation intensity by a factor of 9 in the superconducting state as compared to the normal state[@27]. The difference between the in-plane and out-of-plane susceptibility in the superconducting state arrises from the coherence-factor. Similar theoretical framework is currently used to describe the spin excitation spectrum in spin-singlet HTSC cuprate superconductors[@OP]. Theoretical works show that the opening of a d-wave order gap together with the exchange interaction lead to the appearence of a similar resonant feature below 2$\Delta$ at the antiferronagnetic wavevector. These theories successfully account for the observation by INS of the magnetic resonance peak in the superconducting state. Using unpolarized INS, one measures the superposition of the out-of-plane and in-plane components of the susceptibility (see Eq. \[eq:INSanisotropy\]) and both components are equally weighted, when performing the measurements $\rm Q_l$=0. Thus, the predicted resonance feature should be observable, at the value of $\sim$2$\Delta$, if one obtains an experimental arrangement which allows to study the inelastic magnetic signal in this energy range. Due to the almost linear decrease of $\chi ''({\bf q_i},\omega)$ towards low energies, see Fig. 7, and due to the higher required resolution which implies less neutron flux, these experiments are extremely time demanding. We have analyzed the magnetic excitations in the superconducting phase on the cold triple axis spectrometer IN14 at the ILL using a two crystal assembly, the results are shown in Fig. \[fig9\]. The right part of Fig. \[fig9\] shows a scan across the incommensurate peak at [$\omega$]{}= 4 meV, i.e. the range already studied with thermal neutrons. This signal can be determined with little beam time. Performing the same scan at 1.6 and 0.8 meV requires considerably more time but still exhibits a well defined signal which seems not to experience any change in the superconducting state at T=0.35 K. The results of constant-${\bf Q}$-scans at ${\bf Q}$=(0.7,0.3,0)   are shown in the left part of Fig. \[fig9\]. As there is no visible difference between the results obtained at 0.35 and 0.80 K (both in the SC state), we have added the two scans below $\rm T_c$ in the lower left part of Fig. \[fig9\]. Importantly and despite efforts to get rather high statistics, there is no change visible in the energy spectra above and below the superconducting transition in the energy range of the superconducting gap. The spin susceptibility is not modified appreciably across the superconducting transition; our data even do not show any opening of a gap. One can describe the energy dependence presented in Fig. 9 with the single relaxor using the same fitted parameters as the low temperature data in Fig. \[fig5\].a. However, the detailled shape of the spin suceptibility (Fig. \[fig9\]) does not exactly match such a simple linear behavior but rather seems to indicate some anomaly near 2meV which requires further experimental work. There is actually little known about the value of the superconducting gap in [Sr$_2$RuO$_4$]{} . Laube et al. [@40] have reported an Andreev reflection study where the opening of the gap is clearly visible in an astonishingly large energy range. The quantitative analysis of the spectra is quite involved; assuming a p-wave order parameter Laube et al. obtain 2$\Delta$=2.2 meV which may be compared to the value expected within BCS-theory $2\Delta$= 3.55 k$_B$$T_c$ = 4.97 K = 0.43 meV. Our data shows that there is no change in the excitation spectrum for energies well below the reported value of 2$\Delta$, but it has not been possible to investigate the lowest energies due to the strong elastic incoherent signal. With further increased resolution ($k_f=1.3$Å$^{-1}$) we have scanned the energy range 0.3–0.7 meV at 0.80 K again without evidence for a resonant feature. Furthermore, the comparison of two scans at constant ${\bf Q}$=(1,0,0) did not yield any difference below and above $\rm T_c$ meaning no ferromagnetic spin susceptibility enhancement. The theory presented by Morr et al. [@27] should be considered as being quite reliable in the case of [Sr$_2$RuO$_4$]{}, since the RPA approach to the magnetic excitations is so successful in the normal state, and since [Sr$_2$RuO$_4$]{}  exhibits well defined Fermi-liquid properties at temperatures below 25 K. Therefore, the data in Fig. \[fig9\] gives strong evidence against a simple p-wave order parameter in [Sr$_2$RuO$_4$]{}  with a maximum value of the gap of the order of the reported value [@40]. However, in the meanwhile there are several indications that the order parameter is more complex. The recent specific heat data on the highest quality single crystals [@21] points to the existence of line nodes in the gap function which were then shown to be aligned parallel to the a,b-plane (horizontal line nodes). Such line nodes were explained by Zhitomirsky and Rice through a proximity effect between the active [$\gamma$-band]{}  and the more passive one-dimensional bands [@24]. A modulation of the gap function along the c-direction will wipe out the resonance predicted for the non-modulated p-wave gap, since for the ${\bf Q}$-position analyzed, (0.7 0.3 0), the electron hole excitation involves parts of the Fermi surface which are fully, partially or not gapped at all. In this sense the absence of any temperature dependence in the magnetic excitation spectrum is consistent with the presently most accepted shape of the gap function. Further theoretical as well as experimental studies are required to clarify the possibility of a resonant feature at other positions in $({\bf Q},\omega)$-space. Conclusions =========== Using assemblies of several crystals of [Sr$_2$RuO$_4$]{}  we have analyzed the magnetic excitations by INS. The incommensurate signal arising from the nesting between the one-dimensional bands shows an asymmetry which is well explained by the full RPA analysis as a contribution mainly from the [$\gamma$-band]{}. The energy dependence of the incommensurate signal varies with temperature and exhibits a general softening of the spectrum upon cooling. This behavior indicates that [Sr$_2$RuO$_4$]{}  is approaching the corresponding SDW instability at low temperature even though this compound is not at a quantum critical composition. This interpretation is confirmed by the fact that the generalized susceptibility exhibits some $\omega / T$-scaling only above $\sim$30 K, i.e. in the temperature range where also the transport properties indicate non-Fermi-liquid behavior. The analysis of magnetic excitations besides the nesting ones shows only minor contributions. There is some evidence for additional magnetic scattering closer to the zone-center but still not peaking at the zone-center. This interpretation gets support from the fact that similar scattering is observed in nearly ferromagnetic [Ca$_{1.5}$Sr$_{0.5}$RuO$_4$]{}  and from various RPA calculations which find excitations mainly related to the [$\gamma$-band]{}  in this ${\bf q}$-range. The magnetic excitations in [Sr$_2$RuO$_4$]{}  may be compared to the distinct types of magnetic order which have been induced by substitution. The dominant excitations reflect the SDW reported to occur in [Sr$_2$Ru$_{1-x}$Ti$_x$O$_4$]{}  at small Ti concentrations. The less strong excitations situated more closely to the zone-center and most likely related to the [$\gamma$-band]{}  become enhanced through Ca-substitution which drives the system towards ferromagnetism, but only for rather high Ca-concentration. Upon cooling through the superconducting transition we do not observe any change in the magnetic excitation spectra, which - combined with recent calculations [@27]- indicates that the order-parameter in [Sr$_2$RuO$_4$]{}  does not possess simple p-wave symmetry. These experimental findings are still in agreement with a p-wave order parameter modulated by horizontal line nodes. [**Acknowledgements.**]{} We would like to thank O. Friedt, H.Y. Kee, D. Morr and R. Werner for stimulating discussions and P. Boutrouille (LLB) and S. Pujol (ILL) for technical assistance. Work at Cologne University was supported by the Deutsche Forschungsgemeinschaft through the Sonderforschungsbereich 608. 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--- abstract: 'In this review, we consider the case where electrons, magnetic monopoles, and dyons become massless. Here we consider the ${\cal N} = 2$ supersymmetric Yang-Mills (SYM) theories with classical gauge groups with a rank $r$, $SU(r+1)$, $SO(2r)$, $Sp(2r)$, and $SO(2r+1)$. which are studied by Riemann surfaces called Seiberg-Witten curves. We discuss physical singularity associated with massless particles, which can be studied by geometric singularity of vanishing 1-cycles in Riemann surfaces in hyperelliptic form. We pay particular attention to the cases where mutually non-local states become massless (Argyres-Douglas theories), which corresponds to Riemann surfaces degenerating into cusps. We discuss non-trivial topology on the moduli space of the theory, which is reflected as monodromy associated to vanishing 1-cycles. We observe how dyon charges get changed as we move around and through singularity in moduli space.' address: | Ernest Rutherford Physics Department, McGill University,\ 3600 rue University, Montreal, QC H3A 2T8, Canada\ and\ Centre de recherches mathématiques, Université de Montréal\ C.P. 6128, succ. centre-ville, Montréal, Québec, H3C 3J7, Canada\ [email protected] author: - JIHYE SOFIA SEO bibliography: - 'AScurve.bib' title: | SINGULARITY STRUCTURE OF\ ${\cal N}=2$ SUPERSYMMETRIC YANG-MILLS THEORIES: A REVIEW[^1] --- Motivation ========== Understanding of physics often advances through consideration of extreme and singular situations. We address lots of questions in extreme limits, with very small or very large values of density, temperature, velocity, mass etc. These are not only for theoretical curiosity; instead often it turns out to be a golden mine for new discovery and applications. Neutron stars and big bang are at extremely high density. Superconductivity and superfluidity is studied at extremely low temperature. Some extremes may simplify the situation enough to provide ideal setting to focus on the essence of the system. For example, ideal gas law assumes no interaction among particles. Many freshmen-level classical mechanics problems assume that friction vanishes and spring is massless. Some extremes pose us such a big challenge that it takes a paradigm shift to overcome the huddle. For example, consider the thought experiment in classical mechanics of the escape velocity of the satellite. If the gravity is so strong then the escape velocity approaches the speed of light. This was the first encounter (in thought) with black holes. Very careful consideration and study of the highest possible velocity, the speed of light, gave birth to special relativity, and we learned to think of space and time together as one combined object, spacetime. Even in more contemporary setting, singularities in physics deserves serious attention. It may serve as a warning signal: for example, it may occur when we have integrated out massive fields which are in fact massless. UV divergences in field theory urge us to look for a better theory at higher energy. Understanding singularity is a [*cornerstone*]{} to solving field theory problem, just as imagining an extreme situation gives us an often correct intuition for classical mechanics problem. We often started to think about collision between particles, where their masses are equal or very different. Mathematics, especially geometry has been a faithful and fruitful language in describing physical system. Gravity - Einstein’s General Relativity - is best described in the language of differential geometry. What about other forces in nature? Electromagnetic, weak, and strong forces are formulated in terms of gauge theories with gauge groups $U(1)$, $SU(2)$ and $SU(3)$ respectively. As reflected in Ref. , these gauge theories are well-described in another field of mathematics, so-called fiber bundle theory. In studying the singularity of physical system, geometry is particularly useful. Physical singularity is reflected in geometry as mathematical singularity. There exists a famous dictionary between geometry and physics for gravitational singularity. A black hole in physics will appear as a geometric singularity, that is a puncture in a spacetime fabric. Later we will discuss physical singularity associated with both electrons and magnetic monopoles having zero mass. So far there is no known Lagrangian for this system[@ArgyresDouglas]. Thanks to the close relationship with geometry, however, this can be studied in terms of geometry. Recently Ref. has revealed strange behavior of moduli space near the singularity by careful observation of geometric singularity of Seiberg-Witten curve associated to the physical system. In this article, we will review physical and geometric singularity of Yang-Mills theories, which have close relationship with electromagnetic and nuclear forces, but with multiple supersymmetries. The amount of supersymmetry is denoted by $\cal{N}$, the number of supercharges. Roughly it means that $2^{\cal{N}}$ particles form a set (a supermultiplet, as we will discuss in more detail later.) in which all the physical qualities are identical to one another except for the spin. We will motivate the supersymmetric Yang-Mills theories by looking at ${\cal{N}}=4$ and ${\cal{N}}=2$ theories in this introductory section, and the rest of the article will focus on ${\cal{N}}=2$ supersymmetric Yang-Mills theories. ${\cal{N}}=4$ SYM: gluon scattering amplitudes at hadron colliders ------------------------------------------------------------------ One may ask “Fine. I buy that gauge theories are important because they describe nuclear and electroweak forces. But why should anyone care about [*supersymmetric*]{} gauge theories, when LHC did not observe any superpartners yet?”. First we recall that supersymmetry provides an attractive and graceful exit out of many serious paradoxical situations. It plays a crucial role in resolving the issues of hierarchy problem, unification of coupling constants of gauge theories, mismatch of cosmological constant, etc. In other words, supersymmetry has been a best friend to theorists, who would like to make theoretical sense and feel aesthetic harmony out of observed experimental facts. However, supersymmetry also makes contributions for experiments. One of the most important inputs of supersymmetry recently is computation of scattering amplitudes of gluons. Computation is much easier when theory has supersymmetry. At the tree level, gluon scattering amplitudes agree between supersymmetric and non-supersymmetric theories. Therefore, easier computation in supersymmetric theories can provide useful results for non-supersymmetric and more realistic theories, at least to the leading order. Results on scattering amplitudes in maximally supersymmetric gauge theories (${\cal{N}}=4$), obtained by many string theorists, are implemented into the tools such as BlackHat[@BlackHat], used by experimentalists at hadron colliders. Though massless, gluons are responsible for carrying lots of energy away from the collision process, and it is a big plus to understand their scattering amplitudes. Supersymmetric Yang-Mills theories are even more relevant in this LHC era, with or without supersymmetry detection. In past several years there has been a dramatic progress (almost at an exponential rate) in computation of gluon scattering amplitudes in ${\cal{N}}=4$ (maximal) supersymmetric gauge theories. ${\cal{N}}=4$ supersymmetric Yang-Mills theories are special in that the 3-point function of gluons can be written down purely out of symmetry argument. Having so much supersymmetry, the theory enjoys superconformal symmetry and its dual superconformal symmetry. Conformal symmetry means one can forget about lengths. One does not even need to know the Lagrangian. One can write down S-matrix purely from the symmetry and consistency consideration, with no need for Feynman diagrams. As nicely reviewed in a recent paper Ref. , to build $n$-point function the 3-point function are put together like lego blocks by amalgamation and projection operators. While postponing manifestation of unitarity and locality, scattering amplitudes manifest dual conformal supersymmetry and Yangian symmetry, which would remain opaque in evaluation of each Feynman diagram. Geometry is a bias-free place to look for symmetries in physics. Gluons being massless, their 4-dimensional null (light-like) momenta enjoy amphibian lifestyle: both Lorentzian and twistor spaces[@Penrose:1972ia] provide a natural habitat to describe kinematics. In lieu of Feynman diagrams, scattering amplitude can be organized by much simpler Hodges diagrams[@Hodges:2006tw; @Hodges:2005bf; @Hodges:2005aj] in twistor space, as nicely reviewed in Ref. . Using a higher dimensional version of Cauchy’s theorem, this can be written as partial sum of residues at isolated singularities. Scattering amplitudes in maximally supersymmetric gauge theories are given as a contour integral over a Grassmannian[^2]. Supersymmetric Yang-Mills theories thrive in a close relationship with geometry. Another motivation to consider ${\cal{N}}=4$ SYM is an inseparable bond between supersymmetric gauge theory (SYM) and supergravity (SUGRA). Amplitudes for SYM and SUGRA have tight kinship: supergravity amplitudes can be written in terms of SYM amplitudes, roughly speaking. One of the hot issues is the question of UV finiteness of supergravity: is supergravity a valid theory by itself, or do we must recruit string theory (or other candidates of quantum gravity) to make the supergravity consistent at arbitrarily high energy? The answer has been elusive, but supersymmetric gauge theory might be able to help. More excitement in ${\cal{N}}=4$ supersymmetric Yang-Mills theories can be found in Ref. and its referecences. Now we will switch to less supersymmetric ones, ${\cal{N}}=2$ supersymmetric Yang-Mills theories for the rest of the review. Motivation for ${\cal{N}}=2$ SYM: massless magnetic monopole ------------------------------------------------------------ Fruitful symbiosis between physics and geometry, which we observed for general relativity and ${\cal{N}}=4$ SYM, holds true for ${\cal{N}}=2$ SYM as well. The guest of honor for ${\cal{N}}=2$ supersymmetric Yang-Mills theory is a Riemann surface. Its most famous persona is as a Seiberg-Witten curve[@SeibergWittenNoMatter; @SeibergWittenWithMatter]: in a teamwork with Seiberg-Witten one-form, it encodes lots of information about the physical theories, as we will delve deeper in the rest of the review. This Riemann surface also serves as a spectral curve of integrable system [@dw]. Seen from the 4-dimensional field theory perspective, on which this review will mainly focus, this curve does not live inside the spacetime. One may regard it as an auxiliary object or a bookkeeping device, which happens to encode lots of useful information. This review will focus more on what to learn out of a given SW geometry, rather than how to obtain such geometry to begin with. So far the best way to understand the origin of SW geometry seems to be string theory. In string theory settings (although we won’t discuss them in depth here), the Seiberg-Witten geometry is closely related to extended objects in string theories and M-theory. Ref. interprets that the 4-dimensional field theory comes from wrapping M5-brane on the Seiberg-Witten curve. In M-theory, M5 brane is a solitonic object spanning 5 spatial and 1 temporal directions, carrying a conserved charge. Just as we consider world-line of a point particle traveling in time, we can consider 6-dimensional theory on the world-volume spanned by time evolution of M5-brane. However if we let M5-brane to wrap a 2-dimensional Riemann surface and further assume that the Riemann surface is small compared to other directions in the spacetime, then we will have only 4 remaining directions effectively. This type of 4-dimensional theories are discussed in Refs. . There are also interpretations of Seiberg-Witten curve in terms of non-critical and anti-self dual strings in type IIA, IIB, and heterotic string theories, again appearing as wrapping extended solitonic objects on appropriate cycles, as reviewed in Ref. . Thanks to Seiberg-Witten theory[@SeibergWittenNoMatter; @SeibergWittenWithMatter], many ${\cal{N}}=2$ supersymmetric Yang-Mills theories can be equivalently written as a Riemann surface (written down as a format of hyperelliptic curve) called Seiberg-Witten curve with a one-form. Studying singularity on geometry-side provides us a powerful microscope probing singularity of physical theory. Degeneration and monodromy of hyperelliptic curves translates into massless fields and their dyon charges in Seiberg-Witten theories[^3]. A higher singularity coming from collision of milder singularity gives us an exotic theory with massless electron and massless monopole, so-called Argyres-Douglas theory. It defies Lagrangian description: when Lagrangian mechanics turns its back on us, we have all the more reason to seek the friendship with geometry. Let us pause for a moment and remind ourselves why we need to study magnetic monopoles, especially why the light ones. Despite many experimental claims and findings, magnetic monopole (massive or massless) is something we have not observed in a concrete manner yet. In the set of Maxwell’s equations, magnetic monopoles naturally arise if one tries to manifest the hypothetical electro-magnetic symmetry, and introduces magnetic sources just like electric sources. There are two main reasons why the magnetic monopole must exist beyond a theorist’s fanciful imagination, as pedagogically reviewed in Ref. by D. Milstead and E. J. Weinberg. Grand Unified Theory (GUT) (of electromagnetism and nuclear forces) predicts existence of magnetic monopole as shown in Refs. . The mere existence of magnetic monopole explains and necessitates quantization of electric charge.[@Dirac] For the lack of experimental evidence of monopole, we tend to blame its high mass. In the present Universe, we expect the magnetic monopole to be very heavy - its mass energy is near that of a bacterium or near kinetic energy of a running hippo, which is a lot larger compared to other elementary particles. Big bang also provides an excuse for its absence in experimental data, arguing that the finite number density of magnetic monopole diluted out as the universe evolves. However if we trace back the history of universe, then spontaneously broken symmetries (such as GUT and supersymmetry) are restored and magnetic monopole might have been not so heavy. Particles which are partners under supersymmetry and electromagnetic duality can be thought of babies who were born as identical twins, but as time goes on, who grow into adults with different physical qualities. Supersymmetric Yang-Mills theory is a promising place to learn about quantum field theories. First, supersymmetry allows computation. Second, some properties we find in supersymmetric theories often still hold even in non-supersymmetric quantum field theories. In some sense supersymmetric field theories provide fruitful and fertile toy models to learn about realistic quantum field theories. Plan of the review \[Plan\] --------------------------- In section \[essenSec\], we revisit essential elements of ${\cal{N}}=2$ supersymmetric Yang-Mills theories and Seiberg-Witten geometry, and prepare ourselves with necessary tools for studying singularity. In section \[firstlook\], we study Seiberg-Witten curves for $SU(r+1)$ and $Sp(2r)$ SYM and discuss the root structure of those families of hyperelliptic curve, in preparation for section \[monodromySec\] where we compute dyon charges of massless states. Section \[ArDo\] deals with higher singularity with massless electron and monopole (Argyres-Douglas theories) among ${\cal N} = 2$ supersymmetric Yang-Mills theories with classical gauge groups $SU(r+1)$, $SO(2r)$, $Sp(2r)$, and $SO(2r+1)$. In section \[doublediscsection\], we revisit the tools to capture singularity to learn more about the singularity structure. We conclude with open questions in section \[conclusion\]. Essentials of ${\cal{N}}=2$ SYM and Seiberg-Witten Geometry \[essenSec\] ======================================================================== Many wonderful reviews[@BilalReview; @LercheReview; @AlvarezGaumeHassanReview; @Argyres:1998vt] exist on ${\cal{N}}=2$ SYM, while this review is more focussed on the Seiberg-Witten geometry and singularity of those theories. Here we will only pinpoint the aspects that are necessary for understanding the main idea of the review. The Lagrangian of ${\cal{N}}=2$ supersymmetric Yang-Mills theory can be written elegantly in ${\cal{N}}=1$ or ${\cal{N}}=2$ superspace language, where supersymmetry is more manifest. Here we will write it down in a spelled-out fashion in 4-dimensional spacetime language as below: $$\begin{aligned} {\cal{L}} &=&{\color{red} - \frac{1}{g^2} \int d^4 x {\rm Tr} \left[ \frac{1}{4} F_{\mu \nu} F^{\mu \nu} \right] }+ {\color{purple} \frac{\theta}{32 \pi^2} \int d^4 x {\rm Tr} \frac{i}{4} F_{\mu \nu} \tilde{F}^{\mu \nu}} \nonumber \\ & & {\color{forestgreen} - \int d^4 x \frac{1}{ 2} {\rm Tr} \left[ \phi^+ , \phi \right]^2 }+{\color{blue} {\mathrm{(fermions)}}}. \label{N2Lag}\end{aligned}$$ Here $A_\mu$ and $\phi$ transform as a vector and a scalar (respectively) of Lorentz group $SO(3,1)$ of spacetime. Both are adjoint representation of the gauge group $G$. The choice of gauge group also determines structure constants $f_{abc}$ and generators $T_a$, $$\begin{aligned} A_\mu &=& A^a_\mu T_a, \quad [T_a, T_b]=f_{abc} T_c , \nonumber \\ F_{\mu\nu} &=&\partial_\mu A_\nu -\partial_\nu A_\mu - ig [A_\mu,A_\nu], \quad \tilde{F}^{\mu\nu} =\frac{1}{2} \epsilon^{\mu\nu\rho\sigma} F_{\rho \sigma} \end{aligned}$$ and $g, \theta $ are real-valued coupling constants. If we did not have supersymmetry, then only the [ first term]{} of Eq. would appear in the Lagrangian, as the [Yang-Mills]{} action. For example, the action for weak and strong nuclear forces can be written down by choosing $G=SU(2), SU(3)$ and taking only the first term of Eq. . Having ${\cal{N}}=2$ extended supersymmetry dictates that scalar, spinors, and vector must transform together forming a ${\cal{N}}=2$ supermultiplet[^4]. Instead of keeping track of all component fields in a given supermultiplet, we can save our effort and restrict our attention to the term one of them only. Here it is enough to consider the part with the scalar $\phi$ only which is the [ third term]{} of Eq. , focussing on vacuum structure of a ${\cal{N}}=2$ supersymmetric Yang-Mills theory. In Eq. the [ second term]{} gives [instanton number]{} and the [ last term]{} denote terms involving [fermions]{}. However we won’t make usage of this Lagrangian any more in this review, because Seiberg and Witten proposed another ‘geometric’ way to study ${\cal N} = 2$ SYM theories[@SeibergWittenNoMatter; @SeibergWittenWithMatter], which we now turn to. Review of ${\cal N}=2$ Seiberg-Witten theory and geometry \[ReviewN2\] ---------------------------------------------------------------------- In late 1990s, Seiberg and Witten made a profound discovery on ${\cal N} = 2$, $d = 4$ supersymmetric Yang-Mills theory with gauge group $G=SU(2)$[@SeibergWittenNoMatter; @SeibergWittenWithMatter], giving a huge impact both on physics and mathematics. After two dozen years, Gaiotto blew a new life into research on ${\cal N} = 2$ superconformal theories recently, by discovering a plethora of new theories with often surprising features, which can be all lego-ed from simple building blocks[@gaiotto][^5]. Simply speaking, Seiberg and Witten proposed a powerful dictionary between physics and geometry for ${\cal{N}}=2$ theories. Seiberg-Witten geometry comes in package with Seiberg-Witten (SW) curve and Seiberg-Witten (SW) differential 1-form. The SW curve is a complex curve, or a real 2 dimensional Riemann surface, whose genus is equal to the rank $r$ of the gauge group (such as $SU(r+1)$, $Sp(2r)$, $SO(2r)$, and $SO(2r+1)$) for the supersymmetric Yang-Mills theories (that is, with no matters added). It is also equal to the complex dimension of moduli space. Recall that the moduli are to be understood as parameters controlling the theory and the subsequent SW geometry. If the gauge group had rank 3, then the corresponding SW curve may look like the Riemann surface in Fig. \[nu\], with genus 3. $$\begin{tabular}{|ccc|} \hline Physics & $\leftrightarrow$ & Geometry \\ \hline \hline supersymmetric Yang-Mills theory & $\leftrightarrow$ & Riemann surface \\ \hline rank of gauge group & $\leftrightarrow$ & genus \\ \hline (BPS) particles & $\leftrightarrow$ & (some) 1-cycles \\ \hline \end{tabular} \label{tabledic}$$ ![Various 1-cycles and their symplectic basis for a Riemann surface of genus 3[]{data-label="nu"}](RMsmoothnu.pdf){width="\textwidth"} In pure Seiberg-Witten theory the dimension of the moduli space (or the number of moduli/parameters) is also equal to the genus[^6], which, in turn, is equal to the rank of the gauge group. At a generic point in the moduli space, the SW curve is smooth and all the 1-cycles are non-vanishing as in [Fig. \[nu\]]{}. However, we could move to a less generic location in the moduli space where we have vanishing 1-cycles as in Fig. \[local\] and Fig. \[nonlocal\]. Note that on a SW curve, we can draw various 1-cycles as in [Fig. \[nu\]]{}. Here we have chosen a particular set of symplectic basis 1-cycles, ${\color{blue}\alpha_i}$’s and ${\color{red}\beta_i}$’s. The only rule to keep for the choice of symplectic basis cycles is that the intersection numbers must satisfy: $${\color{blue}\beta_i } \circ {\color{red}\alpha_j} =\delta_{ij}. \label{intersectsymp}$$ The intersection number is an anti-symmetric ($ {\color{red}\alpha_j} \circ {\color{blue}\beta_i } =-\delta_{ij}$) and bilinear (linear dependence on both arguments) operation among 1-cycles[^7]. Each 1-cycle has an orientation (as seen by the arrow in figures), and the intersection number comes with a sign[^8]. As anticipated from Eqn. , 1-cycles of the Riemann surface correspond to physical particles[^9], and a choice of symplectic basis 1-cycles assigns electric and magnetic charges to the particles. For given $i$, ${\color{blue}\alpha_i}$ and ${\color{red}\beta_i}$ denote electric and magnetic charge (respectively) for the $i$’th $U(1)$ inside the gauge group $G$. However, the choice is certainly not unique, and we could modify the choice by $${\color{blue}\alpha_i^{\prime}} \equiv {\color{red}\beta_i}, \qquad {\color{red}\beta_i^{\prime}} \equiv - {\color{blue}\alpha_i} \label{emduality}$$ for given $i$ only, and this still preserves the symplectic property of Eqn. . In physics this corresponds to the electromagnetic duality on $i$’th $U(1)$ charge. Another important fact is that the intersection number (being a scalar) is invariant under electromagnetic dualities of Eq. , and in general, under symplectic transformation (re-choice of symplectic basis 1-cycles). Some of 1-cycles correspond to physical states (stable BPS/supersymmetric dyon), with quantized [electric]{} and [magnetic]{} charges. As shown Fig. \[nu\], any 1-cycle can be written in terms of basis 1-cycles ${\color{blue}\alpha_{i}}$’s and ${\color{red}\beta_i}$’s with integer coefficients, with dyonic charges superposed. These integer coefficients exactly correspond to amount of electric and magnetic charges of each $U(1)$. Two 1-cycles $\nu_{1}$ and $\nu_{2}$ in Fig. \[nu\] can be written as follows: $${\color{forestgreen}\nu_1} = {\color{red}\beta_{1}} +{\color{red}\beta_{2}} , \qquad {\color{forestgreen}\nu_2} = -{\color{blue}\alpha_{3}} +{\color{red}\beta_{3}}. \label{dyonchargecycle}$$ Physical interpretation of this would be that, if they corresponded to BPS dyons, then the first object (${\color{forestgreen}\nu_1}$) behaves as a magnetic monopole for both the first and the second $U(1)$s, and the second object ($ {\color{forestgreen}\nu_2}$) carries the same charge as a bound state of a positron and magnetic monopole of the third $U(1)$. Seiberg-Witten geometry contains lots of (if not all) information about the theory. The physical information is stored not only in SW curves, but also in the SW 1-form. The SW curve and SW 1-form work together, and without each other they lose meaning, just like a needle and a thread. The SW curve provides 1-cycles over which to integrate the SW 1-form. Then we obtain complex number which is meaningful physically (central charge). By integrating Seiberg-Witten differential 1-form $\lambda_{\rm SW}$ over 1-cycle $\nu$, we obtain a complex number. For the purpose of this review, we are only interested in its magnitude, which is the mass of the particle $$M_{\nu}= \left| \oint_\nu \lambda_{\rm SW} \right|. \label{Mlambda}$$ Since we are focussing on physical singularity associated with massless particles, Eqn. provides the most important piece of information for the purpose of this review, among what we learn from the Seiberg-Witten geometry. Assuming $\lambda_{\rm SW}$ is free of delta-function behavior, vanishing of 1-cycle $\nu$ signals existence of massless BPS state (with dyonic charge given by $\nu$) since its mass given in Eqn. vanishes[^10]. Therefore, study of vanishing 1-cycles can teach us about massless BPS states in the system. We therefore assume that: .1in .1in ![Vanishing 1-cycles of genus-3 Riemann surface. All these 3 cycles are mutually local, since intersection numbers all vanish.[]{data-label="local"}](RMlocal.pdf){width="\textwidth"} Now let us imagine tuning various parameters (moduli) for the gauge theory of Fig. \[nu\], to force some 1-cycles to vanish. In Fig. \[local\], we have three vanishing 1-cycles $ {\color{blue}\alpha_1}, {\color{blue}\alpha_2},$ and ${\color{red}\beta_3}$ which do not intersect each other. They correspond to an electron with respect to the first $U(1)$, another electron with respect to the second $U(1)$, and a magnetic monopole with respect to the third $U(1)$. All three particles are massless. If we operate an electromagnetic duality on the third $U(1)$, then the third particle will be renamed into a massless electron with respect to the third $U(1)$. All the massless particles are mutually [*local*]{}, in that they can be treated as pure electrons (carrying no magnetic charge) in some choice of symplectic basis 1-cycles (i.e. after a certain series of performing electromagnetic dualities). This is possible only because (if and only if, in fact) the corresponding 1-cycles have vanishing intersection number with one another. An equivalent mathematical statement is this: if all the 1-cycles in a certain set have zero intersection number with one another, then they can be written in terms of linear combination of $\alpha_i$’s with no need for $\beta_i$ terms. We will soon explain why we call them [*local*]{}, (near Eqn. ) after explaining non-locality now. ![Mutually non-local vanishing cycles of genus-3 Riemann surface. Their intersection number is non-zero.[]{data-label="nonlocal"}](RMnonlocal.pdf){width="\textwidth"} Two 1-cycles ${\color{blue}\alpha_3}$ and ${\color{red}\beta_{3}}$ vanish in [Fig. \[nonlocal\]]{}, and they correspond to an electron and a magnetic monopole, both charged with respect to the third $U(1)$, and with zero mass. No matter how one may try to redefine electric and magnetic charges by electromagnetic dualities and so on, it is never possible to make both of them into electric particles at the same time. That is because the two vanishing 1-cycles have non-zero intersection number $ {\color{blue}\alpha_3} \circ {\color{red}\beta_{3}}=-1\ne 0$, regardless of choice of symplectic bases. If a set of 1-cycles were able to be written as electric particles (in terms of $\alpha_i$’s only), then they must have had zero intersection number with one another. We will pause briefly here to explain naming of locality versus non-locality for vanishing 1-cycles (massless particles, equivalently). As nicely reviewed in Ref. , 1-cycles transform under monodromy action, as one moves around on a non-contractible loop, surrounding a singularity, in moduli space (changing the moduli values accordingly). If the singularity is where a 1-cycle $\nu$ vanishes, then the other 1-cycle $\gamma$ gets transformed according to this Picard-Lefshetz formula, as explained in Ref. $$M_\nu : \gamma \rightarrow \gamma - (\gamma \circ \nu) \nu . \label{PLformula}$$ For a 1-cycle $\gamma$ which does not intersect with $\nu$ i.e. $\gamma \circ \nu=0$, no change will be made on $\gamma$ under monodromy. (However, any 1-cycles which intersect with $\nu$ will be shifted as one goes around the singularity loci associated with vanishing of the 1-cycle $\nu$, as we will explain below.) This gives a motivation for the naming: if two cycles have zero intersection number, when one vanishes, the other cycle does not get affected. Or in physics language, two particles can be written as purely electric ones at the same time. When one becomes massless the other does not change its charges. In some sense, they do not need to care about each other, and they are mutually [*local*]{}. It is possible to write down Lagrangian for those theories, by adding each local pieces. However, now assume that two cycles have non-zero intersection number. When one vanishes, the other cycle receives a monodromic shift. In physics language, the two particles are mutually [*non-local*]{} and they cannot be written as purely electric ones at the same time. When one particle becomes massless, it is ambiguous how to assign charge to the other particle. The dyon charge of the second particle is not a single-valued function of moduli near the singularity locus where the first particle becomes massless. In general, there is no known Lagrangian for these systems. However, in Ref. , this exotic theory (so-called Argyres-Douglas theory) has been discovered and studied, inside moduli space of $SU(r+1)$ SW theories. Since there is no Lagrangian description yet (if not never), studies are conducted by careful analysis of scaling dimensions near the singularity loci of Seiberg-Witten geometry which can be written as a hyperelliptic curve equipped with 1-form. Recent key developments in this direction can be found in Refs. among others. Now we turn to review geometry of hyperelliptic curves. Review of hyperelliptic curves \[GeomReview\] $y^2=f(x)$ -------------------------------------------------------- So far in this section, we discussed Riemann surface with 1-cycles which potentially could collapse. For the purpose of this review, the Riemann surfaces of our interest can be written as an algebraic variety given by $y^2=f(x)$, which include hyperelliptic curves. Since we need to deal with singularity as well, let us begin by recalling a few fundamental facts about singularity of algebraic varieties. Let us consider an algebraic variety given by $F(x,y,z,\ldots)=0$. This is an object embedded inside a bigger space, [*ambient space*]{} whose coordinates are $x,y,z,\ldots$. It is singular if exterior derivative $dF=0$ vanishes, or in other words if all the partial derivatives vanish, namely $\frac{\partial F}{\partial x} = \frac{\partial F}{\partial y}= \cdots=0$. The exterior derivative $d$ is written in terms of the partial derivatives with respect to all the coordinates of the ambient space. Since the Riemann surface is embedded in an ambient space whose coordinates are $x$ and $y$, the exterior derivative is given as $$d= dx \frac{\partial }{\partial x} + dy \frac{\partial }{\partial y}.$$ Later, we will consider algebraic variety embedded inside the moduli space whose coordinates are complex-valued moduli $u_i$’s, then the exterior derivative will accordingly be $d= \sum_i du_i \frac{\partial }{\partial u_i} $. In simpler words, at each point on a given surface (algebraic variety) embedded in a bigger space, [*an ambient space*]{}, we can consider tangent space. However, if the surface develops singularity, then the tangent space suddenly changes there ($dF=0$). Singularity of an algebraic variety (super-elliptic curve) $$F \equiv y^n-f(x)=0, \quad n \ge 2 \label{suel}$$ is given by having $\frac{\partial F}{\partial x} = -\frac{\partial f}{\partial x}=0 $ and $ \frac{\partial F}{\partial y}=n y^{n-1}=0$. Therefore the singularity is at where $y=0=f(x)=\frac{\partial f}{\partial x}$. In order for $f(x)$ and $\frac{\partial f}{\partial x}$ to have a common root, it is equivalent to demanding $f(x)$ to have a degenerate root. We will now see that it happens if and only if $f(x)$ has vanishing discriminant $\Delta_x f =0$. [Discriminant]{} of a polynomial $f_n({\color{red}x}) = \prod_{i=1}^n ({\color{red}x}-e_i) $ is given in terms of its roots as $$\Delta_{\color{red}x} \left( f_n(x) \right) = \prod_{i<j} (e_i-e_j)^2. \label{DiscDef}$$ Vanishing of Eqn. is equivalent to existence of repeated roots. The number of identical roots is called the degeneracy, multiplicity of zero, or order of vanishing. A subscript for the discriminant symbol denotes which variable we take discriminant with respect to. This will be useful when we have a polynomial in multiple variables. For example, we will first discuss discriminant with respect to $x$, in the ambient space whose coordinates are $x, y$, in which Riemann surface is embedded. Next we will discuss algebraic variety defined inside the space of moduli, the parameters which control properties of Riemann surface. Then we will take discriminant with respect to one of the variables in the moduli space. Smooth hyperelliptic curve is defined (similarly to Eqn. ) as a complex curve embedded in an ambient space whose coordinates are two complex variables $x, y \in \mathbb{C}$ satisfying the equation $$y^2 = f_n(x)= \prod_{i=1}^n ({ x}-e_i), \quad n > 4$$ where complex parameters $e_i \in \mathbb{C}$’s are all distinct from each other ($ e_i \ne e_j $ for $i\ne j$). This gives a double-sheet fibration of $x$-plane with multiple ($n$ for even $n$, $n+1$ for odd $n$) separate branch points for a Riemann surface. For generic value of $x$, $y = \pm \sqrt{f(x)}$ has two choices for sign, therefore creating double-sheet. We can choose the upper and lower sheets to satisfy $y = \sqrt{f(x)}$ and $y = - \sqrt{f(x)}$ respectively. If $x=e_i$, then $y=0$ and branch points will be formed. Obviously, $n$ branch points are at each $e_i$’s. If $n$ is odd, then we have an extra branch point at $x=\infty$ because of an extra monodromy of $y^2=f_{{\rm odd}\ n}(x)$ there: As $x$ rotates by $2\pi$, $y$ changes its sign if and only if $n$ is odd. In the double-sheet fibration picture, each upper and lower $x$-plane can be thought as a sphere (with compactification at infinity). Each pair of branch points can be considered as a tube (cylinder) connecting two spheres. Therefore, the genus is $g=\left[ \frac{n-1}{2} \right]$[^11]. When $n=3,4$ as in rank 1 SW curve, then this formula asserts that $g=1$. Recalling the definition of discriminant, demanding $ e_i \ne e_j $ for $i\ne j$ guarantees smoothness. However, we want to consider possibility of singularity and vanishing 1-cycles (for example coming from shrinking of the cylinder connecting two spheres). Therefore for the review, we will extend the definition of hyperelliptic curve as a complex variety given by an equation $$y^2 = f_n(x)= \prod_{i=1}^n ({ x}-e_i), \quad x, y, e_i \in \mathbb{C}, \quad n > 4. \label{hecd}$$ The only difference is that we no longer demand the $e_i$’s to be distinct. Of course physicists are already motivated to look at singular curve due to massless states, as explained near Eqn. . As we explained near Eqn. , even if one is only interested in smooth hyperelliptic curves, the moduli space has nontrivial topology where 1-cycles transform under monodromy as we go around a non-contractible loop in a moduli space. Therefore we consider potentially singular, hyperelliptic curve, as defined in Eqn. , which is a double-sheet fibration of $x$-plane with $2g+2$ branch points. When roots degenerate, the curve degenerates. Because of the squaring in Eqn. , the discriminant is symmetric among the roots $e_i$’s. Therefore, the discriminant can also be expressed in terms of the coefficients of the polynomial $f_n(x)$ (therefore moduli). For an algebraic curve given as $y^2=f(x)$, by a discriminant of the curve, we mean discriminant $\Delta_x f$. Singularity of the curve is captured by colliding roots on the $x$-plane, at vanishing discriminant $\Delta_x f$. Note from right-hand side of Eqn. that $\Delta_x f$ has no dependence on $x$ or $y$. Demanding $\Delta_x f=0$ only gives one complex condition inside moduli space. Therefore vanishing discriminant loci is an algebraic variety inside moduli space with complex codimension one. By [*codimension*]{}, we mean an embedded object has smaller dimension than the ambient space by it, so it is equal to the number of independent conditions imposed. Existence of degenerate root of $f(x)$ signals singularity: there exists a vanishing 1-cycle. Various 1-cycles of $y^2=f(x)$ can vanish as we let the roots of $f(x)$ to degenerate, as depicted in Fig. \[local\]. A ‘donut’ can degenerate into a thin ‘ring’ with shrinking $\alpha$ cycle, or into a fat ‘bagel’ with shrinking $\beta$ cycle. When the degeneracy of the root is high (3 or larger), then it signals that there are multiple vanishing 1-cycles which have non-zero intersection numbers with one another, as depicted in Fig. \[nonlocal\]. Both $\alpha$ and $\beta$ cycles shrink together for the same ‘donut’, forming a cusp (for example a ‘croissant’). This corresponds to an Argyres-Douglas theory, with [*mutually non-local*]{} massless dyons. Each time we demand a branch point to collide with another, we are using up a degree of freedom. Therefore, by demanding 3 branch points to collide all together, we use up two degree of freedom. This means that the Argyres-Douglas theory occurs in complex codimension 2 loci in moduli space. When the degeneracy of branch point is maximized, we call it the maximal Argyres-Douglas point. For pure SYM case with gauge group of rank $r$, we have $r$ degrees of freedom, and we can bring $r+1$ branch points together to form maximal Argyres-Douglas [*isolated*]{} points in the moduli space[^12]. Argyres-Douglas theory contains mutually non-local massless particles, and they occur in a complex codimension two loci of pure Seiberg-Witten theories. Due to presence of massless electron and magnetic monopole, Lagrangian cannot be written down. Here hyperelliptic form of Seiberg-Witten curve forms a cusp-like singularity (or worse), at complex codimension 2 loci of moduli space of ${\cal N} = 2$ theories. First Look at Seiberg-Witten Curves for $SU(r+1)$ and $Sp(2r)$ \[firstlook\] ============================================================================ Among hyperelliptic curves given in Eqn. , here we will consider a few families only, which are Seiberg-Witten curves for SYM with $SU(r+1)$ and $Sp(2r)$ gauge groups. In the parameter space of the hyperelliptic curves, these will form subspaces with dimension almost halved[^13]. Here we will focus on ‘root structure’, in other words, potential degeneracy of branch points. For both cases, we note that the curve is factorized into two polynomials which never share roots. Branch points will be divided into two mutually-exclusive sets where multiplicity may happen only within each group. Each set of branch points will be assigned with a name and a color, therefore enabling bi-coloring (green and purple) of coming figures in this review. Seiberg-Witten geometry for pure ${\mathcal N}=2$ $SU(r+1)$ theories \[suSec\] ------------------------------------------------------------------------------ The SW curve and SW 1-form for pure $SU(r+1)$ of Ref. are rewritten as $$y^{2}=f_{SU(r+1)} = f_{+} f_{ -} , \quad \lambda_{\mathrm SW} = -dx \log \left( - \frac{1}{2}(f_+ + f_-) - \sqrt{f_+ f_-}\right), \label{surcurve1form}$$ where $f_{\pm}$ are given in terms of $r$ gauge invariant complex-valued moduli $u_i$’s as: $$f_{\pm} \equiv x^{r+1} + \sum_{i=1}^r u_i x^{r-i} \pm \Lambda ^{ r+1}. \label{fpm}$$ Note that we are not allowing the full $2r+1$ degrees of freedom of hyperelliptic curves of Eqn. . Instead we get to vary $r$ (same as the rank of the gauge group) moduli $u_i$’s only while $\Lambda$ is a fixed non-zero constant. In discussing singularity of hyperelliptic curve, we will consider collision among the branch points of $f(x)$. Therefore, it is convenient to give names to the roots of $f_{\pm}$, as $$f_{+} \equiv \prod_{i=0}^{r} (x-P_i), \qquad f_{-} \equiv \prod_{i=0}^{r} (x-N_i). \label{fpmPN}$$ Since $\Lambda\ne 0$, $f_{\pm}$ can never vanish at the same time. Therefore $f_+$ and $f_-$ can never share a root, and there is no vanishing 1-cycle mixing these two sets of roots. More explicitly, the discriminant of the $SU(r+1)$ SW curve factorizes into[@SD] $$\label{sufactordisc} \Delta_x f_{SU(r+1)}=(2\Lambda^{ r+1})^{2r+2} \Delta_x f_{+} \Delta_x f_{-}.$$ In other words, in order for $f_{SU(r+1)}(x)$ to have a degenerate root, $f_+$ or $f_-$ itself should have a degenerate root. This justifies binary color coding in figures for branch points and vanishing 1-cycles. On the $x$-plane, only $P_i$’s (or $N_i$’s) can collide among themselves. At discriminant loci $ \Delta_x f_{SU(r+1)}=0$ and near the corresponding vanishing 1-cycle, the 1-form of is regular $$\lambda_{\mathrm SW} = -dx \log \left( \pm \Lambda^{r+1} \right), \qquad {\mathrm{near}} \ f_\pm = \Delta_x f_{\pm}=0,$$ confirming that the singularity of the SW curve is indeed the singularity of the SW theory, as promised earlier above Fig. \[local\]. SW curve for pure $Sp(2r)$ theories and root structure \[sprootstructure\] -------------------------------------------------------------------------- Now consider a slightly different hyperelliptic curve, $$y^{2}=f_{Sp(2r)}= f_C f_Q , \quad \lambda = a \frac{dx}{2\sqrt{x}} \log \left( \frac{ x f_C + f_Q+ 2\sqrt{x} y }{ x f_C + f_Q -2\sqrt{x} y } \right), \label{sprcurve1form}$$ with $f_C$ and $f_Q$ defined as: $$f_C \equiv x^{r } + \sum_{i=1}^r u_i x^{r-i}, \qquad f_Q \equiv x f_C + 16 \Lambda^{2r+2}, \label{fCfQdef}$$ with $r$ (again same as the rank of the gauge group) gauge invariant complex moduli $u_i$’s. This can be easily obtained by taking no-flavor limit of Ref. . Observe in that $f_C=f_Q=0$ is possible only if $\Lambda=0$. In a quantum theory we demand $\Lambda\neq0$, so $f_C$ and $f_Q$ can never share a root. For any choices of moduli, $f_C$ and $f_Q$ can never vanish at the same time. Just similarly to the $SU(r+1)$ case in Eqn. , the discriminant of the $Sp(2r)$ SW curve also factorizes as[@SD] $$\label{spfactordisc} \Delta_x f_{Sp(2r)}=(16\Lambda^{2r+2})^{2r} \Delta_x f_{C} \Delta_x f_{Q}.$$ Again, in order for $f_{Sp(2r)}(x)$ to have a degenerate root, $f_C$ or $f_Q$ itself should have a degenerate root. We can study multiplicity of zeroes for $f_C$ and $f_Q$ separately without worrying about their roots getting mixed. Again, just as in the $SU$ case, when we draw vanishing cycles and collision of branch points, we can use binary coloring. The branch points and vanishing cycles are all grouped into two mutually exclusive groups (for $C$ and $Q$ respectively.). In order to give new names to two sets of branch points, let us introduce $C_i$’s and $Q_i$’s as given in $$f_C = \prod_{i=1}^{r} (x-C_i), \qquad f_Q= \prod_{i=0}^{r} (x-Q_i).$$ At discriminant loci $ \Delta_x f_{Sp(2r)}=0$, near the corresponding vanishing 1-cycle, the 1-form of becomes infinitesimally small [@DSW], far from becoming a delta function. This confirms that a singularity of the SW curve is indeed a singularity of the SW theory, again confirming the claim near Fig. \[local\]. Electric and Magnetic Charges of Massless Particles \[monodromySec\] ===================================================================== So far, we discussed existence of massless particles in ${\cal N}=2$ theories. In this section, we will discuss electric and magnetic (dyonic) charges of these massless particles. Recall that the massless state was associated with a vanishing 1-cycle of Riemann surface, from Eqn. and Eqn. . Dyonic charges can be read off by decomposing a 1-cycle into symplectic basis 1-cycles, as discussed near Eqn. . In the moduli space, there will be complex codimension 1 loci with a vanishing 1-cycle and it creates nontrivial topology on the moduli space with monodromy determined by the dyonic charge of the vanishing 1-cycle, as in Eqn. . In this section, first we will look at dyon charges of the massless particles for the famous and simpler rank 1 case, and then move to higher rank cases reviewing the results obtained in Ref. , with focus on $SU(r+1)$ and $Sp(2r)$ case. Rank 1 examples: how to read off monodromies of the Seiberg-Witten curves \[rank1\] ------------------------------------------------------------------------------------ For pure SYM, the rank of gauge group $r$ equals to the genus of the SW curve and the number of complex moduli $u_i$’s, as one might recall from SW geometry of $SU(r+1)$ and $Sp(2r)$ gauge theory given in Eqns. and . Here we will warm-up by considering their rank 1 cases, which has one complex modulus $u$ for a genus-1 curve. Therefore the moduli space is a complex plane (real 2-dimensional surface), which we denote as $u$-plane. There exists non-trivial topology on the moduli space, created by existence of singular points on it. By singular points on moduli space of SW curve, we mean the values of moduli which make the SW curve singular (i.e. with vanishing 1-cycles). Recall that it is equivalent to having zero discriminant of hyperelliptic curve. Demanding this single complex condition on the moduli space, we will have a complex codimension-1 loci in the moduli space as a solution set. The modulus $u$ being the only parameter controlling the properties of a genus-one SW curve, vanishing discriminant condition will fix $u$ to possible isolated (separated) values. First we will locate the singular points by discriminant condition, and then consider monodromy properties around each of them, by reading off dyon charges of vanishing 1-cycle, in the spirit of Eqn. . Starting from a generic place in moduli space (a reference point $u_\ast$ on the moduli surface), we make non-contractible loops around each singular point (where discriminant vanishes), and consider monodromy along each path, associated with the singularity surrounded inside. Here we will discuss monodromy of rank $r=1$ cases of $SU(r+1)$ SW curve given in Eqn. and $Sp(2r)$ SW curve given in Eqn. , which we will call $SU(2)$ and $Sp(2)$ curves. Since $SU(r+1)$ and $Sp(2r)$ gauge groups are identical at rank 1, these two distinct curves in fact describe the same physical theories and indeed their monodromy properties match up with each other. Historically both curves were called $SU(2)$ SW curves, but we will call one of them $Sp(2)$ curve, because it has nice generalization for $Sp(2r)$ SW theories. After absorbing some powers of two’s into $\Lambda$ for convenience, $Sp(2)$ curve becomes $$y^{2}=x\left( x(x-u)+\frac{1}{4}\Lambda ^{4}\right), \label{sp2SW}$$ as first given in Ref. . On the other hand, $SU(2)$ curve $$y^{2}=(x^{2}-u)^{2}-\Lambda ^{4}=(x^{2}-{u+\Lambda ^{2}})(x^{2}-{u-\Lambda ^{2}}), \label{su2Lerche}$$ follows from Eqn. , as first proposed by Ref. . It is straightforward to check that discriminant vanishes at $u=\pm \Lambda^2$ for both SW curves for pure $SU(2)=Sp(2)$ theory given by and . Now we will turn to finding out which 1-cycle of SW curve vanishes at $u=\pm \Lambda^2$. ### Review of $SU(2)$ monodromy \[su2monodromy\] As explained in Ref. , $SU(2)$ curve has massless dyon and monopole at two different locations in the moduli space[^14]. The curve in Eqn. has four branch points $$N_{0}= -N_{1} = \sqrt{u+\Lambda^2}, \qquad P_{0}= -P_{1}= \sqrt{u-\Lambda^2}.$$ At a generic value of modulus $u$, they are all distinct. As we vary $u$ toward two special values $u\rightarrow \pm \Lambda^2$, different pairs of branch points will collide: $N_0$ and $N_1$ collide as $u\rightarrow -\Lambda^2$ and $P_0$ and $P_1$ collide as $u\rightarrow \Lambda^2$. Fig. \[rank1su\] denotes vanishing 1-cycles, associated to collision of those branch points. The branch points are drawn on the $x$-plane for varying values of the modulus $u$. From the left, $u$ takes the $u \sim \Lambda^2$, $u\sim 0$, $u\sim - \Lambda^2$ in the three figures drawn here. On the left of Fig. \[rank1su\] ($u \sim \Lambda^2$), two purple branch-points $P_0$ and $P_1$ come close to each other. A 1-cycle drawn in purple denotes the corresponding vanishing 1-cycle, which goes through 2 branch cuts. Half of it is solid line, the other half is dashed line. Recalling that hyperelliptic curves are double-sheet fibration over an $x$-plane, we can take solid lines to be on the upper sheet ($y=\sqrt{f(x)}$) and dashed lines to be on the lower sheet ($y=-\sqrt{f(x)}$). Each time a 1-cycle meets a branch-cut, it has to switch from solid to dash and vice versa. Similarly, on the right of Fig. \[rank1su\] ($u\sim - \Lambda^2$), two green branch-points $N_0$ and $N_1$ come close to each other. A 1-cycle drawn in green denotes the corresponding vanishing 1-cycle, which goes through 2 branch cuts. In the center of Fig. \[rank1su\] ($u\sim 0$), all the branch points are separated, however it retains the topological information about 1-cycles which would vanish at $u = \pm \Lambda^2$. By counting their intersection number, we can read off from the figure that these two 1-cycles have mutual intersection number 2. We are allowed to choose an appropriate symplectic basis, so that we can they can be written as $\beta$ and $\beta-2\alpha$ ($\beta \circ (\beta-2\alpha)=-2$). In physics terms, we can perform electric-magnetic dualities to appoint them as a magnetic monopole and a dyon, in agreement with the usual convention. More details about these dyons are reviewed in Ref. for example. The topological information of Fig. \[rank1su\] is summarized in [Fig. \[su2\]]{}. On the left, moduli space is given. Green and purple paths denote the choice of how to vary the modulus $u$. As $u$ varies on the paths given, the branch points will move following the paths drawn on $x$-plane, as on the right of Fig. \[su2\]. For higher ranks of $SU(r+1)$ SW theory in subsection \[SUdyon\], we will omit figures like Fig. \[rank1su\] (which is a procedure how one obtains the information about vanishing 1-cycles) and only display figures similar to Fig. \[su2\], which contains full topological information of vanishing 1-cycles and trajectories in the moduli space. Note that what trajectory each cycle takes does matter. It is important not only which two branch points are connected, but also through what trajectory they are connected. This should be clear from simple counting: with finite number of branch points ($N$), we can choose a finite number of pairs of branch points ($ \frac{N(N-1)}{2}$), however we have infinite (countable) number of distinct 1-cycles (as points on the $(N-2)$-dimensional lattice). As we will discuss later near Fig. \[abc\] in subsection \[Sp4\], a different topological choice of trajectory on moduli space (on the left of Fig. \[su2\]) translates to different dyon charges on the right. Therefore, we should associate each massless dyon not only with a singular loci on the moduli space, but also with topology of trajectory taken on the moduli space. ### $Sp(2)$ monodromy \[sp1monodromy\] Starting from the $Sp(2)$ curve of Eqn. , shift $x$ by $x\rightarrow x+u$ to obtain $$y^{2}=(x+u)\left( x(x+u)+\frac{1}{4}\Lambda ^{4}\right), \label{sp2SWshift}$$ which identifies with expression given in Ref. . Then perform $x\rightarrow 1/x, y\rightarrow x^{-2} y$ transformation to obtain $$y^{2}=x(1+2 u x)\left( 1 +2 ux+ \Lambda ^{4} x^2 \right), \label{sp2SWshift2}$$ whose four branch points are $$\begin{aligned} O_{\infty}=0, \quad C_1=-\frac{1}{2u}, \quad Q_{0 } &=&-\frac{1}{\Lambda^2}\left(u + \sqrt{-\Lambda^2+u^2}\right), \nonumber \\ Q_{ 1}&=&-\frac{1}{\Lambda^2}\left(u- \sqrt{-\Lambda^2+u^2}\right).\label{sp1fourpoints}\end{aligned}$$ At generic value of $u$, all four branch points are separated, but as we vary $u$ into appropriate values, $Q_0$ and $Q_1$ collide with each other. Fig. \[rank1spnoaxes2\] shows a related animation. In the center figure, all branch-points are separated, and two purple 1-cycles (not vanishing here) are drawn for later convenience with two different thickness. On the left and right figures of Fig. \[rank1spnoaxes2\], $Q_0$ and $Q_1$ collide with each other. These figures preserve topological information about branch cuts, branch points, and 1-cycles. On the left, a thick 1-cycle vanishes, which is the same 1-cycle as in the center figure. On the right figure, a thin 1-cycle vanishes, and it is to be identified with the thin 1-cycle in the center figure. Topological information of Fig. \[rank1spnoaxes2\] is summarized in Fig. \[sp1\] along with the choice of trajectory on the moduli space. On $u$-plane, there are two singularities, and depending on which singularity the path surrounds, $Q_0$ and $Q_1$ collide along different path. In Fig. \[sp1\], it is denoted with two different thickness of lines. Again from the trajectories of branch points on the right of Fig. \[sp1\], we observe that the two 1-cycles have mutual intersection number 2. After an appropriate choice of symplectic basis, they can be identified as $\beta$ and $\beta-2\alpha$, or a magnetic monopole and dyon, matching the result for $SU(2)$ case (the same theory physically) above. ### Moving on to the higher rank case Now we move on to cases with higher and arbitrary rank $r$ of pure SYM with $SU(r+1)$ and $Sp(2r)$ gauge groups. Now we have $r$ complex moduli $u_1, \ldots, u_r$. Therefore the moduli space is a real $2r$-dimensional space. Discriminant loci are various real $(2r-2)$-dimensional loci embedded inside the moduli space. Again we employ the same technique as the rank 1 cases: starting from a generic place in moduli space, we make non-contractible loops around each component of discriminant loci, and read off dyon charges of vanishing 1-cycle associated. To render computation manageable, we take a real 2-dimensional slice of moduli space, on which discriminant loci are isolated points. We choose a generic point on that slice, and consider non-contractible loops, on that slice, around each singular point. Since we will be restricted to a modulus plane, the analysis will resemble that of rank 1 case, with the only difference being that we will have much more singular points on the modulus plane. Later in subsection \[Sp4\], we will consider taking multiple moduli slices, so that we can have multi-dimensional viewpoint on the singularity structure of the SW theory. For now, we consider monodromy of $SU(r+1)$ and $Sp(2r)$ SW theories confined on a certain moduli plane, which is chosen for the ease of computation. ${\cal{N}}=2$ $SU(r+1)$ theory \[SUdyon\] ----------------------------------------- Here we discuss monodromy of the $SU(r+1)$ curve given in Eqn. . Again the moduli space has $r$ complex-valued coordinates $u_i$’s. Instead of considering the full real $2r$-dimensional moduli space, we consider its real 2-dimensional slice, a $u_{r-1}$-plane, after fixing all other $r-1$ moduli into constant values. We fix the first $r-2$ moduli to zero, and we also fix $u_r$ to a constant where its phase is chosen carefully as below $$u_1= \cdots = u_{r-2}=0, \qquad u_r/\Lambda^{r+1} \in i {\mathbb{R}}\label{uslice}$$ On such a hyperplane of moduli space, the discriminant simplifies into $$\Delta_x f_{\pm} = (-1)^{\left[ \frac{r}{2} \right]} r^{r} (u_{r-1})^{r+1} + (-1)^{\left[ \frac{r+1}{2} \right]} (r+1)^{r+1} (u_r \pm \Lambda^{r+1})^r, \label{sudet}$$ at whose vanishing loci the SW curve degenerates. Note the symmetries on the moduli slice: there is $\mathbb{Z}_{r+1}$ symmetry from rotation of the phase on $u_{r-1}$. Also there is a $\mathbb{Z}_2$ symmetry for $\Delta_x f_{\pm}$, associated with complex conjugation of all $u_i$’s and switching between $\Delta_x f_{+}$ and $\Delta_x f_{-}$, due to fixed phase of $u_r$ as decided in Eqn. . In Eqn. , note that $u_r$ is a constant and only $u_{r-1}$ is a variable. Since $u_{r-1}$ has power $r+1$, there are $r+1$ solutions to each of $\Delta_x f_{+}=0$ and $\Delta_x f_{-}=0$. Those solutions have equal magnitude, and the phases are equally distanced among one another. If we recall from Eqn. , that $\Delta_x f \sim \Delta_x f_{+} \Delta_x f_{-}$, it follows $\Delta_x f \sim \Delta_x f_{+} \Delta_x f_{-}=0$ has $2r+2$ solutions on $u_{r-1}$-plane. Therefore we can single out $2r+2$ points, arranged on a circle, on $u_{r-1}$-plane, where discriminant of the SW curve vanishes. This is depicted for rank 9 case in Fig. \[rank9SU10u8theta0\] as eighteen marked points in purple and green. Each singular point on $u_{r-1}$-plane is responsible for vanishing of a 1-cycle and associated monodromy of Seiberg-Witten curve. We pick a reference point at origin of $u_{r-1}$-plane, so that we manifest $\mathbb{Z}_{r+1}$ and $\mathbb{Z}_2$ symmetries on $x$-plane as well. When $u_{r-1}=0$, the branch points are spread on a circle on $x$-plane: $N_i$’s (and $P_i$’s) are even distributed on a circle with $\mathbb{Z}_{r+1}$ symmetry among themselves, and there is $\mathbb{Z}_2$ symmetry between $N_i$’s and $P_i$’s because of $\frac{ u_r}{\Lambda^{r+1} }$ is a pure imaginary number (or zero). Rank 9 case is depicted in [Fig. \[rank9SU10xplane\]]{}: branch points on $x$-plane are drawn as green and purple marked points, at a reference point ($u_{r-1}=0$) on the moduli slice given by Eqn. . As we vary $u_{r-1}$, approaching each of $2r+2$ singular points on $u_{r-1}$-plane, then a corresponding 1-cycle will vanish on $x$-plane, and we read off its dyon charge. On the $u_{r-1}$-plane, starting from the origin as the reference point, we make a non-contractible loop surrounding each singular point on $u_{r-1}$-plane (as shown in Fig. \[rank9SU10u8theta0\]). As we vary $u_{r-1}$, we observe the vanishing cycles on the $x$-plane, as in [Fig. \[rank9SU10xplane\]]{} for rank 9 case: the branch cuts are drawn connecting the branch points of corresponding colors. Also, Fig. \[ardyon\] (we stretched out the $x$-plane, opening up the circle on the $x$-plane into a line) is given for general ranks. Vanishing 1-cycles satisfy $${\color{purple}\nu_{i}^P} \circ {\color{purple} \nu_{i+1}^P}= {\color{forestgreen}\nu_{i}^N} \circ {\color{forestgreen}\nu_{i+1}^N}= -1 , \quad {\color{purple} \nu_{i}^P} \circ {\color{forestgreen}\nu_{i}^N}= 2 , \quad {\color{purple}\nu_{i}^P} \circ {\color{forestgreen}\nu_{i+1}^N}=- 2,$$ while all other intersection numbers vanish. To read off the dyon charges of massless states, we choose a set of symplectic bases. All choice is equivalent to up to electromagnetic duality. With a symplectic basis given in [Fig. \[arcyclesbasis\]]{}, we are choosing $\alpha_i$ cycles to go around each branch cut connecting $P_i$ and $N_i$ branch points. We are choosing $\beta_i$ cycles to connect between $P_0$ and $P_r$ branch points. In the [Fig. \[arcyclesbasis\]]{}, to make it convenient to generalize to arbitrary ranks, we rearranged the branch cuts on the $x$-plane into a line. With such a choice of symplectic bases, we can write down the vanishing 1-cycles as: $$\begin{aligned} {\color{purple}\nu_{0}^{P}}&=&\beta_{1}, \qquad {\color{forestgreen}\nu_{0}^{N}}=\beta_{1}+\sum_{i=1}^{r}\alpha_{i}+\alpha_{1}, \qquad {\color{purple} \nu_{r}^{P}}=-\beta_{r}+\sum_{i=1}^{r-1}\alpha_{i}, \qquad {\color{forestgreen} \nu_{r}^{N}}=-\beta_{r}-2\alpha_{r}, \nonumber \\ {\color{purple}\nu_{i}^{P}} &=&\beta_{i+1}-\beta_{i}-\alpha_{i}, \qquad {\color{forestgreen}\nu_{i}^{N}}=\beta_{i+1}-\beta_{i}+\alpha_{i+1}-2\alpha_{i}, \qquad i=1,\cdots,r-1. \label{sucharges} \end{aligned}$$ We will end this subsection by how this analysis (mostly based on Ref. ) fits in the existing literature. Recall that Seiberg-Witten curves and massless dyon charges were much-studied for low rank cases. Original Seiberg-Witten paper[@SeibergWittenNoMatter; @SeibergWittenWithMatter] studied the curves, massless dyons (monodromies), and some aspects of singularity aspects for $SU(2)$ theory with and without matter. Refs. have discussed six vanishing cycles of the SW curve for pure $SU(3)$ on a slightly different moduli slice from the one chosen here. Vanishing 1-cycles of $SU(n)$ SW curve were also studied in Refs. . There are lots of recent developments in obtaining BPS spectra (including massive ones) as discussed in Refs. for example. Monodromies of pure ${\mathcal N}=2$ $Sp(2r)$ theories \[spSec\] ---------------------------------------------------------------- In the last subsection we discussed pure $SU(r+1)$ SW theories and computed their monodromies. Here we will continue the similar analysis for $Sp(2r)$ SW theories. Seiberg-Witten curves for pure $Sp(2r)$ theory is given by[@SD] $$y^2 = x \left(1+\sum_{i=1}^r u_i x^i \right)\left(1+\sum_{i=1}^r u_i x^i +x^{r+1} \right) \label{spcurve}$$ after appropriate coordinate transformations of no-flavor limit of Ref. and setting $\Lambda \ne 0$ to satisfy $16 \Lambda^{2r+2}=1$ without loss of generality. ![Vanishing cycles of $Sp(12)$ curve at a moduli slice, which is given by a $u_1$-plane of $u_2=\cdots=u_{r-1}=0, u_r=1/9$[]{data-label="sp6flower"}](SpFlowerRank6.pdf){width="\textwidth"} Vanishing cycles are computed in Ref. in a certain moduli slice for $r \le 6$ by plotting in Mathematica, and similar form is conjectured for higher ranks. Here we summarize the result. We choose a moduli hyperplane to be a $u_1$-plane given by fixing $u_2=\cdots=u_{r-1}=0$ and setting $u_r$ to be a fixed small number. Choose the origin $u_1=0$ as a reference point. Up to rank 6, if we choose $u_r$ to be small enough then branch points on the $x$-plane are arranged such that all the $Q_i$’s are surrounding origin $O_\infty$ (a branch point at $x=0$ in Eqn. \[spcurve\]), and all the $C_i$’s are surrounding all the $Q$ points. Vanishing 1-cycles have the following non-zero intersection numbers: $$\begin{aligned} \nu_{i}^Q\circ \nu_{i+1}^Q &=& \nu_{i}^C \circ \nu_{i+1}^C= 1,\quad \nu_{i}^Q\circ \nu_{i}^C= \nu_{i}^C \circ \nu_{i-1}^Q=2, \qquad i = 1, \ldots, r ,\nonumber \\ \nu_{r}^Q\circ \nu_{0}^Q&=& 3, \quad \nu_{0}^Q\circ \nu_{r}^C=\nu_{1}^C \circ \nu_{r}^Q=-2. \label{spIntNum} \end{aligned}$$ Rank 6 case is depicted in [Fig. \[sp6flower\]]{}. Unlike the $SU(r+1)$ case, it is difficult to find an exact method to obtain the vanishing cycles for arbitrary high ranks of $Sp(2r)$. Instead, we compute the vanishing cycles in some patches of the moduli space for low ranks, and read off a pattern to conjecture for general ranks. Ref. conjectures that it is always possible to choose $u_r$ to be small enough such that all the $Q_i$ points are inside the $\nu^C$ cycles, such that holds, for any rank $r$. We write down these $2r+1$ vanishing cycles $$\begin{aligned} \nu_{i}^{C}=-\beta_{i}+\beta_{i+1}+\alpha_{i+1} , \quad \nu_{i}^{Q}= \nu_{i}^{C} -\alpha_{i}+ \alpha_{i+1}, \quad i=1,\cdots,r-1 , \nonumber \\ \nu_{r}^{C}=\beta_{1} -\beta_{r} -\sum_{i=2}^{r}\alpha_{i}, \quad \nu_{0}^{Q}=\beta_{1} + 2\alpha_1, \quad \nu_{r}^{Q}= \nu_{r}^{C}+ \beta_1 + \alpha_1 -\alpha_{r}. \label{spcharges} \end{aligned}$$ in terms of symplectic basis given in [Fig. \[sprcyclesbasis\]]{}. We are choosing $\alpha_i$ cycles to go around each branch cut connecting $Q_i$ and $C_i$ branch points. We are choosing $\beta_i$ cycles to connect between $Q_0$ and $Q_r$ branch points. Argyres-Douglas Loci: Massless Electron & Massless Magnetic Monopole \[ArDo\] ============================================================================= In previous section we studied vanishing 1-cycles at a complex codimension 1 loci of the moduli space. Demanding one 1-cycle to vanish takes away 1 complex degree of freedom, and therefore such loci have one dimension less than the full moduli space. In this section, we consider singularity loci of moduli space with codimension 2 (or more), where we demand 2 (or more) 1-cycles to vanish. This section will discuss degeneration of Seiberg-Witten curves so that mutually intersecting 1-cycles vanish at the same time. Such geometry (reviewed in subsection \[ReviewN2\]) corresponds to Argyres-Douglas theories, containing massless dyons which are mutually non-local (they cannot turn into pure electric particles by any electromagnetic dualities) as studied in Ref. . In this review, our first encounter with Argyres-Douglas theory will be in the context of $Sp(4)$ SW theory. Now we take a closer look at various singularity loci with complex-codimensions 1 and 2 inside the moduli space. Singularity structure of $Sp(4)=C_2$: detailed look on BPS spectra \[Sp4\] -------------------------------------------------------------------------- Similarly to the previous section, we compute the dyon charges of vanishing 1-cycles for a SW curve, for $Sp(4)$ gauge group. Instead of confining ourselves on a moduli slice, we will consider a set of moduli slices. We will observe how vanishing cycles change as we change the choice of hypersurface. Since the gauge group has rank 2, $Sp(4)$ SW theory has a 2 complex (4 real) dimensional moduli space whose coordinates are two complex moduli $u\equiv u_1, v\equiv u_2$. Instead of full 4 real dimensional moduli space, we will take a $3$ real dimensional subspace by fixing the phase of $v$ such that $v^3$ is real, as in the left side of [Fig. \[moduli3dcolor\]]{}. This choice was made so that the moduli subspace contains all interesting singular points inside, where multiple 1-cycles vanish at the same time[^15]. ![A subspace inside the moduli space of pure $Sp(4)$ Seiberg-Witten theory. Vanishing discriminant loci $\Sigma_2^Q$ and $\Sigma_{2^\prime}^Q$ are drawn with two different types (big and small) of dashed lines. They get interchanged at a cusp point (red).[]{data-label="moduli3dcolor"}](moduli3dcolor.pdf){width="\textwidth"} As we vary magnitude of $v$, we consider singularity structure on each $u$-plane: we locate where the curve degenerates, and we compute the corresponding dyon charge for vanishing 1-cycle. On the right of [Fig. \[moduli3dcolor\]]{}, each of the five $u$-planes, marked with (a) to (e), is a slice of the moduli space at different magnitude of $v$. The purple and green curves running vertically are denoted by $\Sigma^{C}_i$’s and $\Sigma^{Q}_i$’s. These are singularity loci in the moduli space with complex codimension 1. This is where the corresponding 1-cycle vanishes (i.e. a dyon becomes massless), and it is captured by the vanishing discriminant of the curve (one complex condition). Interesting things happen when discriminant loci $\Sigma$’s intersect inside moduli space, at complex codimension 2 loci, forming a worse singularity. There, two 1-cycles vanish at the same time. In other words, massless dyons coexist. For example, at a blue point labelled ‘node’ in the right of [Fig. \[moduli3dcolor\]]{}, $\Sigma^{Q}_0$ and $\Sigma^{C}_2$ intersect, and that is where 1-cycles denoted by $\nu^{Q}_0$ and $\nu^{C}_2$ degenerate at the same time. When two 1-cycles vanish at the same time, the SW curve (embedded in the ambient space whose coordinates are $x,y$) degenerates into either cusp or node form. The shape of intersection loci of $\Sigma$’s also takes cusp or node form respectively inside the moduli space. Each leads to different kind of singularity, mutually non-local and local massless dyons. This is not tied to $Sp(4)$ gauge group, and similar phenomena occur for pure $SU(3)$ theory, too[@KLYTsimpleADE]. When $\Sigma$’s (discriminant loci) intersect each other, it is seen as colliding of singular points on the corresponding $u$-plane, the moduli slice. For example, on the three $u$-planes marked by (a), (c), and (e), there are five singular points where $\Sigma$’s pierce through. On the two $u$-planes marked by (b) and (d), two of the singular points are on top of each other, so we see only 4 separate points on $u$-plane. Recall from subsection \[GeomReview\] that collision of branch points on the $x$-plane was captured by vanishing of discriminant operator with respect to $x$, $\Delta_x$ acting on $f$. Similarly, when the singular points collide on the $u$-plane, it is captured by another discriminant operator with respect to $u$, $\Delta_u$ acting on $\Delta_x f$. In other words demanding vanishing discriminant $ \Delta_x f =0 $ and double discriminant $ \Delta_u \Delta_x f =0 $ bring out all the candidates for having two vanishing 1-cycles. However among this, only those which satisfy $ d \Delta_x f =0 $ truly qualifies as a singularity loci of discriminant loci of hyperelliptic curve. The points where $ \Delta_u \Delta_x f =\Delta_x f =0 $ and $ d \Delta_x f \ne 0 $ is where the discriminant loci is smooth but it appears singular on a particular slice of moduli space. We will elaborate more near [Fig. \[figure8\]]{} later. In the case of $Sp(4)$ SW theory, vanishing double discriminant gives one constraint as below: $$\begin{aligned} \Delta_u \Delta_x f_{Sp(4)}&=& 2^8 v (v^3-3^3)^3 (2^4 v^3- 3^6)^2 , \label{doublediscsp4} \end{aligned}$$ whose roots $$v=\Bigg\{ 0, 3 \alpha_3^i, \frac{9}{2 \sqrt[3]{2}} \alpha_3^j \Bigg\}$$ are marked by seven dots in the left of the [Fig. \[moduli3dcolor\]]{}. One of the solutions of Eqn. is $v=0$, but it does not translate into having two massless dyons. The other six points correspond to having two massless BPS dyons. In subsection \[exteriordVSdd\] we will discuss how $d\Delta_x f=\Delta_x f=0$ is equivalent to having two massless BPS dyons. $v=0$ does not satisfy that relation, but the rest 6 does (with proper choice of $u$ value). Note the degeneracy of roots to the double discriminant of Eqn. . The roots $3 \alpha_3^i$ and $ \frac{9}{2 \sqrt[3]{2}} \alpha_3^j $ have degeneracy 3 and 2 respectively. In the next section we will review that this is a universal criterion for having mutually non-local and local massless dyons. Actually vanishing of double discriminant is a necessary condition for having multiple massless dyons, but not a sufficient condition as we will see in subsection \[exteriordVSdd\]. For each of $u$-planes marked by (a) to (e) of [Fig. \[moduli3dcolor\]]{}, we have drawn a corresponding $x$-plane in [Fig. \[c2\_10pic\_down\]]{} displaying the vanishing cycles on the $x$-plane, for each moduli slice. Let us have a closer look staring from the top slice marked as (a). ![Singularity structures at slices of the moduli space for pure $Sp(4)$ Seiberg-Witten theory.[]{data-label="c2_10pic_down"}](c2_10pic_down.pdf){height="19.3cm"} (a) : choose a $u$-plane where $|v| > \frac{9}{2 \sqrt[3]{2}}$, then vanishing discriminant loci of the SW curve (marked as $\Sigma^{C}_{i}$’s and $\Sigma^{Q}_{i}$’s in [Fig. \[moduli3dcolor\]]{}) intersect the $u$-plane at five separate points $u^{C}_{i}$’s and $u^{Q}_{i}$’s. Each of the 5 singularity points on the $u$-plane ($u^{C}_{i}$’s and $u^{Q}_{i}$’s) is responsible for one vanishing 1-cycle ($\nu^{C}_{i}$’s and $\nu^{Q}_{i}$’s with corresponding choice of sub-/super- scripts). By observing the relative trajectory of branch points on the $x$-plane (with help of plotting in Mathematica), we can read off the 5 vanishing cycles as below: $$\begin{aligned} \nu_{0}^{Q}&=&\beta_{1} , \quad \nu_{1}^{Q}=-\beta_1 + \beta_{2} +\alpha_{1} +4 \alpha_2 , \quad \nu_{2}^{Q}=-\beta_1-\beta_{2}-\alpha_{1}-2\alpha_{2}, \nonumber \\ \nu_{1}^{C}&=&-\beta_1 + \beta_{2} +2\alpha_{1} +3 \alpha_2 , \quad \nu_{2}^{C}=-\beta_1 - \beta_{2} - \alpha_2 . \label{cycle} \end{aligned}$$ (b) : As we change the moduli $|v|$ so that $|v| = \frac{9}{2 \sqrt[3]{2}}$, singularity points on the $u$-plane, $u^Q_0$ and $u^C_2$, now collide. Corresponding 1-cycles $\nu^Q_0$ and $\nu^C_2$ vanish simultaneously at the point given by $u=u^Q_0 = u^C_2$ and $|v| = \frac{9}{2 \sqrt[3]{2}}$. However these two cycles are mutually local as seen from Eqn. or Fig. \[c2\_10pic\_down\]: responsible four branch points on the $x$-plane collide pairwise ($C_1 \leftrightarrow C_2$, $Q_0 \leftrightarrow Q_1$). The SW curve degenerates into a node form, $y^2 \sim (x-C)^2 (x-Q)^2 \times \cdots$. Singularity loci (of vanishing $\Delta_x f $) $\Sigma_{0}^{Q}$ and $\Sigma_{2}^{C}$ intersect at $|v|=\frac{9}{2\sqrt[3]{3}}$, with node-like crossing. Note that the node-like singularity appears both in the ambient space and the moduli space of SW curve. (c) : We can change the moduli $|v|$ further to enter the range $3 < |v| < \frac{9}{2 \sqrt[3]{2}}$. Unlike (b), $u^Q_0$ and $u^C_2$ are separated again restoring back to (a). Dyon charges of the vanishing cycles for (a), (b), and (c) stay the same as given by Eqn. . (d) : As we change the magnitude of the moduli $v$, to satisfy $ |v|=3$, discriminant loci $\Sigma _{0}^{Q}$ and $\Sigma _{2}^{Q}$ intersect on the $u$-plane, forming a cusp-like singularity inside the moduli space. Two singular points $u_{0}^{Q}$ and $u_{2}^{Q}$ collide on the $u$-plane, and the vanishing cycles $\nu_{0}^{Q}$ and $\nu_{2}^{Q}$ [*merge*]{} into something which is no longer a 1-cycle. Now it is not possible to tell apart vanishing 1-cycles[^16]. Now three branch points $Q_{0}, Q_1,$ and $Q_2$ collide at the same time on the $x$-plane. Two cycles $\nu_{0}^{Q}$ and $\nu_{2}^{Q}$ become massless at the same time, but they are mutually [*non-local*]{} as clear from Eqn. . The SW curve degenerates into a cusp form $y^{2}\sim (x-a)^{3}\times \cdots $, giving Argyres-Douglas theory, with two mutually non-local massless BPS dyons. Note that the cusp-like singularity appears both in the ambient space and the moduli space of SW curve. (e) : After passing Argyres-Douglas loci of (d), we again have 5 separated singular points on the $u$-plane. Especially $u_{0}^{Q}$ and $u_{2}^{C}$ are separated. However note that the BPS dyon charges of vanishing cycles changed from the cases of (a), (b), and (c) given in Eqn. as we go through cusp-like (or Argyres-Douglas) singularity of (d). Instead of $\nu^Q_2$, we have a new vanishing cycle $$\nu_{2^\prime}^{Q}=-\beta_{2}-\alpha_{1}-2\alpha_{2} = \nu_{0}^{Q}+\nu_{2}^{Q}. \label{chargeJump}$$ In observing the right side of Fig. \[c2\_10pic\_down\], we realize that the set of vanishing 1-cycles have changed in going from (c) to (e), through (d). In order to check whether this “jump” is something intrinsic - geometrically and physically - to this system, we can conduct a few tests. One quick test we perform here is to choose alternative paths on moduli slice and check how it may affect vanishing 1-cycles. Instead of the paths on the left of (e) in Fig. \[c2\_10pic\_down\], we can change a new set as given in Fig. \[eprime\]. ![Alternative choice for trajectory in moduli slice[]{data-label="eprime"}](c2_10pic_detail.pdf){width=".5\textwidth"} The new path still surrounds the same singularity, however its relative location with respect to other singularities changed. Two trajectories in the moduli space $\gamma^Q_{2^\prime}$ in Fig. \[c2\_10pic\_down\] and $\gamma^Q_{2^\prime e}$ in Fig. \[eprime\] both surround only $u^Q_{2^\prime}$, a singular point, on the moduli slice. In other words, they form non-contractible loops surrounding a singular locus $\Sigma^Q_{2^\prime}$ in Fig. \[moduli3dcolor\] and nothing else. However the key difference is their relative location with respect to two other singularities $\Sigma^Q_{0}$ and $ \Sigma^C_{2}$ in Fig. \[moduli3dcolor\] (or $\gamma^Q_{0}$ and $ \gamma^C_{2}$ in Fig. \[c2\_10pic\_down\] and Fig. \[eprime\]. In order to answer how this change may affect the assignment of dyon charges of vanishing 1-cycles, let us consider following cartoon in Fig. \[abc\]. On the same moduli slice, we have a few different trajectories drawn, surrounding singular points of vanishing discriminant of the curve. We have drawn two other bunches of singular points in the top and the bottom, to denote other singularities whose spatial relationship with all the trajectories depicted remain unchanged. Both $\gamma_b$ and $\gamma_c$ surround a singular point $u_b$ only. We can associate a monodromy matrix for each closed loop in the moduli space, to denote how the 1-cycles get transformed among themselves, in the spirit of Picard-Lefshetz formula given in Eqn. . Trajectories $\gamma_b$ and $\gamma_a$ can be combined, in that order, to give a trajectory $\gamma_d$, and similarly trajectories $\gamma_a$ and $\gamma_c$ can be combined to give a trajectory $\gamma_d$. Therefore we have the following relations among the monodromy matrices (The matrix on the right will be operated first.) $$M_a M_b = M_c M_a = M_d. \label{Mmulti}$$ For a SW curve of genus 1, the monodromy matrix is given in Ref. as $$M^{(g,q)}=\left( \begin{tabular}{ cc } $1+qg$ & $q^2$ \\$-g^2$ & $ 1-gq $ \end{tabular} \right) \label{monomat}$$ for a closed loop in moduli space, which surrounds a singularity associated with vanishing 1-cycle $g \beta + q \alpha$ (or a massless dyon of charge $(g,q)$). If dyon has vanishing charge or if dyon is absent, then the monodromy matrix reduces to the identity. For higher genus case which we are considering here, we can arrange the dyon charges to contain only $\alpha_1, \beta_1$ with vanishing contribution from other $\alpha_i$’s and $\beta_i$’s, by symplectic transformation (or electromagnetic dualities). In other words, the dyon charge will look like $$\begin{aligned} (\vec{g},\vec{q})& = &(g_1,g_2, \ldots, g_r; q_1,q_2, \ldots, q_r) \nonumber \\ &=& (g_1,0, \ldots, 0; q_1,0, \ldots, 0) = (g,0, \ldots, 0; q,0, \ldots, 0) = g\beta_1 + q \alpha_1\end{aligned}$$ This is equivalent to saying that we can arrange the monodromy matrices $M_a$ and $M_b$ of Fig. \[abc\] to take the following form $$M^{(g,0, \ldots, 0; q,0, \ldots, 0)}=\left( \begin{tabular}{ cc |c } $1+qg$ & $q^2$ & $0$ \\$-g^2$ & $ 1-gq $ & $0$ \\ \hline $0$ & $0$& $0$ \end{tabular} \right), \label{monomatg}$$ which contains Eqn. as a left-top $2 \times 2$ block. All the $0$’s in Eqn. are to be understood as submatrices of appropriate sizes. In the remaining part, we will omit the $0$’s and write down only the first $2 \times 2$ submatrix as the monodromy matrices. Again by symplectic transformation with no loss of generality, we can assign the dyon charges of $\gamma_a$ and $\gamma_b$ to be $(0,a)$ and $(b,0)$, with intersection number $$(0,a) \circ (b,0) = a \alpha \circ b \beta = -a b .$$ Please note we assign the dyon charges to the trajectories $\gamma$’s but not to the singular points $u$’s themselves. From Eqn. and Eqn. , we have monodromy matrices as given below: $$\begin{aligned} M_a &=&M^{ (0,a)}= \left( \begin{tabular}{ cc } $1$ & $a^2$ \\$0$ & $ 1$ \end{tabular} \right)= \left( \begin{tabular}{ cc } $1$ & $-a^2$ \\$0$ & $ 1$ \end{tabular} \right)^{-1}, \nonumber \\ M_b &=& M^{ (b,0)} = \left( \begin{tabular}{ cc } $1$ & $0$ \\$-b^2$ & $ 1$ \end{tabular} \right) =\left( \begin{tabular}{ cc } $1$ & $0$ \\$b^2$ & $ 1$ \end{tabular} \right)^{-1}, \nonumber \\ M_c & = & M_a M_b M_a^{-1} =\left( \begin{tabular}{ cc } $1-a^2 b^2$ & $a^4 b^2$ \\$-b^2 $ & $ 1+a^2 b^2$ \end{tabular} \right) = M^{ \pm (b,- a^2 b)}. \label{Mcshift}\end{aligned}$$ Note that there is an ambiguity for the overall sign of dyon charge associated to $M_c$, because Eqn. is invariant under $(g,q) \rightarrow (-g,-q)$. For fun, we can also examine $M_d$ and attempt (and fail) to interpret it in terms of dyon charges. From Eqn. , monodromy matrix for $\gamma_d$ is given as $$M_d = M_a M_b = \left( \begin{tabular}{ cc } $1-a^2 b^2$ & $a^2$ \\$-b^2$ & $ 1$ \end{tabular} \right) \label{Md}$$ whose trace matches that of Eqn. only for $ab=0$. Therefore $M_d$ cannot be interpreted as a singularity of a single vanishing 1-cycle for $ab \ne 0$. We can interpret the result of Eqn. and Fig. \[abc\] as following: 1. The dyon charge of a singular point $u_b$ depends on choice of trajectory in the moduli slice. 2. If the trajectory passes through another singular point $u_a$ (but not surrounding it by a closed loop), then dyon charge of $u_b$ gets shifted by multiple of dyon charge of $u_a$. 3. If we chose the first sign in Eqn. , then this relation becomes $$(b,0) \rightarrow (b,0) - \left[ (b,0) \circ (0,a) \right] (0,a) = (b, - a^2 b), \label{abrel}$$ with a close agreement with Eqn. . We can now write down general formula for how dyon charge gets changed. In Fig. \[abc\], for each trajectories $\gamma_{a}, \gamma_{b}, \gamma_{c}$, let us say vanishing 1-cycles are $\nu_{a}, \nu_b, \nu_{c}$. Then their relationship is given as similar to Eqn. as $$\nu_c = \nu_b - ( \nu_b \circ \nu_a ) \nu_a. \label{abcmono}$$ Going back to the question of spectra jump for $Sp(4)$ related to Fig. \[eprime\], now we can use the techniques we learned above, especially that of Eqn. . First, by inspecting Fig. \[eprime\] and Fig. \[abc\], we can plug in $\nu_a = \nu_0^Q$, $\nu_b= \nu_{2^\prime}^Q$, $\nu_c = \nu_{2^\prime e}^Q$ into Eqn. to obtain $$\nu_{2^\prime e}^Q =\nu_{2^\prime}^Q - ( \nu_{2^\prime}^Q \circ \nu_0^Q ) \nu_0^Q = \nu_{2^\prime}^Q - \nu_0^Q = \nu_{2}^{Q}, \label{22mono}$$ where the last equality was given from Eqn. . If somehow we can argue that $\gamma^Q_{2^\prime e }$ of Fig. \[eprime\] is more natural choice of trajectory in moduli space than $\gamma^Q_{2^\prime }$ of Fig. \[c2\_10pic\_down\], then we can undo the spectra jump as we move from (c) to (e) in Fig. \[moduli3dcolor\]. It may suggest that we can avoid the spectra jump (in dyon charges of massless states) if we choose a different path on moduli slice. However for the moment we cannot say conclusively what will be the most natural and consistent choice of trajectory paths. It will be interesting to check this by focussing on massless sector of wall-crossing formulas of BPS spectra given in Refs. . Maximal Argyres-Douglas theories and dual Coxeter number \[maxADh\] ------------------------------------------------------------------- So far we considered Argyres-Douglas theories with two massless dyons. Here, we will consider [*maximal*]{} Argyres-Douglas theories, where maximal number of dyons become massless and are mutually non-local. This is achieved by bringing the maximal number of branch points on $x$-plane, so that $f(x)$ will have a root with maximal degeneracy. Let us recall the SW curve for $SU(r+1)$ gauge group given Eqn. and Eqn. . $$y^{2}=f_{SU(r+1)} = f_{+} f_{ -} , \quad f_{\pm} \equiv x^{r+1} + \sum_{i=1}^r u_i x^{r-i} \pm \Lambda ^{ r+1}.$$ If we have $u_r = \mp \Lambda^{r+1}$ while all other $u_i$’s vanish, then $f_{\pm} =x^{r+1}$ holds and $f(x)$ has a root $x=0$ with maximal degeneracy[@ArgyresDouglas; @LercheReview]. It is straightforward to find 2 maximal Argyres-Douglas points in moduli space of pure $SU(r+1)$ and $SO(2r)$ SW theories. Taking no-flavor limit of Ref. , we have SW curve for pure $SO(2r)$ as $$y^2 = C_{SO(2r)}^2 - \Lambda^{2(2r-2 )} x^4 =C_{SO(2r),+} C_{SO(2r),-}$$ with $$\begin{aligned} C_{SO(2r),\pm} & =& C_{SO(2r) } \pm \Lambda^{ (2r-2 )} x^2, \nonumber \\ C_{SO(2r)}(x) & \equiv & x^{2r} + s_2 x^{2r-2} +\cdots + s_{2r-2} x^2 +\tilde{s}_r^2, \end{aligned}$$ in agreement with Ref. . Some of the monodromy properties for pure $SO(2r)$ were studied in Ref. with emphasis on the $SO(8)$ example. Maximal Argyres-Douglas points for the $SO(2r)$ will be two moduli points given by [@EHIY] $$s_{2r-2} = \Lambda^{ \pm (2r-2 )}, \quad s_{2i}=0, ~~i\ne r-1 \label{SOevenMaxAD}$$ which makes $C_{SO(2r),\mp} = x^{2r}$ to have a root with maximal order of vanishing. Just as in the $SU(r+1)$ case, this computation is straightforward, partially thanks to $\mathbb{Z}_2$ symmetric structure between $C_{SO(2r),\pm}$ and between $f_\pm$ in the SW curve, which is lacking in the $Sp(2r)$ and $SO(2r+1)$ cases. Scaling behavior at maximal Argyres-Douglas points for $SU(r+1) $ and $SO(2r)$ SW theory was studied in Ref. recently, with focus on the shape of moduli space in the neighborhood of those theories. More specifically, the quantum Higgs branch appears[^17]. Locating maximal Argyres-Douglas (AD) points for $Sp(2r)$ and $SO(2r+1)$ SYM involves more algebra, and the number of maximal Argyres-Douglas points equals to the dual Coxeter number of the gauge group. For pure $Sp(2r)$ theory, Ref. proposes $r+1$ candidates for maximal Argyres-Douglas theories, while Ref. proposes $2r-1$ candidates for maximal Argyres-Douglas points of pure $SO(2r+1)$ theory. Recall, from Eqn. and Eqn. , the SW curve for $Sp(2r)$ SYM is $$y^{2}=f_{Sp(2r)}= f_C f_Q , \quad f_C = x^r + \sum_{i=1}^r u_i x^{r-i} , \quad f_Q \equiv x f_C + 16 \Lambda^{2r+2}.$$ Maximal AD points occur when we bring all roots of $f_Q=0$ together. This happens at $r+1$ points in moduli space, where the curve develops $A_{r}$ singularity[@DSW]. This occurs when the moduli take the following values: $$\begin{aligned} u_i=& \left( \begin{array} [c]{c}% r+1\\ i \end{array} \right) (-Q)^{i}, \end{aligned}$$ with $Q$ given by $$Q=- \exp\left( \frac{2\pi i}{r+1}k\right) \left( 16\right) ^\frac{1}{r+1} \Lambda^{2}, \qquad k\in \mathbb{Z}. \label{Qvalue}$$ This forces the $f_Q$ to have a root with maximal degeneracy as $$f_Q=(x-Q)^{r+1} . \label{fQQ}$$ It also follows that $C_i$’s, the roots of $f_C$ are given as below: $$\{ C_i \}= \left\{ Q\left( 1-\exp\left( \frac{2\pi i}{r+1}k\right) \right) \right\} , \qquad k \in \mathbb{Z}, \qquad k \notin (r+1) \mathbb{Z}. \label{Cvalues}$$ Given that Eqn. allowed for $r+1$ different choices of phase for $Q$, we have $\mathbb{Z}_{r+1}$ symmetric $r+1$ points on the moduli space, where maximal Argyres-Douglas singularity occurs. Similarly, $2r-1$ maximal Argyres-Douglas points of $SO(2r+1)$ SYM were located in Ref. . The curves for $SO(2r+1)$ and $SO(2r)$ SYM are almost similar, but $SO(2r+1)$ case is much harder to solve for the maximal Argyres-Douglas points. We again take no-flavor limit of the curve of Ref. to obtain the curve for pure SW theory[@KLYTsimpleADE; @SO5DS], $$y^2 = f_{SO(2r+1) }(x)=C_{SO(2r+1) }^2 - \Lambda^{2(2r-1 )} x^2 = C_{SO(2r+1) ,+} C_{SO(2r+1), -} \label{SOoddcurve}$$ with $$\begin{aligned} C_{SO(2r+1) ,\pm} &=& C_{SO(2r+1) } \pm \Lambda^{ (2r-1 )} x ,\nonumber \\ C_{SO(2r+1) }(x) &\equiv& x^{2r} + s_2 x^{2r-2} +\cdots + s_{2r-2} x^2 + {s}_{2r}.\end{aligned}$$ Two polynomials $C_{SO(2r+1) ,\pm}$ will share a root if $x=0$. This, however, does not give much mileage for bringing maximal number of branch points together for $f_{SO(2r+1) }(x)$. This root does not have high enough order of vanishing. Instead we will work on having $C_{SO(2r+1), +}$ (or equivalently $C_{SO(2r+1) , -}$) to have a root with maximal order of vanishing. Since we have only $r$ degrees of freedom, we can only bring together $r+1$ branch points, and the maximal order of vanishing is $r+1$. The best we can do is to bring $C_{SO(2r+1), +}$ into the following form:[@SD] $$C_{SO(2r+1), +} = (x+b)^{r+1} (x^{r-1} + u_1 x^{r-2} + u_2 x^{r-3} + \cdots + u_{r-2} x + u_{r-1} ) \label{CmaxCusp}$$ when the moduli $s_{2i}$’s satisfy $$s_{2k} = (-b^2)^k \frac{(2r-1)}{(2r-2k-1)} \frac{r!}{k! (r-k)!}, \label{svalueformAD}$$ with $b$ given by $$b = \omega_{2r-1}^k \left[ (-1)^{r+1} \frac{(2r-3)!!}{(2r)!!} \right]^{1/(2r-1)}\Lambda , \quad k\in \mathbb{Z}. \label{bvalueformAD}$$ Here $\omega_m$ is $m$’th root of unity. Eqn. allows $2r-1$ possible values of $b$. In turn there are $2r-1$ solutions to Eqn. . Therefore we have $2r-1$ isolated points, which are $\mathbb{Z}_{2r-1}$ symmetric among themselves, in the moduli space of pure $SO(2r+1)$ SYM where maximal Argyres-Douglas theory occurs. Déjà Vu: Singularity Tools: Exterior Derivative & Double Discriminant \[doublediscsection\] =========================================================================================== So far we encountered a few tools to detect singularity of the SW curves. In subsection \[GeomReview\], we discussed various tools for singularity search, namely exterior derivative $d$ in ambient space, and discriminant. In Eqn. , we saw that considering double discriminant $\Delta_u \Delta_x f(x)$ gives candidates for interesting singularities with 2 massless dyons. Now let us write down general rules in a systematic way. We detected singularity of a Riemann surface embedded in the ambient space whose coordinates are $x, y$ by taking an exterior derivative, or by demanding all the partial derivatives with respect to $x$ and $y$ to vanish. Similarly, when an algebraic variety in moduli space forms a singularity, it is captured by demanding the exterior derivative to vanish inside the moduli space. Recall that the vanishing discriminant loci of the SW curve form a complex codimension 1 algebraic variety inside the moduli space, which is given by $\Delta_x f =0$. Therefore it follows that $d \Delta_x f =0$ (where $d$ is taken inside the moduli space) will pinpoint us to where $\Delta_x f =0$ forms a singularity, namely where discriminant loci intersect themselves. That is where we have multiple vanishing 1-cycles (massless BPS dyons). As explained in subsection \[GeomReview\], the exterior derivative $d$ can be written in terms of the partial derivatives with respect to all the coordinates of the ambient space. Therefore the exterior derivative inside the moduli space is given as $d= \sum_{i=1}^{r} du_i \frac{\partial }{\partial u_i} $. One might think that demanding the $d =0$ actually reduces $r$ degrees of freedom, since we demand all the $r$ partial derivatives to vanish. However, $ \Delta_x f=d \Delta_x f=0$ indeed contains codimension 2 solutions (instead of codimension $r+1$). When the $ \Delta_x f=0$ loci become singular (where $ \Delta_x f=d \Delta_x f=0$ holds), double discriminant also vanishes ($\Delta_u \Delta_x f(x)=0$). In other words, singularity is seen from the viewpoint of moduli slices (parallel $u$-planes) as well. However the converse does not hold. Something might appear singular on certain moduli slices but it can be smooth in the full ambient space (moduli space). In other words, $ \Delta_x f= \Delta_u \Delta_x f=0$ is a necessary but not sufficient condition for having $ \Delta_x f=d \Delta_x f=0$. Exterior derivative detects coexistence of multiple massless BPS dyons \[exteriordVSdd\] ---------------------------------------------------------------------------------------- The vanishing discriminant condition of the SW curve $\Delta_x f=0$ defines an algebraic variety $\Sigma$. Since $\Delta_x f$ is written only in terms of moduli $u_i$’s (without $x$ and $y$), $\Sigma$ is an algebraic variety embedded inside the moduli space, denoting moduli loci of massless BPS states. When this algebraic variety $\Sigma$ self-intersects, two or more BPS states become massless, which occurs when we demand $\Delta_x f=d \Delta_x f =0$. An heuristic example is depicted in [Fig. \[figure8\]]{}. Inside a moduli subspace, we draw a figure-eight-like object, which is analogous to $\Delta_x f=0$ loci as in [Fig. \[moduli3dcolor\]]{}. We mark various $u$-planes with (a) to (e). The number of singular points changed on each $u$-planes. When the singular points collide on the $u$-plane we have $\Delta_u\Delta_x f=0$, for example on slices (b) and (d). Slice (b) is true singularity, while (d) is not. Thus we see that $\Delta_x f=\Delta_u \Delta_x f =0$ is a necessary but not sufficient condition to have $\Delta_x f=d \Delta_x f =0$. Fig. \[moduli3dcolor\] and Fig. \[figure8\] look very similar to each other in that it displays the singularity structure inside moduli space. Both shows the collision of discriminant loci and formation of higher singularity - (b) and (d) of Fig. \[moduli3dcolor\] and (b) of Fig. \[figure8\]. The main difference is this: Fig. \[figure8\] contains an example where $\Delta_x f=\Delta_u \Delta_x f =0$ holds but $d \Delta_x f \ne 0$ on its part (d). Factorization of double discriminant, and order of vanishing \[FactorDD\] ------------------------------------------------------------------------- Massless dyons coexist at complex-codimension-2 loci where both discriminant and its exterior derivative vanish. There the curve looks like either of following two:[@SD] - The curve $y^2=(x-a)^3 \times \cdots $ has a cusp-like singularity. Vanishing discriminant $\Delta_x f=0$ locus also intersects with [cusp]{}-like singularity in moduli space. There $\Delta_u \Delta_x f=0$ also holds, with order of vanishing 3. Two massless dyons are mutually non-local. - The curve $y^2=(x-a)^2 (x-b)^2 \times \cdots $ has a node-like singularity. Vanishing discriminant $\Delta_x f=0$ locus also intersects with [node]{}-like singularity. There $\Delta_u \Delta_x f=0$ also holds, with order of vanishing 2. Two massless dyons are mutually local. Order of vanishing of each root of $\Delta_u \Delta_x f$ tells us the type of singularities. In order to justify that, we discuss roots of $\Delta_u \Delta_x f=0$. When $ \Delta_x f=0$ and $d \Delta_x f=0$ hold, each root of vanishing double discriminant $\Delta_u \Delta_x f=0$ corresponds to two massless dyons with appropriate combinatoric meaning. Double discriminants of $SU(r+1)$ and $Sp(2r)$ factorize as:[@SD] $$\begin{aligned} \Delta_u \Delta_x f_{SU(r+1)}= \Delta_u \Delta_x (f_+ f_-) &=&\# \left( v^{(2r+2)^2} + \cdots \right) \nonumber\\ &=& \# \left( v^{(r+1)^2} + \cdots \right)^2 \left( v^{r+1} + \cdots \right)^3 \left( v^{r+1} + \cdots \right)^3 \nonumber\\ & &\times \left( v^{(r+1)(r-2)/2} + \cdots \right)^2 \left( v^{(r+1)(r-2)/2} + \cdots \right)^2 \nonumber\\ & \equiv & \# ({PN_{II}})^2 ({N_{III}})^3 ({P_{III}})^3 ({N_{II}})^2 ({P_{II}})^2, \label{DDsu} \end{aligned}$$ $$\begin{aligned} \Delta_u \Delta_x f_{Sp(2r)} = \Delta_u \Delta_x(f_Q f_C) &=&\# \left( v^{(2r+1)^2} + \cdots \right) \nonumber\\ &=& \# \left( v^{r(r+1)} + \cdots \right)^2 \left( v^{r} + \cdots \right)^3 \left( v^{r+1} + \cdots \right)^3 \nonumber\\ & &\times \left( v^{r(r-3)/2} + \cdots \right)^2 \left( v^{(r+1)(r-2)/2} + \cdots \right)^2 \nonumber\\ & \equiv & \# ({QC_{II}})^2 ({C_{III}})^3 ({Q_{III}})^3 ({C_{II}})^2 ({Q_{II}})^2, \label{DDsp} \end{aligned}$$ where $u\equiv u_1, v\equiv u_2$ denote the two moduli among $r$ complex moduli. The subscripts $II$ and $III$ denote the order of vanishing - each corresponding to node and cusp like singularity. .1in [**Case I:**]{}   As an example, the first factor in second line of Eqn. is $$({PN_{II}}) \equiv \left( v^{(r+1)^2} + \cdots \right). \label{pn2}$$ This corresponds to having two pairs of branch points on the $x$-plane collide each other pairwise, where each pair is $P_i$ type and $N_i$ type respectively. The curve degenerates into a node-like singularity $$y^2 = (x-P_i)^2 (x-N_j)^2 \times \cdots .$$ Number of choices for choosing one pair of $P_i$’s and $N_i$’s is given as: $$\left( \begin{array} [c]{c}% r+1\\ 1 \end{array} \right)^2 = { (r+1)^2},$$ which is exactly the power of $v$ in . .1in [**Case II:**]{}  The first factor in the third line of Eqn. is $$({N_{II}}) \equiv \left( v^{{{}(r+1)(r-2)/2}} + \cdots \right),$$ and this corresponds to the scenario where two pairs of $N$-type branch points on the $x$-plane collide each other pairwise. The curve degenerates into a node-like singularity $$y^2 = (x-{N_i})^2 (x-{N_j})^2 \times \cdots .$$ .1in [**Case III:**]{}  The second factor in the second line of Eqn. $$({N_{III}}) \equiv \left( v^{r+1} + \cdots \right)$$ is related to having three $N$-type branch points on the $x$-plane collide all together. The curve degenerates into a cusp-like singularity $$y^2 = (x-{C_i})^{3} \times \cdots .$$ .1in Similarly, we can understand other factors of Eqn. and Eqn. . Conclusion \[conclusion\] ========================= In this review, we observed the relevance and importance of supersymmetric Yang-Mills theories. We discussed ${\cal N}=2$ SYM with classical gauge groups, with particular attention to their singularity structure associated with massless states. We translated the physics questions into the language of Seiberg-Witten geometry, so that a hyperelliptic curve equipped with a 1-form encodes physical information of SYM. $$\begin{tabular}{|ccc|} \hline Physics & $\leftrightarrow$ & Geometry \\ \hline \hline supersymmetric Yang-Mills theory & $\leftrightarrow$ & Hyperelliptic curve \\ \hline rank of gauge group & $\leftrightarrow$ & genus \\ \hline particle & $\leftrightarrow$ & 1-cycle \\ \hline mass &$\leftrightarrow$ & $ \left| \oint \lambda \right| $ \\ \hline massless states & $\leftrightarrow$ & singularity \\ & & (vanishing 1-cycle at $\Delta_x f=0$) \\ \hline massless & $\leftrightarrow$ & worse singularity \\ $e^-$ \& magnetic monopole & & such as cusp \\ \hline \end{tabular}$$ With this dictionary in mind, we examined singularity loci of families of hyperelliptic curves which are associated with pure SW theories. At discriminant loci $\Delta_x f=0$ of the SW curve, we have vanishing 1-cycles. We identified BPS dyon charges of all the $2r+1$ and $2(r+1)$ vanishing cycles respectively for pure $Sp(2r)$ and $SU(r+1)$ SW curves. When discriminant loci form a singularity inside the moduli space ($d \Delta_x f=0$), multiple massless dyons coexist. Here the ‘double discriminant’ also vanishes ($\Delta_u \Delta_x f=0$). Note however that the converse does not hold. If order of vanishing of roots to the double discriminant is high (equal to $3$), then vanishing 1-cycles coexist and intersect: we are at the Argyres-Douglas loci. On top of many open questions proposed in Ref. , it will be interesting to study the behavior of the SW curve near the Argyres-Douglas loci, extending the works of Ref. , which discusses appearance of quantum Higgs branch at maximal Argyres-Douglas points of SW theories. It will be also interesting to understand the results reviewed here in the context of wall-crossing and quiver mutation in Refs. . Finally, the analysis of monodromy may benefit from making more connection to the braiding procedure in knot theory. Acknowledgments {#acknowledgments .unnumbered} =============== It is a great pleasure to thank Murad Alim, Philip Argyres, Heng-Yu Chen, Keshav Dasgupta, Hoyun Jung, Dong Uk Lee, Andy Neitzke, Jihun Park, Alfred Shapere, Yuji Tachikawa, Donggeon Yhee, and Philsang Yoo for helpful discussions. Long Chen, Pedro Liendo, Chan-Youn Park, and especially Philsang Yoo gave invaluable feedback on the manuscript. The author benefited from encouragement from Howard, Mini, Sun-young Park, and Daniel Tsai. This review is dedicated to the memory of Arthur G. The work is supported in part by NSERC grants. All the figures are created by the author using the program Inkscape with TeXtext. [^1]: Partially based on a seminar given at Imperial College in January 2012 and a colloquium at Gwangju Institute of Science and Technology in April 2013. [^2]: Grassmannian is a manifold which is a generalization of a projective space. A simple example of projective spaces is a sphere. [^3]: A dyon refers to a particle which potentially carries both electric and magnetic charges. [^4]: More specifically it is called ${\cal{N}}=2$ vector multiplet for it contains a vector. The space whose coordinates are the scalar components of vector multiplets is called a Coulomb-branch of moduli space. The scalars of another ${\cal{N}}=2$ supersymmetry representation, hypermultiplet, form a Higgs moduli space. A supermultiplet is a representation of a supersymmetry algebra. More details are given in excellent reviews, Refs. . [^5]: All these methods study theories with low energy effective action. This review also deals with those only. [^6]: One may introduce matters into the Seiberg-Witten theory: By [*pure*]{} SW theory, they mean lack of matter, and it will be our focus on this review. [^7]: For later convenience, the choice of overall sign for intersection number chosen to match that of Ref. and is opposite of that of Ref. . [^8]: This is an algebraic intersection number, as opposed to a geometric intersection number. [^9]: Only some of 1-cycles, which pass the test of wall-crossing formulas, correspond to stable BPS (supersymmetric) particles. Our main focus here is for massless states. With an assumption that the massless states are stable, we can consider 1-1 mapping between 1-cycles and BPS particles, with restriction to the massless sector. [^10]: If the integrand $\lambda_{\rm SW}$ has a delta-function type singularity, integrating it over an infinitesimal interval may give a finite value to Eqn. . [^11]: The square bracket $[ \ ]$ denotes the floor function. $[x]$ is the largest integer which satisfies $[x] \le x$. [^12]: By isolated points, we roughly mean that they form a discrete set and are separated. For example, the points do not congregate to form a line or a plane. \[isolated\] [^13]: Recall from the definition of hyperelliptic curve in Eqn. that we allow the coefficient of each power of $x$ to vary, all independently from each other. For most generic hyperelliptic curve, the overall power of $f(x)$ equals to the number of parameters. By assigning the hyperelliptic curve a role of SW curve for certain gauge groups, we no longer have the full freedom of varying all of them. As we will see soon the number of independent parameters is $r$ while $f(x)$ has the power $2r+1, 2r+2$ etc. [^14]: In fact the moduli space has an extra singular point at $u =\infty$ as reviewed in Ref. . However it is not associated with a particular vanishing 1-cycle, and its monodromy can be inferred from the knowledge of other singular points purely from consistency requirement. Therefore in this review we won’t discuss the singular points at infinity in moduli space. [^15]: As we will discuss later, all solutions to the vanishing double discriminant given in Eqn. satisfy that $v^3$ is real. This is a necessary, but not sufficient, condition for two 1-cycles to vanish. [^16]: Before we identified a vanishing 1-cycle by collision of 2 branch points. Their trajectory formed a 1-cycle, which goes through 2 branch cuts. Half of it was solid line, the other half was dashed line, in figures here. Recall that each time a 1-cycle meets a branch-cut, it has to switch from solid to dash and vice versa. Now we have 3 branch point colliding together. Now we the trajectory has 3 pieces involving 3 branch cuts, and there is no consistent way of assigning dash and solid for the odd number of pieces. [^17]: Some of the best place to learn about Higgs branch are Refs. .

--- abstract: '[The present paper aims at providing a numerical strategy to deal with PDE-constrained optimization problems solved with the adjoint method. It is done through out a unified formulation of the constraint PDE and the adjoint model. The resulting model is a non-conservative hyperbolic system and thus a finite volume scheme is proposed to solve it. In this form, the scheme sets in a single frame both constraint PDE and adjoint model. The forward and backward evolutions are controlled by a single parameter $\eta$ and a stable time step is obtained only once at each optimization iteration. The methodology requires the complete eigenstructure of the system as well as the gradient of the cost functional. Numerical tests evidence the applicability of the present technique.]{} [The adjoint method, PDE-constrained optimal control, hyperbolic conservation laws.]{}' author: - | [Gino I. Montecinos]{}$^*$\ Centro de Modelamiento Matemático (CMM), Universidad de Chile,\ Beauchef 851, Torre Norte, Piso 7, Santiago, Chile\ [Juan C. López-Ríos]{}\ Escuela de Ciencias Matemáticas y Tecnología Informática, YACHAY TECH,\ San Miguel de Urcuquí, Hacienda San José s/n, Ecuador\ [Jaime H. Ortega]{}\ Centro de Modelamiento Matemático (CMM) and Departamento de Ingeniería Matemática, Universidad de Chile,\ Beauchef 851, Torre Norte, Piso 5, Santiago, Chile\ [Rodrigo Lecaros]{}\ Departamento de Matemática, Universidad Técnica Federico Santa María,\ Casilla 110-V, Valparaíso, Chile\ bibliography: - 'ref.bib' title: 'A numerical procedure and unified formulation for the adjoint approach in hyperbolic PDE-constrained optimal control problems' --- Introduction ============ We are concerned with PDE-constrained optimal control problems. In general optimization problems are made up of the following ingredients; i) state variables, ii) design parameters, iii) objectives or cost functionals and iv) constraints that candidate state and design parameters are required to satisfy. The optimization problem is then to find state and design parameters that minimize the objective functional subject to the requirement that constraints are satisfied, see [@Andersson:1998a] for further details. PDE-constrained optimization problems are those where the constraint consists of partial differential equations. These optimization problems arise naturally in control theory and inverse problems. Are usually employed as a methodology to obtain an approximate solution often from numerical simulations, see [@Knopoff:2013a; @Kang:2005a] to mention but a few. Mathematical properties of these problems, as existence and ill-possedness, have been widely studied in the literature, see for instance [@Abergel1990; @Hinze:2008a] and they will not be of interest in this work. In this paper, we are interested on generating a procedure to obtain numerical solutions to be as general as possible. There are several successful strategies to solve these types of minimization problems: i) the one-shot or Lagrange multiplier method ii) based on sensitivity equations and iii) based on adjoint equations. Of interest in this work is the strategy based on adjoint equations. The interested reader can find further information regarding approaches i) and ii) in [@Andersson:1998a; @Gunzburger:2003a] and references therein. One-shot methods cannot be used to solve most practical optimization problems especially involving time-dependent partial differential equations. Methods ii) and iii) are iterative processes involving the gradient of the objective functional, since the functional involves the solution of partial differential equations the task of obtaining the gradient becomes cumbersome, thus a safe option is just the use of approximations of them. However, in these problems even the approximation of gradients is a very challenging issue. Sensitivity equations require the differentiation of the state equation with respect to the design parameters. The simple form to achieve this is using a difference quotient approximation similarly to finite difference approximation as suggested in [@Andersson:1998a]. This strategy can be successfully applied for steady state models and for a finite and small number of design parameters. It becomes prohibitive for time-dependent cases. In this sense, the adjoint method is the suitable alternative. The adjoint method consists of the use of a linearized equation derived from the constraint PDE, it provides a new set of adjoint states, and the gradient of the cost functional can be completely determined in terms of the state variables, design parameters and adjoint states. The equation for the state variables, as well as for the adjoint parameter, is solved only once. It makes the adjoint method to be more efficient than the sensitivity method, see [@Hinze:2008a] for a detailed discussion about the performance of both approaches. In this work, we are going to consider the adjoint approach for solving optimization problems related to hyperbolic type PDE’s. Additionally, for hyperbolic equations, the design parameters, which are parameters of models, can be incorporated into the governing equation as a new variable, and then the problem of finding parameters can be cast into a problem of finding the initial condition. The constraint PDE normally is written in a conservation form and thus suitable schemes are those corresponding to the family of conservative methods. On the other hand, the adjoint models are linearization of the constraint PDE, but this is a quasilinear type model for adjoint variables and thus they are also of hyperbolic type, but these models are non-conservative in a mathematical sense. It is a very important issue from a numerical point of view, the right description of the weave propagation has to be dealt in the frame of the path conservative methods for partial differential equations involving non-conservative products, see [@Pares:2006a; @CastroM:2006a] for a detailed study of non-conservative partial differential equations with a particular emphasis on hyperbolic types. As the constraint PDE and the adjoint model are both hyperbolic, they can be written as a unified model incorporating both the constraint PDE and the adjoint model in a single system of balance laws. In this paper, we back up on the unified formulation to set a numerical methodology able to simultaneously deal with the conservative form of constraint PDE and the adjoint model. The aim is to provide a general frame to deal with optimization problems through the adjoint method. Moreover, the present approach works even if the constraint PDE is non-conservative. In the literature, there exist several works dealing with minimization considering hyperbolic conservation laws, see [@James:1999a; @James:2008a; @Castro:2008a-1; @Lecaros:2014a; @Ulbrich:2002a] to mention but a few. However, to best of our knowledge, the present approach has not been reported previously and this incorporates in a natural way the non-conservative structure of the adjoint model. On the other hand, regarding the solution of hyperbolic conservation laws, one of the successful methods are those based on the finite volume approach, which makes use of numerical fluxes at the interfaces of computational cells, providing a compact one-step evolutionary formula. The accuracy of the approximation via this approach depends on the degree of resolution of numerical fluxes. One approach for achieving high resolution is by mean of the use of the so called Generalized Riemann Problem (GRP), see [@Castro:2008a; @Montecinos:2012a] for further details and comparison among existing GRP solvers. These are building blocks of a class of high-order numerical methods known as ADER schemes [@Toro:2001c; @Toro:2002a]. The numerical scheme presented in this work belong to the class of ADER methods. The scheme uses a modified version of the Osher-Solomon Riemann solver [@Osher:1982a] presented by Dumber and Toro, [@Dumbser:2010b], which is able to account for the non-conservative terms, using the so-called path conservative approach, it is a second order method for hyperbolic problems in conventional finite volume setting and it inherits the properties of stability, well-balance, consistency and robustness. However, the implementation has to be adapted for accounting the correct wave propagation for the evolution of both, the PDE constraint and the adjoint model. In order to assess the performance of the present method, we carry out comparisons between the present scheme and a conventional method based on [@Lellouche:1994a] for solving PDE-constrained optimization problems. We adapt the scheme in [@Lellouche:1994a] implementing an upwind type scheme for the solution of the constraint and for the adjoint model, the same finite difference approach is kept but it is carried out in an explicit time evolution to be consistent with the present method which is globally explicit in time. The rest of the paper is organized as follows. In Section \[PDE-constrint\] the unified formulation of the primal dual strategy for a general system of conservation laws is presented, after the formal derivation of the adjoint system. We also state a general theorem providing the existence of classical solutions for the unified system and study a related optimal control problem. In Section \[section:numericalmethod\], a numerical scheme for solving conservative and non-conservative hyperbolic laws is presented and, as a consequence, a scheme for solving the unified primal-adjoint system. In Section \[sec:numerical-examples\] some specific numerical examples are presented using the numerical scheme from Section \[section:numericalmethod\]. Finally in Section \[sec:conclusions\] some conclusions are drawn. PDE-constrained optimization problem {#PDE-constrint} ==================================== The purpose of this section is to provide a unified formulation of the primal dual strategy, raised from the study of certain optimal control problems concerning the finding of a parameter on a general conservation law. First, we define the hyperbolic PDE to be considered and we prove that the finding of a parameter on this system is reduced to find an initial condition. Next we compute the adjoint system and set the unified formulation. Then we provide some conditions to establish the existence and uniqueness of classical solutions of the unified system. Finally we set the control problem and prove the existence of an optimal solution. We consider the system $$\begin{aligned} \label{eq:primal:0} \left. \begin{array}{c} \partial_t \mathbf{U} + \partial_x \mathbf{R}(\mathbf{ U},\mathbf{b} ) = \mathbf{L} ( \mathbf{U,\mathbf{b}} ) + \tilde{\bf B}(\mathbf{U},\mathbf{b}) \partial_x \mathbf{b} \;, \;t\in [0,T]\;,\\ \mathbf{U}(x,0)= \mathbf{H}(x) \;, \end{array} \right\}\end{aligned}$$ where $T>0$ is a finite time, $\mathbf{H}( x)\in \mathbb{R}^{m}$ is a prescribed initial condition, $\mathbf{U}\in \mathbb{R}^{m}$ is a vector of states, $\mathbf{R}(\mathbf{ U},\mathbf{b} )\in \mathbb{R}^{m}$ a flux function, $\mathbf{L} ( \mathbf{U},\mathbf{b} )\in \mathbb{R}^{m} $ is a source function, $\mathbf{b}(x)\in \mathbb{R}^{n}$ is a vector accounting for model parameters and the expression $\tilde{\bf B}(\mathbf{U},\mathbf{b}) \partial_x \mathbf{b}$ represents source terms which may also include the influence of parameter derivatives. Here, parameters $\mathbf{b}$ only depend on space. Since one main goal is to provide a strategy for finding $\mathbf{b}$ as the minimum of a given functional $J$, let us write $J$ as $$\label{eq:functional:0} J(\mathbf{U},\mathbf{b})=\frac{1}{2}\int_0^T\int_{{\mathbb{R}}}|\psi(\mathbf{U},\mathbf{b})-\bar{\psi}|^2dxdt+\frac{\alpha}{2}\|\mathbf{b}\|_{H^k}^2,$$ where $\psi(\mathbf{U},\mathbf{b}) $ is some scalar field, representing some measurement operator, similarly $\bar{\psi}$ is some available measurement, $\alpha$ is a non-negative constant and $k$ is an exponent to be determined. In an optimization setting we identify the state variables $\mathbf{U}$, design parameters $\mathbf{b}$, objectives or cost functional given by (\[eq:functional:0\]) and constraints consisting of (\[eq:primal:0\]) that candidate state $\mathbf{U}$ and design parameters $\mathbf{b}$ are required to satisfy. Notice that, parameters $ \mathbf{b}$ only depend on space, then $ \partial_t \mathbf{b} (x) = 0$. So, we can include $\mathbf{b}$ as a new variable of system (\[eq:primal:0\]), as follows $$\begin{aligned} \label{eq:primal:1} \left. \begin{array}{c} \partial_t \mathbf{U} + \partial_x \mathbf{R}(\mathbf{ U},\mathbf{b} ) = \mathbf{L} ( \mathbf{U,\mathbf{b}} ) + \tilde{\bf B} (\mathbf{U},\mathbf{b})\partial_x \mathbf{b} \;,\\ \\ \partial_t \mathbf{b}= \mathbf{0}\;. \end{array} \right\}\end{aligned}$$ So, written in the vector form, (\[eq:primal:1\]) becomes $$\begin{aligned} \label{eq:primal:2} \begin{array}{c} \partial_t \tilde{\mathbf{U}} + \partial_x \tilde{\mathbf{R}} (\tilde{\mathbf{ U}}) + \tilde{\bf M} \partial_x \tilde{\mathbf{U}} = \tilde{ \mathbf{L} }( \tilde{\mathbf{U}} ) \;, \end{array}\end{aligned}$$ where $$\begin{aligned} \label{eq:primal:3} \begin{array}{c} \tilde{\mathbf{U}} = \left[ \begin{array}{c} \mathbf{U} \\ \mathbf{b} \end{array} \right] \;, \; \tilde{\mathbf{R}}(\tilde{\mathbf{U}}) = \left[ \begin{array}{c} \mathbf{R}(\mathbf{ U},\mathbf{b} ) \\ \mathbf{0} \end{array} \right]\;, \\ \tilde{\mathbf{L} }(\tilde{\mathbf{U}}) = \left[ \begin{array}{c} \mathbf{L} ( \mathbf{U},\mathbf{b} ) \\ \mathbf{0} \end{array} \right] \;, \; \tilde{ \mathbf{M} } (\tilde{\mathbf{U}}) = \left[ \begin{array}{cc} \mathbf{0} & - \tilde{\bf B} (\mathbf{U},\mathbf{b})\\ \mathbf{0} & \mathbf{0} \\ \end{array} \right] . \end{array}\end{aligned}$$ In this new system, $\mathbf{b}(x)$ is completely fixed once an initial condition is provided, so the task of finding parameters in a model system as (\[eq:primal:0\]) is equivalent to find the initial condition of the enlarged model system (\[eq:primal:1\]), henceforth referred to as [*constraint PDE*]{}. From now on, instead of system we are going to consider system $$\begin{aligned} \label{eq:primal:01} \left. \begin{array}{c} \partial_t \tilde{\mathbf{U}} + \partial_x \tilde{\mathbf{R}} (\tilde{\mathbf{ U}}) + \tilde{\bf M} \partial_x \tilde{\mathbf{U}} = \tilde{ \mathbf{L} }( \tilde{\mathbf{U}} ) \;, \;t\in [0,T]\;,\\ \tilde{\mathbf{U}}(x,0)= \tilde{\mathbf{H}}(x) \;, \end{array} \right\}\end{aligned}$$ where $\tilde{\mathbf{H}}(x)=[\mathbf{H},\mathbf{0}]^T$. The following result provides the conditions on system (\[eq:primal:0\]) which turns system (\[eq:primal:01\]) to be hyperbolic. \[Prop1\] Let (\[eq:primal:0\]) be a hyperbolic system in the variable $\mathbf{U}$, with $ \{\lambda_1, ...,\lambda_m\}$ the corresponding eigenvalues of $\partial \mathbf{R} / \partial \mathbf{U} $, where $\lambda_i \neq 0$. Then (\[eq:primal:01\]) is also a hyperbolic system in the variables $ \tilde{\mathbf{R}}(\tilde{\mathbf{U}}) = [\mathbf{U},\mathbf{b}]^T$. The set of eigenvalues is given by $$\{\lambda_1, ...,\lambda_m, \underbrace{0,...,0}_{n\;times} \}$$ and the corresponding eigenvectors are given by $$\{ [\mathbf{v}_1, \mathbf{0}_{n}]^T, ...,[\mathbf{v}_m, \mathbf{0}_{n}]^T, [\tilde{\bf V}_1, \mathbf{e}_{1} ]^T,...,[\tilde{\bf V}_n, \mathbf{e}_{n} ]^T \} \;,$$ with $\tilde{\bf V}_j = - (\partial \mathbf{R} / \partial \mathbf{U} )^{-1}( \partial \mathbf{R} / \partial \mathbf{b} - \tilde{\bf B} )_j ,$ where $( \partial \mathbf{R} / \partial \mathbf{b} - \tilde{\bf B} )_j$ is the $j-th $ column vector of the matrix $( \partial \mathbf{R} / \partial \mathbf{b} - \tilde{\bf B} ).$ Here, $\mathbf{v}_i$ is an eigenvector associated with $\lambda_i$, $\mathbf{0}_m$ is the zero vector in $\mathbb{R}^{m}$, $\mathbf{e}_i$ is the $it$h canonical vector of $\mathbb{R}^{n}$. The Jacobian matrix of system (\[eq:primal:01\]) is given by $$\begin{aligned} \begin{array}{c} \mathbf{J_R} := \partial \tilde{\mathbf{R}} / \partial \tilde{\mathbf{U}}+ \tilde{\bf M} = \left[ \begin{array}{cc} \partial \mathbf{R} / \partial \mathbf{U} & ( \partial \mathbf{R} / \partial \mathbf{b} - \tilde{\bf B} )\\ \mathbf{0} & \mathbf{0} \end{array} \right]\;. \end{array}\end{aligned}$$ Therefore $$\begin{aligned} \begin{array}{c} det ( \mathbf{J_R} - \lambda \mathbf{I}_{n+m} ) = - \lambda^n det(\partial \mathbf{R} / \partial \mathbf{U} - \mathbf{I}_{m} ) = -\lambda^n p(\lambda) \;, \end{array}\end{aligned}$$ where $\mathbf{I}_{n+m} $ and $\mathbf{I}_{m} $ are the identity matrix in $\mathbb{R}^{n+m}$ and $\mathbb{R}^{m}$, respectively. Here, $p(\lambda)$ is the characteristic polynomial of $\partial \mathbf{R} / \partial \mathbf{U} - \mathbf{I}_{m} $ which has $m$ roots, because (\[eq:primal:0\]) is hyperbolic. This proves $$\{\lambda_1, ...,\lambda_m, \underbrace{0,...,0}_{n\;times} \}$$ are the eigenvalues of $ \mathbf{J_R}$. To complete the proof, we must obtain a set of $m+n$ linearly independent eigenvectors. That means, for each $\mu \in \{\lambda_1, ...,\lambda_m, \underbrace{0,...,0}_{n\;times} \} $ we look for $\mathbf{W}$ such that $$\begin{aligned} \begin{array}{c} \mathbf{J_R} \mathbf{W} = \mu \mathbf{W}\;. \end{array}\end{aligned}$$ Here, we obtain eigenvectors of the form $ \mathbf{W} = [\tilde{\bf V}, \tilde{\bf W}]^T$, so in a block-wise fashion we look for vectors $\tilde{\bf V}$ and $\tilde{\bf W}$ such that $$\begin{aligned} \label{eq:eigenstruct:0} \left. \begin{array}{c} (\partial \mathbf{R} / \partial \mathbf{U} )\tilde{\bf V} +( \partial \mathbf{R} / \partial \mathbf{b} - \tilde{\bf B} ) \tilde{\bf W} = \mu \tilde{\bf V} \;, \\ \mathbf{0} = \mu \tilde{\bf W} \;. \end{array} \right\}\end{aligned}$$ Notice that, setting $\mu = \lambda_i$ and $ \tilde{\bf V} = {\bf v}_i $, then (\[eq:eigenstruct:0\]) is satisfied for $ \tilde{\bf W} = \mathbf{0}_{n}$. It provides $m$ independent vectors $\{ [\mathbf{v}_1, \mathbf{0}_{n}]^T, ...,[\mathbf{v}_m, \mathbf{0}_{n}]^T \}$. On the other hand, if $\mu = 0$ we obtain $$\begin{aligned} \label{eq:eigenstruct:1} \begin{array}{c} (\partial \mathbf{R} / \partial \mathbf{U} )\tilde{\bf V} +( \partial \mathbf{R} / \partial \mathbf{b} - \tilde{\bf B} ) \tilde{\bf W} = \mathbf{0} \;. \end{array}\end{aligned}$$ If we set $\tilde{\bf W} = \mathbf{e}_j$, then $$\begin{aligned} \label{eq:eigenstruct:2} \begin{array}{c} (\partial \mathbf{R} / \partial \mathbf{U} )\tilde{\bf V} = - ( \partial \mathbf{R} / \partial \mathbf{b} - \tilde{\bf B} )_j \;, \end{array}\end{aligned}$$ where $( \partial \mathbf{R} / \partial \mathbf{b} - \tilde{\bf B} )_j$ is the $j-th $ column vector of the matrix $( \partial \mathbf{R} / \partial \mathbf{b} - \tilde{\bf B} ).$ Since $ \partial \mathbf{R} / \partial \mathbf{U} $ is invertible, there exists $\tilde{\bf V}_j\in \mathbb{R}^m$ such that $$\tilde{\bf V}_j = - (\partial \mathbf{R} / \partial \mathbf{U} )^{-1}( \partial \mathbf{R} / \partial \mathbf{b} - \tilde{\bf B} )_j \;.$$ Then the proof holds. Notice that the previous result ensure that extended system (\[eq:primal:01\]) is hyperbolic, whenever system (\[eq:primal:0\]) is. However, it requires (\[eq:primal:0\]) to have an invertible Jacobian matrix. It is not guaranteed in the general case. However, the cases of interest in this study do so. Otherwise, the verification has to be done case by case. As a consequence, the problem of finding a parameter vector for system $(\ref{eq:primal:0})$ becomes an inverse problem where the aim is to find an initial condition for $\mathbf{b}$ which is a conventional variable in a system of hyperbolic balance laws. The adjoint method for solving PDE-constrained optimization problems {#Section2.1} -------------------------------------------------------------------- In this section we derive a system of first order optimality conditions for system , subject to the functional with $\alpha = 0$. Even though these calculations are formal, we will provide the corresponding analysis in sections \[ExSol\] and \[OptCont\]. At this point our main interest is to present the unified primal-dual formulation and the main issues related with. Since it is well known for this strategy [@Hinze:2008a], the so-called adjoint state is, in some sense, a linearization of the constraint PDE. The dependence of the sought parameter $\mathbf{b}$ in is implicit but, as was mentioned above, can be seen as the tracking of an initial condition. As we did in Proposition , for $i=1:m+n$, let us write as $$\begin{aligned} \label{eq:adjoint:-2} \left. \begin{array}{c} \partial_t \tilde{\bf U}_i + \sum_{j = 1}^{m+n} \mathbf{J_R}_{i,j}\partial_x \tilde{\bf U}_j = \tilde{\bf L}_i \;, \; t\in [0,T]\;,\\ \tilde{\mathbf{U}}_i(x,0)= \tilde{\mathbf{H}}_i(x) \;. \end{array} \right\}\end{aligned}$$ Then $$\begin{aligned} \label{eq:adjoint:-1} \begin{array}{c} \partial_t \delta\tilde{\bf U}_i + \sum_{j = 1}^{m+n} \mathbf{J_R}_{i,j}\partial_x \delta\tilde{\bf U}_j + \sum_{j = 1}^{m+n} \sum_{k = 1}^{m+n} \frac{ \partial \mathbf{J_R}_{i,j} }{\partial_x \tilde{\mathbf{U}}_k} \delta\tilde{\bf U}_k \partial_x \tilde{\bf U}_j = \\ \sum_{k = 1}^{m+n} \frac{ \partial \tilde{\bf L}_k }{ \partial \bf \tilde{U}_k} \delta\tilde{\bf U}_k \;. \end{array}\end{aligned}$$ If we denotes the inner product in $L^2({\mathbb{R}})$ by $\langle\cdot,\cdot\rangle$ and multiply last equation by functions $P_i$, such that $ lim_{|x|\rightarrow \infty} P_i(x,t) = 0$, $\forall \;t$, after integrating in space and time, we obtain $$\begin{aligned} \begin{array}{c} \displaystyle \int_{0}^{T} \langle \partial_t \delta\tilde{\bf U}_i ,P_i \rangle + \sum_{j = 1}^{m+n} \displaystyle \int_{0}^{T} \langle \partial_x \delta\tilde{\bf U}_j , \mathbf{J_R}_{i,j} P_i \rangle \\ + \sum_{j = 1}^{m+n} \sum_{k = 1}^{m+n} \displaystyle \int_{0}^{T} \langle \delta\tilde{\bf U}_k , \frac{ \partial \mathbf{J_R}_{i,j} }{\partial \tilde{\mathbf{U}}_k} \partial_x \tilde{\bf U}_j P_i \rangle = \\ \sum_{k = 1}^{m+n} \displaystyle \int_{0}^{T} \langle \delta\tilde{\bf U}_k , \frac{ \partial \tilde{\bf L}_k }{ \partial \bf \tilde{U}_k} P_i \rangle \;. \end{array}\end{aligned}$$ Summing on indices $i$, yields $$\begin{aligned} \begin{array}{c} \displaystyle \int_{0}^{T} \sum_{i=1}^{m+n} \langle \partial_t \delta\tilde{\bf U}_i ,P_i \rangle - \sum_{j = 1}^{m+n} \displaystyle \int_{0}^{T} \langle \delta\tilde{\bf U}_j , \sum_{i=1}^{m+n} \partial_x ( \mathbf{J_R}_{i,j} P_i ) \rangle \\ + \sum_{j = 1}^{m+n} \sum_{k = 1}^{m+n} \displaystyle \int_{0}^{T} \langle \delta\tilde{\bf U}_k , \sum_{i=1}^{m+n} \frac{ \partial \mathbf{J_R}_{i,j} }{\partial \tilde{\mathbf{U}}_k} \partial_x \tilde{\bf U}_j P_i \rangle = \\ \sum_{k = 1}^{m+n} \displaystyle \int_{0}^{T} \langle \delta\tilde{\bf U}_k , \sum_{i=1}^{m+n} \frac{ \partial \tilde{\bf L}_k }{ \partial \bf \tilde{U}_k} P_i \rangle \;. \end{array}\end{aligned}$$ Then integrating by parts, we obtain $$\begin{aligned} \begin{array}{c} \displaystyle \sum_{k=1}^{m+n} \langle \delta\tilde{\bf U}_k ,P_k \rangle |_{0}^{T} - \displaystyle \int_{0}^{T} \sum_{k=1}^{m+n} \langle \delta\tilde{\bf U}_k , \partial_t P_k \rangle - \sum_{k = 1}^{m+n} \displaystyle \int_{0}^{T} \langle \delta\tilde{\bf U}_k , \sum_{i=1}^m \mathbf{J_R}_{i,k} \partial_x P_i \rangle \\ - \sum_{k = 1}^{m+n} \displaystyle \int_{0}^{T} \langle \delta\tilde{\bf U}_k , \sum_{i=1}^m \mathbf{D}_{k,j} \partial_x \tilde{\bf U}_j \rangle = \sum_{k = 1}^{m+n} \displaystyle \int_{0}^{T} \langle \delta\tilde{\bf U}_k , \sum_{i=1}^{m+n} \frac{ \partial \tilde{\bf L}_k }{ \partial \bf \tilde{U}_k} P_i \rangle \;, \end{array}\end{aligned}$$ where $$\begin{aligned} \label{eq:defD} \begin{array}{c} \mathbf{D}_{k,j} = \sum_{i=1}^{m+n} \biggl( \frac{ \partial \mathbf{J_R}_{i,k} }{\partial \tilde{\mathbf{U}}_j} - \frac{ \partial \mathbf{J_R}_{i,j} }{\partial \tilde{\mathbf{U}}_k} \biggr) P_i \;. \end{array}\end{aligned}$$ On the other hand, $\delta J(\mathbf{b}) = \displaystyle \int_{0}^{T} \int_{-\infty }^{\infty} (\psi(\mathbf{U},\mathbf{b}) - \bar{\psi} ) ( \nabla_{\mathbf{U}} \psi ^T \delta \mathbf{U} + \nabla_{\mathbf{b}} \psi ^T \delta \mathbf{b} )$ or $$\delta J(\mathbf{b}) = \sum_{k = 1}^{m+n} \displaystyle \int_{0}^{T} \langle \delta\tilde{\bf U}_k , (\psi - \bar{\psi} ) \frac{ \partial \psi }{ \partial \tilde{\bf U}_k } \rangle \;.$$ Moreover $$\begin{aligned} \begin{array}{c} \delta J = \sum_{j=1}^{n} \langle \nabla J, \delta \mathbf{b} \rangle \;, \end{array}\end{aligned}$$ so, by identifying terms we obtain $$\begin{aligned} \label{FormalGrad:eq-1} \begin{array}{c} \nabla J_j (x) = P_{m+j} (x,0) \;, \end{array}\end{aligned}$$ for all $j=1,...,n$. Imposing $ P_i(x,T) = 0 $ for $i=1,...,n+m$ and $P_i(x,0) = 0 $ for $i=1,...,m$ we obtain the adjoint problem $$\begin{aligned} \begin{array}{c} \partial_t P_k + \sum_{i=1}^{m+n} \mathbf{J_R}_{i,k} \partial_x P_i + \sum_{i=1}^{m+n} D_{k,j} \partial_x \tilde{\bf U}_j = \sum_{i=1}^{m+n} \frac{ \partial \tilde{\bf L}_i }{ \partial \bf \tilde{U}_k} P_i + (\psi - \bar{\psi} ) \frac{ \partial \psi }{ \partial \tilde{\bf U}_k } \;. \end{array}\end{aligned}$$ In a matrix form $$\begin{aligned} \label{eq:adjoint:1} \begin{array}{c} \partial_t \mathbf{P} + \mathbf{J_R}^T \partial_x \mathbf{P} = \tilde{S}(\mathbf{ \tilde{U},P}) \;, \\ \mathbf{P}( x,T) = \mathbf{0} \;, \end{array}\end{aligned}$$ where $ \frac{1}{2}\nabla_{\tilde{\mathbf{U}}} ( (\psi - \bar{\psi} )^2 )_{k} = (\psi - \bar{\psi} ) \frac{ \partial \psi }{ \partial \tilde{\bf U}_k } $ and $\mathbf{P} = [P_1,...,P_{n+m}]^T$. The source term in this case has the form $$\tilde{S}(\mathbf{ \tilde{U},P}) = \frac{\partial \tilde{L}}{\partial \mathbf{\tilde{ U}}}^T \mathbf{P} + \frac{1}{2}\nabla_{\tilde{\mathbf{U}}} ( (\psi - \bar{\psi} )^2 ) - \mathbf{D} \partial_x \mathbf{\tilde{ U}} \;.$$ In the sequel, we shall refer to $\mathbf{P}$ as [*the adjoint state*]{}. Notice that system (\[eq:adjoint:1\]) is also hyperbolic, with the same eigenvalues that system (\[eq:adjoint:-2\]). It is because the Jacobian matrix of the system (\[eq:adjoint:1\]) is $\mathbf{J}_R^T.$ Moreover, notice that system (\[eq:adjoint:1\]), henceforth referred to as [the adjoint model]{}, evolves back in time from $T$ to $0$. The unified formulation ----------------------- In this section, we construct a model system which accounts for both the PDE constraint and the adjoint model, aimed to construct a numerical method, able to solve simultaneously both systems including the treatment of non-conservative products. So, let us consider the following unified primal-dual system $$\begin{aligned} \label{eq:unifiedsystem:1} \begin{array}{c} \partial_t \mathbf{Q} + \partial_x \mathbf{F}(\mathbf{Q} ) + \mathbf{B}(\mathbf{Q}) \partial_x \mathbf{Q} = \mathbf{S} ( \mathbf{Q} ) \;,\\ \end{array}\end{aligned}$$ where $$\begin{aligned} \begin{array}{c} \mathbf{Q} = \left( \begin{array}{c} \mathbf{\tilde{U} } \\ \mathbf{P} \end{array} \right)\;, \; \mathbf{F}(\mathbf{Q} )= \left( \begin{array}{c} \mathbf{\tilde{R} }(\mathbf{ \tilde{U} } ) \\ \mathbf{0} \end{array} \right) \;, \\ \mathbf{B}(\mathbf{Q}) = \left( \begin{array}{cc} \mathbf{0} & \mathbf{0} \\ \mathbf{D} & \mathbf{J_R}^T \end{array} \right) \;, \; \mathbf{S} ( \mathbf{Q}) = \left( \begin{array}{c} \mathbf{ \tilde{L} } \\ - \frac{ \partial \mathbf{\tilde{L}}}{\partial \mathbf{\tilde{U} } }^T \mathbf{P} + \frac{1}{2}\nabla_{\tilde{\mathbf{U}}} ( (\psi - \bar{\psi} )^2 ) \end{array} \right)\;. \end{array}\end{aligned}$$ Notice that the state $\mathbf{Q}\in{\mathbb{R}}^{2(m+n)}$ contains the state variable $\mathbf{U}\in{\mathbb{R}}^m$, the model parameter $\mathbf{b}\in{\mathbb{R}}^n$ and the adjoint state $\mathbf{P}\in{\mathbb{R}}^{m+n}$. Moreover, it is expected that not only system (\[eq:unifiedsystem:1\]) reproduces the evolution of the constraint PDE for marching forward in time, but also it approximates the evolution backward in time of the adjoint model. The aim is to design a numerical solver for the model (\[eq:unifiedsystem:1\]) in order to solve constraint PDE and adjoint model with the same methodology and able to deal with non-conservative products. We are going to prove that in most of the cases the unified system is hyperbolic if constraint PDE as well as the adjoint model, are hyperbolic. In such a case a finite volume scheme is a suitable method for solving this class of problems. Indeed, if (\[eq:primal:0\]) admits a conservative formulation, there exists $\mathbf{\tilde{K}}(\mathbf{\tilde{U}})$ such that $$\begin{aligned} \begin{array}{c} \mathbf{J_R} (\mathbf{\tilde{U}}) = \frac{ \partial \mathbf{\tilde{K}}(\mathbf{\tilde{U}}) }{ \partial \mathbf{\tilde{U}} } \;. \end{array}\end{aligned}$$ Then $\mathbf{D} = \mathbf{0}$ and (\[eq:unifiedsystem:1\]) is hyperbolic with the same eigenvalues that (\[eq:primal:2\]). This is stated in the following \[Prop:unifiedsystem:1\] If there exists a differentiable function $\mathbf{\tilde{K}}(\mathbf{\tilde{U}})$ such that $\mathbf{J_R} (\mathbf{\tilde{U}}) = \frac{ \partial \mathbf{\tilde{K}}(\mathbf{\tilde{U}}) }{ \partial \mathbf{\tilde{U}} } \;,$ then $\mathbf{D} = \mathbf{0}$ and (\[eq:unifiedsystem:1\]) is hyperbolic with eigenvalues $$\{ \lambda_1,....,\lambda_{m+n} \} \;,$$ each one with multiplicity $2$ and eigenvectors $$\{ [\mathbf{v}_1, \mathbf{0}_{m+n} ]^T, [\mathbf{0}_{m+n}, \mathbf{v}^t_1 ]^T,....,[\mathbf{v}_{m+n}, \mathbf{0}_{m+n} ]^T, [\mathbf{0}_{m+n} ,\mathbf{v}^t_{m+n} ]^T \}\;,$$ where $\mathbf{v}_i$ is the eigenvector of $\mathbf{J_R}$ associated with $\lambda_i$ and $\mathbf{v}^t_i$ is the eigenvector of $\mathbf{J_R}^T$ associated with $\lambda_i$. Here $\mathbf{0}_{m+n} $ is the zero vector of $\mathbb{R}^{m+n}.$ Let us write system (\[eq:unifiedsystem:1\]) in the quasilinear form $$\begin{aligned} \label{eq:unifiedsystem:ql-1} \begin{array}{c} \partial_t \mathbf{Q} + \mathbf{A}(\mathbf{Q}) \partial_x \mathbf{Q} = \mathbf{S} ( \mathbf{Q} ) \;,\\ \end{array}\end{aligned}$$ where $$\begin{aligned} \begin{array}{c} \mathbf{A}(\mathbf{Q}) = \left( \begin{array}{cc} \mathbf{\mathbf{J_R}} & \mathbf{0} \\ \mathbf{D} & \mathbf{J_R}^T \end{array} \right) \;. \end{array}\end{aligned}$$ Notice that $ det ( \mathbf{A} - \mu \mathbf{I}_{2(m+n)} ) = det ( \mathbf{J_R} - \mu \mathbf{I}_{(m+n)} ) det ( \mathbf{J_R}^T - \mu \mathbf{I}_{(m+n)} ) = p(\mu)^2 ,$ where $p(\mu)$ is the characteristic polynomial of $\mathbf{J_R}$, from which we obtain that eigenvalues of this system are the same of $\mathbf{J_R}$, each one with at lest an algebraic multiplicity $2$. It is because it may exists indices $i$ and $j$ such that $i\neq j$ and $\lambda_i = \lambda_j$. On the other hand, from the existence of $\mathbf{\tilde{K}}$, we note that $$\begin{aligned} \begin{array}{c} \frac{ \partial \mathbf{J_R}_{i,k} }{\partial \tilde{\mathbf{U}}_j} = \frac{ \partial^2 \mathbf{\tilde{K}}(\mathbf{\tilde{U}}) }{ \partial \mathbf{\tilde{U}}_k \partial \mathbf{\tilde{U}}_j } = \frac{ \partial^2 \mathbf{\tilde{K}}(\mathbf{\tilde{U}}) }{ \partial \mathbf{\tilde{U}}_j \partial \mathbf{\tilde{U}}_k } = \frac{ \partial \mathbf{J_R}_{i,j} }{\partial \tilde{\mathbf{U}}_k} \;, \end{array}\end{aligned}$$ so from (\[eq:defD\]) we have $ \mathbf{D} = \mathbf{0}.$ Thus the Jacobian matrix of the system now takes the form $$\begin{aligned} \begin{array}{c} \mathbf{A}(\mathbf{Q} ) = \left[ \begin{array}{cc} \mathbf{J_R} & \mathbf{0} \\ \mathbf{0} & \mathbf{J_R}^T \end{array} \right] \;. \end{array}\end{aligned}$$ In order to find the set of eigenvectors, let us denote by $\mathbf{v}_i$ the eigenvector of $\mathbf{J_R}$ associated with $\lambda_i$ and $\mathbf{v}^t_i$ by the eigenvector of $\mathbf{J_R}^T$ associated with $\lambda_i$. Then is straightforward to verify that $$\begin{aligned} \begin{array}{c} \left[ \begin{array}{cc} \mathbf{J_R} & \mathbf{0} \\ \mathbf{0} & \mathbf{J_R}^T \end{array} \right] \left[ \begin{array}{c} \mathbf{v}_i \\ \mathbf{0}_{n+m} \end{array} \right] = \lambda_i \left[ \begin{array}{c} \mathbf{v}_i \\ \mathbf{0}_{n+m} \end{array} \right] \end{array}\end{aligned}$$ and $$\begin{aligned} \begin{array}{c} \left[ \begin{array}{cc} \mathbf{J_R} & \mathbf{0} \\ \mathbf{0} & \mathbf{J_R}^T \end{array} \right] \left[ \begin{array}{c} \mathbf{0}_{n+m} \\ \mathbf{v}^t_i \end{array} \right] = \lambda_i \left[ \begin{array}{c} \mathbf{0}_{n+m} \\ \mathbf{v}^t_i \end{array} \right]\;. \end{array}\end{aligned}$$ In this form, a set of eigenvectors is obtained. The verification of the linear independence comes from the fact that $\{\mathbf{v}_1, ..., \mathbf{v}_m\}$ as well as $\{\mathbf{v}^t_1, ..., \mathbf{v}^t_m\}$ are two sets of linearly independent vectors. The previous proposition shows that unified systems coming from conservative hyperbolic equations are always hyperbolic. This does not mean that non-conservative systems are not hyperbolic. So in the case of systems where $\mathbf{D}$ is not the null matrix, the analysis of hyperbolicity has to be done case by case. Existence of solutions, the symetrizable case {#ExSol} --------------------------------------------- In this section we discuss the setting where a well-posedness theorem for system can be formulated. Recall from Proposition \[Prop:unifiedsystem:1\], system can be written in the quasilinear form $$\begin{aligned} \label{eq:existence:1} \begin{array}{c} \partial_t \mathbf{Q} + \mathbf{A}(\mathbf{Q}) \partial_x \mathbf{Q} = \mathbf{S} ( \mathbf{Q} ) \;,\\ \end{array}\end{aligned}$$ with $$\begin{aligned} \begin{array}{c} \mathbf{A}(\mathbf{Q}) = \left( \begin{array}{cc} \mathbf{\mathbf{J_R}} & \mathbf{0} \\ \mathbf{D} & \mathbf{J_R}^T \end{array} \right), \quad \mathbf{S} ( \mathbf{Q}) = \left( \begin{array}{c} \mathbf{ \tilde{L} } \\ - \frac{ \partial \mathbf{\tilde{L}}}{\partial \mathbf{\tilde{U} } }^T \mathbf{P} + \frac{1}{2}\nabla_{\tilde{\mathbf{U}}} ( (\psi - \bar{\psi} )^2 ) \end{array} \right). \end{array}\end{aligned}$$ In [@taylor2010partial], has been made the hypothesis of strict hyperbolicity: that the Jacobian matrix $\mathbf{A}$ has real and distinct eigenvalues. Then, under this assumption, is called a symetrizable hyperbolic system provided there exists $\mathbf{A}_0(\mathbf{Q})$ positive-definite, such that $\mathbf{A}_0\mathbf{A}$ is symmetric. See Proposition 2.2, Chapter 16 in [@taylor2010partial] for further details. The assumption of the symmetry of $\mathbf{A}$ is natural in this context of conservation laws. In [@lax1954weak], Lax introduced a general notion of symmetrizer in the context of pseudodifferential operators to prove that any strictly hyperbolic system is symmetrizable. Also, as was mentioned in [@serre2015relative] under the context of systems of conservation laws having an entropy solution, Godunov [@Godunov:1961a] and Friedrichs and Lax [@Friedrichs:1971a] observed that systems of conservation laws admitting a strongly convex entropy can be symmetrized. It is then under this context of strictly hyperbolic matrices that we state the following theorem for the system . This theorem is the key point to set a descent method to solve optimal control problems related to conservation laws. Even though its proof is classical, see [@taylor2010partial], we outline some key steps for the sake of completeness. Let $I=(-a,b)$ be an interval with $t\in I$. \[Theorem1\] Let $\mathbf{Q}(x,0)=f\in H^k({\mathbb{R}}^d)$, $k>\frac{d}{2}+1$. If $\mathbf{A}_0(t,x,\mathbf{Q})$, $\mathbf{A}(t,x,\mathbf{Q})$ in $C(I,H^k\times H^k({\mathbb{R}}^d))$ and $\mathbf{L}(\mathbf{Q},\mathbf{b})$, $\bar{\psi}$, $\psi(\mathbf{Q},\mathbf{b})$ in $C(I,H^{k+1}\times H^{k}({\mathbb{R}}^d))$, then the initial value problem has a unique solution in $C(I,H^k\times H^k({\mathbb{R}}^d))$. Let $\{J_\epsilon:0<\epsilon\le1\}$ a Friedrichs mollifier. The idea is to obtain a solution to as the limit of unique solutions $\mathbf{Q}_\epsilon$ to the systems of ODEs $$\label{eq:existence:2} \left\{ \begin{aligned} & \mathbf{A}_0(t,x,J_\epsilon \mathbf{Q}_\epsilon)\partial_t\mathbf{Q}_\epsilon+J_\epsilon \mathbf{A}(t,x,J_\epsilon \mathbf{Q}_\epsilon)\partial_xJ_\epsilon \mathbf{Q}_\epsilon=J_\epsilon \mathbf{S}(t,x,J_\epsilon \mathbf{Q}_\epsilon), \\ & \mathbf{Q}_\epsilon(0)=f, \end{aligned} \right.$$ for $t$ close to $0$. We denote $\mathbf{A}_\epsilon=\mathbf{A}(t,x,J_\epsilon \mathbf{Q}_\epsilon)$. Let us consider the $L^2$-inner product $$(w,\mathbf{A}_{0\epsilon} w), \quad \mathbf{A}_{0\epsilon}=\mathbf{A}_0(t,x,J_\epsilon \mathbf{Q}),$$ which, by the positivity of $\mathbf{A}_0$, defines an equivalent $L^2$-norm. Therefore by the symmetry of $\mathbf{A}_0$ and $$\begin{aligned} \frac{d}{dt}(D^\alpha \mathbf{Q}_\epsilon,\mathbf{A}_{0\epsilon}(t)D^\alpha \mathbf{Q}_\epsilon)&=-2(D^\alpha(J_\epsilon \mathbf{A}_{\epsilon}\partial_xJ_\epsilon \mathbf{Q}_\epsilon),D^\alpha \mathbf{Q}_\epsilon)+2(D^\alpha J_\epsilon \mathbf{S},D^\alpha \mathbf{Q}_\epsilon) \\ &\phantom{=} -2([D^\alpha,\mathbf{A}_{0\epsilon}]\partial_t\mathbf{Q}_\epsilon,D^\alpha \mathbf{Q}_\epsilon)+(D^\alpha \mathbf{Q}_\epsilon,\mathbf{A}_{0\epsilon}'(t)D^\alpha \mathbf{Q}_\epsilon).\end{aligned}$$ Then, using Moser–type [@moser1966rapidly] estimates and Sobolev embeddings, for $|\alpha|\le k$, one has $$\label{existence3} \frac{d}{dt}(D^\alpha \mathbf{Q}_\epsilon,\mathbf{A}_{0\epsilon}(t)D^\alpha \mathbf{Q}_\epsilon)\le C_k(\|\mathbf{Q}_\epsilon(t)\|_{C^1})(1+\|J_\epsilon \mathbf{Q}_\epsilon(t)\|_{H^k}^2).$$ Since $\|J_\epsilon \mathbf{Q}_\epsilon(t)\|_{H^k}\le C\|\mathbf{Q}_\epsilon(t)\|_{H^k}$, by , $\mathbf{Q}_\epsilon$ satisfies $$\left\{ \begin{aligned} & \frac{d}{dt}\|\mathbf{Q}_\epsilon(t)\|_{H^k}^2\le E(\|\mathbf{Q}_\epsilon(t)\|_{H^k}^2), \\ & \|\mathbf{Q}_\epsilon(0)\|_{H^k}^2=\|f\|_{H^k}^2. \end{aligned} \right.$$ Thus by Gronwall’s inequality, $\mathbf{Q}_\epsilon$ exists for $t$ in an interval $[0,b)$ independent of $\epsilon$ and $$\label{existence4} \|\mathbf{Q}_\epsilon(t)\|_{H^k}\le K(t)<\infty.$$ Moreover, by the time-reversibility, $\mathbf{Q}_\epsilon$ exists and is bounded on $I=(-a,b)$. By and , $\mathbf{Q}_\epsilon\in C(I,H^k)\cap C^1(I,H^{k-1})$ is bounded and therefore will have a weak limit point $\mathbf{Q}\in L^\infty(I,H^k({\mathbb{R}}^d))\cap\operatorname{Lip}(I,H^{k-1}({\mathbb{R}}^d))$. Moreover, by Ascoli’s Theorem there is a sequence, still denoted $\mathbf{Q}_\epsilon$, such that $\mathbf{Q}_\epsilon\rightarrow \mathbf{Q}$ in $C(I,H^{k-1}({\mathbb{R}}^d))$ and since $k>\frac{d}{2}+1$ $$\mathbf{Q}_\epsilon\rightarrow \mathbf{Q} \quad \text{in} \ C(I,C^1({\mathbb{R}}^d)).$$ Consequently, follows in the limit from . Uniqueness is treated in a similar way to the estimates above. This conclude the proof of the Theorem. \[Cor1\] Let $\mathbf{b}\in H^k({\mathbb{R}}^d)$, $k>\frac{d}{2}+1$. If $\mathbf{Q}\in C(I,H^k({\mathbb{R}}^d))$ is the solution of with initial condition $\mathbf{Q}(0)=\mathbf{b}$ then there exists an open neighborhood $V_\mathbf{b}$ of $\mathbf{b}$ in $H^k({\mathbb{R}}^d)$ such that system with initial condition $\beta\in V_\mathbf{b}$ and right hand side $\mathbf{g}=\partial_\mathbf{Q}\mathbf{S}-\partial_\mathbf{Q}J_R\partial_x\mathbf{Q}\in C(H^k({\mathbb{R}}^d))$ have a solution $\mathbf{Q}_\beta$. Moreover the mapping $G:V_\mathbf{b}\rightarrow C(H^k({\mathbb{R}}^d))$, defined by $G(\beta)=\mathbf{Q}_\beta$, is of class $C^\infty$. Finally, if $V=DG(\beta)\cdot h$, for some $\beta\in V_\mathbf{b}$ and some $h\in H^k({\mathbb{R}}^d)$, then $V$ is the unique solution of the problem $$\partial_t V + J_R(\mathbf{Q}_\beta) \partial_x V = \mathbf{g}V, \quad V(0)=\mathbf{b}.$$ Proceeding as in the proof of Theorem \[Theorem1\], we can consider $$F:C(H^k({\mathbb{R}}^d))\times H^k({\mathbb{R}}^d)\rightarrow C(H^k({\mathbb{R}}^d))\times H^k({\mathbb{R}}^d)$$ the mapping given by $$F(\mathbf{Q},\mathbf{b})=(\partial_t \mathbf{Q}+\mathbf{A}(\mathbf{Q}) \partial_x\mathbf{Q}-\mathbf{S}(\mathbf{Q}),\mathbf{Q}(0)-\mathbf{b}).$$ Then $F$ is of class $C^\infty$ and $$\frac{\partial F}{\partial\mathbf{Q}}(\mathbf{Q},\mathbf{b})\cdot V=(\partial_t V + J_R(\mathbf{Q}_\beta) \partial_x V-\mathbf{g}V,V(0)).$$ Applying Theorem \[Theorem1\] we have that $\frac{\partial F}{\partial\mathbf{Q}}(\mathbf{Q,\mathbf{b}})$ is an isomorphism from $C(H^k({\mathbb{R}}^n))$ onto $C(H^k({\mathbb{R}}^n))\times H^k({\mathbb{R}}^n)$. Now, if $\mathbf{Q}$ is a solution of , then $F(\mathbf{Q},\mathbf{b})=(0,0)$. Therefore, by the implicit function theorem, there exists an open neighborhood $V_\mathbf{b}$ of $\mathbf{b}$ in $H^k({\mathbb{R}}^d)$ and $G:V_\mathbf{b}\rightarrow C(H^k({\mathbb{R}}^d))$ such that $F(G(\beta),\beta)=(0,0)$ for every $\beta\in V_\mathbf{b}$. Moreover, $G$ is also of class $C^\infty$ and $$\frac{\partial F}{\partial \mathbf{Q}}(\mathbf{Q}_\beta,\beta)\circ DG(\beta)\cdot h+\frac{\partial F}{\partial \beta}(\mathbf{Q}_\beta,\beta)\cdot h=(0,0) \quad \forall h\in H^k({\mathbb{R}}^d).$$ Then, if we set $V=DG(\beta)\cdot h$, we get $$\partial_t V + J_R(\mathbf{Q}_\beta) \partial_x V = \mathbf{g}V, \quad V(0)=\mathbf{b}.$$ Note that we have established the results above for $x\in{\mathbb{R}}^d$ and $k>\frac{d}{2}+1$. This is to keep the generality of this results. In this work will be enough to consider $d=1$ and $k>3/2$ as we did in the formulation of system . The optimal control problem {#OptCont} --------------------------- One of our main motivations applying an optimal control strategy to general conservation laws is the possibility of study numerical inverse problems related to this type of systems. Example of them are the bottom detection in water-waves type system, see [@Zuazua:2015a], or the tracking of initial conditions in the presence of shocks [@Castro:2008a-1; @Lecaros:2014a], to mention but a few. We are considering the functional $$J(\mathbf{Q},\mathbf{b})=\frac{1}{2}\int_0^T\int_{{\mathbb{R}}}|\psi(\mathbf{Q},\mathbf{b})-\bar{\psi}|^2dxdt+\frac{\alpha}{2}\|\mathbf{b}\|_{H^k}^2,$$ with $\bar{\psi}(t,x)\in C((0,T),H^k({\mathbb{R}}^d))$, $k>\frac{d}{2}+1$, a given function and $\alpha>0$. We want to find $\mathbf{b}$ such that $J$ is small. The nature of $\psi$ is not specified and it depends on the physical problem under consideration, as well as the itself election of $J$. The admissible control set is a nonempty convex closed subset $U_{ad}$ of $H^k({\mathbb{R}}^d)$. Then the optimal control problem is formulated in the following way $$\label{optProblem} \left. \begin{array}{c} \text{Minimize} \ J(\mathbf{Q},\mathbf{b}),\\ (\mathbf{Q},\mathbf{b})\in H^1(H^k({\mathbb{R}}^d))\times U_{ad} \ \text{verifies} \ \eqref{eq:existence:1}. \end{array} \right\}\tag{P}$$ Now we study the existence of a solution of . \[TheorExiMini\] Let $k>\frac{d}{2}+1$. If there exists a feasible pair $(\mathbf{Q},\mathbf{b})\in U_{ad}\times H^1(H^k({\mathbb{R}}^d))$ satisfying , then there exists at least one optimal solution $(\mathbf{Q}_0,\mathbf{b}_0)$ of . Since there is one feasible pair for the problem and $J$ is bounded below by zero we may take a minimizing sequence $\{(\mathbf{Q}_n,\mathbf{b}_n)\}\subset U_{ad}\times H^1(H^k({\mathbb{R}}^d))$. Then $\frac{\alpha}{2}\|\mathbf{b}_n\|_{H^k}^2\le J(\mathbf{Q}_n,\mathbf{b}_n)<\infty$, which implies that $\{\mathbf{b}_n\}$ is a bounded sequence in $H^k({\mathbb{R}}^d)$. Thus, we can take a subsequence, denoted in the same way, such that $\mathbf{b}_n\rightarrow\mathbf{b}_0$ weakly in $H^k({\mathbb{R}}^d)$. Since $U_{ad}$ is closed and convex, $\mathbf{b}_0\in U_{ad}$. On the other hand, following the proof of Theorem \[Theorem1\], we have that $\{\mathbf{Q}_n\}$ is bounded in $H^1(H^k({\mathbb{R}}^d))$. Therefore we can assume, by taking a subsequence if necessary, that $\mathbf{Q}_n\rightarrow\mathbf{Q}_0$ weakly in $H^1(H^k({\mathbb{R}}^d))$. Using the continuity of the inclusion $H^1(H^k({\mathbb{R}}^d))\subset C(H^{k-1}({\mathbb{R}}^d))$ we can pass to the limit in system satisfied by $(\mathbf{Q}_n,\mathbf{b}_n)$ and to conclude that $(\mathbf{Q}_0,\mathbf{Q}_0)$ also satisfies . Namely, $(\mathbf{Q}_0,\mathbf{b}_0)$ is a feasible pair for problem . Taking into consideration that $J(\mathbf{Q},\mathbf{b})$ is weakly lower semi-continuous, the result follows. Let us remark that the assumption of Theorem \[TheorExiMini\] can be checked by taking $\mathbf{Q}\in H^1(H^k({\mathbb{R}}^d))$ with $\mathbf{Q}(0)=f$. This uniquely defines $\mathbf{b}$ from the partial differential equation as $\mathbf{b}=\mathbf{Q}(0)$. If $\mathbf{b}\in U_{ad}$ then the assumption is satisfied. This is the case if $U_{ad}=H^k({\mathbb{R}}^d)$. Finally we state the following proposition giving the optimality conditions satisfied by the solutions $(\mathbf{Q}_0,\mathbf{b}_0)$ of . Its proof is based on Corollary \[Cor1\] and the results obtained in Section \[Section2.1\]. Let $k>\frac{d}{2}+1$ and assume that $(\mathbf{Q}_0,\mathbf{b}_0)$ is a solution of . Then there exist a unique element $\mathbf{P}_0\in C(H^k({\mathbb{R}}^d))$ such that the following system is satisfied $$\begin{aligned} \begin{array}{c} \partial_t \mathbf{Q}_0 + \mathbf{J}_R \partial_x \mathbf{Q}_0 = \mathbf{L} ( \mathbf{Q}_0 ) \;, \\ \mathbf{Q}_0( x,T) = \mathbf{b}_0 \;, \end{array}\end{aligned}$$ $$\begin{aligned} \begin{array}{c} \partial_t \mathbf{P}_0 + \mathbf{J}_R^T \partial_x \mathbf{P}_0 = \frac{\partial \mathbf{L}}{\partial \mathbf{Q_0}}^T \mathbf{P}_0 + \frac{1}{2}\nabla_{\mathbf{Q}_0} ( (\psi - \bar{\psi} )^2 ) - \mathbf{D} \partial_x \mathbf{Q_0} \;, \\ \mathbf{P}_0( x,T) = \mathbf{0} \;, \end{array}\end{aligned}$$ $$\begin{aligned} \begin{array}{c} \mathbf{P}_0|_{t=\mathbf{0}}. \end{array}\end{aligned}$$ The numerical scheme {#section:numericalmethod} ==================== In this section, we present the scheme for solving simultaneously conservative as well as non-conservative hyperbolic balance laws of the form (\[eq:unifiedsystem:1\]). Cell averages are evolved by following the one-step finite volume formula $$\begin{aligned} \label{one-step:1} \begin{array}{lcl} \mathbf{Q}_i^{n+1} &=& \mathbf{Q}_i^{n} - \eta \frac{\Delta t}{\Delta x} (\mathbf{F}_{i+\frac{1}{2}}-\mathbf{F}_{i-\frac{1}{2}})- \eta\frac{\Delta t}{\Delta x} (\mathbf{D}_{i+\frac{1}{2}} + \mathbf{D}_{i-\frac{1}{2}} ) \\ & &- \frac{\eta}{\Delta x} (\mathbf{B}\partial \mathbf{Q})_i + \eta \Delta t \mathbf{S}_i \;, \end{array}\end{aligned}$$ where $\eta $ controls the evolution of the system. This is because in the context of the adjoint method we need the constraint PDE evolves forward in time up to $T$ and then the adjoint model evolves backward in time from $T$ to $0$. So, $\eta = 1$ makes the forward in time and $\eta = -1 $ makes the backward in time. This formula is a conventional one, available in the literature for solving non-conservative schemes, see for instance [@Dumbser:2014a]. However, we are going to modify the usual implementation of these type of schemes in order to account for the correct wave propagation. By the sake of completeness, we provide a detailed description of the scheme. We have $$\begin{aligned} \label{integrals:eq-1} \begin{array}{c} ( \mathbf{B}\partial \mathbf{Q})_i = \frac{1}{\Delta t \Delta x} \displaystyle \int_{t}^{ t^{n+1} } \displaystyle \int_{x_{ i-\frac{1}{2} }}^{ x_{ i+\frac{1}{2} } } \mathbf{B}( \mathbf{Q}_i (\xi,\tau)) \partial_x \mathbf{Q}_i(\xi,\tau) d\xi d\tau \;, \\ \mathbf{S}_i = \frac{1}{\Delta t \Delta x} \displaystyle \int_{t}^{t^{n+1}} \displaystyle \int_{ x_{ i-\frac{1}{2} }}^{ x_{ i+\frac{1}{2} } } \mathbf{S}( \mathbf{Q}_i (\xi,\tau)) d\xi d\tau \; \end{array}\end{aligned}$$ and $$\begin{aligned} \label{integrals:eq-2} \begin{array}{c} \mathbf{\mathbf{F}}_{i+\frac{1}{2}} = \frac{1}{\Delta t} \int_{t^n}^{t^{n+1}} \tilde{\mathbf{F}}_{h}( \mathbf{Q}_{i+\frac{1}{2}}^{-}(\tau ), \mathbf{Q}_{i+\frac{1}{2}}^{+}(\tau ) ) d\tau \;, \\ \mathbf{D}_{i+\frac{1}{2}} = \frac{1}{\Delta t} \int_{t^{n}}^{t^{n+1}} \tilde{ \bf D}( \mathbf{Q}_{i+\frac{1}{2}}^{-}(\tau ), \mathbf{Q}_{i+\frac{1}{2}}^{+}(\tau ) ) d\tau \;, \end{array}\end{aligned}$$ where $$\begin{aligned} \label{DOT:eq-1} \begin{array}{c} \tilde{\mathbf{F}}_{h}( \mathbf{Q}_{h}^{-}(\tau ), \mathbf{Q}_{h}^{+}(\tau ) ) = \\ \frac{1}{2}( \mathbf{F}( \mathbf{Q}_{h}^{-} ) + (\mathbf{F}( \mathbf{Q}_{h}^{+} ) ) - \eta \displaystyle \frac{1}{2} \int_{0}^{1} | \mathbf{A}(\Psi(s; \mathbf{Q}_{h}^{-},\mathbf{Q}_{h}^{+} )) | \frac{d\Psi}{ds} ds \;. \end{array}\end{aligned}$$ Moreover $\mathbf{A} := \frac{\partial \mathbf{F}}{\partial \mathbf{Q}}$, with $|\mathbf{A}| = \mathbf{R} |\Lambda|\mathbf{R}^{-1} $, where $|\Lambda| = diag( |\lambda_1|,...,|\lambda_m|)$ and then numerical jumps are obtained as $$\begin{aligned} \begin{array}{c} \tilde{ \bf D}( \mathbf{Q}_{h}^{-}(\tau ), \mathbf{Q}_{h}^{+}(\tau ) ) = \eta \cdot \frac{1}{2} \displaystyle \int_{0}^{1} \mathbf{B}( \Psi(s; \mathbf{Q}_{h}^{-}, \mathbf{Q}_{h}^{+} ) \frac{d\Psi}{ds} ds \;, \end{array}\end{aligned}$$ with $$\begin{aligned} \begin{array}{c} \Psi(s; \mathbf{Q}_{h}^{-}, \mathbf{Q}_{h}^{+} ) := \mathbf{Q}_{h}^{-} + s( \mathbf{Q}_{h}^{+} - \mathbf{Q}_{h}^{-} ) \;. \end{array}\end{aligned}$$ Equation (\[DOT:eq-1\]) corresponds to the DOT (Dumbser-Osher-Toro) Riemann solver, introduced in [@Dumbser:2010b]. The integrals are evaluated numerically, by using some quadrature rule. Here $\mathbf{Q}_{i}(\xi,\tau)$ is a predictor inside the computational cell $[x_{i-\frac{1}{2}}, x_{i+\frac{1}{2}}]$ and $ \mathbf{Q}_{i+\frac{1}{2} }^{\pm} (\tau)$ are extrapolations of the solution at both sides of the cell interface position $x = x_{i+\frac{1}{2} }$. They are usually computed by using the so-called Cauchy-Kowalewsky procedure and Taylor series expansions. The predictor step for second order of accuracy ----------------------------------------------- In this section, we deal with the strategy to get predictor inside the computational cell. This stage requires two ingredients, the first one is a polynomial representation of the solution and then a local evolution in time of approximate values within the computational cell at located spatial positions. The second stage is given by the evolution of values within the computational cell, the resulted state of the local evolution is commonly referred to as the predictor. The polynomial representation of the solution is carried in terms of the so called [*reconstruction procedure*]{}, [@Harten:1987a]. Let us define the reconstruction polynomial within cell $[x_{i - \frac{1}{2}}, x_{i + \frac{1}{2}}]$ as $$\begin{aligned} \begin{array}{c} \mathbf{P}_i(\xi) = \mathbf{Q}_i^n + (\xi - \frac{1}{2}) \mathbf{\Delta}_i\;, \end{array}\end{aligned}$$ with $ \mathbf{\Delta}_i$ being a slope endowing the scheme with the Total Variation Diminishing property. In this work we use the so-called Minmod limiter, see the pioneering work [@Toro:2009a] about TVD flux limiters for hyperbolic conservation laws. In a component wise, it has the form $$\begin{aligned} \begin{array}{ccc} \mathbf{\Delta}_{i,j} = \left\{ \begin{array}{ccl} 0 & ,& if\; ( \mathbf{Q}_{i,j}^n - \mathbf{Q}_{i-1,j}^n ) \cdot ( \mathbf{Q}_{i+1,j}^n - \mathbf{Q}_{i,j}^n ) \leq 0 \;, \\ ( \mathbf{Q}_{i,j}^n - \mathbf{Q}_{i-1,j}^n ) & ,& if \; | \mathbf{Q}_{i,j}^n - \mathbf{Q}_{i-1,j}^n | < |( \mathbf{Q}_{i+1,j}^n - \mathbf{Q}_{i,j}^n ) | \;, \\ ( \mathbf{Q}_{i+1,j}^n - \mathbf{Q}_{i,j}^n ) & ,& if \; | \mathbf{Q}_{i+1,j}^n - \mathbf{Q}_{i,j}^n | < |( \mathbf{Q}_{i,j}^n - \mathbf{Q}_{i-1,j}^n ) | \;. \\ \end{array} \right. \end{array}\end{aligned}$$ Polynomial $ \mathbf{P}_i$ is defined in terms of a local variable $\xi\in[0,1]$ and it is related with the computational cell $[x_{i - \frac{1}{2}}, x_{i + \frac{1}{2}}]$ through out the relationship $x = x_{i - \frac{1}{2}} + \xi \Delta x$. The predictor is an approximation of the solution within the cell, which represents the local evolution of the solution at $x$. This evolution in time is given in terms of a Taylor series expansion $$\begin{aligned} \begin{array}{c} \mathbf{Q}_i (\xi,\tau) = \mathbf{Q}(\xi,0_+) + \tau \partial_t \mathbf{Q} (\xi, 0_+)\;. \end{array}\end{aligned}$$ Notice that the series is truncated up to the first spatial derivative. This is enough to produce solutions of second order of accuracy. The time derivative is expressed in terms of purely spatial derivatives. This is called the Cauchy-Kowalewsky procedure and it is based on the use of the governing equation $$\begin{aligned} \begin{array}{c} \tau \partial_t \mathbf{Q} (\xi, \tau)= - \mathbf{A}(\mathbf{Q} (\xi, \tau)) \partial_x \mathbf{Q}(\xi,\tau)\;. \end{array}\end{aligned}$$ Replacing into the expansion, we have $$\begin{aligned} \begin{array}{c}\mathbf{Q}_i (\xi,\tau) = \mathbf{Q}(\xi,0_+) - \tau \mathbf{A}(\mathbf{Q} (\xi, 0_+)) \partial_x \mathbf{Q}(\xi,0_+)\;. \end{array}\end{aligned}$$ Notice that to obtain the evolution at time $\tau$, we require an approximation of the solution at local time $\tau =0$, which is associated with the global time level $ t^n$. As finite volume only provide an approximation of cell averages, the punctual information within cells is not straightforward obtained. In this stage we use the polynomial representation $\mathbf{P}_i(\xi)$ of the solution at time level $t^n$, $\mathbf{Q}_i^n$, so the Taylor series expansion takes the form $$\begin{aligned} \begin{array}{c}\mathbf{Q}_i (\xi,\tau) = \mathbf{P}_i(\xi) - \tau \mathbf{A}(\mathbf{P}_ (\xi) ) \mathbf{P}'_i(\xi)\frac{1}{\Delta x}\;, \end{array}\end{aligned}$$ which is a polynomial in time. Using this polynomial, we evaluate the integrals (\[integrals:eq-1\]) and (\[integrals:eq-2\]). In particular, for the flux evaluation, we require $$\begin{aligned} \begin{array}{lcl} \mathbf{Q}^{-}_{i+\frac{1}{2}}(\tau) &=& \mathbf{Q}_{i+1}(0,\tau) \;, \\ \mathbf{Q}^{+}_{i+\frac{1}{2}}(\tau) &=& \mathbf{Q}_{i}(1,\tau) \;. \end{array}\end{aligned}$$ Treatment for forward and backward evolution in time {#sect:variation} ---------------------------------------------------- The proposed method uses approximations of classical Riemann problem at cell interfaces, for computing the integrals in (\[integrals:eq-2\]). This is carried out for both the forward ($\eta= 1$) and backward ($\eta= -1$) evolution in time, as hyperbolic equations are not reversible in general we may obtain different results and then we must select the right value for the minimization problem. To illustrate this fact, let us consider the Burgers equation in the context of the unified system. So, the governing equation associated to the constraint PDE, has the form $$\begin{aligned} \begin{array}{c} \partial_t q + \eta q\partial_x q = 0 \;, \end{array}\end{aligned}$$ where $\eta =1 $ for forward evolution in time and $\eta = -1$ for backward evolution in time. If we study Riemann problems associated to the unified system but restricted to the state variable, we have $$\begin{aligned} \begin{array}{c} \partial_t q + \eta q\partial_x q = 0 \;, t > 0\;,\\ q(x,0)= \left\{ \begin{array}{c} q_L\;, x<0\;, \\ q_R\;, x\geq 0 \;,\\ \end{array} \right. \end{array}\end{aligned}$$ with $q_L$ and $q_R$ constant states. The Riemann problem solution of this problem is well studied in the literature, see for instance [@Toro:2009a]. Let us assume that $q_L > 0 > q_R$ and $\eta = 1$, so the solution at the interface position is a shock wave propagating with velocity $S = \frac{q_L +q_R}{2}< q_L$. Therefore, at the interface position it has the value $q(0,t) = q_L$. If now, we solve backward in time the unified problem, we set $\eta = -1$, so we have $$\begin{aligned} \begin{array}{c} \partial_t q - q\partial_x q = 0 \;,\\ q(x,0)= \left\{ \begin{array}{c} q_L\;, x<0\;, \\ q_R\;, x\geq 0 \;.\\ \end{array} \right. \end{array}\end{aligned}$$ Then, by using the change of variable $x=-x$, we obtain $$\begin{aligned} \begin{array}{c} \partial_t q + q\partial_x q = 0 \;,\\ q(x,0)= \left\{ \begin{array}{c} q_R\;, x < 0\;, \\ q_L\;, x \geq 0 \;,\\ \end{array} \right. \end{array}\end{aligned}$$ and so the solution corresponds to a rarefaction wave and thus, at the interface position, takes the value $q(0,t) = 0$. On the other hand, the adjoint state in the backward evolution involves $q$ but it requires the value generated in the forward evolution, in this particular case the shock wave and not the right value of $q$ generated in the backward evolution (a rarefaction wave). If we choose the rarefaction wave, then we sub estimate the required values and thus it may have an impact in the minimization procedure. To force the adjoint state to take the wave propagation of interest for us, we record the state variables from the solution obtained in the forward evolution in time and then use it to provide the evolution of the adjoint state in the backward evolution. That means for $\eta = -1$ we only use (\[one-step:1\]) to evolve the variables associated to the adjoint states and we froze the evolution of state variables. The suggested application of the strategy is listed below. - Given $\mathbf{b}^k$, we solve the constraint PDE, using the one-step finite volume formula (\[one-step:1\]) with $\eta = 1$ up to the output time $t = T$ and record from $\mathbf{Q}$ the variables associated to the state variables $\mathbf{U}$. Final time $T$ is reached in a finite number of steps, let say $n_T$. That means, starting from $t^0 = 0$, we do $t^{n+1} = t^{n} + \Delta t$, $n_T$ times. - Solve the adjoint problem. We use the same formula (\[one-step:1\]) with $\eta=-1$ and the initial condition $\mathbf{Q}^{n_T}_{j+m+n} = \mathbf{P}_{j}^n = 0$, but applied in a different way as in step 1). We turn off the evolution associated with the state variables. That means, we do $$\begin{aligned} \label{one-step:1-1} \begin{array}{c} \mathbf{P}_{i,j}^{n} = \mathbf{P}_{i,j}^{n+1}+ \frac{\Delta t}{\Delta x} (\mathbf{F}_{i+\frac{1}{2},j+m+n}-\mathbf{F}_{i-\frac{1}{2},j+m+n}) \\ +\frac{\Delta t}{\Delta x} (\mathbf{D}_{i+\frac{1}{2},j+m+n} + \mathbf{D}_{i-\frac{1}{2},j+m+n} ) + \frac{1}{\Delta x} (\mathbf{B}\partial \mathbf{Q})_{i,j+m+n} - \Delta t \mathbf{S}_{i,j+m+n} \;, \end{array}\end{aligned}$$ for $j=1,...,(m+n)$. Subindex $j+m+n$ in $\mathbf{F}_{i+\frac{1}{2}}$, $\mathbf{D}_{i+\frac{1}{2}}$, $(\mathbf{B}\partial \mathbf{Q})_{i}$ and $\mathbf{S}_{i}$ represent the $(j+m+n)th$ component. For computing the integrals (\[integrals:eq-1\]) and (\[integrals:eq-2\]) we require also the state variables, but those are used from the record carried out in step 1). Index $n+1$ is consistent with the backward evolution in time, that means starting from $t^0 = T$, we do $t^{n+1} = t^n - \Delta t$. So in a finite number of steps, $n_T$, we reach $t^{n_T} = 0$. - Compute the gradient of the cost function (\[eq:functional:0\]) using variables associated to the adjoint variables as well as state variables. In this paper as shown formally in (\[FormalGrad:eq-1\]), they normally have the form $\nabla J_i = \mathbf{P}_i^{n_T}$ (expressed in a discrete form which is normally used in simulations). Notice that this procedure can have more general structures and may also depend on the state variables, so let us express the gradient formally in terms of a functional, $G(\mathbf{U},\mathbf{P},\mathbf{b})$. So the update of the sought parameters has the form $$\begin{aligned} \label{eq:updateB} \mathbf{b}^{k+1}_i = \mathbf{b}^{k}_i - \lambda_{IP} \cdot G(\mathbf{U},\mathbf{P},\mathbf{b}^k)_i\;,\end{aligned}$$ where $\lambda_{IP} $ is a prescribed constant value. Notice that, the procedure is carried out for finding parameters. However, for another type of problems, for example for finding initial conditions, the procedure is quite similar to the present one. - Stop the procedure using some stop criterion. This is normally carried in terms of the relative error between two successive approximations. Numerical examples {#sec:numerical-examples} ================== In this section, we solve PDE-constrained optimization problems using the numerical scheme described in section \[section:numericalmethod\]. In order to assess the performance of the present scheme, we compare with a conventional solver and so we mimic the strategy usually employed for solving this type of problems, see the reference scheme in Appendix \[sec:reference-scheme\]. Simulations carried out in this section are obtained with the stable time step, $\Delta t$, computed as $$\begin{aligned} \label{stable:timestep} \begin{array}{c} \Delta t = C_{cfl} \frac{ \Delta x}{ \lambda_\infty} \;, \end{array}\end{aligned}$$ where $$\begin{aligned} \begin{array}{c} \lambda_{\infty} = \underset{ i}{\max} \underset{j = 1,...,2(m+n)}{ \max } | \lambda_{j} (\mathbf{H}(x_i) ) |\;, \end{array}\end{aligned}$$ with $ \lambda_{j}$ being the $j$th eigenvalue of the Jacobian matrix and $\mathbf{H}(x_i)$ the initial condition evaluated at the barycentre of cell $[x_{i-\frac{1}{2}}, x_{i+\frac{1}{2}}]$, we choose $C_{cfl} = 0.1$. Notice that, if the initial condition $\mathbf{H}(x)$ changes as a consequence of the change in the sought parameters, the time step must also change accordingly to (\[stable:timestep\]). The Burgers equation -------------------- Let us consider the optimal problem consisting of finding the initial condition $h_0(x)$ such that the solution of the following hyperbolic conservation law $$\begin{aligned} \label{eq:burg-forw:1} \begin{array}{c} \partial_t q + \partial_x( \frac{q^2}{2} ) = 0\;, t\in[0,T] \;, \\ q(x,0 ) = h_0(x) \;, \end{array}\end{aligned}$$ minimizes the functional $$\begin{aligned} \label{eq:Jac-bur:1} J(h_0) = \frac{1}{2}\int_{0}^{T} \int_{-\infty}^{\infty} (q(x,t)-\bar{q}(x,t))^2 dx dt \;,\end{aligned}$$ where $\bar{q}$ is some measurement chosen to be the exact solution of the following Burgers equation at $t=T$ $$\begin{aligned} \label{eq:burg-forw:2} \begin{array}{c} \partial_t q + \partial_x( \frac{q^2}{2} ) = 0\;, t\in[0,T] \;, \\ q(x,0 ) = \bar{h}_0(x) \;, \end{array}\end{aligned}$$ where $\bar{h}_0(x)$ is a prescribed function. These problems are endowed with transmissive boundary conditions. The adjoint problem is given by $$\begin{aligned} \label{eq:burg-back:1} \begin{array}{c} \partial_t p + q \partial_x p = \bar{q}- q \;,t\leq T \;, \\ p(x,T) = 0 \;. \end{array}\end{aligned}$$ See [@Lecaros:2014a] for further details concerning the derivation of the adjoint problem. The gradient of the cost function is $$\begin{aligned} \label{burger:jacobian-j} \begin{array}{c} \nabla J(h_0) = p(x,0)\;. \end{array}\end{aligned}$$ The unified formulation, as proposed in this paper takes the form $$\begin{aligned} \label{eq:system:1} \begin{array}{c} \partial_t \mathbf{Q} + \partial_x \mathbf{F}(\mathbf{Q} ) + \mathbf{B}(\mathbf{Q}) \partial_x \mathbf{Q} = \mathbf{S} ( \mathbf{Q} ) \;,\\ \end{array}\end{aligned}$$ with $$\begin{aligned} \begin{array}{c} \mathbf{Q} =\left[ \begin{array}{c} q \\ p \end{array} \right], \; \mathbf{F}( \mathbf{Q} ) =\left[ \begin{array}{c} \frac{q^2}{2} \\ 0 \end{array} \right], \; \mathbf{S}( \mathbf{Q} ) =\left[ \begin{array}{c} 0 \\ \bar{q} - q \end{array} \right], \; \end{array}\end{aligned}$$ and $$\begin{aligned} \begin{array}{c} \mathbf{B}(\mathbf{Q}) = \left[ \begin{array}{cc} 0 & 0 \\ 0 & q \end{array} \right]\;. \end{array}\end{aligned}$$ System (\[eq:system:1\]) is solved from $t=0$ up to $t=T$, thus we need an initial condition at $t = 0$. Notice that $p(x,t)$ has a kind of initial condition only at $t = T$. At $t=0$ we have to set information for $p(x,t)$. However, as $p$ does not influences the evolution of $q$ we set $p(x,0) = p_0 = 0.$ So, the initial condition is $\mathbf{Q}(x,0) = [h_0(x), 0]^T$. On the other hand, at $t= T$, the second component is set to zero which means $$\mathbf{Q}(x,T) =[ q(x,T), 0 ]^T \;.$$ In this way, the evolution of the adjoint variable begins with the right initial condition as required in (\[eq:burg-back:1\]). We are going to implement the procedure for continuous as well as discontinuous initial conditions. In order to assess the performance of the solver we compute the error given by $ Error = max_{i} | h_{0,i} - \bar{h}_0(x_i)|$, where $h_{0,i}$ is the approximated initial condition at the cell $[x_{i-\frac{1}{2}}, x_{i +\frac{1}{2} }]$. ### The Burgers equation with discontinuous initial condition Let us consider the case in which the aim is to find a discontinuous initial condition. So we setup the optimization problem as follows. We set a starting guess for the procedure, in this case we set $h_0(x) = - 0.2 $. We solve both constraint PDE and the adjoint problem on the computational domain $[-3,3]$ using $200$ uniform cells. The expected initial condition is $$\begin{aligned} \begin{array}{c} \bar{h}_0 (x) = \left\{ \begin{array}{c} \frac{1}{2} \;, x < 0 \;, \\ 0 \;, x \geq 0 \;. \\ \end{array} \right. \end{array}\end{aligned}$$ To update the sought initial condition we use (\[eq:updateB\]) with $\lambda_{IP} = 0.7$ and $ G(\mathbf{U},\mathbf{P},\mathbf{b}^k)$ given by (\[burger:jacobian-j\]). We also implement the reference scheme described in appendix \[sec:reference-scheme\] and using the same information described above. The results are depicted in Figure \[fig:disc-Burguers\] at the $80$th iteration. The unified formulation has $ Error= 6.05 \cdot 10^{-2} $ whereas the solution with the reference solution called here as the conventional solver has $Error = 0.11$. Figure \[fig:disc-Burguers-error\] depicts the error at each iteration of the global procedure. We observe that the error is not so good for both methods even when the unified formulation provides the best approximation, it shows undershoot and overshoot at the interface position but the undershoot is less than that resulting from the conventional solver. ![Burgers, discontinuous initial condition. Sought initial condition, results of unified (circles) and conventional (squares) formulations. The constraint PDE problem is evolved up to $T = 0.12$.[]{data-label="fig:disc-Burguers"}](BurguesDiscontinuous.pdf) ![Burgers, discontinuous initial condition. Comparison between errors associated with the unified formulation (squares) and conventional formulation (circles). The constraint PDE problem is evolved up to $T = 0.12$.[]{data-label="fig:disc-Burguers-error"}](ErrorBurguersDicontinuous.pdf) Note that in this test, the interface position is found and also the mean shape of the initial profile. However, we observe that the undershot and also the overshot appear near to the $18$th iteration as is shown in Figure \[fig:disc-Burguers-error\]. Thus it propagates for a while and then it is stabilized. However, these undershoot and overshoot do not disappear. ### The Burgers equation with continuous initial condition Let us consider the case in which the aim is to find a continuous initial condition. So we setup the descent algorithm as follows. The optimization procedure is initialized with $h_0(x) = 0 $. We solve both constraint PDE and the adjoint problem on the computational domain $[0,1]$ using $160$ uniform cells. The expected initial condition is $$\begin{aligned} \begin{array}{c} \bar{h}_0 (x) = sin(2 \pi x) \;. \end{array}\end{aligned}$$ To update the sought initial condition we use (\[eq:updateB\]) with $\lambda_{IP} = 2.7$ and $ G(\mathbf{U},\mathbf{P},\mathbf{b}^k)$ given by (\[burger:jacobian-j\]). We also implement the reference scheme described in appendix \[sec:reference-scheme\]. ![Burgers, smooth initial condition. Sought initial condition, results of unified (circles) and conventional (squares) formulations. The constraint PDE problem is evolved up to $T = 0.1$.[]{data-label="fig:disc-BurguersC"}](BurguesContinuous.pdf) ![Burgers, smooth initial condition. Comparison between errors associated with the unified formulation (squares) and conventional formulation (circles). The constraint PDE problem is evolved up to $T = 0.1$.[]{data-label="fig:disc-BurguersC-error"}](ErrorBurguersContinuous.pdf) The results are depicted in Figure \[fig:disc-BurguersC\] at the $40$th iteration. The unified formulation has $ Error= 6.68 \cdot 10^{-3} $ whereas the solution with the reference scheme has $Error =1.08\cdot 10^{-2} $. Figure \[fig:disc-BurguersC-error\] depicts the error at each iteration of the global procedure. We observe that the error for the present unified strategy is one-order of magnitude more accurate than the reference scheme. If the variant described in section \[sect:variation\] is not included, that means conventional finite volume implementation, the numerical scheme presented in section \[section:numericalmethod\]) is of second order of accuracy. It seems to be that the high accuracy is still present even if the state variable is frozen in the evolution backward in time associated with the adjoint problem. A CPU time measurement gives us that the present scheme is almost three times more expensive than the reference scheme. However, the solution of the constraint PDE, adjoint model and approximation update, take about three seconds, (it depends on $\Delta t$), even so the scheme is very efficient in terms of computational cost. The shallow water equations {#section:sw} --------------------------- In this section we deal with the solution of the shallow water equation written as follows $$\begin{aligned} \label{eq:sw:1} \begin{array}{c} \partial_t \left[ \begin{array}{c} h\\ q \end{array} \right] + \partial_x \left[ \begin{array}{c} \varepsilon q\\ \varepsilon \frac{q^2}{h} + \frac{1}{2}\frac{h^2}{\varepsilon} \end{array} \right] = \left[ \begin{array}{c} 0\\ -h \partial_x b(x) \end{array} \right] \;, \end{array}\end{aligned}$$ where $\zeta(x,t)$ and $b(x)$ are the free surface and bottom parameterization respectively, $h(x,t)= 1 + \varepsilon (\zeta - b)$ is the total depth, $u(x,t)$ is the total depth averaged velocity, $\varepsilon$ is a dimensionless parameter and $q=hu$ [@Lannes:2013a]. There is an extensive literature concerning the general water-waves equations and shallow water approximations regimes [@rs1997modern; @Lannes:2005a; @Lannes:2013a]. Among the many applications to coastal engineering we focus on the bottom detection through measurements on the free surface $\zeta$. This problem has practical interests [@Nersisyan01082015] as well as theoretical challenges [@alazard2015control]. Because of the difficulties arising in the general water-wave system, many asymptotic models are proposed to study this phenomenon [@boussinesq1872theorie; @israwi2011large; @Lannes:2013a; @Peregrine:1967a]. We are going to consider the nonlinear shallow water equations (or Saint-Venant) [@de1871th] for study here the inverse problem consisting of finding the bottom surface $b(x)$ such that a prescribed free surface $ \bar{\zeta}(x,t)$ is achieved. The adjoint problem associated to this problem is $$\begin{aligned} \label{SW:adjoint} \begin{array}{c} \partial_t \left[ \begin{array}{c} \tilde{h}\\ \tilde{q} \end{array} \right] + \left[ \begin{array}{cc} 0 & \frac{h}{\varepsilon} - \varepsilon \frac{ q^2}{h^2} \\ \varepsilon & 2 \varepsilon \frac{ q }{ h} \end{array} \right] \partial_x \left[ \begin{array}{c} \tilde{h}\\ \tilde{q} \end{array} \right] = \left[ \begin{array}{c} \frac{\bar{\zeta} - \zeta}{\varepsilon} + \tilde{q} \partial_x b \\ 0 \end{array} \right] \;, \end{array}\end{aligned}$$ and the gradient of this functional is given by $$\begin{aligned} \label{SW:gradient} \begin{array}{c} \nabla J = \frac{ h -1}{ \varepsilon} + b - \bar{\zeta} + (h\tilde{q} )\;. \end{array}\end{aligned}$$ We neglect the term $(h\tilde{q} )$ and take $\lambda_{IP}\neq 1$. In this way the scheme is forced to do more than one iteration and the computed free surface has to be very close to the expected one. Notice that, the bottom surface is a prescribed function, with $\partial_t \mathbf{b} (x) = 0 $ so it can be included into the model as a non-evolutionary variable, it would allow us to include non-smooth bottom surfaces. So, the model now takes the form $$\begin{aligned} \label{eq:sw:1-with-b} \begin{array}{c} \partial_t \left[ \begin{array}{c} h\\ q \\ b \end{array} \right] + \partial_x \left[ \begin{array}{c} \varepsilon q\\ \varepsilon \frac{q^2}{h} + \frac{1}{2}\frac{h^2}{\varepsilon} \\ 0 \end{array} \right] + \left[ \begin{array}{ccc} 0 & 0 & 0\\ 0 & 0 & h \\ 0 & 0 & 0\\ \end{array} \right] \partial_x \left[ \begin{array}{c} h\\ q \\ b \end{array} \right] = \left[ \begin{array}{c} 0\\ 0\\ 0 \end{array} \right] \;, \end{array}\end{aligned}$$ The unified formulation for this test is given by $$\begin{aligned} \label{eq:system:sw-1} \begin{array}{c} \partial_t \mathbf{Q} + \partial_x \mathbf{F}(\mathbf{Q} ) + \mathbf{B}(\mathbf{Q}) \partial_x \mathbf{Q} = \mathbf{S} ( \mathbf{Q} ) \;,\\ \end{array}\end{aligned}$$ where $$\begin{aligned} \begin{array}{c} \mathbf{Q} \left[ =\begin{array}{c} h \\ q \\ b \\ \tilde{h} \\ \tilde{q} \end{array} \right]\;, \mathbf{F}( \mathbf{Q} ) \left[ =\begin{array}{c} \varepsilon q\\ \varepsilon \frac{q^2}{h} + \frac{1}{2}\frac{h^2}{\varepsilon}\\ 0 \\ 0 \\ 0 \end{array} \right]\;, \\ \mathbf{B}(\mathbf{Q} ) = \left[ \begin{array}{ccccc} 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & h & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{h}{\varepsilon} - \varepsilon \frac{q^2}{h^2} \\ 0 & 0 & -\tilde{q} & \varepsilon & 2\varepsilon \frac{q}{h} \end{array} \right]\;, \mathbf{S}( \mathbf{Q} ) \left[ =\begin{array}{c} 0 \\ 0 \\ 0 \\ \frac{\bar{\zeta}-\zeta}{\varepsilon} \end{array} \right]\;. \end{array}\end{aligned}$$ Notice that the Jacobian matrix of $\mathbf{F}$ with respect to $\mathbf{Q}$, $\mathbf{A}(\mathbf{Q})$ is given by $$\begin{aligned} \begin{array}{c} \mathbf{A}(\mathbf{Q}) = \left[ \begin{array}{ccccc} 0 & \varepsilon & 0 & 0 & 0 \\ \frac{ h }{\varepsilon} - \frac{q^2 }{ h^2} \varepsilon & 2\varepsilon \frac{q}{h} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ \end{array} \right]\;. \end{array}\end{aligned}$$ Moreover, this has the decomposition $\mathbf{A} = \mathbf{R} \mathbf{\Lambda} \mathbf{R}^{-1}\;,$ with $$\begin{aligned} \begin{array}{c} \mathbf{R}(\mathbf{Q}) = \left[ \begin{array}{ccccc} 1 & 1 & 0 & 0 & 0 \\ \frac{ q^2 \varepsilon^2 - h^3 }{ h q \varepsilon^2 + h^{\frac{5}{2}} \varepsilon} & \frac{ q^2 \varepsilon^2 - h^3}{ h q \varepsilon^2 - h^{\frac{5}{2}} \varepsilon} & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ \end{array} \right] \;, \end{array}\end{aligned}$$ $$\begin{aligned} \begin{array}{c} \mathbf{\Lambda}(\mathbf{Q}) = \left[ \begin{array}{ccccc} u \varepsilon - \sqrt{h} & 0 & 0 & 0 & 0 \\ 0 & u \varepsilon + \sqrt{h} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ \end{array} \right] \;, \end{array}\end{aligned}$$ $$\begin{aligned} \begin{array}{c} \mathbf{R}(\mathbf{Q})^{-1} = \left[ \begin{array}{ccccc} \frac{ q^{2} \epsilon^{2} - { h^{3}}}{2 h^{\frac{3}{2}} q \varepsilon - 2 h^{3} } & - \frac{\varepsilon}{2 \sqrt{ h } } & 0 & 0 & 0 \\ - \frac{ q^{2} \epsilon^{2} - { h^{3}}}{2 h^{\frac{3}{2}} q \varepsilon + 2 h^{3} } & \frac{\varepsilon}{2 \sqrt{ h } } & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ \end{array} \right] \;. \end{array}\end{aligned}$$ In this test, the aim is to find the bottom surface $b(x)$ such that the model system (\[eq:sw:1\]) provides a free surface $\zeta(x,t)$ close to the profile $\bar{\zeta}(x,t) = \frac{ 0.3 (x - t) }{ cosh(x - t)^2}$. Here (\[eq:sw:1\]) is endowed with transmissive boundary conditions and initial condition $q(x,t) = 0$, $h(x,0)=1$ and $b(x) = 0.2$, see Figure \[fig:sw:StartingGuess\]. We use the propose methodology for finding the set of condition to find the sought profile up to the final time $t_{out} = 3$. For simulations we have used $300$ cells, $C_{cfl} = 0.1$, $\varepsilon = 0.01$. To update the sought bottom surface we set $\lambda_{IP}= 1.9$. Figure \[fig:sw:Converged\] shows the result of the simulation, as you can see the system achieve the sough profile with a good resolution despite of the few iterations as well. Notice that $\bar{\zeta}$ does not correspond to any exact solution of model (\[eq:sw:1\]) with initial conditions used in this model. So this is not a test using synthetic data. However, the procedure is able to provide an acceptable approximation of the sought profile. Model system consisting of (\[eq:sw:1\]), (\[SW:adjoint\]) and (\[SW:gradient\]), is solved using the scheme of reference, appendix \[sec:reference-scheme\]. Notice also that this strategy is independent of the elected functional $J$ and can be employed to deal with more general shallow water regimes [@israwi2011large; @Lannes:2013a; @Peregrine:1967a], including the possibility of detecting moving bottoms $b(x,t)$. ![Profile $\zeta(x,t)$ of the starting initial condition to initialize the global optimization procedure.[]{data-label="fig:sw:StartingGuess"}](initialcondition.pdf) ![Shallow water. Sought profile for the shallow water, $\varepsilon = 0.01$. Using $300$ cells, $\lambda_{IP}= 1.9$. Bottom profile (thick line), free surface from unified formulation (circle), free surface from reference scheme (square) and expected free surface (continuous line). At $40th$ iteration, unified formulation generates $Error =4.93 \cdot 10^{-3} $ and conventional formulation generate $Error =5.49 \cdot 10^{-3} $. []{data-label="fig:sw:Converged"}](SWplotFreeSurface.pdf) ![Shallow water. Error measured as the distance with respect to the prescribed free surface for both, the unified formulation (square) and conventional formulation (circles).[]{data-label="fig:sw:Error"}](ErrorSW.pdf) Conclusions {#sec:conclusions} =========== In the present paper, we have expressed the adjoint method for PDE-constrained optimization problems in a unified frame. A numerical scheme based on a class of high order finite volume schemes has been adapted for solving the unified system. The scheme is able to deal with non-conservative equations and we have seen that the degree of approximation is at least those of conventional solvers. We have solved two test problems, the first one corresponds to the Burgers equations and the second one, the Nonlinear Shallow Water equations. In the case of Burgers equations, we assess two cases. First, we have solved the problem of finding an initial condition which is a discontinuous function, which generates a shock wave. The method finds the right location and also the main shape of the initial profile. We observed stationary overshot and undershot at the interface position, it is also observed by the reference scheme. Second, we recover a smooth initial condition, the constraint PDE is solved up to $T = 0.12$, before discontinuities start developing, so the solution remains continuous. In this case, the procedure has provided very good agreements in both the present and reference schemes. Regarding the shallow water equations, we also have obtained very good agreements. We have shown the evolution of the error between the computed sought parameters and that obtained in the simulations. We have observed that the present scheme always generated approximations with slightly better approximation than the reference scheme in the case of discontinuous solutions and for smooth cases, it has generated approximations whose accuracy is almost one order of magnitude. Acknowledgements {#acknowledgements .unnumbered} ================ G.I. Montecinos thanks FONDECYT in the frame of the research project FONDECYT Postdoctorado 2016, number 3160743. The numerical scheme of reference {#sec:reference-scheme} ================================= In this section, we modify a strategy available in the literature for solving this type of problem, given by [@Lellouche:1994a]. The original scheme is implemented to solve an inverse problem to find information at boundaries. In this work we are interested in solving initial value problems, boundary conditions in our case are just of the transmissive type. Moreover, we set the strategy for finding model parameters $\mathbf{b}$, through out the adjoint method. So, we provide an initial parameter $\mathbf{b}$ and then we apply the following procedure. - Given $\mathbf{b}^k$, we solve the hyperbolic model (\[eq:primal:0\]). Tests in this paper are conservatives, we neglect source terms $\mathbf{L}$. So, we use the following conservative scheme for marching in time up to the output time $t = T$, which is reached in a finite number of steps, let say $n_T$, that means starting from $t^0 = 0$, we do $t^{n+1} = t^{n} + \Delta t$, $n_T$ times up to $t^{n_T} = T$. $$\begin{aligned} \begin{array}{c} \mathbf{U}^{n+1}_i = \mathbf{U}^{n}_i - \frac{\Delta t}{\Delta x} \left[ \mathbf{R}_{i+\frac{1}{2}} - \mathbf{R}_{i-\frac{1}{2}} \right] + \Delta t \mathbf{L}(\mathbf{U}^{n}_i, \mathbf{b}^{k}_{i} ) \\ + \frac{\Delta t}{2 \Delta x} \tilde{\mathbf{B}}(\mathbf{U}^{n}_i, \mathbf{b}^{k}_{i} ) ( \mathbf{b}^{k}_{i+1} - \mathbf{b}^{k}_{i-1} ) \;, \end{array}\end{aligned}$$ where $ \mathbf{R}_{i+\frac{1}{2}} = \frac{1}{2}( \mathbf{R}( \mathbf{ U}_{i+1}^n,\mathbf{b}^k ) + \mathbf{R}(\mathbf{ U}_{i}^n,\mathbf{b}^k ) ) - \frac{\lambda_{i+\frac{1}{2}} }{2} (\mathbf{ U}_{i+1}^n - \mathbf{ U}_{i}^n ) .$ This corresponds to the well known numerical flux of Rusanov, $ \lambda_{i+\frac{1}{2}} = \max ( \lambda (\mathbf{ U}_{i}^n), \lambda (\mathbf{ U}_{i+1}^n) )$ with $\lambda (\mathbf{ U}) $ being the maximum of the eigenvalues (in magnitude) of the Jacobian matrix of $ \mathbf{R}$ with respect to $\mathbf{U}$. - Once the previous step is completed. We start the backward evolution of the so called adjoint method. We use a finite difference approach. We set the initial condition $\mathbf{P}^{0}_i = 0$. $$\begin{aligned} \begin{array}{c} \mathbf{P}^{n}_i = \mathbf{P}^{n+1}_i + \frac{\Delta t}{2 \Delta x}\mathbf{ J}_R^T( \mathbf{U}^{n+1}_i ) \left[ \mathbf{P}^{n+1}_{i+1} - \mathbf{P}^{n+1}_{i-1} \right] - \Delta t \tilde{\mathbf{S}} ( \mathbf{U}^{n+1}_i ) \;. \end{array}\end{aligned}$$ Index $n+1$ is consistent with the backward evolution in time, that means starting from $t^0 = T$, we do $t^{n+1} = t^n - \Delta t$. So in a finite number of steps, $n_T$, we reach $t^{n_T} = 0$. - Update the parameter $\mathbf{b}^k$. It is carried out using the gradient of the cost functional (\[eq:functional:0\]). The expression for the gradient of this functional depends on the problem at hand. In this paper they normally have the form $\nabla J_i = \mathbf{P}_i^{n_T}$ in terms of the discretization. Notice that this procedure can have more general structures and may also depend on the state variables, so let us express the gradient formally in terms of a functional, $G(\mathbf{U},\mathbf{P},\mathbf{b})$. So the update of the sought parameters has the form $$\begin{aligned} \mathbf{b}^{k+1}_i = \mathbf{b}^{k}_i - \lambda_{IP} \cdot G(\mathbf{U},\mathbf{P},\mathbf{b}^k)_i\;,\end{aligned}$$ where $\lambda_{IP} $ is a prescribed constant value. Notice that, the procedure is carried out for finding parameters. However, for another type of problems, for example for finding initial conditions, the procedure is quite similar to the present one. - Stop the procedure using some stopped criterion. This is normally carried in terms of some relative error. Let us point out that the scheme in [@Lellouche:1994a] is globally implicit, in turns, here we derive the global explicit version of it. This is because the proposed scheme in this paper is globally explicit.

--- author: - | \ Max-Planck-Institut für Radioastronomie, Bonn, Germany\ E-mail: title: Transient sources at the highest angular resolution --- Introduction ============ Radio transients \[1\] are observed from the nearby Sun to cosmological distances. A classification of transients following the physical process in act is complicated: different physical processes, as magnetic reconnection, shocks, electron beams, etc., may work in the same object. A possible classification is that based on the kind of astronomical objects where the transient occur: cool stars, supernovae, pulsars and jets. Straightforward and based in differences in timescale is the classification of radio transients in fast transients and slow transients. Fast radio transients may have timescales of nanoseconds to minutes and they typically are discovered in time-series data. The slow radio transients, variable on timescales of seconds up to years, are typically discovered in images and are those described here. Transients in binary systems, where the interaction between the components is the cause of the transient, are ideal targets for monitoring studies. In the next section I will therefore examine slow radio transients in binary systems. Transients in a young-stellar system ==================================== The binary system V773 Tau A \[2\] shows outstanding magnetic activity demonstrated by radio and X-ray flares and the presence of large, cool, photospheric spots. A periodicity of 51.1 d, the same as the orbital one, has been discovered in the radio flaring activity via long-term monitoring with the Effelsberg 100-m telescope: large flares cluster around periastron passage \[3\]. The orbit, with an eccentricity of $e\,=\,0.27$ \[2\], results at apastron in a distance of 52 stellar radii (1 $R_*$ = 2 solar radii) between the two stars, and at periastron of 30 stellar radii. If the strong flaring activity at periastron is due to interacting coronae, then the magnetic structures should have sizes of at least 15 stellar radii. Observations at 90 GHz around periastron have monitored the onset and decay of a large flare and show that the decay of few hours is consistent with continuous leakage of the emitting relativistic electrons from a magnetic structure of 10–20 stellar radii \[4\]. There exists a close relationship between flares and interaction of magnetic structures. In fact, as observed on the Sun, flares can be triggered by interactions between new and older emergences of magnetic flux in the same area \[5\]. Part of the magnetic energy released during reconnection goes to accelerate a fraction of the thermal electrons trapped in the flaring coronal loop and a population of relativistic electrons is produced \[6,7\]. ![ Phase-referenced images of consecutive (25 February $-$ 2 March 2010) 8.4 GHz VLBA+EB observations of V773 Tau A. Overlaid is the orbit of the binary stellar system; the circle of 4 R$_*$ centered at each star’s position indicates a symmetrical corona of an average size of 3 R$_*$ (the stellar radius corresponds to 0.07 mas at 132.8 pc distance). The restoring beam is shown in the bottom left corner of each map; it is on average of (0.5 $\times$ 1.4) mas. Circularly polarized emission (traced in colour) is associated with the secondary star in E and F with V/I equal to $-4.3\% \pm 0.4\%$, $-8.0\% \pm 0.3\%$ and also to the primary in E, with opposite polarization sense and V/I of $4.0\% \pm 0.4\%$ \[8\].[]{data-label="fig1"}](massifig2a.pdf "fig:"){width=".15\textwidth"} ![ Phase-referenced images of consecutive (25 February $-$ 2 March 2010) 8.4 GHz VLBA+EB observations of V773 Tau A. Overlaid is the orbit of the binary stellar system; the circle of 4 R$_*$ centered at each star’s position indicates a symmetrical corona of an average size of 3 R$_*$ (the stellar radius corresponds to 0.07 mas at 132.8 pc distance). The restoring beam is shown in the bottom left corner of each map; it is on average of (0.5 $\times$ 1.4) mas. Circularly polarized emission (traced in colour) is associated with the secondary star in E and F with V/I equal to $-4.3\% \pm 0.4\%$, $-8.0\% \pm 0.3\%$ and also to the primary in E, with opposite polarization sense and V/I of $4.0\% \pm 0.4\%$ \[8\].[]{data-label="fig1"}](massifig2c.pdf "fig:"){width=".15\textwidth"} ![ Phase-referenced images of consecutive (25 February $-$ 2 March 2010) 8.4 GHz VLBA+EB observations of V773 Tau A. Overlaid is the orbit of the binary stellar system; the circle of 4 R$_*$ centered at each star’s position indicates a symmetrical corona of an average size of 3 R$_*$ (the stellar radius corresponds to 0.07 mas at 132.8 pc distance). The restoring beam is shown in the bottom left corner of each map; it is on average of (0.5 $\times$ 1.4) mas. Circularly polarized emission (traced in colour) is associated with the secondary star in E and F with V/I equal to $-4.3\% \pm 0.4\%$, $-8.0\% \pm 0.3\%$ and also to the primary in E, with opposite polarization sense and V/I of $4.0\% \pm 0.4\%$ \[8\].[]{data-label="fig1"}](massifig2e.pdf "fig:"){width=".15\textwidth"} ![ Phase-referenced images of consecutive (25 February $-$ 2 March 2010) 8.4 GHz VLBA+EB observations of V773 Tau A. Overlaid is the orbit of the binary stellar system; the circle of 4 R$_*$ centered at each star’s position indicates a symmetrical corona of an average size of 3 R$_*$ (the stellar radius corresponds to 0.07 mas at 132.8 pc distance). The restoring beam is shown in the bottom left corner of each map; it is on average of (0.5 $\times$ 1.4) mas. Circularly polarized emission (traced in colour) is associated with the secondary star in E and F with V/I equal to $-4.3\% \pm 0.4\%$, $-8.0\% \pm 0.3\%$ and also to the primary in E, with opposite polarization sense and V/I of $4.0\% \pm 0.4\%$ \[8\].[]{data-label="fig1"}](massifig2g.pdf "fig:"){width=".15\textwidth"} ![ Phase-referenced images of consecutive (25 February $-$ 2 March 2010) 8.4 GHz VLBA+EB observations of V773 Tau A. Overlaid is the orbit of the binary stellar system; the circle of 4 R$_*$ centered at each star’s position indicates a symmetrical corona of an average size of 3 R$_*$ (the stellar radius corresponds to 0.07 mas at 132.8 pc distance). The restoring beam is shown in the bottom left corner of each map; it is on average of (0.5 $\times$ 1.4) mas. Circularly polarized emission (traced in colour) is associated with the secondary star in E and F with V/I equal to $-4.3\% \pm 0.4\%$, $-8.0\% \pm 0.3\%$ and also to the primary in E, with opposite polarization sense and V/I of $4.0\% \pm 0.4\%$ \[8\].[]{data-label="fig1"}](massifig2i.pdf "fig:"){width=".15\textwidth"} ![ Phase-referenced images of consecutive (25 February $-$ 2 March 2010) 8.4 GHz VLBA+EB observations of V773 Tau A. Overlaid is the orbit of the binary stellar system; the circle of 4 R$_*$ centered at each star’s position indicates a symmetrical corona of an average size of 3 R$_*$ (the stellar radius corresponds to 0.07 mas at 132.8 pc distance). The restoring beam is shown in the bottom left corner of each map; it is on average of (0.5 $\times$ 1.4) mas. Circularly polarized emission (traced in colour) is associated with the secondary star in E and F with V/I equal to $-4.3\% \pm 0.4\%$, $-8.0\% \pm 0.3\%$ and also to the primary in E, with opposite polarization sense and V/I of $4.0\% \pm 0.4\%$ \[8\].[]{data-label="fig1"}](massifig2k.pdf "fig:"){width=".15\textwidth"} These relativistic particles gyrating around the magnetic field lines of the loop, where they are confined, generate synchrotron emission in the radio band. Applying this knowledge of solar flares to V773 Tau A, it is clear that in this system the observed relationship between intensity of the flare occurrence and the distance between the two stars indicates another, new mechanism of magnetic interaction, that of interacting coronae. In this case magnetic reconnection would take place far out from the stellar surfaces, where the two coronae interact with each other. Figure 1 shows phase referenced maps of consecutive VLBA+EB observations aimed to spatially trace the flare evolution and polarized emission around periastron passage where the intensity of the flares is highest \[8\]. The visual orbit is derived from an interferometric-spectroscopic orbital model \[2\] and the primary star is overlapped with the North-East feature in the map of run D. By using the images, the total flux density monitoring, model fitting and the information from circular polarization (here in colour) we have a powerful tool for a better understanding of the physical process triggering the transient and of the magnetic field topology of weak-lined T Tauri stars. Radio transients in X-ray binaries ================================== X-ray binaries are stellar systems formed by a compact object (black hole or neutron star) and a normal star. Neutron stars in X-ray binaries may have quite different values of magnetic field, being the range of B about $10^{8}-10^{12}$ G; however, X-ray binaries with a radio emitting jet, systems called “microquasars”, have as compact object either an accreting black hole or an accreting neutron star with low magnetic field ($10^{8}$ G) \[9\]. Radio jets have been imaged at high resolution for the three neutron star systems: Scorpius X-1 \[10\], Circinus X-1 \[11\] and Cygnus X-2 \[12\]. The statistically more powerful radio jets associated to accreting black holes have quite well studied spectral characteristics. There are two kind of jets; the steady jet and the transient jet \[13\]. The origin of a steady jet is relatively well understood as a result of magneto-rotational processes: an initial vertical magnetic field treading the accretion disk is bent by the differentially rotating disk, then magnetic pressure gradient accelerates plasma out of system and magnetic tension pinches and collimates the outflow into a jet \[14\]. Changes in the plasma density and in the strength of the magnetic field along the conical jet create distinct regions of synchrotron emission with each region contributing with a spectrum peaking at a different frequency. Then the overall spectrum, observed with a spatial resolution insufficient to resolve the individual parts of the jet, will be almost flat, i.e., with a spectral index $\alpha \sim 0$, or inverted, i.e., $\alpha > 0$ \[13, 15\] (flux density $S\propto \nu^{\alpha}$). The other type of jet associated to microquasars is the so called transient jet. ![Left: Flat radio spectrum of [LS I +61$^{\circ}$303 ]{}measured with the Effelsberg 100-m telescope at 2.8, 4.85, 8.35, 10.45, 14.3, 23 and 32 GHz \[24\]. Center: Periodogram of GBI radio data. Right: Periodogram of *Fermi*-LAT data (see Sect. 3.1).[]{data-label="fig2"}](spettri13proc.pdf "fig:"){width=".25\textwidth"} ![Left: Flat radio spectrum of [LS I +61$^{\circ}$303 ]{}measured with the Effelsberg 100-m telescope at 2.8, 4.85, 8.35, 10.45, 14.3, 23 and 32 GHz \[24\]. Center: Periodogram of GBI radio data. Right: Periodogram of *Fermi*-LAT data (see Sect. 3.1).[]{data-label="fig2"}](GBIscargle.pdf "fig:"){width=".3\textwidth"} ![Left: Flat radio spectrum of [LS I +61$^{\circ}$303 ]{}measured with the Effelsberg 100-m telescope at 2.8, 4.85, 8.35, 10.45, 14.3, 23 and 32 GHz \[24\]. Center: Periodogram of GBI radio data. Right: Periodogram of *Fermi*-LAT data (see Sect. 3.1).[]{data-label="fig2"}](FermiLAT_scargle.pdf "fig:"){width=".3\textwidth"} It corresponds to an optically thin radio outburst (i.e., $\alpha < 0$) and occurs always [*after*]{} the flat spectrum phase. The transient jet is thought to be associated to shocks where high relativistic plasma catches up with the pre-existing slower-moving material of the steady jet \[13\]. The switch from the self-absorbed jet to the transient one presents differences in the different sources. In the microquasar [GRS 1915+105 ]{}the switch is between a steady, plateau state and the optically thin outburst \[Fig. 1, in 16\], whereas in XTE J1752-223 the switch is between an optically thick outburst and an optically thin outburst \[Fig. 1, in 17\]. A unique case of a radio outburst from an X-ray binary having as compact object a non-accreting neutron star is PSR B1259$-$63. The neutron star is in this case a young pulsar, i.e., having both a strong B ($10^{12}$ G) and a fast rotation (msec period). Around periastron passage the interaction of the pulsar wind and the wind of the companion star generates an optically thin outburst \[18\] mapped at high resolution \[19\]. Latest results on a transient source: The gamma-ray binary [LS I +61$^{\circ}$303]{} ------------------------------------------------------------------------------------ The stellar system [LS I +61$^{\circ}$303 ]{}is formed by a compact object and a Be star in an eccentric orbit, $e = 0.72 \pm 0.15$ \[20\]. High resolution radio images show a structure that not only changes position angle, but it is even sometimes one-sided and at other times two-sided. This suggested the hypothesis of [LS I +61$^{\circ}$303 ]{}being a precessing microquasar \[21\]. A precession of the jet leads to a variation in the angle between the jet and the line of sight, and therefore to variable Doppler boosting. The result is both a continuous variation in the position angle of the radio-emitting structure and its flux density. The rapid variations were alternatively interpreted \[22\] to be due to a young pulsar whose wind enters in collision with the disc/wind of the companion star, i.e., a system as PSR B1259$-$63. \ \ As we saw in the previous section, the radio spectral index for a radio emitting jet of a microquasar and that of a radio emitting nebula associated to interacting winds are quite different. A flat/inverted spectrum is expected for a radio jet, followed by an optically thin outburst in case of a transient. Just an optically thin outburst is expected for the pulsar nebula. It is therefore worth to examine the radio characteristics of [LS I +61$^{\circ}$303 ]{}in detail. ![Long-term modulation in LSI+61303. a) 8 GHz GBI observations (red) and model (black) data \[32\]. b and c) Zoom of two intervals of Fig. 4 a. ](Fig2a.pdf "fig:"){width="35.00000%"} ![Long-term modulation in LSI+61303. a) 8 GHz GBI observations (red) and model (black) data \[32\]. b and c) Zoom of two intervals of Fig. 4 a. ](Fig2b.pdf "fig:"){width="25.00000%"} ![Long-term modulation in LSI+61303. a) 8 GHz GBI observations (red) and model (black) data \[32\]. b and c) Zoom of two intervals of Fig. 4 a. ](Fig2c.pdf "fig:"){width="25.00000%"} VLA observations \[23\] measured a flat spectrum extending from 1.5 GHz to 22 GHz. Effelsberg 100-m telescope observations (Fig. 2), reveal a flat spectrum up to 32 GHz \[24\]. Observations at two frequencies, 2 GHz and 8 GHz of the Green Bank Interferometer (GBI) are given in Fig. 3. As in Fig. 4 of \[25\], it is also evident in Fig. 3 that in [LS I +61$^{\circ}$303 ]{}two consecutive outbursts occur: there is a first outburst either dominating at 8 GHz (i.e., with inverted spectrum, $\alpha > 0$) or with flat spectrum (comparable flux at the two frequencies) and then there follows another outburst, this one clearly dominating at 2 Ghz (i.e., $\alpha <0$). The system [LS I +61$^{\circ}$303 ]{}does not only show the self-absorbed jet but in addition there it follows a transient jet, as typical for microquasars. Finally, following the hypothesis that a short X-ray burst observed in the direction of [LS I +61$^{\circ}$303 ]{}could be attributed to this system and not to the other X-ray source that is located in the same field of view, possible implications for having a magnetar in a binary system have been analysed \[26\]. The presence of another X-ray candidate on the one hand and the radio observations well consistent with those of microquasars on the other hand do corroborate the hypothesis of [LS I +61$^{\circ}$303 ]{}to be a precessing microquasar. Deriving the precessional period from radio images is not straightforward, because the radio structures reflect the variation of the projected angle on the sky plane and therefore a combination of the ejection angle, $\eta$, and inclination. The most powerful tool to determine the precession is the radio astrometry: following the shift of the peak of successive high resolution radio images we determined a period of $27-28$ d for the precession \[27\]. An independent estimate of the precession comes from the timing analysis. In fact, precession implies a periodical Doppler boosting, that is periodical changes in the flux density and this can be revealed by timing analysis. A timing analysis of the 6.7 years of GBI radio data of [LS I +61$^{\circ}$303 ]{}has revealed indeed (Fig. 2) two rather close frequencies: $P_1 = {1\over \nu_1} = 26.49 \pm 0.07$ days and $P_2 = {1\over \nu_2} = 26.92 \pm 0.07$ days \[28\]. The period $P_1$ agrees with the value of $26.4960 \pm 0.0028$ days \[29\] associated to the orbital period of the binary system and corresponds to the predicted periodical accretion peak along an eccentric orbit \[30\]. The period $P_2$ agrees well with the estimate by radio astrometry of $27-28$ days for the precession period. Recently \[31\], Lomb-Scargle analysis of *Fermi*-LAT data around apoastron revealed (Fig. 2) the same periodicities $P_{1_{\gamma}} = 26.48 \pm 0.08$ d, $P_{2_{\gamma}} = 26.99 \pm 0.08$ d. The similar behaviour of the emission at high (GeV) and low (radio) energy towards apoastron is a hint for these emissions to be caused by the same population of electrons in a precessing jet. The beating of the two periodicities, $P_1$ and $P_2$, likely gives rise to the so called “long-term” modulation present in [LS I +61$^{\circ}$303]{}. The peak flux density of the periodical radio outburst exhibits in fact a modulation of 1667$\pm$8 d \[29\]. The beating of the two frequencies determined in the GBI timing analysis gives straightforward the long-term modulation as $1\over {\nu_1} - {\nu_2}$=1667 days \[28\]. Indeed, a physical model \[32\] for [LS I +61$^{\circ}$303 ]{} of synchrotron emission from a precessing ($P_2$) jet, periodically ($P_1$) refilled with relativistic particles, produces a maximum (shown here in Fig. 4 a, b) when the jet electron density is at its maximum and the approaching jet forms the smallest possible angle with the line of sight. This coincidence of the highest number of emitting particles and the strongest Doppler boosting of their emission occurs with the frequency of $\nu_1-\nu_2$ creating the long-term modulation observed in [LS I +61$^{\circ}$303]{}. As one can see the model reproduces in fact also the minimum of the log-term modulation (Fig. 4 c) corresponding at ejections when the approaching jet forms the largest possible angle with the line of sight \[32\]. The model can determine the variation of the ejection angle, $\eta$, at the epochs of VLBI observations. It is therefore of interest to compare predicted variations of $\eta$ with the observed variations in position angle of the radio structures in high resolution images. The trend of $\eta$ vs $\Phi (P_1)$ (orbital phase) is shown in Fig. 5 together with VLBA images \[27,32\]. The plot of $\eta$ vs $\Phi$ reveals that run H was performed at the minimum $\eta$ and D was performed nearly at the maximum $\eta$. These two extreme situations of $\eta$ for the two runs, H and D, are sketched in Fig. 5 (right corner). This important information implies that two runs at similar $\eta$, but one performed before and the other after run D, refer to two jets pointing to opposite directions with respect to the axis of the precession cone (Fig. 5, left corner). If the model is correct, the position angle of the associated radio structures should reflect this different orientation. Indeed, the structure for run E points towards South-West, whereas runs C and J show structures pointing to South-East. Similarly, the structure at run B points towards East whereas that for run F points towards West. Finally, for A, I, H, G our model results in a low $\eta$ angle; this would correspond to a jet pointing closest to the earth (i.e., a micro-blazar). One can see that the related VLBA structures develop indeed a North-South feature. ![ VLBA images and related variations of the ejection angle $\eta$ because of precession (see Sect. 3.1) \[32\] ](fig23.pdf){width="79.00000%"} To conclude this short review, there are different mechanisms producing transients in binary systems. Intrabinary interaction of the magnetic structures of the two stellar objects in [V773TauA]{} is likely the origin of the transient in this system. Variable accretion in an eccentric orbit and consequent refilling of a precessing jet is likely the cause for the modulated transient in the gamma-ray binary [LS I +61$^{\circ}$303]{}. [99]{} J. M. Cordes, T. J. W. Lazio, M. A. 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--- abstract: 'Slow dynamical changes in magnetic-field strength and invariance of the particles’ magnetic moments generate ubiquitous pressure anisotropies in weakly collisional, magnetized astrophysical plasmas. This renders them unstable to fast, small-scale mirror and firehose instabilities, which are capable of exerting feedback on the macroscale dynamics of the system. By way of a new asymptotic theory of the early nonlinear evolution of the mirror instability in a plasma subject to slow shearing or compression, we show that the instability does not saturate quasilinearly at a steady, low-amplitude level. Instead, the trapping of particles in small-scale mirrors leads to nonlinear secular growth of magnetic perturbations, ${{\delta B}}/B \propto t^{2/3}$. Our theory explains recent collisionless simulation results, provides a prediction of the mirror evolution in weakly collisional plasmas and establishes a foundation for a theory of nonlinear mirror dynamics with trapping, valid up to ${{\delta B}}/B ={O(1)}$.' author: - 'F. Rincon' - 'A. A. Schekochihin' - 'S. C. Cowley' title: Nonlinear mirror instability --- #### Introduction. Dynamical, weakly collisional [high-$\beta$]{} plasmas develop pressure anisotropies with respect to the magnetic field as a result of the combination of slow changes in magnetic-field strength $B$ and conservation of the first adiabatic invariant of particles $\mu={v_{\perp}}^2/2B$. This renders them unstable to fast (ion cyclotron timescale $\Omega_i^{-1}$), small-scale (ion gyroscale ${\rho_i}$) firehose, mirror and ion cyclotron instabilities [@rosenbluth56; @chandra58; @parker58; @vedenov58; @rudakov61; @gary92], whose observational signatures have been reported in the solar wind [@hellinger06; @bale09] and planetary magnetosheaths [@kaufmann70; @erdos96; @andre02; @joy06; @genot09; @horbury09; @soucek11]. These instabilities are also thought to be excited in energetic astrophysical environments such as the intracluster medium (ICM) [@fabian94; @carilli02; @govoni04; @schekochihin05; @peterson06], the vicinity of accreting black holes [@quataert01; @narayan05; @blaes13], or the warm ionized interstellar medium [@hall80; @ferriere01], producing strong dynamical feedback at macroscales, with critical astrophysical implications [@chandran98; @sharma06; @sharma07; @kunz11; @schekochihin06; @mogavero14]. A self-consistent description of the multiscale physics of such plasmas requires understanding how these instabilities saturate nonlinearly. Let us consider a typical situation in which slow changes in $B$ due to shearing, compression or expansion of the plasma at large (“fluid”) scales build up ion pressure anisotropy $\Delta_i\equiv ({{p_{i}^{\perp}}}-{{p_{i}^{\parallel}}})/{{p_{i}^{\perp}}}$, driving the plasma through either the ion firehose instability boundary ($\Delta_i < -2/\beta_i$, with $\beta_i=8\pi p_i/B^2$) in regions of decreasing field, or the mirror instability boundary ($\Delta_i \gtrsim 1/\beta_i$) in regions of increasing field. This triggers exponential growth on timescales up to $\Omega_i^{-1}$, much faster than the shearing timescale $S^{-1}$. The separation between these timescales implies that the instabilities always operate close to threshold and regulate the levels of pressure anisotropy in the plasma nonlinearly. However, how they achieve that in the face of the slowly changing $B$ constantly generating more pressure anisotropy, remains an open question. In the simplest case of the parallel firehose instability, the growth of magnetic perturbations ${{\delta {{\ensuremath\mathbf{B}}}}}$ leads to an increase of the average (r.m.s.) field strength and perpendicular pressure, which drives the anisotropy back to marginality, $\Delta_i(t)\rightarrow-2/\beta_i$. If a weakly unstable initial state $\Delta_{io}-2/\beta_i<0$ is postulated with no further driving of $\Delta_i$, quasilinear theory [@shapiro64] predicts saturation at a steady, low amplitude ${{\delta B}}/B \sim \vert\Delta_{io}+2/\beta_i\vert^{1/2}\ll 1$. However, the shearing or expansion process that drove the plasma through the instability boundary in the first place, must ultimately become important again once quasilinear relaxation has pushed the system sufficiently close back to marginality. When such continued driving is accounted for, asymptotic theory [@schekochihin08; @rosin11] predicts secular growth of perturbations as ${{\delta B}}/B\propto t^{1/2}$ up to ${{\delta B}}/B={O(1)}$ (cf. [@matteini06]), a very different outcome from steady-state, low-amplitude saturation. The nonlinear dynamics of the mirror instability [@tajiri67; @hasegawa69; @southwood93; @hellinger07] in a weakly collisional shearing (or compressing) plasma driven through its instability boundary is much more involved and has only recently been explored numerically [@kunz14a; @riquelme14]. In this Letter, we show that weakly nonlinear mirror modes in such conditions do not saturate quasilinearly at a steady, low amplitude either, but continue to grow secularly as ${{\delta B}}/B \propto t^{2/3}$. To do this, we introduce a new asymptotic theory in the spirit of earlier theoretical work [@califano08; @schekochihin08; @rosin11], in which the combined effects of weak collisionality, large-scale shearing, quasilinear relaxation [@shapiro64; @pokhotelov08; @hellinger09], particle trapping [@kivelson96; @pantellini98; @istomin09; @pokhotelov10] and finite ion Larmor radius (FLR) are all self-consistently retained. #### Asymptotic theory. We consider the simplest case of a plasma consisting of cold electrons [^1] and hot ions of mass $m_i$, charge $q_i=Ze$, and thermal velocity ${v_{\mathrm{th}i}}=\sqrt{2\,T_i/m_i}$, coupled to the electromagnetic fields ${{\ensuremath\mathbf{E}}}$ and ${{\ensuremath\mathbf{B}}}$. The dynamics is governed by the non-relativistic Vlasov-Maxwell system, $$\label{eq:vlasov} {{\frac{\partial {f_i}}{\partial {t}}}}+{{\ensuremath\mathbf{v}}}\cdot{{{\ensuremath\mathbf{\nabla}}}{f_i}}+{\frac{q_i}{m_i}}\left({{\ensuremath\mathbf{E}}}+{\frac{{{\ensuremath\mathbf{v}}}\times{{\ensuremath\mathbf{B}}}}{c}}\right)\cdot{{\frac{\partial {f_i}}{\partial {{{\ensuremath\mathbf{v}}}}}}}=C\left[f_i\right]\,,$$ $$\label{eq:maxwell} {{{\ensuremath\mathbf{\nabla}}}\cdot{{{\ensuremath\mathbf{B}}}}}=0~,\quad {{\frac{\partial {{{\ensuremath\mathbf{B}}}}}{\partial {t}}}}=-c\,{{{\ensuremath\mathbf{\nabla}}}\times{{{\ensuremath\mathbf{E}}}}}~,\quad {{\ensuremath\mathbf{j}}}={\frac{c}{4\pi}}{{{\ensuremath\mathbf{\nabla}}}\times{{{\ensuremath\mathbf{B}}}}}~,$$ and Ohm’s law describing the force balance for electrons, $$\label{eq:ohm} {{\ensuremath\mathbf{E}}}+{\frac{{{\ensuremath\mathbf{u}}}_i\times{{\ensuremath\mathbf{B}}}}{c}} ={\frac{\left({{{\ensuremath\mathbf{\nabla}}}\times{{{\ensuremath\mathbf{B}}}}}\right)\times{{\ensuremath\mathbf{B}}}}{4\pi Zen_i}}~.$$ Here, $f_s(t,{{\ensuremath\mathbf{r}}},{{\ensuremath\mathbf{v}}})$, $n_s(t,{{\ensuremath\mathbf{r}}})=\int f_s\,{d}^3{{\ensuremath\mathbf{v}}}$ and ${{\ensuremath\mathbf{u}}}_s(t,{{\ensuremath\mathbf{r}}})=\int{{\ensuremath\mathbf{v}}}f_s\,{d}^3{{\ensuremath\mathbf{v}}}$ are, respectively, the distribution function, number density and mean velocity of species $s=(i,e)$, ${{\ensuremath\mathbf{j}}}=e \,n_e\left({{\ensuremath\mathbf{u}}}_i-{{\ensuremath\mathbf{u}}}_e\right)$ is the total current density given the quasineutrality condition $n_e=Z n_i$. In the following, we use the ion peculiar velocity ${{\ensuremath\mathbf{v}}}'={{\ensuremath\mathbf{v}}}-{{\ensuremath\mathbf{u}}}_i$ as the velocity-space variable and will henceforth drop the primes. Taking the first moment of [Eq. ([\[eq:vlasov\]]{})]{} and using [Eqs. ([\[eq:maxwell\]]{})-(\[eq:ohm\])]{}, we obtain the ion momentum equation $$\label{eq:momentumi} {{\frac{d \,{{\ensuremath\mathbf{u}}}_i}{d t}}} = -{\frac{{{{\ensuremath\mathbf{\nabla}}}\cdot{{\ensuremath\mathbfsf{P}}_i}}}{m_in_i}}+{\frac{\left({{{\ensuremath\mathbf{\nabla}}}\times{{{\ensuremath\mathbf{B}}}}}\right)\times{{\ensuremath\mathbf{B}}}}{4\pi m_in_i}}~,$$ where ${d {}/d {t}}={\partial {}/\partial {t}}+{{\ensuremath\mathbf{u}}}_i\cdot{{{\ensuremath\mathbf{\nabla}}}{}}$ and ${\ensuremath\mathbfsf{P}}_i=m_i\int {{\ensuremath\mathbf{v}}}{{\ensuremath\mathbf{v}}}f_i\,{d}^3{{\ensuremath\mathbf{v}}}$ is the ion pressure tensor. Introducing ${{\hat{{{\ensuremath\mathbf{b}}}}}}\equiv{{\ensuremath\mathbf{B}}}/B$, using [Eqs. ([\[eq:maxwell\]]{})-(\[eq:ohm\])]{}, we obtain the evolution equation for the field strength: $$\label{eq:induction} {{\frac{d \ln B}{d t}}}={{\hat{{{\ensuremath\mathbf{b}}}}}}{{\hat{{{\ensuremath\mathbf{b}}}}}}:{{{\ensuremath\mathbf{\nabla}}}{{{\ensuremath\mathbf{u}}}_i}}-{{{\ensuremath\mathbf{\nabla}}}\cdot{{{\ensuremath\mathbf{u}}}_i}}-{\frac{{{\hat{{{\ensuremath\mathbf{b}}}}}}}{B}}\cdot{{{\ensuremath\mathbf{\nabla}}}\times{\left({\frac{{{\ensuremath\mathbf{j}}}\times{{\ensuremath\mathbf{B}}}}{Zen_i}}\right)}}~.$$ Our derivation is based on an asymptotic expansion of these equations. The separation between the slow magnetic-field-stretching timescale and the fast instability timescale implies that the distance to instability threshold $\Gamma \sim \Delta_i - 1/\beta_i$ must remain small, which provides us with a natural expansion parameter. In order to study the dynamics in this regime, we start from an already weakly unstable situation and order $\Gamma={O({\ensuremath\varepsilon}^2)}$, with ${\ensuremath\varepsilon}\ll 1$. We then construct a “maximal” ordering (summarized in [Eqs. ([\[eq:expansionsummaryfirst\]]{})-(\[eq:expansionsummarylast\])]{}) retaining ion FLR, collisional, quasilinear and trapping effects, as well as the effect of continued slow shearing. Following [@hellinger07], we order the time and spatial scales of mirror modes as $\gamma\sim{\ensuremath\varepsilon}^2{k_{\parallel}}{v_{\mathrm{th}i}}$, ${k_{\perp}}\sim {\ensuremath\varepsilon}^{-1}{k_{\parallel}}$, $\rho_i^{-1}\sim{\ensuremath\varepsilon}^{-2}{k_{\parallel}}$, where $\gamma$ is the instability growth rate, $({{{\ensuremath\mathbf{k}}}_{\perp}},{k_{\parallel}})$ the typical perturbation wavenumbers (defined with respect to the unperturbed field), ${\rho_i}={v_{\mathrm{th}i\perp}}/\Omega_i$, and $\Omega_i^{-1}=(m_ic)/q_i B\sim{\ensuremath\varepsilon}^{-2}{k_{\parallel}}{v_{\mathrm{th}i}}$. The ion distribution function is expanded as $f_i={f_{0i}}+{f_{2i}}+{\delta f}$, where ${f_{0i}}$ provides the required pressure anisotropy to pin the system at the threshold, ${f_{2i}}$ provides an extra $O({\ensuremath\varepsilon}^2)$ anisotropy to drive the system away from it, and ${\delta f}$ contains mirror perturbations. We also expand ${{\ensuremath\mathbf{B}}}={{{{\ensuremath\mathbf{B}}}_0}}+{{\delta {{\ensuremath\mathbf{B}}}}}$ and ${{\ensuremath\mathbf{u}}}_i={{{{\ensuremath\mathbf{u}}}_{0i}}}+ {\delta {{\ensuremath\mathbf{u}}}_i}$, where ${{{{\ensuremath\mathbf{B}}}_0}}$ and ${{{{\ensuremath\mathbf{u}}}_{0i}}}$ have no instability-scale variations, ${{{{\ensuremath\mathbf{u}}}_{0i}}}$ is the slow, large-scale shearing/compressive motion, and ${{\delta {{\ensuremath\mathbf{B}}}}}$ and ${\delta {{\ensuremath\mathbf{u}}}_i}$ are the mirror perturbations. The ordering of ${\delta f}$, ${{\delta {{\ensuremath\mathbf{B}}}}}$, ${\delta {{\ensuremath\mathbf{u}}}_i}$ and of the remaining timescales is guided by physical considerations. The critical pitch-angle $\xi={v_{\parallel}}/v$ below which particles get trapped by magnetic fluctuations is ${{\xi_{\mathrm{tr}}}}=\left({{\delta B}}/{{B_0}}\right)^{1/2}$ and the corresponding bounce frequency is ${\omega_B}\sim{k_{\parallel}}{v_{\mathrm{th}i}}\,{{\xi_{\mathrm{tr}}}}$. To retain their contribution in our calculation, we order ${\omega_B}\sim\gamma\sim{\ensuremath\varepsilon}^2{k_{\parallel}}{v_{\mathrm{th}i}}$, which provides us with the ordering ${{\delta B}}/{{B_0}}={O({\ensuremath\varepsilon}^4)}$ (plus higher-order terms). For consistency of the ${\ensuremath\varepsilon}$ expansion, we must order ${\delta f}={O({\ensuremath\varepsilon}^4)}$, ${\delta {{\ensuremath\mathbf{u}}}_i}={O({\ensuremath\varepsilon}^5)}$ and higher. Averaging [Eq. ([\[eq:induction\]]{})]{} over instability scales and ignoring quadratic nonlinearities, we obtain $$\label{eq:shear} {{\frac{d \ln{{B_0}}}{d t}}} = {{\hat{{{\ensuremath\mathbf{b}}}}_0}}{{\hat{{{\ensuremath\mathbf{b}}}}_0}}:{{{\ensuremath\mathbf{\nabla}}}{{{{{\ensuremath\mathbf{u}}}_{0i}}}}} -{{{\ensuremath\mathbf{\nabla}}}\cdot{{{{{\ensuremath\mathbf{u}}}_{0i}}}}}\,\, \equiv\,\, S~.$$ Subtracting [Eq. ([\[eq:shear\]]{})]{} from [Eq. ([\[eq:induction\]]{})]{}, we find that $$\label{eq:inductionpert} {{\frac{d }{d t}}}{\frac{{{\delta B}}}{{{B_0}}}}= {{\hat{{{\ensuremath\mathbf{b}}}}_0}}{{\hat{{{\ensuremath\mathbf{b}}}}_0}}:{{{\ensuremath\mathbf{\nabla}}}{{\delta {{\ensuremath\mathbf{u}}}_i}}}$$ to all relevant orders [^2]. We order $S\equiv{d {}/d {t}}\,(\ln {{B_0}})$ the same size as ${d {}/d {t}}\,(\delta B/{{B_0}})\sim {\ensuremath\varepsilon}^6{k_{\parallel}}{v_{\mathrm{th}i}}$ so as to be able to investigate how a slow change in field strength affects the dynamics. The aforementioned timescale separation $S/\Omega_i$ is now related to ${\ensuremath\varepsilon}$ through ${\ensuremath\varepsilon}\sim (S/\Omega_i)^{1/8}$ (${\ensuremath\varepsilon}\sim 0.01$ for the ICM [@rosin11]). A separate asymptotic treatment is required for low-pitch-angle resonant particles, which develop a velocity-space boundary layer and evolve into a separate population of trapped particles on the instability timescale (the process is reminiscent of “nonlinear Landau damping” [@istomin09; @dawson61; @oneil65]). As can be seen by considering a simple Lorentz pitch-angle scattering operator [@helander02], $C\left[f_i\right]=({\nu_{ii}}/2)\,{\partial_{\xi} {}}\left[\left(1-\xi^2\right){\partial_{\xi} {f_i}}\right]$, this results in a boost of their effective collisionality, ${\nu_{ii,\mathrm{eff}}}\sim{\nu_{ii}}/{{\xi_{\mathrm{tr}}}}^2\gg {\nu_{ii}}$ for $\xi<{{\xi_{\mathrm{tr}}}}\ll 1$. To retain this effect, we order ${\nu_{ii,\mathrm{eff}}}\sim\gamma\sim{\omega_B}$, or ${\nu_{ii}}\sim {k_{\parallel}}{v_{\mathrm{th}i}}\left({{\delta B}}/{{B_0}}\right)^{3/2}\sim{\ensuremath\varepsilon}^6 {k_{\parallel}}{v_{\mathrm{th}i}}$ (this preserves the low-collisionality assumption in the sense that the rest of the distribution relaxes on a timescale $1/{\nu_{ii}}\gg 1/\gamma$). The maximal mirror ordering is summarized as follows: $$\begin{aligned} & f_i= {f_{0i}}+ {f_{2i}}+ {\delta f_{4i}}+\cdots,\label{eq:expansionsummaryfirst}\\ & \gamma\sim{\ensuremath\varepsilon}^2{k_{\parallel}}{v_{\mathrm{th}i}}~,\ \Omega_i\sim{\ensuremath\varepsilon}^{-2}{k_{\parallel}}{v_{\mathrm{th}i}}~,\ S\sim{\nu_{ii}}\sim{\ensuremath\varepsilon}^6{k_{\parallel}}{v_{\mathrm{th}i}}~,\\ & \rho_i^{-1}\sim{\ensuremath\varepsilon}^{-2}{k_{\parallel}}~,\ {k_{\perp}}\sim {\ensuremath\varepsilon}^{-1}{k_{\parallel}}~, \\ & {{\ensuremath\mathbf{B}}}= {{{{\ensuremath\mathbf{B}}}_0}}+ {{\delta B_4^\parallel}}{{\hat{{{\ensuremath\mathbf{b}}}}_0}}+ {{\delta {{\ensuremath\mathbf{B}}}_5^{\perp}}}+\cdots,\,\ {{\ensuremath\mathbf{u}}}_i = {{{{\ensuremath\mathbf{u}}}_{0i}}}+ {{\delta {{\ensuremath\mathbf{u}}}_{5i}^{\perp}}}+\cdots, \\ & {{\xi_{\mathrm{tr}}}}\sim \left({{\delta B_4^\parallel}}/{{B_0}}\right)^{1/2}\sim {\ensuremath\varepsilon}^2,\,\ {\omega_B}\sim {\nu_{ii,\mathrm{eff}}}\sim \gamma \sim {\ensuremath\varepsilon}^2{k_{\parallel}}{v_{\mathrm{th}i}}~.\label{eq:expansionsummarylast}\end{aligned}$$ Taking the three lowest non-trivial orders of [Eq. ([\[eq:vlasov\]]{})]{}, we first find that ${f_{0i}}$, ${f_{2i}}$, ${\delta f_{4i}}$ are gyrotropic. Expanding and gyroaveraging [Eq. ([\[eq:vlasov\]]{})]{} up to ${O({\ensuremath\varepsilon}^4)}$ then gives ${\delta f_{4i}}$ in terms of the mirror perturbation ${{\delta B_4^\parallel}}/{{B_0}}$, from which the perturbed scalar pressures ${{\delta p_{4i}^{\perp}}}$ and ${{\delta p_{4i}^{\parallel}}}$ are derived (note that resonant/trapped particles are not involved at this stage). Taking the perpendicular projection of [Eq. ([\[eq:momentumi\]]{})]{} at the lowest order ${O({\ensuremath\varepsilon}^3)}$, we obtain the threshold condition for the mirror instability [@hellinger07]: $$\label{eq:Gamma0} \Gamma_0= -{\frac{2\,m_i}{{{p_{0i}^{\perp}}}}}\int{\frac{{v_{\perp}}^4}{4}}\left.{{\frac{\partial {{f_{0i}}}}{\partial {{v_{\parallel}}^2}}}}\right|_{{v_{\perp}}}\!\!\!\!{d}^3{{\ensuremath\mathbf{v}}}-{\frac{2}{{{\beta_{0i}^{\perp}}}}}-2=0~.$$ Next, we expand and gyroaverage [Eq. ([\[eq:vlasov\]]{})]{} to three further orders, up to ${O({\ensuremath\varepsilon}^7)}$. This tedious calculation yields FLR corrections and resonant effects (not shown, see [@califano08] for an almost identical procedure) and provides us with explicit expressions for ${\delta f_{5i}}$ and ${\delta f_{6i}}$, from which we obtain higher-order elements of ${\ensuremath\mathbfsf{P}}_i$, ${{\delta {{\ensuremath\mathbf{p}}}_{5i}^{\perp\parallel}}}\equiv m_i\int {{{\ensuremath\mathbf{v}}}_{\perp}}{v_{\parallel}}\, {\delta f_{5i}}\,{d}^3{{\ensuremath\mathbf{v}}}$ and ${{\delta {\ensuremath\mathbfsf{P}}_{6i}^{\perp\perp}}}$, in terms of ${{\delta B_4^\parallel}}$ and ${{\delta B_6^{\parallel}}}$. No new information arises from [Eq. ([\[eq:momentumi\]]{})]{} at ${O({\ensuremath\varepsilon}^4)}$. Using these results and [Eq. ([\[eq:Gamma0\]]{})]{} in the perpendicular projection of [Eq. ([\[eq:momentumi\]]{})]{} at ${O({\ensuremath\varepsilon}^5)}$, we derive the pressure balance condition: $$\label{eq:momentumiorder5perp} \left[\Gamma_2+{\frac{3}{2}}\,{\rho_*}^2{\nabla_\perp^2{}} -\left({\frac{{{p_{0i}^{\perp}}}-{{p_{0i}^{\parallel}}}}{{{p_{0i}^{\perp}}}}}+{\frac{2}{{{\beta_{0i}^{\perp}}}}}\right) {\frac{{\nabla_\parallel^2{}}}{{\nabla_\perp^2{}}}}\right]{{{{\ensuremath\mathbf{\nabla}}}{}}_\perp{{\frac{{{\delta\tilde{B}_4^\parallel}}}{{{B_0}}}}}}= \displaystyle{{{{{\ensuremath\mathbf{\nabla}}}{}}_\perp{{\frac{{{\delta \tilde{p}_{6i}^{\perp}\phantom{}^{\mathrm{(res)}}}}}{{{p_{0i}^{\perp}}}}}}}}~,$$ where the resonant/trapped particle pressure is $$\label{eq:p6res} {{\delta \tilde{p}_{6i}^{\perp}\phantom{}^{\mathrm{(res)}}}}= m_i\int_{|\xi|<{{\xi_{\mathrm{tr}}}}}\!\!\! {\frac{{v_{\perp}}^2}{2}}\, {\delta \tilde{f}_{4i}^{\mathrm{(res)}}}\,{d}^3{{\ensuremath\mathbf{v}}}~,$$ ${\delta f_{4i}^{\mathrm{(res)}}}\!$ is the resonant part of the perturbed distribution function, the second-order distance to instability threshold is $$\label{eq:Gamma2} \Gamma_2 = -\,{\frac{2\,m_i}{{{p_{0i}^{\perp}}}}}\left(\int{\frac{{v_{\perp}}^4}{4}}\left.{{\frac{\partial {{f_{2i}}}}{\partial {{v_{\parallel}}^2}}}}\right|_{{v_{\perp}}}\!\!\!\!{d}^3{{\ensuremath\mathbf{v}}} +\!\!\int_{|\xi|<{{\xi_{\mathrm{tr}}}}}\!\!\!\!{\frac{{v_{\perp}}^4}{4}}\left.{{\frac{\partial {\overline{{\delta f_{4i}^{\mathrm{(res)}}}}}}{\partial {{v_{\parallel}}^2}}}}\right|_{{v_{\perp}}}\!\!\!\!{d}^3{{\ensuremath\mathbf{v}}}\right)-{\frac{2\,{{p_{2i}^{\perp}}}}{{{p_{0i}^{\perp}}}}}~,$$ and the effective Larmor radius is $${\rho_*}^2 = {\frac{\rho_i^2}{12}}\,{\frac{m_i}{{{p_{0i}^{\perp}}}\,{v_{\mathrm{th}i\perp}}^2}}\int\left(-{v_{\perp}}^6\left.{{\frac{\partial {{f_{0i}}}}{\partial {{v_{\parallel}}^2}}}}\right|_{{v_{\perp}}}-3\,{v_{\perp}}^4\,{f_{0i}}\right)\,{d}^3{{\ensuremath\mathbf{v}}}~.$$ Tildes denote fluctuating (zero field-line average) parts of the perturbed fields and overlines denote line averages. The l.h.s. of [Eq. ([\[eq:momentumiorder5perp\]]{})]{} describes the non-resonant response [^3]. $\Gamma_2$ and ${{\delta \tilde{p}_{6i}^{\perp}\phantom{}^{\mathrm{(res)}}}}$ depend on the regime considered. However, both only involve ${\delta f_{4i}^{\mathrm{(res)}}}$ because restricting the integration to ${{\xi_{\mathrm{tr}}}}$ brings in an extra ${O({\ensuremath\varepsilon}^2)}$ smallness, whereas FLR corrections only start to affect the distribution function at ${O({\ensuremath\varepsilon}^6)}$ within our expansion. Thus, ${\delta \tilde{f}_{4i}^{\mathrm{(res)}}}$ and ${{\delta \tilde{p}_{6i}^{\perp}\phantom{}^{\mathrm{(res)}}}}$ can be directly calculated from the much simpler drift-kinetic equation which, in $(\mu,{v_{\parallel}})$ variables, reads [@kulsrud83]: $$\label{eq:driftkin} {{\frac{d f_i}{d t}}}+{v_{\parallel}}\,{{{{\ensuremath\mathbf{\nabla}}}{}}_\parallel{\,f}}_i= - \mu B\, \left({{{\ensuremath\mathbf{\nabla}}}\cdot{{{\hat{{{\ensuremath\mathbf{b}}}}}}}}\right)\,{{\frac{\partial {f_i}}{\partial {{v_{\parallel}}}}}}+C[f_i]$$ to all orders relevant to our calculation (here $E_\parallel=0$ because the electrons are cold). For resonant particles, ${v_{\parallel}}\sim{\ensuremath\varepsilon}^2{v_{\mathrm{th}i}}$ and ${\partial {{\delta f_{4i}^{\mathrm{(res)}}}}/\partial {{v_{\parallel}}}} \sim ({\ensuremath\varepsilon}^{-2}/{v_{\mathrm{th}i}})\,{\delta f_{4i}^{\mathrm{(res)}}}$, so the expansion of [Eq. ([\[eq:driftkin\]]{})]{} at the first non-trivial order ${O({\ensuremath\varepsilon}^6)}$ is $$\begin{aligned} {{\frac{d {\delta f_{4i}^{\mathrm{(res)}}}}{d t}}}+{v_{\parallel}}\,{{{{\ensuremath\mathbf{\nabla}}}{}}_\parallel{\,{\delta f_{4i}^{\mathrm{(res)}}}}} & = & \mu {{B_0}}\,\left({\frac{{{{{\ensuremath\mathbf{\nabla}}}{}}_\parallel{\,{{\delta B_4^\parallel}}}}}{{{B_0}}}}\right)\left({{\frac{\partial {{f_{0i}}}}{\partial {{v_{\parallel}}}}}}+{{\frac{\partial {{\delta f_{4i}^{\mathrm{(res)}}}}}{\partial {{v_{\parallel}}}}}}\right) \nonumber \\ && +C[{\delta f_{4i}^{\mathrm{(res)}}}] \label{eq:driftkinexpand}~.\end{aligned}$$ We have omitted the collision term $C[{f_{0i}}]$: it gives a ${O({\ensuremath\varepsilon}^4)}$ correction to the line-averaged pressure on the instability timescale that can be absorbed into $\Gamma_0$. #### Linear and quasilinear regimes. Neglecting the nonlinear and collision terms in [Eq. ([\[eq:driftkinexpand\]]{})]{} and taking its space-time Fourier transform, we obtain the linear solution: $$\label{eq:f4itilde} {\delta {\hat{f}_{4i{{\ensuremath\mathbf{k}}}}^{\mathrm{(res)}}}}= {\frac{\mu\,i{k_{\parallel}}\,{{\delta\hat{B}_{4{{\ensuremath\mathbf{k}}}}^\parallel}}}{{\gamma_{{{\mbox{\tiny $L$}}}}}+ i\,{k_{\parallel}}{v_{\parallel}}}}{{\frac{\partial {{f_{0i}}}}{\partial {{v_{\parallel}}}}}}~,$$ where ${\gamma_{{{\mbox{\tiny $L$}}}}}$ is the linear instability growth rate. Using [Eq. ([\[eq:p6res\]]{})]{} to compute ${{\delta \tilde{p}_{6i}^{\perp}\phantom{}^{\mathrm{(res)}}}}$ and substituting the result into [Eq. ([\[eq:momentumiorder5perp\]]{})]{}, the classical linear mirror dispersion relation [@hellinger07] is recovered: $$\label{eq:growthrate} {\gamma_{{{\mbox{\tiny $L$}}}}}\!=\!\sqrt{{\frac{2}{\pi}}}|{k_{\parallel}}|{v_*}\left[\Gamma_2-{\frac{3}{2}}\,{\rho_*}^2{k_{\perp}}^2-\left({\frac{{{p_{0i}^{\perp}}}-{{p_{0i}^{\parallel}}}}{{{p_{0i}^{\perp}}}}}+{\frac{2}{{{\beta_{0i}^{\perp}}}}}\right){\frac{{k_{\parallel}}^2}{{k_{\perp}}^2}}\right]~,$$ with the effective thermal speed $${v_*}^{-1}=-\sqrt{2\pi}\,{\frac{2 m_i}{{{p_{0i}^{\perp}}}}}\int{\frac{{v_{\perp}}^4}{4}}\left.{{\frac{\partial {{f_{0i}}}}{\partial {{v_{\parallel}}^2}}}}\right|_{{v_{\perp}}}\!\!\!\!\!\delta({v_{\parallel}})\,{d}^3{{\ensuremath\mathbf{v}}}~.$$ Because of the resonance, ${\delta \tilde{f}_{4i}^{\mathrm{(res)}}}$ develops a velocity-space boundary layer on the timescale $\sim 1/{\gamma_{{{\mbox{\tiny $L$}}}}}$, resulting in a correction to the line-averaged distribution function that satisfies: $$\label{eq:driftkinlineav} {{\frac{d {\overline{{\delta f_{4i}^{\mathrm{(res)}}}}}}{d t}}}=-\mu {{B_0}}\, {\overline{\left({\frac{{{{{\ensuremath\mathbf{\nabla}}}{}}_\parallel{\,{{\delta\tilde{B}_4^\parallel}}}}}{{{B_0}}}}\right){{\frac{\partial {{\delta f_{4i}^{\mathrm{(res)}}}}}{\partial {{v_{\parallel}}}}}}}}+C[{\overline{{\delta f_{4i}^{\mathrm{(res)}}}}}]~.$$ Assuming a monochromatic perturbation for simplicity and using [Eq. ([\[eq:f4itilde\]]{})]{} to calculate the line-averaged nonlinear term on the r.h.s. of [Eq. ([\[eq:driftkinlineav\]]{})]{}, we recover the resonant quasilinear diffusion equation [@shapiro64]: $$\label{eq:quasilinear} {{\frac{\partial {{\overline{{\delta f_{4i}^{\mathrm{(res)}}}}}}}{\partial {t}}}}\!=\!{{\frac{\partial {}}{\partial {{v_{\parallel}}}}}}\left[{\frac{2\left(\mu{{B_0}}\right)^2{k_{\parallel}}^2{\gamma_{{{\mbox{\tiny $L$}}}}}}{{\gamma_{{{\mbox{\tiny $L$}}}}}^2+\left({k_{\parallel}}{v_{\parallel}}\right)^2}}{\overline{\left({\frac{{{\delta\tilde{B}_4^\parallel}}}{{{B_0}}}}\right)^2}}{{\frac{\partial {{f_{0i}}}}{\partial {{v_{\parallel}}}}}}\right]+C[{\overline{{\delta f_{4i}^{\mathrm{(res)}}}}}]~.$$ The effect of the first term on the r.h.s. of [Eq. ([\[eq:quasilinear\]]{})]{} is to relax $\Gamma_2$ (see [Eq. ([\[eq:Gamma2\]]{})]{}) by flattening the total averaged distribution function at low $\xi$, thereby decreasing the growth rate [@pokhotelov08; @hellinger09]. #### Trapping regime. Quasilinear relaxation ceases to be the dominant saturation mechanism once particle trapping becomes dynamically significant (${\omega_B}\sim {k_{\parallel}}{v_{\mathrm{th}i}}\,({{\delta B}}/B)^{1/2}\sim {\gamma_{{{\mbox{\tiny $L$}}}}}$). Indeed, due to the growth of ${{\delta B}}/B$ and quasilinear reduction of the growth rate ${\partial {}/\partial {t}}\ll {\gamma_{{{\mbox{\tiny $L$}}}}}$ for $t\gg 1/{\gamma_{{{\mbox{\tiny $L$}}}}}$, (i) the system eventually reaches a bounce-dominated regime, ${\omega_B}\gg {\partial {}/\partial {t}}$, and (ii) collisional and shearing effects, however slow their timescales are, compared to the initial linear instability timescale, inevitably become important after a few instability times (hence the maximal ordering [Eqs. ([\[eq:expansionsummaryfirst\]]{})-(\[eq:expansionsummarylast\])]{}). To elicit these effects, we rewrite [Eq. ([\[eq:driftkin\]]{})]{} in $(\mu,E=v^2/2)$ variables: $$\label{eq:driftkinEmu} {{\frac{d f_i}{d t}}}\pm{\sqrt{2(E\!-\!\mu B)}}\,{{\frac{\partial {f_i}}{\partial {\ell}}}}= -\mu{{\frac{d B}{d t}}}{{\frac{\partial {f_i}}{\partial {E}}}}+C[f_i]~,$$ where $\ell$ is the distance along the perturbed field line. Expanding [Eq. ([\[eq:induction\]]{})]{} and [Eq. ([\[eq:driftkinEmu\]]{})]{} at ${O({\ensuremath\varepsilon}^6)}$, we obtain $$\begin{aligned} \label{eq:driftkinexpandEmu} {{\frac{d {\delta f_{4i}^{\mathrm{(res)}}}}{d t}}}\pm{\sqrt{2(E\!-\!\mu B)}}\,{{\frac{\partial {{\delta f_{4i}^{\mathrm{(res)}}}}}{\partial {\ell}}}} & = & -\mu{{B_0}}{{\frac{d }{d t}}}{\frac{{{\delta B_4^\parallel}}}{{{B_0}}}}{{\frac{\partial {{f_{0i}}}}{\partial {E}}}} \\ & & \hspace{-2.5cm} -\mu{{B_0}}\left({{\frac{d \ln{{B_0}}}{d t}}} +{{\frac{d }{d t}}}{\frac{{{\delta B_4^\parallel}}}{{{B_0}}}}\right){{\frac{\partial {{\delta f_{4i}^{\mathrm{(res)}}}}}{\partial {E}}}} + C[{\delta f_{4i}}]~.\nonumber\end{aligned}$$ Here $C[{f_{0i}}]$ and $-\mu{{B_0}}\,\left({d {\ln{{B_0}}}/d {t}}\right) ({\partial {{f_{0i}}}/\partial {E}})$ have been discarded for the same reason as in [Eq. ([\[eq:driftkinexpand\]]{})]{}. Note that both ${\partial {{\delta f_{4i}^{\mathrm{(res)}}}}/\partial {E}}$ and ${\partial {{\delta f_{4i}^{\mathrm{(res)}}}}/\partial {\mu}}$ are ${O(1)}$ because of the velocity-space boundary layer in $|\xi|<{{\xi_{\mathrm{tr}}}}={O({\ensuremath\varepsilon}^2)}$. We anticipate that magnetic fluctuations will grow secularly as ${{\delta B_4^\parallel}}\sim{{\delta B_4^\parallel}}(t_{{{\mbox{\tiny $L$}}}})(t/t_{{{\mbox{\tiny $L$}}}})^s$, with $s>0$, for $t\gg t_{{{\mbox{\tiny $L$}}}} \sim 1/{\gamma_{{{\mbox{\tiny $L$}}}}}\,(\sim 1/{\omega_B})$, and introduce a secondary ordering parameter $\chi=(t_{{\mbox{\tiny $L$}}}/t)^{s/2}\ll 1$, so now ${{\delta B_4^\parallel}}/{{B_0}}={O({\ensuremath\varepsilon}^4/\chi^2)}$ and ${{\xi_{\mathrm{tr}}}}\sim \left({{\delta B_4^\parallel}}/{{B_0}}\right)^{1/2}= {O({\ensuremath\varepsilon}^2/\chi)}\gg{\ensuremath\varepsilon}^2$. The instantaneous nonlinear growth rate is ${\gamma_{{{\mbox{\tiny $NL$}}}}}\sim {\partial {}/\partial {t}}\sim 1/t={O({\ensuremath\varepsilon}^2\chi^{2/s})}$, so the new ordering guarantees ${\omega_B}\sim{k_{\parallel}}{v_{\mathrm{th}i}}\,{{\xi_{\mathrm{tr}}}}\gg {\gamma_{{{\mbox{\tiny $NL$}}}}}$. For trapped particles to play a role in the nonlinear evolution, their pressure in [Eq. ([\[eq:momentumiorder5perp\]]{})]{} must be taken to be of the same order as the instability-driving term, $\Gamma_2\,({{\delta B_4^\parallel}}/{{B_0}}) \sim {{\delta p_{6i}^{\perp}\phantom{}^{\mathrm{(res)}}}}/{{p_{0i}^{\perp}}}= {O({\ensuremath\varepsilon}^2/\chi^2)}$. Given that ${{\delta p_{6i}^{\perp}\phantom{}^{\mathrm{(res)}}}}\sim{{\xi_{\mathrm{tr}}}}{\delta f_{4i}^{\mathrm{(res)}}}$, we must therefore order ${\delta f_{4i}^{\mathrm{(res)}}}={O({\ensuremath\varepsilon}^4/\chi)}$. Expanding [Eq. ([\[eq:driftkinexpandEmu\]]{})]{} to lowest order ${O({\ensuremath\varepsilon}^6/\chi^2)}$, we find that the distribution function of the trapped particles is homogenized along the field lines within the traps: ${\partial {{\delta f_{4i}^{\mathrm{(res)}}}}/\partial {\ell}}=0$. Therefore, ${\delta f_{4i}^{\mathrm{(res)}}}={\left<{{\delta f_{4i}^{\mathrm{(res)}}}}\right>}+{{\delta f_{4i}^{\mathrm{(res)}}}}'$, where ${{\delta f_{4i}^{\mathrm{(res)}}}}'\ll{\left<{{\delta f_{4i}^{\mathrm{(res)}}}}\right>}$ and ${\left<{\bullet}\right>}=\oint\,\bullet\, {d}\ell$ denotes a bounce average between bounce points $\ell_1$ and $\ell_2$ defined by the relation $E=\mu B(\ell_1)=\mu B(\ell_2)$. Looking at the next orders of [Eq. ([\[eq:driftkinexpandEmu\]]{})]{}, we find that the first term on the r.h.s. (the betatron term linear in perturbations) is ${O({\ensuremath\varepsilon}^6\chi^{2/s-2})}$, while the time derivative on the l.h.s. and the r.h.s. term quadratic in perturbations are ${O({\ensuremath\varepsilon}^6\chi^{2/s-1})}$, so quasilinear effects are subdominant. The terms involving ${d {\ln {{B_0}}}/d {t}}$ and collisions are ${O({\ensuremath\varepsilon}^6\chi)}$. For [Eq. ([\[eq:driftkinexpandEmu\]]{})]{} to have a solution at ${O({\ensuremath\varepsilon}^6\chi)}$, we see that $s=2/3$ is required, so ${{\delta B_4^\parallel}}/{{B_0}}\propto t^{2/3}$. The resulting equation for ${{\delta f_{4i}^{\mathrm{(res)}}}}'$ is $$\begin{aligned} \pm{{\frac{\partial {{{\delta f_{4i}^{\mathrm{(res)}}}}'}}{\partial {\ell}}}} &\! = & \!- {\frac{\mu{{B_0}}}{{\sqrt{2(E\!-\!\mu B)}}}}\left({{\frac{d }{d t}}}{\frac{{{\delta B_4^\parallel}}}{{{B_0}}}}{{\frac{\partial {{f_{0i}}}}{\partial {E}}}}\!+\! {{\frac{d \ln{{B_0}}}{d t}}} {{\frac{\partial {\,{\left<{{\delta f_{4i}^{\mathrm{(res)}}}}\right>}}}{\partial {E}}}}\right)\nonumber\\ & &\!+C[{\left<{{\delta f_{4i}^{\mathrm{(res)}}}}\right>}]~. \label{eq:deltaequation}\end{aligned}$$ Using a Lorentz operator and bounce averaging, we obtain $$\begin{aligned} {\left<{{\frac{\mu{{B_0}}}{{\sqrt{2(E\!-\!\mu B)}}}}{{\frac{d }{d t}}}{\frac{{{\delta B_4^\parallel}}}{{{B_0}}}}}\right>}{{\frac{\partial {{f_{0i}}}}{\partial {E}}}} & = & \label{eq:trappedequation} \\ && \hspace{-2cm} - {{\frac{d \ln{{B_0}}}{d t}}} {\left<{{\frac{\mu{{B_0}}}{{\sqrt{2(E\!-\!\mu B)}}}}}\right>}{{\frac{\partial {\,{\left<{{\delta f_{4i}^{\mathrm{(res)}}}}\right>}}}{\partial {E}}}}\nonumber \\ & &\hspace{-2cm} +{\frac{{\nu_{ii}}}{{{B_0}}}}{{\frac{\partial {}}{\partial {\mu}}}}\left(\mu\,{\left<{{\sqrt{2(E\!-\!\mu B)}}}\right>}{{\frac{\partial {\,{\left<{{\delta f_{4i}^{\mathrm{(res)}}}}\right>}}}{\partial {\mu}}}}\right)~.\nonumber\end{aligned}$$ #### Physical behavior and temporal evolution. This equation has taken some effort to derive but is fairly transparent physically. It represents a competition between perpendicular betatron cooling of the equilibrium distribution due to the local decrease of the magnetic field in the deepening mirror traps (the l.h.s. of [Eq. ([\[eq:trappedequation\]]{})]{}), the perpendicular betatron heating of the trapped-particle population associated with the increasing mean field ${{B_0}}$ (the first term on the r.h.s.), and their collisional isotropization (the second term on the r.h.s.). In the weakly collisional, unsheared regime (${\nu_{ii}}\neq 0$, ${d {\ln{{B_0}}}/d {t}} \equiv S = 0$), the balance is between betatron cooling and collisions. In the collisionless, shearing regime (${\nu_{ii}}=0$, $S > 0$), it is instead between betatron cooling (of the bulk distribution in mirror perturbations) and heating (of the perturbed distribution in the growing mean field). In order for the system to stay marginal in the face of continued driving and/or collisional relaxation, magnetic perturbations have to continue growing. A simple physical interpretation of the collisionless case is that trapped particles regulate the evolution so as to “see” effectively a constant total magnetic field. Solutions of [Eqs. ([\[eq:momentumiorder5perp\]]{})-(\[eq:p6res\])-(\[eq:trappedequation\])]{} can be found in the form ${{\delta B_4^\parallel}}/{{B_0}}=\mathcal{A}(t)\mathcal{B}({{\ensuremath\mathbf{\ell}}})$. Using [Eq. ([\[eq:trappedequation\]]{})]{}, this implies ${\left<{{\delta f_{4i}^{\mathrm{(res)}}}}\right>}=\alpha\,\mathcal{A}\,\left({d {\mathcal{A}}/d {t}}\right)\,\mathcal{H}[(E-\mu {{B_0}})/\mathcal{A}(t)]\,{\partial {{f_{0i}}}/\partial {E}}$, where $\alpha=1/S$ if ${\nu_{ii}}=0$ and $1/{\nu_{ii}}$ if $S=0$ ($\mathcal{H}$ also depends on the functional form of $\mathcal{B}({{\ensuremath\mathbf{\ell}}})$). Then, from [Eq. ([\[eq:p6res\]]{})]{}, ${{\delta p_{6i}^{\perp}\phantom{}^{\mathrm{(res)}}}}=\alpha\, \mathcal{A}^{3/2}\left({d {\mathcal{A}}/d {t}}\right)\,\mathcal{F}({{\ensuremath\mathbf{\ell}}})$. [Equation ([\[eq:momentumiorder5perp\]]{})]{} will have solutions if $\alpha\, \mathcal{A}^{1/2} \left({d {\mathcal{A}}/d {t}}\right) = \Lambda \Gamma_2$, with $\Lambda$ a constant of order unity. As anticipated in our discussion of the secondary ordering (in $\chi$), this implies that perturbations grow secularly as $$\mathcal{A}(t) = (\Lambda \Gamma_2 S t)^{2/3} \ \mathrm{and}\ \mathcal{A}(t) = (\Lambda\Gamma_2{\nu_{ii}}t)^{2/3}$$ in the shearing-collisionless regime and unsheared-collisional regime, respectively. This result is formally valid for times $St, {\nu_{ii}}\, t ={O({\ensuremath\varepsilon}^4)}$ [^4]. The $t^{2/3}$ time dependence also holds in mixed regimes (${\nu_{ii}}\neq 0$, $S\neq 0$). $\Lambda$ and $\mathcal{B}({{\ensuremath\mathbf{\ell}}})$ (which is not sinusoidal) are obtained by solving a nonlinear eigenvalue equation involving trapping integrals. A detailed classification of the solutions of this equation lies beyond the scope of this Letter. #### Conclusion. [Equation ([\[eq:trappedequation\]]{})]{} and the secular growth of the nonlinear mirror instability, ${{\delta B}}/B\propto t^{2/3}$, in both collisional and collisionless regimes, are the main results of this work. Thus, we appear to be approaching a theory in which trapping effects, higher amplitude nonlinearities [@kuznetsov07; @califano08; @istomin09; @pokhotelov10] and relaxation of anisotropy through anomalous particle scattering [@kunz14a] blend together harmoniously. The results make manifest the importance of particle trapping [@kivelson96; @pantellini98]. Numerical simulations [@kunz14a] confirm $t^{2/3}$ secular growth of mirror perturbations in a collisionless, shearing plasma, with saturation amplitudes ${{\delta B}}/B={O(1)}$ independent of $S$ (cf. [@riquelme14]). The weakly collisional, weakly shearing regimes studied in this Letter occur in many natural environments [@quataert01; @bale09; @rosin11] and are increasingly the focus of attention in the context of high-energy astrophysical plasmas [@schekochihin08; @rosin11; @kunz14a; @riquelme14]. The emergence of finite-amplitude magnetic mirrors with scales smaller than the mean free path lends credence to the idea that microscale instabilities regulate processes such as heat conduction [@chandran98], viscosity, heating [@sharma06; @sharma07; @kunz11] and dynamo [@schekochihin06; @mogavero14] in such plasmas, and therefore profoundly alter their large-scale energetics and dynamics. #### Acknowledgements. The authors thank I. 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[^1]: Formally, we first expand the electron Vlasov equation in $\sqrt{m_e/m_i}$ and then take the limit $T_e\ll T_i$, so electrons still stream along the field very fast. This is purely a matter of analytical convenience: our results also hold for hot, isothermal electrons. [^2]: A Hall term $-\left({\rho_i}{v_{\mathrm{th}i\perp}}/{{\beta_{0i}^{\perp}}}\right){{\hat{{{\ensuremath\mathbf{b}}}}_0}}\cdot\left[{{{{{\ensuremath\mathbf{\nabla}}}{}}_\perp{}}\times{\,\left({\nabla_\parallel{\,{{\delta {{\ensuremath\mathbf{B}}}_5^{\perp}}}/{{B_0}}}}\right)}}\right]$ is formally present in the induction equation at the maximum order ${O({\ensuremath\varepsilon}^6)}$ needed here. However, it can be proven to be zero because of the particular polarization of mirror modes [@califano08] and has therefore been summarily discarded in [Eq. ([\[eq:inductionpert\]]{})]{} to simplify the algebra. [^3]: Nonlinearities quadratic in ${{\delta B}}/B$ are negligible here because of the weakly nonlinear ordering ${{\delta B}}/B\sim\Gamma^2$, which differs from ${{\delta B}}/B\sim\Gamma$ used in [@califano08]. [^4]: These results do not apply to the frequently discussed “idealized” case ${\nu_{ii}}=S=0$, because continued growth of fluctuations was assumed to derive [Eq. ([\[eq:trappedequation\]]{})]{}. In that regime, fluctuations should instead settle in a steady state ${{\delta B}}/B \sim \Gamma^2$ through quasilinear relaxation, possibly after a few transient bounce oscillations [@istomin09]. However, only a small amount of collisions or continued shearing/compression is required for the present theory to apply: these effects are bound to become dynamically important after a few instability times, once quasilinear relaxation has reduced the instability drive $\Gamma$ sufficiently.

--- abstract: 'We perform Hartree calculations of symmetric and asymmetric semi-infinite nuclear matter in the framework of relativistic models based on effective hadronic field theories as recently proposed in the literature. In addition to the conventional cubic and quartic scalar self-interactions, the extended models incorporate a quartic vector self-interaction, scalar-vector non-linearities and tensor couplings of the vector mesons. We investigate the implications of these terms on nuclear surface properties such as the surface energy coefficient, surface thickness, surface stiffness coefficient, neutron skin thickness and the spin–orbit force.' --- = 22.5cm = 16.4cm = -0.4cm = -0.5cm addtoreset[equation]{}[section]{} tempcntc citex\[\#1\]\#2[@filesw auxout tempcnta@tempcntb@neciteacite[forciteb:=\#2citeo]{}[\#1]{}]{} citeo[tempcnta&gt;tempcntbciteacitea[,]{} tempcnta=tempcntbtempcnta]{} [**Nuclear surface properties in\ relativistic effective field theory**]{}\ M. Del Estal, M. Centelles, X. Viñas\ [*Departament d’Estructura i Constituents de la Matèria, Facultat de Física,\ Universitat de Barcelona, Diagonal [*647*]{}, E-[*08028*]{} Barcelona, Spain*]{} [*PACS:*]{}  21.60.-n, 21.30.-x, 21.10.Dr, 21.65.+f [*Keywords:*]{}  Nuclear surface properties; spin–orbit potential; semi-infinite nuclear matter; non-linear self-interactions; Quantum Hadrodynamics; effective field theory. Introduction ============ Quantum hadrodynamics (QHD) and the relativistic treatment of nuclear systems has been a subject of growing interest during recent years [@Ser86; @Cel86; @Rei89; @Ser92; @Ser97]. The model of Walecka [@Ser86] and its non-linear extensions with cubic and quartic self-interactions of the scalar-meson field [@Bog77] have been widely used to this end. This model contains Dirac nucleons together with neutral scalar and vector mesons as well as isovector-vector $\rho$ mesons. At the mean field (Hartree) level, it already includes the spin–orbit force, the finite range and the density dependence which are essential ingredients of the nuclear interaction. This simple model has become very popular in relativistic calculations and describes successfully many properties of the atomic nucleus. From a theoretical point of view, the non-linear model with cubic and quartic scalar self-interactions was classed within renormalizable field theories which can be characterized by a finite number of coupling constants. However, very recently, generalizations of this model that include other non-linear interactions among the meson fields and tensor couplings have been presented on the basis of effective field theories by Serot et al.[@Ser97; @Fur96; @Mul96; @Fur97]. The effective theory contains many couplings of non-renormalizable form that are consistent with the underlying symmetries of QCD. Consequently, one must find some suitable expansion parameters and develop a systematic truncation scheme. For this purpose the concept of naturalness has been employed: it means that the unknown couplings of the theory should all be of the order of unity when written in appropriate dimensionless form using naive dimensional analysis [@Ser97; @Fur96; @Mul96; @Fur97]. Then, one can estimate the contributions coming from different terms by counting powers in the expansion parameters and truncating the Lagrangian at a given level of accuracy. One important fact is the observation that at normal nuclear densities the scalar and vector meson fields, denoted by $\Phi$ and $W$, are small as compared with the nucleon mass $M$ and that they change slowly in finite nuclei. This implies that the ratios $\Phi/M$, $W/M$, $|\mbox{\boldmath $\nabla$}\Phi|/M^2$ and $|\mbox{\boldmath $\nabla$} W|/M^2$ are useful expansion parameters when the effective field theory is applied to the nuclear many-body problem. From this viewpoint, if all the terms involving scalar and meson self-interactions are retained in the Lagrangian up to fourth order, one recovers the well-known non-linear model plus some additional terms [@Ser97; @Fur96; @Fur97]. For the truncation to be consistent, the corresponding coupling constants should exhibit naturalness and cannot be arbitrarily dropped out without an additional symmetry argument. The effective Lagrangian truncated at fourth order contains thirteen free parameters that have been fitted to reproduce twenty-nine finite nuclei observables [@Ser97; @Fur97]. Remarkably, the fitted parameters turn out to be natural and the results are not dominated by the last terms retained. This evidence confirms the utility of the principles of naive dimensional analysis and naturalness and shows that truncating the effective Lagrangian at the first lower orders is justified. The term with a vector-meson quartic self-interaction has been considered previously in relativistic mean field (RMF) calculations from a phenomenological point of view. Bodmer [@Bod91] considered this coupling to avoid the negative coefficient of the quartic scalar self-interaction that appears in many non-linear parametrizations that correctly describe the atomic nucleus [@Bod89]. In some special situations this negative term can lead to a pathological behaviour of the scalar potential. On the other hand, the equation of state is softened at moderate high densities when the vector non-linearity is taken into account. The quartic vector self-interaction has also been phenomenologically used by Gmuca [@Gmu92a; @Gmu92b] in a non-linear model for parametrizing Dirac–Brueckner–Hartree–Fock calculations of nuclear matter. The same idea was developed by Toki et al. and applied to study finite nuclei [@Sug94] and neutron stars [@Sum95]. Recently, the properties of high-density nuclear and neutron matter have been analyzed in the RMF approach taking into account scalar and vector non-linearities [@Mul96]. The tensor couplings of the vector $\omega$ and $\rho$ mesons to the nucleon were investigated by Reinhard et al. [@Rei89; @Ruf88] as an extension of the RMF model, and more recently by Furnstahl et al.[@Fur97; @Fur98] from the point of view of relativistic effective field theory. In these works it was shown that the tensor coupling of the $\omega$ meson has an important bearing on the nuclear spin–orbit splitting. The surface properties of nuclei play a crucial role in certain situations. This is the case, for instance, of saddle-point configurations in nuclear fission or fragment distributions in heavy-ion collisions. Within a context related to the liquid droplet model (LDM) and the leptodermous expansion [@Mye69], the surface properties can be extracted from semi-infinite nuclear matter calculations either quantally or semiclassically (though the total curvature energy coefficient can only be computed semiclassically [@Cen96]). In the non-relativistic case most of the calculations of the surface properties have been carried out using Skyrme forces, quantally [@Bra85] or semiclassically with the help of the extended Thomas–Fermi (ETF) method [@Bra85; @Sto88]. In the relativistic case the nuclear surface has been analyzed within the model since a long time ago. The calculations have been performed semiclassically using the relativistic Thomas–Fermi (TF) method or its extensions (RETF), for symmetric [@Ser86; @Bog77; @Sto91; @Cen93; @Spe93] and asymmetric [@Von94a; @Cen98] matter, and also in the quantal Hartree approach [@Hof89; @Von94b; @Von95]. In the framework of the relativistic model and effective field theory, the main purpose of the present work is to carefully analyze the influence on surface properties of the quartic vector non-linearity, of the newly proposed scalar-vector self-interactions and of the tensor coupling. We shall investigate quantities such as the surface energy coefficient, surface thickness, spin–orbit strength, surface stiffness coefficient and neutron skin thickness obtained from Hartree calculations of symmetric and asymmetric semi-infinite nuclear matter. The paper is organized as follows. Section 2 is devoted to the basic theory. The results on the surface properties of symmetric matter are discussed in Section 3. Section 4 addresses the case of asymmetric systems. The summary and conclusions are given in the last section. Mean field equations for symmetric semi-infinite nuclear matter =============================================================== Following Ref. [@Fur96], to derive the mean field equations one starts from an energy functional containing Dirac baryons and classical scalar and vector mesons. The energy functional can be obtained from the effective Lagrangian in the Hartree approach using many-body techniques [@Ser97; @Fur97]. However, this energy functional can also be considered as an expansion in $\Phi/M$, $W/M$, $|\mbox{\boldmath $\nabla$}\Phi|/M^2$ and $|\mbox{\boldmath $\nabla$} W|/M^2$ of a general energy density functional that contains all the correlation effects. The theoretical basis of this functional lies on the extension of the Hohenberg–Kohn theorem [@Hoh64] to QHD [@Spe92]. Using the Kohn–Sham scheme [@Koh65] with the mean fields playing the role of Kohn–Sham potentials, one finds similar mean field equations to those obtained from the Lagrangian [@Spe92], but including effects beyond the Hartree approach through the non-linear couplings [@Ser92; @Fur96; @Mul96; @Fur97]. A semi-infinite system of uncharged nucleons corresponds to a one-dimensional geometry where half the space is filled with nuclear matter at saturation and the other half is empty, so that a surface develops around the interface. The fields and densities change only along the direction perpendicular to the medium. Specifying the energy density functional considered in Refs. [@Ser97] and [@Fur97] to symmetric semi-infinite nuclear matter with the surface normal pointing into the $z$ direction one has (z) & = & \_\_\^(z) { -i + + W(z) - W(z) } \_(z)\ & & + ( 1 + \_1) ( (z))\^2 + ( + + ) \^2(z)\ & & - ( 1 +\_2) ( W(z) )\^2 - W\^4 (z)\ & & - (1 + \_1 + ) W\^2 (z) , \[eq1\] where the index $\alpha$ runs over all occupied states of the positive energy spectrum, $\Phi \equiv g{_{\rm s}} \phi_0$ and $ W \equiv g{_{\rm v}} V_0$ (notation as in Ref. [@Ser86]). Except for the terms with $\alpha_1$ and $\alpha_2$, the functional [(\[eq1\])]{} is of fourth order in the expansion. We retain the fifth-order terms $\alpha_1$ and $\alpha_2$ because in Refs. [@Ser97] and [@Fur97] they have been estimated to be numerically of the same magnitude as the quartic scalar term in the nuclear surface energy. The mean field equations are obtained by minimizing with respect to $\varphi^\dagger_\alpha$, $\Phi$ and $W$: { -i + + W(z) - W(z) } \_(z) = \_ \_(z) , \[eq2\] -(z) + m[\_[s]{}]{}\^2 (z) & = & g[\_[s]{}]{}\^2 [\_[s]{}]{}(z) -[m[\_[s]{}]{}\^2M]{}\^2 (z) ([\_32]{}+[\_43!]{}[(z)M]{} )\ & & +[g[\_[s]{}]{}\^2 2M]{} (\_1+\_2[(z)M]{}) [ m[\_[v]{}]{}\^2g[\_[v]{}]{}\^2]{} W\^2 (z)\ & & +[\_1 2M]{}\[ ((z))\^2 +2(z)(z) \] + [\_2 2M]{} [g[\_[s]{}]{}\^2g[\_[v]{}]{}\^2]{} (W(z))\^2 , \[eq3\]\ -W(z) + m[\_[v]{}]{}\^2 W(z) & = & g[\_[v]{}]{}\^2 ( (z) + [\_[T]{}]{}(z) ) -( \_1+[\_22]{}[(z)M]{} )[(z)M]{} m[\_[v]{}]{}\^2 W(z)\ & & -[13!]{}\_0 W\^3(z) +[\_2 M]{} \[(z) W(z) +(z)W(z)\] . \[eq4\] The baryon, scalar and tensor densities are respectively (z) & = & \_\_\^(z) \_(z) , \[eq4a\]\ [\_[s]{}]{}(z) & = & \_\_\^(z) \_(z) , \[eq4b\]\ [\_[T]{}]{}(z) & = & \_ . \[eq4c\] The expression of the four-component spinors $\varphi_\alpha(z)$ in the semi-infinite medium was given by Hofer and Stocker in Ref.[@Hof89]. In a semi-infinite nuclear matter calculation the sum over the single-particle states is replaced by an integration over momenta: \_2 \_ d[**[k]{}**]{} , \[eq5\] where $\Omega$ stands for the volume of the box, the factor 2 takes into account the isospin degree of freedom and $\lambda$ describes the spin orientation of the nucleons. Introducing the Fermi momentum $k{_{\rm F}}$, the integration domain is restricted to $k_{x}^2 + k_{y}^2 + k_{z}^2 = k_{\bot}^2 + k_{z}^2 \leq k{_{\rm F}}^2$, with $k_{z} \geq 0$ if the bulk nuclear matter is located at $z=-\infty$. Following the method outlined in Ref. [@Hof89] one finds two sets ($\lambda = \pm1$) of first-order differential equations for the orbital part of the upper and lower components of the Dirac spinors: - G\_a (z) & = & F\_a (z) , \[eq6a\]\ - - F\_a (z) & = & G\_a (z) , \[eq6b\]where $a = (k_{z},k{_{\rm \bot}},\lambda)$ and $M^{*}(z) = M - \Phi(z)$ is the Dirac effective mass of the nucleons. From the asymptotic behaviour at $z=-\infty$ (bulk nuclear matter), the condition on the energy eigenvalues is $\varepsilon_a = \sqrt{k_{\bot}^2 + k_{z}^2 + {M^{*}_{\infty}}^{2}} + W_{\infty}$, with $M^{*}_{\infty}$ and $W_{\infty}$ being the nuclear matter values of $M^{*}$ and $W$. The densities for each spin orientation $\lambda= \pm1$ read \^(z) & = & \_[0]{}\^[ k [\_[F]{}]{}]{} dk\_[z]{} \_[0]{}\^ dk\_ k\_ ( |G\_a (z) |\^2 + |F\_a (z) |\^2 ) , \[eq7\]\ [\_[s]{}]{}\^(z) & = & \_[0]{}\^[ k [\_[F]{}]{}]{} dk\_[z]{} \_[0]{}\^ dk\_ k\_ ( |G\_a (z) |\^2 - |F\_a (z) |\^2 ) , \[eq8\]\ [\_[T]{}]{}\^(z) & = & \_[0]{}\^[ k [\_[F]{}]{}]{} dk\_[z]{} \_[0]{}\^ dk\_ k\_ ( F\_a(z) G\_a(z) ) , \[eq9\] and the total densities are given by (z) = \_\^(z) , [\_[s]{}]{}(z) = \_[\_[s]{}]{}\^(z) , [\_[T]{}]{}(z) = \_[\_[T]{}]{}\^(z) . \[eq9b\] Using the equations of motion the energy density of the semi-infinite nuclear matter system can be written as follows: (z) & = & \_ \_[0]{}\^[k [\_[F]{}]{}]{} dk\_[z]{} \_[0]{}\^ dk\_ k\_ ( +W\_ ) (| G\_a (z) |\^2 + |F\_a (z) |\^2 )\ & & + (z)[\_[s]{}]{}(z) - W(z) ( (z) + [\_[T]{}]{}(z) ) - (+ ) \^2 (z)\ & & + ( \_1 +\_2 ) W\^2 (z) + W\^4 (z)\ & & - ( (z) )\^2 + ( W(z) )\^2 . \[eq10\] Finally, the surface energy coefficient $E{_{\rm s}}$ is obtained from the expression [@Mye69] E[\_[s]{}]{} = 4r\_[0]{}\^2 \^\_[-]{} dz , \[eq11\] where $a{_{\rm v}}$ is the energy per particle in bulk nuclear matter and $r_0$ is the nuclear radius constant: $r_0 = (3/4 \pi \rho_0)^{1/3}$, with $\rho_0$ the nuclear matter density. Another important quantity in the study of the nuclear surface structure is the spin–orbit interaction. By elimination of the lower spinor in terms of the upper spinor, one obtains a Schrödinger-type equation with a term $V{_{\rm so}}(z)$ that has the structure of the single-particle spin–orbit potential for the non-relativistic case [@Hof89; @Von95]. In our present model the orbital part of the spin–orbit potential reads as V[\_[so]{}]{}(z) = . \[eq12\]In the non-relativistic limit, by means of a Foldy–Wouthuysen reduction, Eq. [(\[eq12\])]{} becomes V[\_[so]{}]{}\^[FW]{}(z) = \[eq13\]and the nucleons are then moving in a central potential of the form V[\_[c]{}]{}(z) = W(z) - (z) . \[eq14\] Surface properties in the symmetric case ======================================== Although previous works [@Cen93; @Spe93; @Von94a; @Cen98; @Hof89; @Von94b; @Von95] have thoroughly investigated the properties of the nuclear surface in the standard non-linear model, for which $\kappa_3$ and $\kappa_4$ are the only non-linearities in [(\[eq1\])]{}, here we wish to enlarge this study by including the additional non-linear and tensor couplings considered in Refs. [@Ser97; @Fur96; @Fur97]. In concrete, we want to study the role of the quartic vector self-interaction $\zeta_0$ that has been used after the work of Bodmer [@Bod91], the role of the terms with $\eta_1$ and $\eta_2$ that couple the scalar and vector fields, of the terms with $\alpha_1$ and $\alpha_2$ that imply the gradients of the fields, and of the tensor coupling $f{_{\rm v}}$ of the vector $\omega$ meson to the nucleon. While $\zeta_0$, $\eta_1$ and $\eta_2$ can be classified as volume contributions, the couplings $\alpha_1$, $\alpha_2$ and $f{_{\rm v}}$ are genuine surface terms. On the basis of the concept of naturalness there is no reason to omit any of these terms in the energy density functional [(\[eq1\])]{}, unless there exists a symmetry principle to forbid it. However, to clarify the impact on the surface properties of the aforementioned couplings we will analyze each one separately, as it has been similarly done in Ref.[@Fur96]. In this section we shall study symmetric systems, while in Section 4 we shall address the case of asymmetric matter. Effect of the quartic vector self-interaction --------------------------------------------- In the conventional non-linear model the value of the coefficients $g{_{\rm s}}^2/m{_{\rm s}}^2$, $g{_{\rm v}}^2/m{_{\rm v}}^2$, $\kappa_3$ and $\kappa_4$ can be univocally obtained by imposing that for nuclear matter at saturation the density $\rho_0$, energy per particle $a{_{\rm v}}$, effective mass $M^{*}_{\infty}/M$ and incompressibility modulus $K$ take given values. When the vector-meson quartic self-interaction is switched on, the Dirac equation for the baryons and the Klein–Gordon equation for the vector field in infinite nuclear matter become (throughout this subsection we set $\eta_1=\eta_2=\alpha_1=\alpha_2=f{_{\rm v}}=0$): a[\_[v]{}]{} & = & + W\_ - M , \[eq3.1\]\ m[\_[v]{}]{}\^2 W\_ & = & g[\_[v]{}]{}\^2 \_0 - \_0 W\^3\_ . \[eq3.2\] The saturation density $\rho_0$ and the Fermi momentum $k{_{\rm F}}$ are related as usual by $\rho_0= 2k{_{\rm F}}^3/3\pi^2$. Specifying $\rho_0$, $a{_{\rm v}}$ and $M^{*}_{\infty}/M$, from the above equations one extracts the coupling constant $g{_{\rm v}}$ as a function of $\zeta_0$ (the nucleon and $\omega$ masses take their empirical values: $M= 939$ MeV and $m{_{\rm v}}= 783$ MeV). The steps to calculate $g{_{\rm s}}^2/m{_{\rm s}}^2$, $\kappa_3$ and $\kappa_4$ are then the same as when $\zeta_0= 0$, see e.g. Refs. [@Bod91] and [@Bod89], but now these coefficients become functions of $\zeta_0$. The reader will find a detailed study of the implications of the vector non-linearity $\zeta_0$ in nuclear matter in Refs. [@Fur96] and [@Bod91]. The assumption of naturalness requires that the couplings $g{_{\rm s}}/4\pi$, $g{_{\rm v}}/4\pi$, $\kappa_3$, $\kappa_4$ and $\zeta_0$ should all be roughly of the order of unity. Figure 1 illustrates the variation of these couplings as a function of the non-dimensional parameter \_0 = \[eq3.2.b\] used by Bodmer[^1] in Ref. [@Bod91]. With $m{_{\rm s}}= 490$ MeV, the figure presents the results for four sets of nuclear matter properties: $\rho_0 = 0.152$ fm$^{-3}$ ($k{_{\rm F}}= 1.31$ fm$^{-1}$), $a{_{\rm v}} = -16.42$ MeV, $K= 200$ and 350 MeV, and $M_{\infty}^{*}/M= 0.6$ and 0.7. When $K= 200$ MeV and $M_{\infty}^{*}/M= 0.6$ the equilibrium properties of the interaction and the scalar mass $m{_{\rm s}}$ are very close to those of the non-linear parametrization NL1 [@Rei86]. We realize that $g{_{\rm s}}$ and $g{_{\rm v}}$ are only weakly affected by $\eta_0$ (the change is not appreciable in the scale of Figure 1). However, $\eta_0$ has a direct effect on the scalar-meson quartic self-interaction $\kappa_4$, moving it from a negative value when the quartic vector term is absent ($\eta_0 =\infty$) to a desirable positive value when $\eta_0 \sim 2$. The change of sign of $\kappa_4$ takes place at larger values of $\eta_0$ for larger $K$ and $M^{*}_{\infty}/M$; in fact, $\kappa_4$ is already positive at $\eta_0= \infty$ if $K = 350$ MeV and $M^{*}_{\infty}/M = 0.7$. Both non-linearities of the scalar field $\kappa_3$ and $\kappa_4$ remain in the natural zone for $2 \leq \eta_0 \leq \infty$ approximately, excepting the case of $K = 350$ MeV and $M^{*}_{\infty}/M = 0.7$, but they start to depart appreciably from their natural values when $\eta_0 \leq 2$. These trends fairly agree with the assumption of naturalness: $\eta_0 \geq 2$ corresponds to the region where $\zeta_0$ can be considered as natural (see the figure), and for this range of $\eta_0$ values also the rest of coupling constants are natural. The behaviour gleaned from Figure 1 is rather independent of the saturation density $\rho_0$ and energy $a{_{\rm v}}$, and e.g. we have found similar trends with the specific nuclear matter properties used by Bodmer in Ref. [@Bod91]. Next we turn our attention to the surface properties. In Figures 2 and 3 we have plotted, respectively, the surface energy coefficient $E{_{\rm s}}$ and the surface thickness $t$ of the semi-infinite density profile (the 90%–10% fall-off distance) against the $\eta_0$ parameter. The selected values of $\eta_0$ are those employed in Table 1 of Ref. [@Bod91]. With $\rho_0 = 0.152$ fm$^{-3}$ and $a{_{\rm v}} = -16.42$ MeV, we have performed the calculations for a few incompressibilities ($K= 125$, 200 and 350 MeV) and effective masses ($M^{*}_{\infty}/M= 0.6$ and 0.7). Furthermore, we have considered two values of the mass of the scalar meson ($m{_{\rm s}}= 490$ and 525 MeV). This quantity governs the range of the attractive interaction and determines the surface fall-off: a larger $m{_{\rm s}}$ results in a steeper surface and a reduction of $E{_{\rm s}}$ and $t$. From Figures 2 and 3 we realize that the quartic vector self-interaction scarcely alters the values of the surface energy coefficient and of the surface thickness if $\zeta_0$ remains in the natural domain, i.e., if $2 \leq \eta_0 \leq \infty$. At $K= 350$ MeV and $M^{*}/M \geq 0.6$, $E{_{\rm s}}$ is raised by decreasing $\eta_0$. This tendency may be inverted at smaller values of the incompressibility, depending also on the value of the nucleon effective mass, but the global trends are practically independent of the mass of the scalar meson. The thickness $t$ exhibits a more monotonous behaviour: in all cases it stays almost equal to its value at $\eta_0= \infty$ and goes down slightly for $\eta_0 \leq 2$. Noticeable departures of the surface energy coefficient and thickness from the $\eta_0 = \infty$ values can be found only if $\eta_0$ is decreased beyond the natural limit, a situation where the interaction is mainly ruled by the vector-meson quartic self-interaction. Even though the impact of the vector-meson quartic self-interaction is small, it can help to find parameter sets for which both the surface energy coefficient and the surface thickness lie in the empirical region. This fact is illustrated in Figure 4, where $E{_{\rm s}}$ and $t$ are drawn versus $M^{*}_{\infty}/M$ for $\eta_0= \infty$, 2 and $0.5$ and for $m{_{\rm s}}= 450$, 500 and 550 MeV, using the nuclear matter conditions of Ref. [@Bod91]: $\rho_0 = 0.1484$ fm$^{-3}$, $a{_{\rm v}} = -15.75$ MeV and $K=200$ MeV. (We also performed the calculations for $\eta_0= 5$ and $\eta_0= 1$ to confirm the trends we discuss below.) The horizontal dashed lines in Figure 4 serve to indicate the empirical region for the surface energy and thickness. If at $\eta_0= \infty$ we concentrate, for instance, on the parametrizations with $m{_{\rm s}}=450$ MeV we see that the one with $M^{*}_{\infty}/M= 0.7$ yields, simultaneously, $E{_{\rm s}}$ and $t$ within the empirical region. The agreement between the calculated values with $m{_{\rm s}}=450$ MeV and the empirical region improves for $\eta_0= 2$, where both $E{_{\rm s}}$ and $t$ are acceptable for practically all the values of the effective mass considered. For $\eta_0 \geq 2$ the dependence of $E{_{\rm s}}$ and $t$ upon the nucleon effective mass is similar to that found in the usual model without a quartic vector self-interaction [@Cen93; @Von94b]. The general tendencies start to change when $\eta_0$ is lowered and leaves the natural region. As $\eta_0$ becomes smaller Figure 4 shows that the slope of the curves of $E{_{\rm s}}$ and $t$ as a function of $M^{*}_{\infty}/M$ changes, and that the curves for the different $m{_{\rm s}}$ come closer together. To get more insight about the influence of the vector-meson quartic self-interaction, we display in Figure 5 the profiles of the baryon density $\rho(z)$ and of the surface tension density (Swiatecki integrand) $\sigma(z)= {\cal E}(z) - (a{_{\rm v}}+M) \rho(z)$, Eq.[(\[eq11\])]{}. In turn, we have represented in Figure 6 the orbital part of the spin–orbit potential $V{_{\rm so}}(z)$, Eq. [(\[eq12\])]{}, at the Fermi surface (i.e., evaluated at $k= k{_{\rm F}}$) and the central mean field $V{_{\rm c}}(z)$ defined in Eq. [(\[eq14\])]{}. The properties of the interactions used in these figures are $\rho_0= 0.152$ fm$^{-3}$, $a{_{\rm v}}= -16.42$ MeV, $K= 200$ MeV, $M^{*}_{\infty}/M= 0.6$ and 0.7, and $m{_{\rm s}}= 490$ MeV. Results are shown for $\eta_0= \infty$, 2 and 0.5. The corresponding values of the surface energy coefficient and surface thickness can be read from Figures 2 and 3. It can be seen that the local quantities depicted in Figures 5 and 6 oscillate as functions of $z$ (Friedel oscillations). Both the surface tension density and the single-particle spin–orbit potential are confined to the surface and average to zero inwards as the bulk matter is approached. The local profiles for $\eta_0= 2$, which somehow marks the limit of naturalness as we have commented, are almost equal to those obtained in the absence of the quartic vector self-interaction. In agreement with Figures 2 and 3, only when $\eta_0$ is decreased to non-natural values one can notice some changes in the profiles, which are more visible for $M^{*}_{\infty}/M = 0.6$ than for $M^{*}_{\infty}/M = 0.7$. Decreasing $\eta_0$ makes the surface steeper and the thickness $t$ smaller, produces an enhancement in the surface region of the density $\rho(z)$ and of the mean field $V{_{\rm c}}(z)$, and builds up Friedel oscillations in $\sigma(z)$ and in the spin–orbit potential $V{_{\rm so}}(z)$. It is well known that the experimental spin–orbit splittings require within narrow bounds a Dirac effective mass $M^{*}_{\infty}/M$ around 0.6 in the conventional relativistic model [@Bod89; @Fur96]. Figure 6 shows that introducing a quartic vector self-interaction makes the spin–orbit well deeper. However, it is only a minor effect: with $M^{*}_{\infty}/M= 0.7$ it is not possible to reproduce the spin–orbit interaction of the case $M^{*}_{\infty}/M= 0.6$ at $\eta_0= \infty$, not even if one sets $\eta_0= 0.5$ (which in addition brings about an unreallistically small $t$, see Figure 3). A similar conclusion was drawn in Ref. [@Bod91] from an analysis in nuclear matter. We also have computed the non-relativistic limit $V{_{\rm so}}^{\rm FW}(z)$ of the spin–orbit potential given by the expression [(\[eq13\])]{}. In agreement with Ref. [@Von95] we have found that while $V{_{\rm so}}^{\rm FW}(z)$ qualitatively reproduces the behaviour of $V{_{\rm so}}(z)$, it strongly underestimates the quantitative depth of the fully relativistic spin–orbit strength (by $\sim 20$% for $M^{*}_{\infty}/M= 0.6$ and by $\sim 15$% for $M^{*}_{\infty}/M= 0.7$). Influence of the volume cubic and quartic scalar-vector interactions -------------------------------------------------------------------- The next bulk terms in the energy density [(\[eq1\])]{} that contain non-linear meson interactions are - ( \_1 + ) W\^2 . \[eq3.7\] In analogy to Figure 1 for $\eta_0$, in Figure 7 we study the change of the couplings $g{_{\rm s}}/4\pi$, $g{_{\rm v}}/4\pi$, $\kappa_3$ and $\kappa_4$ with the parameters $\eta_1$ and $\eta_2$ (introducing each one separately) for some specific equilibrium properties. With $\rho_0 = 0.152$ fm$^{-3}$, $a{_{\rm v}} = -16.42$ and $m{_{\rm s}}= 490$ MeV, in part (a) of Figure 7 it is $K= 200$ MeV and $M_{\infty}^{*}/M= 0.6$, while in part (b) it is $K= 350$ MeV and $M_{\infty}^{*}/M= 0.7$. For $K= 200$ MeV and $M_{\infty}^{*}/M= 0.6$ both $\eta_1$ and $\eta_2$ have a considerable effect on the scalar non-linearities $\kappa_3$ and $\kappa_4$. The coupling $\kappa_4$ changes sign for $\eta_2 \approx 1$, but it remains negative in the interval of $\eta_1$ values used. To keep all the coupling constants within natural values we see that the range for $\eta_1$ and $\eta_2$ (when introduced separately) is restricted to run roughly from $-0.5$ to $2.5$. Consider now different values of the incompressibility and effective mass, as in part (b) of Figure 7. The dependence of the couplings on $\eta_2$ is not significantly altered. However, increasing either $K$ or $M_{\infty}^{*}/M$ results in a smoother slope of $\kappa_3$ with $\eta_1$, while it makes $\kappa_4$ grow steadily with $\eta_1$ and become positive at some value of this parameter. We have checked for $K= 200$ MeV and $M_{\infty}^{*}/M= 0.55$, and for $K= 125$ MeV and $M_{\infty}^{*}/M= 0.6$, that $\kappa_4$ remains negative at all values of $\eta_1$ and that $\kappa_3$ grows with $\eta_1$ faster than in part (a) of Figure 7. The influence of the non-linear interactions $\eta_1$ and $\eta_2$ on the surface properties is analyzed in Figure 8. With $\zeta_0= \alpha_1= \alpha_2= f{_{\rm v}}= 0$, in this figure we have computed $E{_{\rm s}}$ and $t$ taking into account the terms [(\[eq3.7\])]{}. The saturation conditions of the interaction and the scalar mass are the same as in part (a) of Figure 7. The results are displayed in the plane $\eta_1$–$\eta_2$ in the form of contour plots of constant $E{_{\rm s}}$ (solid lines) and of constant $t$ (dashed lines). The range of variation of $\eta_1$ and $\eta_2$ lies in the region imposed by naturalness, and yields values of $E{_{\rm s}}$ and $t$ within reasonable limits. As it can be inferred from the nearly vertical lines in the $\eta_1$–$\eta_2$ plane, the surface energy coefficient and thickness depend mostly on $\eta_1$ and are rather independent of $\eta_2$. The consequence of increasing $\eta_1$ is a reduction of the values of $E{_{\rm s}}$ and $t$. The lines of constant $t$ turn out to be, roughly speaking, parallel to the lines of constant $E{_{\rm s}}$. This means that from the interplay of the parameters $\eta_1$ and $\eta_2$ it is not possible to change the value of $t$ relative to that of $E{_{\rm s}}$ (for example, we see in Figure 8 that $t \sim 2.2$ fm if $E{_{\rm s}} = 18$ MeV). We have calculated the spin–orbit potential $V{_{\rm so}}(z)$ at the Fermi surface for several values of $\eta_1$ and $\eta_2$. We have found that these couplings have a marginal effect on the spin–orbit strength, as it happened to be the case with the other bulk non-linearity $\eta_0$. To get some information about the incidence on the surface energy and thickness of all the volume non-linear meson interactions together, we have repeated the calculations in the $\eta_1$–$\eta_2$ plane setting $\eta_0= 2$ for the quartic vector self-interaction. One finds similar features to those of Figure 8. The effect of $\eta_0= 2$ is just shifting $E{_{\rm s}}$ and $t$ towards smaller values as compared with the case $\eta_0= \infty$ ($\zeta_0 = 0$), which is in accordance with what was found in Figures 2 and 3 at $K= 200$ MeV and $M_{\infty}^{*}/M= 0.6$. Influence of the non-linear terms with gradients ------------------------------------------------ Now we discuss the non-linear interactions \[eq3.8\] that vanish in infinite nuclear matter. We recall that these terms are actually of order 5 in the expansion of the effective Lagrangian but, following Refs. [@Ser97] and [@Fur97], we include them because they can be relevant in the surface due to their gradient structure. Using the same saturation properties and scalar mass of Figure 8, in Figure 9 we have calculated $E{_{\rm s}}$ and $t$ for several values of $\alpha_1$ and $\alpha_2$ with $\zeta_0= \eta_1= \eta_2= f{_{\rm v}}= 0$. One observes that the curves of constant $E{_{\rm s}}$ are projected onto the plane $\alpha_1$–$\alpha_2$ as almost parallel straight lines (at least in the analyzed region, corresponding to natural values of $\alpha_1$ and $\alpha_2$). The same happens to the curves of constant $t$. But in contrast with the situation found in the plane $\eta_1$–$\eta_2$ (Figure 8), the slope of the lines of constant $t$ is different from that of the lines of constant $E{_{\rm s}}$. This means that by varying $\alpha_1$ and $\alpha_2$ one can achieve some modification on the surface thickness while keeping the same surface energy. For example, if we consider the contour line of $E{_{\rm s}}= 18$ MeV we find that for $\alpha_2= 2.0$ it is $t \sim 2.05$ fm, whereas for $\alpha_2= -1.5$ it is $t \sim 2.25$ fm. From Figure 9 we also see that increasing $\alpha_1$ at constant $\alpha_2$ brings about larger values of $E{_{\rm s}}$ and $t$, and that the opposite happens if one increases $\alpha_2$ at constant $\alpha_1$. We have repeated the calculations of Figure 9 ($K= 200$ MeV, $M_{\infty}^{*}/M= 0.6$) for $K= 350$ MeV and for $M_{\infty}^{*}/M= 0.7$, to verify to which extent the behaviour in the $\alpha_1$–$\alpha_2$ plane is affected by the incompressibility and effective mass of the interaction. Certainly, the contour lines of $E{_{\rm s}}$ and $t$ are shifted with respect to Figure 9, but the trends with $\alpha_1$ and $\alpha_2$ turn out to be qualitatively the same. The range of variation of the surface energy and thickness in the $\alpha_1$–$\alpha_2$ region we are considering is shorter when $M^{*}_\infty/M= 0.7$, while it is more or less the same when $K= 350$ MeV. To assess the importance of the bulk non-linear meson interactions on our study on $\alpha_1$ and $\alpha_2$, we have performed calculations as in Figure 9 but setting $\eta_0= 2$ with $\eta_1= \eta_2= 0$, and setting $\eta_1= 1$ with $\eta_0= \infty$ and $\eta_2= 0$ (as indicated, the effect of $\eta_2$ is much smaller than that of $\eta_1$). The results show a completely similar behaviour to Figure 9. Even the slope of the contour lines of $E{_{\rm s}}$ and $t$ in the $\alpha_1$–$\alpha_2$ plane changes only slightly. Comparing with Figure 9, when $\eta_0= 2$ one finds that $E{_{\rm s}}$ is shifted by approximately $-1$ MeV, and that when $\eta_1= 1$ then $E{_{\rm s}}$ is shifted by around $-3$ MeV. The shifts of the surface thickness $t$ are less regular and their magnitude depends on the value of $\alpha_1$ and $\alpha_2$. In order to investigate the impact of the gradient interactions $\alpha_1$ and $\alpha_2$ on the spin–orbit potential, in Figure 10 we have plotted $V{_{\rm so}}(z)$ at the Fermi surface for a few selected values of $\alpha_1$ and $\alpha_2$. The nuclear matter properties and the scalar mass are the same as in Figure 6, where we studied the dependence of $V{_{\rm so}}(z)$ on $\eta_0$. One can see that the meson interaction with $\alpha_1= 1$ and $\alpha_2= 0$ reduces the strength of $V{_{\rm so}}(z)$ and shifts the position of the minimum slightly to the exterior. On the contrary, the interaction with $\alpha_1= 0$ and $\alpha_2= 1$ makes the potential well deeper. The combined effect is probed in the case $\alpha_1= \alpha_2= 1$. Since in the relativistic model the spin–orbit force is strongly correlated with the Dirac effective mass, we compare in Figure 10 the situation at $M^{*}_{\infty}/M = 0.6$ and at $M^{*}_{\infty}/M = 0.7$. We realize that the incidence of $\alpha_1$ on $V{_{\rm so}}(z)$ is weaker for $M^{*}_{\infty}/M = 0.7$. The small perturbations arising from the gradient interactions when $M^{*}_{\infty}/M = 0.7$ are not sufficient to produce a spin–orbit strength equivalent to that of the case $M^{*}_{\infty}/M = 0.6$. Role of the tensor coupling of the omega meson ---------------------------------------------- To conclude this section we investigate the influence of the $\omega$ tensor coupling \_\_\^(z) \_(z) \[eq3.9\] which adds some momentum and spin dependence to the interaction. The natural combination for this coupling is $f{_{\rm v}}/4$. Well known from one-boson-exchange potentials (where $f{_{\rm v}}$ above is commonly written as $f{_{\rm v}}/g{_{\rm v}}$), the tensor coupling was included in the fits to nuclear properties of Refs.[@Rei89; @Ruf88] (conventional QHD) and [@Ser97; @Fur97] (effective field theory), and in the study of the nuclear spin–orbit force in chiral effective field theories carried out in Ref.[@Fur98]. These works noticed the existence of a trade-off between the size of the $\omega$ tensor coupling and the size of the scalar field. In other words, the tensor coupling breaks the tight connection existing in relativistic models between the value of the nucleon effective mass at saturation and the empirical spin–orbit splitting in finite nuclei (which constrains $M^{*}_{\infty}/M$ to lie between 0.58 and 0.64 [@Fur96]). Including a tensor coupling the authors of Refs. [@Ser97; @Fur97; @Fur98] were able to obtain natural parameter sets that provide excellent fits to nuclear properties and spin–orbit splittings with an equilibrium effective mass remarkably higher ($M^{*}_{\infty}/M \sim 0.7$) than in models that ignore such coupling. We want to analyze the nature of this effect in the simpler but more transparent framework of semi-infinite nuclear matter. In Figure 11 we have drawn the surface energy coefficient and the surface thickness as functions of $f{_{\rm v}}$ in the range $[-0.6,0.9]$ for two values of the effective mass and of the incompressibility, having set $m{_{\rm s}}= 490$ MeV, $\rho_0 = 0.152$ fm$^{-3}$ and $a{_{\rm v}} = -16.42$ MeV. To exemplify the incidence of $f{_{\rm v}}$ on the spin–orbit potential, Figure 12 displays $V{_{\rm so}}(z)$ at the Fermi surface for a few of the cases of Figure 11. We also performed the calculations for $m{_{\rm s}}= 525$ MeV: $E{_{\rm s}}$ and $t$ are shifted downwards with respect to Figure 11 and $V{_{\rm so}}(z)$ is deeper than in Figure 12, but the global trends with $f{_{\rm v}}$ are the same. Figure 11 shows the strong reduction of $E{_{\rm s}}$ and $t$ as $f{_{\rm v}}$ increases (the slope of the curves is milder for $M^{*}_{\infty}/M= 0.7$ than for $M^{*}_{\infty}/M= 0.6$). Figure 12 reveals that this fact is associated with a deeper and wider spin–orbit potential. This agrees with the results of Hofer and Stocker [@Hof89] who showed in the standard RMF model that the spin–orbit coupling reduces the surface energy and thickness. At variance with the individual values of $E{_{\rm s}}$ and $t$, the ratio $E{_{\rm s}}/t$ stays to a certain extent constant with $f{_{\rm v}}$. Figure 12 evinces the sensitivity of $V{_{\rm so}}(z)$ to $f{_{\rm v}}$. The lower the nucleon effective mass is, the larger the effect. For $M^{*}_{\infty}/M= 0.7$ we realize that with positive values of $f{_{\rm v}}$ ($\sim 0.3$ in the present case) one can get a spin–orbit strength comparable, or even stronger, to that of the case $M^{*}_{\infty}/M= 0.6$ and $f{_{\rm v}}= 0$, something that could not be achieved with natural values of the couplings studied in the previous sections. Since our parametrization with $M^{*}_{\infty}/M= 0.7$ and $K= 200$ MeV at $f{_{\rm v}}= 0$ already has reasonable surface energy and thickness ($E{_{\rm s}}= 16.6$ MeV and $t= 1.97$ fm), increasing $f{_{\rm v}}$ results in smaller values of $E{_{\rm s}}$ and $t$. This should be compensated with the other couplings (especially $\alpha_1$ and $\alpha_2$) that modify the spin–orbit strength to a lesser degree than $f{_{\rm v}}$, or the starting point should have other values of the incompressibility $K$ and the scalar mass $m{_{\rm s}}$. The spin–orbit effect has to do with the explicit dependence of the nucleon orbital wave functions on the spin orientation $\lambda$. As described in Ref. [@Hof89] nucleons with $\lambda= +1$ feel an attractive spin–orbit potential and are pushed to the exterior of the surface, whereas the spin–orbit force is repulsive for nucleons with $\lambda= -1$ which are pushed to the interior. As a consequence of this a depletion of particles with $\lambda= -1$ occurs at the surface. This behaviour is contrasted in Figure 13 for $f{_{\rm v}}= 0$ and $f{_{\rm v}}= 0.6$ in the case $M^{*}_{\infty}/M= 0.7$. The figure depicts the profiles of the total baryon and tensor densities as well as those of their spin components $\rho^\lambda (z)$ and $\rho{_{\rm T}}^\lambda (z)$ for $\lambda= \pm1$, Eqs.[(\[eq7\])]{}–[(\[eq9b\])]{}. When the spin–orbit strength is large, attraction dominates over repulsion and more particles accumulate at the surface than particles are removed from it. Then the total baryon density is enhanced at the surface region and it falls down more steeply. Surface properties in the asymmetric case ========================================= We briefly recall some basic definitions concerning nuclear surface symmetry properties (further details on the relativistic treatment of asymmetric infinite and semi-infinite nuclear matter can be found in Refs. [@Von94a; @Cen98; @Von95]). For a bulk neutron excess $\delta_0= (\rho{_{\rm n0}} - \rho{_{\rm p0}}) / (\rho{_{\rm n0}} + \rho{_{\rm p0}})$ (i.e., the asymptotic asymmetry far from the surface), a surface energy coefficient can be computed as E[\_[s]{}]{} (\_0) = 4r\_[0]{}\^2 \^\_[-]{} dz , \[eq4.3\]where ${\cal E}(z)$ is the total energy density of the system of neutrons and protons, $a{_{\rm v}}(\delta_0)$ denotes the energy per particle in nuclear matter of asymmetry $\delta_0$, and $\rho(z)= \rho{_{\rm n}}(z) + \rho{_{\rm p}}(z)$ with $\rho{_{\rm n}}$ and $\rho{_{\rm p}}$ referring to the neutron and proton densities, respectively. According to the liquid droplet model (LDM) [@Mye69], for small values of the neutron excess $E{_{\rm s}} (\delta_0)$ can be expanded as follows: E[\_[s]{}]{}(\_0) = E[\_[s]{}]{} + ( + ) \_0\^2 + . \[eq4.4\]In this equation $J$ stands for the bulk symmetry energy coefficient, $L$ reads for the LDM coefficient that expresses the density dependence of the symmetry energy, and $Q$ is the so-called surface stiffness coefficient that measures the resistance of the system against pulling the neutron and proton surfaces apart. All of these macroscopic coefficients are familiar from semi-empirical LDM mass formulae. Another quantity of interest is the neutron skin thickness $\Theta$, namely the separation between the neutron and proton surface locations: = \_[-]{}\^dz . \[eq4.5\]In finite nuclei $\Theta$ would correspond to the difference between the equivalent sharp radii of the neutron and proton distributions. In the small asymmetry limit the LDM predicts a linear behaviour of $\Theta$ with $\delta_0$: = \_0 . \[eq4.6\] For calculations of finite nuclei of small overall asymmetry $I = (N-Z)/A$, the LDM expansion of the energy can be written as E = ( a[\_[v]{}]{} + J I\^2 ) A + A\^[2/3]{} + a[\_[C]{}]{} Z\^2 A\^[-1/3]{} + , \[eq4.7\]where $a{_{\rm C}}$ is the Coulomb energy coefficient. Notice that $I \neq \delta_0$ in finite nuclei. To describe asymmetric matter in the relativistic approach we need to generalize the energy density [(\[eq1\])]{} by including the isovector $\rho$ meson. In terms of the mean field $R= g_\rho b_0$, with $b_0$ the time-like neutral component of the $\rho$-meson field, the additional contributions to Eq. [(\[eq1\])]{} read & & \_\_\^(z) \_(z) + R(z)\ & & - ( R(z) )\^2 - ( 1 + \_ ) R\^2(z) . \[eq4.1\]The symmetry energy coefficient turns out to be J = + . \[eq4.2\]In the conventional model one has $f_\rho= \eta_\rho= 0$. The isovector tensor coupling $f_\rho$ was included in the calculations of Refs. [@Rei89; @Ser97; @Fur97; @Ruf88]. The new non-linear coupling $\eta_\rho$ between the $\rho$- and $\sigma$-meson fields is of order 3 in the expansion and it has been introduced in Refs.[@Ser97; @Fur97]. We will not consider higher-order non-linear couplings involving the $\rho$ meson since the expectation value of the $\rho$ field is typically an order of magnitude smaller than that of the $\omega$ field [@Ser97; @Fur97]. For example, in calculations of the high-density nuclear equation of state, Müller and Serot [@Mul96] found the effects of a quartic $\rho$ meson coupling ($R^4$) to be only appreciable in stars made of pure neutron matter. On the other hand, in analogy to the couplings $\alpha_1$ and $\alpha_2$ for the $\sigma$ and $\omega$ fields, we also tested a surface contribution $-\alpha_3 \Phi \, ( \mbox{\boldmath$\nabla$} R )^2 /(2 g_\rho^2 M)$ and found that the impact it has on the properties we will study in this section is absolutely negligible. As we have seen, the quantity that governs the surface properties in the regime of low asymmetries is the surface stiffness $Q$. Table 1 analyzes the effect on $Q$ and $L$ of the couplings discussed in the preceding sections and of the $f_\rho$ and $\eta_\rho$ parameters. On the basis of Eq. [(\[eq4.6\])]{}, we have extracted $Q$ from a linear regression in $\delta_0$ to fit our results for $\Theta$ up to $\delta_0= 0.1$. We have set the equilibrium properties to $\rho_0= 0.152$ fm$^{-3}$, $a{_{\rm v}}= -16.42$ MeV, $K= 200$ MeV, $M^{*}_{\infty}/M= 0.6$ and $J= 30$ MeV, and have used a scalar mass $m{_{\rm s}}= 490$ MeV and a $\rho$-meson mass $m_\rho= 763$ MeV. Though here we are interested in tendencies rather than in absolute values, for comparison we mention that NL1 has (units in MeV) $J= 43.5$, $L= 140$ and $Q= 27$ [@Von95], the sophisticated droplet-model mass formula FRDM [@Mol93] implies $J= 33$, $L= 0$ and $Q= 29$, and the ETFSI-1 mass formula [@Abo92] based on microscopic forces predicts $J= 27$, $L= -9$ and $Q= 112$. Table 1 shows that the influence on the surface stiffness of the volume self-interactions $\eta_0$, $\eta_1$ and $\eta_2$ is not very large for natural values of these couplings. In the present case $Q$ is slightly increased by decreasing $\eta_0$ (i.e., by increasing the quartic vector coupling $\zeta_0$). For $\eta_1= 1$ we find a non-negligible increase of $Q$, which signals a larger rigidity of the nuclear system against the separation of the neutron and proton surfaces. The effect of $\eta_2$ is again moderate as compared to that of $\eta_1$. $Q$ is augmented by a positive $\eta_\rho$ coupling, while a negative $\eta_\rho$ induces a lower value of $Q$. Some visible changes in $Q$ take place when the $\alpha_1$ and $\alpha_2$ gradient interactions are taken into account. Due to the opposite behaviour of $Q$ with $\alpha_1$ and $\alpha_2$, the tendencies compensate in a case like $\alpha_1= \alpha_2= 1$, but the net effect is reinforced e.g. if $\alpha_1= -\alpha_2= 1$. As one could expect the isoscalar tensor coupling $f{_{\rm v}}$ has a notable effect on $Q$, even for the relatively small value $f{_{\rm v}}=0.3$ that we have used in Table 1. On the contrary, $Q$ is virtually insensitive to the isovector tensor coupling $f_\rho$. The reason is that the derivative of the $R(z)$ field is much smaller than that of the $W(z)$ field. In the least-square fits to ground-state properties of Refs. [@Rei89; @Ruf88] nothing was gained by the $\rho$ tensor coupling. In any case, the best fits of Refs.[@Ser97; @Fur97] have $f_\rho \approx 4$. From Table 1 we recognize that the main changes in the coefficient $L$ arise from the $\eta_\rho$ coupling. As a rule of thumb, increasing values of $Q$ are associated with decreasing values of $L$ for the bulk couplings $\eta_0$, $\eta_1$, $\eta_2$ and $\eta_\rho$. Since $L$ is a bulk quantity, it is not modified by the surface interactions. Figures 14 and 15 illustrate the dependence on asymmetry of the neutron skin thickness, surface energy and surface thickness for some of the cases considered in Table 1. The figures extend up to $\delta{_{\rm 0}}= 0.3$, which widely covers the range relevant for laboratory nuclei ($\delta_0 \leq 0.2$). As the system becomes neutron rich we can appreciate how a neutron skin develops and $\Theta$ grows from its vanishing value at $\delta_0= 0$. For small asymmetries the growth is linear in $\delta_0$, as predicted by the LDM. The surface energy coefficient grows quadratically with increasing neutron excess and the LDM equation [(\[eq4.4\])]{} is clearly a good approximation. In general, the interactions having thicker neutron skins (smaller values of $Q$) also have larger surface energies. The parameter $\eta_\rho$ can be used for the fine tuning of the symmetry properties of the interaction without spoiling the predictions for symmetric systems. If in the conventional ansatz $g_\rho$ is fixed by the value of the symmetry energy $J$, in the extended model $J$ depends on a combination of $g_\rho$ and $\eta_\rho$, Eq. [(\[eq4.2\])]{}. Therefore, $\eta_\rho$ provides in practice a mechanism that can help to simultaneously adjust $Q$ (to get the required neutron skin $\Theta$) and $J$ (to keep the fit to the masses) preserving the symmetric surface properties. Summary ======= Within relativistic mean field theory, we have investigated the influence on nuclear surface properties of the non-linear meson interactions and tensor couplings recently considered in the literature. These interactions, beyond standard QHD, are based on effective field theories. The effective field theory approach allows one to expand the non-renormalizable couplings, which are consistent with the underlying QCD symmetries, using naive dimensional analysis and the naturalness assumption [@Ser97; @Fur96; @Mul96; @Fur97]. The quartic vector self-interaction $\zeta_0$ makes it possible to obtain a desirable positive value of the coupling constant $\kappa_4$ of the quartic scalar self-interaction, for realistic nuclear matter properties and within the bounds of naturalness. This $\zeta_0$ coupling has only a slight impact on the surface properties. Nevertheless, it helps to find parametrizations where both the surface energy coefficient $E{_{\rm s}}$ ant the surface thickness $t$ lie in the empirical region. The $\zeta_0$ vector non-linearity makes the spin–orbit potential well deeper, although the effect is almost negligible. Concerning the volume non-linear couplings $\eta_1$ and $\eta_2$, they also allow one to obtain positive values of $\kappa_4$ in the region of naturalness, depending somewhat on the saturation properties (incompressibility and effective mass). The surface properties are not much affected by these bulk terms either, and it turns out that $\eta_2$ has a marginal effect as compared to that of $\eta_1$. The equilibrium properties do not depend on the couplings $\alpha_1$ and $\alpha_2$ that involve the gradients of the fields. Thus, these couplings serve to improve the quality of the surface properties without changing the bulk matter. In the conventional model the only parameter not fixed by the saturation conditions is the mass of the scalar meson. In the $\alpha_1$–$\alpha_2$ plane the lines of constant $E{_{\rm s}}$ have a different slope than those of constant $t$. It is then possible to keep a fixed value of $E{_{\rm s}}$ and to modify the value of $t$ by choosing $\alpha_1$ and $\alpha_2$ appropriately. The range of variation of $E{_{\rm s}}$ and $t$ with $\alpha_1$ and $\alpha_2$ is wider than with the volume couplings. This justifies including these gradient terms in the energy functional in spite of being of order 5 in the expansion. The $\alpha_1$ and $\alpha_2$ surface meson interactions also influence the spin–orbit potential, but the effect is not extremely significant. The effective model is augmented with a tensor coupling of the $\omega$ meson to the nucleon. An outstanding feature is the drastic consequences it has for the spin–orbit force. We have emphasized how inclusion of $f{_{\rm v}}$ permits to obtain a spin–orbit strentgh similar to that of $M^{*}_{\infty}/M \sim 0.6$ with larger values of the equilibrium nucleon effective mass, contrary to the phenomenology known from models without such a coupling. We have discussed the implications of the extra couplings of the extended model on various surface symmetry properties. We have restricted ourselves to the regime of low asymmetries, where the liquid droplet model can be applied and the surface stiffness coefficient $Q$ is the key quantity. In particular we have pointed out the role that the non-linearity $\eta_\rho$ of the isovector $\rho$-meson field may play in the details of the symmetry properties. Acknowledgements {#acknowledgements .unnumbered} ================ The authors would like to acknowledge support from the DGICYT (Spain) under grant PB95-1249 and from the DGR (Catalonia) under grant GR94-1022. M. 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Swiatecki, [At. Data Nucl. Data Tables 59]{} (1995) 185. Y. Aboussir, J.M. Pearson, A.K. Dutta and F. Tondeur, [Nucl. Phys. A 549]{} (1992) 155. Table captions {#table-captions .unnumbered} ============== Table 1. : The surface stiffness coefficient $Q$ and the coefficient $L$ for several values of the couplings analyzed in the text. We have set $\rho_0= 0.152$ fm$^{-3}$, $a{_{\rm v}}= -16.42$ MeV, $K= 200$ MeV, $M^{*}_{\infty}/M= 0.6$, $J= 30$ MeV and $m{_{\rm s}}= 490$ MeV. Table 1 {#table-1 .unnumbered} ======= $\eta_0$ $\eta_1$ $\eta_2$ $\eta_\rho$ $\alpha_1$ $\alpha_2$ $f{_{\rm v}}$ $f_\rho$ $Q$ (MeV) $L$ (MeV) ---------- ---------- ---------- ------------- ------------ ------------ --------------- ---------- ----------- ----------- $\infty$ 0 0 0 0 0 0 0 21 96 5 21.5 95 2 22 90 $\infty$ 1 0 0 0 0 0 0 24.5 91 0 1 22 93 1 1 25.5 89 $\infty$ 0 0 1 0 0 0 0 23 87 $-1$ 18 119 $\infty$ 0 0 0 1 0 0 0 17 96 0 1 25 96 1 1 19 96 1 $-1$ 16 96 $\infty$ 0 0 0 0 0 0.3 0 24 96 $-0.3$ 19 96 $\infty$ 0 0 0 0 0 0 5 21 96 Figure captions {#figure-captions .unnumbered} =============== Figure 1. : The couplings $g{_{\rm s}}/4\pi$, $g{_{\rm v}}/4\pi$, $\kappa_3$, $\kappa_4$ and $\zeta_0$ versus the parameter $\eta_0$ defined in Eq.[(\[eq3.2.b\])]{}. The naturalness assumption requires all these couplings to be of order unity. We have taken $\rho_0 = 0.152$ fm$^{-3}$ ($k{_{\rm F}}= 1.31$ fm$^{-1}$), $a{_{\rm v}} = -16.42$ MeV and $m{_{\rm s}}= 490$ MeV. Figure 2. : Surface energy coefficient $E{_{\rm s}}$ for several values of the parameter $\eta_0$, $K$, $M^{*}_{\infty}/M$ and $m{_{\rm s}}$, with $\rho_0 = 0.152$ fm$^{-3}$ and $a{_{\rm v}} = -16.42$ MeV. Figure 3. : Surface thickness $t$ of the baryon density profile for several values of the parameter $\eta_0$, $K$, $M^{*}_{\infty}/M$ and $m{_{\rm s}}$, with $\rho_0 = 0.152$ fm$^{-3}$ and $a{_{\rm v}} = -16.42$ MeV. Figure 4. : Surface energy coefficient $E{_{\rm s}}$ and surface thickness $t$ for several values of the parameter $\eta_0$, $M^{*}_{\infty}/M$ and $m{_{\rm s}}$, with $\rho_0 = 0.1484$ fm$^{-3}$, $a{_{\rm v}} = -15.75$ MeV and $K=200$ MeV (Ref. [@Bod91]). Figure 5. : Baryon density $\rho(z)$ and surface tension density $\sigma(z)= {\cal E}(z) - (a{_{\rm v}}+M) \rho(z)$ of semi-infinite nuclear matter for some values of the parameter $\eta_0$. It is $\rho_0= 0.152$ fm$^{-3}$, $a{_{\rm v}}= -16.42$ MeV, $K= 200$ MeV and $m{_{\rm s}}= 490$ MeV. Figure 6. : Orbital part of the spin–orbit potential $V{_{\rm so}}(z)$ at the Fermi surface and central mean field $V{_{\rm c}}(z)$, Eqs. [(\[eq12\])]{} and [(\[eq14\])]{} respectively, for some values of the parameter $\eta_0$. It is $\rho_0= 0.152$ fm$^{-3}$, $a{_{\rm v}}= -16.42$ MeV, $K= 200$ MeV and $m{_{\rm s}}= 490$ MeV. Figure 7. : The couplings $g{_{\rm s}}/4\pi$, $g{_{\rm v}}/4\pi$, $\kappa_3$ and $\kappa_4$ against the parameters $\eta_1$ (left) and $\eta_2$ (right). With $\rho_0 = 0.152$ fm$^{-3}$, $a{_{\rm v}} = -16.42$ MeV and $m{_{\rm s}}= 490$ MeV, results are shown for $K= 200$ MeV and $M^{*}_{\infty}/M= 0.6$ in part (a), and for $K= 350$ MeV and $M^{*}_{\infty}/M= 0.7$ in part (b). Figure 8. : Level curves in the plane $\eta_1$–$\eta_2$ of the surface energy coefficient $E{_{\rm s}}$ (in MeV, solid lines) and of the surface thickness $t$ (in fm, dashed lines), with $\zeta_0= \alpha_1= \alpha_2= f{_{\rm v}}= 0$. The point $\eta_1= \eta_2= 0$ is marked by a cross. It is $\rho_0 = 0.152$ fm$^{-3}$, $a{_{\rm v}} = -16.42$ MeV, $K= 200$ MeV, $M_{\infty}^{*}/M= 0.6$ and $m{_{\rm s}}= 490$ MeV. Figure 9. : Same as Figure 8 in the plane $\alpha_1$–$\alpha_2$, with $\zeta_0= \eta_1= \eta_2= f{_{\rm v}}= 0$. The point $\alpha_1= \alpha_2= 0$ is marked by a cross. Figure 10. : Orbital part of the spin–orbit potential $V{_{\rm so}}(z)$ at the Fermi surface for some values of the couplings $\alpha_1$ and $\alpha_2$. The equilibrium properties of nuclear matter and the scalar mass are the same of Figure 6. Figure 11. : Surface energy coefficient $E{_{\rm s}}$ and surface thickness $t$ as functions of the strength $f{_{\rm v}}$ of the $\omega$-meson tensor coupling, with $\zeta_0= \eta_1= \eta_2= \alpha_1= \alpha_2= 0$. We have set $\rho_0 = 0.152$ fm$^{-3}$, $a{_{\rm v}} = -16.42$ MeV and $m{_{\rm s}}= 490$ MeV. Figure 12. : Orbital part of the spin–orbit potential $V{_{\rm so}}(z)$ at the Fermi surface for some values of the tensor coupling $f{_{\rm v}}$. The equilibrium properties of nuclear matter and the scalar mass are the same of Figures 6 and 10. Figure 13. : Total baryon density $\rho(z)$, total tensor density $\rho{_{\rm T}}(z)$ and their components $\rho^\lambda (z)$ and $\rho_{\rm T}^\lambda (z)$ for the two spin orientations $\lambda= \pm 1$. They have been calculated for $f{_{\rm v}}= 0$ and $f{_{\rm v}}= 0.6$, with $M^{*}_{\infty}/M= 0.7$, $K= 200$ MeV and $m{_{\rm s}}= 490$ MeV. Figure 14. : Neutron skin thickness $\Theta$ as a function of the bulk neutron excess $\delta_0$. The solid line is the result of the conventional model ($\zeta_0= \eta_1= \eta_2= \eta_\rho= \alpha_1= \alpha_2= f{_{\rm v}}= f_\rho= 0$). The other lines differ from the latter in the indicated parameter. Figure 15. : Same as Figure 14 for the surface energy coefficient $E{_{\rm s}}$ and the surface thickness $t$ as functions of the bulk neutron excess squared $\delta_0^2$. [^1]: Notice that in Ref. [@Bod91] the parameter $\eta_0$ was called $z$.

--- abstract: 'We study the global influence of curvature on the free energy landscape of two-dimensional binary mixtures confined on closed surfaces. Starting from a generic effective free energy, constructed on the basis of symmetry considerations and conservation laws, we identify several model-independent phenomena, such as a curvature-dependent line tension and local shifts in the binodal concentrations. To shed light on the origin of the phenomenological parameters appearing in the effective free energy, we further construct a lattice-gas model of binary mixtures on non-trivial substrates, based on the curved-space generalization of the two-dimensional Ising model. This allows us to decompose the interaction between the local concentration of the mixture and the substrate curvature into four distinct contributions, as a result of which the phase diagram splits into critical sub-diagrams. The resulting free energy landscape can admit, as stable equilibria, strongly inhomogeneous mixed phases, which we refer to as “antimixed” states below the critical temperature. We corroborate our semi-analytical findings with phase-field numerical simulations on realistic curved lattices. Despite this work being primarily motivated by recent experimental observations of multi-component lipid vesicles supported by colloidal scaffolds, our results are applicable to any binary mixture confined on closed surface of arbitrary geometry.' author: - Piermarco Fonda - Melissa Rinaldin - 'Daniela J. Kraft' - Luca Giomi title: Thermodynamic equilibrium of binary mixtures on curved surfaces --- Introduction ============ Two-dimensional fluids represent a special class of materials, whose mechanical and thermodynamical properties are simultaneously simple and exotic. Their dynamics and thermodynamics can be considerably less involved compared to three-dimensional counterparts (see e.g. Ref. [@batchelor1967introduction]). Yet, being lower dimensional systems embedded in higher dimensional space, their geometry and topology may be non-trivial. This gives rise to a variety of phenomena where the static and dynamical configurations of the fluid conspire with the shape of the underlying substrate, resulting in a wealth of complex mechanical and thermodynamical behaviours, ranging from the proliferation of defects in two-dimensional liquid crystals and superfluids [@Bowick2009; @Turner2010] to the emergence of topologically protected oceanic waves [@Delplace2017]. Lipid membranes represent one of the most relevant and largely studied realizations of two-dimensional fluids confined on curved surfaces. Artificial lipid membranes, i.e. *in vitro* bilayers which have been purified from other components, have served for decades as fruitful model systems to investigate the stability and material properties of self-assembled biological lipid structures (see e.g. [@lipowsky1995structure; @david2004statistical]). This is especially true in the case of artificial bilayers consisting of multiple lipid components (see e.g. [@Feigenson2009] and references therein), where the heterogeneity of the system shortens the gap between artificial and cellular membranes, despite maintaining a physically tractable complexity. It is well-known that multi-component mixtures of phospholipids and cholesterol have rich phase diagrams, including two different types of liquids known as the (cholesterol-rich) liquid-ordered (LO) and liquid-disordered (LD) phases. While binary lipid mixtures, which provide the simplest example of a multi-component membrane, clearly exhibit coexistence between liquid and solid phases [@mouritsen2015life], there is still lack of conclusive evidence in support of a genuine LO/LD coexistence in mixtures of saturated lipids and cholesterol [@Marsh2010]. For this reason, and because liquid/liquid phase separation is believed to be very relevant for biological systems [@Stillwell2013], most literature shifted the attention toward ternary membranes, usually featuring saturated and unsaturated lipids and cholesterol, where the critical nature of the phase separation is unquestioned. The LO/LD coexistence has so far been realized in several experimental set-ups which have also shown a a correlation between geometry and chemical composition: giant unilamellar vesicles (GUVs) [@Baumgart2003; @Hess2007; @Semrau2009; @Sorre2009; @Heinrich2010], supported lipid bilayers (SLBs) [@Parthasarathy2006; @Subramaniam2010] and scaffolded lipid vesicles (SLVs) [@rinaldin2018geometric]. Here we focus on SLVs, since it is the only experimental set-up that is simultaneously closed (i.e. as in GUVs, there is no exchange of lipids with the surrounding solvent) and of prescribed shape (a property shared with SLBs). We stress, nonetheless, that the results of the present work apply, in principle, to any generic two-dimensional liquid mixture confined on a curved substrate. SLVs have typical size of a few micrometers [@rinaldin2018geometric], whereas a single lipid molecule occupies an area on the membrane of order $\sim 1$ [@mouritsen2015life]: therefore, the number of constituents is approximately in the millions. With such a high number of molecules, it is natural to describe the membrane as a single smooth surface where the local composition is a continuous space-dependent field. A satisfactory physical description can be attained by a coarse-grained two-dimensional scalar field theory, with the fields representing the concentration of the various molecule types. For incompressible liquids, an $n$-component mixture is described by $n-1$ fields. Much of this work will focus on the ability of a single scalar field, $\phi$, to describe the curvature-composition interactions. Despite being appropriate for binary systems only, a single scalar degree of freedom can capture, at least qualitatively, the effect of geometry on the structure of the free energy landscape and the resulting phase behaviour. Furthermore, focusing on a single field has numerous advantages, as it roots in the classical theory of phase separations and was first used to model the interaction between curvature and lipid lateral organization by Markin [@Markin1981] and Leibler [@Leibler1986]. In the latter work, the interplay between the membrane chemical composition and geometry was modelled in terms as a concentration-dependent spontaneous mean curvature, leading to a linear coupling in the effective free energy, analogous to that between an order parameter and an external ordering field. Such coupling breaks the reflection symmetry along the membrane mid-surface, since the mean curvature is sensitive to the surface orientation. This type of interaction was adopted by many subsequent works (see e.g. Refs. [@Leibler1987; @Taniguchi1996; @Jiang2000; @Gozdz2006; @Singh2012; @Zhu2012; @schick2012membrane; @Rautu2015]), whereas others (e.g. Refs. [@McWhirter2004; @Baumgart2011; @Gozdz2012]) considered also linear couplings with the squared mean curvature, which is better suited to describe symmetric bilayers. Conversely, other works did not introduce any interaction terms, but rather studied the effects of a non-trivial intrinsic geometry [@novick1991stable; @Rubinstein1992; @Gomez2015]. Explicit intrinsic couplings were considered in Ref. [@paillusson2016phase], with a direct coupling to the Gaussian curvature, and in Ref. [@adkins2017geodesic], where the notion of spontaneous geodesic curvature was introduced. Note that, because of the Gauss-Bonnet theorem, a direct coupling between the Gaussian curvature and the concentration is irrelevant for chemically homogeneous membranes, and likely for this reason it has often been disregarded. Couplings quadratic and cubic in $\phi$ were considered in other works (e.g. in Refs. [@Jiang2008; @Wang2008; @Lowengrub2009; @Elliott2010; @Jiang2012; @Choksi2013; @Helmers2013; @hausser2013thermodynamically], and also by us in Ref. [@rinaldin2018geometric]) and appears to be the most popular choice within the mathematics-oriented literature. There is no general consensus on how to choose neither the type nor the functional form of the couplings between the shape and the concentration. Although linear terms are the natural choice from a field-theoretic point of view, it is not clear how model-specific will be the results obtained, and thus it is hard do assess their general validity. Furthermore, most of the cited works focus on the local and dynamical effect of given couplings in an open setting. However, vesicle-shaped objects are inherently constrained systems, being topologically spherical and with no relevant exchange with the surrounding environment: the total number of molecules is an externally fixed parameter. For these reasons, we try to have a more systematic approach and explore the all the possible equilibrium configurations of closed two-dimensional systems. For sake of conciseness, we ignore the role of fluctuations (but see e.g. Ref. [@david2004statistical]). The paper is organized as follows. In Sec. \[sec:effective\] we develop an effective scalar field theory on curved backgrounds, using only symmetry and scaling arguments as guiding principles. We highlight a few possible general phenomena, such as local shifts of the binodal concentrations and a curvature-dependent line tension for interfaces separating different phases, and highlight the regimes where Jülicher’s and Lipowsky’s sharp interface theory [@Julicher1993] can be recovered from our diffuse interface model. In Sec. \[sec:model\] we explore in great detail a specific geometry, the asymmetric dumbbell, and a specific microscopic model, consisting of a curved-space generalization of the mean-field two-dimensional Ising model. In the continuum limit, we derive a functional form of the concentration-dependent coefficients of the free energy, linking them to four specific types of microscopic interactions. Within this framework we can compute analytically the general quantities defined in Sec. \[sec:effective\]. By approximating the dumbbell with two disjoint spheres able to exchange molecules, we construct temperature-concentration phase diagrams for any value of the curvature couplings. Interestingly, we are able to give a precise, model-independent definition of the antimixed state, which we observed experimentally in Ref. [@rinaldin2018geometric]. Lastly, we prove numerically that our results, and in particular the existence of the antimixed state, are robust and continue to apply also to more realistic geometries. Mixing and demixing on curved surfaces {#sec:effective} ====================================== Effective free energies for inhomogeneous systems {#sec:effective_intro} ------------------------------------------------- We consider a two-dimensional binary fluid and assume that all the relevant degrees of freedom can be captured by a single, generally space-dependent, scalar order parameter $\phi=\phi(\bm{r})$. If the fluid is incompressible and the average area per molecule is the same for both components, $\phi$ can be interpreted as the absolute concentration of either one of the two components, e.g.: $$\phi = \frac{[A]}{[A]+[B]}\;,$$ where $[\ldots]$ indicate the concentration of the $A$ and $B$ molecules. By construction, $0\le \phi \le 1$ and any value other than $\phi=0$ or $\phi=1$ indicates local mixing of the two components. The system is defined on an arbitrarily curved surface $\Sigma$. Crucially, we assume $\Sigma$ fixed so that the local geometry can influence the configuration of the order parameter $\phi$, but not vice versa. The most general free energy functional of such a system will then be of the form: F = \_[[d]{}]{}A(,, ) , \[effective F\] where $\mathcal{F}$ is a free energy density depending on $\phi$, its surface-covariant gradient $\nabla \phi$ and on the shape of the surface. Here ${{\rm d}}A = {{\rm d}}x^{1} {{\rm d}}x^{2} \sqrt{\det h}$, with $\{x^{1},x^{2}\}$ local coordinates, is the surface area element and $h_{ij}$ ($i,\,j=1,\,2$) is the metric tensor on $\Sigma$. In practice, the explicit form of $\mathcal{F}$ can be obtained upon coarse-graining a microscopic model over a mesoscopic portion of $\Sigma$. Such portion should be small compared to the size of the whole system and yet large compared to typical molecular length-scales, which we call $a$. Alternatively, as in most cases of practical interest, $\mathcal{F}$ is constructed phenomenologically, on the basis of symmetry arguments and physical insight. Because the order parameter is generally non-uniform across the surface, the gradient $\nabla \phi$ introduces new length scales in the system. Here we assume that the spatial variation of the order parameter occurs on a length scale much larger than the molecular size, namely: $|\nabla\phi| \sim \xi^{-1} \ll a^{-1}$. Moreover, at physical equilibrium, gradients are always negligible with the only possible exception for isolated quasi-one-dimensional regions where the spatial variation of the order parameter can be more pronounced. As we will explain later, these regions correspond to diffuse interfaces between bulk phases and, being lower dimensional structures, do not affect the bulk value of the free energy. Since integrated variations have to be finite, $\xi$ also sets the typical thickness of these interfaces. The symmetries of Eq. dictate how $\mathcal{F}$ can depend on the shape of $\Sigma$. If the fluid is isotropic (i.e. molecules do not have a specific direction on the tangent plane of $\Sigma$), $\mathcal{F}$ depends on the surface either intrinsically, through the Gaussian curvature $K$, or extrinsically, through the mean curvature $H$. Furthermore, if the molecules are insensitive to the orientation of the surface (i.e. they do not discriminate convex from concave shapes), $\mathcal{F}$ must be invariant for $H\rightarrow-H$, since, on orientable surfaces, the sign of $H$ depends uniquely on the choice of the normal direction. Thus $\mathcal{F}$ depends on the curvature only through $H^2$, $K$ and, in principle, their derivatives. Non-vanishing curvatures introduce further length scales in the system, which we collectively denote as $R$ and assume larger or equal to $\xi$, thus $R \geq \xi \gg a$. Now, expanding Eq. to the second order in the gradients and the curvatures (thus with respect to $a/\xi$ and $a/R$) yields: ||\^[2]{} + f() + k() H\^2 + |[k]{}() K + , \[F expansion\] where $D$, $f$, $k$ and $\bar{k}$ are the resulting coefficients in the Taylor expansion and the dots indicate higher order terms. These coefficients depend, in general, on the local order parameter $\phi$ and cannot be determined from symmetry arguments. To render Eq. dimensionless, we rescale all the terms by a constant energy density, in such a way that $f$ is dimensionless, whereas $D$, $k$ and $\bar{k}$ have dimensions of area. The physical meaning of the various terms in Eq. is intuitive and has been thoroughly discussed in the literature of phase field models [@Lowengrub2009; @Elliott2010] and lipid membranes [@Lazaro2015]. To have an energy bounded from below requires $D \ge 0$, so that the first term promotes uniform configurations of the order parameter. This term originates from the short-range attractive interactions between molecules and gives rise to a concentration-dependent diffusion coefficient (see e.g. [@crank1979mathematics; @Rubinstein1992; @Schoenborn1997; @Jiang2012]). Notice that $D$ does not depend on the curvatures, because of the quadratic truncation underling Eq. . Higher order terms coupling the order parameter gradients and the curvature tensor have been discussed elsewhere (see e.g. [@Goetz1996; @Fournier1997; @Siegel2010; @Deserno2015]) and will not be considered here. The function $f$ is the local thermodynamic free energy in flat space. This includes both energetic and entropic contributions, promoting phase separation and phase mixing respectively. In the case of fluctuating surfaces, such as lipid membranes, $f$ could be interpreted as a concentration-dependent surface tension. Finally, $k$ and $\bar{k}$ are, respectively, the bending and saddle splay moduli of the mixture, expressing the energetic cost, or gain, of having a given configuration of the field $\phi$, in a curved region of the surface. Analogously to $f$, for a fluctuating surface these terms could be interpreted as a curvature-dependent contributions to the surface tension, introducing a departure for the flat-space value. The length scale associated to these deviations is commonly known as Tolman length [@Tolman1949]. A generic surface may have up to two independent Tolman lengths. For systems sensitive to the orientation of the surface, such as Langmuir monolayers and asymmetric lipid bilayers, the expansion is not required to be invariant for $H \rightarrow -H$ and can feature linear contributions of the form $c H$, with $c=c(\phi)$ a coupling coefficient, equivalent to a concentration-dependent spontaneous curvature $H_{0}=-c/(2k)$. For simplicity, we will ignore this contribution, even if most of our results can be easily extended to this case. Equilibrium configurations are defined as the minima of the free energy functional Eq. . Here we focus on closed systems, where the order parameter is globally conserved. Thus: = \_ [[d]{}]{}A= [const]{}, \[Phi\] with $A_{\Sigma}$ the area of the surface. The problem then reduces to finding the function $\phi$ minimizing the constrained free energy: G = F - , \[grand canonical G\] where $\hat{\mu}$ is the Lagrange multiplier enforcing the constraint . For homogeneous systems, $\hat{\mu}$ is the chemical potential, thermodynamic conjugate of the concentration. The first functional derivative of $G$ yields the equilibrium condition $$\begin{gathered} f'(\phi) + k'(\phi) H^2 + \bar{k}'(\phi)K\\[5pt] = D(\phi) \nabla^{2} \phi + \frac{1}{2} D'(\phi) |\nabla\phi|^{2} + \mu \,, \label{G variation}\end{gathered}$$ where the prime indicates differentiation with respect to $\phi$ (e.g. $f'=\partial f/\partial \phi$), $\nabla^{2}=h^{ij}\nabla_{i}\nabla_{j}$ is the Laplace-Beltrami operator on $\Sigma$ and $\mu=\hat{\mu}/A_\Sigma$ is the chemical potential density. Eq. is too generic to draw specific conclusions, unless the $\phi-$dependence of the various coefficients is specified. In Sec. \[sec:model\] we consider a specific lattice model, but, before then, it is useful to review the case of homogeneous potentials and make some general consideration on the linearization of inhomogeneous terms. Review of homogeneous potentials {#sec:review} -------------------------------- ![When the system phase-separates on a SLV [@rinaldin2018geometric], the surface $\Sigma$ (here shown as a generic closed surface) is partitioned into two regions $\Sigma_\pm$. The thin interface $\gamma$ separating them is a curved strip of finite geodesic width $\sim 2 \xi$, shown in red. In reality we require $\xi$ to be much smaller than any macroscopic length scale. In order to study the behaviour of $\phi(x)$ near the interface, we need to construct an adapted geodesic frame, spanned by the coordinates $s$, the arc-length parameter of the sharp interface (shown in black), and by the normal arc-length coordinate $z=w/\xi$. Constant $s$ lines are geodesics of $\Sigma$. []{data-label="fig:figure 1"}](figure_1.pdf){width="\columnwidth"} In this Section we review the classical theory of phase coexistence and of thin interfaces for binary mixtures in homogeneous backgrounds. For further references see e.g. Refs. [@Elder2001; @provatas2011phase]. We now consider a flat and compact surface $\Sigma$, such as a rectangular domain with periodic boundaries (i.e. a flat torus). Thus $H^{2}=K=0$, while the total area $A_\Sigma$ is finite. Since $D \geq 0$, the homogeneous configuration is a trivial minimizer of the free energy . In most physical systems at equilibrium, field variations occur in almost-negligible portions of $\Sigma$, so that, as first crude approximation, gradient terms in $G$ can be ignored. Then, Eq. reduces to the classical equilibrium condition f’() =. \[equilibrium 1\] If $f$ is convex, the single homogeneous phase $\phi=\Phi$ is a solution of Eq. , corresponding to a stable thermodynamic state, where the two components of the mixture are homogeneously mixed with one another. We refer to this configuration as the mixed phase. Consistently we must have $$\mu=f'(\Phi)\;,\qquad \frac{G}{A_\Sigma} = f(\Phi)-f'(\Phi)\Phi\;.$$ If, on the other hand, $f$ is concave for some $\phi$ values (i.e. $f''<0$), then the mixed phase might become unstable and it is energetically favourable to split the system into (at least) two regions where $\phi$ takes different values, say $\phi_-$ and $\phi_+$ (without loss of generality we choose $\phi_- < \phi_+$). We refer to this configuration as demixed (or phase-separated) phase: () = { [ll]{} \_+, & \_+\ \_-, & \_- ., \[sharp interface\] with $\Sigma_\pm$ the two domains into which $\Sigma$ partitions (see Fig. \[fig:figure 1\]). Now, calling $A_\pm=\int_{\Sigma_{\pm}} {{\rm d}}A\;$ the respective areas and $x_\pm = A_\pm/ A_\Sigma$ their relative area fraction, with $x_{+}+x_{-}=1$, the total fixed concentration is = x\_+ \_+ + x\_- \_- . \[total phi\] Since $\phi$ is assumed to vary smoothly over a region of negligible area, it is possible to formally integrate Eq. with respect to $\phi$ and obtain the set of equilibrium conditions = f’(\_) = , \[maxwell\] known as Maxwell common-tangent construction, see Fig. \[fig:binodal\]. The interval of $\Phi$ values for which the demixed phase, Eq. , is the true minimum of the free energy is always strictly larger than the interval where $f(\Phi)$ is concave. Thus, a mixed phase with total concentration $\Phi$ in the interval $\phi_-< \Phi <\phi_+$, but such that $f''(\Phi)>0$, is metastable, since such phase can still resist small perturbations. The field values $\phi_\pm$ are known as binodal points, the interval $[\phi_-,\phi_+]$ is known as the miscibility gap, whereas the concentrations for which $f''(\phi)=0$ are known as spinodal points (see e.g. [@Safran1994]). ![For concave free energies the thermodynamic minimum is attained by demixed configurations when the total concentration $\Phi$ lies within the miscibility gap. We show respectively in black and gray the binodal and spinodal points relative to $f(\phi)$. The diagonal dashed line is the common tangent which defines, via Eq. , the binodal points. For a given $\Phi$, the area fractions $x_\pm$ of the $A$ and $B$ components are found with the lever rule, i.e. by solving combined with $x_++x_-=1$.[]{data-label="fig:binodal"}](binodal.pdf){width="\columnwidth"} A further layer of complexity is added if one allows for smooth spatial variations of the order parameter $\phi$. In this case, the gradient terms in $G$ becomes relevant, but, because of the scale separation postulated in Sec. \[sec:effective\_intro\], $D|\nabla \phi|^{2} \ll f$, almost everywhere. Since $|\nabla \phi| \sim \xi^{-1}$, by construction, and $D$ has dimensions of area in our units, this inequality implies $D \sim \xi^{2}$. We assume that $D$ - which relates to both compressibility and diffusion - does not depend strongly on the local concentration (for instance, this is certainly the case for lipid mixtures [@Machan2010], where all molecules in the mixture are roughly of the same size) and can be effectively treated as a constant. Without loss of generality, one can then set $D=\xi^{2}$, so that Eq. reduces to partial differential equation: f’() = + \^[2]{} \^[2]{} . \[equilibrium 2\] Since $f'$ is, in general, a non-linear function of $\phi$, Eq. is often analytically intractable. However, as long as $\xi$ is much smaller than the system size, Eq. , is still a valid solution over large portions of $\Sigma$. Globally, the solution can then be constructed upon matching homogeneous configurations of the field over different domains of $\Sigma$ via perturbative solutions of Eq. within the boundary layers at the interface between neighbouring domains (see e.g. Ref. [@provatas2011phase]). This is a standard technique which can be easily generalized to the case of curved environments, see also [@Rubinstein1992]. The effect of curvature {#sec:curvature} ----------------------- We now consider the more generic case in which $\Sigma$ has non-vanishing curvatures $H$ and $K$, but no explicit coupling with the order parameter (a similar situation in dynamical contexts was considered in Refs. [@Rubinstein1992; @Gomez2015]), by setting $k=\bar{k}=0$ in Eq. . This scenario occurs, for instance, in mixtures whose components are equally compliant to bending, thus there is no energetic preference for the order parameter $\phi$ to adjust to the underlying curvature of the surface. Yet, as any interface in the configuration of the field $\phi$ costs a finite amount of energy, roughly proportional to the interface length, the shape of $\Sigma$ indirectly affects the spatial organization of the binary mixture via the geometry of interfaces. In Ref. [@fonda2018interface], we have discussed this and other related phenomena in the framework of the sharp interface limit (i.e. with $\xi = 0$). Here we show how the present field-theoretical approach enables one to recover and further extend these results. Upon demixing, the system drives the formation of interfaces. This means that in regions of thickness $\approx \xi$ the field $\phi$ is smoothly interpolating between the bulk values of the two phases. Since we are in a regime where this thickness is much smaller than the size of the system, we can take Eq. and expand it in powers of $\xi$. As shown in Fig. \[fig:figure 1\], in the proximity of $\gamma$ we need to adapt the coordinate system to take into account both the curvature of the interface, as a strip embedded on the surface, and of the intrinsic curvature of the surface itself. We explain in detail how to build such frame in Appendix \[app:normal\]. Then, we can treat the scalar field as a function of coordinates in this frame, $\phi=\phi(s,z)$, where $s$ is the arc-length parameter of the sharp interface $\gamma$ (the black curve in Fig. \[fig:figure 1\]) and $z$ is the normal geodesic distance from the curve. Furthermore, variations along $z$ happen on a scale $\sim \xi$, while variations along $s$ become relevant only at macroscopic distances. This implies that $\phi$ is a function of only the normal coordinate $z$ up to at least order $\xi^2$, and we can rescale the variable $z \to w/\xi$ so that the values $w \to \pm \infty$ correspond to the bulk phases. With this construction at hand, we collect the various terms in , order by order in $\xi$, and solve iteratively the differential equation. At $O(1)$ we find the so-called profile equation, which, after matching with the bulk values of $\phi$ away from the interface, reads \_[w]{}\^2 = g() , \[equipartition\] where $\varphi(w)=\phi(z/\xi)$ is the order parameter expressed as function of the rescaled normal coordinate $w$, and $g$ is the shifted potential g() = f() + , \[shifted g potential\] which has the properties $g(\phi_\pm)=g'(\phi_\pm)=0$ and $g''(\varphi)=f''(\varphi)$, i.e. $g$ shares the same binodal points with $f$. Typically, solutions of Eq. decay exponentially towards the bulk phases and interpolate monotonically between the two phases. As we shall later see, this will not necessarily be the case for non-homogeneous systems. Solving Eq. at $O(\xi)$ is slightly more involved (see Appendix \[app:thin\] for more details), but leads to a series of simple and interesting results. First, in regions where $\xi^2 K$ is small, the equilibrium interface must obey \_g = [const]{}, \[CGC\] with $\kappa_g$ the geodesic curvature of the interface $\gamma$ (see Appendix \[app:normal\] for definitions). Eq. is the simplest two-dimensional version of the Young-Laplace equation on a curved geometry. The value of the constant, which is proportional to the lateral pressure difference on the two sides of the interface, sets the radius of curvature of the interface. While on a flat plane constant $\kappa_g$ lines are circles (and geodesics are straight lines), on an arbitrary surface they can have significantly less trivial shapes. We explored this subject in much more detail in [@fonda2018interface], and refer the interested reader there. Note that does not constrain the topology of $\gamma$: in principle it could consist of many simple curves, provided they all have the same curvature. In this case, the constraint on $\kappa_g$ is non-local [@Rubinstein1992]. From this it can be shown that a non geodesic interface induces a modification of the equilibrium chemical potential = + , \[sigma oxi\] where we introduced the interfacial line tension $\sigma$, defined as = \_[\_-]{}\^[\_+]{} [[d]{}]{} . \[line tension\] Eq. implies that, for non-geodesic interfaces, equilibrium bulk concentrations slightly deviate from the Maxwell values. This phenomenon is entirely absent in phase separations of open systems, where instead the bulk phases concentrations are not affected by the interface curvature. Such an effect is manifest also when evaluating the equilibrium free energy up to $O(\xi)$. Namely, we find F = \_+ \_[=]{} A\_. \[thin F\] The above relation shows that $\sigma$ is precisely the coefficient that couples to the interface length, $\ell_\gamma$, and hence is a proper interfacial tension. Furthermore, since the two-dimensional lateral pressures are defined as p\_= , we see that the pressure difference $\Delta p = p_+-p_-$ does indeed depend on the interfacial curvature. Although small - it is an $O(\xi)$ correction - this contribution is always present in phase coexistence of closed systems. It was first derived by Kelvin [@Thomson1872] from the Young-Laplace equation. Coupling mechanisms between curvature and order parameter {#sec:coupling} --------------------------------------------------------- Here we consider the most generic scenario in which all terms in Eqs. and , including $k$ and $\bar{k}$, are non-vanishing. In this case the local curvature affects directly the magnitude of the order parameter $\phi$, instead of just indirectly influencing lateral displacement through non-trivial topology and intrinsic geometry. Without specifying the shape of $\Sigma$ nor the functional form of $k(\phi)$ and $\bar{k}(\phi)$ it is hard to make precise predictions. We will deal with a specific model and specific geometries in the next Section. Here we instead consider an approximately flat membrane, so we can treat the curvature terms as perturbations. If $k(\phi) H^2$ and $\bar{k}(\phi) K$ are much smaller than $f$, we get that the binodal points of the free energy are shifted by a small, curvature-dependent, amount. More precisely (see also Appendix \[app:linear\]), we have that the Maxwell values are shifted as $\phi_\pm \to \phi_\pm + \delta \phi_\pm$, with \_= - , \[delta phi HK\] where $\Delta k = k(\phi_+) - k(\phi_-)$ and $\Delta \bar{k} = \bar{k}(\phi_+) - \bar{k}(\phi_-)$ are the differences between the bending moduli evaluated on the homogeneous binodal concentrations. Eq. shows how the equilibrium bulk phases are directly influenced by local curvature. Since $\xi$ is smaller than any other scale, we can still assume that the interface separating the two phases lies entirely in a region where curvature can be considered to be constant along the $z$ geodesic normal direction. This implies that we can use again Eq. to compute the line tension using the shifted binodal values , finding a curvature-dependent line-tension $\tilde{\sigma}$ + \_k H\^2 + \_[|[k]{}]{} K + … \[sigma HK\] where the dots stand for higher order terms in the curvatures. The two coefficients $\delta_{k,\bar{k}}$ are defined as integrals over the homogeneous miscibility gap \_[k,|[k]{}]{} = \_[-]{}\^[\_+]{}[[d]{}]{} , \[delta kkb\] where $g$ is defined in and $g_{k,\bar{k}}$ are defined in a similar manner, i.e. $g_{k,\bar{k}}(\phi_\pm)=g_{k,\bar{k}}'(\phi_\pm)=0$ and $g_{k,\bar{k}}''(\varphi)$ coincides with the second derivative of the bending moduli (see the derivation of equation for more details). Interestingly, the terms $\delta_k/\sigma$ and $\delta_{\bar{k}}/\sigma$ in Eq. resemble one-dimensional analogues of the Tolman lengths (see Sec. \[sec:effective\_intro\] and Ref. [@Tolman1949]). If instead the curvature couplings are so small that they enter in the effective free energy $\mathcal{F}$ as $O(\xi)$ terms, they have a different effect. Formally, this can be achieved by replacing $k(\phi) H^2+\bar{k}(\phi) K$ with $\xi (k(\phi) H^2+\bar{k}(\phi) K)$ in Eq. and Eq. . This means that, contrary to the case we just discussed, the curvature interactions will not affect the interface profile Eq. , nor they will influence the line tension or the bulk phase values $\phi_\pm$. Rather, they will only affect equilibrium at $O(\xi)$, thus they will contribute to the determination of interface position. It is easy to show that in this case is equation that needs to be modified to \_g -k H\^2 - |[k]{} K = . \[JL interface\] Not surprisingly, this equation is precisely the one obtained by the first functional variation of the Jülicher-Lipowsky sharp interface model [@Julicher1993], which we treated in detail in [@fonda2018interface]. This latter result hints at a more general concept. When adding environmental couplings to sharp interface models there is an implicit assumption about the subleading character of the interactions - relatively to an expansion in the interface thickness -, since they can affect the position of the interface but not its inner structure. Physical interfaces have however finite thickness, and thus any coupling with other degrees of freedom will naturally influence the interface as a diffuse thermodynamic entity, rather than just as a geometric submanifold. For this reason thin interface models, where $\xi$ is small but non-zero, can produce more physically reliable results. A simple model {#sec:model} ============== {width="\linewidth"} The rich phenomenology of binary mixtures on curved surfaces has much more to offer than the general results outlined in Sec. \[sec:effective\]. To draw more precise conclusions, however, it is indispensable to make the $\phi-$dependence of the functions $D$, $f$, $k$ and $\bar{k}$ in Eq. explicit, and thus focus our analysis on a specific subset of possible material properties. Whereas this operation can be performed in multiple ways (see Sec. \[sec:effective\_intro\]), here we propose a simple and yet insightful strategy based on a curved-space generalization of the most classic microscopic model of phase separation, namely the lattice-gas model. To this purpose, we discretize $\Sigma$ into a regular lattice, with coordination number $q$ and lattice spacing $a$, this being defined as the geodesic distance between neighbouring sites. We ignore the fact that, for closed surfaces with genus $g\neq 1$, there are topological obstructions to construct regular lattices and point defects (i.e. isolated sites where the coordination number differs from $q$) become inevitable. We assume that these isolated points give a negligible contribution to the free energy in the continuum limit. Each site is characterized by a binary spin $s_{i}=\pm 1$, serving as a label for either one of the molecular components (e.g. $s_{i}=+1$ indicates that the $i-$th site is occupied by a molecule of type $A$, while $s_{i}=-1$ in case the molecule is of type $B$). Because of the short range interactions between the molecules, the total energy of the system is computed via the Ising Hamiltonian: = - \_[ij ]{} J\_[ij]{} s\_[i]{} s\_[j]{} - \_[i]{} h\_i s\_i , \[hamiltonian\] where $i=1,\,2\ldots\,N$ and $\langle ij \rangle$ indicates a sum over all the pairs of nearest neighbours in the lattice. Finally, conservation of the total number of molecules implies: \_i () = N. Now, in the classic lattice-gas model, the coupling constant $J_{ij}$ and the external field $h_{i}$ are uniform across the system. Here, we allow them to depend on the local geometry of $\Sigma$. Using the same assumptions underlying the expansion , augmented by the additional symmetry $J_{ij}=J_{ji}$, yields: \[ising\_parameters\] $$\begin{aligned} J_{ij} &= \frac{1}{4}\,\left(J + {Q_k}\,\frac{H_{i}^2+H_{j}^2}{2} + {Q_{\bar{k}}}\,\frac{K_{i}+K_{j}}{2}\right) \;, \label{ising quadratic} \\ h_{i} &= -\frac{1}{2}\,\left({L_k}H_{i}^2 + {L_{\bar{k}}}K_{i}\right) \;,\end{aligned}$$ where $H_{i}$ and $K_{i}$ are respectively the mean and the Gaussian curvature evaluated at the $i-$th lattice site. The $Q-$couplings modulate the relative strength of the attraction/repulsion between molecules, reflecting that both the distance and relative orientation of neighbouring molecules vary across the surface. Similarly, the $L-$couplings measure the propensity of a molecule to adapt to the local curvature. In particular, we note that ${L_{\bar{k}}}$ is exactly the only curvature coupling employed in Ref. [@paillusson2016phase] to describe the interaction of binary mixtures with minimal surfaces. We stress that, in order for the Hamiltonian to admit phase separation, $J_{ij}>0$. As the local Gaussian curvature can be both positive and negative, this is not necessarily true for a generic surface and an arbitrary choice of the constants $J$, ${Q_k}$ and ${Q_{\bar{k}}}$. In the following, we assume that $J>0$ is sufficiently large to prevent $J_{ij}$ from changing sign. Furthermore, we assume for simplicity all the other constants in Eqs. to be positive. The latter assumption is not indispensable and has not qualitative effects on the structure of the free-energy landscape and on the phase diagram. The free energy of the mixture can now be easily calculated using the mean-field approximation, upon assuming the variables $s_{i}$ to be spatially uncorrelated (i.e. $\langle s_{i}s_{j} \rangle = \langle s_{i} \rangle \langle s_{j} \rangle$, with $\langle\cdots\rangle$ the ensemble average). Thus, letting $$P(s_{i})=\phi_{i}\delta_{s_{i},1} + (1-\phi_{i})\delta_{s_{i},-1}\;,$$ the probability associated with finding a molecule of type $A$ or type $B$ at $i-$th site, yields, after standard algebraic manipulations (see e.g. Ref. [@parisi1988statistical]), $$\begin{gathered} \label{eq:mean_field} F = -\sum_{\langle ij \rangle} J_{ij}(2\phi_{i}-1)(2\phi_{j}-1)+\sum_{i}h_{i}(2\phi_{i}-1)\\ + T \sum_{i}\left[\phi_{i}\log \phi_{i}+(1-\phi_{i})\log(1-\phi_{i})\right]\;,\end{gathered}$$ with $T$ the temperature in units of $k_{B}$. Coarse-graining Eq. over the length scale $\xi$, finally yields Eqs. and , with \[eq:lattice\_gas\] $$\begin{aligned} \label{ising D} D(\phi) &= \xi^2 J \,, \\ \label{ising f} f(\phi) &= T \mathcal{S}(\phi) + q J \phi (1-\phi) \,,\\ \label{ising k} k(\phi) &= q {Q_k}\phi (1-\phi) + {L_k}\phi \,, \\ \label{ising kb} \bar{k}(\phi) &= q {Q_{\bar{k}}}\phi (1-\phi) + {L_{\bar{k}}}\phi \,,\end{aligned}$$ \[ising MF\] where $\mathcal{S}(\phi) = \phi \ln \phi + (1-\phi)\ln (1-\phi)$ is the mixing entropy and we dropped $\phi$-independent terms from the bending moduli. The symmetry $\phi \leftrightarrow 1-\phi$ is explicitly broken only by linear $L-$couplings. Note that because of the total constraint on $\Phi$ we can disregard homogeneous terms linear in $\phi$, but we are not allowed to do the same for linear terms which depend on local geometry. Consistently with the assumptions about the separation of scales outlined in Sec. \[sec:effective\] (i.e. $\xi^{2}H^{2} \sim \xi^{2} K \approx 0$), we have dropped curvature-dependent terms in the expression of $D$. Surfaces of constant curvature {#sec:constant_curvature} ------------------------------ With Eqs. in hand, we are now ready to fully explore the phase diagram of binary mixtures on curved surfaces. As a starting point, we consider the case of surfaces with constant curvatures, such as the sphere or the cylinder. In this case, the coupling of the order parameter with the curvatures, embodied by the third and second term in Eq. , merely results in a renormalization of the critical temperature. In fact, if $T > T_c$, with T\_c = ( J + [Q\_k]{}H\^2 + [Q\_[|[k]{}]{}]{}K ) , \[ising Tc\] the free energy density $f(\phi)+k(\phi) H^2 + \bar{k}(\phi)K$ is always convex, and thus the homogeneously mixed configuration, $\phi=\Phi$, is the only stable equilibrium. Evidently, the linear terms in Eqs. do not affect the convexity of the free energy, thus do not contribute to the critical temperature. Despite the known limitations of mean-field theory in two dimensions - here further corroborated by the experimental evidence that lipid mixtures belong to the same universality class as the two-dimensional Ising model [@Veatch2008; @Honerkamp-Smith2008; @Honerkamp-Smith2009] - it is nonetheless instructive to see how the generic picture illustrated in Sec. \[sec:curvature\] specializes for the choice of potentials given by Eqs. when $T \lesssim T_c$ (which is the case for the majority of experiments on lipid membranes at room temperature). At the first order in the Ginzburg-Landau expansion, the binodal concentrations are \_ (1 ) . From these we can compute the shifted potential $g(\varphi)$ of Eq. g() (-\_+)\^2(-\_-)\^2 , which is, as expected, a symmetric double-well quartic polynomial potential with minima at the binodal points. From here we can explicitly solve the interface profile equation , finding the well-known hyperbolic tangent kink (w) + ( w ) , \[kink\] where the zero of the geodesic normal coordinate $w$ (see the inset of Fig. \[fig:figure 1\]) has been chosen such that the integral of the difference $|\varphi-\phi_-|$ for $w<0$ matches the integral of $|\varphi - \phi_+|$ for $w>0$ (this is the definition of the Gibbs sharp interface, see Eq. ). The interface width, defined as the length scale over which $\phi$ changes from $\phi_-$ to $\phi_+$, scales as $\sim \xi J^{1/2} (T-T_c)^{-1/2}$ and diverges for $T \to T_c$. On the other hand the line tension can be computed to be (T\_c-T)\^[3/2]{} , \[ising sigma tilde\] which instead vanishes at the critical temperature. With these results we can compute explicitly the quantities discussed in Section \[sec:coupling\] when the curvatures are small. In particular, the curvature-dependent line tension can be evaluated using Eq. - or equivalently by substituting Eq. into Eq. and expanding for small curvatures, finding q Q\_[k,|[k]{}]{} (T\_c- T)\^[-1]{} . \[ising delta k\] Since this ratio is diverging for $T \to T_c$, it implies that curvature-dependent effects to the line tension, in our mean-field model, become more relevant near the critical temperature. Similarly, if the curvature couplings are $O(\xi)$ and thus do not influence the interface profile nor the homogeneous binodal points, then the bending moduli differences of the Jülicher-Lipowsky model - as defined in Eq. - are $$\begin{aligned} \label{MF Delta k} \Delta k \simeq {L_k}\sqrt{\frac{3}{T}} (T_c-T)^{1/2}\,, \\ \label{MF Delta kbar} \Delta \bar{k} \simeq {L_{\bar{k}}}\sqrt{\frac{3}{T}} (T_c-T)^{1/2} \,,\end{aligned}$$ \[MF Delta\] which vanish at the critical temperature and depend only on the $L-$couplings since only terms that break the symmetry $\phi \leftrightarrow 1-\phi$ can produce a bending moduli difference. More generally, since the linear couplings $L_{k,\bar{k}}$ give no contribution to the redefinition of the critical temperature, Eq. , nor to the line tension, Eq. , it might appear that they play no role in shaping the equilibrium phase diagram of the binary mixture. One would expect that adding a linear interaction term has no effect on the global thermodynamic stability of the system. In the next Section we will show how this is not the case when inhomogeneous surfaces are considered. ![ Lines of equilibrium. Solutions of Eq. for two spheres with radii ratio $R_2/R_1=2/3$ at sub-critical temperature $T=0.45\,qJ$. **a)** lines have all $T_L^{(a)}=0$ but regularly increasing $T_c^{(a)}$ from $1/2\,qJ$ (red) to $(1/2+1/10 R_a^{-2})\,qJ$ (blue). **b)** lines have all $\Delta T_c=0$ while $T_L^{(a)}$ increases from $0$ (red) to $1/20 \,R_a^{-2} qJ$ (blue). The black, thick, dashed line corresponds to the infinite $T_L^{(a)}$ limit. Note that the homogeneous solution $\phi_a=\Phi$ (the diagonal red line in both panels) is possible only in the absence of direct curvature couplings. Diagonal dashed lines are of constant $\Phi$. Notice that for a given $\Phi$ there can be multiple equilibrium solutions. []{data-label="fig:loeLM"}](loe_LM_h.pdf){width="\linewidth"} The phase diagram of disjoint spheres {#sec:disjoint spheres} ------------------------------------- In order to gain a deeper understanding of the role of curvature on the thermodynamics of phase separation, we need to consider a specific inhomogeneous shape. Building on our recent experimental results on SLVs [@rinaldin2018geometric], we focus on asymmetric dumbbell-shaped substrates, as the one depicted in Fig. \[fig:spheres\]a. In this case, $\Sigma$ consists approximately of two spherical caps connected to each other. We call the portion of the surface where the two spheres are in contact the “neck region”. While the principal curvatures on the caps are approximately constant and proportional to their inverse radius, on the neck they reach higher values, so that both the mean and the (negative) Gaussian curvatures are significantly larger [^1]. In terms of area, however, the neck occupies a relatively small portion of the whole surface. For the latter reason, in this Section we trade an accurate depiction of the geometry for analytic tractability and make the strong assumption that the neck will play a minor role in determining the equilibrium phase diagram of dumbbell-shaped two-dimensional liquid mixtures. Under this assumption, we approximate $\Sigma$ with a closed system consisting of two disjoint spheres, $S_1$ and $S_2$, of different radii, allowed to exchange molecules with one another, as shown in Fig. \[fig:spheres\]b. Thus the total concentration can be expressed as = \_[a=1,2]{} x\_a \_a, \[xi Phi\] where $ \phi_a = 1/A_a \int_{S_a} {{\rm d}}A\,\phi\,, $ is the average concentration over the $S_{a}$ sphere ($a=1,\,2$), with $A_a = 4 \pi R_a^2$ the sphere area and $R_{a}$ the radius. Analogously, $x_a=A_a/A_\Sigma$, represents the area fraction of each sphere. Eq. can now be solved using the mean-field parameter given by Eqs. , averaged over each sphere. Since for spheres $H^2=K=R^{-2}$, the four geometric couplings of Eqs. become pairwise equivalent, thus reducing the number of independent parameters to two: a symmetry-preserving quadratic term and a symmetry-breaking linear term. To see this explicitly we first minimize the free energy separately on each sphere, which gives the equations 1- 2\_a = , \[two sphere eq\] where we defined the local critical temperature by means of Eq. T\_c\^[(a)]{} = (J + ) , \[Tci\] and we introduced the curvature-dependent energy scale associated with the linear coupling T\_L\^[(a)]{} = . \[Tmi\] Constructing the equilibrium phase diagram of this system is a two-step process. First, one must find the values $\phi_a$ satisfying Eq. and the constraint . Once these have been found, one must check the stability of each average concentration against spontaneous phase separation, i.e. verify whether $\phi_a$ lies within the local miscibility gap $[\phi_-^{(a)},\phi_+^{(a)}]$ on each sphere. For fixed values of temperature and curvature couplings, the solutions of Eq. define a family of curves in the $\{\phi_1,\phi_2\}$ plane, as the total concentration $\Phi$ is smoothly changed from $0$ to $1$. We refer to these curves as “lines of equilibrium” and show some examples of them in Fig. \[fig:loeLM\]. Although smooth, these lines do not need to be connected. Mathematically, they correspond to the set of points in concentration space where the gradient of the free energy is proportional to the vector $\{1,1\}$. Fig. \[fig:loeLM\]a shows the effect of varying the local critical temperature $T_c^{(a)}$ on each sphere. Since the free energy is still symmetric under the exchange $\phi \leftrightarrow 1-\phi$, the lines of equilibrium are invariant under the mapping $\phi_a \to 1-\phi_a$. Different colours correspond to different ${Q_k}+{Q_{\bar{k}}}$ values in Eq. , ranging from $0$ (red) to $1/10\;qJR_1^{2} $ (blue). All curves pass through $\phi_1=\phi_2=1/2$. Fig. \[fig:loeLM\]b shows the effect of the linear $L-$couplings: ${L_k}+{L_{\bar{k}}}$ is increased from $0$ (red) to $1/20\;qJR_1^{2}$ (blue). In both panels the temperature is $T=9/2\;qJ$, and the spheres have radii $R_1=1$ and $R_2=2/3$. It is instructive to compare, in closer detail, these results with those obtained in the absence of explicit coupling between the order parameter and the curvature, namely: $T_L^{(a)}=0$ and $T_c^{(1)}=T_c^{(2)}$ (the red-most lines in both panels). In this case, the lines of equilibrium consist of two mutually intersecting curves: a diagonal straight line $\phi_1=\phi_2=\Phi$, corresponding to the usual homogeneously mixed phase, and a second oval-shaped closed curve. The latter curve implies the existence of a second branch of solutions, where the amount of order parameter on each sphere is different from the total average. This result might be surprising, given that in this case the free energy density is homogeneous. However, it can be easily argued that this is an artefact of our model, originating from the following two arguments. First, the geometry we are considering is exceptional: the two spheres are not in direct contact and having $\phi_1 \neq \phi_2$ does not cost any extra interfacial energy, as it would be the case for a single connected surface. In fact, non-zero gradients would be strongly disfavoured. Secondly, it can be verified that the oval always lies within the miscibility gap of the potential and, therefore, even if mathematically possible, these extra solutions are thermodynamically metastable at best. This case alone shows another, rather general, fact: for a given set of external parameters, there can be multiple pairs of solutions of Eq. , each corresponding to a possible (meta-)stable equilibrium state. Spatial curvature changes this picture by introducing a smooth deformation of the lines of equilibrium. In Fig. \[fig:loeLM\]a the straight line and the oval merge together into a single S-shaped connected curve, while in Fig. \[fig:loeLM\]b one portion of the oval and of the straight line merge into a single line, and the rest splits into a closed curve. The latter becomes smaller and smaller as $T_L^{(a)}$ increases, and eventually disappears, leaving a single branch of equilibrium solutions. Our sign choices are such that it is thermodynamically preferable to first build-up non-zero $\phi$ on the largest sphere up to its maximum capacity (i.e. $\phi_{1}\approx\Phi$ and $\phi_{2}\approx 0$), rather than keeping the concentration everywhere uniform. Hence, at small $\Phi$, the lines of equilibrium bend towards the lower-right half of the diagram. For the linear coupling, this trend continues until the larger sphere is almost saturated. Then the concentration starts increasing on the small sphere too (so that the closed curves in the top left of Fig. \[fig:loeLM\]b are always metastable). For the quadratic coupling the situation is more symmetric, in such a way that, for larger $\Phi$ values, it is more convenient to have a higher concentration on the small sphere. Note that, because of the classic double-well structure of the thermodynamics potentials, for a given $\Phi$ value there can be up to three different equilibrium solutions. Regardless of these quantitative differences, the main qualitative feature of the toy-model described in this Section is that, as a consequence of the influence of curvature on the free energy landscape of the binary mixture, the two disjoint spheres exhibit different concentrations despite being still in the “mixed” phase, i.e. without developing any interface. Interestingly, this phenomenon has some similarity with the thermodynamics of lipid membranes adhering onto non-homogeneous flat substrates [@Lipowsky2013]. {width="\linewidth"} Fig. \[fig:phase diagrams\] shows the phase diagram of the two-spheres system, obtained upon varying the temperature $T$ and the total concentration $\Phi$, while keeping $T_{c}^{(a)}$ and $T_L^{(a)}$ fixed. To highlight the specific role of each of these couplings, we isolate the effect of the quadratic coupling in Fig. \[fig:phase diagrams\]a and that of the linear couplings in Fig. \[fig:phase diagrams\]b, by setting $T_L^{(1)}=T_L^{(2)}$ and $T_{c}^{(1)}=T_{c}^{(2)}$ respectively. We see that there are essentially three stable phases (for the sake of simplicity, we focus only on stable phases and ignore metastable states): there is a mixed phase with no interfaces (red/yellow shades), there is a partially demixed phase with interfaces only on one sphere (lighter gray), and finally there is fully demixed phase with phase separation occurring on both spheres (darker gray). To better characterize the mixed phase we introduce the difference = \_1 - \_2 , \[Delta phi\] which quantifies the departure of the concentration on a single sphere from the total average. A completely homogeneous mixed phase would then have $\Delta \phi=0$. The different shades of red/yellow in Fig. \[fig:phase diagrams\] indicate different values of $\Delta \phi$, as shown in the legend. From the diagrams it is clear that, even in absence of genuine phase separation, one needs to relax and generalize the notion of mixing in order to grasp the complexity of the current scenario in comparison to the traditional picture. In fact, outside local miscibility gaps the “mixed” phase has a non-zero $\Delta \phi$. This effect is enhanced when there is a linear coupling, as in Fig. \[fig:phase diagrams\]b, especially below $T_c$ and for concentrations close to the relative area ratio of the two spheres, $\Phi \sim x_1$ (which is equal to $\sim 0.69$ in the Figure). Before dwelling into a detailed description of this phenomenon, let us emphasize that what we call here inhomogeneous mixing, is not a new thermodynamic phase, but rather the generalization of mixing to macroscopically non-homogeneous closed systems. In fact, the effect of inhomogeneities is smoothly smeared out at high temperatures, where the usual homogeneous mixing is always the true equilibrium. To see this, consider the limit where $T \gg T_c^{(a)}$ and $T \gg T_L^{(a)}$. We can then linearise the curvature couplings in Eq. and solve the equilibrium equation perturbatively. To this purpose, let us introduce the average critical temperature \_c = , \[hat Tc\] and the two energy scale differences T\_c = , T\_L = . \[Delta TcM\] By expanding Eq. at first order in $\Delta T_c$ and $\Delta T_L$, we get the deviation of the local concentrations from the total average = C\_Q(,\_c/T) - C\_L(,\_c/T) , \[small Dphi\] where $C_Q$ and $C_L$ are derived exactly in Appendix \[app:CLCM\]. Their only relevant property is that they take finite values in the large $T$ limit, namely $$\begin{aligned} \label{CL T infinity} C_Q(\Phi,0) &= 4\Phi (1-\Phi) (2\Phi-1)\,, \\ \label{CM T infinity} C_L(\Phi,0) &= 2 \Phi (1-\Phi)\,.\end{aligned}$$ \[CLM T infinity\] Eq. clearly shows that, regardless of the magnitude of the curvature couplings, homogeneous mixing is always restored at high temperature. Furthermore, as the free energy is a continuous function of the concentrations $\phi_a$, such a crossover between inhomogeneous and homogeneous mixing occurs continuously, i.e. without passing through a first order phase transition. This argument can straightforwardly be extended to any arbitrary perturbative order in $\Delta T_{c}$ and $\Delta T_L$, demonstrating that equilibria with $\Delta \phi \neq 0$ and $\Delta \phi = 0$ corresponds to different states of the same phase. Despite of spatial curvature not giving rise to additional thermodynamic phases, its effect below the critical temperature is nonetheless dramatic as indicated by Fig. \[fig:phase diagrams\]b. In this region of the phase diagram, the binodal line splits into two disconnected regions, separated by an intermediate continuum of states where $\Delta \phi$ is large and positive, hence the concentration on the two spheres is highly non-homogeneous. In a previous work, we have reported a direct experimental observation of these type of states and named the phenomenon “antimixing” [@rinaldin2018geometric]. An intuitive understanding can be achieved by considering the limiting case in which $|\Delta T_L|$ overweights any other energy scale. Since the linear interaction breaks the $\phi \to 1-\phi$ symmetry, the energetic cost of having low or high concentrations of the order parameter become highly uneven and position-dependent. Specifically, if ${L_k}+{L_{\bar{k}}}$ is large and positive, with $R_1 > R_2$, having $\phi_2 \neq 0$ will cost much more energy than a non-zero concentration on $S_1$. Thus, any increment of the total concentration $\Phi$ will be first accommodated by $S_{1}$ until saturation (i.e. $\phi_{1}=1$) and only later the order parameter will start propagating on $S_{2}$. The corresponding lines of equilibrium associated with this scenario are represented as thick dashed black lines in Fig. \[fig:loeLM\]b and consist of two perpendicular segments. The horizontal segment, i.e. $\{0\le \phi_{1} \le 1$,$\phi_{2}=0\}$, represents the build-up of order parameter on the sphere $S_{1}$, whereas the vertical segment, i.e. $\{\phi_{1}=1,\,0\le \phi_{2} \le 1\}$., indicates the subsequent build-up of order parameter on the sphere $S_{2}$. ![The equilibrium phase diagram in the strong linear coupling limit. This figure is analogous to Fig. \[fig:phase diagrams\], with non-zero linear and quadratic couplings: we set ${Q_k}+{Q_{\bar{k}}}=1/5 \, R_1^2 qJ$ and ${L_k}+{L_{\bar{k}}}=7/4 \, R_1^2 qJ$. The critical temperatures on each sphere are $T_c^{(1)}=0.55\, qJ$ and $T_c^{(2)}=0.65\, qJ$. The green lines are the analytic binodal lines obtained from Eq. . In the lighter gray region where phase separation happens on $S_1$, we have $\phi_2=0$. Conversely, in the region where phase separation happens on $S_2$, we have $\phi_1=1$. The two black dots are the critical points relative to each sphere, with critical concentrations given by Eq. .[]{data-label="fig:antimixing"}](antimixing.pdf){width="\linewidth"} In this limit, the overall phase diagram is simply a disjoint union of the phase diagrams of each subsystem, given the simple mapping between $\phi_a$ and $\Phi$. This is illustrated in Fig. \[fig:antimixing\]. Analyzing the stability of the mixed phase on each sphere is straightforward and leads to the conclusion that global phase separation is impossible in such a large $\Delta T_L$ limit, since there is no overlap between miscibility gaps of the two spheres. Moreover, the binodals of each sphere (the green lines in Fig. \[fig:antimixing\]) can be analytically derived: $$\begin{aligned} T_{\rm binodal}^{(1)} &= \mathcal{T}\left(\frac{\Phi}{x_1}\right) \,,\\ T_{\rm binodal}^{(2)} &= \mathcal{T}\left( \frac{\Phi-x_1}{x_2}\right) \,,\end{aligned}$$ \[T anitimixing binondal\] with $\mathcal{T}(y) = (1-2y)/\mathrm{arctanh}(1-2y)$. Clearly, there are two distinct critical points of the system, specific for each sphere, located at $\{\Phi_c^{(a)},T_c^{(a)}\}$ in the phase diagram. The associated two critical temperatures are given by Eq. , while the critical concentrations are \_c\^[(1)]{} = ,\_c\^[(2)]{} = x\_1 + . \[critical P antimixing\] With this analytical results it is then possible to give a precise definition of the antimixing phenomenon first reported in Ref. [@rinaldin2018geometric]: we define as [*antimixed*]{} the mixed phase of an inhomogeneous binary fluid at sub-critical temperature with non-overlapping local miscibility gaps. Now, from a strictly technical point of view, it may be argued that our treatment of the substrate geometry is oversimplified, as we approximate the dumbbell-shaped membrane of Fig. \[fig:spheres\]a with the two disjoint spheres of Fig. \[fig:spheres\]b. Evidently, a real membrane is a single structure, and having $\Delta \phi \neq 0$ will inevitably induce gradients in the neck region that interpolates between the two lobes. Could these interfacial effects destroy the antimixed state? This question is addressed in the following Section. Numerical results on more general surfaces {#sec:numerics} ------------------------------------------ In this Section we test whether our predictions on the existence of inhomogeneous mixing and antimixing hold for more realistic geometries. In particular, we must verify whether these bulk equilibrium states are compatible with the existence of concentration gradients. Therefore, let us now consider a new axisymmetric approximation of Fig. \[fig:spheres\]a, i.e. the rotationally invariant surface of Fig. \[fig:spheres\]c. Its radial profile has been obtained by joining two circular arcs by an interpolating polynomial of degree eight, chosen such that the neck interpolation and the circular arc match smoothly up to the fourth derivative at each of the two gluing points. Our general strategy to find the equilibria is to implement an evolution equation that lets an arbitrary configuration smoothly flow towards minima of Eq. . Inspired by [@Rubinstein1992] we choose to implement gradient flow with conserved global order parameter: \_t = - = D \^2 - f’() - k’() H\^2 - |[k]{}’() K + , \[explicit flow\] where $\phi=\phi(\bm{r},t)$ is now a function of both space and flow parameter $t$. We stress that the $L^2$-gradient flow generated by Eq. is purely fictitious and does not reflect the actual coarsening dynamics the binary fluid. However, this approach offers a practical way to generate stable equilibrium configurations for arbitrary geometries. We then solve Eq. numerically using a finite difference scheme on unstructured triangular meshes. More details about our numerical methods can be found in Ref. [@rinaldin2018geometric] and our code is available for download on GitHub [@githubMembrane]. Meshes are constructed using the software package *Gmsh* [@geuzaine2009gmsh]. As in the case of planar droplets on the plane, the rotational symmetry of the substrate is not necessarily inherited by the minimizers of the Gibbs free energy $G$, thus it is necessary to solve the full two-dimensional problem. ![ Equilibrium states of the axisymmetric geometry of Fig. \[fig:spheres\]c, obtained from numerical solutions of Eq. with mean-field potentials from at sub-critical temperature $T=.45\,qJ$. **a)** Lines of equilibrium for different $L-$couplings: the solid blue squares have ${L_k}=1/40\,qJR_1^2$, the empty red square have ${L_{\bar{k}}}=1/40\,qJR_1^2$ and the empty green circles have no direct interactions. The dashed black line is the line of equilibrium obtained as in Fig. \[fig:loeLM\] for the two-spheres geometry with ${L_k}+{L_{\bar{k}}}=1/40\,qJR_1^2$. The solid vertical and horizontal lines are the binodal concentrations $\phi_\pm$ at zero coupling. The inset shows what the four different equilibria look like at the same total concentration $\Phi=0.55$ (shown as a dashed gray line in the main plot). **b)** Concentration profiles as function of the arc-length axisymmetric coordinate $z$, for the three dumbbells shown in the inset of *a)*. The thin black line in the background shows the radial profile of the surface (in cylindrical arc-length coordinates, the profile of a sphere looks like a trigonometric sine). The two horizontal dashed lines correspond to the Maxwell values $\phi_\pm$. In all simulations we set $\xi=0.024 \, R_1$. []{data-label="fig:comparison"}](comparison.pdf){width="\linewidth"} Our main numerical results are shown in Fig. \[fig:comparison\]a. We focus on the linear couplings that explicitly break the $\phi \rightarrow 1-\phi$ symmetry of the free energy, since they offer the most interesting phenomenology. In all the simulations summarized in Fig. \[fig:comparison\], we set the temperature to $T=0.9 T_c$ with $T_c = qJ/2$ uniform over all $\Sigma$. As a guide to the eye, the numerical data are superimposed to the stable branch of the lines of equilibrium associated with the two disconnected spheres (see also Fig. \[fig:loeLM\]b), with ${L_k}+{L_{\bar{k}}}=1/40 \, q J R_1^2$. This value is almost an order of magnitude lower than the one used to construct the phase diagram of Fig. \[fig:phase diagrams\]b, yet it can be shown that it retains antimixed states as equilibrium solutions. Each data point is obtained upon averaging the numerically found stationary solutions of Eq. over ten random initial field configurations. To facilitate the comparison, the $\phi_a$ values are computed by integrating $\phi$ over axisymmetric regions which have the same area fraction $x_1$ as the one occupied by $S_1$ in the case of the two disjoint spheres. The solid horizontal (vertical) lines correspond to the Maxwell values $\phi_\pm$ on the small (large) sphere. In general, if the local concentrations take the binodal values, $\phi_a = \phi_\pm$, it means that the system is likely phase separated, with the interface lying in only one sub-region of $\Sigma$. The differently colours denote different values of ${L_k}$ and/or ${L_{\bar{k}}}$, while keeping the $Q-$couplings to zero. Different data points with the same colour correspond to different values of $\Phi$. The green circles corresponds to the homogeneous case, where also $L_{k,\bar{k}}=0$, and demixing occurs uniformly over the entire surface. Outside of the binodal interval, i.e. for either $\Phi < \phi_-$ or $\Phi > \phi_+$, the equilibrium state is homogeneously mixed with $\phi_{1}=\phi_{2}=\Phi$ and the data points are aligned along the diagonal. Conversely, phase separation occurs when $\phi_{-}\le \Phi \le \phi_{+}$, for which the data points depart from the diagonal and either $\phi_1$ or $\phi_2$ - the one containing no interfaces - coincide with $\phi_\pm$. The square dots correspond to either ${L_k}$ (full blue) or ${L_{\bar{k}}}$ (empty red) equal to $1/40 \,q J R_1^2$, with all other couplings set to zero. In both cases we find that the numerical results follow qualitatively the dashed line of equilibrium. The coupling with the squared mean curvature, ${L_k}$, seems to be the one that follows the two disjoint sphere results more closely, and is the only one of the two data sets that features configurations with $\phi_1 > \phi_+$ and $\phi_2 < \phi_-$ (see the bottom-right corner of Fig. \[fig:comparison\]a). Interestingly, for some $\Phi$, the equilibrium concentrations depart from a line of equilibrium and follow the horizontal (or vertical) binodal line, although only in a specific range of parameters (e.g. red dots, with $\Phi<0.5$) the data exhibit $\phi$ values that approximate the binodal value $\phi_{-}$ with reasonable accuracy. In all other cases, $\phi_{a}$ relaxes toward different $\Phi$-independent values. This behaviour likely originates from one or both of the following features of our model. First, the interpretation of $\phi_{a}$ is less stringent when applied to a connected dumbbell, where the two lobes are not geometrically distinct regions. Second, there might be additional contributions resulting from the finite thickness of the interface ($\xi=0.024R_1$ in Fig. \[fig:comparison\]). In general, these observation indicate that, in non-homogeneous spaces, the definition itself of phase-separation requires special care. This latter statement can be made more precise by considering the inset of Fig. \[fig:comparison\]a and Fig. \[fig:comparison\]b. In both plots, $\Phi=0.55$, corresponding to the dashed diagonal line in Fig. \[fig:comparison\]a. This value lies within the miscibility gap, thus, in the absence of an explicit coupling with the curvature, the system phase separates, and since $\Phi \neq x_a$, the expected areas occupied by the two phases do not match the relative size of the two lobes, so that the interface will lie away from the dumbbell’s neck. The snapshots in the inset of Fig. \[fig:comparison\]a are color-coded based on the local $\phi$ value, with $\phi=0$ in magenta, $\phi=1$ in green and $\phi=1/2$ in white. The rightmost snapshot illustrates the case of homogeneous phase-separation with the associated interface lying along a constant geodesic curvature line, as predicted by Eq. for homogeneous potentials. Fig. \[fig:comparison\]b shows a plot of $\phi$ along a meridian as a function of the arc-length $z$ from the equator of the larger sphere. The green dots show the interfacial profile of the classical phase-separated configuration, i.e. the hyperbolic tangent kink, given by Eq. , interpolating between $\phi_+$ and $\phi_-$. The dots are not perfectly aligned since the interface itself is not axisymmetric, so the arc-length $z$ does not match exactly the geodesic normal coordinate we employed in Sec. \[sec:curvature\]. When either ${L_k}$ (blue dots) or ${L_{\bar{k}}}$ (red dots) are switched on, the configuration of the phase field $\phi$ changes dramatically. The field now interpolates between values which are [*not*]{} the binodal values - shown as two horizontal dashed lines in the plot - a signature of the fact that the curvature affects the bulk concentration, even away from high curvature regions. The influence of the curvature becomes particularly striking in the neck region, where the Gaussian and mean curvature couplings give rise to opposite effects. Since $H^2$ is always positive, the coupling ${L_k}\phi H^2>0$ favours small $\phi$ values in regions of high curvature, as demonstrated by the prevalence of magenta tones around the neck. Conversely, since $K<0$, the coupling ${L_{\bar{k}}}\phi K$ is negative and favours higher $\phi$ values, as indicated by the prevalence of green tones. This behaviour is reversed in case ${L_{\bar{k}}}<0$, but, given the arbitrariness in the definition of the field $\phi$, this does not change the qualitative picture. Notice that in both cases the interface is slightly shifted away from the neck, a phenomenon which is reminiscent of the behaviour of phase domains in axisymmetric sharp interface models of free-standing lipid membranes [@Baumgart2005]. Interestingly, both the blue and green profiles are not monotonic functions of $z$. Finally, note that all three cases interpolate between different values in the two bulk regions: this proves that, in general, the distinction between inhomogeneous mixing and demixing is not well-defined: we choose to interpret the blue curve as the realization of the antimixed state (since the profile interpolates between values which are outside the local miscibility gap) on a single, connected, smooth geometry. Discussion ========== In this work we investigated the thermodynamic equilibrium of two-dimensional fluids confined on closed spatially curved substrates. Our model is primarily intended to describe self-organization in scaffolded lipid vesicles (SLVs) [@rinaldin2018geometric], i.e. self assembled lipid bilayers supported by arbitrarily shaped colloidal particles. Our results, however, are also immediately applicable to any other mixture forced to lay on a curved surface, such as in the case of coating and adsorption phenomena at liquid interfaces. We considered a binary mixture that can be characterized by a single scalar order parameter $\phi$. The generalization of the phenomena discussed here to the case of $n$-nary fluids is relatively straightforward. Crucially, we focused on closed thermodynamical systems, i.e. systems where there is no exchange of $\phi$ with the surrounding environment. This implies that the average total concentration, $\Phi$, is an externally fixed parameter. Equilibrium states are found from minimization of the Gibbs free energy $G=F-\hat{\mu}\Phi$, where the chemical potential $\hat{\mu}$ is here set by the constraint on the total concentration. In Sec. \[sec:effective\_intro\] we constructed the most general form for $F$, using only symmetry and scaling arguments, and identified four $\phi-$dependent parameters that, together with the total concentration $\Phi$, determines the equilibrium state of the system. These are: the compressibility $D$, the homogeneous free energy density $f$ and the two bending moduli $k$ and $\bar{k}$. In Sec. \[sec:review\] we reviewed the classical theory of phase separation for coexisting liquids. In Sec. \[sec:curvature\] and Appendix \[app:thin\] we reviewed the boundary layer analysis of the thin interface limit of two-dimensional phase-field models on curved surfaces, without direct curvature interactions. We derived the two-dimensional versions of the Young-Laplace and of the Kelvin equations. In Sec. \[sec:coupling\] and Appendix \[app:linear\] we considered the case where the bending moduli are small and yet non-vanishing. Depending on their scaling with respect to $D$, they produce very different effects. In case $k,\bar{k} \sim O(\sqrt{D})$, the Young-Laplace equation is changed to the equilibrium equation of the Jülicher-Lipowsky model [@Julicher1993], which we studied in detail in Ref. [@fonda2018interface]. If, on the other hand, the bending moduli are of the same order of $f$, curvature effects become more dramatic and can result in local shifts of the binodal concentrations and a spatial dependence in the line tension $\sigma$. The deviation of $\sigma$ from its flat space value is parametrized by two length-scales, which are the one-dimensional analogues of the Tolman lengths [@Tolman1949] for three-dimensional droplets. Although very general, the results of Sec. \[sec:effective\], have limited predictive power, since the $\phi-$dependence of the phenomenological parameters is left unspecified. In order to overcome this limitation, in Sec. \[sec:model\], we derived these parameters from the mean-field approximation of a microscopic lattice-gas model with curvature dependent interactions \[see Eqs. \]. We found that the curvature of the substrate directly affects the structure of the free energy landscape via four non-equivalent couplings, which either break or preserve the symmetry of the free energy under exchange of the two phases (i.e. $\phi\to1-\phi$). We refer to them respectively as $Q-$ and $L-$interactions. Motivated by the experimental results we reported in Ref. [@rinaldin2018geometric], we applied our model to dumbbell-shaped membranes, as shown in Fig. \[fig:spheres\]. For simplicity, we first approximated this surface as consisting of two disjointed spheres, allowed to exchange order parameter, but otherwise isolated from the environment (see Sec. \[sec:disjoint spheres\]). We found that $L-$interactions, which linearly couple with the order parameter $\phi$, favour inhomogeneous mixing, i.e. a single phase with non-uniform concentration across the system. For our simple two-sphere geometry, this implies that each sphere is characterized by a distinct $\phi$ value, depending upon the strength of the $Q-$ and $L-$couplings and the local curvature radius. Exceptionally, for certain specific $\Phi$ values, such an inhomogeneously mixed phase remains stable even [*below*]{} the critical temperature. In this regime, the inhomogeneity becomes more severe and the two spheres exhibit a stark concentration difference, even though phase separation has not occurred. We named this peculiar phenomenon, that was observed in Ref. [@rinaldin2018geometric] experimenting with scaffolded lipid vesicles (SLVs), [*antimixing*]{}, to stress that, albeit still in the mixed phase, the equilibrium concentrations split on the two opposite sides of a local miscibility gap. Surprisingly, this behaviour depends on the linear couplings between the concentration and the local curvature (i.e. the $L-$coupling, in our notation), despite these not altering the Maxwell construction and being thermodynamically irrelevant in binary membranes confined on homogeneous substrates. This originates from the fact that, in the presence of sufficiently large geometrical inhomogeneities and sufficiently strong symmetry-breaking coupling with the curvature, the phase diagram partitions into two sub-diagrams, each with its own distinct critical point (see Fig. \[fig:antimixing\]). Lastly, in Sec. \[sec:numerics\], we verified that inhomogeneous mixing and antimixing persist also on more realistic dumbbell-shaped substrates, obtained by connecting two spherical caps with a smooth neck (see Fig. \[fig:spheres\]c). In this case, inhomogeneous mixing demands the occurrence of sharp concentration gradients, whose structure is substantially different than that of standard interfacial profiles. Most importantly, the average concentrations on the spherical lobes, i.e. the regions away from the neck, differ from the binodal values, even if the thermodynamic potential has the same Maxwell concentrations everywhere. This phenomenon is somewhat similar to the change in the bulk lateral pressure due to curved interfaces, as predicted by the Kelvin equation, whereas now the ambient curvature is inducing this change. Finally, we found that the ${L_k}$, i.e. the linear interaction with the squared mean curvature, produces equilibrium concentrations which match very closely the values found from the two-spheres simplified geometry, thus indirectly confirming that the antimixed state is a valid concept also for connected geometries. Beyond mixing and demixing {#beyond-mixing-and-demixing .unnumbered} -------------------------- We have demonstrated that the thermodynamics of mixtures confined on inhomogeneous [*closed*]{} substrates, entails a spectrum of interesting phenomena that is, perhaps, broader than initially thought. In particular, the importance of closeness (i.e. the fact that a mixture cannot exchange material with the external environment), might have been overlooked in the past, even though, after the seminal work by Baumgart [*et al.*]{} [@Baumgart2003], the interplay between geometry and chemical composition in multicomponent membranes has become a subject of thorough theoretical and experimental investigations. One of the most fundamental outcomes of our analysis is that curvature inhomogeneities force to relax the usual distinction between mixed and demixed phases, since now concentration gradients and interface-like structures can be induced by curvature rather than spinodal instabilities. The very existence of antimixing, on dumbbell-shaped substrates, provides a prominent example of stable equilibrium states which have features of both phases. From a model-building perspective, this implies that extreme care must be used in choosing the functional form of the bending moduli profiles $k(\phi)$ and $\bar{k}(\phi)$, since, even the simplest interaction term (e.g. the linear coupling introduced by Markin [@Markin1981]), can produce highly non-trivial effects to the equilibrium phase diagram of closed systems. Furthermore, slightly different choices can lead to very different phenomenologies, thus negatively affecting the validity of a given model. Some of our predictions appear amenable to a reasonably viable experimental verification. First, we have shown that $Q-$interactions may induce a curvature-dependent line tension and critical temperature. Even experiments on multicomponent spherical vesicles can potentially test this effect by searching for a possible dependence of $\sigma$ and $T_{c}$ on the vesicle’s radius. Furthermore, we recall that curvature terms were neglected in deriving $D$ from the lattice-gas model \[see Eq. \] to comply with the general assumptions of Sec. \[sec:effective\]. Lifting these assumptions yields in fact: D= \^2 (J + [Q\_k]{}H\^2 + [Q\_[|[k]{}]{}]{}K ) , which reveals a curvature dependence exactly analogous to $T_c$ in Eq. , since the mean-field value of the nearest-neighbour interaction simultaneously affects both quantities. Note that this effect does not have any implications when only $L-$interactions are considered, thus our conclusions on the antimixed state are unchanged. However, this relation does predict that not only compressibility, but also the effective diffusion (e.g. as measured from photobleaching experiments) of lipids on a vesicle might depend on the vesicle size. To the best of our knowledge, neither one of these phenomena has yet been experimentally investigated. #### Acknowledgements {#acknowledgements .unnumbered} This work was supported by the Netherlands Organisation for Scientific Research (NWO/OCW), as part of the Frontiers of Nanoscience program (MR) and the VIDI scheme (LG,PF). Geodesic normal coordinates {#app:normal} =========================== If the equilibrium configuration is in a phase-separated state, then $\phi$ develops linear interfaces. In this Appendix we show how to construct a set of coordinates which is adapted to the arbitrary shape of the system. We suppose that locally the effective free energy is homogeneous, so the two Maxwell values of the pure phases $\phi_\pm$ are well defined. We then define the interface $\gamma$ (the black curve in Fig. \[fig:figure 1\]) as the level set = { : () = }. \[def interface\] The fact that we choose the average value between the two pure phase concentrations $\phi_\pm$ as defining the interface is purely conventional and does not carry any special meaning. Any other level set would work equally well. Note that $\gamma$ in general will consist of multiple curves, which we assume to be not mutually intersecting. If $\partial\Sigma = \varnothing$, the curves will be closed. From now on we restrict to the case of the interface consisting of a single, closed and simple curve, although the generalization to multiple interfaces is straightforward. We can parametrize $\gamma$ by its arc-length $s$, and define the tangent two-vector $T^i(s)\equiv\partial_s x^i(s)$ ($i=1,\,2$) and fix the normal $N^i(s)$ to consistently point in the $\Sigma_+$ domain. The geodesic curvature of the curve is defined as \_g = T\^i T\^j \_i N\_j , \[geodesic curvature\] with $\nabla_i$ the covariant derivative on $\Sigma$ (see the Appendices of [@fonda2018interface] for much more detail on the theory of curves applied to linear interfaces). The arc-length condition implies that the norm of the tangent vector is constant when parallel-transported along $\gamma$, i.e. $T^i T^j \nabla_i T_j = 0$. The fact that also $N^i$ is of unit norm along $\gamma$ implies that $T^i N^j \nabla_i N_j = 0$. Orthogonality to $T^i$ implies $\kappa_g = - T^i N^j \nabla_i T_j$. Note that $T^i$ and $N^i$ are two-vector fields which are defined only along the curve, so that we are allowed to take derivatives of them only along $T^i$, and not in directions normal to the curve. To this purpose, we need to extend the coordinate system away from $\gamma$. The most natural way to do so is to use geodesic normal coordinates. In a sufficiently small neighbourhood of $\gamma$ we associate to every point $P$ in $\Sigma$ the coordinate pair $(s,z)$, where $z$ represent the length of the (unique) geodesic segment starting from $P$ and intersecting $\gamma$ orthogonally. The point where this intersection occurs defines the value of $s$ (see the inset of Fig. \[fig:figure 1\]). To these coordinates we associate the two vector fields $t^i=\partial_s$ and $n^i=\partial_z$ with the defining properties t\^i\_[z=0]{} = T\^i ,n\^i\_[z=0]{} = N\^i, and $n_i t^i = 0$ in the whole neighbourhood. Note however that $y^i$ is not unit normalized outside of $\gamma$: $s$ is the arc-length only of the $z=0$ line. On the other hand $n^i n_i=1$ throughout the whole patch. The induced metric on $\Sigma$ is diagonal h\_[ij]{} = t\_i t\_j + n\_i n\_j . With these definitions, the gradient of the scalar field $\phi(s,z)$ is \_i = t\_i \_s + n\_i \_z , where $\phi_s = \partial_s \phi$ and $\phi_z = \partial_z \phi$. We are finally able to compute the action of the Laplace-Beltrami operator on a scalar function in the adapted frame \^2 = t\^i t\_i \_[ss]{} + \_[zz]{} + \_[z]{} , \[adapted LB\] with the vector norm computed with respect to the induced metric. There is no mixed term $\phi_{sz}$ because of the orthogonality of the coordinates and there is no $\phi_s$ term because of the geodesicity of $\partial_z$. Here $\kappa= t^i t^j \nabla_i n_j$ is the geodesic curvature of the $z=\rm{const}$ lines and satisfies \_[z=0]{} = \_g . The $z$-dependence of $\kappa$ is non-trivial and for arbitrary geometries it cannot be computed explicitly. In the neighbourhood of the interface we can expand for small $z$ and use standard formulas for normal variations of geometric invariants (see e.g. Ref. [@Fonda:2016ine]), finding = \_g - z (\_g\^2 + K) + O(z\^2) . \[kappa K\] In general higher order terms in $z$ become increasingly complicated, depending on derivatives of $K$ along $z$. However, in case $K=\rm const$ it is possible derive exact results. For instance, for a flat surface with $K=0$ it is easy to prove (see also e.g. Appendix A of Ref. [@Elder2001]) = , whereas for a sphere of radius $R$ (thus with $K=1/R^2$), we get = . Thin interface limit {#app:thin} ==================== In this Appendix we review the technical details of the thin interface approximation $D = \xi^2 \ll A_\Sigma$, for equations of the form given by Eq. . Since the diffusive length is small, we can look for perturbative solutions. Being the thickness of interface also $O(\xi)$, there are essentially two regimes to consider: the bulk phases where gradients are mild (the so-called outer region) and the interface itself where derivatives are unbounded (the inner region). At each order in $\xi$, the outer expansion provides the correct boundary conditions for the inner expansion. We focus now on the inner expansion. Since the interface $\gamma$ is assumed to be a smooth curve, we can use the adapted coordinate system outlined in the previous Appendix. The normal coordinate will lie in an interval of the order $z \in [-\xi, \xi]$, with positive (negative) $z$ pointing along $\Sigma_+$ ($\Sigma_-$) domains. In this approximation, we expand the scalar field and the chemical potential as $$\begin{aligned} \phi(s,z) &= \phi^{(0)}(z) + \xi \phi^{(1)}(z) + \dots \,, \label{phi xi}\\ \hat{\mu} &= \mu^{(0)} + \xi \mu^{(1)} + \dots \,, \label{mu xi}\end{aligned}$$ \[xi expansion\] where the dots stand for $O(\xi^2)$ terms. As specified in the main text, we assume that over the interface the $z$-derivatives scale at most as $\xi^{-1}$. The $O(1)$ inner expansion of the equilibrium condition is thus f’(\^[(0)]{}) = \^[(0)]{} + \^2 \^[(0)]{}\_[zz]{} . \[EL order 0\] Asymptotic matching with the bulk boundary conditions shows unsurprisingly that $\mu^{(0)}$ is precisely the chemical potential obtained by the common tangent construction, while $\phi^{(0)}$ approaches the bulk values $\phi_\pm$. We can rescale the geodesic normal distance by $\xi$ so that the variable $w=z/\xi$ spans approximately the full real line $w \in [-\infty,\infty]$. Defining $\varphi(w)\equiv \phi^{(0)}(w \xi)$ we can rewrite as \_[ww]{} = g’() , \[profile equation\] where $g$ is the physically equivalent, shifted potential g() = f() - + , \[shifted g\] which satisfies the properties $g(\phi_\pm)=g'(\phi_\pm)=0$ and $g''(\varphi)=f''(\varphi)$. Equation can be multiplied by $\varphi_w$ and integrated - the choice of $g$ is such that the integration constant is zero - and one obtains the equipartition relation in the main text, namely Eq. , whose solutions are one-dimensional kinks. Without specifying $f$ it is not possible to solve further, however note that since $g$ and its first derivative approach zero in the limit $w \to \pm \infty$, we have that the decay towards $\phi_\pm$ of $\varphi$ is always exponential $|\varphi-\phi_\pm| \underset{w \to \pm \infty}{\sim} e^{\mp \lambda_\pm w}$ with decay lengths \_= , \[lambdas\] which are diverging at critical points. We now consider the next term in the inner expansion. Equation at order $O(\xi)$, upon the substitution $z\to w/\xi$, reads f”() \^[(1)]{} = \^[(1)]{} + \^[(1)]{}\_[ww]{} + \_g \_[w]{} . \[EL order 1\] We now multiply this equation by $\varphi_w$ and integrate over $w$. By using the identity $$\begin{gathered} \int_{-\infty}^{+\infty} {{\rm d}}w \left(\phi^{(1)}_{ww} - f''(\varphi) \phi^{(1)} \right) \varphi_w =\\ = \int_{-\infty}^{+\infty} {{\rm d}}w \left(\varphi_{www} - f''(\varphi) \right) \phi^{(1)} = 0 \,,\end{gathered}$$ which follows from and $\varphi_w(\pm \infty)=\varphi_{ww}(\pm \infty)=0$, we find the relation \^[(1)]{} = , which proves that equilibrium interfaces must be curves of constant geodesic curvature. In the above expression we defined Z = \_[-]{}\^[+]{} [[d]{}]{}w (\_w)\^2 = \_[\_-]{}\^[\_+]{} [[d]{}]{} , where the last equality follows from . By taking the limit $w \to \pm \infty$ of , one finds the asymptotic relation for $\phi^{(1)}$ \^[(1)]{}() = = . This result (which was also derived e.g. in [@Rubinstein1992]) is in striking contrast with the usual $O(\xi)$ matching condition for non-conserved order parameters: the chemical potential renders $\phi^{(1)}$ non-zero also in the bulk phases. Having specified how to expand perturbatively in powers of $\xi$ and having solved the equations at $O(1)$ and at $O(\xi)$, the last step is to evaluate the free energy on the equilibrium solutions. To this purpose we assume that, at finite $\xi$, $\Sigma$ is partitioned into three distinct regions: a strip $\gamma^{(\xi)}$ centered at $\gamma$ and of width $\sim 2\xi$ separating the two bulk domains $\Sigma_\pm^{(\xi)}$ which consist of $\Sigma_\pm$ with the half-strip region removed. The area of the strip is $\approx 2 \xi \ell_\gamma$ with $\ell_\gamma$ the length of the interface. The area of the two bulk domains is (\_\^[()]{}) = x\_A\_- \_+ O(\^2) . Now integrals over the whole surface can be split into the sum of three terms: if $\mathcal{G}(\phi)$ is an arbitrary function of $\phi$ and its derivatives, its integral over $\Sigma$ can be computed as $$\begin{gathered} \frac{1}{A_\Sigma} \int_\Sigma {{\rm d}}A \mathcal{G}(\phi) = x_+ \mathcal{G}(\phi_+) + x_- \mathcal{G}(\phi_-)- \\ -\frac{\xi \ell_\gamma}{A_\Sigma} \left(\mathcal{G}(\phi_+)+ \mathcal{G}(\phi_-) - \lim_{\xi\to 0}\frac{1}{\xi} \int_{-\xi}^{+\xi} {{\rm d}}z \mathcal{G}(\phi^{(0)}) \right) + \\ + \xi \mu^{(1)} \left( x_+ \frac{\mathcal{G}'(\phi_+)}{f''(\phi_+)} + x_- \frac{\mathcal{G}'(\phi_-)}{f''(\phi_-)} \right) + O(\xi^2) \,. \label{average Oxi}\end{gathered}$$ There are two contributions of $O(\xi)$: one from the integration of $O(1)$ terms on the strip, the other from the $O(\xi)$ corrections to the bulk integrals. The integral in the second line can be evaluated by substituting $dz = \xi dw$ and integrating over the real line. By picking $\mathcal{G}(\phi)=1$ one immediately recovers the general condition $x_++x_-=1$, which obviously does not take any correction. Instead, by picking $\mathcal{G}(\phi)=\phi$ one computes the total average concentration. In this case the second line of vanishes, because of \_[0]{} \_[-]{}\^[+]{} [[d]{}]{}z \^[(0)]{}(z) = 2 \^[(0)]{}(0) , \[Gibbs 1\] and of the definition of $\gamma$, . This result contains however some degree of arbitrariness, since we defined the limit in in a symmetric manner: any other choice of the location of the interface within the strip $\gamma^{(\xi)}$ would have led to a different value. This ambiguity is fixed in general by an appropriate shift of the zero-point of the geodesic normal coordinate in such a way that the following equality holds \_0\^[+]{} [[d]{}]{}w (-\_+) + \_[-]{}\^0 [[d]{}]{}w (- \_-) = 0, \[Gibbs 2\] which defines the so-called Gibbs interface. This condition states that the integrated difference between inner and outer concentrations should match on both sides of the $z=0$ line. Formally, we should replace definition with , even if nothing of the following results depends on this choice. We finally find that = x\_+ \_+ + x\_- \_- + \^[(1)]{} ( x\_+ \_+ + x\_- \_- ) , i.e. the total concentration does indeed pick a contribution from the interface and deviates from the homogeneous relation . The extra factor depends on penetration depths defined in , and vanishes for geodesics. By plugging $\mathcal{G}(\phi)=f'(\phi)$ into , we precisely re-obtain the chemical potential expansion . Instead, by choosing $\mathcal{G}(\phi)=f(\phi)$ one finds $$\begin{gathered} \int_\Sigma {{\rm d}}A f(\phi) = \frac{1}{2} \sigma \ell_\gamma + \\ + A_\Sigma \sum_{\alpha=\pm} x_\alpha \left( f(\phi_\alpha) + \sigma \kappa_g \frac{f'(\phi_\alpha)}{(\phi_+-\phi_-)f''(\phi_\alpha)} \right)\,,\end{gathered}$$ with $\sigma \equiv \xi Z$. Finally, with $\mathcal{G}(\phi)=\xi^2/2\nabla_i\phi \nabla^i \phi$ one finds \_[[d]{}]{}A \_i\^i = \_. Combining the last two expression we obtain the $O(\xi)$ expansion for the total free energy, Eq. . Linear corrections to the Maxwell construction {#app:linear} ============================================== In this Appendix we show how to compute the linear corrections when the free energy $f(\phi)$ of Eq. is modified by a small perturbation () = f() + h() , with $\varepsilon \ll 1$. In the following, we will keep only first order corrections in $\varepsilon$. The Maxwell common tangent condition reads f’(\_) + h’(\_) = , where $\tilde{\phi}_\pm = \phi_\pm + \varepsilon \delta\phi_\pm + \dots$ are the shifted values of the bulk phases. The $O(\varepsilon)$ solution to these equations gives \_= - . We now compute the $O(\varepsilon)$ correction to the line tension . First, it is easy to see that the shifted potential $g$ defined in becomes () = g() + g\_h(), with g\_h()= h() + , which remarkably does not depend on $f(\phi)$ nor on its derivatives. Substituting the above expression in and expanding again, one finds = + \_h , \[eps sigma 1\] where \_h = \_[-]{}\^[\_+]{}[[d]{}]{} . \[Zh 1\] Note that in there are no endpoint contributions at order $O(\varepsilon)$ since $g(\phi_\pm)=0$. The integral in the above expression can be rewritten by means of the Gibbs condition . Formally we can compute the integral of the linear and constant terms as $$\begin{aligned} \int_{-\infty}^{+\infty} {{\rm d}}w\frac{h(\phi_-)(\phi_+-\varphi)}{\phi_+-\phi_-} &= \int_{-\infty}^0 {{\rm d}}w h(\phi_-)\,, \label{eps Gibbs 1}\\ \int_{-\infty}^{+\infty} {{\rm d}}w\frac{h(\phi_+)(\varphi-\phi_-)}{\phi_+-\phi_-} &= \int_0^{+\infty} {{\rm d}}w h(\phi_+) \,, \label{eps Gibbs 2}\end{aligned}$$ \[eps Gibbs\] where the integration limits should be thought as momentarily regularized. Plugging this into , we find $$\begin{gathered} \delta_h = \xi \int_0^{+\infty} {{\rm d}}w (h(\varphi)-h(\phi_+)) +\\+ \xi \int_{-\infty}^0 {{\rm d}}w (h(\varphi)-h(\phi_-)) \,, \label{Zh 2}\end{gathered}$$ which shows how the first correction to the line tension is due to the integrated difference between the zero-th order inner and outer values of $h(\phi)$, evaluated on either side of the Gibbs interface. Any term which is symmetric with respect to the exchange $z\to -z$, such as constant and linear terms, will give a vanishing contribution to $\delta_h$. In the main text, we replace $\varepsilon$ by $H^2$ or $K$, and $h(\phi)$ by either $k(\phi)$ or $\bar{k}(\phi)$. High temperature expansion of the inhomogeneous mixing {#app:CLCM} ====================================================== Given the definition of $\Delta \phi$ in Eq. , we can rewrite the local concentrations as $$\begin{aligned} \label{Delta phi1 } \phi_1 &= \Phi + x_2 \Delta \phi \,, \\ \label{Delta phi2} \phi_2 &= \Phi - x_1 \Delta \phi\,,\end{aligned}$$ \[Delta phi12\] and the local quadratic and linear couplings as T\_[c,M]{}\^[(a)]{}= \_[c,M]{} + (-1)\^a \_[c,M]{} , where $\hat{T}_c$ is defined in Eq. , $\hat{T}_L$ has an obvious analogous definition and $\Delta T_{c,M}$ are defined by Eqs. . By plugging these expressions into and expanding for small differences we get the equation - 4 \_c + 4 (1-2) T\_c + 2 T\_L = 0 , whose solution is of the form with coefficients C\_Q(, \_c/T ) = 4 , and C\_L(, \_c/T ) = 2 . These are always finite quantities whenever $T> 2 \hat{T}_c$, thus expansion can be trusted only in the high temperature limit, where they approach the values of . [^1]: In principle, the neck region could consist of a *catenoid-*like shape, with $H \sim 0$. It is however sufficient a small deviation from this ideal case for $H^2$ to be much larger than $1/R_a^2$, the curvatures of the spherical caps.

--- abstract: 'The Berry phase acquired by an electromagnetic field undergoing an adiabatic and cyclic evolution in phase space is a purely quantum-mechanical effect of the field. However, this phase is usually accompanied by a dynamical contribution and can not be manifested in any light-beam interference experiment because it is independent of the field state. We here show that such a phase can be produced using an atom coupled to a quantized field and driven by a slowly changing classical field, and it is manifested in the atomic Ramsey interference oscillations. We also show how this effect may be applied to one-step implementation of multi-qubit geometric phase gates, which is impossible by previous geometric methods. The effects of dissipation and fluctuations in the parameters of the pump field on the Berry phase and visibility of the Ramsey interference fringes are analyzed.' address: | Department of Physics\ Fuzhou University\ Fuzhou 350002, P. R. China author: - 'Shi-Biao Zheng' title: Manifestation of nonclassical Berry phase of an electromagnetic field in atomic Ramsey interference --- 0.5cm INTRODUCTION ============ When the Hamiltonian of a quantum system, depending on a set of parameters, is adiabatically changed along a closed curve in parameter space, then the quantum system in an eigenstate of the Hamiltonian will acquire a purely geometric phase in addition to the usual dynamical phase \[1\]. Compared with the dynamic phase, the Berry phase is given by a circuit integral in parameter space and is independent of energy and time. During the past few decades, Berry phase has been the subject of a variety of theoretical and experimental investigations. Besides the fundamental interest, Berry phase has many important applications, ranging from optics and molecular physics to quantum computation by geometric means \[2-6\]. Since the Berry phase only depends upon the geometry of the evolution path, it is robust against fluctuation perturbations that affect the dynamical phase \[7\]. This feature makes quantum logic gates based on geometric phases have potential fault-tolerance in the presence of noise perturbation. The geometric phase has been generalized to the case of nonadiabatic, noncyclic, and nonunitary evolution of a quantum system \[8,9\]. The Berry phase and its robustness against noise perturbations has been experimentally tested in various two-state systems \[10-14\]. Optical experiments have been performed to observe Berry phase of light beams \[15-17\] that can be understood as a classical effect following the Maxwell equations \[18\]. The observation of this effect at the single-photon level has also been reported \[19\], but the Berry phase without classical origin has not been directly measured for any quantum harmonic oscillator in continuous-variable (infinite-dimensional) states. The cyclic and adiabatic displacement in phase space is the simplest quantum-mechanical transformation that can produce a nonclassical Berry phase for a continuous-variable system. Unfortunately, no scheme has been proposed for realizing such transformation in a realistic physical system without introducing the dynamical phase. Furthermore, the Berry phase acquired through such a transformation cannot be manifested in any optical interference experiment. This is due to the fact that the interference of light fields is fundamentally different from that of particles. It is the relative phase of states that manifests in the latter case, while it is the relative phase of the electric amplitudes in the former case. Thus, the geometric phase measured in a light-beam interference experiment is the Hannay angle rather than the Berry phase \[20\]. For a cyclic displacement evolution, the acquired Berry phase is independent of the field state and the Hannay angle is zero. Here we show that the Berry phase of a quantized field, associated with a cyclic and adiabatic displacement in phase space, can be produced and measured with an atomic quantum bit (qubit) that is coupled to the quantized field and driven by a classical pump field. By means of variation of the parameters of the pump field the quantized field undergoes an adiabatic and cyclic evolution in phase space, conditional upon the state of the qubit. The two qubit states correspond to two evolution paths in the Hilbert space, one correlated with the adiabatic displacement of the quantized field and the other correlated with free evolution. The Berry phase of the quantized field is manifested in the interference of the atomic qubit, other than in the interference of the field itself. As far as we know, our proposal is the first realistic one for directly measuring the Berry’s adiabatic geometric phase for infinite-dimensional field states. When multiple qubits are involved, the acquired Berry phase leads to one-step implementation of important geometric quantum gates, which can not be achieved by previous methods. The effects of dissipation and the fluctuations in the parameters of the pump field on this phase and the atomic coherence are analyzed. The required qubit-boson coupling can be realized in cavity or circuit QED systems with atomic or superconducting qubits coupled to a resonator. The paper is organized as follows. In Sec.2 we propose a scheme for measuring the Berry phase of a quantized field using the Ramsey interference of an atomic qubit coupled to the quantized field and driven by an external field. We show that the acquired Berry phase can be used for one-step implementation of $n$-qubit quantum phase gates in Sec.3. In Sec.4 we investigate the geometric phase and visibility of the Ramsey interference fringes with the field decay and atomic spontaneous emission being included. Sec. 5 sees an analysis of the effect of the fluctuations in the parameters of the pump field on the Berry phase and atomic coherence. Finally we dissuss the physical implementation of the model in microwave cavity QED and generalization of the ideas to the ion trap system, and we present a summary of our results in Sec. 6. MEASUREMENT OF THE BERRY PHASE OF THE QUANTIZED FIELD ===================================================== We first consider a two-level atom resonantly interacting with a single-mode quantized electromagnetic field and driven by a classical field. The dynamics of the whole system is described by the driven Jaynes-Cummings model \[21\]. We will label the upper and lower states of the two-level atom as $\left| e\right\rangle $ and $\left| g\right\rangle $ and describe its dynamics in terms of the Pauli operators $\sigma _z$, $\sigma ^{\pm }=(\sigma _x\pm i\sigma _y)/2$. Then the Hamiltonian, in the interaction picture, is (assuming $\hbar =1$) $$H=\lambda (a^{\dagger }\sigma ^{-}+a\sigma ^{+})+\Omega (\sigma ^{-}e^{-i\phi }+\sigma ^{+}e^{i\phi }),$$ where $a^{\dagger }$ and $a$ are the creation and annihilation operators for the quantized field, $\lambda $ is the atom-cavity coupling strength, and $% \Omega $ and $\phi $ are the Rabi frequency and phase of the pump field. The first part of the Hamiltonian, describing the coherent energy exchange between the two-level atom and the quantized field, corresponds to the normal Jaynes-Cummings model \[22\]. The second part represents the coupling between the atomic transition $\left| g\right\rangle \leftrightarrow \left| e\right\rangle $ and the classical field. The eigenenergies of the driven Jaynes-Cummings model are the same as those of the normal Jaynes-Cummings model, while the field parts of the eigenstates are displaced in phase space with the displacement parameter determined by the Rabi frequency and phase of the pump field \[21\]. This allows one to adiabatically drive the quantized field to undergo a cyclic displacement evolution by slowly changing the parameters of the pump field. The Berry phase gained this way is determined by the area enclosed by the phase-space displacement trajectory, which is independent of the state of the system. To illustrate the idea clearly, we first consider the evolution of the dark state (the eigenstate with zero eigenenergy) of the Hamiltonian $H$. Such a state is $\left| \psi _0(\alpha )\right\rangle =\left| g\right\rangle \left| \alpha \right\rangle $, where $% \left| \alpha \right\rangle $ is the coherent state of the quantized field with the parameter $\alpha =-\Omega e^{i\phi }/\lambda $. Adiabatic following of the dark state renders the phase of the quantized field to be opposite to that of the pump field. When the parameters of the Hamiltonian complete a cyclic evolution after the duration $T$, the dark state describes a displacement loop of the quantized field and acquires a Berry phase $$\begin{aligned} \beta &=&i\int_0^Tdt\left\langle \psi _0(\alpha )\right| \frac d{dt}\left| \psi _0(\alpha )\right\rangle \\ \ &=&\frac i2\oint (\alpha ^{*}d\alpha -\alpha d\alpha ^{*})=\pm 2A, \nonumber\end{aligned}$$ where $A$ is the area enclosed by the phase-space loop. The $\pm $ sign depends on whether the sense of rotation is clockwise or counter-clockwise. Suppose that the Rabi frequency $\Omega $ of the pump field is kept constant and the phase $\phi $, serving as the control parameter, is slowly varied from $0$ to $2\pi $. Then the state of the quantized field is moved around a circle with radius $\left| \alpha \right| $ in phase space and the acquired Berry phase is given by $-2\pi \left| \alpha \right| ^2$. During the adiabatic evolution, the atom remains in the lower state $\left| g\right\rangle $ and the Berry phase completely originates from the cyclic quantum-mechanical displacement of the quantized field induced by the adiabatic variation of the Hamiltonian. This phase is significantly different from the nonadiabatic geometric phase produced by a coherent displacement force \[23\] in that the latter involves a nonzero dynamical component that is proportional to the total phase and thus cannot be removed \[8, 24\]. We note that the displacement evolution path of the quantized field is not affected if a term proportional to $\sigma _z$ is added to the Hamiltonian. This implies that the acquired Berry phase is immune from the fluctuation in the qubit transition frequency as compared with the geometric phase of a qubit coupled only to a classical field. In order to interpret the acquired Berry phase in the parameter space we rewrite the Hamiltonian as $H=\lambda [(a^{\dagger }\sigma ^{-}+a\sigma ^{+})+{\bf B}\cdot {\bf \sigma }]$, where ${\bf \sigma =\{\sigma }_x$, ${\bf % \sigma }_y$, ${\bf \sigma }_z{\bf \}}$ and ${\bf B}=(\Omega \cos \phi /\lambda $, $\Omega \sin \phi /\lambda $, $0)$ is the dimensionless effective vector field which acts as the control parameter. By cyclically changing $\Omega $ and/or $\phi $ the Hamiltonian describes a closed path in the two-dimensional parameter space $\left\{ {\bf B}\right\} $. When the system is initially in the dark state of the Hamiltonian, the state of the system will follow the effective field and after a closed cycle it gains a purely geometric phase proportional to the area enclosed by the circuit traversed by the effective field. Note that, although the two-level system is coupled to the slowly changing effective vector field, its state is not varied when the system is initially in the dark state because the transition paths induced by the two fields with opposite phases interfere destructively. It is the quantized field whose state adiabatically follows the effective field ${\bf B}$ and eventually acquires the Berry phase which depends upon the global property of the evolution path in the parameter space of the effective field. The eigenstates of the Hamiltonian $H$ with nonzero eigenvalues are $\left| \psi _{n+1,\pm }(\alpha )\right\rangle =D(\alpha )(\left| g\right\rangle \left| n+1\right\rangle \pm \left| e\right\rangle \left| n\right\rangle )/% \sqrt{2}$, where $D(\alpha )=e^{\alpha a^{\dagger }-\alpha ^{*}a}$ is the displacement operator, and $n+1$ is the quantum number of the displaced field state. After a complete evolution cycle all the eigenstates gain the same Berry phase $\beta $, which makes it impossible to observe the Berry phase in an interference experiment by initially preparing a superposition of different eigenstates as no relative geometric phase can be obtained. Neither can this phase be observed in optical interference experiments because it does not depend on the state of the quantized field. In the following, we show such a phase can be manisfested in the interference between the probability amplitudes associated with the atomic state $\left| g\right\rangle $ and an auxiliary state $\left| f\right\rangle $ which is neither coupled to the pump field nor to the quantized field mode. The atom is initially driven to the superposition state $\frac 1{\sqrt{2}}(\left| g\right\rangle +\left| f\right\rangle )$ from $\left| f\right\rangle $ using a classical pulse, which is analogous to the splitting of the photon beam at the first beam splitter of a Mach-Zehnder interferometer. The quantized field, initially prepared in the coherent state $\left| -\Omega /\lambda \right\rangle $, acts as the dephasing element in one arm of the interferometer. After adiabatically dragging the Hamiltonian of Eq. (1) along a closed loop the quantized field undergoes a conditional cyclic displacement in phase space and introduces a relative phase between the two atomic states $\left| g\right\rangle $ and $\left| f\right\rangle $, leading to the superposition state $\frac 1{\sqrt{2}}(e^{i\beta }\left| g\right\rangle +\left| f\right\rangle )$. This implies that the Berry phase acquired by the quantized field is encoded in the probability amplitude for finding the atom in the state $\left| g\right\rangle $. The interference between the probability amplitudes of the two superposed atomic states can be achieved through the transformations $\left| g\right\rangle \longrightarrow \frac 1{\sqrt{2}}(\left| g\right\rangle -\left| f\right\rangle )$ and $\left| f\right\rangle \longrightarrow \frac 1{\sqrt{2}% }(\left| f\right\rangle +\left| g\right\rangle )$, which are analogous to the recombination of the photon beams at the second beam splitter of the Mach-Zehnder interferometer. Finally, the atomic state is measured. The probabilities of finding the atom in the states $\left| g\right\rangle $ and $\left| f\right\rangle $ are given by $$P_{g,f}=\frac 12(1\pm \cos \beta ),$$ which is independent of the atom-field interaction time. By varying the Berry phase $\beta $ the probability of finding the atom in a definite state exhibits Ramsey interference fringes. Therefore, the Berry phase of light field is manisfested in the atomic Ramsey interference. With the choice $% \Omega =\lambda /\sqrt{2}$ and $\left| \alpha \right| =1/\sqrt{2}$, an adiabatic and cyclic evolution from $\phi =0$ to $\phi =2\pi $ achieves the Berry phase $\beta =-\pi $. Due to the presence of the Berry phase the atom is finally in the state $\left| g\right\rangle $. On the other hand, if no Berry phase is present, the atom is finally in the state $\left| f\right\rangle $. Therefore, the detection of the final state of the atom unambigually distinguishes whether the Berry phase is present or not. It is important to note that even if the quantized field is in a macroscopic coherent state with $\left| \alpha \right| \gg 1$ the adiabatic and cyclic evolution of this field can be engineered by changing the Hamiltonian around a suitable circuit in parameter space, enabling the nonclassical Berry phase of light to be tested at the macroscpic level. A modification of the interferometer can be applied to measure the Berry phase for any initial field state, for example, a thermal state. If the system is not initially in the dark state a purely geometric phase can be observed by applying a phase kick $-\sigma _z$ to the atom at time $T/2$. The product of the atomic state $\left| g\right\rangle $ with any displaced number state $D(\alpha )\left| n+1\right\rangle $ can be expressed as a superposition of the two eigenstates $\left| \psi _{n+1,+}(\alpha )\right\rangle $ and $\left| \psi _{n+1,-}(\alpha )\right\rangle $. During the adiabatic evolution these two eigenstates acquire opposite dynamical phases. The phase kick inverts these two eigenstates, effectively inverting the dynamical phases accumulated from time $0$ to $T/2$. At time $T$ the system completes a nontrivial cyclic evolution and the dynamical phase is completely canceled. The whole procedure results in a purely geometric phase $\beta $, which is determined by the area of the circuit followed by the effective field ${\bf B}$ in parameter space and independent of the initial field state. This implies that the quantized field does not need to be prepared in a specific state since any field state can be expressed in terms of coherent state $\left| \alpha \right\rangle $ and displaced number states $D(\alpha )\left| n+1\right\rangle $. After the cyclic evolution all components acquire the same geometric phase $\beta $ and the dynamical phases associated with all displaced number states $D(\alpha )\left| n+1\right\rangle $ are removed via the application of the atomic phase kick. The atomic interferometer can also be used to measure the geometric phase for the quantized field undergoing a noncyclic evolution. In this case the geometric phase can be expressed as the minus double of the area enclosed by the displacement trajectory and the straight line (the null phase curve) connecting the starting and ending points in phase space \[25\]. The geometric phase is directly related to the shift of the Ramsey interference fringes \[26\]. ONE-STEP IMPLEMENTATION OF MULTI-QUBIT PHASE GATES ================================================== Besides being of fundamental interest, geometric phases can be applied to design of quantum logic gates that have an intrinsic resilience against noise perturbations. We note that the conditional phase gates for $n$ atomic qubits can be produced via a single conditional adiabatic displacement of the quantized field. The computational basis for each qubit is formed by the two states $\left| g\right\rangle $ and $\left| f\right\rangle $. If the transition $\left| g\right\rangle \leftrightarrow \left| e\right\rangle $ of each qubit is resonantly coupled to a quantized field and driven by a pump field, the interaction Hamiltonian is $$H_n=\sum_{j=1}^n[\lambda _j(a^{\dagger }\sigma _j^{-}+a\sigma _j^{+})+\Omega _j(e^{-i\phi }\sigma _j^{-}+e^{i\phi }\sigma _j^{+})],$$ where the subscript $j$ labels the qubits. Under the condition $\Omega _1/\lambda _1=\Omega _2/\lambda _2=...=\Omega _n/\lambda _n=r$ the Hamiltonian $H_n$ has dark states of the form $\left| \phi _a\right\rangle \left| -re^{i\phi }\right\rangle $, where $\left| \phi _a\right\rangle $ can be any computational basis state except $\left| f_1f_2...f_n\right\rangle $. If the system is initially in the state $\left| \phi _a\right\rangle \left| -r\right\rangle $, slow variation of the phase $\phi $ from $0$ to $2\pi $ produces a Berry phase $\beta $. On the other hand, the state $\left| f_1f_2...f_n\right\rangle \left| -r\right\rangle $ is completely decoupled from the Hamiltonian $H_n$. As the state $\left| f\right\rangle $ is not coupled to the fields, transitions between degenerate dark states do not occur. Then the evolution of the qubit system proceeds as $\left| \phi _a\right\rangle \rightarrow e^{i\beta }\left| \phi _a\right\rangle $ and $% \left| f_1f_2...f_n\right\rangle \rightarrow e^{i\beta }e^{-i\beta }\left| f_1f_2...f_n\right\rangle $. Discarding the trivial common phase factor $% e^{i\beta }$, this is equivalent to an $n$-qubit controlled phase gate $$U_n=e^{-i\beta \left| f_1f_2...f_n\right\rangle \left\langle f_1f_2...f_n\right| },$$ in which if and only if all the qubits are in the state $\left| f\right\rangle $ the system undergoes a phase shift $-\beta $. This gate is essential to implementation of Grover’s algorithms \[27\] and quantum Fourier transform \[28\]. Though any quantum computational network can be decomposed into a series of two- plus one-qubit logic gates, it may be extremely complex to construct an $n$-qubit phase gate using these elementary gates as the number of required operations increases exponentially with $n$. Thus, direct realization of $n$-qubit phase gates would greatly simplify practical implementation of certain quantum computational tasks. We note that the nonadiabatic geometric means \[23\] can be directly used for implementation of the $n$-qubit entangling gate $U_n^{^{\prime }}=\exp (i\theta J_z^2)$ where $% J_z=\frac 12\sum_{j=1}^n(\left| f_j\right\rangle \left\langle f_j\right| -\left| g_j\right\rangle \left\langle g_j\right| )$, but it does not allow one-step implementation of the phase gate $U_n$. Another important feature of the present gate operation is that it does not require the qubit-resonator coupling strengths $\lambda _j$ to be identical. Furthermore, the conditional phase shift is not affected even if the transition frequencies of the qubits are different because the evolution of the quantized field is not changed when we add to the Hamiltonian $H_n$ the terms $\sum_{j=1}^n\delta _j\sigma _{z,j}$, where $\delta _j$ is detuning between the transition frequency of qubit $j$ and the field frequency. The tolerance to the nonuniformity of the qubits is important for the solid-state implementation of quantum computation since the parameters of artificial atomic qubits are usually not uniform. The geometric phase gate can also be implemented with the quantized field being initially in any state by applying the phase kick to all the atoms at time $T/2$. THE EFFECT OF DISSIPATION ========================= Now let us derive the geometric phase and the visibility of the Ramsey interference fringes with the dissipation being included. The evolution of the system is governed by the master equation $$\stackrel{.}{\rho }=-i[H,\rho ]+{\cal L}_\gamma \rho +{\cal L}_\kappa \rho ,$$ where $$\begin{aligned} {\cal L}_\gamma \rho &=&\frac \gamma 2(2a\rho a^{\dagger }-\rho a^{\dagger }a-a^{\dagger }a\rho ), \nonumber \cr {\cal L}_\kappa \rho &=&\sum_j\sum_k\frac{\kappa _{j,k}}2(2S_{j,k}^{-}\rho S_{j,k}^{+}-\rho S_{j,k}^{+}S_{j,k}^{-}-S_{j,k}^{+}S_{j,k}^{-}\rho ),\\\end{aligned}$$ $S_{j,k}^{+}=\left| j\right\rangle \left\langle k\right| $, $S^{-}=\left| k\right\rangle \left\langle j\right| $, $\gamma $ is the decay rate for the quantized field, and $\kappa _{j,k}$ is the rate for the atomic spontaneous emission $\left| j\right\rangle \rightarrow \left| k\right\rangle $ ($\left| k\right\rangle $ being the levels lower than $\left| j\right\rangle $). The evolution of the system during the infinitesimal interval \[$t$, $t+dt$\] is \[29\] $$\rho (t)\rightarrow \rho (t+dt)=e^{{\cal L}_\kappa dt}e^{{\cal L}_\gamma dt}% {\cal U}(t,dt)\rho (t),$$ where $${\cal U}(t,dt)\rho (t)=U(t,dt)\rho (t)U^{\dagger }(t,dt),$$ with $U(t,dt)$ being the unitary evolution operator governed by the slowly changing Hamiltonian $H$ during \[$t$, $t+dt$\]. In the coherent state basis the action of the superoperator $e^{{\cal L}_\gamma dt}$ on the elements of the density matrix is given by \[30\] $$e^{{\cal L}_\gamma dt}\left| \alpha _1\right\rangle \left\langle \alpha _2\right| =\left\langle \alpha _2\right| \left. \alpha _1\right\rangle ^{\gamma dt}\left| \alpha _1e^{-\gamma dt}\right\rangle \left\langle \alpha _2e^{-\gamma dt}\right| .$$ We here have discarded the atomic part in the density matrix element, which is not affected by $e^{{\cal L}_\gamma dt}$. The action of the superoperator $e^{{\cal L}_\kappa dt}$ is given by $$\begin{aligned} e^{{\cal L}_\kappa dt}\left| g\right\rangle \left\langle g\right| &=&e^{-\kappa _gdt}\left| g\right\rangle \left\langle g\right| +\sum_j\frac{% \kappa _{g,j}}{\kappa _g}(1-e^{-\kappa _gdt})\left| j\right\rangle \left\langle j\right| , \nonumber \\ e^{{\cal L}_\kappa dt}\left| f\right\rangle \left\langle f\right| &=&e^{-\kappa _fdt}\left| f\right\rangle \left\langle f\right| +\sum_k\frac{% \kappa _{f,j}}{\kappa _f}(1-e^{-\kappa _fdt})\left| k\right\rangle \left\langle k\right| , \nonumber \\ e^{{\cal L}_\kappa dt}\left| g\right\rangle \left\langle f\right| &=&e^{-(\kappa _g+\kappa _f)dt/2}\left| g\right\rangle \left\langle f\right| , \nonumber \\ e^{{\cal L}_\kappa dt}\left| f\right\rangle \left\langle g\right| &=&e^{-(\kappa _g+\kappa _f)dt/2}\left| f\right\rangle \left\langle g\right| ,\end{aligned}$$ where $\kappa _g=\sum_j\kappa _{g,j}$, $\kappa _f=\sum_k\kappa _{f,k}$, and $% \left| j\right\rangle $ and $\left| k\right\rangle $ are levels lower than $% \left| g\right\rangle $ and $\left| f\right\rangle $, respectively. The dissipation makes the system deviate from the dark state of the Hamiltonian and acquire a dynamical phase, which can be removed by frequently performing the atomic phase kick $-\sigma _z$ during the course of the evolution of the system ($M$ times with $M\gg 1$). When $M/T\gg \gamma $, $\kappa _e$, and $% \kappa _g$, the dynamical effect is canceled before dissipation can affect it, where $\kappa _e=\sum_j\kappa _{e,j}$. When the dynamical effect is removed the action of the unitary evolution operator on the off-diagonal matrix elements is $$\begin{aligned} &&\ \ \ U(t,dt)\left| g\right\rangle \left\langle f\right| \otimes \left| \alpha _1\right\rangle \left\langle \alpha _2\right| U^{\dagger }(t,dt) \nonumber \\ \ &=&e^{-iIm(\alpha _1^{*}d\alpha _0)}\left| g\right\rangle \left\langle f\right| \otimes \left| \alpha _1+d\alpha _0\right\rangle \left\langle \alpha _2\right| ,\end{aligned}$$ where $\alpha _0=-\Omega e^{i\phi }/\lambda $. Suppose that the Rabi frequency $\Omega $ of the pump field is kept constant and the phase $\phi $ is slowly varied from $0$ to $2\pi $ with the constant angular velocity $\omega =2\pi /T$ during the interval \[$0$, $T$\]. For the initial state $\frac 1{\sqrt{2}}(\left| g\right\rangle +\left| f\right\rangle )\left| -\Omega /\lambda \right\rangle $, the state of the system at time $T$ is $$\begin{aligned} \rho (T) =\frac 12(e^{-\kappa _gT}\left| g\right\rangle \left\langle g\right| \otimes \left| \alpha _g\right\rangle \left\langle \alpha _g\right| +e^{-\kappa _fT}\left| f\right\rangle \left\langle f\right| \cr \otimes \left| \alpha _f\right\rangle \left\langle \alpha _f\right| \ \ +e^{-\Gamma +i\theta }\left| g\right\rangle \left\langle f\right| \otimes \left| \alpha _g\right\rangle \left\langle \alpha _f\right|\cr +e^{-\Gamma -i\theta }\left| f\right\rangle \left\langle g\right| \otimes \left| \alpha _f\right\rangle \left\langle \alpha _g\right| ) \ \ \cr+\frac 12\sum_j\left| j\right\rangle \left\langle j\right| \otimes \int_0^T\kappa _{g,j}e^{-\kappa _gt}\left| \alpha _g^{^{\prime }}\right\rangle \left\langle \alpha _g^{^{\prime }}\right| dt \nonumber \cr \ \ +\frac 12\sum_k\frac{\kappa _{f,k}}{\kappa _f}(1-e^{-\kappa _fT})\left| k\right\rangle \left\langle k\right| \otimes \left| \alpha _f\right\rangle \left\langle \alpha _f\right| ,\\\end{aligned}$$ where $$\begin{aligned} \alpha _g &=&\frac r{\gamma +i\omega }(i\omega +\gamma e^{-\gamma T}), \nonumber \\ \alpha _g^{^{\prime }} &=&\frac r{\gamma +i\omega }(i\omega e^{i\omega t}+\gamma e^{-\gamma t})e^{-(T-t)}, \nonumber \\ \alpha _f &=&re^{-\gamma T}, \nonumber \\ \Gamma &=&(\kappa _g+\kappa _f)T/2+\frac{r^2\omega ^2}{\gamma ^2+\omega ^2}[% \frac 12\gamma T+\frac 14(1-e^{-2\gamma T})], \nonumber \cr \theta &=&-r^2[2\pi -\frac{\omega \gamma ^2T}{\gamma ^2+\omega ^2}-\frac{% \omega \gamma (\omega ^2-2\gamma ^2)}{(\gamma ^2+\omega ^2)^2}(1-e^{-2\gamma T})],\\\end{aligned}$$ $r=\Omega /\lambda $. We here have discarded the events that the atom spontaneously emits two or more photons. After the transformations $\left| g\right\rangle \longrightarrow \frac 1{\sqrt{2}}(\left| g\right\rangle -\left| f\right\rangle )$ and $\left| f\right\rangle \longrightarrow \frac 1{% \sqrt{2}}(\left| f\right\rangle +\left| g\right\rangle )$, the probabilities of finding the atom in the states $\left| g\right\rangle $ and $\left| f\right\rangle $ are given by $$P_{g,f}=\frac 14[e^{-\kappa _gT}+e^{-\kappa _fT}+\frac{\kappa _{g,f}}{\kappa _g}(1-e^{-\kappa _gT})\pm 2\nu \cos \beta ],$$ where $$\begin{aligned} \nu &=&\exp [-\Gamma +\frac{r^2\omega ^2}{\gamma ^2+\omega ^2}(e^{-\gamma T}-% \frac 12e^{-2\gamma T}-\frac 12)], \nonumber \\ \beta &=&\theta -\frac{r^2\omega \gamma }{\gamma ^2+\omega ^2}e^{-\gamma T}.\end{aligned}$$ We here have assumed that the level $\left| f\right\rangle $ is lower than $% \left| g\right\rangle $. Under the condition $\kappa _g$, $\kappa _f$, $% \gamma \ll 1/T$, to first order correction the visibility of the interference fringes and geometric phase are approximately $\nu \simeq 1-r^2\gamma T-\kappa _{g,f}T/2$ and $\beta \simeq -\left| \alpha _0\right| ^2(2\pi +\gamma /\omega )$. The spontaneous emissions $\left| g\right\rangle \rightarrow \left| j\right\rangle $ ($j\neq f$) and $\left| f\right\rangle \rightarrow \left| k\right\rangle $ decrease the probabilities of finding the atom in the states $\left| g\right\rangle $ and $\left| f\right\rangle $, but do not affect the visibility of the interference fringes since the sum of the diagonal matrix elements and the sum of the off-diagonal matrix elements of the reduced density oprator of the atom are shrunk by the same factor due to these spontaneous emissions. It should be noted that the second order spontaneous emissions $\left| j\right\rangle \rightarrow \left| k\right\rangle $ following $\left| g\right\rangle \rightarrow \left| j\right\rangle $ ($j\neq f$) or $\left| f\right\rangle \rightarrow \left| j\right\rangle $ affect neither the probabilities of finding the atom in the states $\left| g\right\rangle $ and $\left| f\right\rangle $ nor the visibility of the interference fringes. Due to the second order spontaneous emission $\left| f\right\rangle \rightarrow \left| k\right\rangle $ following $\left| g\right\rangle \rightarrow \left| f\right\rangle $ the correction to the visibility of the interference fringes is on the order of $% \kappa _{g,f}\kappa _fT^2$. THE EFFECT OF FLUCTUATIONS IN THE PARAMETERS OF THE PUMP FIELD ============================================================== We now analyze the effect of the fluctuations in the parameters of the pump field on the Berry phase and atomic coherence. Set the fluctuations of the Rabi frequency and phase of the pump field to be $\delta \Omega (t)$ and $% \delta \phi (t)$. Due to the presence of the fluctuation noise, the dark state of the system is $\left| g\right\rangle \left| \alpha (t)\right\rangle $, where $\alpha (t)=-[r_0(t)+\delta r(t)]e^{i\phi (t)}$, $\phi (t)=\phi _0(t)+\delta \phi (t)$, $r_0(t)=\Omega _0(t)/\lambda $, and $\delta r(t)=\delta \Omega (t)/\lambda $. Here $\Omega _0(t)$ and $\phi _0(t)$ are the Rabi frequency and phase of the pump field in the absence of noise. If the system is initially in the dark state $\left| g\right\rangle \left| \alpha (0)\right\rangle $ and the adiabatic condition is satisfied the final state is $e^{i\theta (T)}\left| g\right\rangle \left| \alpha (T)\right\rangle $, where $\theta (T)=-\int_{\phi (0)}^{\phi (T)}(r_0+\delta r)^2d\phi $. To the first order correction the geometric phase is $$\beta \simeq \beta _0-\int_0^T2r_0\delta r\stackrel{\cdot }{\phi }% _0dt-\int_0^Tr_0^2\delta \stackrel{\cdot }{\phi }dt+r_0^2(0)\sin \delta \phi (T),$$ where $\beta _0$ is the Berry phase in the absence of noise. We here have assumed $\delta r(0)=0$ and $\delta \phi (0)=0$. Without loss of the generality, we again consider the case that $\Omega _0$ is kept constant and $\phi _0$ undergoes an adiabatic cyclic evolution from $0$ to $2\pi $ with the constant change rate $\stackrel{\cdot }{\phi }_0=2\pi /T$. Then the expression reduces to $\beta \simeq \beta _0-\frac{4\pi }Tr_0\int_0^T\delta rdt$. We here assume that the fluctuations are Gaussian and Markovian processes with the bandwidth $\Gamma _\Omega $ ($\Gamma _\phi $) and intensity $\sigma _\Omega ^2$ ($\sigma _\phi ^2$) for $\delta \Omega $ ($% \delta \phi $). Under the condition $\Gamma _\Omega $, $\Gamma _\phi \ll \lambda $, the pump field fluctuations are adiabatic with respect to the Rabi frequency of the dressed atom-field system \[7\]. Consequently, the geometric phase is Gaussian distributed with the mean value equal to the noiseless Berry phase. Its variance is given by $$\sigma _\beta ^2=16\pi \left| \beta _0\right| \frac{\sigma _\Omega ^2}{% (\lambda \Gamma _\Omega T)^2}(\Gamma _\Omega T-1+e^{-\Gamma _\Omega T}).$$ In the limit of the evolution being much slower than the fluctuation of the Rabi frequency ($\Gamma _\Omega T\gg 1$), the variance approximates $\sigma _\beta ^2=16\pi \left| \beta _0\right| \sigma _\Omega ^2/(\lambda ^2\Gamma _\Omega T)$, which vanishes in the limit $(\Gamma _\Omega T)^{-1}\rightarrow 0$. When $\Gamma _\Omega T\ll 1$, the variance reduces to $\sigma _\beta ^2=8\pi \left| \beta _0\right| \sigma _\Omega ^2/\lambda ^2$. This implies that the geometric phase is sensitive to slow fluctuations, with the variance being path-dependent like the geometric phase itself, coinciding with the geometric dephasing for a spin 1/2 in a slowly changing magnetic field \[7\]. As long as the adiabatic condition is satisfied, the fluctuation does not induce any correction to the dynamical phase as the spectrum of the Hamiltonian does not depend on the parameters of the pump field. Thus the dephasing of this system is different from that of the spin 1/2 to which the main contribution has the dynamical origin \[7\]. Let us consider the effect of fluctuations in the classical control parameters on the atomic Ramsey interference. In the presence of the noise, after the conditional displacement evolution the system is in a mixed state, whose density operator is given by $$\rho =\int \left| \psi \right\rangle \left\langle \psi \right| P(\beta )P(\delta r(T))P(\delta \phi (T))d\beta d\delta r(T)d\delta \phi (T),$$ where $$\left| \psi \right\rangle =\frac 1{\sqrt{2}}[e^{i\theta (T)}\left| g\right\rangle \left| \alpha (T)\right\rangle +\left| f\right\rangle \left| \alpha (0)\right\rangle )],$$ $P(\beta )$, $P(\delta r(T))$, and $P(\delta \phi (T))$ are Gaussian distribution functions for $\beta $, $\delta r(T)$, and $\delta \phi (T)$, respectively. The coherence between the two atomic states is shrunk by a factor $F=e^{-\sigma _\beta ^2/2-\sigma _\Omega ^2/(2\lambda ^2)-\left| \beta _0\right| \sigma _\phi ^2/(4\pi )}$ due to random distributions in the Berry phase and the noncyclic evolution of the field state correlated with the atomic state $\left| g\right\rangle $. Now we analyze how the classical noises affect the fidelity of the $n$-qubit quantum gates. Set the initial state of the qubit system to be $c_1\left| \psi _a\right\rangle +c_2$ $\left| f_1f_2...f_n\right\rangle $, where $% \left| c_1\right| ^2+\left| c_2\right| ^2=1$ and $\left| \psi _a\right\rangle $ can be any normalized superposition of the computational basis states except $\left| f_1f_2...f_n\right\rangle $. Due to the fluctuation perturbation the density operator of the whole system again has the form of Eq. (19), where $$\left| \psi \right\rangle =c_1e^{i\theta (T)}\left| \psi _a\right\rangle \left| \alpha (T)\right\rangle +c_2\left| f_1f_2...f_n\right\rangle \left| \alpha (0)\right\rangle .$$ The infidelity caused by the fluctuations is $\epsilon =\left| c_1c_2\right| ^2(1-F)$. The result is valid even if these qubits are entangled with other qubits not invovled in the gate operation. DISCUSSION AND CONCLUSION ========================= For potential physical implementation of the model, we consider microwave cavity QED experiments with circular Rydberg atoms and a superconducting millimeter-wave cavity, which has a remarkably long damping time $T_c=0.13$ s \[31\]. The states $\left| f\right\rangle $, $\left| g\right\rangle $, and $% \left| e\right\rangle $ are the circular states with principal quantum numbers 49, 50, and 51, respectively. The corresponding atomic radiative time is about $T_r=3\times 10^{-2}s$. The transition $\left| g\right\rangle \longleftrightarrow \left| e\right\rangle $ is strongly coupled to the cavity mode with the coupling strength $\lambda =2\pi \times 25$ kHz. The pump field can be provided by a classical microwave source. The adiabatic approximation is valid under the condition that the time scale $T$ of the variation of the control parameter is longer than the inverse of the energy gap $\delta E=\lambda $ between the dark state and the nearest bright states with nonzero eigenenergies, i.e., the dynamical time scale. If we set $% T=20/\lambda $, then the leakage error to the excited eigenstates is on the order of $1/(\lambda T)^2=2.5\times 10^{-3}$. Suppose that the Rabi frequency $\Omega $ of the pump field is kept constant and the phase $\phi $ is slowly varied from $0$ to $2\pi $. Then due to dissipation the visibility of interference fringes is approximately reduced by $(\Omega /\lambda )^2T/T_c+\kappa _{g,f}T/2$, which is on the order of $10^{-3}$. The correction to the geometric phase is $-2\pi (\Omega /\lambda )^2T/T_c$, also on the order of $10^{-3}$. Therefore, the adiabatic condition can be perfectly satisfied and the influence of decoherence is negligible. We note that the atomic phase kick has been experimentally achieved by applying a fast electric field pulse \[32\], which allows the measurement of the geometric phase for the cavity field even if the system is not in the dark state. In experiment, it may be more convenient to keep the phase of the pump field unchanged, but detune its frequency from the frequency of the quantized field by an amount $\delta $ with $\delta \ll \lambda $. Then $% t\delta $ takes the role of the slowly varying phase $\phi $. An alternative physical system to implement the required Hamiltonian is the circuit QED setup, in which superconducting qubits are strongly coupled to the resonator field. In fact, the Hamiltonian of Eq. (1) has been experimentally realized using a phase qubit coupled to a superconducting coplanar waveguide resonator and driven by an external microwave pulse \[33\]. The above ideas can be directly applied to the ion trap system. Consider an ion trapped in a harmonic potential and driven by two laser beams tuned to the carrier and the first lower vibrational sideband with respect to the electronic transition $\left| g\right\rangle \rightarrow \left| e\right\rangle $ and the vibrational mode. In the Lamb-Dicke limit, the coupling between the internal and external degrees of freedom of the trapped ions is described by the Hamiltonian (1), with the Rabi frequency and phase of the laser for the carrier excitation serving as the control parameters. The Berry phase acquired by the vibrational mode after an adiabatic displacement evolution in phase space is directly manifested in the Ramsey interference between $\left| g\right\rangle $ and an auxiliary state $\left| f\right\rangle $ decoupled from the Hamiltonian. It is worthwhile to note that a scheme has been proposed for measuring the geometric phase in the vibrational mode of a trapped ion by adiabatically varying the squeezing parameter in the engineered squeezing operator \[34\]. However, the scheme requires squeezing transformations with opposite squeezing parameters before and after the adiabatic evolution, discrimination between two nonorthogonal coherent states, and techniques to cancel the accumulated dynamical phase. We have shown how to produce and observe the nonclassical Berry phase of an electromagnetic field in a generic qubit-boson system involving a qubit coupled to a quantized field and driven by a classical pump field. The adiabatic variation of the parameters of the pump field forces the quantized field mode to displace along a loop in phase space, producing a purely geometric phase. The origin of the geometric phase is the quantum nature of the field mode without any classical counterpart. When the system is initially in the dark state, the geometric phase can be directly detected using the atomic Ramsey interferometer. Otherwise, one should apply a phase kick to the atom to cancel dynamical contributions. Besides fundamental interest, geometric manipulation of the quantized field opens new possibilities for robust implementation of important quantum phase gates in a single step. The effects of both the quantum and classical noises on the Berry phase and visibility of the Ramsey interference fringes are analyzed. Note added. Since completion of this work, two preprints by Vacanti et al. \[35\] and Pechal et al. \[36\] investigating the measurement of the geometric phase of a quantum harmonic oscillator using the interference of a qubit have appeared. The first one proposed a scheme to measure the nonadiabatic geometric phase produced by a coherent displacement force. 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--- abstract: 'Consider a $q$-Weil polynomial $f$ of degree $2g$. Using an equidistribution assumption that is too strong to be true, we define and compute a product of local relative densities of matrices in $\operatorname{GSp}_{2g}({\mathbb F_\ell})$ with characteristic polynomial $f\mod\ell$ when $g$ is an odd prime. This infinite product is closely related to a ratio of class numbers. When $g=3$ we conjecture that the product gives the size of an isogeny class of principally polarized abelian threefolds.' address: - 'James Madison University, Harrisonburg, VA 22807' - 'James Madison University, Harrisonburg, VA 22807' author: - Jonathan Gerhard - Cassandra Williams bibliography: - 'bibliography.bib' title: Local heuristics and an exact formula for abelian varieties of odd prime dimension over finite fields --- Introduction ============ This paper is a direct generalization of the work of Achter-Williams [@achterwilliams2015] to abelian varieties of odd prime dimension, and is guided in philosophy by both Gekeler [@gekeler03] and Katz [@katz-lt]. We begin by considering abelian varieties over a finite field ${\mathbb F_q}$, where $q$ is a power of a prime. To each such variety $X$, we can associate a characteristic polynomial of Frobenius $f_X(T) \in {\mathbb Z}[T]$. A theorem of Tate [@tate66] tells us that two varieties are isogenous if and only if their characteristic polynomials are equal. Let ${\mathcal A}_g(k)$ denote the moduli space of principally polarized abelian varieties of dimension $g$ over a field $k$, where each variety is weighted by the size of its automorphism group. Also define ${\mathcal A}_g({\mathbb F_q}; f)$ to be those members of ${\mathcal A}_g({\mathbb F_q})$ with characteristic polynomial $f(T)$. Then ${\mathcal A}_g({\mathbb F_q}; f)$ denotes the set of isomorphism classes of principally polarized abelian varieties of dimension $g$ over ${\mathbb F_q}$ with characteristic polynomial of Frobenius $f$, weighted inversely by the size of the automorphism group, and computing $\#{\mathcal A}_g({\mathbb F_q};f)$ gives the number of (isomorphism classes of) abelian varieties over ${\mathbb F_q}$ in a particular isogeny class. The Frobenius endomorphism gives an automorphism of the Tate module $T_{\ell}X$ (for $\ell$ not dividing $q$), so it has a representation as an element of the matrix group $\operatorname{GSp}_{2g}({\mathbb F_\ell})$. For each $\ell$, we define a term $\nu_{\ell}(f)$ measuring the relative frequency of $f \mod \ell$ as the characteristic polynomial for an element of $\operatorname{GSp}_{2g}({\mathbb F_\ell})$, as well as an archimedean term $\nu_{\infty}(f)$. Since (as much as possible) Frobenius elements are equidistributed in $\operatorname{GSp}_{2g}({\mathbb F_\ell})$, it seems possible that the product of these local densities could at least estimate the value of $\#{\mathcal A}_g({\mathbb F_q};f)$. This, of course, is a ridiculous tactic, as the $\bmod$ $\ell$ Frobenius elements are only equidistributed when $q\gg_g\ell$. However, as in [@achterwilliams2015], we again show that this local data does apparently control isogeny class size. Our main result is as follows. Consider a particular class of $q$-Weil polynomials $f$ of degree $2g$ (see the next section for details). Let $K$ be the splitting field of $f$ over ${\mathbb Q}$ and $K^+$ be its maximal totally real subfield, with class numbers $h_K$ and $h_{K^+}$ respectively. A theorem of Everett Howe in the preprint [@howepreprint], together with the conditions stated in the next section, implies that $$\#{\mathcal A}_{g}({\mathbb F_q}; f)=\frac{h_K}{h_{K^+}}$$ (see Theorem \[Thm:Howe\] and Corollary \[CorToHowe\]). Then an immediate corollary of this theorem and our work is that $$\nu_{\infty}(f)\prod_{\ell}\nu_{\ell}(f) = \#{\mathcal A}_{g}({\mathbb F_q}; f)$$ when $g = 3$ (see Corollary \[Cor:g3UseHowe\]). (For $g>3$ an odd prime, there is only a minimal obstruction to this corollary which is explained at the end of section \[SectionCentralizerOrders\] and in Remark \[RemarkNonSSWillMatch\].) Abelian varieties and Weil polynomials {#Sec:WeilPolys} ====================================== Let $X/{\mathbb F}_q$ be an abelian variety of dimension $g$ over a finite field of $q=p^a$ elements, and let $f(T)\in{\mathbb Z}[T]$ be the characteristic polynomial of its Frobenius endomorphism. Then $f(T)$ is a $q$-Weil polynomial, a polynomial of degree $2g$ with complex roots $\alpha_1,\dots,\alpha_{2g}$ with ${\left\vert\alpha_j\right\vert}=\sqrt{q}$ for every $j$, where the ordering can be chosen so that $\alpha_j\alpha_{g+j}=q$ for $1\leq j \leq g$. Each $q$-Weil polynomial $f(T)$ corresponds to a (possibly empty) isogeny class $\mathcal{I}_f$ of abelian varieties of dimension $g$ over ${\mathbb F}_q$. Following [@achterwilliams2015], we will assume: 1. ([*ordinary*]{}) \[enord\] the middle coefficient of $f$ is relatively prime to $p$; 2. ([*principally polarizable*]{}) \[enpp\] there exists a principally polarized abelian variety of dimension $g$ with characteristic polynomial $f$; 3. ([*cyclic*]{}) \[encyclic\] the polynomial $f(T)$ is irreducible over ${\mathbb Q}$, and $K_f := {\mathbb Q}[T]/f(T)$ is Galois, cyclic, and unramified at $p$; 4. ([*maximal*]{}) \[enmax\] for $\pi_f$ a (complex) root of $f(T)$, with complex conjugate $\bar\pi_f$, then ${\mathcal{O}}_f := {\mathbb Z}[\pi_f,\bar\pi_f]$, [*a priori*]{} an order in $K_f$, is actually the maximal order ${\mathcal{O}}_{K_f}$. Assumptions , , and are identical to those in [@achterwilliams2015]. The condition is similar; we want to assume $K_f$ is abelian and Galois. In [@achterwilliams2015 (W.3)], the authors only assumed that $K_f$ is Galois, but as $K_f$ was a number field of degree 4 this also guaranteed it to be abelian. Many of the results proven in the present work require only that $K_f$ is abelian and Galois; however, we will assume that $g$ is an odd prime in many of our major results, and thus $K_f$ will be cyclic (with $\operatorname{Gal}(K_f/{\mathbb Q})\cong {\mathbb Z}/2g{\mathbb Z}$). It should be noted that is in fact a serious restriction, as number fields $K_f$ with a cyclic Galois group are quite rare among all number fields of degree $2g$. Our method does work in broader contexts; for example, in [@rauchthesis] the author proves analogous results to those in [@achterwilliams2015] and our own for degree 4 fields with a nonabelian Galois group. It seems likely that our methods readily generalize to any abelian Galois extension $K_f$, at the cost of more elaborate and extensive computations as the factorization of $g$ becomes more complex. Thus, in this work, we will restrict to cyclic Galois groups in the hope of demonstrating our method and heuristic in the simplest generalized situation, rather than distracting the reader with details that provide no new insight into the problem. Note that $K_f$ is a CM field, and as such it comes equipped with an intrinsic complex conjugation $\iota\in\operatorname{Gal}(K_f/{\mathbb Q})$. Also, the isomorphism class of ${\mathcal{O}}_f$ (as an abstract order) is independent of the choice of $\pi_f$. The polynomial $f(T)=T^6 + 10 T^5 + 48 T^4 + 151 T^3 + 336 T^2 + 490 T + 343$ is a 7-Weil polynomial that meets all of the assumptions - when $g=3$. The Weil polynomial $f(T)$ factors as $f(T)=\prod_{j = 1}^{g} (T - \sqrt{q}e^{i\theta_j})(T - \sqrt{q}e^{-i\theta_j})$; then under our assumptions the polynomial $f^+(T)= \prod_{j = 1}^{g} (T - 2\sqrt{q}\cos(\theta_j))$ is the minimal polynomial of $\pi_f+\bar\pi_f$ and $K^+_f={\mathbb Q}[T]/f^+(T)$ is the maximal totally real subfield of $K_f$. Note that ${\mathbb Z}[\pi_f]\cong {\mathbb Z}[T]/f(T) \subset {\mathcal{O}}_f$ and define the conductor of $f$, $\operatorname{cond}(f)$ as the index $[{\mathcal{O}}_f:{\mathbb Z}[\pi_f]]$. We will denote the discriminants of the polynomials $f$ and $f^+$ as $\operatorname{disc}(f)$ and $\operatorname{disc}(f^+)$, respectively, while $\Delta_{\mathcal{O}}$ will represent the discriminant of an order ${\mathcal{O}}$. Note that $\Delta_{{\mathbb Z}[\pi_f]}=\operatorname{disc}(f)$ and $\Delta_{{\mathcal{O}}_{K^+_f}}=\operatorname{disc}(f^+)$. In the following technical lemma, we give explicit forms for $\operatorname{disc}(f)$ and $\operatorname{disc}(f^+)$ for any positive integer $g$. \[discs\] Let $f$ be a $q$-Weil polynomial of degree $2g$ with $g \geq 1$. Then 1. $\begin{displaystyle}\operatorname{disc}(f) = (-1)^g2^{2g^2}q^{2g^2 - g}\left(\prod_{j = 1}^g \sin^2(\theta_j)\right)\left(\prod_{1 \leq k < t \leq g} (\cos(\theta_k) - \cos(\theta_t))^2\right)^2 \end{displaystyle}$ and 2. $\begin{displaystyle} \operatorname{disc}(f^+) = 2^{g(g-1)}q^{\frac{g(g-1)}{2}}\prod_{1 \leq k < t \leq g} (\cos(\theta_k) - \cos(\theta_t))^2 \end{displaystyle}.$ Recall that the roots of $f(T)$ are of the form $\sqrt{q} e^{\pm i\theta_j}$ for $1\leq j\leq g$. Then the proof of part (1) of the lemma proceeds by induction on $g$ using elementary methods and is omitted here. A direct computation of the discriminant of $f^+$, which has roots $\sqrt{q}e^{i\theta_j} + \sqrt{q} e^{-i\theta_j}=2\sqrt{q}\cos(\theta_j)$ for $1\leq j \leq g$, proves part (2). See [@howepreprint Theorem 4.3] for a similar computation relating the Frobenius angles $\theta_i$ to (in our notation) $$\sqrt{\Delta_{{\mathcal{O}}_{K_f}}/\Delta_{{\mathcal{O}}_{K_f^+}}}.$$ The explicit forms of Lemma \[discs\] will be helpful in proving the following lemma, as well as for defining local factors in section \[Sec:LocalFactorsf\]. \[lemconductor\] The index of ${\mathbb Z}[\pi_f]$ in ${\mathcal{O}}_f$ is $q^{\frac{g(g-1)}{2}}$. From [@howe95], we have $\Delta_{{\mathcal{O}}_f} = (-1)^g\operatorname{disc}(f^+)^2 \, N_{K_f/{\mathbb Q}}(\pi_f - \bar{\pi}_f)$ and $\operatorname{disc}(f) = \operatorname{cond}(f)^2 \, \Delta_{{\mathcal{O}}_f}.$ Then $$\operatorname{cond}(f)^2 = (-1)^g\frac{\operatorname{disc}(f)}{\operatorname{disc}(f^+)^2 N_{K_f/{\mathbb Q}}(\pi_f - \bar{\pi}_f)}.$$ Without loss of generality, choose $\pi_f = \sqrt{q}e^{i\theta_1}$. Then $\pi_f - \bar{\pi}_f = 2i\sqrt{q}\sin(\theta_1)$ and all of its Galois conjugates are of the form $\pm 2i\sqrt{q}\sin(\theta_j)$ for $j \in \{1,2, \dots, g\}$. Therefore, $$N_{K_f/{\mathbb Q}}(\pi_f - \bar{\pi}_f) = \prod_{j = 1}^g (2i\sqrt{q}\sin(\theta_j))(-2i\sqrt{q}\sin(\theta_j)) = 4^g q^{g} \prod_{j=1}^g \sin^2(\theta_j).$$ Applying $\operatorname{disc}(f)$ and $\operatorname{disc}(f^+)$ from Lemma \[discs\], we find $\operatorname{cond}(f)^2 = q^{g(g-1)}$, and the lemma follows. \[lemalmostmaximal\] If $\ell\not = p$, then ${\mathcal{O}}_{K_f}\otimes {\mathbb Z}_{(\ell)} \cong {\mathbb Z}_{(\ell)}[T]/f(T)$. Corollary \[lemalmostmaximal\] is proved, independent of the dimension of the abelian variety, in [@achterwilliams2015] and so its proof is omitted here. Lastly, we prove that ${\mathbb Z}[T]/f^+(T)$ is the maximal order of $K_f^+$. \[lemmaximalrealorder\] The order ${\mathbb Z}[T]/f^+(T)$ is the maximal order ${\mathcal{O}}_{K_f^+}$. Condition implies that ${\mathcal{O}}_f\cap K_f^+={\mathcal{O}}_{K_f}\cap K_f^+={\mathcal{O}}_{K_f^+}$. Certainly ${\mathbb Z}[T]/f^+(T)={\mathbb Z}[\pi_f+\bar\pi_f] \subseteq {\mathcal{O}}_f\cap K_f^+$. Let $\alpha\in K_f^+={\mathbb Q}[T]/f^+(T)={\mathbb Q}(\pi_f+\bar\pi_f)$, which has dimension $g$ over ${\mathbb Q}$. Then $$\alpha=a_0+a_1(\pi_f+\bar\pi_f) + a_2(\pi_f+\bar\pi_f)^2+\cdots+a_{g-1}(\pi_f+\bar\pi_f)^{g-1}$$ with all $a_i\in{\mathbb Q}$. Suppose $\alpha$ is also an element of ${\mathcal{O}}_f$, so $\alpha\in{\mathcal{O}}_f\cap K^+_f$. It is straightforward to show that the set $${\left\{1, \pi_f, \pi_f^2,\ldots,\pi_f^g,\bar\pi_f,\bar\pi_f^2,\ldots,\bar\pi_f^{g-1}\right\}}$$ forms a basis for ${\mathcal{O}}_f={\mathbb Z}[\pi_f,\bar\pi_f]$. Recall that $\pi_f\bar\pi_f=q$; expand $\alpha$ and collect powers of $\pi_f$ and $\bar\pi_f$. Notice that the coefficient of $\pi_f^{g-1}$ in $\alpha$ is exactly $a_{g-1}$, and so $a_{g-1}\in{\mathbb Z}$. Using back substitution on the coefficients of powers of $\pi_f$ and $\bar\pi_f$, we find that $a_{g-2}\in{\mathbb Z}$, $a_{g-3}\in{\mathbb Z}$, and so on. Therefore, all $a_i\in{\mathbb Z}$ and $\alpha\in{\mathcal{O}}_f\cap K_f^+$ is such that $\alpha\in{\mathbb Z}[\pi_f,\bar\pi_f]$ and ${\mathcal{O}}_f\cap K_f^+\subseteq {\mathbb Z}[\pi_f+\bar\pi_f].$ Therefore, ${\mathbb Z}[\pi_f+\bar\pi_f]={\mathcal{O}}_{K_f^+}$. Conjugacy classes in $\operatorname{GSp}_{2g}({\mathbb F}_\ell)$ {#Sec:cclasses} ================================================================ Symplectic groups and conjugacy ------------------------------- The symplectic group $\operatorname{GSp}_{2g}({\mathbb F_\ell})$ is the subgroup of $\operatorname{GL}_{2g}({\mathbb F_\ell})$ preserving an antisymmetric bilinear form $J$ up to a scalar multiple. We choose $$J = \begin{bmatrix} 0 & {\rm I}_g \\ -{\rm I}_g & 0 \end{bmatrix}$$ but note that different choices of $J$ produce isomorphic copies of $\operatorname{GSp}_{2g}({\mathbb F_\ell})$. Explicitly, $$\operatorname{GSp}_{2g}({\mathbb F_\ell}) = \{ M \in \operatorname{GL}_{2g}({\mathbb F_\ell}) \mid MJM^T = mJ \text{ for some $m \in {\mathbb F_\ell}^{\times}$}\}.$$ The value $m$ is called the *multiplier* of $M$. All matrices in $\operatorname{GSp}_{2g}({\mathbb F_\ell})$ have the property that there exists a pairing (dictated by the choice of antisymmetric bilinear form) of its eigenvalues such that each pair has product $m$. In [@shinoda80 Theorem 1.18], Shinoda parametrizes the set of conjugacy classes of $\operatorname{GSp}_{2g}({\mathbb F_\ell})$. For our purposes, we do not need their parametrization in full generality, and will describe the relevant portions in our own notation. Let $$f(T) = T^d + c_{d-1}T^{d-1} + \dots + c_1T + c_0$$ be a polynomial in ${\mathbb F_\ell}[T]$. Define the *dual* of $f(T)$ with respect to the multiplier $m$ to be $$\bar{f}^m(T) = \frac{T^d}{c_0}f(mT^{-1}).$$ (We will occasionally omit the $m$ on the left hand side for notational convenience.) Then we have three types of polynomials: 1. *Root polynomials* are polynomials of the form $f(T) = T^2 - m$ when $m$ is not a square, or either of $f(T) = T \pm \sqrt{m}$ when $m$ is a square. (It is easy to check that all root polynomials satisfy $\bar{f}^m(T)=f(T).$) 2. *$\alpha$ pairs* are pairs of polynomials $(f(T), \bar{f}^m(T))$ such that $ \bar{f}^m(T)\neq f(T)$. 3. *$\beta$ polynomials* are polynomials $f(T)$ such that $\bar{f}^m(T)=f(T)$ and $f(T)$ is not a root polynomial. Note that any linear polynomial satisfying $\bar{f}^m(T)=f(T)$ is a root polynomial. One consequence of these definitions is as follows. \[NoOddBetas\] There are no irreducible $\beta$ polynomials of odd degree. That is, for all irreducible nonlinear polynomials $f(T)$ of odd degree and all multipliers $m\in{\mathbb F_q}^\times$, $$\bar{f}^m(T)\neq f(T).$$ Let $f(T) = T^d + c_{d-1}T^{d-1} + \dots + c_1T + c_0$ be an irreducible polynomial with $d\geq 3$ odd and suppose $\bar{f}^m(T)=f(T)$. This implies $c_0^2 = m^d$, but if $m$ is non-square then $m^d$ is non-square since $d$ is odd, which is a contradiction. If instead $m$ is a square, then $-\sqrt{m} \in {\mathbb F_\ell}$ is a root of $f(T)$, contradicting the fact that $f(T)$ is irreducible. The general theory of conjugacy classes in $\operatorname{GL}_n$ as well as the additional intricacies of conjugacy in $\operatorname{GSp}_n$ are given (briefly) in [@achterwilliams2015 Section 3.2], and the reader is encouraged to revisit this section if needed. For our current purposes, recall that to each irreducible factor of the characteristic polynomial of a matrix, we associate a partition of its multiplicity; the characteristic polynomial together with its partition data determine conjugacy in $\operatorname{GL}_n$. We also remind the reader that *cyclic* matrices are those for which the characteristic and minimal polynomials coincide, and thus are those where all partitions are maximal (consist of only one part). Then characteristic polynomials for conjugacy classes of cyclic matrices in $\operatorname{GSp}_{2g}({\mathbb F_\ell})$ are constructed as follows. \[ConjClassCharacterization\] The following characteristic polynomials uniquely determine a conjugacy class of cyclic matrices in $\operatorname{GSp}_{2g}({\mathbb F_\ell})$, where, for each $m$, the first product is over all $\alpha$ pairs and the second product is over all $\beta$ polynomials. 1. For square $m \in {\mathbb F_\ell}^{\times}$, $$f(T) = (T - \sqrt{m}\,)^{e_{R_1}}(T + \sqrt{m}\,)^{e_{R_2}}\prod_{\alpha}(f_\alpha(T)\bar{f_\alpha}(T))^{e_\alpha} \prod_{\beta} (f_\beta(T))^{e_\beta}$$ for a choice of $e_{R_1}, e_{R_2}, e_\alpha,$ and $e_\beta$ such that any odd parts in the partitions of $e_{R_1}$ and $e_{R_2}$ have even multiplicity and $$e_{R_1} + e_{R_2} + 2\sum_{\alpha}\deg(f_\alpha)e_\alpha + \sum_\beta \deg(f_\beta)e_\beta = 2g.$$ 2. For non-square $m \in {\mathbb F_\ell}^{\times}$, $$f(T) = (T^2 - m)^{e_{R}}\prod_{\alpha}(f_\alpha(T)\bar{f_\alpha}(T))^{e_\alpha} \prod_{\beta} (f_\beta(T))^{e_\beta}$$ for a choice of $e_R, e_\alpha,$ and $e_\beta$ such that any odd parts in the partition of $e_{R}$ have even multiplicity and $$2e_R + 2\sum_{\alpha}\deg(f_\alpha)e_\alpha + \sum_\beta \deg(f_\beta)e_\beta = 2g.$$ For each exponent $e_k$ defined above, the only allowable partition is $[e_k]$. Shinoda’s parameterization ([@shinoda80 Theorem 1.18]) also includes sets of (nondegenerate symmetric) bilinear forms with ranks relating to the number of parts of even sizes in the partitions of any root polynomials present in the factorization of the characteristic polynomial. The number of equivalence classes of these bilinear forms detect when a characteristic polynomial (and associated set of partitions) gives rise to two conjugacy classes in $\operatorname{GSp}_{2g}({\mathbb F}_\ell)$, indexed by ${\left\{+,-\right\}}$. We omit this component of Shinoda’s parameterization because, when $g$ is odd, all of the characteristic polynomials that we will declare as *relevant* in the next section give rise to only one conjugacy class. Relevant conjugacy classes of $\operatorname{GSp}_{2g}({\mathbb F}_\ell)$ {#conjclasses} ------------------------------------------------------------------------- For the remainder of section \[Sec:cclasses\], let $g$ be an odd prime. We want to identify the possible shapes (factorization structures) of characteristic polynomials which correspond to conjugacy classes of cyclic matrices in $\operatorname{GSp}_{2g}({\mathbb F_\ell})$ which respect the assumptions -. Therefore, a *relevant* characteristic polynomial is one such that - the factorization is as in Theorem \[ConjClassCharacterization\], - the degrees of all irreducible factors are equal, and - the multiplicities of each irreducible factor are equal. The first condition forces our matrices to be cyclic elements of $\operatorname{GSp}_{2g}({\mathbb F_\ell})$, and the other two correspond with the requirement that $K_f$ be Galois from . Let $[d]$ denote a monic irreducible degree $d$ polynomial in ${\mathbb F_\ell}[T]$, and assume that $[d]_i\neq[d]_j$ if $i\neq j$. Then the following are the shapes of all relevant characteristic polynomials. ---------------------- ------------------- $[1]_1\dots[1]_{2g}$ $[1]^{2g}$ $[2]_1\dots[2]_g$ $[1]_1^g[1]_2^g $ $[g]_1[g]_2$ $[2]^g$ $[2g]$ ---------------------- ------------------- We exclude the shape $[g]^2$ by Lemma \[NoOddBetas\]. Additionally, we exclude $[1]_1^2\dots[1]_g^2$; since there are an odd number of factors, it must be that one of these linear factors is a root polynomial while the rest are $\alpha$ pairs. Then the Galois group of $K_f$ cannot act transitively on the roots of such an $f$. The seven relevant conjugacy class shapes fall into two categories: regular semisimple and non-semisimple. A regular semisimple conjugacy class contains elements with a squarefree characteristic polynomial (and thus are cyclic by definition). A class is not semisimple when the characteristic polynomial of its elements is not squarefree. We list the relevant conjugacy classes by the shape of their characteristic polynomial in Tables \[Tab:RegSSCharPolShape\] (regular semisimple) and \[Tab:NonSSCharPolShape\] (non-semisimple). Each table also contains information about the multiplier $m$ and the type of the irreducible factors. [\*3c]{} Char. Pol. Shape & Valid $m$ & Polynomial type\ $[1]_1\dots[1]_{2g}$ & All $m$ & $\alpha$ pairs\ $[2]_1\dots[2]_g $ & All $m$ & $\beta$ polynomials\ $[g]_1[g]_2$ & All $m$ & $\alpha$ pair\ $[2g]$ & All $m$ & $\beta$ polynomial\ [\*3c]{} Char. Pol. Shape & Valid $m$ & Polynomial type\ $[1]^{2g}$ & Square $m$ & Root polynomial\ $[1]_1^g[1]_2^g $ & All $m$ & $\alpha$ pair\ $[2]^g$ & All $m$ & $\beta$ polynomial\ Centralizer orders {#SectionCentralizerOrders} ------------------ Denote by ${\mathcal{C}}$ a conjugacy class of matrices in $\operatorname{GSp}_{2g}({\mathbb F}_\ell)$ with characteristic polynomial $f_{\mathcal{C}}(T)$. For each relevant conjugacy class, we will find its order by instead finding the order of its centralizer. Let ${\mathcal Z}_{\operatorname{GSp}_{2g}({\mathbb F_\ell})}({\mathcal{C}})$ be the centralizer in $\operatorname{GSp}_{2g}({\mathbb F_\ell})$ of an element of ${\mathcal{C}}$. Since we need only the size of the centralizer, the choice of this element is arbitrary. \[RegCent\] Let $\mathcal{C}$ be a regular semisimple conjugacy class with characteristic polynomial having one of the shapes listed in Table \[Tab:RegSSCharPolShape\]. Then we have $$\#{\mathcal Z}_{\operatorname{GSp}_{2g}({\mathbb F_\ell})}({\mathcal{C}}) = \begin{cases} (\ell-1)^{g + 1} & \text{ if $f_{{\mathcal{C}}}(T)$ has shape $[1]_1 \dots [1]_{2g}$}, \\ (\ell^2 - 1)(\ell+1)^{g-1} & \text{ if $f_{{\mathcal{C}}}(T)$ has shape $[2]_1\dots[2]_g$}, \\ (\ell^g - 1)(\ell-1) & \text{ if $f_{{\mathcal{C}}}(T)$ has shape $[g]_1[g]_2$}, \\ (\ell^g + 1)(\ell - 1) & \text{ if $f_{{\mathcal{C}}}(T)$ has shape $[2g]$}. \end{cases}$$ Since each class is regular and semisimple, their centralizers are tori. For example, consider the case where $f_{{\mathcal{C}}}(T)$ has the shape $[2g]$. Then $f_{{\mathcal{C}}}(T)$ has roots $\alpha,\alpha^\ell,\ldots,\alpha^{\ell^{2g-1}}$ in ${\mathbb F}^\times_{\ell^{2g}}$ in a single orbit under the action of Galois. Since ${\mathcal{C}}\subseteq \operatorname{GSp}_{2g}({\mathbb F_\ell})$, it must be that the elements of ${\mathcal{C}}$ have multiplier $m = \alpha^{\ell^{i}}\alpha^{\ell^{g + i}}$ for $i \in \{0, 1, \dots, g - 1\}$. Thus, elements of the centralizer of ${\mathcal{C}}$ are those where roots of the characteristic polynomial are elements of ${\mathbb F}^\times_{\ell^{2g}}$ with an ${\mathbb F}_{\ell^g}$-norm lying in ${\mathbb F}^\times_\ell$. There are $\frac{\ell^{2g} - 1}{\ell^g - 1} = \ell^g + 1$ elements of ${\mathbb F}^\times_{\ell^{2g}}$ with such a norm for a fixed $m$. Since we have $\ell - 1$ choices for the multiplier, $\#{\mathcal Z}_{\operatorname{GSp}_{2g}({\mathbb F_\ell})}({\mathcal{C}}) = (\ell^g + 1)(\ell - 1).$ The centralizer sizes for the other cases are computed similarly. In the case where ${\mathcal{C}}$ is not semisimple, the process for determining the order of its centralizer is significantly more challenging. In these cases, we must construct an explicit matrix $\gamma$ which is a representative of ${\mathcal{C}}$, and verify that $\gamma$ has the correct characteristic polynomial, that $\gamma$ is cyclic, and that $\gamma\in\operatorname{GSp}_{2g}({\mathbb F_\ell})$. Then we must find an explicit matrix $C$ which is a generic member of the centralizer of that $\gamma$ (also in $\operatorname{GSp}_{2g}({\mathbb F_\ell})$) and use it to count the number of possible elements of ${\mathcal Z}_{\operatorname{GSp}_{2g}({\mathbb F_\ell})}({\mathcal{C}})$. For any particular $g$, this process is possible. (As an example, the centralizer orders of the non-semisimple classes for $g=3$ are given in Proposition \[NonSSCent\].) However, we have not yet constructed representatives for all three non-semisimple classes for a general (odd prime) $g$ and thus do not have formulae for their centralizer orders. We hope, in future work, to address this gap. Local factors for $f$ {#Sec:LocalFactorsf} ===================== In this section we define local factors $\nu_\ell(f)$ for each finite rational prime $\ell$ and one for the archimedean prime, $\nu_\infty(f)$. For all $\ell\neq p$, this local factor is given by the density of elements of $\operatorname{GSp}_{2g}({\mathbb F}_\ell)$ with a fixed multiplier and characteristic polynomial $f$ with respect to the “average" frequency. We also define $\nu_p(f)$ and $\nu_\infty(f)$ based on the same notions. These definitions are in direct analogue with those of [@achterwilliams2015], and are thus philosophically guided by [@gekeler03] as well. $\nu_\ell(f)$ ------------- Suppose $\ell\neq p$ is a rational prime and consider a principally polarized abelian variety $X/{\mathbb F_q}$ of dimension $g$. The Frobenius endomorphism $\pi_{X/{\mathbb F_q}}$ of $X$ acts as an automorphism of $X_\ell$, and scales by a factor of $q$ the symplectic pairing on $X_\ell$ induced by the polarization. Thus, we can consider $\pi_{X/{\mathbb F_q}}$ as an element of $\operatorname{GSp}_{2g}({\mathbb F}_\ell)^{(q)}$ (the set of elements of $\operatorname{GSp}_{2g}({\mathbb F}_\ell)$ with multiplier $q$). There are $\ell^g$ possible characteristic polynomials for an element of $\operatorname{GSp}_{2g}({\mathbb F}_\ell)^{(q)}$, and so the average frequency of a particular polynomial occurring as the characteristic polynomial of an element of the group (with respect to all such polynomials) is given by $$\#\operatorname{GSp}_{2g}({\mathbb F}_\ell)^{(q)} / \ell^g.$$ Then for primes $\ell$ unramified in $K_f$, we define $\nu_\ell(f)$ as $$\label{definenuellf} \nu_\ell(f) = \frac{\#{\left\{\gamma\in\operatorname{GSp}_{2g}({\mathbb F}_\ell)^{(q)} \mid \operatorname{charpol}(\gamma)\equiv f \bmod \ell\right\}}}{\#\operatorname{GSp}_{2g}({\mathbb F}_\ell)^{(q)}/\ell^g}.$$ (See for a definition for all $\ell\neq p$.) $\nu_p(f)$ ---------- The definition of $\nu_p(f)$ is similar to but more intricate than . Under our assumptions, $X/{\mathbb F}_q$ is an ordinary abelian variety of (odd prime) dimension $g$ with characteristic polynomial of Frobenius $$f_X(T)=T^{2g} + c_1 T^{2g-1} + \dots +c_g T^g+qc_{g-1} T^{g-1} + \dots +q^{g-1} c_1 T + q^g.$$ Thus, as in [@achterwilliams2015], we have a canonical decomposition of the $p$-torsion group scheme into étale and toric components $X[p]\cong X[p]^{\text{et}}\oplus X[p]^{\text{tor}}$. By ordinarity, $X[p]^{\text{et}}(\bar{\mathbb F}_q)\cong ({\mathbb Z}/p)^g$ and $(X[p]^{\text{tor}})^*(\bar{\mathbb F}_q)\cong ({\mathbb Z}/p)^g$, and the $q$-power Frobenius $\pi_{X/{\mathbb F_q}}$ acts invertibly on $X[p](\bar{\mathbb F}_q)$. This action of $\pi_{X/{\mathbb F_q}}$ on both the étale and toric components of $X[p]$ has characteristic polynomial $g_X(T)=T^g + c_1 T^{g-1} + \dots +c_g \bmod p$, and must preserve the decomposition of $X[p]$. Let $m_g$ be the multiplier of $g_X$ (so $m_g$ is a $g^{\text{th}}$ root of $c_g^2$). Thus, we set $$\label{definenupf} \nu_p(f)=\frac{\#{\left\{\gamma\in\operatorname{GSp}_{2g}({\mathbb F_p})^{(m_g)} \mid \operatorname{charpol}(\gamma)\equiv (g_X)^2 \bmod p \text{ and $\gamma$ semisimple}\right\}}}{\#\operatorname{GSp}_{2g}({\mathbb F_p})^{(m_g)}/p^g}.$$ $\nu_\infty(f)$ --------------- Lastly, we define an archimedean term, which is related to the Sato-Tate measure. As stated in [@achterwilliams2015], the Sato-Tate measure on abelian varieties conjecturally explains the distribution of Frobenius elements, and is a pushforward of Haar measure on the space of “Frobenius angles", $0\leq \theta_1\leq \dots \leq \theta_g \leq \pi$. The Weyl integration formula [@weyl p218, 7.8B] gives the Sato-Tate measure on abelian varieties of dimension $g$ explicitly as $$\mu_{ST}(\theta_1, \dots, \theta_g) = 2^{g^2}\left( \prod_{j < k} (\cos(\theta_j) - \cos(\theta_k))^2 \right)\prod_{i = 1}^g \left(\frac{1}{\pi}\sin^2(\theta_i)d\theta_i\right).$$ Fixing a particular $q$, the set of angles ${\left\{\theta_1, \dots ,\theta_g\right\}}$ gives rise to a $q$-Weil polynomial. We use the induced measure on the space of all such polynomials to define the archimedean term $\nu_\infty(f)$. To derive this induced measure, we first write the polynomial in terms of its roots and in terms of its coefficients as $$\begin{aligned} f(T) &= \prod_{j = 1}^g (T - \sqrt{q}e^{i\theta_j})(T - \sqrt{q}e^{-i\theta_j}) \\ &= T^{2g} + c_1 T^{2g - 1} + \dots + c_gT^g + c_{g - 1}qT^{g - 1} + \dots + c_1q^{g-1}T + q^g,\end{aligned}$$ and perform a change of variables. Thus, we find our induced measure on the space of all $q$-Weil polynomials to be $$\mu(c_1, \dots, c_g) = \frac{1}{q^{g^2}(2\pi)^g}\sqrt{{\left\vert\frac{\operatorname{disc}(f)}{\operatorname{disc}(f^+)}\right\vert}} \, dc_1 \dots dc_g.$$ Note that there are approximately $q^{\dim \mathcal{A}_g} = q^{\frac{g(g+1)}{2}}$ principally polarized abelian varieties over ${\mathbb F_q}$, so $q^{\frac{g(g+1)}{2}}\mu(c_1, \dots, c_g)$ can be thought of as an archimedean predictor for $\#{\mathcal A}_g({\mathbb F_q}; f)$. Then we define $$\label{nuinfinity} \nu_\infty(f) = \frac{1}{\operatorname{cond}(f) (2\pi)^g} \sqrt{{\left\vert\frac{\operatorname{disc}(f)}{\operatorname{disc}(f^+)}\right\vert}}.$$ (We note that definition holds for any $g$, not just odd prime $g$.) Polynomials and primes in $K$ ============================= Fix a $q$-Weil polynomial $f(T)$ which satisfies conditions -. For the remainder of the paper, we write $K$ for $K_f$, ${\mathcal{O}}_K$ for ${\mathcal{O}}_f$, $K^+$ for $K^+_f$, $\Delta_K$ for $\Delta_{{\mathcal{O}}_K}$, and $\Delta_{K^+}$ for $\Delta_{{\mathcal{O}}_{K^+}}$. Let $\kappa_\ell={\mathcal{O}}_K\otimes{\mathbb F}_\ell$, a $2g$-dimensional vector space over ${\mathbb F}_\ell$. Our goal in this section is to relate the polynomial $f(T) \bmod \ell$ to (a representative of) one of the conjugacy classes defined in section \[conjclasses\]. There are two lenses through which we can consider such a correspondence, as outlined in [@achterwilliams2015 Section 5]. Regardless of the perspective, we will use the factorization of $f(T)\bmod\ell$ to determine a cyclic element of $\operatorname{GSp}_{2g}({\mathbb F}_\ell)$ whose semisimplification is conjugate to $\gamma_\ell$, the image of the action of $\pi_f$ on $\kappa_\ell$. Then we define $$\label{redefinenuellf} \nu_\ell(f)= \frac{\#{\left\{\gamma\in\operatorname{GSp}_{2g}({\mathbb F}_\ell) \mid \gamma \text{ is cyclic with semisimplification } \gamma_\ell\right\}}}{\#\operatorname{GSp}_{2g}({\mathbb F_\ell})^{(q)}/\ell^g}.$$ If $\ell\nmid p\Delta_K$, then definitions and coincide. If $\ell\nmid p\Delta_K$ then $\ell\nmid\operatorname{disc}(f)$ and so $f(T)\bmod\ell$ has distinct roots. Under condition , any factorization of $f(T)\bmod\ell$ with distinct roots appears in Table \[Tab:RegSSCharPolShape\], and so any element with characteristic polynomial $f(T)\bmod\ell$ is conjugate to $\gamma_\ell$. All regular semisimple elements are cyclic, so the lemma is proven. Note that $\operatorname{charpol}(\gamma_\ell)$ is precisely $f(T)\bmod\ell$. Also note that $\kappa_\ell={\mathcal{O}}_K/\ell\cong {\mathbb F}_\ell[T]/f(T)$ by Corollary \[lemalmostmaximal\], so the factorization of $f(T)\bmod\ell$ is determined by the splitting of $\ell$ in ${\mathcal{O}}_K$. That is, if $f(T)\mod\ell=\prod_{1\leq j\leq r} g_j(T)^{e_j}$, then $\ell=\prod_{1\leq j\leq r} \lambda_j^{e_j}$ for primes $\lambda_j$ of ${\mathcal{O}}_K$ where the residue degree of $\lambda_j$ equals the degree of the irreducible polynomial $g_j(T)$. Because of Condition , $K/{\mathbb Q}$ is a finite Galois extension with $\operatorname{Gal}(K/{\mathbb Q})\cong {\mathbb Z}/2g{\mathbb Z}$. Then the residue degrees of the $\lambda_j$ are all equal to a common value $\frak{f}$, and the ramification degrees of the $\lambda_j$ are all equal to a common value $e$. In particular $2g=e\frak{f}r$. (Notice that this restriction is precisely how we identified relevant class shapes in Section \[conjclasses\].) Without loss of generality, let $\lambda$ be a prime of ${\mathcal{O}}_K$ over $\ell$. Let $D(\ell)$ and $I(\ell)$ denote the decomposition and inertia groups, respectively, of a rational prime $\ell$. Lastly, if $\ell\nmid \operatorname{disc}(f)$, then $\operatorname{Gal}(\kappa(\lambda)/{\mathbb F}_\ell)$ (the residue field of $\lambda$) is cyclic. Let $\operatorname{Frob}_K(\ell)\in\operatorname{Gal}(K/{\mathbb Q})$ be the element which induces the generator of this group, and call it the Frobenius endomorphism of $\lambda$ over $\ell$. Let $\operatorname{Gal}(K/{\mathbb Q})={\left\langle\sigma\right\rangle}$ so that complex conjugation is given by $\iota=\sigma^g$. We classify the splitting of rational primes of $K$ by enumerating the possibilities for $D(\ell)$ and $I(\ell)$. \[lemFrob\] Suppose $f$ satisfies Conditions -. Let $\ell\neq p$ be a rational prime. The cyclic shape of $\gamma_\ell$ is determined by the decomposition and inertia groups $D(\ell)$ and $I(\ell)$ as in Table \[Tab:Frob\]. [\*5l]{} $D(\ell)$ & $I(\ell)$ & $\operatorname{Frob}_K(\ell)$ & $(e,\frak{f},r)$ & Class shape\ ${\left\{1\right\}}$ &${\left\{1\right\}}$ & $1$ & $(1, 1, 2g)$ & $[1]_1\dots[1]_{2g}$\ ${\left\langle\sigma^g\right\rangle}$ &${\left\{1\right\}}$ & $\sigma^g$ & $(1,2,g)$ & $[2]_1\dots[2]_g$\ ${\left\langle \sigma^2 \right\rangle}$ &${\left\{ 1 \right\}}$ & $\sigma^2$ & $(1, g, 2)$ & $[g]_1[g]_2$\ ${\left\langle \sigma^2 \right\rangle}$ & ${\left\langle \sigma^2 \right\rangle}$ & - & $(g, 1, 2)$ & $[1]_1^g[1]_2^g$\ ${\left\langle \sigma \right\rangle}$ & ${\left\{ 1 \right\}}$ & $\sigma^i$ for $(i,g)=1$ & $(1, 2g, 1)$ & $[2g]$\ ${\left\langle \sigma \right\rangle}$ & ${\left\langle \sigma^2 \right\rangle}$ & - & $(g,2,1)$ & $[2]^g$\ ${\left\langle \sigma \right\rangle} $ & ${\left\langle \sigma \right\rangle}$ & - & $(2g, 1, 1)$ & $[1]^{2g}$\ Note that in every case, the data $D(\ell)$ and $I(\ell)$ determines a unique conjugacy class from those given in Tables \[Tab:RegSSCharPolShape\] and \[Tab:NonSSCharPolShape\]. In Table \[Tab:Frob\], we enumerated all possibilities for pairs of subgroups $I(\ell)\subseteq D(\ell) \subseteq \operatorname{Gal}(K/{\mathbb Q})$. In every case, there are $r=\#\operatorname{Gal}(K/{\mathbb Q})/\#D(\ell)$ distinct irreducible factors of $f(T)\bmod\ell$, each with degree $\frak{f}=\#D(\ell)/\#I(\ell)$ and multiplicity $e=\#I(\ell)$. In all cases, this factorization pattern exactly determines the conjugacy class for which $\gamma_\ell$ is a representative. Local factors for $K$ {#Sec:LocalFactorsK} ===================== Recall that $K/{\mathbb Q}$ is a finite Galois extension with $\operatorname{Gal}(K/{\mathbb Q})\cong {\mathbb Z}/2g{\mathbb Z}$. Then $K$ has two unique subfields, the maximal totally real field $K^+$ of index 2 in $K$, and a complex quadratic extension of ${\mathbb Q}$ which we will call $K_1$ in what follows. Let $X(K)$ be the character group of the Galois group of $K$. For $\chi \in X(K)$, let $K^\chi$ be the fixed field of $\ker(\chi)$. For a rational prime $\ell$, define $$\chi(\ell) = \begin{cases} \chi(\operatorname{Frob}_{K^\chi}(\ell)) & \text{if $\ell$ is unramified in $K^\chi$,} \\ 0 & \text{otherwise.} \end{cases}$$ Let $S(K) = X(K) \smallsetminus X(K^+)$ and define $$\label{Eqn:definenuellK} \nu_{\ell}(K) = \prod_{\chi \in S(K)}\left(1-\frac{\chi(\ell)}{\ell}\right)^{-1}.$$ Recall that $\sigma$ is a generator of $\operatorname{Gal}(K/{\mathbb Q})$ and let $\chi$ be a generator of $X(K)$. Then $\langle \sigma^2 \rangle = \operatorname{Gal}(K^+/{\mathbb Q})$ and $\langle \chi^2 \rangle = X(K^+)$, so $S(K) = \{\chi, \chi^3, \dots, \chi^{2g - 1}\}.$ A quick computation shows $K^{\chi^g} = K_1$ and $K^{\chi^i} = K$ for all other $\chi^i \in S(K)$. We have $\chi^{i}(\ell) = (\chi(\ell))^i$ for all odd $i \neq g$, and so to compute $\chi^i(\ell)$ for odd $i$, we only need to know $\chi(\ell)$ and $\chi^g(\ell).$ The values of $\chi(\ell)$ and $\chi^g(\ell)$ are determined by $D(\ell)$ and $I(\ell)$, as given in Table \[Tab:Chars\]. The values in the table follow from the definitions of $\chi$ and $\chi^g$ above, the Frobenius elements given in Table \[Tab:Frob\], and the fact that $\operatorname{Gal}(K/{\mathbb Q})$ and $X(K)$ are cyclic. Then when $\operatorname{Frob}_K(\ell)$ generates $\operatorname{Gal}(K/{\mathbb Q})$, $\chi(\ell)$ is a primitive $2g^{th}$ root of unity. In particular, the values in Table \[Tab:Chars\] are independent of the choice of generator for each of $D(\ell)$, $I(\ell)$, and $X(K)$. [\*4l]{} $D(\ell)$ & $I(\ell)$ & $\{\chi(\ell),\chi^g(\ell)\}$ & Class shape\ ${\left\{1\right\}}$ &${\left\{1\right\}}$ & $\{1,1\}$ & $[1]_1\dots[1]_{2g}$\ ${\left\langle\sigma^g\right\rangle}$ &${\left\{1\right\}}$ & $\{-1,-1\}$ & $[2]_1\dots[2]_g$\ ${\left\langle\sigma^2\right\rangle}$ &${\left\{1\right\}}$ & $\{e^{2\pi i/g},1\}$ & $[g]_1[g]_2$\ ${\left\langle\sigma^2\right\rangle}$ & ${\left\langle\sigma^2\right\rangle}$ & $\{0 ,1\}$ & $[1]_1^g[1]_2^g$\ ${\left\langle\sigma\right\rangle}$ & ${\left\{1\right\}}$ & $\{e^{\pi i/g} ,-1\}$ & $[2g]$\ ${\left\langle\sigma\right\rangle}$ & ${\left\langle\sigma^2\right\rangle}$ & $\{0 ,-1\}$ & $[2]^g$\ ${\left\langle\sigma\right\rangle}$ & ${\left\langle\sigma\right\rangle}$ & $\{0,0\}$ & $[1]^{2g}$\ Matching ======== In this section, we will prove a series of propositions to establish equalities between the local factor defined for $f$ in Section \[Sec:LocalFactorsf\] and the local term intrinsic to $K = K_f$ defined in Section \[Sec:LocalFactorsK\] both for general odd prime $g$ and for the specific case when $g=3$. General case ------------ Let $g$ be an odd prime. In Proposition \[PropMatchGeneralCaseForEll\], we must restrict to regular semisimple conjugacy classes because we only have $\#{\mathcal Z}_{\operatorname{GSp}_{2g}({\mathbb F_\ell})}({\mathcal{C}})$ for all odd prime $g$ in those cases. \[PropMatchGeneralCaseForEll\] Suppose $f$ is a $q$-Weil polynomial of degree $2g$ such that $f\mod\ell=f_{\mathcal{C}}$ for one of the conjugacy class shapes in Table \[Tab:RegSSCharPolShape\]. If $\ell\neq p$ then $\nu_\ell(f)=\nu_\ell(K)$. Let ${\mathcal{C}}$ be a conjugacy class from Table \[Tab:RegSSCharPolShape\]. Using the orbit-stabilizer theorem, we can rewrite $\nu_\ell(f)$ in terms of $\#{\mathcal Z}_{\operatorname{GSp}_{2g}({\mathbb F_\ell})}({\mathcal{C}})$ as $$\nu_\ell(f) = \frac{\#{\mathcal{C}}}{\#\operatorname{GSp}_{2g}({\mathbb F}_\ell)^{(q)}/\ell^g} = \frac{\#\operatorname{GSp}_{2g}({\mathbb F_\ell})/\#{\mathcal Z}_{\operatorname{GSp}_{2g}({\mathbb F_\ell})}({\mathcal{C}})}{\#\operatorname{GSp}_{2g}({\mathbb F}_\ell)^{(q)}/\ell^g} = \frac{\ell^g(\ell - 1)}{\#{\mathcal Z}_{\operatorname{GSp}_{2g}({\mathbb F_\ell})}({\mathcal{C}})}.$$ Since we found the centralizer orders for each regular semisimple class in Theorem \[RegCent\], we use this expression for $\nu_\ell(f)$ to calculate the third column of Table \[Tab:Match\]. Additionally, from we have $$\nu_{\ell}(K) = \prod_{\chi \in S(K)}\left(1-\frac{\chi(\ell)}{\ell}\right)^{-1} = \left(\frac{\ell}{\ell - \chi(\ell)} \right) \left(\frac{\ell}{\ell - \chi^3(\ell)} \right) \cdots \left(\frac{\ell}{\ell - \chi^{2g-1}(\ell)} \right).$$ The fourth column of Table \[Tab:Match\] contains the relevant values from Table \[Tab:Chars\] from which we compute $\nu_\ell(K)$ in the fifth column. To compute the third and fourth rows of the fifth column of Table \[Tab:Match\], recall that if $z$ is a primitive $(2g)^{th}$ root of unity, then $z^2$ is a primitive $g^{th}$ root of unity. Then since $(\ell^{2g} - 1) = (\ell^g - 1)(\ell^g + 1)$, $$\ell^{2g} - 1 = \prod_{j = 1}^{2g} (\ell - z^j), \ \text{and} \ \ell^g - 1 = \prod_{j = 1}^{g} (\ell - z^{2j}),$$ we must have $$(\ell^g + 1) = \prod_{j = 1}^{g} (\ell - z^{2j - 1}).$$ We see the third and fifth columns of Table \[Tab:Match\] match, so the proposition is proved. [\*5c]{} Class Shape & $\#{\mathcal Z}({\mathcal{C}})$ & $\nu_{\ell}(f)$ & $\{\chi(\ell),\chi^g(\ell)\}$ & $\nu_\ell(K)$\ $[1]_1\dots[1]_{2g}$ & $(\ell-1)^{g + 1}$ & $\frac{\ell^g}{(\ell-1)^g}$ & $\{1,1\}$ & $\left(\frac{\ell}{\ell - 1}\right)^g$\ $[2]_1\dots[2]_g $ & $(\ell^2 - 1)(\ell+1)^{g-1}$ & $\frac{\ell^g}{(\ell + 1)^g}$ & $\{-1,-1\}$ & $\left(\frac{\ell}{\ell + 1}\right)^g$\ $[g]_1[g]_2$ & $(\ell^g - 1)(\ell-1)$ & $\frac{\ell^g}{\ell^g - 1}$ & $\{e^{2\pi i/g}, 1\}$ & $ \frac{\ell^g}{\ell^g - 1}$\ $[2g]$ & $(\ell^g + 1)(\ell - 1)$ & $\frac{\ell^g}{\ell^g + 1}$ & $\{e^{\pi i/g}, -1\}$ & $\frac{\ell^g}{\ell^g + 1}$\ In order to show that $\nu_p(f)=\nu_p(K)$, we first must determine the possible factorizations of $p$ in ${\mathcal{O}}_K$. \[lemprimesplitsKplustoK\] Let $K/{\mathbb Q}$ be a degree $2g$ CM number field, and let $X$ be an ordinary abelian variety of dimension $g$ over ${\mathbb F_q}$. Suppose $K$ acts on $X$. Then any prime of $K^+$ over $p$ splits in $K$. Recall that $K^+={\mathbb Q}(\pi+\bar{\pi})$, and that $\pi,\bar\pi$ are roots of $T^2-(\pi+\bar\pi)T+q$ over $K$. Let $\frak{p}^+$ be a prime of $K^+$ over $p$; since $\pi\bar\pi=q$, at least one of $\pi$ or $\bar\pi$ lies in $\frak{p}$, a prime of $K$ over $\frak{p}^+$. However, if $\pi+\bar\pi \in \frak{p}^+$, then both $\pi$ and $\bar\pi$ are in $\frak{p}$. This contradicts the assumption that $f$ was ordinary, so $\pi+\bar\pi \notin \frak{p}^+$. Then $$\begin{aligned} T^2-(\pi+\bar\pi)T+q &\equiv T^2 - uT \bmod \frak{p}^+ \\ &\equiv T(T-u) \bmod \frak{p}^+\end{aligned}$$ where $u \in \bmod \frak{p}^+$ is nonzero. Therefore, $\frak{p}^+$ splits in $K$. \[PropMatchGeneralCaseForP\] Suppose $f$ is as in Proposition \[PropMatchGeneralCaseForEll\]. Then $\nu_p(f)=\nu_p(K)$. We assumed in that $p$ is unramified in $K$ and $\operatorname{Gal}(K/{\mathbb Q})={\mathbb Z}/2g{\mathbb Z}$, and by Lemma \[lemprimesplitsKplustoK\] all primes of $K^+$ over $p$ split in $K$. Then the only possible factorizations of $p$ in ${\mathcal{O}}_K$ are that $p$ splits completely, or that $p{\mathcal{O}}_K=\mathfrak{p}_1 \mathfrak{p}_2$. If $p$ splits completely, then $g_X(T)$ (from ) factors as a product of linear polynomials over ${\mathbb F_p}$ and the set of *semisimple* matrices with characteristic polynomial $(g_X(T))^2$ has the same cardinality as $[1]_1\dots [1]_{2g}$. Then $\nu_p(f)=\nu_p(K)$ by the first row of Table \[Tab:Match\]. If instead $p{\mathcal{O}}_K= \mathfrak{p}_1\mathfrak{p}_2$ (so $p$ is inert in ${\mathcal{O}}_{K^+}$), then $g_X(T)$ is irreducible so the set of *semisimple* matrices with characteristic polynomial $(g_X(T))^2$ has the same cardinality as $[g]_1[g]_2$. Then $\nu_p(f)=\nu_p(K)$ by the third row of Table \[Tab:Match\]. The following proposition is true for any value of $g$. Let $f$ be a $q$-Weil polynomial of degree $2g$ with splitting field $K$ and ${\mathcal{O}}_K={\mathbb Z}[\pi,\bar\pi]$. Then $$\nu_{\infty}(f) = \frac{1}{(2\pi)^g}\sqrt{{\left\vert\frac{\Delta_K}{\Delta_{K^+}}\right\vert}}$$ This follows from the fact that $\operatorname{disc}(f) = \operatorname{cond}(f)^2\Delta_K$ and $\operatorname{disc}(f^+) = \Delta_{K^+}$. Specific case ($g=3$) --------------------- In the particular case when $g=3$, we can also prove that $\nu_\ell(f)$ and $\nu_\ell(K)$ match in the case when $f\bmod\ell = f_{\mathcal{C}}$ for a non-semisimple conjugacy class ${\mathcal{C}}$. We begin by giving the orders of the centralizers for the non-semisimple conjugacy classes in $\operatorname{GSp}_6({\mathbb F_\ell})$. \[NonSSCent\] Let $\mathcal{C}$ be one of the non-semisimple conjugacy classes of matrices in $\operatorname{GSp}_6({\mathbb F_\ell})$ with characteristic polynomial in one of the shapes listed in Table \[Tab:NonSSCharPolShape\]. Then we have $$\#{\mathcal Z}_{\operatorname{GSp}_{6}({\mathbb F_\ell})}({\mathcal{C}}) = \begin{cases} \ell^3(\ell - 1) & \text{ if $f_{{\mathcal{C}}}(T)$ has shape $[1]^6$}, \\ \ell^2(\ell - 1)^2 & \text{ if $f_{{\mathcal{C}}}(T)$ has shape $[1]_1^3[1]_2^3$}, \\ \ell^2(\ell^2 - 1) & \text{ if $f_{{\mathcal{C}}}(T)$ has shape $[2]^3$}. \\ \end{cases}$$ Refer back to the end of section \[Sec:cclasses\] for an outline of the general method for determining centralizer orders for non-semisimple classes. We demonstrate this method for $g = 3$ in the case where $f_{\mathcal{C}}(T)$ has the shape $[1]_1^3[1]_2^3.$ Suppose $f_{\mathcal{C}}(T) = (T - a)^3(T - b)^3$ with $a,b \in {\mathbb F_q}^{\times}.$ Since these are $\alpha$ polynomials (by Table \[Tab:NonSSCharPolShape\]), $m = ab$. A representative of the class ${\mathcal{C}}$ is $$\gamma = \begin{bmatrix} a & -a & 0 & 0 & 0 & 0 \\ 0 & a & -a & 0 & 0 & 0 \\ 0 & 0 & a & 0 & 0 & 0 \\ 0 & 0 & 0 & b & b & b \\ 0 & 0 & 0 & 0 & b & b\\ 0 & 0 & 0 & 0 & 0 & b\\ \end{bmatrix},$$ where we verify that $\operatorname{charpol}(\gamma)=f_{\mathcal{C}}(T)$, $\gamma\in\operatorname{GSp}_6({\mathbb F_\ell})$ with multiplier $ab$, and $\operatorname{minpol}(\gamma)=\operatorname{charpol}(\gamma)$ (so that $\gamma$ is cyclic). Then a generic element of ${\mathcal Z}_{\operatorname{GSp}_{6}({\mathbb F_\ell})}({\mathcal{C}})$ has the form $$C = \begin{bmatrix} c_1 & y_1 & y_2 & 0 & 0 & 0 \\ 0 & c_1 & y_1 & 0 & 0 & 0 \\ 0 & 0 & c_1 & 0 & 0 & 0 \\ 0 & 0 & 0 & c_2 & c_3 & c_4 \\ 0 & 0 & 0 & 0 & c_2 & c_3\\ 0 & 0 & 0 & 0 & 0 & c_2\\ \end{bmatrix}$$ where $$y_1 = -\frac{c_1c_3}{c_2} \hspace{5mm} \text{and} \hspace{5mm} y_2 = c_1\left(\frac{c_3^2 - c_2c_4}{c_2^2}\right).$$ Since $\det C=c_1^3 c_2^3$, it must be that $c_1,c_2\in{\mathbb F_\ell}^\times$, while $c_3,c_4\in{\mathbb F_\ell}$. Then $$\#{\mathcal Z}_{\operatorname{GSp}_{6}({\mathbb F_\ell})}({\mathcal{C}}) = \ell^2(\ell-1)^2.$$ The other cases require similar computations. \[PropMatchSpecificCaseForEll\] Suppose $f$ is a $q$-Weil polynomial of degree $6$ such that $f\bmod\ell=f_{\mathcal{C}}$ for one of the conjugacy classes in Table \[Tab:NonSSCharPolShape\]. If $\ell\neq p$ then $\nu_\ell(f)=\nu_\ell(K)$. This proof is identical to the proof of Proposition \[PropMatchGeneralCaseForEll\], but for non-semisimple classes. The second column of Table \[Tab:NonSSMatch\] comes from Proposition \[NonSSCent\], which we use to compute the third column. The fourth column comes from Table \[Tab:Chars\], which we use to compute the fifth column. Seeing that the third and fifth columns match, we are done. [\*5c]{} Class Shape & $\#{\mathcal Z}({\mathcal{C}})$ & $\nu_{\ell}(f)$ & $\{\chi(\ell),\chi^g(\ell)\}$ & $\nu_\ell(K)$\ $[1]^6$ & $\ell^3(\ell - 1)$ & 1 & $\{0,0\}$ & $\left(\frac{\ell}{\ell - 1}\right)^g$\ $[1]_1^3[1]_2^3$ & $\ell^2(\ell-1)^{2}$ & $\frac{\ell}{\ell - 1}$ & $\{0,1\}$ & $\frac{\ell}{\ell - 1}$\ $[2]^3$ & $\ell^2(\ell^2-1)$ & $\frac{\ell}{\ell + 1}$ & $\{0,-1\}$ & $\frac{\ell}{\ell + 1}$\ Main results ============ This section generalizes many of the results of [@achterwilliams2015 Section 7]. We define an infinite product of numbers $\{a_\ell\}$ indexed by finite primes by $$\prod_{\ell} a_\ell = \lim_{B \rightarrow \infty} \prod_{\ell < B}a _\ell$$ so that $\prod_\ell a_\ell \prod_\ell b_\ell = \prod_\ell (a_\ell b_\ell)$. For a number field $L$, let $h_L$, $\omega_L$, and $R_L$ denote the class number, number of roots of unity, and regulator of $L$ respectively. \[PropClassNumberRatio\] Let $f$ be a degree $2g$ $q$-Weil polynomial that is ordinary, principally polarizable, cyclic, and maximal. Let $K$ be the splitting field of $f$ over ${\mathbb Q}$ and let $K^+$ be its maximal totally real subfield. Then $$\label{Eq:classnumberratio} \frac{h_K}{h_{K^+}} = \omega_K \nu_\infty(f) \prod_\ell \nu_\ell(K).$$ By the analytic class number formula, the ratio of class numbers on the left side of is $$\frac{h_K}{h_{K^+}} = \lim_{s \rightarrow 1} \frac{(s-1)\zeta_K(s)}{(s-1)\zeta_{K^+}(s)}\frac{\sqrt{{\left\vert\Delta_K\right\vert}}2^g\omega_K R_{K^+}}{\sqrt{{\left\vert\Delta_{K^+}\right\vert}}(2\pi)^g \omega_{K^+} R_{K}}.$$ For a finite abelian extension $L/{\mathbb Q}$, we have $$\lim_{s \rightarrow 1}(s-1)\zeta_L(s) = \prod_{\chi \in \operatorname{Gal}(L/{\mathbb Q})^*\backslash \text{id}} L(1, \chi)$$ where we interpret the Dirichlet $L$-function as the conditionally convergent Euler product $$L(1, \chi) = \lim_{B \rightarrow \infty} \prod_{\ell < B} \frac{1}{1 - \chi(\ell)/\ell}.$$ We then see that $$\begin{aligned} \lim_{s \rightarrow 1}\frac{ (s - 1)\zeta_K(s)}{(s - 1)\zeta_{K^+}(s)} &= \frac{\prod_{\chi \in \operatorname{Gal}(K/{\mathbb Q})^*\backslash \text{id}}\left(\prod_{\ell} \frac{1}{1 - \chi(\ell)/\ell}\right)}{\prod_{\chi \in \operatorname{Gal}(K^+/{\mathbb Q})^*\backslash \text{id}}\left(\prod_{\ell} \frac{1}{1 - \chi(\ell)/\ell}\right)}\\ &= \prod_{\chi \in S(K)} \prod_{\ell}\frac{1}{1 - \chi(\ell)/\ell}\\ &= \prod_{\ell} \nu_{\ell}(K)\end{aligned}$$ where $S(K)$ is as in Section \[Sec:LocalFactorsK\]. By [@Wash97 Proposition 4.16] $R_K = \frac{1}{Q}2^{g-1}R_{K^+}$, where $Q$ is Hasse’s unit index. Since $K/{\mathbb Q}$ is cyclic, $Q=1$ by [[@furuya_1977 Theorem 3]]{}. Therefore, $R_K=2^{g-1} R_{K^+}$. Lastly, $\omega_{K^+} = 2$ since $K^+$ is totally real, so $$\begin{aligned} \frac{h_K}{h_{K^+}} &= \sqrt{{\left\vert\frac{\Delta_K}{\Delta_{K^+}}\right\vert}}\frac{\omega_K R_{K^+}}{2\pi^g R_K}\prod_{\ell}\nu_{\ell}(K) \\ &= \sqrt{{\left\vert\frac{\Delta_K}{\Delta_{K^+}}\right\vert}}\frac{\omega_K}{(2\pi)^g}\prod_{\ell}\nu_{\ell}(K) \\ &= {\omega_K} \nu_\infty(f) \prod_{\ell}\nu_{\ell}(K).\end{aligned}$$ \[RemarkNonSSWillMatch\] Propositions \[PropMatchGeneralCaseForEll\] and \[PropMatchGeneralCaseForP\] tell us that in many cases the $\nu_\ell(K)$ in the previous proposition is in fact equal to $\nu_\ell(f)$. While we did not give the orders of the centralizers of the non-semisimple conjugacy classes for all odd prime $g$ in this paper, we have partial progress which suggests that Proposition \[PropMatchSpecificCaseForEll\] will also generalize to all odd prime $g$. On the assumption that Proposition \[PropMatchSpecificCaseForEll\] in fact generalizes to all odd prime $g$, we make the following conjecture. \[ConjMainThmGeneral\] Let $g$ be an odd prime, and let $f$ be a degree $2g$ $q$-Weil polynomial that is ordinary, principally polarizable, cyclic, and maximal. Let $K$ be the splitting field of $f$ over ${\mathbb Q}$ and let $K^+$ be its maximal totally real subfield. Then $$\nu_\infty(f)\prod_{\ell}\nu_\ell(f) = \frac{1}{\omega_K}\frac{h_{K}}{h_{K^+}}.$$ For any primes $\ell$ where $f_{\mathcal{C}}\cong f \bmod\ell$ has one of the regular semisimple shapes in Table \[Tab:RegSSCharPolShape\], Propositions \[PropMatchGeneralCaseForEll\] and \[PropMatchGeneralCaseForP\] give that $\nu_\ell(f)=\nu_\ell(K)$. Assume that $\nu_\ell(f)=\nu_\ell(K)$ for primes $\ell$ where $f_{\mathcal{C}}\cong f \bmod\ell$ is non-semisimple for all odd prime $g$. (That is, assume a version of Proposition \[PropMatchSpecificCaseForEll\] is true for all odd prime $g$.) Then for all primes $\ell$ we have $\nu_\ell(f)=\nu_\ell(K)$, and so by Proposition \[PropClassNumberRatio\] the conjecture would be true. In the case where $g=3$, we can say more. \[Thm:Main\] Let $f$ be a degree $6$ $q$-Weil polynomial that is ordinary, principally polarizable, cyclic, and maximal. Let $K$ be the splitting field of $f$ over ${\mathbb Q}$ and let $K^+$ be its maximal totally real subfield. Then $$\label{Eq:Main} \nu_\infty(f)\prod_{\ell}\nu_\ell(f) = \frac{1}{\omega_K}\frac{h_{K}}{h_{K^+}}.$$ Combine Proposition \[PropMatchSpecificCaseForEll\] with Propositions \[PropMatchGeneralCaseForEll\], \[PropMatchGeneralCaseForP\], and \[PropClassNumberRatio\] for $g=3$, and the theorem is proven. While Theorem \[Thm:Main\] and Conjecture \[ConjMainThmGeneral\] are interesting in their own right as an unexpected equality of a product of local densities of matrices to a ratio of class numbers, there is an interpretation of this quantity related to isogeny classes of abelian varieties via results in a preprint of Everett Howe [@howepreprint]. We begin with some notation. Given a Weil polynomial $f\in{\mathbb Z}[T]$, let $K$, $\pi_f$, $\bar\pi_f$, $K^+$, and ${\mathcal{O}}_f$ be defined as in section \[Sec:WeilPolys\] and let ${\mathcal{O}}_{f^+}={\mathbb Z}[\pi_f+\bar\pi_f]$. Let $U$ be the unit group of ${\mathcal{O}}_f$ and let $U^+_{>0}$ be the group of totally positive units in ${\mathcal{O}}_{f^+}$. Denote the narrow class number of $K^+$ by $h^+_{K^+}$. \[Thm:Howe\] Let $\mathcal{I}$ be an isogeny class of simple ordinary abelian varieties over ${\mathbb F_q}$ corresponding to an irreducible Weil polynomial $f\in{\mathbb Z}[T]$. Using the notation above, suppose that ${\mathcal{O}}_f$ is the maximal order of $K$, and suppose that $K$ is ramified over $K^+$ at a finite prime. Then the number of abelian varieties in $\mathcal{I}$ that have a principal polarization is equal to $\frac{h_K}{h^+_{K^+}}$, and each such variety has (up to isomorphism) exactly $[U^+_{>0}:N(U)]$ principal polarizations, where $N$ is the norm map from $U$ to $U^+_{>0}$. We note that the conditions in Theorem \[Thm:Howe\] are indeed met under our conditions; we assume that the abelian varieties are ordinary in , the polynomial $f$ is irreducible in , and ${\mathcal{O}}_f$ is the maximal order of $K$ in . By [@howe95 Lemma 10.2], $K$ is ramified over $K^+$ at a finite prime whenever $g$ is odd. \[CorToHowe\] Let everything be as in Theorem \[Thm:Howe\]. If in addition the unit groups of $K$ and $K^+$ are equal, then the total number of principally polarized varieties lying in the isogeny class $\mathcal{I}$ is equal to $h_K/h_{K^+}$. If the unit groups of $K$ and $K^+$ are equal, then the ratio $\frac{h^+_{K^+}}{h_{K^+}}$ is equal to $[U^+_{>0}:N(U)]$ and the result follows from Theorem \[Thm:Howe\]. Corollary \[CorToHowe\] generalizes the classical result that the number of elliptic curves in a fixed isogeny class is given by the class number of an appropriate imaginary quadratic field. We assume in that $K/{\mathbb Q}$ is cyclic, which implies that Hasse’s unit index is 1 (see the proof of Proposition \[PropClassNumberRatio\]). Thus, under our conditions, the unit groups of $K$ and $K^+$ are equal, and so we can apply Corollary \[CorToHowe\]. Recall that ${\mathcal A}_g({\mathbb F_q}; f)$ denotes the set of isomorphism classes of principally polarized abelian varieties of dimension $g$ over ${\mathbb F_q}$ with characteristic polynomial of Frobenius $f$, weighted inversely by the size of the automorphism group. Then we have the following corollary of Theorem \[Thm:Main\]. \[Cor:g3UseHowe\] Let $f$ be as in Theorem \[Thm:Main\]. Then $$\nu_{\infty}(f)\prod_{\ell}\nu_{\ell}(f) = \#\mathcal{A}_3({\mathbb F}_q;f).$$ From Corollary \[CorToHowe\], the (unweighted) size of the isogeny class of principally polarized abelian threefolds with characteristic polynomial $f(T)$ is $\frac{h_{K}}{h_{K^+}}$. The size of the automorphism group of any element of that isogeny class is $\omega_K$, so $$\frac{1}{\omega_K} \frac{h_K}{h_{K^+}} = \#\mathcal{A}_3({\mathbb F}_q;f)$$ and the corollary is proven. If we also assume Conjecture \[ConjMainThmGeneral\], then we can state the following corollary, which generalizes the previous result. Assume that Conjecture \[ConjMainThmGeneral\] is true and let $f$ be as in that conjecture. Then $$\nu_{\infty}(f)\prod_{\ell}\nu_{\ell}(f) = \#\mathcal{A}_g({\mathbb F}_q;f).$$ Recent work of Marseglia gives an algorithm for computing isomorphism classes of abelian varieties in certain isogeny classes [@marseglia2018]. One possible application of the algorithm, according to the author, is to provide computational evidence for the main formulas in [@achterwilliams2015], [@achtergordon2017], and the present work. Acknowledgments {#acknowledgments .unnumbered} --------------- We thank Everett Howe for sharing with us his work related to isogeny classes of principally polarized abelian varieties, and Jeff Achter for very helpful conversations. We also thank the referee for useful suggestions and insightful questions.

--- abstract: 'In this article, we classify all algebraic Ricci solitons on five-dimensional nilmanifolds.' address: | Department of Pure Mathematics\ Faculty of Mathematics and Statistics\ University of Isfahan\ Isfahan\ 81746-73441-Iran. author: - Hamid Reza Salimi Moghaddam title: 'Left invariant Ricci solitons on five-dimensional nilmanifolds' --- **Introduction** {#Introduction} ================ Suppose that $(M,g)$ is a complete Riemannian manifold and $\textsf{ric}_g$ denotes its Ricci tensor. If, for a real number $c$ and a complete vector field $X$, the Riemannian metric $g$ satisfies in the equation $$\label{Main Ricci soliton equation} \textsf{ric}_g=c g+\textsf{L}_Xg,$$ then $g$ is named a Ricci soliton. In this equation, if $c>0$, $c=0$ or $c<0$, then $g$ is named shrinking, steady, or expanding Ricci soliton.\ Although the above definition is a generalization of Einstein metrics but the main motivation for considering Ricci solitons is the Ricci flow equation $$\label{Ricci flow equation} \frac{\partial}{\partial t}g_t=-2\textsf{ric}_{g(t)},$$ where $\phi_t$ is a one-parameter group of diffeomorphisms. A Riemannian metric $g$ is a Ricci soliton on $M$ if and only if the following one-parameter family of Riemannian metrics is a solution of (\[Ricci flow equation\]) (see [@Lauret1] and [@Lauret2]), $$\label{family of Riemannian metrics} g_t=(-2c t+1)\phi_t^\ast g.$$ A very interesting case happens when we study Ricci solitons on Lie groups. Suppose that $G$ is a Lie group, $\frak{g}$ is its Lie algebra and $g$ is a left invariant Riemannian metric on $G$. If for a real number $c$ and a derivation $D\in\textsf{Der}(\frak{g})$, the $(1,1)$-Ricci tensor $\textsf{Ric}_g$ of $g$ satisfies the equation $$\label{Algebraic Ricci soliton equation} \textsf{Ric}_g=c.Id+D,$$ then $g$ is called an algebraic Ricci soliton. It is shown that all algebraic Ricci solitons are Ricci soliton (for more details see [@Jablonski1] and [@Lauret3]).\ In the year 2001, Lauret proved that on nilpotent Lie groups a left invariant Riemannian metric $g$ is a Ricci soliton if and only if it is an algebraic Ricci soliton. In fact for left invariant Riemannian metrics on nilpotent Lie groups the equations (\[Main Ricci soliton equation\]) and (\[Algebraic Ricci soliton equation\]) are equivalent (see [@Lauret3] and [@Jablonski2]).\ Naturally, one can generalize the concept of algebraic Ricci soliton to homogeneous spaces. In 2014, Jablonski showed that any homogeneous Ricci soliton is an algebraic Ricci soliton (see [@Jablonski2]). So the classification of left invariant Ricci solitons on Lie groups reduces to the classification of algebraic Ricci solitons on them.\ Suppose that $g$ is a left invariant Riemannian metric on a Lie group $G$ and $\alpha_{ijk}$ are the structural constants of the Lie algebra $\frak{g}$, with respect to an orthonormal basis $\{E_1,\cdots,E_n\}$, defined by the following equations: $$\label{structural constants} [E_i,E_j]=\sum_{k=1}^n\alpha_{ijk}E_k.$$ In an earlier paper (see [@Salimi]), we proved that $g$ is an algebraic Ricci soliton if and only if there exists a real number $c$ such that, for any $t, p, q=1,\cdots,n$ the structural constants satisfies in the following equation: $$\begin{aligned} \label{Main formula} c\alpha_{qpt}+\frac{1}{4} \sum_{i=1}^n\sum_{j=1}^n\sum_{r=1}^n &&\hspace{-0.6cm} 2\alpha_{rjj}\Big{(}\alpha_{iqt}(\alpha_{pri}+\alpha_{ipr}-\alpha_{rip}) -\alpha_{ipt}(\alpha_{qri}+\alpha_{iqr}-\alpha_{riq})\Big{)}\nonumber\\ &&\hspace{-0.6cm} +2(\alpha_{rji}+\alpha_{irj}-\alpha_{jir})(\alpha_{ipt}\alpha_{qjr}-\alpha_{iqt}\alpha_{pjr})\nonumber\\ &&\hspace{-0.6cm} +(\alpha_{jri}+\alpha_{ijr}-\alpha_{rij})\Big{(}\alpha_{ipt}(\alpha_{qjr}+\alpha_{rqj}-\alpha_{jrq}) -\alpha_{iqt}(\alpha_{pjr}+\alpha_{rpj}-\alpha_{jrp})\Big{)}\\ &&\hspace{-0.6cm} -2\alpha_{pqi}\alpha_{rjj}(\alpha_{irt}+\alpha_{tir}-\alpha_{rti})+2\alpha_{pqi}\alpha_{ijr}(\alpha_{rjt}+\alpha_{trj}-\alpha_{jtr})\nonumber\\ &&\hspace{-0.6cm} +\alpha_{pqi}(\alpha_{ijr}+\alpha_{rij}-\alpha_{jri})(\alpha_{jrt}+\alpha_{tjr}-\alpha_{rtj})=0. \nonumber\end{aligned}$$ In this case we showed that, for any $i=1\cdots n$, the derivation $D$ satisfies in the following equation: $$\begin{aligned} \label{Derivation equation} D(E_i) &=& -cE_i+\frac{1}{4} \sum_{l=1}^n\Big{\{} \sum_{j=1}^n\sum_{r=1}^n2\alpha_{rjj}(\alpha_{irl}+\alpha_{lir}-\alpha_{rli})\nonumber \\ &&\hspace{4cm}-2\alpha_{ijr}(\alpha_{ril}+\alpha_{lrj}-\alpha_{jlr})\\ &&\hspace{4cm}-(\alpha_{ijr}+\alpha_{rij}-\alpha_{jri})(\alpha_{jrl}+\alpha_{ljr}-\alpha_{rlj})\Big{\}}E_l.\nonumber\end{aligned}$$ In [@Salimi], using the above equations, we classified all algebraic Ricci solitons on three-dimensional Lie groups.\ In this article, using the equations (\[Main formula\]) and (\[Derivation equation\]), we explicitly give all algebraic Ricci solitons on five-dimensional nilmanifolds. We remember that in this method, against some recent works (such as [@Di; @Cerbo]) which have studied the Ricci solitons only on the set of left invariant vector fields, we consider the Ricci solitons on all vector fields.\ A Riemannian nilmanifold is a connected Riemannian manifold $(M,g)$ which there exists a nilpotent Lie subgroup of its isometry group $I(M)$ such that acts transitively on $M$. Wilson, in [@Wilson], proved that if $M$ is a homogeneous Riemannian nilmanifold then there is a unique nilpotent Lie subgroup of $I(M)$ which acts simply transitively on $M$. Also, he showed that this Lie subgroup is normal in $I(M)$. Thus, we can assume a homogeneous Riemannian nilmanifold as a nilpotent Lie group $N$ equipped with a left invariant Riemannian metric $g$.\ In 2006, Homoloya and Kowalski, up to isometry, classified five-dimensional two-step nilpotent Lie groups equipped with left invariant Riemannian metrics (see [@Homolya-Kowalski]). In a paper written by Figula and Nagy in 2018, up to isometry, the classification of five-dimensional nilmanifolds has been completed by the classification of five-dimensional nilpotent Lie algebras of nilpotency classes three and four equipped with inner products. We mention that this classification does not contain the Lie algebras which are direct products of Lie algebras of lower dimensions (see [@Figula-Nagy]).\ In the next section we will classify all left invariant Ricci solitons on all ten classes of five-dimensional nilmanifolds. **The classification of algebraic Ricci solitons on five-dimensional nilmanifolds** {#The classification} =================================================================================== In [@Homolya-Kowalski], all five-dimensional two-step nilmanifolds up to isometry have been classified by Homolya and Kowalski. Recently, in [@Figula-Nagy], Figula and Nagy have completed the classification of five-dimensional nilmanifolds up to isometry by classification of five-dimensional nilpotent Lie groups of nilpotency class grater than two. In this section using formula (\[Main formula\]) and the above classifications we classify all algebraic Ricci solitons on simply connected five-dimensional nilmanifolds.\ Suppose that $N$ is an arbitrary simply connected five-dimensional nilmanifold then, up to isometry, its Lie algebra is one of the following ten cases. In all cases the set $\{E_1, \cdots, E_5\}$ is an orthonormal basis for the Lie algebra and in any case we give only non-vanishing commutators. **Two-step nilpotent Lie algebra with one-dimensional center.** --------------------------------------------------------------- Let $N$ be the two-step nilpotent Lie group with one-dimensional center and $\frak{n}$ denotes its Lie algebra. By [@Homolya-Kowalski], the non-zero Lie brackets are as follows: $$[E_1,E_2]=sE_5, \ \ \ \ [E_3,E_4]=mE_5,$$ where $s\geq m >0$. So, in this case, the structure constants with respect to the orthonormal basis $\{E_1, \cdots, E_5\}$ are of the following forms: $$\alpha_{125}=-\alpha_{215}=s, \ \ \ \ \alpha_{345}=-\alpha_{435}=m.$$ Easily, using the formula (\[Main formula\]), we see that $\frak{n}$ is an algebraic Ricci soliton if and only if $$\left\{ \begin{array}{ll} cs+\frac{1}{2}sm^2+\frac{3}{2}s^3=0, & \\ cm+\frac{1}{2}ms^2+\frac{3}{2}m^3=0. & \end{array} \right.$$ A direct computation shows that the above equations hold if and only if $m=s$. So for the constant $c$ we have $c=-2m^2$. Now, the equation (\[Derivation equation\]) shows that for the matrix representation of $D$ we have $$D= \left( \begin{array}{ccccc} \frac{3}{2}m^2 & 0 & 0 & 0 & 0 \\ 0 & \frac{3}{2}m^2 & 0 & 0 & 0 \\ 0 & 0 & \frac{3}{2}m^2 & 0 & 0 \\ 0 & 0 & 0 & \frac{3}{2}m^2 & 0 \\ 0 & 0 & 0 & 0 & 3m^2 \\ \end{array} \right).$$ **Two-step nilpotent Lie algebra with two-dimensional center.** --------------------------------------------------------------- The second case is two-step nilpotent Lie group $N$ with two-dimensional center. In [@Homolya-Kowalski], it is shown that there are real numbers $m\geq s >0$ such that the non-zero Lie brackets are $$[E_1,E_2]=mE_4, \ \ \ \ [E_1,E_3]=sE_5.$$ In this case for the structure constants we have: $$\alpha_{124}=-\alpha_{214}=m, \ \ \ \ \alpha_{135}=-\alpha_{315}=s.$$ Now, the equation (\[Main formula\]) shows that $\frak{n}$ satisfies the algebraic Ricci soliton equation (\[Algebraic Ricci soliton equation\]) if and only if $$\left\{ \begin{array}{ll} cm+\frac{3}{2}m^3+\frac{1}{2}s^2m=0, & \\ cs+\frac{3}{2}s^3+\frac{1}{2}m^2s=0. & \end{array} \right.$$ Easily we can see, for the solutions we have $m=s$ and $c=-2m^2$. Now, by equation (\[Derivation equation\]), for the matrix representation of $D$ in the basis $\{E_1, \cdots, E_5\}$ we have: $$D= \left( \begin{array}{ccccc} m^2 & 0 & 0 & 0 & 0 \\ 0 & \frac{3}{2}m^2 & 0 & 0 & 0 \\ 0 & 0 & \frac{3}{2}m^2 & 0 & 0 \\ 0 & 0 & 0 & \frac{5}{2}m^2 & 0 \\ 0 & 0 & 0 & 0 & \frac{5}{2}m^2 \\ \end{array} \right).$$ **Two-step nilpotent Lie algebra with three-dimensional center.** ----------------------------------------------------------------- The third Lie group which we have considered is the two-step nilpotent Lie group $N$ with three-dimensional center. This is the last five-dimensional two-step nilpotent Lie group. Based on [@Homolya-Kowalski], the non-zero Lie bracket of this case is $$[E_1,E_2]=mE_3,$$ where $m$ is a positive real number. So, the non-zero structure constants are $$\alpha_{123}=-\alpha_{213}=m.$$ It is easy to show that the equation (\[Main formula\]) holds if and only if $cm+\frac{2}{3}m^3=0$. The last equation shows that $c=-\frac{3}{2}m^2$. So, by (\[Derivation equation\]), in the basis $\{E_1, \cdots, E_5\}$, the derivation $D$ has the following matrix representation: $$D= \left( \begin{array}{ccccc} m^2 & 0 & 0 & 0 & 0 \\ 0 & m^2 & 0 & 0 & 0 \\ 0 & 0 & 2m^2 & 0 & 0 \\ 0 & 0 & 0 & \frac{3}{2}m^2 & 0 \\ 0 & 0 & 0 & 0 & \frac{3}{2}m^2 \\ \end{array} \right).$$ **Four-step nilpotent Lie algebra $\frak{l}_{5,7}$, (case A)** -------------------------------------------------------------- In [@Figula-Nagy], Figula and Nagy have proven that the five-dimensional nilpotent Lie algebras $\frak{n}$ of nilpotency class greater than two are of the forms $\frak{l}_{5,7}$, $\frak{l}_{5,6}$, $\frak{l}_{5,5}$ and $\frak{l}_{5,9}$. In this subsection and the next subsection, we study the necessary and sufficient conditions for the Lie algebra $\frak{l}_{5,7}$ to be an algebraic Ricci soliton. By [@Figula-Nagy], the non-vanishing Lie brackets of $\frak{l}_{5,7}$ are $$[E_1,E_2]=mE_3+sE_4+uE_5, \ \ \ [E_1,E_3]=vE_4+wE_5, \ \ \ [E_1,E_4]=xE_5,$$ where for the real numbers $m,s,u,v,w,x$ we have two cases: (case A: $m,v,x>0$, $s=0$ and $w\geq 0$) and (case B: $m,v,x>0$ and $s>0$). In this subsection we study the case A. In the case of $\frak{l}_{5,7}$ (in two cases A and B) for the structure constants we have: $$\begin{aligned} &&\alpha_{123}=-\alpha_{213}=m, \ \alpha_{124}=-\alpha_{214}=s, \ \alpha_{125}=-\alpha_{215}=u, \\ &&\alpha_{134}=-\alpha_{314}=v, \ \alpha_{135}=-\alpha_{315}=w, \ \alpha_{145}=-\alpha_{415}=x.\end{aligned}$$ The equation (\[Main formula\]) together with some computations shows that $\frak{l}_{5,7} (case A)$, satisfies the algebraic Ricci soliton equation (\[Algebraic Ricci soliton equation\]) if and only if $$x=m, \ \ c=-2m^2, \ \ u=w=s=0, \ and \ \ v=\frac{2}{\sqrt{3}}m.$$ Now, using (\[Derivation equation\]), the matrix representation of $D$ is of the form: $$D= \left( \begin{array}{ccccc} \frac{1}{3}m^2 & 0 & 0 & 0 & 0 \\ 0 & \frac{3}{2}m^2 & 0 & 0 & 0 \\ 0 & 0 & \frac{11}{6}m^2 & 0 & 0 \\ 0 & 0 & 0 & \frac{13}{6}m^2 & 0 \\ 0 & 0 & 0 & 0 & \frac{5}{2}m^2 \\ \end{array} \right).$$ **Four-step nilpotent Lie algebra $\frak{l}_{5,7}$, (case B)** -------------------------------------------------------------- As we mentioned above, in this case (case B) for the same structure constants of case A, we have $m,v,x>0$ and $s>0$. We see that the equation (\[Main formula\]) deduces to a system of equations with no solution. So the Lie algebra $\frak{l}_{5,7}$, (case B) does not admit algebraic Ricci soliton structure. **Four-step nilpotent Lie algebra $\frak{l}_{5,6}$, (case A)** -------------------------------------------------------------- During this subsection and the next subsection, we have considered the Lie algebra $\frak{l}_{5,6}$. In two cases (cases A and B), by [@Figula-Nagy], the Lie brackets are $$[E_1,E_2]=mE_3+sE_4+uE_5, \ \ \ [E_1,E_3]=vE_4+wE_5, \ \ \ [E_1,E_4]=xE_5, \ \ \ [E_2,E_3]=yE_5,$$ where $m,s,u,v,w,x,y$ are real numbers such that $m,v,x,y\neq0$. For the structure constants we have: $$\begin{aligned} &&\alpha_{123}=-\alpha_{213}=m, \ \alpha_{124}=-\alpha_{214}=s, \ \alpha_{125}=-\alpha_{215}=u, \ \alpha_{134}=-\alpha_{314}=v,\\ &&\alpha_{135}=-\alpha_{315}=w, \ \alpha_{145}=-\alpha_{415}=x, \ \alpha_{235}=-\alpha_{325}=y.\end{aligned}$$ Similar to the Lie algebra $\frak{l}_{5,7}$ we have two cases, case A and B. In the case A we have $s=0$ and $w\geq 0$ and for the case B $s>0$. A direct computation, using (\[Main formula\]), shows that the Lie algebra $\frak{l}_{5,6} (case A)$, is an algebraic Ricci soliton if and only if $$u=w=s=0, m=\pm\sqrt{\frac{3}{2}}x, y=\pm x, v=\pm\sqrt{\frac{3}{2}}x \ and \ c=-\frac{11}{4}x^2.$$ The equation (\[Derivation equation\]) shows that for the matrix representation of $D$ we have: $$D= \left( \begin{array}{ccccc} \frac{3}{4}x^2 & 0 & 0 & 0 & 0 \\ 0 & \frac{3}{2}x^2 & 0 & 0 & 0 \\ 0 & 0 & \frac{9}{4}x^2 & 0 & 0 \\ 0 & 0 & 0 & 3x^2 & 0 \\ 0 & 0 & 0 & 0 & \frac{15}{4}x^2 \\ \end{array} \right).$$ **Four-step nilpotent Lie algebra $\frak{l}_{5,6}$, (case B)** -------------------------------------------------------------- In the case B for the same structure constants of the above case, we have $s>0$. Then we see that the system of equations defined by (\[Main formula\]) has no solution. Therefore similar to the case $\frak{l}_{5,7}$ (case B), the Lie algebra $\frak{l}_{5,6}$ (case B), does not admit an algebraic Ricci soliton structure. **Three-step nilpotent Lie algebra $\frak{l}_{5,5}$** ----------------------------------------------------- This section is devoted to $\frak{l}_{5,5}$ which is a three-step nilpotent Lie algebra with one dimensional center. For the real numbers $s,u\geq0$, and $m,v,w>0$, the non-vanishing Lie brackets are as follows: $$[E_1,E_2]=mE_4+sE_5, \ \ \ [E_1,E_3]=uE_5, \ \ \ [E_1,E_4]=vE_5, \ \ \ [E_2,E_3]=wE_5.$$ Hence for the structure constants we have: $$\begin{aligned} &&\alpha_{124}=-\alpha_{214}=m, \ \alpha_{125}=-\alpha_{215}=s, \ \alpha_{135}=-\alpha_{315}=u, \\ &&\alpha_{145}=-\alpha_{145}=v, \ \alpha_{235}=-\alpha_{325}=w.\end{aligned}$$ Now substituting the above structure constants in the equation (\[Main formula\]), leads us to a system of equations with the following solution, $$u=s=0, v=m, w=\frac{\sqrt{2}}{2}m, c=-\frac{7}{4}m^2.$$ So the Lie algebra $\frak{l}_{5,5}$ admits an algebraic Ricci soliton structure if and only if the above equations hold. In this case, by (\[Derivation equation\]), the matrix representation of $D$ in the basis $\{E_1, \cdots, E_5\}$, is of the following form: $$D= \left( \begin{array}{ccccc} \frac{3}{4}m^2 & 0 & 0 & 0 & 0 \\ 0 & m^2 & 0 & 0 & 0 \\ 0 & 0 & \frac{3}{2}m^2 & 0 & 0 \\ 0 & 0 & 0 & \frac{7}{4}m^2 & 0 \\ 0 & 0 & 0 & 0 & \frac{5}{2}m^2 \\ \end{array} \right).$$ **Three-step nilpotent Lie algebra $\frak{l}_{5,9}$, (case A)** --------------------------------------------------------------- For the non-vanishing Lie brackets of this case we have: $$[E_1,E_2]=mE_3+sE_4+uE_5, \ \ \ [E_1,E_3]=vE_4, \ \ \ [E_2,E_3]=wE_5,$$ where $m>0$, $w>v>0$ and $s,u\geq0$. So we have: $$\begin{aligned} &&\alpha_{123}=-\alpha_{213}=m, \ \alpha_{124}=-\alpha_{214}=s, \ \alpha_{125}=-\alpha_{215}=u, \\ &&\alpha_{134}=-\alpha_{314}=v, \ \alpha_{235}=-\alpha_{325}=w.\end{aligned}$$ Easily we see that the system of equations which is defied by (\[Main formula\]) has no solution. Therefore, the case A of the Lie algebra $\frak{l}_{5,9}$ does not admit Algebraic Ricci soliton structure. **Three-step nilpotent Lie algebra $\frak{l}_{5,9}$, (case B)** --------------------------------------------------------------- In the case B of the Lie algebra $\frak{l}_{5,9}$, for the same Lie algebra brackets, we have $m,v>0$, $v=w$ and $s\geq0$. Then, the equation (\[Main formula\]), leads us to the following system of equations: $$\left\{ \begin{array}{l} mvs=0,\\ cm+\frac{3}{2}(m^3+s^2m)=0, \\ cs+\frac{3}{2}(s^3+m^2s)+2v^2s=0,\\ cv+\frac{3}{2}s^2v+2v^3=0,\\ cv+\frac{1}{2}s^2v+2v^3=0. \end{array} \right.$$ The solution of the above system of equations is as follows: $$s=0, v=\frac{\sqrt{3}}{2}m, c=-\frac{3}{2}m^2.$$ So the case B of the Lie algebra $\frak{l}_{5,9}$ admits an algebraic Ricci soliton structure with the following derivation: $$D= \left( \begin{array}{ccccc} \frac{5}{8}m^2 & 0 & 0 & 0 & 0 \\ 0 & \frac{5}{8}m^2 & 0 & 0 & 0 \\ 0 & 0 & \frac{5}{4}m^2 & 0 & 0 \\ 0 & 0 & 0 & \frac{15}{8}m^2 & 0 \\ 0 & 0 & 0 & 0 & \frac{15}{8}m^2 \\ \end{array} \right).$$ [99]{} L. F. Di Cerbo, *Generic properties of homogeneous Ricci solitons*, Adv. Geom., **14** (2014), 225-237. A. Figula and P. T. Nagy, *Isometry classes of simply connected nilmanifolds*, J. Geom. Phys., **132**(2018), 370–381. S. Homolya and O. Kowalski, *Simply connected two-step homogeneous nilmanifolds of dimension 5*, Note Math., **26**(2006), 69–77. M. Jablonski, *Homogeneous Ricci solitons*, J. Reine Angew. Math., **699** (2015), 159-182. M. Jablonski, *Homogeneous Ricci solitons are algebraic*, Geom. Topol., **18**(2014), 2477-2486. J. Lauret, *Ricci soliton solvmanifolds*, J. Reine Angew. Math., **650** (2011), 1-21. J. Lauret, *Einstein solvmanifolds and nilsolitons*, Contemp. Math., **491** (2009), 1-35. J. Lauret, *Ricci soliton homogeneous nilmanifolds*, Math. Ann., **319** (2001), 715-733. H. R. Salimi Moghaddam, *Left invariant Ricci solitons on three-dimensional Lie groups*, J. Lie Theory **29** (2019), No. 4, 957–968. E. N. Wilson, *Isometry groups on homogeneous nilmanifolds*, Geom. Dedicata, **12**(1982), 337–346.

--- abstract: 'In this paper we propose a higher dimensional Cosmology based on FRW model and brane-world scenario. We consider the warp factor in the brane-world scenario as a scale factor in 5-dimensional generalized FRW metric, which is called as *bulk scale factor*, and obtain the evolution of it with space-like and time-like extra dimensions. It is then showed that, additional space-like dimensions can produce exponentially bulk scale factor under repulsive strong gravitational force in the empty universe at a very early stage.' author: - 'M. Mohsenzadeh$^{1}$[^1] and E. Yusofi$^{2}$[^2]' title: '*Bulk Scale Factor* at Very Early Universe ' --- 23 cm -0.5cm -0.15cm =cmcsc10 =cmti10 Keywords: FRW Model; Scale Factor; de-Sitter space-time; Extra Dimensions Introduction and Motivation =========================== In 1917, a year after Einestein introduced general relativity, he derived a static cosmological model which was closed [@c1]. His motivation in driving this model was the strong believe that the universe is static. For this reason, he introduced cosmological constant in the field equation which is, \_[0]{}=8G \_[0]{} Where $ G $ is Newton’s constant of gravitation and $ \rho_{0} $ is density of the whole universe. In the same year, W. de-Sitter obtained another solution to the modified Einstein’s field equation [@c2]. The de-Sitter solution reads as equations [@c3] ds\^[2]{}=dt\^[2]{}-a\^[2]{}(t)\[dx\_[1]{}\^[2]{}+dx\_[2]{}\^[2]{}+dx\_[3]{}\^[2]{}\]=dt\^[2]{}-e\^[2Ht]{}\[dx\_[1]{}\^[2]{}+dx\_[2]{}\^[2]{}+dx\_[3]{}\^[2]{}\] Where $1$,$2$,$3$ indices refer to the three spatial dimensions and $ a(t) $ is scala factor and $ H=\sqrt{\frac{\Lambda}{3}} $. The de-Sitter metric is a solution to homogenous Einstein’s equation, [*i.e.*]{} for empty universe and in the past two decades, observational data have shown that our universe might be in de-Sitter phase. The most successful model, having wondrous predictive power for cosmology, was obtained by Friedmann in 1922 [@c4], as well as by Robertson and Walker in 1935 and 1936,respectively [@c5; @c6], which is called as Friedmann-Robertson-Walker ( FRW ) model. This expanding and centrally symmetric homogeneous model obey the cosmological principle. The successfulness of this model is revealed by the fact that the big-bang model is based on this model [@c7; @c8]. After Hubble’s observation, in 1929, Einstein realized that he committed a mistake by introducing the cosmological constant. Due to this epoch-making observation, cosmologists were prompted to think for expanding models of the universe more seriously, though de-Sitter and Friedmann had already derived non-static models by this time. So, it was natural to think that if the present universe is so large, it would have been very small and dense in the extreme past. In 1940, George Gamow addressed to this question and proposed that, in the beginning of its evolution, universe was like an extremely hot ball [@c9]. He called this ball as primeval atom and this model is popularly known as the big-bang model. On the other hand, the idea that our universe may consist of more dimensions than the usual 4 dimensional space-time, was first considered by T. Caluza in the 1919 [@c10; @c11; @c12]. His aim was to unify gravity and electromagnetism. Later in 1999, the 5D (5-dimensional) warped geometry theory, which is a brane-world theory, developed by Lisa Randall and Ramam Sundrum, while trying to solve the hierarchy problem of the Standard Model, which is called RS model [@c13; @c14]. They considered one extra “non-factorizable” dimension and the metric was found for RS model to be given by [@c15; @c16; @c17]ds\^[2]{}=g\_a\^[2]{}(y)dx\^dx\^-dy\^[2]{}=g\_ e\^[-2ky]{} dx\^dx\^-dy\^[2]{} Where Greek indices $\mu,\nu=(0,1,2,3)$ are refer to the usual four observable dimensions, and $\textit{y}$ signifies the coordinate on the additional dimension of length $\textit{r}$ and $ a(y) $ is called the warp factor and $ k $ is of order of the Plank mass. Here it is assumed to have two branes. One at $ y=0 $ called Planck-brane ( where gravity is relatively a strong force ) and another at $ y=\pi r $ called the TeV-brane ( where gravity is relatively a weak force ). In that case, by warping any lagrangian mass parameter which is naturally $ \approx{M_{PL}} $ ( Planck mass ), will appear to us in the 4D space-time on the TeV-brane to be $ \approx{TeV} $ (Tera electron-volt). For example, lets see how this works for the Higgs field on the TeV-brane. First, we right down the action for the Higgs field in 5D S=d\^[4]{}x dy \[ g\^\_\^\_-(\^[2]{}-\_[0]{}\^[2]{})\^[2]{}\] (y-r)Where $ \nu_{0} $ is the vacuum expectation value of the Higgs field and it is of order of the Planck mass, and $ \hat{H} $ is Higgs field operator. The integration over extra dimension is easy to calculate S=d\^[4]{}x\[ e\^[-2kr ]{}\_\^\^-e\^[-4kr ]{} (\^[2]{}-\_[0]{}\^[2]{})\^[2]{}\]If we re-scale the Higgs field, $ \hat{H}\rightarrow e^{kr \pi}\hat{H} $, this gives:S=d\^[4]{}x\[ \_\^\^-(\^[2]{}-\_[0]{}\^[2]{}e\^[-2kr]{})\^[2]{}\]Thus the Higgs field is still seen as a Higgs field in four dimensional space-time, but the vacuum expectation value is now\^[2]{}=\_[0]{}\^[2]{}e\^[-2kr]{} Which is at the TeV-scale. Thus by considering warped extra dimensions one is able to solve the hierarchy problem. This means that transition from 4D world to 5D world, exponentially shrink sizes and grows mass and energy [@c18; @c19; @c20]. The possibility of the existence of extra dimensions has opened up new and exciting avenues of research in quantum gravity and quantum cosmology ( for instance see recently works [@c21; @c22; @c23; @c24; @c25] ). Since a higher dimensional world has more energy ( the range of Planck-scale ) relative to 4D world ( the range of TeV-scale ), in this paper, similar to Gamow idea, we suggest higher dimensional world as primeval atom. In contrast to warped geometry model, which makes transition from 4D world to the 5D world, we go from 5D world to the 4D world. By considering the symmetry, an expansion in size in our scenario is expected. Indeed, we intend to propose a higher dimensional Cosmology based on FRW model and brane-world scenario. For this purpose, We consider generalized form of FRW metric in 5D space-time and by solving of the Einstein equation for this 5D space-time, we obtain the evolution of the bulk scale factor. Thus, the outline of paper is as follows: In section 2, the scale factor in 4D de-Sitter space-time is recalled briefly. In section 3, we consider the evolution of bulk scale factor in 5D space-time with space-like and time-like extra dimensions. Some conclusions are given in final section. Scale factor in 4D de-Sitter space-time ======================================= The standard FRW space-time is[@c7] ds\^[2]{}=dt\^[2]{}-a\^[2]{}(t)\[dx\_[1]{}\^[2]{}+dx\_[2]{}\^[2]{}+dx\_[3]{}\^[2]{}\]. Where $1$,$2$,$3$ indices refer to the three spatial dimensions. In this case, for empty universe, $ T_{\mu\nu}=0 $, and cosmological constant, $ \Lambda\neq 0 $, which is the source of gravitation in place of a massive object, the Enistein’s field equation reads R\_=g\_. Then we have:-=. and a+2\^[2]{}=(-a\^[2]{}). Then from (10) and (11) we have:$$(\frac{\dot{a}}{a})^{2}=H^{2}=\frac{\Lambda}{3}.$$ Which integrates to: a(t)=a\_[0]{}e\^[Ht]{}=a\_[0]{}e\^[t]{}. Where $ a_{0} $ is the scale factor at the end of period inflation in 4D de-Sitter space-time. So, non-static form of de-Sitter space-time with exponentially scale factor is obtained asds\^[2]{}=dt\^[2]{}-a\_[0]{}\^[2]{}e\^[2Ht]{}\[dx\_[1]{}\^[2]{}+dx\_[2]{}\^[2]{}+dx\_[3]{}\^[2]{}\]. Evolution of Bulk scale factor ============================== By considering the standard FRW metric given by (8) and the metric (3), generalized form RS brane-world scenario [@c15], we suggest the generalized form of 5D FRW metric as ds\^[2]{}=g\_[ab]{}dx\^[a]{}dx\^[b]{}=\_a\^[2]{}(y)dx\^dx\^+r\^[2]{}dy\^[2]{}, Where g\_[ab]{}=*diag*(a\^[2]{}(y),-a\^[2]{}(y),-a\^[2]{}(y),-a\^[2]{}(y),r\^[2]{}).Which $a,b=(0,1,2,3,4)$ and Greek indices $\mu,\nu=(0,1,2,3)$, refer to the four observable dimensions, and $\textit{y}$ signifies the coordinate on the additional dimension of length $\textit{r}$, and scale factor $\tilde{a}=a(y) $, is a ${\textit{bulk scale factor}}$, which only respect to the extra dimension coordinate $\textit{y}$. For space-like extra dimensions (**SLED**), $ r^{2}=-1 $ and for time-like extra dimensions (**TLED**), $ r^{2}=+1 $ [@c26; @c27; @c28]. Similar to section 2, for empty universe which $ T_{ab}=0 $, and bulk cosmological constant, $ \widetilde{\Lambda}\neq 0 $, the Enistein’s field equation reads as R\_[ab]{}= g\_[ab]{}. Then for **TLED** we have:-3\^[2]{}-= \^[2]{}, Which $ab=(00,11,22,33)$, and for $ ab=44 $, we have =.Where dot denotes derivative with respect to $y$ as a **TLED**. From Eq.(17) together with Eq.(18), we have: $$(\frac{\dot{\tilde{a}}}{\tilde{a}})^{2}=\widetilde{H}^{2}=\frac{-5\widetilde{\Lambda}}{12}$$ So we obtain, =\_[0]{}e\^[y]{}.Where $ \tilde{H} $ is 5D Hubble parameter, and $ \tilde{a_{0}} $ is the bulk scale factor at the end of its expansion in 5D space-time. In this case, $ \widetilde{\Lambda} < 0 $, and we have an expanding space-time with harmonically scale factor (19) as:ds\^[2]{}=\_[0]{}\^[2]{}e\^[2y]{}\[dt\^[2]{}-(dx\_[1]{}\^[2]{}+dx\_[2]{}\^[2]{}+dx\_[3]{}\^[2]{})\]+dy\^[2]{}. On the other hand, the Enistein’s field equation, $ R_{ab}=\widetilde{\Lambda} g_{ab} $, for **SLED** yields to:3’\^[2]{}+’= \^[2]{}.where $ ab=(00,11,22,33) $, and for $ ab=44 $ we have: = (-1).Where prime denotes derivative with respect to $ y $ as a **SLED**. Then from (21) and (22), we have: $$(\frac{\tilde{a}'}{\tilde{a}})^{2}=\widetilde{H}^{2}=\frac{\widetilde{\Lambda}}{4}$$ Which integrates to: =\_[0]{}e\^[y]{}=\_[0]{}e\^[y]{}Where $ \widetilde{H} $ is 5D Hubble parameter, and $ \tilde{a}_{0} $ is the bulk scale factor at the end of its expansion in 5D space-time. In this case, $ \widetilde{\Lambda} > 0 $, and we have an expanding space-time with exponentially scale factor (23) as:ds\^[2]{}=\_[0]{}\^[2]{}e\^[2y]{}\[dt\^[2]{}-(dx\_[1]{}\^[2]{}+dx\_[2]{}\^[2]{}+dx\_[3]{}\^[2]{})\]-dy\^[2]{}.Though, in the brane picture, the electromagnetism and the weak and strong nuclear forces are localized on the TeV-brane, but gravity has no such constraint and so much of its attractive power “leaks” into the bulk and the force of gravity should appear significantly stronger on small scales. So, the form of expansion (24), can be as a consequent of repulsive of this strong gravitational force in the very early universe with extra dimensions. Discussions and Conclusions =========================== We considered the generalized form of FRW metric similar to generalized RS brane-world scenario, which admits both **SLED** and **TLED**. We solved the Einstein equation for an empty 4D and 5D space-time and demonstrated that the evolution of the bulk scale factor in 5D space-time for **SLED** case (24), is exponential form similar to scale factor in 4D space-time (13). Besides, the following results and points can be obtained from this work: $\bullet$ In this scenario, due attention to hierarchy problem and brane-world Cosmology, it seems more logical that, the 5D very early universe to have further energy in comparison with the 4D early universe. $\bullet$ In the higher dimensional very early universe, incorporation of hidden extra dimensions and the visible 4D space-time, the unification of gravitational force and gauge forces seems to be more possible. $\bullet$ In this scenario, for TLED we have an harmonically bulk scale factor (20) and the strong gravitational force is likely harmonic ( similar to non-static form of anti de-Sitter 5D space-time ). $\bullet$ In this scenario, for SLED we have an exponentially bulk scale factor (24) and the strong gravitational force is likely repulsive ( similar to non-static form of de-Sitter 5D space-time ). On the other hand, in the brane-world model, by warping any lagrangian mass, shrink size and the strong gravitational force is likely attractive. Consequently, in the bulk and in the Planck scale of energy, gravitation can be attractive as well as repulsive. Thus, the quantum effects of gravitation are of great importance [@c24]. $\bullet$ This mechanism of expansion emanate of transition from unstable and dense state of 5D world ( where is situate on the exited mode of Kaluza-Klein model and under effects of strong gravitational force ) to the 4D world ( where is under effects of weak gravitational force with lesser energy ) [@c29]. These highlight points, will motivate us to suggest another physics for very early universe in the higher dimensional Cosmology. [**[Acknowlegements]{}**]{}: This work has been supported by the Islamic Azad University-Qom Branch, Qom, Iran. [a]{} A.Einstein,(1917), *Preuss.Akad.Wiss.Berlin, Sitzber.*,142. W.de Sitter,(1917), *Proc. Koninkl. Akad. Weteusch. Amsterdam 19,1217.* A.Einstein and W.de Sitter,(1932), *Proc.Nat.Acad.Sci.(USA)18,213.* A.Z. Friedmann, Phys, 10, 377 (1922). H.P. Robertson, Astrophys. J., 82, 284 (1935). A.G.Walker, Proc. London Math. Soc. 42 (2), 90 (1936). S. M. Carroll: An Introduction to General Relativity: Spacetime and Geometry. Addison Wesley, Reading (2004). S.K. Srivastava: General Relativity and Cosmology. PHI (2008). Gamow, G (1948), *Nature,***162,**680. T.Kaluza, *Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.) **K1**,966(1921)*. O. Klein, *Z. Phys.***37**, 895(1926)\[*Surveys High Energy. Phys. 5,241(1926)\]*. O. Klein, Nature **118**,516(1926). L. Randal, and L. Sundrum: Phys. Rev. Lett. 83, 4690(1999) \[arxiv:hep-th/9906064\]. N. Arkani-Hamed, S. Dimopoulos, N. Kalpor and R. Sundrum: Phys. Lett. B480, 193(2000)\[arXiv:hep-th/0001197\]. S. Das, D. Maity, and S. SenGupta: \[arxiv:hep-th /0711.1744v1\]. V. I. Afonso, D. Bazeia, R. Menezes, and A. Yu. Petrov: \[arxiv:hep-th/0710.3790v\]. J. Sadeghi, and A. Mohammadi :2007 Eur. Phys. J. C 49, 859-864. K. R. Dienes, Tasi lectures 2002. T. G. Rizzo:\[hep-ph/0409309\]. F. J. Yndurain: Phys. Lett. B256, 15-16 (1991). M. Holthausen and R. Takahashi: Phys. Lett. B691 (2010), p. 56. V. Sahni and Y. Shtanov,arXiv:0811.3839. P. J. Steinhardt and D. Wesley:Phys.Rev.D79:104026,2009. K. Nozari and S. H. Mehdipour,Int.J.Mod.Phys.A21:4979-4992,2006. T.R. Mongan,Gen. Rel. Grav33,1415-1424(2001). M. Chaichian and A. B. Kobakhidze,Phys. Lett. B 488 (2000) 117 \[arXiv:hep-th/0003269\]. Z. Berezhiani, M. Chaichian and A. B. Kobakhidze and Z. H. Yu, Phys. Lett. B 517 (2001) 387 \[arXiv:hep-th/0102207\]. Tianjun Li,Phys.Lett. B503 (2001) 163-172. Hsin-Chia Cheng:\[hep-ph/1003.1162V1\]. [^1]: e-mail: [email protected] [^2]: e-mail: [email protected]

--- abstract: 'We investigate the critical current density and vortex dynamics in single crystals of Ca(Fe$_{1-x}$Co$_{x}$)$_2$As$_2$ ($x$ = 0.051, 0.056, 0.065, and 0.073). The samples exhibit different critical temperatures and superconducting phase fractions. We show that in contrast to their Ba-based counterpart, the crystals do not exhibit a second peak in the field dependence of magnetization. The calculated composition-dependent critical current density ([$J_{\rm{C}}$]{}) increases initially with Co doping, maximizing at $x$ = 0.065, and then decreases. This variation in [$J_{\rm{C}}$]{} follows the superconducting phase fractions in this series. The calculated [$J_{\rm{C}}$]{} shows strong temperature dependence, decreasing rapidly upon heating. Magnetic relaxation measurements imply a nonlogarithmic dependence on time. We find that the relaxation rate is large, reflecting weak characteristic pinning energy. The analysis of temperature- and field- dependent magnetic relaxation data suggests that vortex dynamics in these compounds is consistent with plastic creeping rather than the collective creep model, unlike other 122 pnictide superconductors. This difference may cause the absence of the second peak in the field dependent magnetization of Ca(Fe$_{1-x}$Co$_{x}$)$_2$As$_2$.' author: - 'A. K. Pramanik' - 'L. Harnagea' - 'S. Singh' - 'S. Aswartham' - 'G. Behr' - 'S. Wurmehl' - 'C. Hess' - 'R. Klingeler' - 'B. Büchner' title: 'Critical current and vortex dynamics in single crystals of Ca(Fe$_{1-x}$Co$_{x}$)$_2$As$_2$' --- Introduction ============ The recent discovery of superconductivity (SC) in Fe-based pnictides with chemical formula LaO$_{1-x}$F$_x$FeAs (Ref. ) has attracted a lot of research activities in the field of condensed matter physics in general and superconductivity in particular. [@takahashi; @zhao; @luetkens; @christ; @grafe; @klauss; @qazi; @singh] Similar to cuprates, pnictide superconductors also exhibit high critical temperature ([$T_{\rm{C}}$]{}) and type-II nature. Moreover, SC arises from the layers, i.e., Fe-As layers in this case. In type-II superconductors, magnetic fields above the lower critical field ([$H_{\rm{c1}}$]{}) penetrate the bulk of the superconductor in the form of flux lines or vortices, which in presence of an external current and/or magnetic field move, thus causing finite dissipation to the transport current. This motion of vortices is further assisted by thermal fluctuations. However, vortex movement can be hampered due to the pinning barriers arising from disorders present in system. The measurement of isothermal magnetization ($M$) vs. field ($H$) and the magnetic relaxation are the most extensively used tools to study the vortex dynamics in a variety of superconducting materials. [@yeshurun; @blatter] Commonly, the appearance of a second peak (SP) in field dependent magnetization loop is believed to be directly associated with the nature of pinning and thereof depending vortex creep mechanism. [@abulafia; @giller] Therefore, understanding the vortex dynamics is one of the central issue in high-[$T_{\rm{C}}$]{} superconductors pertaining to both basic science as well as technological applications. In the case of the cuprates, high anisotropy and small coherence lengths ($\xi$) render the pinning energy weak which in combination with their high working temperature results in giant flux creep and large magnetic relaxation. [@yeshurun; @blatter] Although Fe-based superconductors share many aspects with the cuprates but they have less anisotropy and larger $\xi$. [@can-aniso] Therefore, it remains a matter of interest to understand the vortex dynamics in these materials. Here, we study the vortex dynamics by means of isothermal $M$($H$) as well as magnetic relaxation measurements on single crystals of Ca(Fe$_{1-x}$Co$_{x}$)$_2$As$_2$ ($x$ = 0.051, 0.056, 0.065, and 0.073). All samples are superconductors having varying [$T_{\rm{C}}$]{} and superconducting phase fractions. The parent compound CaFe$_2$As$_2$ belongs to the 122-family of Fe-based pnictides which has the generic formula AFe$_2$As$_2$ where A is a alkaline earth element (i.e., Ba, Sr, Ca). Unlike its 1111 analogue, 122 compounds are oxygen free and easier to grow with substantial crystal size. [@ni; @can-Ca] At room temperature, all 122 compounds crystallize in the ThCr$_2$Si$_2$-type tetragonal structure, however, the significant mismatch in ionic radii of atoms at the A-site will presumably create local distortion, which will vary in different systems. Around $T \simeq 170$ K, CaFe$_2$As$_2$ exhibits a first order structural phase transition from a high-temperature tetragonal to a low-temperature orthorhombic symmetry which is accompanied by formation of a spin density wave (SDW)-type antiferromagnetic (AFM) order. [@ronning; @can-Ca] Partial doping with 3d elements like Co and Ni at the Fe-site or with Na and K at the A-site yields electron or hole doping, respectively, in the system. Both the doping suppresses the structural and SDW-transitions, and induces SC. [@rotter; @sasmal; @dong; @neeraj1; @neeraj2; @rudi-Ca] The vortex dynamics in Ba-122 superconducting compounds has been investigated extensively and described within the framework of a collective pinning scenario. These studies yield the presence of a SP in $M$($H$), a moderate critical current density [$J_{\rm{C}}$]{} and strong vortex pinning. [@prozo-Ba07; @yang; @prozo-Ba; @nakajima; @kim; @sun; @shen; @eskildsen] Moreover, the general appearance of the SP in [Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$]{} remains irrespective of the crystal growth conditions. Interestingly, the SP manifestation is less pronounced in strongly under- and over-doped [Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$]{} where pinning is relatively weak as compared to the optical doping. [@shen] In order to examine whether the presence of the SP is common to the whole 122-family, we have studied the vortex dynamics in Co doped Ca-122 compounds which exhibit similar [$T_{\rm{C}}$]{} as their Ba-122 counterparts but differ in terms of ionic radii of Ca$^{2+}$ and Ba$^{2+}$. [@ni; @rudi-Ca] To the best of our knowledge, the present study is the first of its kind in Ca-122 compounds. Our study shows the absence of the SP in $M$($H$) and we calculate relatively low critical currents [$J_{\rm{C}}$]{}. Both findings are in contrast to previous results on Ba-122. While our data imply a nonlogarithmic time dependence of the magnetic relaxation, however, the relaxation rate is large indicating weak pinning potential. From the detailed analysis of relaxation data we infer that vortex movement in Ca-122 occurs through the plastic creeping which may explain the absence of the SP in Ca(Fe$_{1-x}$Co$_{x}$)$_2$As$_2$. Experimental Details ==================== Single crystals of Ca(Fe$_{1-x}$Co$_{x}$)$_2$As$_2$ with $x$ = 0.051, 0.056, 0.065, and 0.073 have been grown from Sn-flux. The details of sample preparation and characterization will be presented in elsewhere. [@surjeet] Note, that in the present work the same crystals as in Ref.  have been used. The crystal structure was investigated by means of powder x-ray diffraction (XRD) which implies impurity concentrations of less than 3%. The compositions have been determined by means of energy dispersive x-ray spectroscopy (EDX). The EDX analysis is performed at different places of each sample. The variation in Co distribution is found within the limit of $x$ $\pm$ 0.003. The above mentioned Co concentration ($x$) is the average value obtained after several individual measurements. For the studies at hand, crystals of rectangular shape have been used with typical dimensions of (1-2) $\times$ (1-2) $\times$ (0.3-0.4) mm$^3$. Magnetization measurements have been performed in a Quantum Design VSM-SQUID. Magnetic isotherms have been measured after cooling the samples in zero magnetic field to the specific temperatures. Before collecting each $M$($H$) plot, the samples have been heated much above their respective [$T_{\rm{C}}$]{}’s and care has been taken for proper thermal stabilization of the samples before measuring the isotherms. The typical field sweep rate for $M$($H$) studies was 80 Oe/s. For magnetic relaxation measurements, the samples were zero field cooled from well above $T_C$ to the selected temperatures ($T_i$) and after proper thermal stabilization the desired field ($H_a$) has been applied and magnetization has been measured as a function of time ($t$) for about 12000 s. Similar to the $M$ vs. $H$ measurements, the samples have been warmed considerably above [$T_{\rm{C}}$]{}before collecting $M$($t$). Results and discussions ======================= Fig. 1 presents the temperature dependence of the volume susceptibility ([$\chi_{\rm{vol}}$]{}) for Ca(Fe$_{1-x}$Co$_{x}$)$_2$As$_2$ with $x$ = 0.051, 0.056, 0.065 and 0.073, respectively. [$\chi_{\rm{vol}}$]{} has been deduced from dc-magnetization data measured at $H = 20$ Oe following the zero field cooling (ZFC) protocol. The field has been applied parallel to the crystallographic $c$-axis. The data have been corrected for the demagnetization effect where the demagnetization factor has been estimated from the physical dimensions of the samples. [@osborn] [Fig.\[fig:Fig1\]]{} clearly shows SC for all samples at low temperature. Upon progressively increasing the Co-content $x$, the superconducting transition becomes sharper till $x$ = 0.065, and then it again broadens. The lower susceptibility and broad transition at $x$ = 0.051 and 0.073 suggest that a larger amount of inhomogeneity is associated with these samples. This inhomogeneity is intrinsic to the samples as we do not find other chemically inhomogeneous phases within the resolution limit of XRD. This inhomogeneity may be associated with the coexistence of superconducting and SDW phases arising from the spontaneous electronic phase separation. [@chu; @park] We mention that, our EDX results imply reasonable homogeneous distribution of Co in macroscopic scale, and thus render a chemical inhomogeneity an unlikely explanation. The current measurements are not sufficient to determine the nature of inhomogeneity in these samples. A small kink in [$\chi_{\rm{vol}}$]{}($T$) at low temperature around 3.5 K is most probably associated with a small fraction of included Sn and is common to similar samples made from Sn-flux. [@ronning] We note that the superconducting phase fractions inferred from [Fig.\[fig:Fig1\]]{} are very similar to what has been reported for Ba(Fe$_{1-x}$Co$_{x}$)$_2$As$_2$. [@ni] The superconducting transition temperatures amount to 18.5, 19.0, 17.2, and 16.8 K for $x$ = 0.051, 0.056, 0.065, and 0.073, respectively (see the inset of Fig. 1). Though there is only a small change in [$T_{\rm{C}}$]{} within the studied compositions, the superconducting volume fraction exhibits a clear doping dependence with a maximum around $x=0.065$. [@surjeet] ![(Color online) Volume susceptibility [$\chi_{\rm{vol}}$]{} of Ca(Fe$_{1-x}$Co$_{x}$)$_2$As$_2$ deduced from dc magnetization measured in $H=20$ Oe with field parallel to the $c$-axis as a function of temperature. The data have been obtained following zero field cooling protocol and were corrected by the demagnetization factor. Inset: [$T_{\rm{C}}$]{} vs. Co-content $x$. Line is guide to the eyes.[]{data-label="fig:Fig1"}](Fig1.eps){width="0.95\columnwidth"} Isothermal magnetization curves ($M$ vs. $H \parallel c$) collected at temperatures $\approx$0.5T$_C$ are shown in [Fig.\[fig:Fig2\]]{}. The data exhibit a central peak at zero magnetic field and then magnetization decreases continuously with increasing magnetic fields. The sharp peak around $H=0$ is similarly observed for other materials in the 122-family. [@prozo-Ba07; @kim; @shen] Note, however, that the central peak appears to be slightly sharper than found recently in [Ba(Fe$_{0.93}$Co$_{0.07}$)$_2$As$_2$]{}. [@prozo-Ba07] Remarkably, there is no SP in $M$($H$) within the measured field range for all samples which is in stark qualitative contrast to previous results for [Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$]{}. Note, that a SP is also missing when measuring $M$ vs. $H$ with field perpendicular to the $c$-axis for $x$ = 0.056 (data not shown). Moreover, the $M$($H$) plots are rather symmetric with respect to the polarity of applied field. From the irreversible part of the M vs H loop we have calculated the superconducting critical current [$J_{\rm{C}}$]{} exploiting the Bean critical state model which describes the penetration of magnetic field into type-II superconductors:[@bean] ![(Color online) (a) The isothermal magnetization measured at temperature $\approx$ 0.5T$_C$ as a function of field is plotted for Ca(Fe$_{1-x}$Co$_{x}$)$_2$As$_2$ series. The field is applied parallel to crystallographic c axis. (b) The critical current density is plotted against field for Ca(Fe$_{1-x}$Co$_{x}$)$_2$As$_2$ series with H $\parallel$ c axis. Inset shows composition dependence of critical current density at H = 0 obtained from main panel. The line is a guide to the eyes.[]{data-label="fig:Fig2"}](Fig2.eps){width="0.95\columnwidth"} $$\begin{aligned} J_c = 20 \frac{\Delta M}{a \left(1-\frac{a}{3b} \right)}\end{aligned}$$ where $\Delta M = M_{\rm dn} - M_{\rm up}$, $M_{\rm up}$ and $M_{\rm dn}$ is the magnetization measured with increasing and decreasing field, respectively, $a$ and $b$ ($b > a$) are the dimensions of the rectangular cross-section of the crystal normal to the applied field. [@footnote] The unit of $\Delta M$ is in emu/cm$^3$, $a$ and $b$ are in cm and the calculated [$J_{\rm{C}}$]{} is in A/cm$^2$. As expected from $M$($H$) in Fig. 2a, [$J_{\rm{C}}$]{} also does not show a SP, and decreases monotonically with field. Interestingly, it exhibits a pronounced doping dependence but its values are more than one order of magnitude smaller than in the Ba-122 counterparts. [@prozo-Ba07; @yang; @prozo-Ba; @nakajima; @kim; @sun; @shen] This low [$J_{\rm{C}}$]{} indicates weak pinning in [Ca(Fe$_{1-x}$Co$_x$)$_2$As$_2$]{}. The inset of Fig. 2b displays [$J_{\rm{C}}$]{} at $H = 0$ for the different doping levels $x$ under study. [$J_{\rm{C}}$]{} initially increases with Co concentration showing a maximum at $x = 0.065$ and a decrease upon further increase of the Co content. Since [$J_{\rm{C}}$]{} is associated with the pinning of the vortices our data hence imply that pinning becomes stronger upon initial doping. In particular, it appears that the variation of [$J_{\rm{C}}$]{}($x$) and thus of the pinning strength do not follow the the variation of [$T_{\rm{C}}$]{}($x$) (see Fig. 1). The dome-like shape of [$J_{\rm{C}}$]{}(x) in Fig. 2b is similarly observed in superconducting [Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$]{} where it was shown that [$J_{\rm{C}}$]{} enhances due to an increase in intrinsic pinning arising from the domain walls of the coexisting AFM/orthorhombic phase. These walls act as effective extended pinning centers which upon suppression of the related ordering phenomena by doping become more fine and intertwined, thus giving rise to enhanced [$J_{\rm{C}}$]{}when increasing $x$ in the underdoped regime. [@prozo-Ba] Though our preliminary analysis shows that the variation in [$J_{\rm{C}}$]{}($x$) follows the [$T_{\rm{C}}$]{}($x$) in Ca(Fe$_{1-x}$Co$_{x}$)$_2$As$_2$, it needs to be further investigated whether such phase inhomogeneities play a crucial role in controlling [$J_{\rm{C}}$]{} in these samples. ![(Color online) The field dependence of (a) magnetization and (b) critical current density at different temperatures for [Ca(Fe$_{0.944}$Co$_{0.056}$)$_2$As$_2$]{}. Inset: Temperature dependence of the critical current density [$J_{\rm{C}}$]{} at $H = 0$ for the same sample. The line is a guide to the eyes.[]{data-label="fig:Fig3"}](Fig3.eps){width="0.95\columnwidth"} In order to study the temperature variation of [$J_{\rm{C}}$]{} we have measured the magnetization as a function of field at different temperatures for [Ca(Fe$_{0.944}$Co$_{0.056}$)$_2$As$_2$]{} which is the material with the highest [$T_{\rm{C}}$]{} in our study. Fig. 3a shows the recorded $M$($H$) curves at 2, 5, 10, 15 and 18 K, i.e. within the superconducting regime. The width of the hysteresis increases with lowering the temperature which suggests that the pinning characteristics becomes stronger at low temperature. However, it is worth noting that we do not find the SP in $M$($H$) at any temperature below [$T_{\rm{C}}$]{}. In order to assess the pinning strength, we have again calculated [$J_{\rm{C}}$]{}($H$) following Eq. 1 from the magnetization data (see Fig. 3b). With increasing temperature, [$J_{\rm{C}}$]{} decreases as is can already be inferred from the reduced $\Delta M$ in Fig. 3a. A strong temperature dependence of [$J_{\rm{C}}$]{}($H=0$) is demonstrated by the inset of Fig. 3b. ![(Color online) Time dependence of the ZFC magnetization as normalized at $t = 0$ recorded at various temperatures. The magnetization is measured in $B$ = 10 kOe applied parallel to the $c$-axis after cooling the sample in zero field to the desired temperature. In the inset, the best fit of $M$($t$) at 10 K using Eq. 2 is shown with the obtained fitting parameters.[]{data-label="fig:Fig4"}](Fig4.eps){width="0.95\columnwidth"} Although the critical currents in [Ca(Fe$_{1-x}$Co$_x$)$_2$As$_2$]{} are relatively low, there is no SP effect visible. In order to elucidate this observation, we have studied the vortex dynamics in further detail by means of temperature- and field-dependent magnetic relaxation measurements in one of the samples, i.e., [Ca(Fe$_{0.944}$Co$_{0.056}$)$_2$As$_2$]{}. The results are displayed in the main panel of Fig. 4. Here, $M$($t$) is shown at different temperatures which is normalized to $M$ at $t=0$ in order to compare the relaxation. We have used [$H_{\rm{a}}$]{} = 10 kOe (parallel to the $c$-axis) and [$T_{\rm{i}}$]{} = 2, 5, 10 and 12.5 K. In order to measure the field dependence, we fixed the temperature at 10 K, and varied [$H_{\rm{a}}$]{}, with [$H_{\rm{a}}$]{} = 5, 7.5, 10, 15 and 20 kOe. The data in [Fig.\[fig:Fig4\]]{} show that the magnetization increases continuously with time in the complete measurement period of 12000 s. For all temperatures, we observe significant increase of the magnetization demonstrating high relaxation rates. For instance, at 10 K the magnetization increases by almost 26% within the measuring time span. However, it is evident in figure that magnetization shows fast relaxation within initial time period of roughly 1000 s, and then relaxation is relatively show. In addition, the figure shows that with increasing temperature the relaxation rate becomes larger which indicates that the relaxation process is thermally activated. ![(a) The parameter $\mu$ as obtained from fitting $M$($t$) by means of Eq. 2 as a function of temperature. (b) The magnetic relaxation rate S calculated using Eq. 3 at $t = 1000$ s vs. temperature (see the text). Lines are guides to the eyes.[]{data-label="fig:Fig5"}](Fig5.eps){width="0.95\columnwidth"} Magnetic relaxation in superconductors arises due to the nonequilibrium spatial distribution of flux lines which are trapped at the pinning wells created by disorders. In the presence of an externally applied magnetic field or current, flux lines experience a Lorentz force which drives the vortices out of the pinning centers. In this scenario, [$J_{\rm{C}}$]{} corresponds to the (critical) current density at which the Lorentz force equals the maximum pinning force. However, this experimentally obtained supercurrent density, $J$, is always lower than [$J_{\rm{C}}$]{} due to the thermal fluctuations. The redistribution or creeping of flux lines causes the change (or relaxation) of the magnetic moment with time. The magnetic relaxation is basically determined by two competing factors: one is the localizing effect of vortices arising from the pinning potential and another is the delocalizing effect realized by the Lorentz force and the thermal depinning. The higher the pinning strength, the lower is the relaxation of the magnetization. In the initial Anderson-Kim model [@anderson] the barrier energy $U$ is assumed to depend linearly on the current density as $U = U_0$(1 - $J/J_{\rm c}$), which implies a logarithmic time dependence of the magnetization. However, more realistic descriptions of energy barriers follow a nonlinear dependence on the current density. One of such functional forms is the inverse power law barrier which has emerged from the collective vortex pinning theory and is defined as $U = U_0$\[($J_{\rm c}/J)^\mu - 1$\]. This approach is commonly known as ‘interpolation formula’ according to which time dependence of the magnetization behaves as: [@feigel] $$\begin{aligned} M(t) = M_0 \left[1 + \frac{\mu k_{\rm B}T}{U_0} \ln \left(\frac{t}{t_0}\right)\right]^{-1/\mu}\end{aligned}$$ where $k_{\rm B}$ is the Boltzmann constant, $U_0$ is the barrier height in absence of a driving force, $t_0$ is the characteristic relaxation time (10$^{-6}$ - 10$^{-12}$ s), and $\mu$ is the field-temperature dependent parameter whose value is predicted to depend on the dimensionality of the system as well as on the size of vortex bundles. [@feigel] In 3 dimensional lattice, $\mu$ = 1/7, 3/2 and 7/9 for single vortex, small bundles and large bundles, respectively. Positive values of $\mu$ indicate a collective creep barrier while negative ones signal plastic creep. [@prozo-Ba07; @griessen] In the former case, the thermally activated creep occurs through jumps of vortex bundles whereas in the latter one creep is driven by the sliding of vortex lattice dislocations. From the Eq. 2, the normalized magnetization relaxation rate $S$ \[=(1/$M$)d$M$/d$\ln$($t$)\] can be deduced as following: [@yeshurun] $$\begin{aligned} S(t) = \frac{k_{\rm B}T}{U_0 + \mu k_{\rm B}T \ln(t/t_0)}\end{aligned}$$ In order to exploit this model, we have measured magnetic relaxation at different temperatures and fields. The measured magnetic relaxation data do not show a logarithmic dependence on time which is expected in the pure Anderson-Kim model. However, we can describe the $M$($t$) data well in terms of Eq. 2. One representative fit for data taken at $T=10$ K on [Ca(Fe$_{0.944}$Co$_{0.056}$)$_2$As$_2$]{} is shown in the inset of [Fig.\[fig:Fig4\]]{}. Our analysis implies a negative value of the parameter $\mu$, thereby indicating the plastic creep nature of vortices. Interestingly, the extracted value of $\mu$ is significantly larger than predicted by theory [@feigel]. However, higher values of $\mu$ than the theoretical prediction have also been observed in both cuprate and pnictide superconductors. [@abulafia; @prozo-Ba07] The high value of $\mu$ within the plastic creep scenario signifies a high relaxation rate (Eq. 3) which is consistent with data in the Fig. 4. This high $\mu$ and the initial fast increase of magnetic relaxation in Fig. 4 probably suggest that in crystal there is a inhomogeneous distribution of pinning centers which render locally correlated vortex bundles. As soon as magnetic field is applied, these bundles of flux lines slide through the crystal giving rise to plastic creep. Once, however, this redistribution of vortex bundles is settled down, the individual or bundle of vortex perform slow creeping, thus exhibiting relatively slow relaxation. The temperature dependence of parameter $\mu$ (cf. [Fig.\[fig:Fig5\]]{}) exhibits a significant increase of $|\mu|$ upon heating at low temperatures while for $T\gtrsim 5$ K there is only a very weak variation with a negative slope. The relaxation rate $S$ which is calculated exploiting Eq. 3 is plotted as a function of temperature in Fig. 5b at $t = 1000$ s. Even at low temperature, the calculated value of $S$ is much larger than in conventional superconductors but comparable to cuprate and Ba-122 superconductors. [@yeshurun; @blatter; @prozo-Ba07] With increasing temperature, $S$ exhibits a significant monotonic increase instead of a plateau as it would be expected in collective pinning theories of vortices. [@yeshurun; @blatter] The high value of $S$ is indicative of giant flux motion which is in a qualitative agreement to the observed low [$J_{\rm{C}}$]{} in our materials. On the other hand, the positive slope d$S$/d$T$ is consistent with the scenario of plastic motion of flux lines. [@prozo-Ba07] ![Field dependence of (a) the parameter $\mu$ (see Eq. 2) and (b) the magnetic relaxation rate at $t = 1000$ s (see Eq. 3) for [Ca(Fe$_{0.944}$Co$_{0.056}$)$_2$As$_2$]{}. The inset of (a) displays the barrier energy $U_0$ obtained by means of Eq. 2 vs. magnetic field (see the text). Lines are guides to the eyes.[]{data-label="fig:Fig6"}](Fig6.eps){width="0.95\columnwidth"} In order to further confirm the plastic nature of flux creeping we have measured $M$($t$) with varying [$H_{\rm{a}}$]{} at 10 K. Similar to Fig. 4, the data depict an increase of the magnetic relaxation rate for higher [$H_{\rm{a}}$]{} (not shown). Analysis of the data in terms of Eq. 2 again implies negative $\mu$ for all applied fields. As seen in [Fig.\[fig:Fig6\]]{}a, there is a maximum in $|\mu |$ at $H_{\rm m} \approx 7.5$ kOe. The field dependence of the barrier energy $U_0$ follows a very similar behavior. Noteworthy, the decrease of $\mu$ and U$_0$ at $H_{\rm a} > H_{\rm m}$ contradicts the collective creep model which implies $U_0 \propto H^\nu$, where $\nu$ is a positive exponent. [@abulafia] In contrast, it agrees well to the plastic creep model in which the barrier energy depends on the field as $\propto$ 1/$\sqrt{H}$, thus decreasing with field. Further information is provided by estimating the magnetic relaxation rate using Eq. 3 (Fig. 6b). $S$($H$) exhibits a considerable monotonic increase which again corroborates a plastic nature of the creep vortex dynamics in these compounds. [@prozo-Ba07] Thus, the above experimental signatures strongly suggest that the vortex dynamics in [Ca(Fe$_{1-x}$Co$_x$)$_2$As$_2$]{} is determined by the plastic creep of flux lines. The scenario of plastic vortex creep was invoked in cuprate superconductors due to failure of the collective creep model in explaining the experimental observations in high fields above the SP. [@abulafia; @giller] To be specific, it is argued that the SP in $M$($H$) is related to the crossover from collective to plastic creeping of vortices. This crossover is attributed to different magnetic field effects on the plastic and the collective energy barrier, respectively, the former being smaller in high fields and thereby controlling the the vortex dynamics in this field regime. Following this scenario, the absence of the SP in [Ca(Fe$_{1-x}$Co$_x$)$_2$As$_2$]{} can be explained by the weak pinning energy and the absence of collective pinning, which is evident from our analysis above. To be specific, weak pinning forces suggest that dislocations in the vortex lattice can easily move, thereby causing dissipation and leading to plastic motion. However, we remind the contrasting feature at low field in Fig. 6a, where both $\mu$(H) and U$_0$(H) increases. This observation might suggest that the field associated with SP in these compounds is very low so that the pronounced central peak just masks the SP. Note, that similar scenarios are discussed for very underdoped and overdoped Ba(Fe$_{1-x}$Co$_{x}$)$_2$As$_2$ series where the weak pinning energy leads to less pronounced SP in these samples. [@shen] The observed differences in vortex dynamics for [Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$]{} and [Ca(Fe$_{1-x}$Co$_x$)$_2$As$_2$]{} are quite intriguing. Both the series differ in ionic radii of Ba$^{2+}$ and Ca$^{2+}$, otherwise they have comparable $T_C$ and superconducting phase fractions. The Ca-122 compounds are relatively less anisotropic both in terms of resistivity and upper critical field $H_{c2}$ which is expected to yield high pinning energy in contrast to our observed behavior. [@neeraj2; @sun; @tanatar-rho] However, during our crystal growing studies, we observe that Ba-122 crystals can support As deficiency whereas Ca-122 materials do not allow such deficiency. This might create added pinning centers in Ba-122 crystals. It has been recently suggested for [Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$]{} that the structural domain walls in coexisting AFM/orthorhombic phases create effective pinning centers. The presence of these structural domains are common in 122 family. [@tanatar-dom] With increasing $x$, it is shown that these walls become finer and denser, and form interwoven pattern giving rise to stronger pinning. These are favored by the suppression of orthorhombic distortion $\delta$ \[= (a-b)/(a+b)\] with increasing Co concentration. For comparison, we have calculated $\delta$ for the parent compounds ($x$ = 0) with values 0.36% and 0.67% for BaFe$_2$As$_2$ and CaFe$_2$As$_2$, respectively. [@rotter1; @krey] With Co doping, this trend will presumably be retained. Due to higher degree of orthorhombic distortion such domain walls will differ in Ca- and Ba-122 crystals, and this might explain the different pinning in these compounds. Nonetheless, the effects of crystallographic parameters on vortex pinning is quite significant for superconductors in general, and such pinning due to twin boundaries is observed in YBa$_2$Cu$_3$O$_{7-y}$ (Ref. 43) and RNi$_2$B$_2$C (R = Er and Ho) (Ref. 44). Conclusion ========== In summary, we have studied the vortex dynamics in single crystals of Ca(Fe$_{1-x}$Co$_{x}$)$_2$As$_2$ ($x$ = 0.051, 0.056, 0.065 and 0.073) superconductors. Unlike Ba-based 122 pnictides, we observe no SP in the field dependent magnetization loops. The critical current density [$J_{\rm{C}}$]{} calculated in the frame of Bean’s model is low compared to other 122-family members. With Co doping, [$J_{\rm{C}}$]{} initially increases and then decreases which variation appears to be consistent with that of the superconducting phase fractions. Moreover, the estimated [$J_{\rm{C}}$]{} shows a strong temperature dependence, decreasing rapidly upon heating. The magnetic relaxation exhibits a nonlogarithmic time-dependence, however, the data are explained well by means of the interpolation model. High magnetic relaxation rates imply weak pinning energy. Furthermore, from the analysis of magnetic relaxation data we conclude that vortex dynamics in Co doped Ca-122 superconductors is rather of plastic than collective nature. The comparison with other 122 superconductors might indicate that the observed vortex creep behavior is connected with the strength of pinning energy which varies with the A-site elements of 122 pnictides. However, to generalize this result similar studies are needed in different samples including variety of doping species.\ We acknowledge valuable discussions with V. Grinenko. We thank M. Deutschmann, S. Müller-Litvanyi, R. Müller, J. Werner, K. Leger, and S. Ga[ß]{} for technical support. Work was supported by the DFG through project BE 1749/13. Y. Kamihara, T. Watanabe, M. Hirano, and H. Hosono, J. Am. Chem. 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--- abstract: 'A truncation scheme for the Dyson–Schwinger equations of QCD in Landau gauge is presented which implements the Slavnov–Taylor identities for the 3–point vertex functions. Neglecting contributions from 4–point correlations such as the 4–gluon vertex function and irreducible scattering kernels, a closed system of equations for the propagators is obtained. For the pure gauge theory without quarks this system of equations for the propagators of gluons and ghosts is solved in an approximation which allows for an analytic discussion of its solutions in the infrared: The gluon propagator is shown to vanish for small spacelike momenta whereas the ghost propagator is found to be infrared enhanced. The running coupling of the non–perturbative subtraction scheme approaches an infrared stable fixed point at a critical value of the coupling, $\alpha_c \simeq 9.5$. The gluon propagator is shown to have no Lehmann representation. The gluon and ghost propagators obtained here compare favorably with recent lattice calculations. Results for the quark propagator in the quenched approximation are presented.' author: - | Reinhard Alkofer, Steven Ahlig\ Universität Tübingen, Institut für Theoretische Physik,\ Auf der Morgenstelle 14, 72076 Tübingen, Germany\ E-mail: [email protected]\ [email protected]\ and Lorenz von Smekal\ Universität Erlangen–Nürnberg,\ Institut für Theoretische Physik III,\ Staudtstr. 7, 91058 Erlangen, Germany\ E-mail: [email protected] title: | [UNITU–THEP–1/1999 FAU–TP3–99/1 hep-ph/9901322 ]{}\ The Infrared Behavior of Gluon, Ghost, and Quark Propagators in Landau Gauge QCD --- =cmr8 1.5pt PACS: 02.30.Rz 11.10.Gh 12.38.Aw 14.70.Dj Introduction ============ Despite the remarkable success of perturbative QCD the description of hadronic states and processes based on the dynamics of confined quarks and gluons remains the outstanding challenge of strong interaction physics. Especially, one has to explain why only hadrons are produced from processes involving hadronic initial states, and that the only thresholds in hadronic amplitudes are due to the productions of other hadronic states. To this end one would like to understand how singularities appear in the Green’s functions of composite hadron fields where, on the other hand, they have to disappear in colored correlations functions. To study these aspects of QCD amplitudes non–perturbative methods are required, and, since infrared divergences are anticipated, a formulation in the continuum is desirable. Both of these are provided by studies of truncated systems of Dyson–Schwinger equations (DSEs), the equations of motion of QCD Green’s functions. Typically, for their truncation, additional sources of information like the Slavnov–Taylor identities, entailed by gauge invariance, are used to express vertex functions in terms of the elementary two–point functions, i.e., the quark, ghost and gluon propagators. Those propagators can then be obtained as selfconsistent solutions to non–linear integral equations representing a closed set of truncated DSEs. Some systematic control over the truncating assumptions can be obtained by successively including higher $n$–point functions in selfconsistent calculations, and by assessing their influence on lower $n$–point functions in this way. Until recently all solutions to truncated DSEs of QCD in Landau gauge, even in absence of quarks, relyed on neglecting ghost contributions completely [@Man79; @Atk81; @Bro89; @Hau96]. In addition to providing a better understanding of confinement based on studies of the behavior of QCD Green’s functions in the infrared, DSEs have proven successful in developing a hadron phenomenology which interpolates smoothly between the infrared non–perturbative and the ultraviolet perturbative regime [@Dub98], for recent reviews see, [*e.g.*]{}, [@Tan97; @Rob94]. In particular, a dynamical description of spontaneous breaking of chiral symmetry from studies of the DSE for the quark propagator is well established in a variety of models for the gluonic interactions of quarks [@Miranski]. For a sufficiently large low–energy quark–quark interaction quark masses are generated dynamically in the quark DSE in some analogy to the gap equation in superconductivity. This in turn leads naturally to the Goldstone nature of the pion and explains the smallness of its mass as compared to all other hadrons. In this framework a description of the different types of mesons is obtained from Bethe–Salpeter equations for quark–antiquark bound states [@Mun92]. Recent progress towards a solution of a fully relativistic three–body equation extends this consistent framework to baryonic bound states, see [*e.g.*]{} [@Oet98] and references therein. Here a simultaneous solution of a truncated set of DSEs for the propagators of gluons and ghosts in Landau gauge is presented [@Sme97; @Sme98]. An extension of this selfconsistent framework to include quarks is subject to on–going research [@Ahl99]. Preliminary results for the quark propagator in the quenched approximation have been obtained and will be shown. The behavior of the solutions in the infrared, implying the existence of a fixed point at a critical coupling $\alpha_c \approx 9.5$, is obtained analytically. The gluon propagator is shown to vanish for small spacelike momenta in the present truncation scheme. This behavior, though in contradiction with previous DSE studies [@Man79; @Atk81; @Bro89; @Hau96], can be understood from the observation that, in our present calculation, the previously neglected ghost propagator assumes an infrared enhancement similar to what was then obtained for the gluon. In the meantime such a qualitative behavior of gluon and ghost propagators is supported by investigations of the coupled gluon ghost DSEs using bare vertices [@Atk97; @Atk98]. As expected, however, the details of the results depend on the approximations employed. The set of truncated gluon and ghost DSEs ========================================= Besides all elementary 2–point functions, i.e., the quark, ghost and gluon propagators, the DSE for the gluon propagator also involves the 3– and 4–point vertex functions which obey their own DSEs. These equations involve successively higher n–point functions. The gluon equation is truncated by neglecting all terms with 4–gluon vertices. These are the momentum independent tadpole term, an irrelevant constant which vanishes perturbatively in Landau gauge, and explicit 2–loop contributions to the gluon DSE. For all details regarding this truncation scheme we refer the reader to [@Sme98]. The ghost and gluon propagators are parameterized by their respective renormalization functions $G$ and $Z$, $$D_G(k) = -\frac{G(k^2)}{k^2} \; , \quad D_{\mu\nu}(k) = \bigg( \delta_{\mu\nu} - \frac{k_\mu k_\nu}{k^2}\bigg) \frac{Z(k^2)}{k^2} \; . \label{rfD}$$ In order to arrive at a closed set of equations for the functions $G$ and $Z$, we use a form for the ghost–gluon vertex which is based on a construction from its Slavnov–Taylor identity (STI) which can be derived from the usual Becchi–Rouet–Stora invariance neglecting irreducible 4–ghost correlations in agreement with the present level of truncation [@Sme98]. This together with the crossing symmetry of the ghost–gluon vertex fully determines its form at the present level of truncation: $$G_\mu(p,q) = i q_\mu \, \frac{G(k^2)}{G(q^2)} + i p_\mu \, \biggl( \frac{G(k^2)}{G(p^2)} - 1 \biggr) . \label{fvs}$$ With this result, we can construct the 3–gluon vertex according to procedures developed and used previously [@Bar80], for details see [@Sme98]. We have solved the coupled system of integral equations of the present truncation scheme numerically using an angle approximation. The infrared behavior of the propagators can, however, be deduced analytically. To this end we make the Ansatz that for $x := k^2 \to 0$ the product $Z(x)G(x) \to c x^\kappa$ with $\kappa \not= 0$ and some constant $c$. The special case $\kappa = 0$ leads to a logarithmic singularity for $x \to 0$ which precludes the possibility of a selfconsistent solution. In order to obtain a positive definite function $G(x)$ for positive $x$ from an equally positive $Z(x)$, as $x\to 0$, we obtain the further restriction $0 < \kappa < 2$. The ghost DSE then yields, $$\begin{aligned} G(x) \to \left( g^2\gamma_0^G \left(\frac{1}{\kappa} - \frac{1}{2} \right) \right)^{-1} c^{-1} x^{-\kappa} \quad \Rightarrow Z(x) \to \left( g^2\gamma_0^G \left(\frac{1}{\kappa} - \frac{1}{2} \right) \right) \, c^{2} x^{2\kappa} \; , \nonumber\end{aligned}$$ where $\gamma_0^G = 9/(64\pi^2)$ is the leading order perturbative coefficient of the anomalous dimension of the ghost field. Using these relations in the gluon DSE, we find that the 3–gluon loop contributes terms $\sim x^\kappa$ to the gluon equation for $x \to 0$ while the dominant (infrared singular) contribution arises from the ghost–loop, $$Z(x) \to g^2\gamma_0^G \, \frac{9}{4} \left(\frac{1}{\kappa} - \frac{1}{2} \right)^2 \! \left( \frac{3}{2}\, \frac{1}{2-\kappa} - \frac{1}{3} + \frac{1}{4\kappa} \right)^{-1}\!\! c^2 x^{2\kappa} .$$ Requiring a unique behavior for $Z(x)$ we obtain a quadratic equation for $\kappa$ with a unique solution for the exponent in $0 < \kappa < 2$: $$\kappa = \frac{61-\sqrt{1897}}{19} \simeq 0.92 \; .$$ The leading behavior of the gluon and ghost renormalization functions and thus of their propagators is entirely due to ghost contributions. The details of the approximations to the 3–gluon loop have no influence on the above considerations. Compared to the Mandelstam approximation, in which the 3–gluon loop alone determines the infrared behavior of the gluon propagator and the running coupling in Landau gauge [@Man79; @Atk81; @Bro89; @Hau96], this shows the importance of ghosts. The result presented here implies an infrared stable fixed point in the non–perturbative running coupling of our subtraction scheme, defined by $$\alpha_S(s) = \frac{g^2}{4\pi} Z(s) G^2(s) \quad \to \quad \frac{16\pi}{9} \left(\frac{1}{\kappa} - \frac{1}{2}\right)^{-1} \approx 9.5 \; ,$$ for $s\to 0$. This is qualitatively different from the infrared singular coupling of the Mandelstam approximation [@Hau96]. Comparison to lattice results ============================= It is interesting to compare our solutions to recent lattice results available for the gluon propagator [@Der98] and for the ghost propagator [@Sum96] using lattice versions to implement the Landau gauge condition. We would like to refer the reader to ref. [@Hau98] where this has been done in some detail. It is very encouraging to observe that our solution fits the lattice data at low momenta rather well, especially for the ghost propagator. We therefore conclude that present lattice calculations confirm the existence of an infrared enhanced ghost propagator of the form $D_G \sim 1/(k^2)^{1+\kappa}$ with $0 < \kappa < 1$. This is an interesting result for yet another reason: In the calculation of [@Sum96] the Landau gauge condition was supplemented by an algorithm to select gauge field configurations from the fundamental modular region which is to avoid Gribov copies. Thus, our results suggest that the existence of such copies of gauge configurations might have little effect on the solutions to Landau gauge DSEs. Here we want to add a remark concerning the comparison of the running coupling obtained in our calculation to lattice results. Recent lattice calculations of the running coupling are reported in Refs. [@All97; @Bou98] based on the 3–gluon vertex, and Ref. [@Sku98] on the quark–gluon vertex. The non–perturbative definitions of these couplings are related but manifestly different from the one adopted here. One of the most recent results from the 3–gluon vertex is shown in the left graph of Fig. \[lat\_alpha\] and compared to the three–loop expression which is for the momenta displayed almost identical to our expression (4) for the running coupling. This lattice result is obtained from an asymmetric momentum subtraction scheme. This corresponds to a definition of the running coupling $\bar g^2_{3GVas}$ which can explicitly be related to the present one ($\bar g^2(t,g)$ with $t = \ln \mu'/\mu$ and $g := g(\mu)$), $$\bar g^2(t,g^2)_{3GVas} \, = \, \bar g^2(t,g^2) \, \lim_{s \to 0} \, \frac{G^2(s)}{G^2({\mu'}^2)} \, \left( 1 \, - \, \frac{\beta(\bar g(t,g))}{\bar g(t,g)} %{\mu'}^2 \frac{d}{d{\mu'}^2} \, %\ln\big( G^2({\mu'}^2)Z({\mu'}^2)\big) \, \right)^2 \; . \label{comp_rc}$$ An inessential difference in these two definitions of the running coupling is the last factor in brackets in eq. (\[comp\_rc\]) which can be easily accounted for in comparing the different schemes. However, the crucial difference is the ratio of ghost renormalization functions $G(s\to 0)/G({\mu'}^2)$. These considerations show that the asymmetric scheme can be extremely dangerous if infrared divergences occur in vertex functions as our calculation indicates. Clearly, from the infrared enhanced ghost renormalization function this scale dependence could account for the infrared suppressed couplings which seem to be found in the asymmetric schemes. Similarly, the results from the quenched calculation of the quark–gluon vertex of Ref. [@Sku98] which are compared in the right graph of Fig. \[lat\_alpha\] to our solution are obtained from an analogous asymmetric scheme. It is thus expected to have the same problems in taking the possible infrared divergences of the vertices into account which arise in both, the 3–gluon and the quark–gluon vertex, as a result of the infrared enhancement of the ghost propagator. Furthermore, definitions of the coupling which lead to extremas at finite values of the scale correspond to double valued $\beta$–functions with artificial zeros. If the maxima in the couplings of the asymmetric schemes at finite scales are no lattice artifacts, these results seem to imply that the asymmetric schemes are less suited for a non–perturbative extension of the renormalization group to all scales. Indeed, the results for the running coupling from the 3–gluon vertex obtained for the symmetric momentum subtraction scheme in Ref. [@Bou98] differ from those of the asymmetric scheme, in particular, in the infrared. These results would be better to compare to the DSE solution, however, they unfortunately seem to be much noisier thus far (see Ref. [@Bou98]). The ultimate lattice calculation to compare to the present DSE coupling would be obtained from a pure QCD calculation of the ghost–gluon vertex in Landau gauge with a symmetric momentum subtraction scheme. This is unfortunately not available yet. .6cm -.6cm 1.7cm -.4cm Quark Propagator ================ We have solved the quark DSE in quenched approximation [@Ahl99]. In a first step we have specified the quark–gluon vertex from the corresponding Slavnov–Taylor identity. It contains explicitely a ghost renormalization function, $$\Gamma^\mu (p,q) = G(k^2) \Gamma^\mu _{\rm CP} (p,q)$$ where $\Gamma^\mu _{\rm CP}$ is the Curtis–Pennington vertex (for its definition see [*e.g.*]{} [@Rob94]). It is obvious that this leads to an effective coupling very different from the one in Abelian approximation, especially in the infrared: This effective coupling vanishes in the infrared and is similar to the lattice result of Ref. [@Sku98] shown in the right graph of Fig. \[lat\_alpha\] the main difference being that the maximum occurs at lower scale, $\mu \approx 220$MeV. This leads to a kernel in the quark DSE which is only very slightly infrared divergent. This allows, [*e.g.*]{}, to use the Landshoff–Nachtmann model for the pomeron in our approach. With our solution we obtain as Pomeron intercept 2.7/GeV as compared to the value 2/GeV deduced from phenomenology, see [*e.g.*]{} [@Cud89]. We have found dynamical chiral symmetry breaking in the quenched approximation. Using a current mass, $m(1{\rm GeV})=6$MeV we obtain a constituent mass of approximately 170 MeV. In the Pagels–Stokar approximation the calculated value for the pion decay constant is 50 MeV. These numbers are quite encouraging, especially for proceeding with the self–consistent inclusion of the quark DSE into the gluon–ghost system. Considering the quark loop in the gluon DSE one realizes that the quark loop will produce an infrared divergence which is, however, subleading as compared to the one generated by the ghost loop. In the latter there appear three ghost renormalization functions in the numerator and one in the denominator leading effectively to an infrared divergence of the order $(k^2)^{-2\kappa}$. In the quark loop term there is only one factor $G$ and thus a divergence of type $(k^2)^{-\kappa}$. Due to this subleading divergence the infrared analysis has to be redone completely before one is able to draw conclusions whether or not and how quark confinement is implemented in our set of truncated DSEs. Summary ======= In summary, we presented a solution to a truncated set of coupled Dyson–Schwinger equations for gluons and ghosts in Landau gauge. The infrared behavior of this solution, obtained analytically, represents a strongly infrared enhanced ghost propagator and an infrared vanishing gluon propagator. The Euclidean gluon correlation function presented here can be shown to violate reflection positivity [@Sme98], which is a necessary and sufficient condition for the existence of a Lehmann representation. We interpret this as representing confined gluons. In order to understand how these correlations can give rise to confinement of quarks, it will be necessary to redo the infrared analysis including self–consistently the quark propagator. Nevertheless, we found dynamical chiral symmetry breaking in the quenched approximation. The existence of an infrared fixed point for the coupling is in qualitative disagreement with previous studies of the gluon DSE neglecting ghost contributions in Landau gauge [@Man79; @Atk81; @Bro89; @Hau96]. On the other hand, our results for the propagators, in particular for the ghost, compare favorably with recent lattice calculations [@Der98; @Sum96]. This shows that ghosts are important, in particular, at low energy scales relevant to hadronic observables. Acknowledgments {#acknowledgments .unnumbered} =============== R. A. wants to thank the organizers of the conference, and especially Dubravko Klabucar, for their help and the warm hospitality extended to him. 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--- abstract: 'We analyze doping dependent spectral intensities and Fermi surface maps obtained recently in Nd$_{2-x}$Ce$_x$CuO$_{4\pm\delta}$ (NCCO) via high resolution ARPES measurements, and show that the behavior of this electron-doped compound can be understood as the closing of a Mott (pseudo) gap, leading to a quantum critical point just above optimal doping. The doping dependence of the effective Hubbard $U$ adduced by comparing theoretical and experimental spectra is in resonable accord with various estimates and a simple screening calculation.' address: '1. Physics Department, Northeastern University, Boston MA 02115, USA; 2. Institute of Physics, Tampere University of Technology, 33101 Tampere, Finland' author: - 'C. Kusko$^1$, R.S. Markiewicz$^1$, M. Lindroos$^{1,2}$, and A. Bansil$^1$' title: 'Quantum Critical Point in Electron-Doped Cuprates' --- Angle-resolved photoemission spectra (ARPES) follow a remarkably different route with doping in the [*hole-doped*]{} La$_{2-x}$Sr$_x$CuO$_4$ (LSCO)[@ZZX] compared to the [*electron-doped*]{} NCCO[@nparm]. Starting from a Mott insulator near $x=0$, LSCO exhibits the appearance of dynamic or static stripes[@ZZX], with a concomittant discontinuity in the chemical potential[@AIP]. In sharp contrast, in NCCO, characteristic signatures of stripe order, e.g. the 1/8 anomaly and the NQR wipeout, appear greatly attenuated if not absent[@AKIS], and the Fermi level shifts smoothly into the upper Hubbard band[@AIP]. Here we show how the salient features of the most recent high resolution ARPES data in NCCO[@nparm] can be understood in terms of the behavior of a uniformly doped Mott insulator. The doping dependence of the effective Hubbard $U$ parameter adduced from the experimental spectra is in reasonable accord with various estimates, including a computation in which we screen the bare $U$ via interband excitations. A quantum critical point (QCP) where the Mott gap closes is predicted just beyond optimal doping. Finally, we have carried out extensive first-principles simulations of photointensities in order to ascertain that the key spectral features discussed in this article (e.g. the appearance and growth of the $(\pi/2,\pi/2)$ centered hole orbit with doping) are genuine effects of electron correlations beyond the conventional LDA framework and not related to the energy and k-dependencies of the ARPES matrix element[@matri]. Our analysis of the doping dependence of the Mott gap – perhaps better referred to as a pseudogap[@MW; @mw], invokes a mean field solution to the one band[@3b; @MK2] Hubbard model. The mean field approach provides a good description of not only the pseudogap in the 1D charge density wave (CDW) systems[@LRA], but also of the undoped insulator in the 2D Hubbard problem, including the presence of the ‘remnant Fermi surface’ (rFS)[@rfs] seen in ARPES experiments[@CCOC] and Monte Carlo simulations[@BSW]; the spin density wave (SDW) excitations are described by fluctuations about the mean field[@SWZ]. The mean field theory correctly predicts a (stripe) phase instability (negative compressibility) associated with hole doping[@SiTe]; whereas assuming a finite second-neighbor hopping $t'<0$, for electron doping the compressibility is positive[@MK1], suggesting that a uniformly doped AFM state should be stable. The preceding considerations argue that the mean field would provide a reasonable model for discussing the doping dependence of the pseudogap. This viewpoint is further supported by mode-coupling calculations[@MK2], which have been applied previously to approximately describe stripes in terms of a CDW in hole-doped cuprates[@Cast2; @RM5]. Concerning methodology, we specifically consider the one-band Hubbard model where the Mott gap $\Delta$ is well known to arise (in the mean field SDW formulation[@SWZ]) from a finite expectation value of the magnetization $m_{\vec Q}$ at the wave vector $\vec Q=(\pi,\pi)$. The self-consistent gap equation is $$1=U\sum_k {f(E^{v}_{\vec k})-f(E^{c}_{\vec k})\over E_{0\vec k}}, \label{eq:1}$$ where $f$ is the Fermi function and $$E^{c,v}_{\vec k}={1\over 2}(\epsilon_{\vec k}+\epsilon_{\vec k+\vec q}\pm E_ {0\vec k}) \label{eq:2}$$ Here, the superscript $c$ refers to the upper Hubbard band (UHB) and goes with the + sign on the right side while $v$ goes with the - sign and the lower Hubbard band (LHB)[@ZR1]. $E_{0\vec k}=\sqrt{(\epsilon_{\vec k}-\epsilon_{\vec k+q})^2+4\Delta^2}$, and the independent particle dispersion is: $\epsilon_k=-2t(c_x+c_y)-4t'c_xc_y$, $c_i=\cos{k_ia}$. We study Eq. \[eq:1\] self-consistently as a function of electron doping, assuming that the mean-field transition temperature corresponds to the experimentally observed pseudogap. $U$ is thus treated as an effective parameter $U_{eff}$ to fit the experimental data. The Green’s function in the antiferromagnetic (AFM) ground state is given by $$G(\vec {k},\omega)={u^2_{\vec {k}}\over {\omega-E^{c}_{\vec {k}}}}+{v^2_{\vec {k}}\over {\omega-E^{v}_{\vec {k}}}} \label{eq:3}$$ where $$u^2_{\vec k}={1\over 2} (1+{\epsilon_{\vec k}-\epsilon_{\vec k+\vec Q}\over 2E_{0\vec k}}), v^2_{\vec k}={1\over 2} (1-{\epsilon_{\vec k}-\epsilon_{\vec k+\vec Q} \over 2E_{0\vec k}}) \label{eq:4}$$ are the coherence factors. Using Eqs. 1-4, we discuss spectral density $A(\vec k, E)$ (given by the imaginary part of the Green’s function) and the related energy dispersions for various doping levels. The FS maps are obtained by taking appropriate cuts through $A(\vec k, E)$. Although such theoretical maps do not account for the effects of the ARPES matrix element[@matri], these maps are relevant nevertheless in gaining a handle on the FS topology expected in the correlated system from ARPES experiments. In any event, we have also carried out first-principles simulations based on the conventional LDA picture where the ARPES matrix element is properly treated and the photoemission process is modeled including full crystal wavefunctions in the presence of the surface – see Refs.  for details of the methodology. Fig. \[fig:1\] displays the doping dependence of the spectral weight in the vicinity of the $(\pi/2, \pi/2)$ and $(\pi ,0)$ points. Fig. \[fig:1\](a) shows that around $(\pi/2, \pi/2)$, with increasing doping, the spectral weight shifts rapidly towards the $E_F$ as the gap between the LHB and UHB decreases. At $x=0.15$, the gap is quite small and the LHB and UHB overlap. In contrast, in the momentum region near $(\pi,0)$, Fig. \[fig:1\](b), both LHB and UHB are more directly involved. Even at small doping levels ($x=0.04$), $E_F$ intersects the bottom of the UHB. With increasing electron concentration, the UHB moves to lower energies while the LHB shifts closer to the $E_F$ as the gap decreases. These results are remarkably consistent with the corresponding ARPES data; Fig. 2a of Ref.  shows a rapid movement of spectral weight at $(\pi /2,\pi /2)$ from $~1.3$ eV to around 0.3 eV binding energy close to $E_F$. This effect, although somewhat less clear, is seen at $(\pi,0)$ as well (Fig. 2b of Ref. ): the weight near $E_F$ grows faster with doping, but the LHB is at a lower binding energy, and is less clearly resolved from the background. This doping dependence suggests that electrons first enter the UHB near $(\pi,0)$. The energy dispersions of Fig. 2 are useful not only in understanding the spectra of Fig. 1, but also provide a handle on the FS maps expected in ARPES experiments. The $E_F$ is seen in Fig. 2 to rise smoothly with respect to the UHB with increasing electron doping, whereas in the hole doped cuprates, the $E_F$ gets pinned by the stripes near the mid-gap region over a large doping range. We should keep in mind that various bands do not possess the same spectral weight – this point is emphasized by depicting the coherence factors of Eq. 4 via the width of the bands in Fig. 2. As the Mott gap nearly collapses, the thick lines in Fig. 2(d) essentially present the appearance of the uncorrelated band structure. Fig. 3 illustrates the evolution of the FS corresponding to the band structure of Fig. 2. For low $x$, $E_F$ lies near the bottom of the UHB and gives rise to electron pockets centered at $(\pi,0)$ and $(0,\pi)$ points in Fig. \[fig:3\](a). The shadow segments of the FS are clearly evident in Fig. \[fig:3\](b) where half of the electron pocket gains spectral weight at the expense of the other half due to the decreasing intensity in the magnetic Brillouin Zone (BZ) via the coherence factors; a weak imprint of the magnetic BZ boundary (the diagonal line connecting $(0,\pi)$ and $(\pi,0)$ points) can also be seen. By $x=0.15$, the gap shrinks considerably, and the $E_F$ begins intersecting the LHB around $(\pi/2,\pi/2)$. The FS now consists of three sheets: electron-like sheets near $(\pi,0)$ and $(0,\pi)$ and a hole-like sheet around $(\pi/2,\pi/2))$ separated by a residual gap located at the ‘hot-spots’[@HlR] along the BZ diagonal. Interestingly, in this doping regime, transport studies find evidence for two band conduction, and a change in the sign of the Hall coefficient[@nctran]; the hole pocket associated with the LHB can explain both these effects. We stress that the FS evolves following a very different route in the hole-doped case for the $t-t'$ one band Hubbard model[@ChubMorr] (neglecting stripes). At low doping levels, the FS consists of small hole pockets around $(\pi/2,\pi/2)$ points. With increasing hole concentration, these pockets increase in size and merge to yield a large $(\pi,\pi)$ centered FS satisfying the volume constraints of the Luttinger theorem. Figure 4 directly compares the theoretical FS maps against the corresponding experimental results. Here we have included a small second neighbor hopping parameter $t''=0.1t$ in the computations in order to account for the slight shift of the center of the hole pocket away from $(\pi/2,\pi/2)$ in the experimental data, even though the maps of Fig. 3 with only $t-t'$ already provide a good overall description (after resolution broadening) of the measurements. The agreement between theory and experiment in Fig. 4 is remarkable: Both sets of maps show the electron pockets at low doping ($x=0.04$); the beginning of the hole pocket at $x=0.10$, together with the presence of shadow features around electron pockets and the appearance of intensity around the magnetic BZ; and finally, at $x=0.15$, the evidence of a well formed hole pocket and a three-sheeted FS which has begun to resemble the shape of a large $(\pi,\pi)$ centered hole sheet. Fig. 4 displays discrepancies as well (e.g. the hole pocket related feature appears double peaked in experiments, but not in theory, and other details of spectra around the $E_F$), but this is not surprising since we are invoking a rather simple one-band Hubbard model and the effects of the ARPES matrix element are missing in these calculations. The parameters used in the computations of Fig. 4 are as follows.[@ueff] For the half-filled case, the values are identical to those used previously[@OSP]: $t=0.326 eV$, $t'/t=-0.276$, $U=6t$. For finite doping $x$, the only change is that $U$ is assumed to be $x$-dependent, $U=U_{eff}(x)$, and $t''=0.1t$. The actual values of $U_{eff}$ used (solid dots in Fig. 5(a)), are in reasonable accord with various estimates shown in Fig. 5(a) which are: the approximate value from Kanamori (arrow) for a nearly empty band[@Kana], and Monte Carlo results[@BSW2; @ChAT], for $t'=0$ (star). Finally, we have carried out a computation of [*screened*]{} $U$ using $U_{eff}=U/(1+<P>U)$, taking $P$ as the charge susceptibility which includes only interband contributions with bare $U=6.75t$. The $U_{eff}$ so obtained for electrons (solid line) and holes (dashed) is shown[@Chen0]. Fig. 5(b) considers the behavior of the staggered magnetization, $m_{\vec Q}(x)$, for electrons and holes using the computed values of $U_{eff}$ for [*holes*]{} given by dashed line in Fig. 5(a).[@calc] We see that $m_{\vec Q}(x)$ and hence the pseuodgap, which is proportional to $m_{\vec Q}(x) $, vanishes slightly above optimal doping, yielding a QCP. Notably, superconductivity near an AFM QCP has been reported in a number of systems[@MaLo]. However, the present case is different in that we have a mean-field QCP associated directly with short-range AFM fluctuations and the [*closing of the Mott gap*]{}. Since there is no interfering phase separation instability, the bulk Néel transition persists out to comparable, but clearly lower dopings, with $T_N\rightarrow 0$ near $x=0.13$. We have carried out extensive first-principles simulations of the ARPES intensities in NCCO within the LDA framework for different photon energies, polarizations and surface terminations, in order to ascertain the extent to which the characteristics of measured FS maps could be confused with the effects of the ARPES matrix element[@matri] missing in the computations of Figs. 3 and 4. The computed FS maps including full crystal wavefunctions of the initial and final states in the presence of the Nd-CuO$_2$-Nd-O$_2$-terminated surface at 16 eV are shown in Fig. 6 for two different polarizations, and are typical. The intensity is seen to undergo large variations as one goes around the $(\pi,\pi)$ centered hole orbit, and to nearly vanish along certain high symmetry lines in some cases. Nevertheless, we do not find any situation which resembles the doping dependencies displayed in Figs. 3 and 4. It is clear that strong correlation effects beyond the conventional LDA-based picture are needed to describe the experimental ARPES spectra and that the $t-t'-t''$ Hubbard model captures some of the essential underlying physics. In conclusion, this study indicates that the electron doped NCCO is an ideal test case for investigating how superconductivity arises near a QCP in doped Mott insulators, untroubled by complications of stripe phases. The doping dependence of $U_{eff}$ adduced in this work has implications in understanding the behavior of the cuprates more generally since the pseudogap in both the electron and the hole doped systems must arise from the same Mott gap at sufficiently low doping. We thank N.P. Armitage and Z.-X. Shen for sharing their data with us prior to publication. This work is supported by the U.S.D.O.E. 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--- abstract: | The entanglement of graph states up to eight qubits is calculated in the regime of iteration calculation. The entanglement measures could be the relative entropy of entanglement, the logarithmic robustness or the geometric measure. All 146 local inequivalent graphs are classified as two categories: graphs with identical upper LOCC entanglement bound and lower bipartite entanglement bound, graphs with unequal bounds. The late may displays non-integer entanglement. The precision of iteration calculation of the entanglement is less than $10^{-14}$. PACS number(s): 03.67.Mn, 03.65.Ud, 02.10.Ox Keyword(s): graph state; iteration; multipartite entanglement author: - 'Xiao-yu Chen\' title: Entanglement of graph states up to 8 qubits --- Introduction ============ Multipartite entanglement plays an important role in quantum error correction and quantum computation. The quantification of multipartite entanglement is still open even for a pure multipartite state. Until now, a variety of different entanglement measures have been proposed for multipartite setting. Among them are the *(Global) Robustness of Entanglement* [@Vidal]* ,* the *Relative Entropy of Entanglement*[@Vedral1] [@Vedral2]* ,* and *the Geometric Measure* [@Wei1]*.* The robustness measures the minimal noise (arbitrary state) that we need to added to make the state separable. The geometric measure is the distance of state to the closest product state in terms of the fidelity. The relative entropy of entanglement is a valid entanglement measure for multipartite state, it is the relative entropy of the state under consideration to the closest fully separable state. The quantification of multipartite entanglement is usually very difficult as most measures are defined as the solutions to difficult variational problems. Even for pure multipartite state, the entanglement can only be obtained for some special scenario. Fortunately, due to the inequality on the logarithmic robustness, relative entropy of entanglement and geometric measure of entanglement[@Wei2] [@Hayashi1] [@Wei3], these entanglement measures are all equal for stabilizer states [@Hayashi2]. It is known that the set of graph states is a subset of the set of stabilizer states. Thus these entanglement measures are all equal for graph states. The paper is organized as follows: In section 2, we overview the concept of graph state and graph state basis, the entanglement inequalities, the bounds of the entanglement. In section 3, we describe the iteration method of finding the closest product state. In section 4, we classify all the inequivalent graphs according to the entanglement bounds up to 8 qubits and propose the precise values of entanglement. Section 5 is devoted to the precision of the iteration. Conclusions are given in section 6. Preliminary =========== Graph state ----------- A graph $G=(V;\Gamma )$ is composed of a set $V$ of $n$ vertices and a set of edges specified by the adjacency matrix $\Gamma $, which is an $n\times n$ symmetric matrix with vanishing diagonal entries and $\Gamma _{ab}$ $=1$ if vertices $a,b$ are connected and $\Gamma _{ab}$ $=0$ otherwise. The neighborhood of a vertex $a$ is denoted by $N_a$ $=\{v\in V\left| \Gamma _{av}=1\right. \}$, i.e, the set of all the vertices that are connected to $% a $. Graph states [@Hein1] [@Sch]\] are useful multipartite entangled states that are essential resources for the one-way computing [@Raus] and can be experimentally demonstrated [@Walther]. To associate the graph state to the underlying graph, we assign each vertex with a qubit, each edge represents the interaction between the corresponding two qubits. More physically, the interaction may be Ising interaction of spin qubits. Let us denote the Pauli matrices at the qubit $a$ by $X_a,Y_a,Z_a$ and identity by $I_a$. The graph state related to graph $G$ is defined as $$\left| G\right\rangle =\prod_{\Gamma _{ab}=1}U_{ab}\left| +\right\rangle _x^V=\frac 1{\sqrt{2^n}}\sum_{\mathbf{\mu }=\mathbf{0}}^{\mathbf{1}% }(-1)^{\frac 12\mathbf{\mu }\Gamma \mathbf{\mu }^T}\left| \mathbf{\mu }% \right\rangle _z \label{wave0}$$ where $\left| \mathbf{\mu }\right\rangle _z$ is the joint eigenstate of Pauli operators $Z_a$ ($a\in V$) with eigenvalues $(-1)^{\mu _a}$, $\left| +\right\rangle _x^V$ is the joint +1 eigenstate of Pauli operators $X_a$ ( $% a\in V$) , and $U_{ab}$ ($U_{ab}=diag\{1,1,1,-1\}$ in the $Z$ basis) is the controlled phase gate between qubits $a$ and $b$. Graph state can also be viewed as the result of successively performing 2-qubit Control-Z operations $U_{ab}$ to the initially unconnected $n$ qubit state $\left| +\right\rangle _x^V$. It can be shown that graph state is the joint $+1$ eigenstate of the $% n$ vertices stabilizers $$K_a=X_a\prod_{b\in N_a}Z_b:=X_aZ_{N_a},\text{ }a\in V.$$ Meanwhile, the graph state basis are $\left| G_{k_1,k_2,\cdots k_n}\right\rangle $ $=\prod_{a\in V}Z_a^{k_a}\left| G\right\rangle ,$ with $% k_a=0,1.$ Thus $$K_a\left| G_{k_1,k_2,\cdots k_n}\right\rangle =(-1)^{k_a}\left| G_{k_1,k_2,\cdots k_n}\right\rangle .$$ Since all of the graph basis states are local unitary equivalent, they all have equal entanglement, so we only need to determine the entanglement of graph state $\left| G\right\rangle $. Once the entanglement of a graph state is obtained, the entanglement of all the graph basis states are obtained. A widely used local operation in dealing with graph states is the so-called local complementation (LC)[@Hein2], which is a multi-local unitary operation $V_a$ on the $a-th$ qubit and its neighbors, defined as $V_a=\sqrt{% K_a}$ $=$ $\exp (-i\frac \pi 4X_a)\prod_{b\in N_a}\exp (i\frac \pi 4Z_b)$. LC centered on a qubit $a$ is visualized readily as a transformation of the subgraph of $a$-th qubit’s neighbours, such that an edge between two neighbours of $a$ is deleted if the two neighbours are themselves connected, or an edge is added otherwise. Entanglement inequalities ------------------------- The global robustness of entanglement [@Vidal] is defined as $$R(\rho )=\min_\omega t$$ such that there exists a state $\omega $ such that $(\rho +t\omega )/(1+t)$ is separable. The logarithmic robustness [@Cavalcanti] is $$LR(\rho )=\log _2(1+R(\rho )).$$ The relative entropy of entanglement is defined as the ”distance” to the closest separable state with respect to the relative entropy [@Vedral2], $$E_r(\rho )=\min_{\omega \in Sep}S\left( \rho \right\| \left. \omega \right) ,$$ where $S\left( \rho \right\| \left. \omega \right) =-S(\rho )-$ $tr\{\rho log_2\omega \}$ is the relative entropy, $S(\rho )$ is the von Neumann entropy, and $Sep$ is the set of separable states. The minimum is taken over all fully separable mixed states $\omega $. The geometric measure of entanglement for pure state $\left| \psi \right\rangle $, is defined as $$E_g(\left| \psi \right\rangle )=\min_{\left| \phi \right\rangle \in \mathit{% Pro}}-\log _2\left| \left\langle \phi \right| \left. \psi \right\rangle \right| ^2,$$ where $Pro$ is the set of product states. Hayashi *et al* [@Hayashi1] has been shown that the maximal number $N$ of pure states in the set $\{\left| \psi _i\right\rangle |i=1,...,N\}$, that can be discriminated perfectly by LOCC is bounded by the amount of entanglement they contain: $$\log _2N\leq n-\overline{LR(\left| \psi _i\right\rangle )}\leq n-\overline{% E_r(\left| \psi _i\right\rangle )}\leq n-\overline{E_g(\left| \psi _i\right\rangle )}, \label{wee}$$ where $n=\log _2D_H,$ $D_H$ is the total dimension of the Hilbert space, and $\overline{x}=\frac 1N\sum_{i=1}^Nx_i$ denotes the ”average”. For stabilizer state $\left| S\right\rangle $, it has been shown that[@Hayashi2] $$LR(\left| S\right\rangle )=E_r(\left| S\right\rangle )=E_g(\left| S\right\rangle )$$ The technique is to prove $LR(\left| S\right\rangle )=E_g(\left| S\right\rangle )$ and utilizing inequality (\[wee\]).The entanglement can be written as $$E(\left| S\right\rangle )=\min_\phi -\log _2\left| \left\langle S\right| \left. \phi \right\rangle \right| ^2, \label{wave1}$$ where $\left| \phi \right\rangle =\prod_j(\sqrt{p_j}\left| 0\right\rangle +% \sqrt{1-p_j}e^{i\varphi _j}\left| 1\right\rangle )$ is the product pure state. Entanglement bounds ------------------- The entanglement is upper bounded by the local operation and classical communication (LOCC) bound $E_{LOCC}=n-\log _2N$ , and lower bounded by some bipartite entanglement deduced from the state, that is, the ’matching’ bound $E_{bi}$ [@Markham]. It is well known that all graph states are stabilizer states, so the inequality for the entanglement of a graph state is $$E_{bi}\leq E\leq E_{LOCC}.$$ If the lower bound coincides with the upper bound, the entanglement of the graph state can be obtained. This is the case for ’2-colorable’ graph states such as multipartite GHZ states, Steane code, cluster state, and state of ring graph with even vertices. For a state of ring graph with odd $n $ vertices, we have $\left\lfloor \frac n2\right\rfloor \leq E\leq \left\lceil \frac n2\right\rceil $ [@Markham]$.$ The fidelity $F_\phi =\left| \left\langle G\right| \left. \phi \right\rangle \right| ^2$plays a crucial rule in calculating the entanglement. For a graph state, we have $$E=\min_{\phi \in \mathit{Pro}}-\log _2\left| \left\langle G\right| \left. \phi \right\rangle \right| ^2=-\log _2(\max_{\phi \in \mathit{Pro}}F_\phi ). \label{wave2}$$ Denote $F=\max_{\phi \in \mathit{Pro}}F_\phi $ as the fidelity between the graph state and the closest pure separable state. The upper LOCC bound for a graph state is $$E\leq n-\left| A\right| , \label{wave3}$$ since the largest number of entanglement basis states is $2^{\left| A\right| },$where $\left| A\right| $ is the largest number of vertices with any two of the vertices being not adjacent [@Markham]. The entanglement is lower bounded by the entanglement of a bipartition of the graph. The lower bound can be found by ”matching”[@Markham]. A convenient way of finding the lower bound of the entanglement is to find the largest set of non-adjacent edges first, then assign the two vertices of each edge to two parties to form a bipartition of the graph. The vertices that are not assigned can be assigned to either parties. To verify if these edges are the last Bell pairs that can be obtained, one can apply local Control-Z and LC to delete the redundant adjacent edges. It can be verified that all the graphs up to $8$ qubits in the literatures can be treated in this manner to obtain the lower bound of the entanglement. This is not a difficult task since most of the graphs in the literatures are already in the simplest LC equivalent form. Thus, at least for graph states up to $8$ qubits, the lower bound of the entanglement can be obtained by counting the largest number of non-adjacent edges. Iterative method for the closest product states =============================================== If the upper LOCC bound coincides with the lower bipartition bound, the entanglement of the graph state can be determined and equals to the bounds. Still there are graph states that the two bounds do not meet. We need a systematical method to calculate the entanglement of such graph states according to Eq. (\[wave2\]). The product pure state $\left| \phi \right\rangle $ can be denoted as $$\left| \phi \right\rangle =\prod_j(x_j\left| 0\right\rangle +y_j\left| 1\right\rangle )$$ where $x_j$ and $y_j$ are complex numbers subjected to $\left| x_j\right| ^2+\left| y_j\right| ^2=1.$ Denote $f=\left\langle G\right| \left. \phi \right\rangle ,$ then $$f=\frac 1{\sqrt{2^n}}\sum_{\mathbf{\mu =0}}^{\mathbf{1}}(-1)^{\frac 12% \mathbf{\mu \Gamma \mu }^T}\prod_j(x_j^{1-\mu _j}y_j^{\mu _j}).$$ Let $L=\left| f\right| ^2-\sum_j\lambda _j(\left| x_j\right| ^2+\left| y_j\right| ^2-1)$, where $\lambda _j$ are the Lagrange multipliers. Then $% \frac{\partial L}{\partial x_j}=0$ and $\frac{\partial L}{\partial y_j}=0$ lead to $$\begin{aligned} \frac{\partial f}{\partial x_j}f^{*}-\lambda _jx_j^{*} &=&0, \\ \frac{\partial f}{\partial y_j}f^{*}-\lambda _jy_j^{*} &=&0.\end{aligned}$$ The two equations are combined to $$y_j^{*}\frac{\partial f}{\partial x_j}-x_j^{*}\frac{\partial f}{\partial y_j}% =0. \label{wave4}$$ The left hand of Eq. (\[wave4\]) is $\left\langle G\right| \left. \phi _j\right\rangle ,$ with $$\begin{aligned} \left| \phi _j\right\rangle &=&\prod_{k=1}^{j-1}(x_k\left| 0\right\rangle +y_k\left| 1\right\rangle )(y_j^{*}\left| 0\right\rangle -x_j^{*}\left| 1\right\rangle ) \nonumber \\ &&\times \prod_{m=j+1}^n(x_m\left| 0\right\rangle +y_m\left| 1\right\rangle ).\end{aligned}$$ Thus Eq. (4) is to say that the graph state is orthogonal to all $\left| \phi _j\right\rangle $ ($j=1,\ldots ,n$ ) when $\left| \phi \right\rangle $ is the closest product state. It is clear that $\left| \phi \right\rangle $ is orthogonal to all $\left| \phi _j\right\rangle $ ($j=1,\ldots ,n$ ) too. Denote $z_j=y_j/x_j,$the derivatives are $$\begin{aligned} \frac{\partial f}{\partial x_j} &=&\frac 1{\sqrt{2^n}}\sum_{\mathbf{\nu =0}% }^{\mathbf{1}^{\prime }}(-1)^{\frac 12\mathbf{\nu \Gamma \nu }% ^T}\prod_{k\neq j}(x_k^{1-\mu _k}y_k^{\mu _k}) \nonumber \\ &=&\frac{\prod_{k\neq j}x_j}{\sqrt{2^n}}\sum_{\mathbf{\nu =0}}^{\mathbf{1}% ^{\prime }}(-1)^{\frac 12\mathbf{\nu \Gamma \nu }^T}\prod_{k\neq j}z_k^{\mu _k}, \\ \frac{\partial f}{\partial y_j} &=&\frac 1{\sqrt{2^n}}\sum_{\mathbf{\nu }% ^{\prime }\mathbf{=0}^{\prime }}^{\mathbf{1}}(-1)^{\frac 12\mathbf{\nu }% ^{\prime }\mathbf{\Gamma \nu }^{\prime T}}\prod_{k\neq j}(x_k^{1-\mu _k}y_k^{\mu _k}) \nonumber \\ &=&\frac{\prod_{k\neq j}x_j}{\sqrt{2^n}}\sum_{\mathbf{\nu }^{\prime }\mathbf{% =0}^{\prime }}^{\mathbf{1}}(-1)^{\frac 12\mathbf{\nu }^{\prime }\mathbf{% \Gamma \nu }^{\prime T}}\prod_{k\neq j}z_k^{\mu _k}.\end{aligned}$$ where $\mathbf{\nu =\{}\mu _1,\ldots ,\mu _{j-1},0,\mu _{j+1},\ldots ,\mu _n\},$ $\mathbf{\nu }^{\prime }\mathbf{=\{}\mu _1,\ldots ,\mu _{j-1},1,\mu _{j+1},\ldots ,\mu _n\},$ and $\mathbf{1}^{\prime }=\{1,\ldots ,1,0,1,\ldots ,1\},$ $\mathbf{0}^{\prime }=\{0,\ldots ,0,1,0,\ldots ,0\}.$ The binary vector $\mathbf{1}^{\prime }$ has all its entries being $1$ except the $j-th$ entry being $0.$ $\mathbf{0}^{\prime }$ is the logical NOT of $\mathbf{1}% ^{\prime }$. From Eq. (\[wave4\]) we obtain the iterative equations for $% z_j,$$$z_j^{*}=\frac{\sum_{\mathbf{\nu }^{\prime }\mathbf{=0}^{\prime }}^{\mathbf{1}% }(-1)^{\frac 12\mathbf{\nu }^{\prime }\mathbf{\Gamma \nu }^{\prime T}}\prod_{k\neq j}z_k^{\mu _k}}{\sum_{\mathbf{\nu =0}}^{\mathbf{1}^{\prime }}(-1)^{\frac 12\mathbf{\nu \Gamma \nu }^T}\prod_{k\neq j}z_k^{\mu _k}}. \label{wave5}$$ We consider the change of fidelity in one step of iteration, that is, we only renew $z_j$ according to Eq. (\[wave5\]) while keeping all the other $% z_k$ $(k\neq j)$ invariant in the step. Let $h_j=\frac{\partial f}{\partial x_j},$ $g_j=\frac{\partial f}{\partial y_j},$ then $h_j$ and $g_j$ are invariant in the step, $f=x_jh_j+y_jg_j.$ Forget the iteration equation for a while, we seek the maximization of the fidelity with respect to $x_j=\cos \theta ,y_j=\sin \theta e^{i\varphi }.$ The maximal fidelity should be $% \left| f\right| ^2=\left| h_j\right| ^2+\left| g_j\right| ^2,$which is achieved when $$z_j=\frac{y_j}{x_j}=\frac{g_j^{*}}{h_j^{*}}. \label{wee1}$$ The condition (\[wee1\]) of maximal fidelity is just the iterative equation (\[wave5\]). Thus in each step of the iteration, the fidelity does not decrease. The fidelity increases abruptly or keeps unchanged in one step. In fact in each step, the fidelity can increase continuously from its initial value to its final value by changing $(\theta ,\varphi )$ continuously. Starting with any initial complex random vector $\mathbf{z=}(z_1,\ldots ,z_n) $, the iterative equation renews each $z_j$ successively, the fidelity increases (or does not change). After all $z_j$ are renewed, a new round of iteration starts. The whole picture of iteration can be seen as a discrete process of the fidelity. The fidelity increases in each step until it does not increase any more. There may be the case that the fidelity reaches its local maximum. To find the global maximum, we run the iterative algorithm many times with random initial $\mathbf{z}$. Moreover, we calculate the fidelity of the graph state with respect to random separate states for a million times to determine roughly the possible range of the fidelity before the iteration calculation. Classification of the graph state up to 8 qubits ================================================ The LC inequivalent graphs up to $7$ qubits are all plotted in [@Hein1] and numbered. There are a two qubit graph (No.1), a three qubit graph (No.2), $2$ of four qubit graphs (No.3 and No.4), $4$ of five qubit graphs (No.5 to No.8), $11$ of six qubit graphs (No.9 to No.19), $26$ of seven qubit graphs (No.20 to No.45). In [@Cabello], the authors plotted all $% 101$ LC inequivalent graphs of $8$ qubits. The graphs of $8$ qubits are numbered from No.46 to No.146. Graph states with equal lower and upper bounds ---------------------------------------------- The entanglement of graph states with equal lower and upper bounds can be calculated with the methods in Ref. [@Markham]. It is listed in Table 1 and Table 2 for completeness. The graphs that are ”2-colorable” up to $8$ qubits are listed in Table $1.$ It is a well known fact in graph theory that a graph is 2-colorable iff it does not contain any cycles of odd length. The LOCC upper bound and the lower bipartite bound of the entanglement for ”2-colorable” graph state can be obtained by the methods described in Ref. [@Markham]. For each of these ”2-colorable” graph states, it has been found that the two bounds coincide with each other $E_{LOCC}=E_{bi}=E_r$, and the relative entropy of entanglement is equal to the entanglement in Schmidt measure [@Hein1], $% E_r=E_S=E$. Table $1$ shows the results. [**Table 1**]{}\ E No. --- -------------------------------------------- 1 1,2,3,5,9,20,46 2 4,6,7,10-12,15,21-24,31,47-51,69-70 3 13-14,18,25-30,34,38,43,52-63,74,76-77,81, 3 83-84,103-104,122 4 64-68,87,89,91,95,99-100,120,128,143 The LOCC bound for a ”non 2-colorable” graph can be obtained with the largest set of non-adjacent vertices. The lower bipartition bound can be found by first searching for the largest set of non-adjacent edges, then verifying the candidate Bell pairs with local Control-Z and LC. Thus for ”non 2-colorable” graph, the entanglement bounds of graph state can be obtained with ”balls” (vertices) and ”sticks” (edges) in the graph. When $E_{LOCC}=E_{bi}=E_r$, the graph states are shown in Table $2$ (with $% E=E_r=E_S$). No.101 graph is special for the graph state has the relative entropy of entanglement $E_r=4$, the Schmidt measure $E_S=3$. [**Table 2**]{}\ E No. --- ----------------------------------------------- 3 16-17,32-33,35-37,71-73,75,78-80,82,102,121 4 86,88,90,92-94,96-98,106-119,123-127,129-132, 4 135,136$^{*}$,144 \ \*the LC equivalent of No.136 Graph states with unequal bounds --------------------------------- Up to $8$ qubits, what left are the ”non 2-colorable” graph states whose upper entanglement bound $E_{LOCC}$ ($E_u$) and lower bound $E_{bi}$ ($E_l$) do not coincide. We utilize Eq. (\[wave5\]) to iteratively calculate the entanglement and find the closest product state with random initial complex numbers for $z_j$ ($j=1,\ldots ,n$). The values of relative entropy of entanglement are listed in Table $3.$ A detail comparison of computed closest product states of No.8, No.39, No.41, No.45, No.85, No.105, No.134,No.137,No.138, No.140 shows that all these closest states have a substructure of the closest product state of ring $5$ graph (No.8), although graph No.140 does not contain ring $5$ graph explicitly, graph No.45 seems to contain graph No.19 as its subgraph. Ring $% 5 $ graph is essential to all these graph states with entanglement $k+0.9275$ (integer $k$)$.$ In Ref. [@Chen] an identical product closest state is supposed for ring $5$ graph state, and it has been shown that the entanglement of ring $5$ graph state is $$E_{ring5}=1+\log _23+\log _2(3-\sqrt{3})\approx 2.9275.$$ Denote $\left| \Phi _j\right\rangle =\sqrt{p}\left| 0\right\rangle +\sqrt{1-p% }e^{i\varphi _j}\left| 1\right\rangle ,$($j=1,\ldots ,4$), with $\sqrt{p}=% \sqrt{\frac 12(1-\frac 1{\sqrt{3}})}\approx 0.4597,$ $\varphi _1=\frac \pi 4,\varphi _2=-\frac \pi 4,\varphi _3=\frac{3\pi }4,\varphi _4=-\frac{3\pi }% 4. $ Typically, the closest product state of ring $5$ graph state is $$\left| \phi _{ring5}\right\rangle =\left| \Phi _1\right\rangle ^{\otimes 5},$$ The other graph states may have their closest product states $$\begin{aligned} \left| \phi _{No.39}\right\rangle &=&\left| -\right\rangle \left| 0\right\rangle \left| \Phi _3\right\rangle \left| \Phi _2\right\rangle ^{\otimes 4}, \\ \left| \phi _{No.41}\right\rangle &=&\left| \Phi _4\right\rangle \left| \Phi _1\right\rangle ^{\otimes 3}\left| \Phi _4\right\rangle \left| 0\right\rangle \left| -\right\rangle , \\ \left| \phi _{No.45}\right\rangle &=&\left| -\right\rangle )\left| 0\right\rangle \left| \Phi _4\right\rangle \left| \Phi _3\right\rangle ^{\otimes 3}\left| \Phi _4\right\rangle , \\ \left| \phi _{No.85}\right\rangle &=&\left| -\right\rangle ^{\otimes 2}\left| \Phi _1\right\rangle ^{\otimes 4}\left| \Phi _4\right\rangle \left| 0\right\rangle , \\ \left| \phi _{No.105}\right\rangle &=&\left| +\right\rangle ^{\otimes 2}\left| \Phi _1\right\rangle ^{\otimes 5}\left| 1\right\rangle , \\ \left| \phi _{No.134}\right\rangle &=&\left| -\right\rangle ^{\otimes 2}\left| \Phi _4\right\rangle \left| \Phi _3\right\rangle ^{\otimes 3}\left| \Phi _4\right\rangle \left| 0\right\rangle , \\ \left| \phi _{No.137}\right\rangle &=&\left| +\right\rangle \left| 0\right\rangle \left| \Phi _4\right\rangle \left| \Phi _1\right\rangle ^{\otimes 3}\left| \Phi _4\right\rangle \left| 0\right\rangle , \\ \left| \phi _{No.138}\right\rangle &=&\left| \Phi _2\right\rangle \left| \Phi _3\right\rangle ^{\otimes 2}\left| 0\right\rangle \left| +\right\rangle \left| 0\right\rangle \left| \Phi _3\right\rangle ^{\otimes 2}, \\ \left| \phi _{No.140}\right\rangle &=&\left| 1\right\rangle \left| \Phi _2\right\rangle ^{\otimes 2}\left| \Phi _4\right\rangle ^{\otimes 3}\left| 0\right\rangle \left| -\right\rangle ,\end{aligned}$$ where $\left| \pm \right\rangle =\frac 1{\sqrt{2}}(\left| 0\right\rangle \pm \left| 1\right\rangle ).$ The next graph set (No.19, No.139, No.141 ) with non-integer entanglement ( $% k+0.5850$) graph states is specified by No.19 (\[\[6,0,4\]\] stabilizer state). The entanglement of No.19 is $$E_{No.19}=2+\log _23\approx 3.5850.$$ Typically, the closest product state is $$\left| \phi _{No.19}\right\rangle =\left| \Phi _3\right\rangle ^{\otimes 3}\left| \Phi _4\right\rangle ^{\otimes 3},$$ The closest states for No.139 and No.141 graph state can be $$\begin{aligned} \left| \phi _{No.139}\right\rangle &=&\left| -\right\rangle \left| 1\right\rangle \left| \Phi _4\right\rangle \left| \Phi _3\right\rangle ^{\otimes 3}\left| \Phi _4\right\rangle \left| \Phi _1\right\rangle . \\ \left| \phi _{No.141}\right\rangle &=&\left| -\right\rangle \left| 0\right\rangle \left| \Phi _2\right\rangle \left| \Phi _4\right\rangle ^{\otimes 3}\left| \Phi _3\right\rangle \left| \Phi _2\right\rangle .\end{aligned}$$ No.139 and No.141 graphs have No.19 as their subgraph. The entanglement of No.133 graph state is $$\begin{aligned} E_{No.133} &=&2+3\log _23+\log _2(2-\sqrt{3}) \\ &\approx &4.8549,\end{aligned}$$ its closest product state can be $$\left| \phi _{No.133}\right\rangle =\left| \Phi _4\right\rangle \left| \Phi _1\right\rangle ^{\otimes 2}\left| \Phi _4\right\rangle ^{\otimes 2}\left| \Phi _1\right\rangle ^{\otimes 2}\left| \Phi _4\right\rangle .$$ The entanglement of No.44 is $E_{No.44}=4,$ its closest product state can be $$\left| \phi _{No.44}\right\rangle =\left| \bigcirc \right\rangle ^{\otimes 2}\left| +\right\rangle ^{\otimes 3}\left| \bigcirc \right\rangle \left| -\right\rangle .$$ where $\left| \bigcirc \right\rangle =\frac 1{\sqrt{2}}(\left| 0\right\rangle -i\left| 1\right\rangle ).$ The closest product states for No.40,42,142,145,146 can also be obtained, the iteration calculation should be modified as explained in the next section. [**Table 3**]{}\ No. $E_u$ $E_l$ $E_r$ $E_S$ $P_{s}$ ----- ------- ------- -------- ------- ---------- 8 3 2 2.9275 2-3 0.997 19 4 3 3.5850 3-4 1.000 39 4 3 3.9275 3-4 0.790 40 4 3 4 3-4 0.241(1) 41 4 3 3.9275 3-4 0.264 42 4 3 4 3-4 0.950(1) 44 4 3 4 3-4 0.432 45 4 3 3.9275 3-4 0.967 85 4 3 3.9275 3-4 0.689 105 4 3 3.9275 3-4 0.658 133 5 4 4.8549 4-5 0.969 134 4 3 3.9275 3 0.646 137 5 4 4.9275 4-5 0.917 138 5 4 4.9275 4-5 0.617 139 5 4 4.5850 4-5 0.999 140 5 4 4.9275 4-5 0.571 141 5 4 4.5850 4 0.935 142 5 4 5 4-5 0.281(1) 145 5 4 5 4-5 0.870(2) 146 5 4 5 4-5 0.501(1) \ $E_{u}=E_{LOCC},E_{l}=E_{bi}$ Precision of iteration ====================== We concentrate on the precision of iteration for calculating the entanglement of graph state whose lower and upper bounds do not meet. Let $% \Delta =\left| E_{numeric}-E_{theory}\right| $ be the computational error of the iteration, where $E_{numeric}$ is the entanglement determined by iteration, $E_{theory}$ $(=E_r)$ is the entanglement proposed in the former section . We use the exact value of $E_{theory}$ rather than its approximation. For simplicity, we just give the successful probabilities of achieving the precision within $\Delta \leq 10^{-14}$ for some reasonable rounds of iteration with random initial conditions. From the actual numerical calculations, we can see that a precision of $10^{-14}$ is limited by the computer for our iterative algorithm (without double precision calculation). For all graph states presented in Table 3 except No.40, 42, 142, 145, 146, the algorithm can be applied directly. The successful probabilities ($P_s$ ) are listed in Table 3. The round of the iteration is set to $150$ except for No.140, whose round of iteration is $300$. We renew $\mathbf{z}$ after each round instead of renewing $z_j$ after each step of the round in the actual calculation for the reason of programming. In order to calculate the successful probability, we run the algorithm $1000$ times for each graph state to count the number of algorithm that achieves the precision within $% \Delta \leq 10^{-14}$. For No.40, 42, 142, 145, 146 graph states, direct application of iterative algorithm fails. The numerical results of entanglement are all greater than the values given in Table 3, but the precision of the calculation is far from satisfactory. The precision can be $10^{-4}$ or so. A detail analysis of the separable state which gives best numerical value of entanglement shows us that the iterative equations (\[wave5\]) are correlated. The common figure of these nonlinear correlations of equations can be illustrated by applying iterative algorithm to the simplest graph state, the No.1 graph state (Bell pair). The iterative equations should be $$\begin{aligned} z_1^{*} &=&\frac{1-z_2}{1+z_2},\text{ } \label{wave6} \\ z_2^{*} &=&\frac{1-z_1}{1+z_1}. \label{wave7}\end{aligned}$$ Substituting Eq.(\[wave7\]) into Eq.(\[wave6\]), we obtain the identity $% z_1^{*}=z_1^{*}.$ Thus the two equations are correlated. The correlation of equations leads to the fail of iteration. We can delete one of Eq.(\[wave6\]) into Eq.(\[wave7\]) to solve the problem. The fidelity is $$\left| f\right| ^2=\frac{\left| 1+z_1+z_2-z_1z_2\right| ^2}{4(1+\left| z_1\right| ^2)(1+\left| z_1\right| ^2)}$$ Applying Eq.(\[wave6\]) and ignoring Eq.(\[wave7\]), we obtain the correct maximal fidelity $\left| f\right| ^2=\frac 12$. Thus to obtain the maximal fidelity, we should omit some of the equations and use the remain equations for iteration. For No.40, 42,142 and 146, we omit one of the equations, indicated in Table 3 with notation (1) behind the successful probabilities. For No.145, we omit two of the equations, indicated in Table 3 with notation (2) behind the successful probability. In the numerical calculations, we set one or two $z_i$ to random numbers that do not change in the iteration, respectively. Since we do not know if all the equations are correlated or only some of them are correlated, we calculate all possible choices of fixing $z_i$. For a given graph state, some of the choices of fixing $z_i$ may not lead to sufficiently high successful probabilities or simply fail. The successful probabilities shown in Table 3 are the best. We can see that the entanglement of all graph states in Table 3 can be efficiently calculated by iterative algorithm with very high precision. Most of them can be calculated directly, five of them can be calculated with modified iterative algorithm. A heuristic point of view is that we can set the fixed $z_i$ to be $0$. For an $n$ vertices graph state $\left| G_n\right\rangle $, the closest separable state should be $\left| \phi _n\right\rangle =$ $\left| \phi _{n-1}\right\rangle \left| 0\right\rangle $ when we set $z_n$ $=$ $0$ without loss of generality. Denote the $n$ bit binary vector $\mathbf{\mu }% _n $ as $(\mathbf{\mu }_{n-1},\mu _n),$ and the $n\times n$ adjacent matrix $% \Gamma _n$ as $\left( \begin{array}{ll} \Gamma _{n-1} & c^T \\ c & 0 \end{array} \right) ,$then $$\frac 12(\mathbf{\mu }_{n-1},0)\Gamma _n(\mathbf{\mu }_{n-1},0\mathbf{)}% ^T=\frac 12\mathbf{\mu }_{n-1}\Gamma _{n-1}\mathbf{\mu }_{n-1}^T.$$ Since $\left\langle \phi _n\right. \left| \mathbf{\mu }_n\right\rangle =\left\langle \phi _{n-1}\right| \left\langle 0\right| \left. \mathbf{\mu }% _n\right\rangle =\left\langle \phi _{n-1}\right| \left. \mathbf{\mu }% _{n-1}\right\rangle \delta _{0\mu _n},$ from the definition of graph state, we have $$\left\langle G_n\right. \left| \phi _{n-1}\right\rangle \left| 0\right\rangle =\frac 1{\sqrt{2}}\left\langle G_{n-1}\right| \left. \phi _{n-1}\right\rangle . \label{wave8}$$ Where $G_{n-1}$ is the subgraph of $G_n$. From Eq.(\[wave8\]), a general relation for entanglement of any graph state and its subgraph state follows $$E_n\leq E_{n-1}+1. \label{wave9}$$ The equality holds for the case when $\left| \phi _n\right\rangle =$ $\left| \phi _{n-1}\right\rangle \left| 0\right\rangle $ is the closest separable state. For graph states No.40, 42,142,145,146, we can choose $\left| \phi _n\right\rangle =$ $\left| \phi _{n-1}\right\rangle \left| 0\right\rangle $ as the closest separable state when $\left| \phi _{n-1}\right\rangle $ is the closest separable state of the subgraph state, the calculation of the entanglement can be reduced, we have $$E_n=E_{n-1}+1.$$ for these graph states. The subgraphs of No.40 and No.42 belong to the set $% \{No.13,No.14,No.17,No.18\},$the entanglement of all the subgraph states is $% 3$. Thus the entanglement of No.40 and No.42 is $4.$ The subgraphs of No.142, No.145 and No.146 belong to the set $\{No.40,No.42,No.44\},$the entanglement of all the subgraph states is $4$. Thus the entanglement of No.142, No.145 and No.146 is $5.$ Conclusions =========== The entanglement of graph states measured in terms of the relative entropy of entanglement (also with logarithmic robust, the geometric measure) is given up to eight qubits. We use the iterative method to calculate the fidelity of graph state with respect to closest separable pure state. The iterative equations are results of the maximization of the fidelity. The equations have a clear meaning that graph state should be orthogonal to all other separable states that are orthogonal to the closest separable state. We have proved that in each step of a round of iteration the fidelity does not decrease. The iteration calculation has a very high efficiency if the resultant closest state does not contain $\left| 0\right\rangle $ or $\left| 1\right\rangle $ at all qubits (No.8, No.19, No.133). The precision of the iteration calculation can be less than $10^{-14}$ for all graph states up to $8$ qubits with unequal lower and upper bounds of entanglement. To avoid possible missing of the global maximum, with random initial parameters, we calculate the fidelity for each graph state a million times without iteration to determine its rough range, and calculate the maximal fidelity 1000 times with iteration. Iterative method brings us with the exact entanglement value of the graph state if we substitute the numerical closest separable state with its nearest exact one. For a given graph state, there are many local equivalent closet separable states, they all lead to the same exact value of the entanglement. The precision of the numerical calculation is defined as the difference of the numerical and the exact entanglement. For some of the graph states, the iterative equations may correlate with each other. 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--- author: - 'Hazim Shakhatreh, Ahmad Sawalmeh, Ala Al-Fuqaha, Zuochao Dou, Eyad Almaita, Issa Khalil, Noor Shamsiah Othman, Abdallah Khreishah, Mohsen Guizani [^1] [^2] [^3] [^4] [^5] [^6]' bibliography: - 'Ref5.bib' title: 'Unmanned Aerial Vehicles: A Survey on Civil Applications and Key Research Challenges' --- Abstract {#abstract .unnumbered} ======== The use of unmanned aerial vehicles (UAVs) is growing rapidly across many civil application domains including real-time monitoring, providing wireless coverage, remote sensing, search and rescue, delivery of goods, security and surveillance, precision agriculture, and civil infrastructure inspection. Smart UAVs are the next big revolution in UAV technology promising to provide new opportunities in different applications, especially in civil infrastructure in terms of reduced risks and lower cost. Civil infrastructure is expected to dominate the more that $\$45$ Billion market value of UAV usage. In this survey, we present UAV civil applications and their challenges. We also discuss current research trends and provide future insights for potential UAV uses. Furthermore, we present the key challenges for UAV civil applications, including: charging challenges, collision avoidance and swarming challenges, and networking and security related challenges. Based on our review of the recent literature, we discuss open research challenges and draw high-level insights on how these challenges might be approached. UAVs, Wireless Coverage, Real-Time Monitoring, Remote Sensing, Search and Rescue, Delivery of goods, Security and Surveillance, Precision Agriculture, Civil Infrastructure Inspection. Introduction ============ UAVs can be used in many civil applications due to their ease of deployment, low maintenance cost, high-mobility and ability to hover [@hayat2016survey]. Such vehicles are being utilized for real-time monitoring of road traffic, providing wireless coverage, remote sensing, search and rescue operations, delivery of goods, security and surveillance, precision agriculture, and civil infrastructure inspection. The recent research literature on UAVs focuses on vertical applications without considering the challenges facing UAVs within specific vertical domains and across application domains. Also, these studies do not discuss practical ways to overcome challenges that have the potential to contribute to multiple application domains. The authors in [@hayat2016survey] present the characteristics and requirements of UAV networks for envisioned civil applications over the period 20002015 from a communications and networking viewpoint. They survey the quality of service requirements, network-relevant mission parameters, data requirements, and the minimum data to be transmitted over the network for civil applications. They also discuss general networking related requirements, such as connectivity, adaptability, safety, privacy, security, and scalability. Finally, they present experimental results from many projects and investigate the suitability of existing communications technologies to support reliable aerial networks.  -------------------------- -------------------- ---------------- ------------ ------------ ------------- -------------- ------------- ---------------- Reference Providing wireless Remote sensing Real-Time Search and Delivery of Surveillance Precision Infrastructure coverage monitoring Rescue goods agriculture inspection [@hayat2016survey] [@gupta2016survey] [@motlagh2016low] [@mozaffari2018tutorial] [@khawaja2018survey] [@aja2018] This work -------------------------- -------------------- ---------------- ------------ ------------ ------------- -------------- ------------- ---------------- -------------------------- ----------- -------- ------------ ----------- ---------- ----- ----- ------------ Reference Collision mmWave Free space Cloud Machine NFV SDN Image avoidance optical computing learning processing [@hayat2016survey] [@gupta2016survey] [@motlagh2016low] [@mozaffari2018tutorial] [@khawaja2018survey] [@aja2018] This work -------------------------- ----------- -------- ------------ ----------- ---------- ----- ----- ------------ In [@gupta2016survey], the authors attempt to focus on research in the areas of routing, seamless handover and energy efficiency. First, they distinguish between infrastructure and ad-hoc UAV networks, application areas in which UAVs act as servers or as clients, star or mesh UAV networks and whether the deployment is hardened against delays and disruptions. Then, they focus on the main issues of routing, seamless handover and energy efficiency in UAV networks. The authors in [@bekmezci2013flying] survey Flying Ad-Hoc Networks (FANETs) which are ad-hoc networks connecting the UAVs. They first clarify the differences between FANETs, Mobile Ad-hoc Networks (MANETs) and Vehicle Ad-Hoc Networks (VANETs). Then, they introduce the main FANET design challenges and discuss open research issues. In [@zeng2016wireless], the authors provide an overview of UAV-aided wireless communications by introducing the basic networking architecture and main channel characteristics. They also highlight the key design considerations as well as the new opportunities to be explored. The authors of [@kumbhar2017survey] present an overview of legacy and emerging public safety communications technologies along with the spectrum allocation for public safety usage across all the frequency bands in the United States. They conclude that the application of UAVs in support of public safety communications is shrouded by privacy concerns and lack of comprehensive policies, regulations, and governance for UAVs. In [@chmaj2015distributed], the authors survey the applications implemented using cooperative swarms of UAVs that operate as distributed processing system. They classify the distributed processing applications into the following categories: 1) general purpose distributed processing applications, 2) object detection, 3) tracking, 4) surveillance, 5) data collection, 6) path planning, 7) navigation, 8) collision avoidance, 9) coordination, 10) environmental monitoring. However, this survey does not consider the challenges facing UAVs in these applications and the potential role of new technologies in UAV uses. The authors of [@motlagh2016low] provide a comprehensive survey on UAVs, highlighting their potential use in the delivery of Internet of Things (IoT) services from the sky. They describe their envisioned UAV-based architecture and present the relevant key challenges and requirements. In [@mozaffari2018tutorial], the authors provide a comprehensive study on the use of UAVs in wireless networks. They investigate two main use cases of UAVs; namely, aerial base stations and cellular-connected users. For each use case of UAVs, they present key challenges, applications, and fundamental open problems. Moreover, they describe mathematical tools and techniques needed for meeting UAV challenges as well as analyzing UAV-enabled wireless networks. The authors of [@khawaja2018survey] provide a comprehensive survey on available Air-to-Ground channel measurement campaigns, large and small scale fading channel models, their limitations, and future research directions for UAV communications scenarios. In [@aja2018], the authors provide a survey on the measurement campaigns launched for UAV channel modeling using low altitude platforms and discuss various channel characterization efforts. They also review the contemporary perspective of UAV channel modeling approaches and outline some future research challenges in this domain. UAVs are projected to be a prominent deliverer of civil services in many areas including farming, transportation, surveillance, and disaster management. In this paper, we review several UAV civil applications and identify their challenges. We also discuss the research trends for UAV uses and future insights. The reason to undertake this survey is the lack of a survey focusing on these issues. Tables \[tableint1\] and \[tableint2\] delineate the closely related surveys on UAV civil applications and demonstrate the novelty of our survey relative to existing surveys. Specifically, the contributions of this survey can be delineated as: - Present the global UAV payload market value. The payload covers all equipment which are carried by UAVs such as cameras, sensors, radars, LIDARs, communications equipment, weaponry, and others. We also present the market value of UAV uses. - Provide a classification of UAVs based on UAV endurance, maximum altitude, weight, payload, range, fuel type, operational complexity, coverage range and applications. - Present UAV civil applications and challenges facing UAVs in each application domain. We also discuss the research trends for UAV uses and future insights. The UAV civil applications covered in this survey include: real-time monitoring of road traffic, providing wireless coverage, remote sensing, search and rescue, delivery of goods, security and surveillance, precision agriculture, and civil infrastructure inspection. - Discuss the key challenges of UAVs across different application domains, such as charging challenges, collision avoidance and swarming challenges, and networking and security challenges. Our survey is beneficial for future research on UAV uses, as it comprehensively serves as a resource for UAV applications, challenges, research trends and future insights, as shown in Figure \[hazoverall\]. For instance, UAVs can be utilized for providing wireless coverage to remote areas such as Facebook’s Aquila UAV [@zuckerberg2014connecting]. In this aplication, UAVs need to return periodically to a charging station for recharging, due to their limited battery capacity. To overcome this challenge, solar panels installed on UAVs harvest the received solar energy and convert it to electrical energy for long endurance flights [@sun2018resource]. We can envisage Laser power-beaming as a future technology to provide supplemental energy at night when solar energy is not available or is minimal at high latitudes during winter. This would enable such UAVs to fly day and night for weeks or possibly months without landing [@sheet2014beamed]. The rest of the survey is organized as follows. We start with an overview of the global UAV market value and UAV classification in Part I of this survey (Sections II and III). In Part II (Sections IV-XI), we present UAV civil applications and the challenges facing UAVs in each application domain, and we also discuss the research trends and future insights for UAV uses. In Part III (Sections XII and XIII), we discuss the key challenges of UAV civil applications and conclude this study. Finally, a list of the acronyms used in the survey is presented in Section XIV. To facilitate reading, Figure \[enable2\] provides a detailed structure of the survey.  PART I: MARKET OPPORTUNITY and UAV Classification {#part-i-market-opportunity-and-uav-classification .unnumbered} ================================================== MARKET OPPORTUNITY {#Market} ================== UAVs offer a great market opportunity for equipment manufacturers, investors and business service providers. According to the PwC report [@pwc18], the addressable market value of UAV uses is over $\$127$ billion as shown in Figure \[ma2\]. Civil infrastructure is expected to dominate the addressable market value of UAV uses, with market value of $\$45$ billion. A report released by the Association for Unmanned Vehicle Systems International, expects more than $100,000$ new jobs in unmanned aircrafts by $2025$ [@Tiffany]. To know how to operate a UAV and meet job requirements, a person should attend training programs at universities or specialized institutes. Global UAV payload market value is expected to reach $\$3$ billion by $2027$ dominated by North America, followed by Asia-Pacific and Europe [@SDI]. The payload covers all equipment which are carried by UAVs such as cameras, sensors, radars, LIDARs, communications equipment, weaponry, and others [@grandviewresearch]. Radars and communications equipment segment is expected to dominate the global UAV payload market with a market share of close to $80\%$, followed by cameras and sensors segment with around over $11\%$ share and weaponry segment with almost $9\%$ share [@SDI] as shown in Figure \[ma1\]. Business Intelligence expects sales of UAVs to reach $\$12$ billion in $2021$, which is up by a compound annual growth rate of $7.6\%$ from $\$8.5$ billion in $2016$ [@DivyaJoshi]. This future growth is expected to occur across three main sectors: 1) Consumer UAV shipments which are projected to reach $29$ million in $2021$; 2) Enterprise UAV shipments which are projected to reach $805,000$ in $2021$; 3) Government UAVs for combat and surveillance. According to Bard Center for the Study of UAVs, U.S. Department of Defense allocated a budget of $\$4.457$ billion for UAVs in $2017$ [@gettinger2016drone]. All these statistics show the economic importance of UAVs and their applications in the near future for equipment manufacturers, investors and business service providers. Smart UAVs will provide a unique opportunity for UAV manufacturers to utilize new technological trends to overcome current challenges of UAV applications. To spread UAV services globally, a complete legal framework and institutions regulating the commercial use of UAVs are needed [@pwc17]. ![ Predicted Value of UAV Solutions in Key Industries (Billion).[]{data-label="ma2"}](srv_sec2_f1.png) ![ Global UAV Payload Market Predictions 2027.[]{data-label="ma1"}](srv_sec2_f2.png) UAV Classification {#S:1} ================== Unmanned vehicles can be classified into five different types according to their operation. These five types are unmanned ground vehicles, unmanned aerial vehicles, unmanned surface vehicles (operating on the surface of water), unmanned underwater vehicles, and unmanned spacecrafts. Unmanned vehicles can be either remote guided or autonomous vehicles [@Uncrewed-vehicle:Online]. There has been many research studies on unmanned vehicles that reported progress towards *autonomic systems* that do not require human interactions. In line with the concept of autonomy with respect to humans and societies, technical systems that claim to be autonomous must be able to make decisions and react to events without direct interventions by humans [@sebbane2015smart]. Therefore, some fundamental elements are common to all autonomous vehicles. These elements include: ability of sensing and perceiving the environment, ability of analyzing, communicating, planning and decision making using on-board computers, as well as acting which requires vehicle control algorithms. UAV features may vary depending on the application in order for them to fit their specific task. Therefore, any classification of UAVs needs to take into consideration their various features as they are widely used for a variety of civilian operations [@korchenko2013generalized]. The use of UAVs as an aerial base station in communications networks, can be categorized based on their operating platform, which can be a Low Altitude Platform (LAP) [@al2014modeling; @valcarce2013airborne; @reynaud2012deployable] or High Altitude Platform (HAP) [@tozer2001high; @thornton2001broadband; @karapantazis2005broadband]. LAP is a quasi-stationary aerial communications platform that operates at an altitude of less than $10$ km. Three main types of UAVs that fall under this category are vertical take-off and landing (VTOL) vehicles, aircrafts, and balloons. On the other hand, HAP operates at very high altitude above $10$ km, and vehicles utilizing this platform are able to stay for a long time in the upper layers of the stratosphere. Airships, aircrafts, and balloons are the main types of UAVs that fall under this category. These two categories of aerial communications platforms are shown in Figure  \[fig:ahmad1-1\]. More specifically, in each platform the main UAV types and examples of each type are illustrated in this figure. for tree=[ align=center, parent anchor=south, child anchor=north, font=, edge=[thick, -[Stealth\[\]]{}]{}, l sep+=10pt, edge path=[ (!u.parent anchor) – +(0,-10pt) -| (.child anchor); ]{}, if level=0[ inner xsep=0pt, tikz=[(.south east) – (.south west);]{} ]{} ]{} \[ UAV classification based on\ communication platform \[LAP \[Balloon \[Tethered Helikite\ [@chandrasekharan2016designing; @bucaille2013rapidly]\] \] \[VTOL \[Quadrotor \[The FALCON [@Sentinel:Online]\] \] \] \[Aircraft \[Viking aircraft\ [@matolak2014initial; @sun2017air; @matolak2017air]\] \] \] \[HAP \[Aircraft \[Helios\ (NASA) [@tozer2001high]\ Heliplat(Europe)\ [@thornton2001broadband; @grace2003european]\] \] \[Balloon \[Project Loon\ Balloon\ (Google)\ [@katikala2014google] \] \] \[Airship \[Zeppelin\ NT\ [@tozer2001high]\] \] \] \] A comparison between HAP and LAP is presented in Table \[table:ahmad1-2\] . This table also summarizes the main UAV types for each platform, and its performance parameters. Specifically, these parameters are UAV endurance, maximum altitude, weight, payload, range, deployment time, fuel type, operational complexity, coverage range, applications and examples for each UAV type is also shown in this table.\ \[table1\] [width=1.0]{} Issues ----------------- --------------------------------------- -------------------------------- --------------------------------------- -------------------------- -------------------- ----------------------------------------------- Type Airship Aircraft Balloon VTOL Aircraft Balloon Endurance long endurance - 15-30 hours JP-fuel Long endurance Few hours Few hours From 1 day - $>$7 days Solar Up to 100 days To few days Max. Altitude Up to 25 km 15-20 km 17-23 km Up to 4 km Up to 5 km Up to 1.5 km Payload (kg) Hundreds of kg’s Up to 1700 kg Tens of kg’s Few kg’s Few kg’s Tens of kg’s Flight Range Hundreds of km’s From 1500 to Up to Tens of km’s Less than 200 km Tethered Balloon 25000 km 17 million km Deployment time Need Runway Need Runway custom-built Easy to deploy Easy to launched Easy to deploy Auto launchers by catapult 10-30 minutes Fuel type Helium Balloon JP-8 jet fuel Helium Balloon Batteries Fuel injection Helium Solar panels Solar panels Solar panels engine Operational Complex Complex Complex Simple Medium Simple complexity Coverage area Hundreds of km’s Hundreds of km’s Thousands of km’s Tens of km’s Hundreds of km’s Several tens of km’s UAV Weight Few hundreds Few thousands Tens of kg’s Few of kg’s Tens of kg’s Tens of kg’s of kg’s of kg’s Public safety Considered safe Considered safe Need global regulations Need safety regulations Safe Safe Applications Testing environmental GIS Imaging Internet Delivery Internet Delivery Agriculture Aerial effects application base station Examples HiSentinel80 [@smith2011hisentinel80] Global Hawk [@gupta2013review] Project Loon LIDAR [@Sentinel:Online] EMT Luna Desert Star Balloon (Google)[@katikala2014google] X-2000 [@Luna2000] 34cm Helikite [@chandrasekharan2016designing] \[table:ahmad1-2\] PART II: UAV Applications {#part-ii-uav-applications .unnumbered} ========================= Search and Rescue (SAR) ======================== In the wake of new scientific developments, speculations shot up with regard to the future potential of UAVs in the context of public and civil domains. UAVs are believed to be of immense advantage in these domains, especially in support of public safety, search and rescue operations and disaster management. In case of natural or man-made disasters like floods, Tsunamis, or terrorist attacks, critical infrastructure including water and power utilities, transportation, and telecommunications systems can be partially or fully affected by the disaster. This necessitates rapid solutions to provide communications coverage in support of rescue operations [@hayat2016survey]. When the public communications networks are disrupted, UAVs can provide timely disaster warnings and assist in speeding up rescue and recovery operations. UAVs can also carry medical supplies to areas that are classified as inaccessible. In certain disastrous situations like poisonous gas infiltration, wildfires, avalanches, and search for missing persons, UAVs can be used to play a support role and speed up SAR operations [@silvagni2017multipurpose]. Moreover, UAVs can quickly provide coverage of a large area without ever risking the security or safety of the personnel involved. UAV-Based SAR System -------------------- SAR operations using traditional aerial systems (e.g., aircrafts and helicopters) are typically very costly. Moreover, aircrafts require special training, and special permits for taking off and landing areas. However, using UAVs in SAR operations reduces the costs, resources and human risks. Unfortunately, every year large amounts of money and time are wasted on SAR operations using traditional aerial systems [@silvagni2017multipurpose]. UAVs can contribute to reduce the resources needed in support of more efficient SAR operations. There are two types of SAR systems, single UAV systems, and Multi-UAV systems. A single UAV system is illustrated in Figure \[sar1\]. In the first step, the rescue team defines the search region, then the search operation is started by scanning the target area using a single UAV equipped with vision or thermal cameras. After that, real-time aerial videos/images from the targeted area are sent to the Ground Control System (GCS). These videos and images are analyzed by the rescue team to direct the SAR operations optimally [@doherty2007uav]. ![Use of Single UAV Systems in SAR Operations, Mountain Avalanche Events. []{data-label="sar1"}](srv_SAR_f1.png) In Multi-UAV systems, UAVs with on-board imaging sensors are used to locate the position of missing persons. The following processes summarize the SAR operations that use these systems. Firstly, the rescue team conduct path planning to compute the optimal trajectory of the SAR mission. Then, each UAV receives its assigned path from the GCS. Secondly, the search process is started. During this process, all UAVs follow their assigned trajectories to scan the targeted region. This process utilizes object detection, video/image transmission and collision avoidance methods. Thirdly, the detection process is started. During this process, a UAV that detects an object hovers over it, while the other UAVs act as relay nodes to facilitate coordination between all the UAVs and communications with the GCS. Afterword, UAVs switch to data dissemination mode and setup a multi-hop communications link with the GCS. Finally, the location of the targeted object, and related videos and images are transmitted to the GCS. Figure \[sar-steps\], illustrates the use of multi-UAV systems in support of SAR operations [@scherer2015autonomous]. (module4) |- (\[yshift=-10mm\]module1.south) – (additional-module1); How SAR Operations Utilize UAVs ------------------------------- SAR operations are one of the primary use-cases of UAVs. Their use in SAR operations attracted considerable attention and became a topic of interest in the recent past. SAR missions can utilize UAVs as follows: 1. Taking high resolution images and videos using on-board cameras to survey a given target area *(stricken region)*. Here, UAVs are used for post disaster aerial assessment or damage evaluation. This helps to evaluate the magnitude of the damage in the infrastructure caused by the disaster. After assessment, rescue teams can identify the targeted search area and commence SAR operations accordingly [@ruiz2017unmanned]. 2. SAR operations using UAVs can be performed autonomously, accurately and without introducing additional risks [@silvagni2017multipurpose]. In the Alcedo project [@ALCEDO:Online], a prototype was developed using a lightweight quadrotor UAV equipped with GPS to help in finding lost persons. In a Capstone project [@joern2015examining], using UAVs in support of SAR operations in snow avalanche scenarios is explored. The used UAV utilizes thermal infrared imaging and Geographic Information System (GIS) data. In such scenarios, UAVs can be utilized to find avalanche victims or lost persons. 3. UAV can be also used to deliver food, water and medicines to the injured. Although the use of UAVs in SAR operations can help to present potential dangers to crews of the flight, UAVs still suffer from capacity scale problems and limitations to their payloads. In [@jo2017development], a UAV with vertical takeoff and landing capabilities was designed. Even, a high power propellant system has been added to allow the UAV to lift heavy cargo between $10$-$15$ Kg, which could include medicine, food, and water. 4. UAVs can act as aerial base stations for rapid service recovery after complete communications infrastructure damage in disaster stricken areas. This helps in SAR operations as illustrated in Figures \[SAR-out\] and  \[SAR-in\] for outdoor and indoor environments, respectively. ![Use UAV to Provide Wireless Coverage for Outdoor Users. []{data-label="SAR-out"}](srv_SAR_f2.png) ![Use UAV to Provide Wireless Coverage for Indoor Users.[]{data-label="SAR-in"}](srv_SAR_f3.png) Challenges ----------- ### Legislation In the United States, the FAA does not currently permit the use of swarms of autonomous UAVs for commercial applications. But it is possible to adjust the regulations to allow this type of use. Swarms of UAVs can be used to coordinate the operations of SAR teams [@smith2015regulating]. ### Weather Weather conditions pose a challenge to UAVs as they might result in deviations in their predetermined paths. In cases of natural or man-made disasters, such as Tsunamis, Hurricanes, or terrorist attacks, weather becomes a tough and cardinal challenge. In such scenarios, UAVs may fail in their missions as a result of the detrimental weather conditions [@jordan2015bird]. ### Energy Limitations Energy consumption is one of the most important challenges facing UAVs. Usually, UAVs are battery powered. UAV batteries are used for UAV hovering, wireless communications, data processing and image analysis. In some SAR operations, UAVs need to be operated for extended periods of time over disaster stricken regions. Due to the power limitations of UAVs, a decision must be taken on whether UAVs should perform data and image analysis on-board in real-time, or data should be stored for later analysis to reduce the consumed power [@vergouw2016drone; @gupta2016survey].  Research Trends and Future Insights ----------------------------------- ### Image Processing SAR operations using UAVs can employ image processing techniques to quickly and accurately find targeted objects. Image processing methods can be used in autonomous single and multi-UAV systems to locate potential targets in support of SAR operations. Moreover, location information can be augmented to aerial images of target objects [@macke2013systems]. UAVs can be integrated with target detection technologies including thermal and vision cameras. Thermal cameras (e.g., IR cameras) can detect the heat profile to locate missing persons. Two stage template based methods can be used with such cameras [@rudol2008human]. Vision cameras can also help in the detection process of objects and persons [@hernandez2012detecting; @mikolajczyk2004human]. Methods that utilize a combination of thermal and vision cameras have been reported in the literature as in [@rudol2008human]. In SAR operations, image processing can be done either at the GCS, post target identification, or at the UAV itself, using on-board processors with real-time image processing capabilities. In [@sun2016camera], the authors implemented a target identification method using on-board processors, and the GCS. This method utilizes image processing techniques to identify the targeted objects and their coarse location coordinates. Using terrestrial networks, the UAV sends the images and their GPS locations to the GCS. Another possible approach requires the UAV to capture and save high resolution videos for later analysis at the GCS. ### Machine Learning Machine learning techniques can be applied on images captured by UAVs to help in SAR operations [@giusti2016machine; @bejiga2017convolutional]. In [@bejiga2017convolutional], the authors propose machine learning techniques applied to images captured by UAVs equipped with vision cameras. In their study, pre-trained Convolutional Neural Network (CNN) with trained linear Support Vector Machine (SVM) is used to determine the exact image/video frame at which a lost person is potentially detected. Figure \[SAR-ML\]. illustrates a block diagram of a SAR system utilizing UAVs in conjunction with machine learning technology [@bejiga2017convolutional]. SAR operations employing machine learning technology face many challenges. UAVs are battery powered so there is significant limitation on their on-board processing capability. Protection against adversarial attacks on the employed machine learning techniques pose another important challenge Reliable and real-time communications with the GCS given QoS and energy constraints is another challenge [@bejiga2017convolutional]. ### Future Insights Based on the reviewed current literature focusing on SAR scenarios using UAVs, we believe there is a need for more research on the following: - Data fusion and decision fusion algorithms that integrate the output of multiple sensors. For example, GPS can be integrated with Forward looking infrared (FLIR) sensors and thermal sensors to realize more accurate detection solution [@rudol2008human]. - While traditional machine learning techniques have demonstrated their success on UAVs, deep learning techniques are currently off limits because of the limitations on the on-board processing capabilities and power resources on UAVs. Therefore, there is a need to design and implement on-board, low power, and efficient deep learning solutions in support of SAR operations using UAVs [@carrio2017review]. - Design and implementation of power-efficient distributed algorithms for the real-time processing of UAV swarm captured videos, images, and sensing data [@bejiga2017convolutional]. - New lighter materials, efficient batteries and energy harvesting solutions can contribute to the potential use of UAVs in long duration missions [@vergouw2016drone]. - Algorithms that support UAV autonomy and swarm coordination are needed. These algorithms include: flight route determination, path planning, collision avoidance and swarm coordination [@alexopoulos2013comparative; @pham2015survey]. - In Multi-UAV systems, there are many coordination and communications challenges that need to be overcome. These challenges include QoS communications between the swarm of UAVs over multi-hop communications links and with the GCS [@scherer2015autonomous]. - More accurate localization and mapping systems and algorithms are required in support of SAR operations. Nowadays, GPS is used in UAVs to locate the coordinates UAVs and target objects but GPS is known to have coverage and accuracy issues. Therefore, new algorithms are needed for data fusion of the data received from multiple sensors to achieve more precise localization and mapping without coverage disruptions. - The use of UAVs as aerial base stations is still nascent. Therefore, more research is needed to study the use of such systems to provide communications coverage when the public communications network is disrupted or operating above its maximum capacity [@al2014modeling]. Remote Sensing ============== UAVs can be used to collect data from ground sensors and deliver the collected data to ground base stations [@tuyishimire2017cooperative]. UAVs equipped with sensors can also be used as aerial sensor network for environmental monitoring and disaster management [@quaritsch2010networked]. Numerous datasets originating from UAVs remote sensing have been acquired to support the research teams, serving a broad range of applications: crop monitoring, yield estimates, drought monitoring, water quality monitoring, tree species, disease detection, etc [@vito]. In this section, we present UAV remote sensing systems, challenges during aerial sensing using UAVs, research trends and future insights.  for tree=[ align=center, parent anchor=south, child anchor=north, font=, edge=[thick, -[Stealth\[\]]{}]{}, l sep+=10pt, edge path=[ (!u.parent anchor) – +(0,-10pt) -| (.child anchor); ]{}, if level=0[ inner xsep=0pt, tikz=[(.south east) – (.south west);]{} ]{} ]{} \[ UAV remote sensing systems \[Active sensing systems \[ \] \[ \] \[ \] \[ \] \[ \] \[ \] \] \[Passive sensing systems \[ \] \[ \] \[ \] \[ \] \[ \] \[ \] \[ \] \] \] Remote Sensing Systems ---------------------- There are two primary types of remote sensing systems: active and passive remote sensing systems [@NASA]. In active remote sensing system, the sensors are responsible for providing the source of energy required to detect the objects. The sensor transmits radiation toward the object to be investigated, then the sensor detects and measures the radiation that is reflected from the object. Most active remote systems used in remote sensing applications operate in the microwave portion of the electromagnetic spectrum and hence it makes them able to propagate through the atmosphere under most conditions [@NASA]. The active remote sensing systems include laser altimeter, LiDAR, radar, ranging instrument, scatterometer and sounder. In passive remote sensing system, the sensor detects natural radiation that is emitted or reflected by the object as shown in Figure \[ha1\]. The majority of passive sensors operate in the visible, infrared, thermal infrared, and microwave portions of the electromagnetic spectrum [@NASA]. The passive remote sensing systems include accelerometer, hyperspectral radiometer, imaging radiometer, radiometer, sounder, spectrometer and spectroradiometer. In Figure \[ha2\], we show the classifications of UAV aerial sensing systems. In Table \[table1Remote\], we make a comparison among the UAV remote sensing systems based on the operating frequency and applications. The common active sensors in remote sensing are LiDAR and radar. A LiDAR sensor directs a laser beam onto the surface of the earth and determines the distance to the object by recording the time between transmitted and backscattered light pulses. A radar sensor produces a two-dimensional image of the surface by recording the range and magnitude of the energy reflected from all objects. The common passive sensor in remote sensing is spectrometer. A spectrometer sensor is designed to detect, measure, and analyze the spectral content of incident electromagnetic radiation. Image Processing and Analysis ----------------------------- The image processing steps for a typical UAV mission are described in details by the authors in [@hugenholtz2013geomorphological] and [@w2013low]. The process flow is the same for most remotely sensed imagery processing algorithms. First, the algorithm utilizes the log file from the UAV autopilot to provide initial estimates for the position and orientation of each image. The algorithm then applies aerial triangulation process in which the algorithm reestablishes the true positions and orientations of the images from an aerial mission. During this process, the algorithm generates a large number of automated tie points for conjugate points identified across multiple images. A bundle-block adjustment then uses these automated tie points to optimize the photo positions and orientations by generating a high number of redundant observations, which are used to derive an efficient solution through a rigorous least-squares adjustment. To provide an independent check on the accuracy of the adjustment, the algorithm includes a number of check points. Then, the oriented images are used to create a digital surface model, which provides a detailed representation of the terrain surface, including the elevations of raised objects, such as trees and buildings. The digital surface model generates a dense point cloud by matching features across multiple image pairs [@w2013low]. At this stage, a digital terrain model can be generated, which is referred to as a bare-earth model. A digital terrain model is a more useful product than a surface model, because the high frequency noise associated with vegetation cover is removed. After the algorithm generates a digital terrain model, orthorectification process can then be performed to remove the distortion in the original images. After orthorectification process, the algorithm combines the individual images into a mosaic, to provide a seamless image of the mission area at the desired resolution [@whitehead2014remote]. Figure \[ha3\] summarizes the image processing steps for remotely sensed imagery. Type of remote sensing Operating Frequency Type of sensor Applications ------------------------ ---------------------------- -------------------------- -------------------------------------------------------------------------- Laser altimeter It measures the height of a UAV with respect to the mean Earth’s surface to determine the topography of the underlying surface.     LiDAR It determines the distance to the object by recording the time   between transmitted and backscattered light pulses.                  Radar It produces a two-dimensional image of the surface by recording Active Microwave portion of the the range and magnitude of the energy reflected from all objects. electromagnetic spectrum Ranging Instrument It determines the distance between identical microwave             instruments on a pair of platforms.                                      Scatterometer It derives maps of surface wind speed and direction by measuring backscattered radiation in the microwave spectral region.             Sounder It measures vertical distribution of precipitation, temperature,      humidity, and cloud composition.                                       Accelerometer It measures two general types of accelerometers: 1) The           translational accelerations (changes in linear motions); 2) The     angular accelerations (changes in rotation rate per unit time).      Hyperspectral radiometer It discriminates between different targets based on their spectral  response in each of the narrow bands.                                Imaging radiometer It provides a two-dimensional array of pixels from which         an image may be produced.                                            Passive Visible, infrared, thermal Radiometer It measures the intensity of electromagnetic radiation in some    infrared, and microwave bands within the spectrum.                                             portions of the Sounder It measures vertical distributions of atmospheric parameters    electromagnetic spectrum such as temperature, pressure, and composition from            multispectral information.                                            Spectrometer It designs to detect, measure, and analyze the spectral content of incident electromagnetic radiation.                                Spectroradiometer It measures the intensity of radiation in multiple wavelength   bands. It designs for remotely sensing specific geophysical    parameters.                                                             Flight Planning --------------- Although each UAV mission is unique in nature, the same steps and processes are normally followed. Typically, a UAV mission starts with flight planning [@hugenholtz2013geomorphological]. This step depends on specific flight-planning algorithm and uses a background map or satellite image as a reference to define the flight area. Extra data is then included, for example, the desired flying altitude, the focal length and orientation of the camera, and the desired flight path. The flight-planning algorithm will then find an efficient way to obtain overlapping stereo imagery covering the area of interest. During the flight-planning process, the algorithm can adjust various parameters until the operator is satisfied with the flight plan. As part of the mission planning process, the camera shutter speed settings must satisfy the different lighting conditions. If exposure time is too short, the imagery might be too dark to discriminate among all key features of interest, but if it is too long, the imagery will be blurred or will be bright. Next, the generated flight plan is uploaded to the UAV autopilot. The autopilot uses the instructions contained in the flight plan to find climb rates and positional adjustments that enable the UAV to follow the planned path as closely as possible. The autopilot reads the adjustments from the global navigation and satellite system and the initial measurement unit several times per second throughout the flight. After the flight completion, the autopilot download a log file, this file contains information about the recorded UAV 3D placements throughout the flight, as well as information about when the camera was triggered. The information in log file is used to provide initial estimates for image centre positions and camera orientations, which are then used as inputs to recover the exact positions of surface points [@whitehead2014remote]. Challenges ---------- ### [Hostile Natural Environment]{} UAVs can be utilized to study the atmospheric composition, air quality and climate parameters, because of their ability to access hazardous environments, such as thunderstorms, hurricanes and volcanic plumes [@austin2011unmanned]. The researchers used UAVs for conducting environmental sampling and ocean surface temperature studies in the Arctic [@villa2016overview]. The authors in [@curry2004applications] modify and test the Aerosonde UAV in extreme weather conditions, at very low temperatures (less than $-20\,^{\circ}\mathrm{C}$) to ensure a safe flight in the Arctic. The aim of the work was to modify and integrate sensors on-board an Aerosonde UAV to improve the UAV’s capability for its mission under extreme weather conditions such as in the Arctic. The steps to customize the UAV for the extreme weather conditions were: 1) The avionics were isolated; 2) A fuel-injection engine was used to avoid carburetor icing; 3) A servo-system was adopted to force ice breaking over the leading edge of the air-foil. In Barrow, Alaska, the modified UAVs successfully demonstrated their capabilities to collect data for 48 hours along a 30 $km^{2}$ rectangular geographical area. When a UAV collects data from a volcano plume, a rotary wing UAV was particularly beneficial to hover inside the plume [@mcgonigle2008unmanned]. On the other hand, a fixed wing UAV was suitable to cover longer distances and higher altitudes to sense different atmospheric layers [@saggiani2007uav]. In [@lin2008eyewall], the authors presented a successful eye-penetration reconnaissance flight by Aerosonde UAV into Typhoon Longwang (2005). The 10 hours flight was split into four flight legs. In these flight legs, the UAV measured the wind field and provided the tangential and radial wind profiles from the outer perimeter into the eye of the typhoon at the 700 hPa layer. The UAV also took a vertical sounding in the eye of the typhoon and measured the strongest winds during the whole flight mission [@villa2016overview]. ### [Camera Issues]{} The radiometric and geometric limitations imposed by the current generation of lightweight digital cameras are outstanding issues that need to be addressed. The current UAV digital cameras are designed for the general market and are not optimized for remote sensing applications. The current commercial instruments tend to be too bulky to be used with current lightweight UAVs, and for those that do exist, there is still a question of calibration process with conventional sensors. Spectral drawbacks include the fact that spectral response curves from cameras are usually poorly calibrated, which makes it difficult to convert brightness values to radiance. However, even cameras designed specifically for UAVs may not meet the required scientific benchmarks [@whitehead2014remote]. Another drawback is that the detectors of camera may also become saturated when there are high contrasts, for example when an image covers both a dark forest and a snow covered field. Another drawback is that many cameras are prone to vignette, where the centres of images appear brighter than the edges. This is because rays of light in the centres of the image have to pass through a less optical thickness of the camera lens, and are thus low attenuated than rays at the edges of the image. There are a number of techniques that can be taken into account to improve the quality of image: 1) micro-four-thirds cameras with fixed interchangeable lenses can be used instead of having a retractable lens, which allows for much improved calibrations and image quality; 2) a simple step that can make a big difference in the processing stage is to remove images that are blurred, under or overexposed, or saturated [@whitehead2014remote]. ### [Illumination Issues]{} The shadows on a sunny day are clear and well defined. These weather conditions can cause critical problems for the automated image matching algorithms used in both triangulation process and digital elevation model generation [@whitehead2014remote]. When clouds move rapidly, shaded areas can vary between images obtained during the same mission, therefore the aerial triangulation process will fail for some images, and also will result in errors for automatically generated digital elevation models. Furthermore, the automated color balancing algorithms used in the creation of image mosaics may be affected by the patterns of light and shade across images. This can cause mosaics with poor visual quality. Another generally observed illumination effect is the presence of image hotspots, where a bright points appear in the image. Hotspots occur at the antisolar point due to the effects of bidirectional reflectance, which is dependent on the relative placement of the image sensor and the sun [@whitehead2014remote]. Research Trends and Future Insights ----------------------------------- ### [Machine Learning]{} In remote sensing, the machine learning process begins with data collection using UAVs. The next step of machine learning is data cleansing, which includes cleansing up image and/or textual-based data and making the data manageable. This step sometimes might include reducing the number of variables associated with a record. The third step is selecting the right algorithm, which includes getting acquainted with the problem we are trying to solve. There are three famous algorithms being used in remote sensing:1) Random forest; 2) Support vector machines; 3) Artificial neural networks. An algorithm is selected depending on the type of problem being solved. In some scenarios, where there are multiple features but limited records, support vector machines might work better. If there are a lot of records but less features, neural networks might yield a better prediction/classification accuracy. Normally, several algorithms will be applied on a dataset and the one that works best is selected. In order to achieve a higher accuracy of the machine learning results, a combination of multiple algorithms can also be employed, which is referred to as ensemble. Similarly, multiple ensembles will need to be applied on a dataset, in order to select the ensemble that works the best. It is practical to choose a subset of candidate algorithms based on the type of problem and then use the narrowed down algorithms on a part of the dataset and see which one performs best. The first challenge in machine learning is that the training segment of the dataset should have an unbiased representation of the whole dataset and should not be too small as compared to the testing segment of the dataset. The second challenge is overfitting which can happen when the dataset that has been used for algorithm training is used for evaluating the model. This will result in a very high prediction/classification accuracy. However, if a simple modification is performed, then the prediction/classification accuracy takes a dip [@pythontips]. The machine learning steps utilized by UAV remote sensing are shown in Figure \[ha4\]. ### [Combining Remote Sensing and Cloud Technology]{} Use of digital maps in risk management as well as improving data visualization and decision making process has become a standard for businesses and insurance companies. For instance, the insurance company can utilize UAV to generate a normalized difference vegetation index (NDVI) map in order to have an overview of the hail damage in corn. The geographic information system (GIS) technology in the cloud utilizes the NDVI map generated from UAV images to provide an accurate and advanced tool for assisting with crop hail damage insurance settlements in minimal time and without conflict, while keeping expenses low [@giscloud]. ### [Free Space Optical]{} FSO technology over an UAV can be utilized in armed forces, where military wireless communications demand for secure transmission of information on the battlefield. Remote sensing UAVs can utilize this technology to disseminate large amount of images and videos to the fighting forces, mostly in a real time. Near Earth observing UAVs can be utilized to provide high resolution images of surface contours using synthetic aperture radar and light detection and ranging. Using FSO technology, aerial sensors can also transmit the collected data to the command center via on-board satellite communication sub-system [@kaushal2017optical]. ### Future Insights Some of the future possible directions for this application are: - Camera stabilization during flight [@whitehead2014remote] is one of the issues that needs to be addressed, in the employment of UAV for remote sensing. - Battery weight and charging time are critical issues that affect the duration of UAV missions [@madden15future]. The development and incorporation of lightweight solar powered battery components of UAV can improve the duration of UAV missions and hence it reduces the complexity of flight planning. - The temporal digital surface models produced from aerial imagery using UAV as the platform, can become practical solution in mass balance studies for example mass balance of any particular metal in sanitary landfills, chloride in groundwater and sediment in a river. More specifically, the UAV-based mass balance in a debris-covered glacier was estimated at high accuracy, when using high-resolution digital surface models differencing. Thus, the employment of UAV save time and money when compared with the traditional method of stake drilling into the glaciers which was labor-intensive and time consuming  [@immerzeel2014high; @bhardwaj2016uavs]. - The tracking methods utilized on high-resolution images can be practical to find an accurate surface velocity and glacier dynamics estimates. In [@sam2016remote], the authors suggested a differential band method for estimating velocities of debris and non-debris parts of the glaciers. The on-demand deployment of UAV to obtain high-resolution images of a glacier has resulted in an efficient tracking methods when compared with satellite imagery which depends on the satellite overpass [@bhardwaj2016uavs]. - UAV remote sensing can be as a powerful technique for field-based phenotyping with the advantages of high efficiency, low cost and suitability for complex environments. The adoption of multi-sensors coupled with advanced data analysis techniques for retrieving crop phenotypic traits have attracted great attention in recent years [@yang2017unmanned]. - It is expected that with the advancement of UAVs with larger payload, longer flight time, low-cost sensors, improved image processing algorithms for Big data, and effective UAV regulations, there is potential for wider applications of the UAV-based field crop phenotyping [@yang2017unmanned]. - UAV remote sensing for field-based crop phenotyping provides data at high resolutions, which is needed for accurate crop parameter estimations. The derivation of crop phenotypic traits based on the spectral reflection information using UAV as the platform has shown good accuracy under certain conditions. However, it showed a low accuracy in the research on the non-destructive acquisition of complex traits that were indirectly related to the spectral information [@yang2017unmanned]; - Image processing of UAV imagery faces a number of challenges, such as variable scales, high amounts of overlap, variable image orientations, and high amounts of relief displacement arising from the low flying altitudes relative to the variation in topographic relief [@whitehead2014remote]. Researchers need to find efficient ways to overcome these challenges in future studies. Construction & Infrastructure Inspection ======================================== As already mentioned in the market opportunity section \[Market\], the net market value of the deployment of UAV in support of construction and infrastructure inspection applications is about 45% of the total UAV market. So there is a growing interest in UAV uses in large construction projects monitoring [@liu2014review] and power lines, gas pipelines and GSM towers infrastructure inspection [@deng2014unmanned; @mohamadi2014vertical]. In this section, we first present a literature review. Then, we show the uses of UAVs in support of infrastructure inspection. Finally, we present the challenges, research trends and future insights. Literature Review ----------------- In construction and infrastructure inspection applications, UAVs can be used for real-time monitoring construction project sites [@gheisari2014uas4safety]. So, the project managers can monitor the construction site using UAVs with better visibility about the project progress without any need to access the site [@liu2014review]. Moreover, UAVs can also be utilized for high voltage inspection of the power transmission lines. In [@jones2005power; @luque2014power; @sampedro2014supervised; @li2010towards], the authors used the UAVs to perform an autonomous navigation for the power lines inspection. The UAVs was deployed to detect, inspect and diagnose the defects of the power line infrastructure. In [@larrauri2013automatic], the authors designed and implemented a fully automated UAV-based system for the real-time power line inspection. More specifically, multiple images and data from UAVs were processed to identify the locations of trees and buildings near to the power lines, as well as to calculate the distance between trees, buildings and power lines. Furthermore, TIR camera was employed for bad conductivity detection in the power lines. UAVs can also be used to monitor the facilities and infrastructure, including gas, oil and water pipelines. In [@mohamadi2014vertical], the authors proposed the deployment of small-UAV (sUAV) equipped with a gas controller unit to detect air and gas content. The system provided a remote sensing to detect gas leaks in oil and gas pipelines. Table \[cons\_t1\] summarizes some of the construction and infrastructure inspection applications using UAVs. More specifically, this table presents several types of UAV used in construction and infrastructure inspection applications, as well as the type of sensors deployed for each application and the corresponding UAV specifications in terms of payload, altitude and endurance. [width=1.0]{} to 1.0 **UAV Type** & **Applications**& [Payload]{}**/**[Altitude]{}**/**[Endurance]{}&**Sensor Type**&**References**\ AR.Drone French Company Parrot. & Use UAVs to enhance the safety on on construction sites by providing a real-time visual view for these sites. & No / 50 $m$ / 12 min.& On-board HD camera, Wi-Fi connection. &[@gheisari2014uas4safety]\ A Multi Rotor UAV. & Use UAVs with image processing methods for crack detection and assessment of surface degradation. & 100 g / LAP / 20 min.& Color imaging sensors. &[@sankarasrinivasan2015health]\ MikroKopter L4-ME Quadcopter [@MikroKopter-L4-ME:Online]. & Use UAVs for vertical inspection for high rise infrastructures such as street lights, GSM towers or high rise buildings. & Up to 500 g / Up to 247 $m$ / 13-20 min. [@MikroKopter-L4-ME:Online]& Laser scanner. &[@sa2014vertical]\ Quadrotor Helicopter, UAV. & Inspection of the high voltage of power transmission lines. & Less than 1 kg/ LAP/ Less than 1 hour. & Color and TIR cameras, GPS, IMU. &[@luque2014power]\ Fixed Wing Aircraft, UAV. & Sketchy inspection, identifying the defects of the power transmission lines. & Less than 3 kg/ Up to 500 $m$/ Up to 50 min (50 $km$). & HD ultra-wide angle video camera. &[@deng2014unmanned]\ Quadrotor, UAV. & Use cooperative UAVs platform for inspection and diagnose of the power lines infrastructure. & Less than 6 kg/ Up to 200 $m$/ Up to 25 min (10 $km$). & TIR cameras, GPS. &[@deng2014unmanned]\ Quadrotor (VTOL), sUAV. & provide a remote sensing to detect gas leaks in gas pipelines. & NA/ LAP/ 30-50 min. & Gas controller unit, GPS. &[@mohamadi2014vertical; @bretschneider2015uav]\ \[table:ahmad1-4\] The Deployment of UAVs for Construction & Infrastructure Inspection Applications -------------------------------------------------------------------------------- In this section, we present several specific example of the deployment of UAV for construction and infrastructure inspection. Figure \[uses\] illustrates the classification of these deployments. (b1) [Oil/gas and wind turbine inspection ]{}; (b2) at (b1.east) [Critical land building inspection (e.g., cell tower) ]{}; (b3) at (b2.east) [Infrastructure internal inspection (e.g., pipe) ]{}; (b4) at (b3.east) [Extreme condition inspection ]{}; (top) at ($(b2.north)!.5!(b3.north)$) ; (atop) at ($(top.south) + (0,-5pt)$); (btop) at ($(b3.south) + (0,-5pt)$); (top.south) – (atop) (b1.north) |- (atop) -| (b4.north) (b2.north) |- (atop) -| (b3.north); - [Oil/gas and wind turbine inspection:]{} In 2016, it was reported that Pacific Gas and Electric Company (PG&E) performed drone tests to inspect its electric and gas services for better safety and reliability with the authorization from Federal Aviation Administration (FAA) [@peg2016]. The inspections focused on hard-to-reach areas to detect methane leaks across its 70,000-square-mile service area. In the future, PG&E plans to extend the drone tests for storm and disaster response. Cyberhawk is one of world’s premier oil and gas company that uses UAV for inspection[@Cyberhawk2008]. Furthermore, it has completed more than 5,000 structural inspections, including: 1) oil and gas; 2) wind turbine; and 3) live flare. UAV inspection conducted by Cyberhawk provides a bunch of photos, as well as conducts close visual and thermal inspections of the inspected assets. Industrial SkyWorks [@SkyWorks2018] uses drones for building inspections and oil/gas inspections in North America. Furthermore, a powerful machine learning algorithm, BlueVu, has been developed to efficiently handle the captured data. To sum up, it provides the following solutions: - Asset inspections and data acquisition; - Advanced data processing with 2D and 3D images; - Detailed reports of the inspected asset (i.e., annotations, inspector comments, and recommendations). - [Critical land building inspection (e.g., cell tower):]{} AT&T owns about 65,000 cell towers that need to be inspected, repaired or installed. The video analytic team at AT&T Labs has collaborated with other forces (e.g., Intel, Qualcomm, etc.) to develop faster, better, more efficient, and fully automated cell tower inspection using UAVs [@att2017]. One of the approaches is to employ deep learning algorithm on high definition (HD) videos to detect defects and anomalies in real time. Honeywell InView inspection service has been launched to provide industrial critical structure inspections [@honeywell2017]. It combines the Intel® Falcon™ 8+ UAV system with Honeywell aerospace and industrial technology solutions. Specifically, the Honeywell InView inspection service can achieve: 1) safety of the workers; 2) improved efficiency; and 3) advanced data processing. - [Infrastructure internal inspection:]{} Maverick has provided industrial UAV inspection services for equipment, piping, tanks, and stack internals in western Canada since 1994 [@Maverick2018]. It provides a dedicated services for assets internal inspection using Flyability ELIOS. Maverick also provides post data processing that analyzes data using measurement software and CAD modeling. - [Extreme condition inspection:]{} Bluestream offers UAV inspection services for onshore and offshore assets [@Alongside2018]. It services are particularly suitable for: 1) onshore and offshore live flare inspections; 2) topside, splash zone and under deck inspections; and 3) hard to access infrastructure inspection. Challenges ---------- There are several challenges in utilizing UAVs for construction and infrastructure inspection: - Some of the challenges in using UAVs for infrastructure inspection are the limited energy, short flight time and limited processing capabilities [@gheisari2014uas4safety]. - Limited payload capacities for sUAVs is a big challenge. The on-board loads could include optical wavelength range camera, TIR camera, color and stereo vision cameras, different types of sensors such as gas detection, GPS, etc., [@bretschneider2015uav]. - There is a lack of research attention to multi-UAV cooperation for construction and infrastructure inspection applications. Multi-UAV cooperation could provide wider inspection scope, higher error tolerance, and faster task completion time. - Another challenge is to allow autonomous UAVs that can maneuver an indoor environment with no access to GPS signals [@dupont2017potential]. Research Trends and Future Insights ----------------------------------- ### Machine Learning Machine learning has become an increasingly important artifical intelligence approach for UAVs to operate autonomously. Applying advanced machine learning algorithms (e.g., deep learning algorithm) could help the UAV system to draw better conclusions. For example, due to its improved data processing models, deep learning algorithms could help to obtain new findings from existing data and to get more concise and reliable analysis results. UAV inspection program at AT&T uses deep learning algorithms on HD videos to detect defects and anomalies in real time [@att2017]. Industrial SkyWorks introduces advanced machine learning algorithms to process 2D and 3D images [@SkyWorks2018]. More specifically, deep learning is useful for feature extraction from the raw measurements provided by sensors on-board a UAV (details about UAV sensor technology is presented in the Precision Agriculture section). Convolutional Neural Networks (CNNs) is one of the main deep learning feature extractors used in the area of image recognition and classification which has been proven very effective [@carrio2017review]. Figure \[cnn\] illustrates one example of how CNNs works. ![Illustration of How CNNs Work [@carrio2017review].[]{data-label="cnn"}](srv_CIS_f1.png) ### Image Processing Construction and infrastructure inspection using UAVs equipped with an on-board cameras and sensors, can be efficiently operated when employing image processing techniques. The employment of image processing techniques allows for monitoring and assessing the construction projects, as well as performing inspection of the infrastructure such as surveying construction sites, work progress monitoring, inspection of bridges, irrigation structures monitoring, detection of construction damage and surfaces degradation. [@sankarasrinivasan2015health; @ham2016visual]. In [@sankarasrinivasan2015health], presented an integrated data acquisition and image processing platform mounted on UAV. It was proposed to be used for the inspection of infrastructures and real time structural health monitoring (SHM). In the proposed framework, a real time image and data will be sent to the GCS. Then the images and data will be processed using image processing unit in the GCS, to facilitate the diagnosis and inspection process. The authors proposed to combine HSV thresholding and hat transform for cracks detection on the concrete surfaces. Figure \[crack\] presents the crack detection algorithm block diagram proposed in [@sankarasrinivasan2015health]. ![Proposed Approach for Crack Detection Algorithm [@sankarasrinivasan2015health].[]{data-label="crack"}](srv_CIS_f2.png) In [@luque2014power], the authors proposed for the deployment of UAV with a vision-based system that consists of a color camera, TIR camera and a transmitter to send the captured images to the GCS. Then these images will be processed in the GCS, to be used in the inspection and estimation of the real temperature of the power lines joints. In [@larrauri2013automatic], a fully automatic system to determine the distance between power lines and the trees, buildings and any others obstacles was proposed. The authors designed and developed vision-based algorithms for processing the HD images that were obtained using HD camera equipped on a UAV. The captured video was sent to the computer in the GCS. At the GCS, the video was converted into the consecutive images and were further processed to calculate the distance between the power line and the obstacles. ### Future Insights {#cons-ins} Based on the reviewed articles on construction and infrastructure inspection applications using UAVs, we suggest these future possible directions: - More research is required to improve UAVs battery life time to allow longer distance and to increase the UAV flight time [@gheisari2014uas4safety]. - For future research, it is important to propose and develop an accurate, autonomous and real-time power lines inspection approaches using UAVs, including ultrasonic sensors, TIR or color cameras, image processing and data analysis tools. More specifically, to propose and develop techniques to monitor, detect and diagnose any power lines defects automatically [@pagnano2013roadmap]. - More advanced data collection, sharing and processing algorithms for multi-UAV cooperation are required, in order to achieve faster and more efficient inspections. - The researchers should also focus on the improvement of the autonomy and safety for UAVs to maneuver in the congested and indoor environment with no or weak GPS signals [@dupont2017potential]. Precision Agriculture ===================== UAVs can be utilized in precision agriculture (PA) for crop management and monitoring [@huang2013development; @muchiri2016review], weed detection [@kazmi2011adaptive], irrigation scheduling [@gonzalez2013using], disease detection [@garcia2013comparison], pesticide spraying [@huang2013development] and gathering data from ground sensors (moisture, soil properties, etc.,) [@mathur2016data]. The deployment of UAVs in PA is a cost-effective and time saving technology which can help for improving crop yields, farms productivity and profitability in farming systems. Moreover, UAVs facilitate agricultural management, weed monitoring, and pest damage, thereby they help to meet these challenges quickly [@primicerio2012flexible]. In this section, we first present a literature review of UAVs in PA. Then, we show the deployment of UAV in PA. Moreover, we present the challenges, as well as research trends future insights. Literature Review ----------------- UAVs can be efficiently used for small crop fields at low altitudes with higher precision and low-cost compared with traditional manned aircraft. Using UAVs for crop management can provide precise and real time data about specific location. Moreover, UAVs can offer a high resolution images for crop to help in crop management such as disease detection, monitoring agriculture, detecting the variability in crop response to irrigation, weed management and reduce the amount of herbicides [@muchiri2016review; @jensen2003assessing; @huang2013development; @hunt2010acquisition; @sullivan2007evaluating]. In Table \[PA-t0\], a comparison between UAVs, traditional manned aircraft and satellite based system is presented in terms of system cost, endurance, availability, deployment time, coverage area, weather and working conditions, operational complexity, applications usage and finally we present some examples from the literature. **Issues** UAVs Manned Aircraft Satellite System -------------------- -------------------------------------------- ------------------- ---------------------- Cost Low High Very High Endurance Short-time Long-time All the times Availability When needed Some times All the times Deployment time Easy Need runway Complex Coverage area Small Large Very large Weather and Sensitive Low sensitivity Require clear sky working conditions for imaging Payload Low Large Large Operational Simple Simple Very complicated complexity Applications Carry small Spraying UAV high resolution and usage digital, thermal system pesti- images for cameras & sensors cide spraying specific-area Examples [@muchiri2016review; @jensen2003assessing] [@akesson1974use] [@reed1994measuring] : []{data-label="PA-t0"} Table \[PA\_t1\] summarizes some of the precision agriculture applications using UAVs. More specifically, this table presents several types of UAV used in precision agriculture applications, as well as the type of sensors deployed for each application and the corresponding UAV specifications in terms of payload, altitude and endurance. [width=1.0]{} to 1.0 **UAV Type** & **Applications**& [Payload]{}**/**[Altitude]{}**/**[Endurance]{}&**Sensor Type**&**References**\ Yamaha Aero Robot "R-50. & Monitoring Agriculture, spraying UAV systems. & 20 kg / LAP / 1 hour.& Azimuth and Differential Global Positioning System (DGPS) sensor system. &[@muchiri2016review]\ Yanmar KG-135, YH300 and AYH3. & Pesticide spraying over crop fields. & 22.7 kg / 1500m / 5 hours.& Spray system with GPS sensor system. &[@huang2013development]\ RC model fixed-wing airframe. & Imaging small sorghum fields to assess the attributes of a grain crop. & Less than 1kg / LAP / less than 1 hour.& Image sensor digital camera. &[@jensen2003assessing; @huang2013development]\ Vector-P UAV. & Crop management (e.g. winter wheat) for site-specific agriculture, a correlation is investigated between leaf area index and the green normalized difference vegetation index (GNDVI). & Less than 1kg /105m-210m/1-6 hours deepening on the payload.& Digital color-infrared camera with a red-light-blocking filter. &[@hunt2010acquisition]\ Fixed-wing UAV. & Detect the variability in crop response to irrigation (e.g. cotton). & Lightweight camera/ 90m / Less than 1 hour.& Thermal camera , Thermal Infrared (TIR) imaging sensor. &[@sullivan2007evaluating]\ Multi-rotor micro UAV. & Agricultural management, disease detection for citrus (citrus greening, Huanglongbing (HLB)). & Less than 1kg / 100 m / 10-20 min.& Multi-band imaging sensor, 6-channel multispectral camera. &[@garcia2013comparison]\ Vario XLC helicopter. & Weed management, reduce the amount of herbicides using aerial images for crop. & 7 kg / LAP / 30 min.& Advanced vision sensors for 3D and multispectral imaging. &[@kazmi2011adaptive]\ VIPtero UAV. & Crop management, they used UAV to acquire high resolution multi-spectral images for vineyard management. & 1 Kg / 150 m / 10 min.& Tetracam ADC-lite camera, GPS. &[@primicerio2012flexible]\ Fieldcopter UAV. & Water assessment. UAVs was used for acquiring high resolution images and to assess vineyard water status, which can help for irrigation processes. & Less than 1 Kg / LAP /NA.& Multispectral and thermal cameras on-board UAV. &[@baluja2012assessment]\ Multi-rotor hexacopter ESAFLY A2500-WH & Cultivations analysis, processing multi spectral data of the surveyed sites to create tri-band ortho-images used to extract some Vegetation Indices (VI). & Up to 2.5 kg Kg / LAP /12-20 min.& Tetracam camera on-board UAV. &[@candiago2015evaluating]\ \[table:ahmad1-4\] The Deployment of UAV in Precision Agriculture ---------------------------------------------- In [@khanal2017overview], the authors presented the deployment of UAVs in precision agriculture applications as summarized in Figure \[pa1\]. The deployment of UAV in precision agriculture are discussed in the following: - [Irrigation scheduling:]{} There are four factors that needs to be monitored, in order to determine a need for irrigation: 1) Availability of soil water; 2) Crop water need, which represents the amount of water needed by the various crops to grow optimally; 3) Rainfall amount; 4) Efficiency of the irrigation system [@natinoal]. These factors can be quantified by utilizing UAVs to measure soil moisture, plant-based temperature, and evapotranspiration. For instance, the spatial distribution of surface soil moisture can be estimated using high-resolution multi-spectral imagery captured by a UAV, in combination with ground sampling [@hassan2015assessment]. The crop water stress index can also be estimated, in order to determine water stressed areas by utilizing thermal UAV images [@gonzalez2013using]. - [Plant disease detection:]{} In the U.S., it is estimated that crop losses caused by plant diseases result in about \$33 billion in lost revenue every year [@pimentel2005update]. UAVs can be used for thermal remote sensing to monitor the spatial and temporal patterns of crop diseases pre-symptomatically during various disease development phases and hence farmers may reduce the crop losses. For instance, aerial thermal images can be used to detect early stage development of soil-borne fungus [@calderon2013high]. - [Soil texture mapping:]{} Some soil properties, such as soil texture, can be an indicative of soil quality which in turn influences crop productivity. Thus, UAV thermal images can be utilized to quantify soil texture at a regional scale by measuring the differences in land surface temperature under a relatively homogeneous climatic condition [@de2012mapping; @wang2015retrieval]. - [Residue cover and tillage mapping:]{} Crop residues is essential in soil conservation by providing a protective layer on agricultural fields that shields soil from wind and water. Accurate assessment of crop residue is necessary for proper implementation of conservation tillage practices [@Department]. In [@sullivan2004evaluation], the authors demonstrated that aerial thermal images can explain more than 95% of the variability in crop residue cover amount compared to 77% using visible and near IR images. - [Field tile mapping:]{} Tile drainage systems remove excess water from the fields and hence it provides ecological and economic benefits [@hofstrand2015economics]. An efficient monitoring of tile drains can help farmers and natural resource managers to better mitigate any adverse environmental and economic impacts. By measuring temperature differences within a field, thermal UAV images can provide additional opportunities in field tile mapping [@mapping]. - [Crop maturity mapping:]{} UAVs can be a practical technology to monitor crop maturity for determining the harvesting time, particularly when the entire area cannot be harvested in the time available. For instance, UAV visual and infrared images from barley trial areas at Lundavra, Australia were used to map two primary growth stages of barley and demonstrated classification accuracy of 83.5% [@jensen2009crop]. - [Crop yield mapping:]{} Farmers require accurate, early estimation of crop yield for a number of reasons, including crop insurance, planning of harvest and storage requirements, and cash flow budgeting. In [@swain2010adoption], UAV images were utilized to estimate yield and total biomass of rice crop in Thailand. In [@geipel2014combined], UAV images were also utilized to predict corn grain yields in the early to mid-season crop growth stages in Germany. The authors in [@sankaran2015low] presented several types of sensor that were used in UAV-based precision agriculture, as summarized in Table \[pa2\]. (b1) Irrigation scheduling ; (b2) at (b1.east) [Plant disease detection ]{}; (b3) at (b2.east) Soil texture mapping ; (b4) at (b3.east) [Residue cover and tillage mapping ]{}; (b5) at (b4.east) [Field tile mapping ]{}; (b6) at (b5.east) [Crop maturity mapping ]{}; (b7) at (b6.east) [Crop yield mapping ]{}; (top) at ($(b2.north)!.5!(b6.north)$) ; (atop) at ($(top.south) + (0,-5pt)$); (btop) at ($(b5.south) + (0,-5pt)$); (top.south) – (atop) (b1.north) |- (atop) -| (b7.north) (b2.north) |- (atop) -| (b6.north); Challenges ---------- There are several challenges in the deployment of UAVs in PA: - Thermal cameras have poor resolution and they are expensive. The price ranges from \$2000-\$50,000 depending on the quality and functionality, and the majority of thermal cameras have resolution of 640 pixels by 480 pixels [@khanal2017overview]. - Thermal aerial images can be affected by many factors, such as the moisture in the atmosphere, shooting distance, and other sources of emitted and reflected thermal radiation. Therefore, calibration of aerial sensors is critical to extract scientifically reliable surface temperatures of objects [@khanal2017overview]. - Temperature readings through aerial sensors can be affected by crop growth stages. At the beginning of the growing season, when plants are small and sparse, temperature measurements can be influenced by reflectance from the soil surface [@khanal2017overview]. - In the event of adverse weather, such as extreme wind, rain and storms, there is a big challenge of UAVs deployment in PA applications. In these conditions, UAVs may fail in their missions. Therefore, small UAVs cannot operate in extreme weather conditions and even cannot take readings during these conditions. - One of the key challenges is the ability of lightweight UAVs to carry a high-weight payload, which will limit the ability of UAVs to carry an integrated system that includes multiple sensors, high-resolution and thermal cameras[@anderson2013lightweight]. - UAVs have short battery life time, usually less than 1 hour. Therefore, the power limitations of UAVs is one of the challenges of using UAVs in PA. Another challenge, when UAVs are used to cover large areas, is that it needs to return many times to the charging station for recharging. [@garcia2013comparison; @jensen2003assessing; @huang2013development]. Research Trends and Future Insights ----------------------------------- ### Machine Learning The next generation of UAVs will utilize the new technologies in precision agriculture, such as machine learning. Hummingbird is a UAV-enabled data and imagery analytics business for precision agriculture [@hummingbirdtech]. It utilizes machine learning to deliver actionable insights on crop health directly to the field. The process flow begins by performing UAV surveys on the agricultural land at critical decision-making points in the growing season. Then, UAV images is uploaded to the cloud, before being processed with machine learning techniques. Finally, the mobile app and web based platform provides farmers with actionable insights on crop health. The advantages of utilizing UAVs with machine learning technology in precision agriculture are: 1) Early detection of crop diseases; 2) Precision weed mapping; 3) Accurate yield forecasting; 4) Nutrient optimization and planting; 5) Plant growth monitoring [@hummingbirdtech]. Type of sensor Operating Frequency Applications Disadvantages ---------------------- ------------------------------ ------------------------------------------------------- ------------------------------------------------------------------ Digital camera Visible region Visible properties, outer defects, greenness, growth. - Limited to visual spectral bands and properties.          Multispectral camera Visible-infrared region Multiple plant responses to nutrient deficiency, - Limited to few spectral bands.                              water stress, diseases among others. Hyperspectral camera Visible-infrared region Plant stress, produce quality, and safety control. - Image processing is challenging.                           - High cost sensors.                                             Thermal camera Thermal infrared region Stomatal conductance, plant responses to - Environmental conditions affect the performance.        water stress and diseases. - Very small temperature differences are not detectable.  - High resolution cameras are heavier.                       Spectrometer Visible-near infrared region Detecting disease, stress and crop responses. - Background such as soil may affect the data quality.   - Possibilities of spectral mixing.                            - More applicable for Ground sensor systems.              3D camera Infrared laser region Physical attributes such as plant height - Lower accuracies.                                             and canopy density. - Limited field applications.                                   LiDAR Laser region Accurate estimates of plant/tree height - Sensitive to small variations in path length.               and volume. SONAR Sound propagation Mapping and quantification of the canopy volumes, - Sensitivity limited by acoustic absorption, background digital control of application rates in sprayers or noise, etc.                                                    fertilizer spreader. - Lower sampling rate than laser-based sensing.           ### Image Processing UAV-based systems can be used in PA to acquire high-resolution images for farms, crops and rangeland. It can also be utilized as an alternative to satellite and manned aircraft imaging system. Processing of these images is one of the most rapidly developing fields in PA applications. The Vegetation Indices (VI) can be produced using image processing techniques for the prediction of the agricultural crop yield, agricultural analysis, crop and weed management and in diseases detection. Moreover, the VIs can be used to create vigor maps of the specific-site and for vegetative covers evaluation using spectral measurements [@candiago2015evaluating; @bannari1995review]. In [@bannari1995review; @glenn2008relationship; @candiago2015evaluating; @hunt2010acquisition], most of the VIs found in the literature have been summarized and discussed. Some of these VIs are: - Green Vegetation Index (GVI). - Normalized Difference Vegetation Index (NDVI). - Green Normalized Difference Vegetation index (GNDVI). - Soil Adjusted Vegetation Index (SAVI). - Perpendicular Vegetation Index (PVI). - Enhanced Vegetation Index (EVI). Many researchers utilized VIs that are obtained using image processing techniques in PA. The authors in [@candiago2015evaluating] presented agricultural analysis for vineyards and tomatoes crops. A UAV with Tetracam multi-spectral camera was deployed to take aerial image for crop. These images were processed using PixelWrench2 (PW2) software which came with the camera and it will be exported in a tri-band TIFF image. Then from the contents of this images VIs such as NDVI [@rouse1974monitoring], GNDVI [@gitelson1996use], SAVI [@huete1988soil] can be extracted. In [@laliberte2010acquisition], the authors used UAVs to take aerial images for rangeland to identify rangeland VI for different types of plant in Southwestern Idaho. In the study, image processing and analysis was performed in three steps as shown in Figure \[PA-image\]. More specifically, the three steps were: ![Steps of Image Processing and Analysis for Identifying Rangeland VI [@laliberte2010acquisition].[]{data-label="PA-image"}](srv_PA_f1.png) - Ortho-Rectification and mosaicing of UAV imagery [@laliberte2010acquisition]. A semi-automated ortho-rectification approach were developed using PreSync procedure [@laliberte2008procedure]. - Clipping of the mosaic to the 50$m$$\times$50$m$ plot areas measured on the ground. In this step, image classification and segmentation was performed using an object-based image analysis (OBIA) program with Definiens Developer 7.0 [@definiens2007definiens], where the acquisition image was segmented into homogeneous areas [@laliberte2010acquisition]. - Image classification: In this step, hierarchical classification scheme along with a rule based masking approach were used [@laliberte2010acquisition]. ### Future Insights {#future-insights-2} Based on the reviewed articles focusing on PA using UAVs, we suggest these future possible directions: - With relaxed flight regulations and improvement in image processing, geo-referencing, mosaicing, and classification algorithms, UAV can provide a great potential for soil and crop monitoring [@khanal2017overview; @zhang2012application]. - The next generation of UAV sensors, such as 3p sensor [@SLANTRANGE], can provide on-board image processing and in-field analytic capabilities, which can give farmers instant insights in the field, without the need for cellular connectivity and cloud connection [@next]. - More precision agricultural researches are required towards designing and implementing special types of cameras and sensors on- board UAVs, which have the ability of remote crop monitoring and detection of soil and other agricultural characteristics in real time scenarios [@primicerio2012flexible]. - UAVs can be used for obtaining high-resolution images for plants to study plant diseases and traits using image processing techniques [@patil2011advances]. Delivery of Goods ================= UAVs can be used to transport food, packages and other goods  [@fandetti2015method; @pwc; @redstagfulfillment; @tworedstagfulfillment] as shown in Figure \[dg1\]. In health-care field, ambulance drones can deliver medicines, immunizations, and blood samples, into and out of unreachable places. They can rapidly transport medical instruments in the crucial few minutes after cardiac arrests. They can also include live video streaming services allowing paramedics to remotely observe and instruct on-scene individuals on how to use the medical instruments [@wikipedia]. In July 2015, the Federal Aviation Administration (FAA) approved the first delivery of medical supplies using UAVs at Wise, Virginia [@howell2016first]. With the rapid demise of snail mail and the massive growth of e-Commerce, postal companies have been forced to find new methods to expand beyond their traditional mail delivery business models. Different postal companies have undertaken various UAV trials to test the feasibility and profitability of UAV delivery services [@unmannedcargo]. In this section, we present the UAV-based goods delivery system and its challenges as shown in Figure \[dg2\].  ![UAV Delivery Applications and Challenges.[]{data-label="dg2"}](srv_DoG_f2.png) ![UAV Delivery of Goods System.[]{data-label="dg3"}](srv_DoG_f3.png) UAV-Based Goods Delivery System ------------------------------- In UAV-based goods delivery system, a UAV is capable of traveling between a pick up location and a delivery location. The UAV is equipped with control processor and GPS module. It receives a transaction packet for the delivery operation that contains the GPS coordinates and the identifier of a package docking device associated with the order. Upon arrival of a UAV at the delivery location, the control processor checks if the identifier of a package docking device matches the device identifier in the transaction packet, performs the package transfer operation, and sends confirmation of completion of the operation to an originator of the order [@hoareau2017package]. If the identifier of a package docking device at the delivery point does not match the device identifier in the transaction packet, the UAV communication components transmit a request over a short-range network such as bluetooth or Wi-Fi. The request may contain the device identifier, or network address of the package docking device. Under the assumption that the package docking device has not moved outside of the range of UAV communication, the package docking device having the network address transmits a signal containing the address of a new location. The package docking device may then transmit updated GPS coordinates to the UAV. The UAV is re-routed to the new address based on the updated GPS location [@hoareau2017package]. In Figure \[dg3\], we present the flowchart of UAV delivery system. Challenges ---------- ### Legislation In the United States, the FAA regulation blocked all attempts at commercial use of UAVs, such as the Tacocopter company for food delivery [@gilbert2012tacocopter]. As of 2015, delivering of packages with UAVs in the United States is not allowed [@mashable]. Under current rules, companies are permitted to operate commercial UAVs in the United States, but only under certain conditions. Their UAVs must be flown within a pilot’s line of sight and those pilots must get licenses. Commercial operators also are restricted to fly their UAVs during daylight hours. Their UAVs are limited in size, altitude and speed, and UAVs are generally not allowed to fly over people or to operate beyond visual line of sight [@recode]. ### Liability Insurance Some UAVs can weigh up to 25 kg and travel at speeds approaching 45 m/s. A number of media reports describes severe lacerations, eye loss, and soft tissue injuries caused by UAV accidents. In addition to the risk of injuries or property damage from a UAV crash, ubiquitous UAV uses also create other types of accidents, such as automobile accidents due to distraction from low-flying UAVs, injuries caused by dropped cargo, liability for damaged goods, or accidents resulting from a UAV’s interference with aircraft. Liability for UAV use, however, is not limited to personal injury or property damage claims, UAVs present an enormous threat to individual privacy [@law].  ### Theft The main concerns of utilizing UAVs for data gathering and wireless delivery, are cyber liability and hacking. UAVs that are used to gather sensitive information might become targets for malicious software seeking to steal data. A hacker might even usurp control of the UAV itself for the purpose of illegal activities, such as theft of its cargo or stored data, invasion of privacy, or smuggling. Liability for utilizing a UAV does not fit neatly into the coverage offered by the types of liability insurance policies that most individuals and businesses currently possess [@law]. ### Weather Similar to light aircrafts, UAVs cannot hover in all weather conditions. The capability to resist certain weather conditions is determined by the specifications of the UAV [@fornace2014mapping]. In pre-flight planning, it is clear that advanced weather data will play an essential role in ensuring that UAVs can fly their weather-sensitive missions safely and efficiently to deliver commercial goods. During flight operations, weather data affects flight direction, path elevation, operation duration and other in-flight variables. Wind speeds in particular are an essential component for a smooth UAV-based operations and thus should be factored in the operation planning and deployment phases. In post-flight analysis, by analyzing data through advanced weather visualization dashboards, we can improve the UAV flight operations to ensure future mission success [@unmanned]. ### Air Traffic Control Air traffic control is an essential condition for coordinating large fleets of UAVs, where regulators will not permit large-scale UAV delivery missions without such systems in place [@businessinsider]. Amazon designs an airspace model for the safe integration of UAV systems as shown in Figure \[dg4\]. In this proposed model, the “low-speed localized traffic” will be reserved for: 1) Terminal non-transit operations such as surveying, videography and inspection; 2) Operations for lesser-equipped UAVs, e.g. ones without sophisticated sense-and-avoid technology. The “high-speed transit” will be reserved for well equipped UAVs as determined by the relevant performance standards and rules. The “no fly zone” will serve as a restricted area in which UAV operators will not be allowed to fly, except in emergencies. Finally, the “predefined low risk locations” will include areas like designated academy of model aeronautics airfields, altitude and equipage restrictions in these locations will be established in advance by aviation authorities [@air2015revising]. Research Trends and Future Insights ----------------------------------- ### [Machine Learning]{} With machine learning, UAVs can fly autonomously without knowing the objects they may encounter which is important for large-scale UAV delivery missions. Qualcomm’s tech shows more advanced computing that can actually understand what the UAV encountered in mid-air and creates a flight route. They show how the UAV processing and decision-making technology is nimble enough to allow UAVs to operate in unpredictable settings without using any GPS. All of the UAV computational tasks, like the machine learning and flight control, happens on 12 grams processor without any off-board computing [@recodejaz]. Some of challenges are : 1) We need to put a lot of effort to find efficient methods to do unsupervised learning, where collecting large amounts of unlabeled data is nowadays becoming economically less expensive [@carrio2017review]; 2) Real-world problems with high number of states can turn the problem intractable with current techniques, severely limiting the development of real applications. An efficient method for coping with these types of problems remains as an unsolved challenge [@carrio2017review]. ### [Navigation System]{} Researchers are developing navigation systems that do not utilize GPS signals. This could enable UAVs to fly autonomously over places where GPS signals are unavailable or unreliable. Whether delivering goods to remote places or handling emergency tasks in hazardous conditions, this type of capability could significantly expand UAVs’ usefulness [@theconversation]. Researchers from GPU maker Nvidia are currently working on a navigation system that utilizes visual recognition and computer learning to make sure UAVs don’t get lost. The team believes that the system has already managed the most stable GPS-free flight to date [@newatlas]. Some of challenges are: 1) The design of computing devices with low-power consumption, particularly GPUs, is a challenge and active working field for embedded hardware developers [@carrio2017review]; 2) While a few UAVs can already travel without a UAV operators directing their routes, this technology is still emerging. Over the next few years, system-failure responses, adaptive routing, and handoffs between user and UAV controllers should be improved [@mckinsey]. ### Future Insights {#future-insights-3} Some of the future possible directions for this application are: - The energy density of lithium-ion batteries is improving by 5%-8% per year, and their lifetime is expected to double by 2025. These improvements will make commercial UAVs able to hover for more than an hour without recharging, enabling UAVs to deliver more goods [@mckinsey]. - Detect-and-avoid systems which help UAVs to avoid collisions and obstacles, are still in development, with strong solutions expected to emerge by 2025 [@mckinsey]. - UAVs currently travel below the height of commercial aircraft due to the collision potential. The methods that can track UAVs and communicate with air-traffic-control systems for typical aircraft are not expected to be available before 2027, making high-altitude missions impossible until that time [@mckinsey]. - To make UAV delivery practical, automation research is required to address UAVs design. UAVs design covers creating aerial vehicles that are practical, can be used in a wide range of conditions, and whose capability rivals that of commercial airliners; this is a significant undertaking that will need many experiments, ingenuity and contributions from experts in diverse areas [@d2014guest]. - More research is needed to address localization and navigation. The localization and navigation problems may seem like simple problems due to the many GPS systems that already exist, but to make drone delivery practical in different operating conditions, the integration of low cost sensors and localization systems is required [@d2014guest]. - More research is needed to address UAVs coordination. Thousands of UAV operators in the air, utilizing the same resources such as charging stations and operating frequency, will need robust coordination which can be studied by simulation [@d2014guest]. Real-Time Monitoring of Road Traffic ==================================== Automation of the overall transportation system cannot be automated through vehicles only[@menouar2017uav]. In fact, other components of the end-to-end transportation system, such as tasks of field support teams, traffic police, road surveyors, and rescue teams, also need to be automated. Smart and reliable UAVs can help in the automation of these components. UAVs have been considered as a novel traffic monitoring technology to collect information about traffic conditions on roads. Compared to the traditional monitoring devices such as loop detectors, surveillance video cameras and microwave sensors, UAVs are cost-effective, and can monitor large continuous road segments or focus on a specific road segment [@ke2017real]. Data generated by sensor technologies are somewhat aggregated in nature and hence do not support an effective record of individual vehicle tracks in the traffic stream. This restricts the application of these data in individual driving behavior analysis as well as calibrating and validating simulation models [@GUIDO2016136]. Moreover, disasters may damage computing, communications infrastructure or power systems. Such failures can result in a complete lack of the ability to control and collect data about the transportation network [@Leitloff_2014]. Literature Review {#related-work} ----------------- UAVs are getting accepted as a method to hasten the gathering of geographic surveillance data [@smartcities]. As autonomous and connected vehicles become popular, many new services and applications of UAVs will be enabled [@menouar2017uav]. Recognition of moving vehicles using UAVs is still a challenging problem. Moving vehicle detection methods depend on the accuracy of image registration methods, since the background in the UAV surveillance platform changes frequently. Accurate image registration methods require extensive computing power, which affecta the real-time capability of these methods [@qu2016moving]. In [@qu2016moving], Qu et al. studied the problem of moving vehicle detection using UAV cameras. In their proposed approach, they used convolutional neural networks to identify vehicles more accurately and in real-time. The proposed approach consists of three steps to detect moving vehicles: First, adjacent frames are matched. Then, frame pixels are classified as background or candidate targets. Finally, a deep convolutional neural network is trained over candidate targets to classify them into vehicles or background. They achieved detection accuracy of around $90\%$ when evaluating their method using the CATEC UAV dataset. In [@WANG2016294], the authors introduce a vehicle detection and tracking system based on imagery data collected by a UAV. This system uses consecutive frames to generate the vehicle’s dynamic information, such as positions and velocities over time. Four major modules have been developed in this study: image registration, image feature extraction, vehicle shape detecting, and vehicle tracking. Some unique features have been introduced into this system to customize the vehicle and traffic flow and use them together in multiple contiguous images to increase the system’s accuracy of detecting and tracking vehicles. A framework is presented in [@ke2017real] to support real-time and accurate collection of traffic flow parameters, including speed, density, and volume, in two travel directions simultaneously. The proposed framework consists of the following four features: (1) A framework for estimating multi-directional traffic flow parameters from aerial videos (2) A method combining the Kanade–Lucas–Tomasi (KLT) tracker, k-means clustering, and connected graphs for vehicle detection and counting (3) Identifying traffic streams and extracting traffic information in a real-time manner (4) The system works in daytime and nighttime settings, and is not sensitive to UAV movements (i.e., regular movement, vibration, drifting, changes in speed, and hovering). A challenge that this framework faces is that their algorithm sometimes recognizes trucks, buses, and other large/heavy vehicles as multiple passenger cars. A real-time framework for the detection and tracking of a specific road segment using low and mid-altitude UAV video feeds was presented in [@zhou2015efficient]. This framework can be used for autonomous navigation, inspection, traffic surveillance and monitoring. For road detection, they utilize the GraphCut algorithm abecause of its efficient and powerful segmentation performance in 2-D color images. For road tracking, they develop a tracking technique based on homography alignment to adjust one image plane to another when the moving camera takes images of a planar scene. In [@Leitloff_2014], the authors develop a processing procedure for fast vehicle detection, which consists of three stages; pre-classification with a boosted classifier, blob detection, and final classification using SVM. In [@puri2007statistical], the authors propose to integrate collected video data from UAVs with traffic simulation models to enhance real-time traffic monitoring and control. This can be performed by transforming collected video data into ‘useful traffic measures’ to generate essential statistical profiles of traffic patterns, including traffic parameters such as mean-speed, density, volume, turning ratio, etc. However, a main issue with this approach is the limitation of flying time for UAVs which could hover to obtain data for a few hours a day. The work in [@reshma2016security] addresses the security issues of road traffic monitoring systems using UAVs. In this work, the role of a UAV in a road traffic management system is analyzed and various situational security issues that occur in traffic management are mitigated. In their proposed approach, the UAV’s intelligent systems analyze real-time traffic as well as security issues and provide the appropriate mitigation commands to the traffic management control center for re-routing. Instead of image processing, the authors used sensor networks and graph theory for representing the road network. They also devised different situational security scenarios to assess the road traffic management.For example, a car without an RFID tag that enters a defense area or government building area is considered a potential security risk. Based on a research study by Kansas Department of Transportation (KDOT) [@Kansas], the use of UAVs for KDOT’s operations could lead to improved safety, efficiency, as well as reduced costs. The study also recommends the use of UAVs in a range of applications including bridge inspection, radio tower inspection, surveying, road mapping, high-mast light tower inspection, stockpile measurement, and aerial photography. However, the study indicates that UAVs are not recommended to replace the current methods of traffic data collection in KDOT operations which rely on different kinds of sensors (e.g., weight, loop, piezo). However, UAVs can complement existing data collection projects to gather data in small increments of time in certain traffic areas. Based on their survey, battery life and flight time of UAVs may limit the samples of traffic data. For example, current UAV technology cannot collect 24-hour continuous data. Yu Ming Chen et al. [@chen2007real] proposed a video relay scheme for traffic surveillance systems using UAVs. The proposed communications scheme is straightforward to implement because of the typical availability of mobile broadband along highways. Their results show that the proposed scheme can transfer quality videos to the traffic management center in real-time. They also implemented two types of data communications schemes to transmit captured videos through existing public mobile broadband networks: (1) Video stream delivered directly to clients. (2) Video stream delivered to clients through a server. In their experiments, they were able to transmit video signals with an image size of $320\times280$, at a rate of $112$ Kbps, and $15$ frames per second. In [@ro2007lessons], a platform which operates autonomously and delivers high-quality video imagery and sensor data in real-time is utilized. In their scenario, the authors employ a $10$ lbs aircraft to fly up to $6$ hours with a telemetry range of $1$ mile and payload capacity of $4$ lbs. Their system consists of five components: (1) a GPS signal receiver (2) a radio control transmitter (3) a modem for flight data (4) a PC to display the UAV on a map. (5) a real-time video down-link. Apeltauer et al. [@apeltauer2015automatic] present an approach for moving vehicle detection and tracking through the intersection of aerial images captured by UAVs. Overall, the system follows three steps: pre-processing, vehicle detection, and tracking. For pre-processing, images are undistorted geo-registered against a user-selected reference frame. For the detection step, the boosting technique is used to improve the training phase which employs Multi-scale Block Local Binary Patterns (MB-LBP). Finally for tracking, the system uses a set of Bootstrap particle filters, one per vehicle. An improved vehicle detection method based on Faster R-CNNs is proposed in [@tang2017vehicle]. The overall vehicle detection method is illustrated in Figure \[fig:HRPN\]. For training, the method crops the original large-scale images into segments and augments the number of image segments with four angles (i.e., [0]{}, [90]{}, [180]{}, and [270]{}). Then, all the training image blocks that constitute the HRPN input are processed to produce candidate region boxes, scores and corresponding hyper features. Finally, the results of the HRPN are used to train a cascade of boosted classifiers, and a final classifier is obtained. For testing, a large-scale testing image is cropped into image blocks. Then, HRPN takes these image blocks as input and generates potential outputs as well as hyper feature maps. The final classifier checks these boxes using hyper features. Finally, all the detection results of segments are gathered to integrate the original image. The authors tested their method on UAV images successfully. {width="100.00000%"} [|l|l|l|l|l|]{} **Project** & **Goal** & **Hardware** & **Dataset** & **Link to dataset**\ Qu et al. [@qu2016moving] & Moving vehicle detection & UAV + Camera & CATEC UAV & -\ Wang et al. [@WANG2016294] & ----------------------- Vehicle detection and tracking ----------------------- & UAV + Camera & not avaiable & -\ Ke et al. [@ke2017real] & ----------------------- Extraction of traffic flow parameters ----------------------- & UAV + Camera & -------------------- Taken from Beihang University -------------------- & -\ Zhou et al. [@zhou2015efficient] & -------------------------- Real-time road detection and tracking -------------------------- & UAV + camera & Self-collected & https://sites.google.com/site/hailingzhouwei\ Leitloff et al. [@Leitloff_2014] & Fast vehicle detection & UAV + camera & Self-collected &-\ Puri et al. [@puri2007statistical] & ------------------------ real-time traffic monitoring and control ------------------------ & UAV + camera & Self-collected & -\ Reshma et al. [@reshma2016security] & ----------------------------- address the security issues of road traffic monitoring ----------------------------- & UAV + RFID & ----------- Proteus simulator ----------- & -\ M-Chen et al. [@chen2007real] & traffic surveillance & UAV + camera & self-collected & -\ Tang et al. [@tang2017vehicle] & vehicle detection & UAV + camera & Munich vehicle dataset & ------------------------------------- http://pba-freesoftware.eoc.dlr.de/ 3K\_VehicleDetection\_dataset.zip ------------------------------------- \ Apeltauer et al. [@apeltauer2015automatic] & vehicle trajectory extraction & UAV + camera & self-collected & -\ Use Cases --------- The major applications of UAVs in transportation include security surveillance, traffic monitoring, inspection of road construction projects, and survey of traffic, rivers, coastlines, pipelines, etc. [@6851187]. Some of the ITS applications that can be enabled by UAVs are as follows: - Flying Accident Report Agents: Rescue teams can use UAVs to quickly reach accident locations. Flying accident report agents can also be used to deliver first aid kits to accident locations while waiting for rescue teams to arrive [@menouar2017uav]. - Flying Police Eyes: UAVs can be used to fly over different road segments in order to stop vehicle for traffic violations. The UAV can change the traffic light in front of the vehicle to stop it or relay a message to a specific vehicle to stop [@menouar2017uav] [@menouar2017uav]. - Flying Roadside Unit: A UAV can be complemented with DSRC to enable a flying RSU. The flying RSU can fly to a specific position to execute a specific application. For example, consider an accident on the highway at a specific segment that is not equipped with any RSU. Then the traffic management center can activate a UAV to fly to the accident location and land at the proper location to broadcast the information and warn all approaching vehicles about a specific incident [@menouar2017uav]. - Behavior Recognition Method: UAVs can be used to recognize suspicious or abnormal behavior of ground vehicles moving along with the road traffic [@oh2014behaviour]. - Monitor Pedestrian Traffic: Sutheerakul et al. [@SUTHEERAKUL20171717] used UAVs as an alternative data collection technique to monitor pedestrian traffic and evaluate demand and supply characteristics. In fact, they classified their collected data into four areas: the measurement of pedestrian demand, pedestrian characteristics, traffic flow characteristics, and walking facilities and environment. - Flying Dynamic Traffic Signals [@menouar2017uav]. UAVs may also be employed for a wide range of transportation operations and planning applications such as following [@puri2005survey]: - Incident response. - Monitor freeway conditions. - Coordination among a network of traffic signals. - Traveler information. - Emergency vehicle guidance. - Measurement of typical roadway usage. - Monitor parking lot utilization. - Estimate Origin-Destination (OD) flows. Figures \[fig:use-caseA\], \[fig:use-caseB\], and \[fig:use-caseC\] illustrate different applications use-cases of UAVs in smart cities. ![\[fig:use-caseA\]A UAV is Used By Police to Catch Traffic Violators [@menouar2017uav].](srv_RTM_f2){width="45.00000%"} ![\[fig:use-caseC\]A UAV is Used to Provide the Rescue Team an Advance Report Prior to Reaching the Incident Scene [@menouar2017uav].](srv_RTM_f3){width="45.00000%"} ![\[fig:use-caseC\]A UAV is Used to Provide the Rescue Team an Advance Report Prior to Reaching the Incident Scene [@menouar2017uav].](srv_RTM_f4){width="45.00000%"} Legislation ----------- The Federal Aviation Administration (FAA) approves the civil use of UAVs [@FAA]. They can be utilized for public use provided that the UAVs are flown at a certain altitude. For maintaining the safety of manned aircrafts and the public, the FAA in the United States has developed rules to regulate the use of small UAVs [@vattapparamban2016drones]. For example, the FAA requires small Unmanned Aircraft Systems (UAS) that weigh more than $0.55$ lbs and below $55$ lbs to be registered in their system. Regulations are broadly organized into two categories; namely, prescriptive regulations and performance-based regulations (PBRs). Prescriptive regulations define what must not be done whereas PBRs indicate what must be attained. $20\%$ of FAA regulations are PBRs [@NAKAMURA2017]. Table X describes the rules for operating UAS in the US[@FAA]. [|l| p[4.75cm]{}|]{} & **Fly for work** \ Pilot Requirements & Must have Remote Pilot Airman Certificate Must be 16 years old Must pass TSA vetting\ Aircraft Requirements & Must be less than 55 lbs. Must be registered if over 0.55 lbs. (online) Must undergo pre-flight check to ensure UAS is in condition for safe operation\ Location Requirements & Class G airspace\*\ Operating Rules & Must keep the aircraft in sight (visual line-of-sight)\* Must fly under 400 feet\* Must fly during the day\* Must fly at or below 100 mph\* Must yield right of way to manned aircraft\* Must NOT fly over people\* Must NOT fly from a moving vehicle\*\ Example Applications & Flying for commercial use (e.g. providing aerial surveying or photography services).incidental to a business (e.g. doing roof inspections or real estate photography)\ Legal or Regulatory Basis & Title 14 of the Code of Federal Regulation (14 CFR) Part 107\ When violating the FAA regulations, owners of drones can face civil and criminal penalties [@vattapparamban2016drones]. The FAA has also provided a smartphone application B4UFLY5 which provides drone users important information about the limitations that pertain to the location they where drone is being operated. The “Know Before You Fly” campaign by the FAA aims to educate the public about UAV safety and responsibilities. For civil operations, the FAA authorization can be received either through Section 333 Exemption (i.e., by issuing a COA), or through a Special Airworthiness Certificate (SAC) in which applicants describe their design, software development, control, along with how and where they intend to fly. The FAA also enforces their regulations along with Law Enforcement Agencies (LEAs) to deter, detect, investigate, and stop unauthorized and unsafe UAV operations [@vattapparamban2016drones]. Overall, the FAA regulations for small UAVs require flying under $400$ feet without obstacles in their vicinity, such that operators maintain a line of sight with the operated UAV at all times. It also requires UAVs not to fly within $5$ miles from an airport unless permission is received from the airport and the control tower; thus, avoiding the endangerment of people and aircrafts. Other FAA regulations require drones not to be operated over public infrastructure (e.g., stadiums) as that may pose dangers to the public. [The Federal UAS regulations final rule requires drone pilots to keep unmanned aircrafts within visual line of sight and operations are only allowed during daylight and during half-light if the drone is equipped with “anti-collision lights.” The new regulations also initiate height and speed constraints and other operational limits, such as prohibiting flights over unprotected people on the ground who aren’t directly participating in the UAS operation. There is a process through which users can apply to have some of these restrictions waived, while those users currently operating under section 333 exemptions (which allowed commercial use to take place prior to the new rule) are still able to operate depending on their exemptions [@regulations].]{} In Section 107.51 of the FAA regulations [@federalregulations], it is mentioned that a remote pilot in command and the person manipulating the flight controls of the small unmanned aircraft system must consent with all of the following operating limitations when operating a small unmanned aircraft system: - The ground speed of a small unmanned aircraft may not exceed $87$ knots (i.e., $100$ miles per hour). - The altitude of the small unmanned aircraft cannot be higher than $400$ feet above ground level, unless the small unmanned aircraft: (1) Is flown within a $400$-foot radius of a structure; and (2) Does not fly higher than $400$ feet above the structure’s immediate uppermost limit. - The minimum flight visibility, as observed from the location of the control station must be no less than $3$ statute miles. For critical infrastructure, a legislation has been developed to protect such infrastructure from rogue drone operators. UAVs should not be close to such places when these critical infrastructure exist. The classification of critical infrastructure differs by state, but generally includes facilities such as petroleum refineries, chemical manufacturing facilities, pipelines, waste water treatment facilities, power generation stations, electric utilities, chemical or rubber manufacturing facilities, and other similar facilities [@regulationsncls]. Challenges and Future Insights ------------------------------ There are several challenges and needed future extensions to facilitate the use of UAVs in support of ITS applications: - One of the important challenges is to preserve the privacy of sensitive information (e.g., location) from other vehicles and drones [@menouar2017uav]. Since usually there is no encryption on UAVs on-board chips, they can be hijacked and subjected to man-in-middle attacks originating up to two kilometers away [@vattapparamban2016drones]. - countries should devise registration mechanisms for UAVs operated on their geographic areas. Commercial airplanes and their navigation might be affected by UAVs. So countries must implement rules and regulations for their proper use [@mohammed2014uav]. - Developing precise coordination algorithms is one of the challenges that need to be considered to enable ITS UAVs [@menouar2017uav]. - Wireless sensors can be utilized to smooth the operations of UAVs. For example, surveillance and live feeds from wireless sensors can be developed for control traffic systems [@mohammed2014uav]. - Data fusion of information from diverse sensors, automating image data compression, and stitching of aerial imagery are required techniques[@mohammed2014uav]. - Enabling teams of operators to control the UAV and retrieve imagery and sensor information in real-time. To achieve this goal, the development of network-centric infrastructure is required [@mohammed2014uav]. - Limited energy, processing capabilities, and signal transmission range [@menouar2017uav] are also some of the main issues in UAVs that require more development to contribute to the maturity of the UAV technology - UAVs have slower speeds compared to vehicles driving on highways. However, a possible solution might entail changing the regulations to allow UAVs to fly at higher altitudes. Such regulations would allow UAVs to benefit from high views to compensate the limitation in their speed. Finding the optimal altitude change in support of ITS applications is a research challenge [@menouar2017uav]. - Battery technology that allow UAVs to achieve long operational times beyond half an hour is another challenge. Several UAVs can fly together and form a swarm, by which they could overcome their individual limitations in terms of energy efficiency through optimal coordination algorithms [@menouar2017uav]. Another alternative is for UAVs to use recharge stations. UAVs can be recharged while on the ground, or their depleted batteries can be replaced to minimize interruptions to their service. To this end, deployment of UAVs, recharge stations, and ground RSUs jointly becomes an interesting but complicated optimization problem [@menouar2017uav]. - The detection of multiple vehicles at the same time is another challenge. In [@zhang2007video], Zhang et al. developed several computer-vision based algorithms and applied them to extract the background image from a video sequence, identify vehicles, detect and remove shadows, and compute pixel-based vehicle lengths for classification. - Truly autonomous operations of UAV swarms are a big challenge, since they need to recognize other UAVs, humans and obstacles to avoid collisions. Therefore, the development of swarm intelligence algorithms that fuse data from diverse sources including location sensors, weather sensors, accelerometers, gyoscopes, RADARs, LIDARs, etc. are needed. Surveillance Applications of UAVs ================================= In this section, we first present a detailed literature review of surveillance applications of UAVs. Then, based on the review, we summarize the advantages, disadvantages and important concerns of UAV uses in surveillance. Also, we discuss several takeaway lessons for researchers and practitioners in the area, which we believe will help guide the development of effective UAV surveillance applications. Literature review {#literature-review-2} ----------------- The report in [@haddal2010homeland] discusses the advantages and disadvantages of employing UAVs along US borders for surveillance and introduces several important issues for the Congress. The advantages include: (1) The usage of UAVs for border surveillance improves the coverage along the remote border sections of the U.S.; and (2) The UAVs provide a much wider coverage than current approaches for border surveillance (e.g., border agents on patrol, stationary surveillance equipment, etc.). On the other hand, the disadvantages include: (1) The high accident rates of UAVs (e.g., inclement weather conditions); and (2) The significant operating costs of a UAV, which are often more than double compared to the costs of a manned aircraft. The report also highlights other issues of UAVs that should be considered by the Congress including: UAV effectiveness, lack of information, coordination with USBP agents, safety concerns, and implementation details. This report provides a clear view of the advantages, disadvantages and important concerns for the border surveillance using UAVs. However, it lacks details/examples on each aspect discussed (i.e., only high level summaries are included). Also, other important aspects are missing, for example, the robustness benefits (e.g., human error tolerance) of using the UAVs compared to manned aircraft. The authors in [@maza2011experimental] present a multi-UAV coordination system in the framework of the AWARE project (distributed decision-making architecture suitable for multi-UAV coordination). Generally speaking, a set of tasks are generated and allocated to perform a given mission in an efficient manner according to planning strategies. These tasks are sub-goals that are necessary for achieving the overall goal of the system, and that can be achieved independent of other sub-goals. The key issues include: - [Task allocation]{}: determines the place (i.e., in which UAV node) that each task should be executed to optimize the performance and to ensure appropriate co-operation among different UAVs. - [Operative perception]{}: generates and maintains a consistent view of all operating UAVs (e.g., number of sensors equipped). In addition, several types of field experiments are conducted, including: (1) Multi-UAV cooperative area surveillance; (2) Wireless sensor deployment; (3) Fire threat confirmation and extinguishing; (4) Load transportation and deployment with single and multiple UAVs; and (5) People tracking. To sum up, this paper presents details of the proposed algorithm and shows the results of several real filed experiments. However, it lacks detailed analysis or formal proof of the proposed approach to demonstrate its correctness and effectiveness. In [@wall2011surveillance], the authors perform analysis for UAVs deployments within war-zones (i.e., Afghanistan, Iraq, and Pakistan), border-zones and urban areas in the USA. The analysis highlights the benefits of UAVs in such scenarios which include: - [Safety]{}: Drones insulates operators and allies from direct harm and subjects targets to ’precise’ attack. - [Robustness]{}: Drones reduce human errors (e.g., moral ambiguity) from political, cultural, and geographical contexts. Most importantly, the analysis uncovers a major limitation in UAV surveillance practices. The usage of drones for surveillance has difficulties in exact target identification and control in risk societies. Potential blurred identities include: (1) insurgent and civilian; (2) criminal and undocumented migrant; and (3) remotely located pilot and front-line soldier. To sum up, this paper puts more focus on safety and robustness benefits of using UAVs in battle fields and urban areas. Also, it presents the limitation of exact target identification and control. However, it overlooks the coverage benefits and the deployment/implementation limitations of using UAV systems for surveillance. In [@finn2012unmanned], the authors show the impact of UAV-based surveillance in civil applications on privacy and civil liberties. First, it states that current regulatory mechanisms (e.g., the US Fourth amendment, EU legislation and judicial decisions, and UK legislation) do not adequately address privacy and civil liberties concerns. This is mainly because UAVs are complex, multi-modal surveillance systems that integrate a range of technologies and capabilities. Second, the inadequacy of current legislation mechanisms results in disproportionate impacts on civil liberties for already marginalized population. In conclusion, multi-layered regulatory mechanisms that combine legislative protections with a bottom-up process of privacy and ethical assessment offer the most comprehensive way to adequately address the complexity and heterogeneity of unmanned aircraft systems and their intended deployments. To sum up, this paper focuses on the law enforcement of privacy and civil liberty aspects of using UAVs for surveillance. However, the regulation recommendation provided is high level and lacks sufficient details and convincing analysis. In [@kingston2008decentralized], the authors introduce a cooperative perimeter surveillance problem and offer a decentralized solution that accounts for perimeter growth (expanding or contracting) and insertion/deletion of team members. The cooperative perimeter surveillance problem is defined to gather information about the state of the perimeter and transmit that data back to a central base station with as little delay and at the highest rate possible. The proposed solution is presented in Algorithm \[alg1\]. The proposed scheme is described comprehensively and a simple formal proof is provided. However, there is no related works or comparative evaluations provided to justify the effectiveness of the proposed scheme. [|l|l|c|c|]{} & & **Research Focus** & **Category**\ [@kingston2008decentralized] & 2008 & UAVs cooperative perimeter surveillance & Multi-UAV cooperation\ [@haddal2010homeland] & 2010 & UAV with border surveillance & Homeland security concerns\ [@maza2011experimental] & 2011 & ------------------------------------------------ Multi-UAV coordination for disaster management and civil security applications ------------------------------------------------ & Multi-UAV cooperation\ [@wall2011surveillance] & 2011 & Concerns of UAVs in battle filed & Military concerns\ [@birk2011safety] & 2011 & -------------------------------------- UAV system filed tests with improved photo mapping algorithm -------------------------------------- & Algorithm improvement\ [@finn2012unmanned] & 2012 & ------------------------------------------------ UAV surveillance in civil applications impacts upon privacy and other civil liberties ------------------------------------------------ & Privacy and civil liberties\ [@wada2015surveillance] & 2015 & A small UAV system with related technologies & Product introduction\ [@motlagh2017uav] & 2017 & UAV-based IoT platform for crowd surveillance & New use case\ , (1) Calculate shared border position ; (2) Travel with neighbor to shared border ; (3) Set direction to monitor own segment;      [**If** Reached perimeter endpoint:    **Then**]{}      Reverse direction.      [**ENDIF**]{} The authors of [@birk2011safety] present the results of using a UAV in two field tests, the 2009 European Land Robot Trials (ELROB-2009) and the 2010 Response Robot Evaluation Exercises (RREE-2010), to investigate different realistic response scenarios. They transplant an improved photo mapping algorithm which is a variant of the Fourier Mellin Invariant (FMI) transform for image representation and processing. They used FMI with the following two modifications: - A logarithmic representation of the spectral magnitude of the FMI descriptor is used; - A filter on the frequency where the shift is supposed to appear is applied. To sum up, this report mainly discusses two field tests with a description of a transplanted improved photo mapping algorithm. However, the proof/analysis of the improved algorithm is missing. In [@motlagh2017uav], the authors discuss a potential application of UAVs in delivering IoT services. A high-level overview is presented and a use case is introduced to demonstrate how UAVs can be used for crowd surveillance based on face recognition. They also study the offloading of video data processing to an MEC (Mobile Edge Computing) node compared to the local processing of video data on board UAVs. The results demonstrate the efficiency of the MEC-based offloading approach in energy consumption, processing time of recognition, and in promptly detecting suspicious persons. To sum up, this paper successfully demonstrates the benefits of using a single UAV system with IoT devices for crowd surveillance with the introduced video data processing offloading technique. However, further analysis of the coordination of multiple UAVs is missing. In [@wada2015surveillance], the authors present the use of small UAVs for specific surveillance scenarios, such as assessment after a disaster. This type of surveillance applications utilizes a system that integrates a wide range of technologies involving: communication, control, sensing, image processing and networking. This article is a product technical report and the corresponding manufacturer successfully applied multiple practical techniques in a real UAV surveillance product. Table \[t1\] summarizes the eight papers we have reviewed including their research focus and category. Discussion ---------- ### State-of-the-art Research {#state-of-the-art-research .unnumbered} Table \[t2\] summarizes the advantages, disadvantages and important concerns for the UAV surveillance applications based on our literature review. We classify the UAV surveillance research works into six categories based on their focus and content: - Multi-UAV cooperation: this line of research focuses on developing new orchestration algorithms or architectures in the presence of multiple UAVs. - Homeland/military security concerns: this line of research provides analysis and suggestions for the authorities with the considerations for Homeland/military security. - Algorithm improvement: this line of research develops new algorithms for more efficient post-processing of the data (e.g., videos or photos captured by the UAVs). - New use case: this line of research integrates UAV system into new application scenarios for potential extra benefits. - Product introduction: this line of research describes mature techniques leveraged by a real UAV surveillance product. [|l|l|]{} **Advantages** & --------------------------------------------- surveillance coverage and range improvement better safety for human operators robustness and efficiency in surveillance --------------------------------------------- : []{data-label="t2"} \ **Disadvantages** & ------------------------------------------------- high accidental rate high operating costs difficulties in surveillance of risk societies. ------------------------------------------------- : []{data-label="t2"} \ & ---------------------------------------- co-operation of multi-UAVs post-processing algorithm improvements privacy concerns law enforcement ---------------------------------------- : []{data-label="t2"} \ Research Trends and Future Insights ----------------------------------- ### Research Trends Based on the reviewed literature and our analysis, we identify that the use of research trends (e.g., machine learning algorithms, nano-sensors [@chen2013infrared], short-range communications technologies, etc.) could bring further benefits in the area of UAV surveillance applications: - There are several preferred features of UAV surveillance applications, especially for some military or homeland security use cases: (1) small size: this feature not only reduces the physical attack surface of the UAV but also achieves more secrete UAV patrolling. (2) high resolution photos: this feature helps the data post processing algorithms to draw more accurate conclusions or obtain more insight findings; (3) low energy consumption: this feature allows the UAV to operate for a longer time of period such that to achieve more complex tasks. Therefore, to capture more accurate data in a more efficient and secret way, advanced sensor technologies should be considered. Such as nano-sensors (small size), ultra-high-resolution image sensors (accurate data) [@jiang2015ultrahigh], and energy efficient sensors (low energy consumption) [@folea2015low]. - To develop more efficient and accurate multi-UAV cooperation algorithms or data post-processing algorithms, advanced machine learning algorithms (e.g., deep learning [@lecun2015deep]) could be utilized to achieve better performance and faster response. Specifically, multi-UAV cooperation brings more benefits than single UAV surveillance, such as more wider surveillance scope, higher error tolerance, and faster task completion time. However, multi-UAV surveillance requires more advanced data collection, sharing and processing algorithms. Processing of huge amount data is time consuming and much more complex. Applying advanced machine learning algorithms (e.g., deep learning algorithm) could help the UAV system to draw better conclusions in a short period of time. For example, due to its improved data processing models, deep learning algorithms could help to obtain new findings from existing data and to get more concise and reliable analysis results. ### [Future Insights]{} {#future-insights-4} based on our literature review and analysis, to develop or deploy effective UAV surveillance applications, we suggest the following: - Select/design appropriate UAV system based on the state/federal laws and individual budget. For example, the design of UAVs surveillance systems should take the privacy and security requirements enforced by the local laws; - Develop efficient customized algorithms (e.g., co-operation algorithm of multi-UAVs, photo mapping algorithms, etc.) based on system requirements. Multi-UAVs co-operation algorithm could achieve more efficient surveillance (e.g., optimum patrolling routes with minimum power consumption). Also, advanced data post-processing algorithms help the users to draw more accurate conclusions; - Conduct various field experiments to verify newly proposed systems. Extensive field experiments in various surveillance conditions (e.g., day time, night time, sunshine day, cloudy day, etc.) should be conducted to verify the effectiveness and robustness of the UAVs systems. Providing Wireless Coverage {#comm} =========================== UAVs can be used to provide wireless coverage during emergency cases where each UAV serves as an aerial wireless base station when the cellular network goes down [@bupe2015relief]. They can also be used to supplement the ground base station in order to provide better coverage and higher data rates for users [@bor2016efficient]. In this section, we present the aerial wireless base stations use cases, UAV links and UAV channel characteristics. We also show the path loss models for UAVs, classify them based on environment, altitude and telecommunication link, and present some challenges facing these path loss models. Moreover, we present UAV deployment strategies that optimize several objective functions of interest. Then, we discuss the challenges facing UAV interference mitigation techniques. Finally, we present the research trends and future insights for aerial wireless base stations. ![Typical Use Cases of UAV-Aided Wireless Communications.[]{data-label="haz9"}](srv_WC_f1.eps) for tree=[ align=center, parent anchor=south, child anchor=north, font=, edge=[thick, -[Stealth\[\]]{}]{}, l sep+=10pt, edge path=[ (!u.parent anchor) – +(0,-10pt) -| (.child anchor); ]{}, if level=0[ inner xsep=0pt, tikz=[(.south east) – (.south west);]{} ]{} ]{} \[ UAV links \[Control links \[Command and\ control from\ GCS to UAVs \] \[UAV status\ report from\ UAVs to GCS \] \[Sense and avoid\ information\ among UAVs \] \] \[Data links \[Direct ground\ mobile-UAV\ communication \] \[UAV-base station\ and UAV-gateway\ wireless backhaul \] \[UAV-UAV\ wireless backhaul \] \] \] Type of link Security requirement Latency requirement Capacity requirement Frequency bands -------------- ---------------------- --------------------- ---------------------- ----------------------- Control link High High Low L-band (960-977MHz) C-band (5030-5091MHz) Data link Low Low High 450 MHz to mmWave ![Basic Networking Architecture of UAV-Aided Wireless Communications.[]{data-label="haz7"}](srv_WC_f2.eps) for tree=[ align=center, parent anchor=south, child anchor=north, font=, edge=[thick, -[Stealth\[\]]{}]{}, l sep+=10pt, edge path=[ (!u.parent anchor) – +(0,-10pt) -| (.child anchor); ]{}, if level=0[ inner xsep=0pt, tikz=[(.south east) – (.south west);]{} ]{} ]{} \[ Classification of Path loss Models for UAVs \[Altitude of UAV \[LAP \] \[HAP \] \] \[Environment \[Outdoor-ground users \] \[Indoor users \] \[UAV-UAV \] \] \[Telecommunication link \[Downlink \] \[Uplink \] \] \] Aerial Wireless Base Stations Use Cases --------------------------------------- The authors in [@zeng2016wireless; @jawhar2017communication; @zhao2017beam] present the typical use cases of aerial wireless base stations which are discussed in the following: - [UAVs for ubiquitous coverage:]{} UAVs are utilized to assist wireless network in providing seamless wireless coverage within the serving area. Two example scenarios are rapid service recovery after disaster situations, and when the cellular network service is not available or it is unable to serve all ground users as shown in Figure \[haz9\].a. - [UAVs as network gateways: ]{}In remote geographic or disaster stricken areas, UAVs can be used as gateway nodes to provide connectivity to backbone networks, communication infrastructure, or the Internet. - [UAVs as relay nodes:]{} UAVs can be utilized as relay nodes to provide wireless connectivity between two or more distant wireless devices without reliable direct communication links as shown in Figure \[haz9\].b. - [UAVs for data collection:]{} UAVs are utilized to gather delay-tolerant information from a large number of distributed wireless devices. An example is wireless sensors in precision agriculture applications as shown in Figure \[haz9\].c. - [UAVs for worldwide coverage:]{} UAV-satellite communication is an essential component for building the integrated space-air-ground network to provide high data rates anywhere, anytime, and towards the seamless wide area coverage. UAV Links and Channel Characteristics -------------------------------------- ### [Control Links]{} The control links are fundamental to guarantee the safe operation of all UAVs. These links must be low-latency, highly reliable, and secure two-way communications, usually with low data rate requirement for safety-critical information exchange among UAVs, and additionally between the UAV and ground control stations. The control links information flow can be classified into three types: 1) command and control from ground control stations to UAVs; 2) UAV status report from UAVs to ground; 3) sense-and-avoid information among UAVs. Also for autonomous UAV, which can fulfill missions depending on intelligent devices without real-time human control, the control links are important in case of emergency when human intervention is needed [@zeng2016wireless]. There are two frequency bands allocated for the control links, namely the L-band (960-977MHz) and the C-band (5030-5091MHz) [@matolak2015unmanned]. For delay reasons, we always prefer the primary control links between ground control stations to UAVs, but the secondary control links via satellite could also be utilized as a backup to enhance reliability and robustness. Another key necessity for the control links is the high security. Specifically, efficient security mechanisms should be utilized to avoid the so-called ghost control scenario, a possibly disastrous circumstance in which the UAVs are controlled by unapproved operators via spoofed control or navigation signals. Accordingly, practical authentication techniques, perhaps supplemented by the emerging physical layer security techniques, should be applied for control links. Compared to data links, the control links usually have lower tolerance in terms of security and latency requirements [@zeng2016wireless]. ### [Data Links]{} The purpose of using data links is to support task-related communications for the ground terminals which include ground base stations, mobile terminals, gateway nodes, wireless sensors, etc. The data links of UAVs need to provide the following communication services: 1) Direct mobile-UAV communication; 2) UAV-base station and UAV-gateway wireless backhaul; 3) UAV-UAV wireless backhaul. The capacity requirement of these data links depends on the communication services, ranging from low capacity (kbps) in UAV-sensor links to high speed (Gbps) in UAV-gateway wireless backhaul. The UAV data links could utilize the existing band assigned for the particular communication services to enhance performance, e.g., using millimeter wave band [@rappaport2014millimeter] and free space optics [@alzenad2016fso] for high capacity UAV-UAV wireless backhaul [@zeng2016wireless]. In Figure \[haz6\], we show the applications of UAV links. In Table \[tablehaz3\], we make a comparison between control and data links. ### [UAV Channel Characteristics]{} Both control and data channels in UAV communication networks consist of two primary types of channels, UAV-ground and UAV-UAV channels as shown in Figure \[haz7\]. These channels have several unique characteristics compared with the characteristics of terrestrial communication channels. While line of sight links are expected for UAV-ground channels in most cases, they could also be occasionally blocked by obstacles such as buildings. For low-altitude UAVs, the UAV-ground channels may also suffer a number of multipath components due to reflection, scattering, and diffraction by buildings, ground surface, etc. The UAV-UAV wireless channels are line of sight dominated and thus the effect of multipath is minimal compared to that experienced in UAV-ground or ground-ground channels. Due to the continuous movements of UAVs with different velocities, the UAV-to-UAV wireless channels will have high Doppler frequencies, especially the fixed wing UAVs. On one hand, we can utilize the dominance of the line of sight channels to achieve high-capacity for emerging mmWave communications. On the other hand, due to the continuous movements of UAVs with different velocities coupled with the higher carrier frequency in the mmWave band, the doppler shift will increase [@zeng2016wireless]. Path Loss Models ---------------- Path loss is an essential in the design and analysis of wireless communication channels and represents the amount of reduction in power density of a transmitted signal. The characteristics of aerial wireless channels are different than the terrestrial wireless channels due to the variations in the propagation environments and hence the path loss models for UAVs are also different than the traditional path loss models for terrestrial wireless channels. We classify the UAV path loss models based on environment, altitude and telecommunication link as shown in Figure \[haz1\] ### [Air-to-Ground Path Loss for Low Altitude Platforms]{} The authors in [@al2014modeling] present a statistical propagation model for predicting the path loss between a low altitude UAV and a Ground terminal. In this model, the authors assume that a UAV transmits data to more than 37000 ground receivers and use the Wireless InSite ray tracing software to model three types of rays (Direct, Reflected and Diffracted). Based on the simulation results, they divide the receivers to three groups. The first group corresponds to receivers that have Line-of-Sight or near-Line-of-Sight conditions. The second group corresponds to receivers with non Line-of-Sight condition, but still receiving coverage via strong reflection and refraction. The third group corresponds to receivers that have deep fading conditions resulting from consecutive reflections and diffractions, the third group only represents 3% of receivers. Therefore, based on the first and second groups, the authors present the path loss model as a function of the altitude $h$ and the coverage radius $R$ as shown in Figure \[haz2\] and it is given as follows: $$\begin{split} L(h,R)=P(LOS)\times L_{LOS}+P(NLOS)\times L_{NLOS} \end{split}$$ $$\begin{split} P(LOS)=\dfrac{1}{1+\alpha.exp(-\beta[\frac{180}{\pi}\theta -\alpha] )} \end{split}$$ $$\begin{split} L_{LOS}(dB)=20log(\dfrac{4\pi f_cd}{c})+\zeta_{LOS} \end{split}$$ $$\begin{split} L_{NLOS}(dB)=20log(\dfrac{4\pi f_cd}{c})+\zeta_{NLOS} \end{split}$$ where $P(LOS)$ is the probability of having line of sight (LOS) connection at an elevation angle of $\theta$, $P(NLOS)$ is the probability of having non LOS connection and it equals (1- $P(LOS)$), $L_{LOS}$ and $L_{NLOS}$ are the average path loss for LOS and NLOS paths. In equations (2), (3) and (4), $\alpha$ and $\beta$ are constant values which depend on the environment, $f_c$ is the carrier frequency, $d$ is the distance between the UAV and the ground user, $c$ is the speed of the light, $\zeta_{LOS}$ and $\zeta_{NLOS}$ are the average additional loss which depends on the environment. {width="\textwidth"} {width="\textwidth"} In this path loss model, the authors assume that all users are outdoor and the location of each user can be represented by an outdoor 2D point. These assumptions limit the applicability of this model when one needs to consider indoor users. The authors of [@mozaffari2015drone] describe the tradeoff in this model. At a low altitude, the path loss between the UAV and the ground user decreases, while the probability of line of sight links also decreases. On the other hand, at a high altitude line of sight connections exist with a high probability, while the path loss increases. ### [Outdoor-to-Indoor Path Loss Model]{} The Air-to-Ground path loss model presented in [@al2014modeling] is not appropriate when we consider wireless coverage for indoor users, because this model assumes that all users are outdoor and located at 2D points. In [@haz2017efficient], the authors adopt the Outdoor-Indoor path loss model, certified by the International Telecommunication Union (ITU) [@series2009guidelines] to provide wireless coverage for indoor users using UAV as shown in Figure \[haz3\]. The path loss is given as follows: $$\begin{split} L=L_F+L_B+L_I=(wlog_{10}d_{out}+wlog_{10}f+g_1)\\ +(g_2+g_3(1-cos\theta_i)^2)+(g_4d_{in}) \end{split}$$ where $L_F$ is the free space path loss, $L_B$ is the building penetration loss, and $L_I$ is the indoor loss. Also, $d_{out}$ is the distance between the UAV and indoor user, $\theta_i$ is the incident angle, and $d_{in}$ is the indoor distance of the user inside the building. In this model, we also have $w=20$, $g_1$=32.4, $g_2$=14, $g_3$=15, $g_4$=0.5 and $f$ is the carrier frequency. The authors of [@haz2017efficient] describe the tradeoff in the above model when the horizontal distance between the UAV and a user changes. When this horizontal distance increases, the free space path loss (i.e.,$L_F$) increases as $d_{out}$ increases, while the building penetration loss (i.e., $L_B$) decreases as the incident angle (i.e., $\theta_i$) decreases. ### [Cellular-to-UAV Path Loss Model]{} In [@al2017modeling], the authors model the statistical behavior of the path loss from a cellular base station towards a flying UAV. They present the path loss model based on extensive field experiments that involve collecting both terrestrial and aerial coverage samples. They report the value of the path loss as a function of the depression angle and the terrestrial coverage beneath the UAV as shown in Figure \[haz4\]. The path loss is given as follows: $$\begin{split} L(d,\theta)=L_{ter}(d)+\eta(\theta)+X_{uav}(\theta)=10\alpha log(d)+\\A(\theta -\theta_o)exp(-\dfrac{\theta -\theta_o}{B})+\eta_o+N(0,a\theta +\sigma_o) \end{split}$$ where $L_{ter}(d)$ is the mean terrestrial path-loss at a given terrestrial distance $d$ from the cellular base station, $\eta (\theta)$ is the excess aerial path-loss, $X_{uav}(\theta)$ is a Gaussian random variable with an angle-dependent standard deviation $\sigma_{uav}(\theta)$ representing the shadowing component, $\theta$ is depression angle between the UAV and the cellular base station, $\alpha$ is the path-loss exponent. Also, $A, B, \theta_o$ and $\eta_o$ are fitting parameters. ### [Air-to-Ground Path Loss for High Altitude Platforms]{} The authors in [@holis2008elevation] present an empirical propagation prediction model for mobile communications from high altitude platforms in built-up areas, where the frequency band is $2\textendash6$ GHz. The probability of LOS paths between the UAV and a ground user and the additional shadowing path loss for NLOS paths are presented as a function of the elevation angle $(\theta)$. The path loss model is defined for four different types of built-up area (Suburban area, Urban area, Dense urban area and Urban high-rise area). The path loss is given as follows: $$\begin{split} L=P(LOS)\times L_{LOS}+P(NLOS)\times L_{NLOS} \end{split}$$ $$\begin{split} P(LOS)=a-\dfrac{a-b}{1+(\frac{\theta-c}{d})^e} \end{split}$$ $$\begin{split} L_{LOS}(dB)=20log(d_{km})+20log(f_{GHz})+92.4+\zeta_{LOS} \end{split}$$ $$\begin{split} L_{NLOS}(dB)=20log(d_{km})+20log(f_{GHz})+92.4+L_s\\+\zeta_{NLOS} \end{split}$$ In equation (7), $P(LOS)$ is the probability of having line of sight (LOS) connection at an evaluation angle of $\theta$, $P(NLOS)$ is the probability of having non LOS connection and it equals (1- $P(LOS)$), $L_{LOS}$ and $L_{NLOS}$ are the average path loss for LOS and NLOS paths. In equation (8), $a, b, c, d$ and $e$ are the empirical parameters. In equations (9) and (10), $d_{km}$ is the distance between the UAV and the ground user in km, $f_{GHz}$ is the frequency in GHz, $L_s$ is a random shadowing in dB as a function of the elevation angle $(\theta)$, $\zeta_{LOS}$ and $\zeta_{NLOS}$ are the average additional losses which depend on the environment. ### [UAV to UAV path Loss Model]{} In general, the UAV-to-UAV wireless channels are line of sight dominated, so that the free space path loss can be adopted for the aerial channels. Due to the continuous movements of UAVs with different velocities, the UAV-to-UAV wireless channels will have high Doppler frequencies (especially the fixed wing UAVs). In Table \[tablehaz1\], we make a comparison among the path loss models based on operating frequency, altitude, environment, type of link, type of experiments and challenges. Path loss model Frequency band Altitude Environment Type of link Type of experiments Challenges -------------------------- ---------------- ---------- ---------------------- -------------- --------------------- ---------------------------------------------------------  [@al2014modeling] 2 GHz LAP Outdoor-ground users Downlink Simulations 1\) The authors didn’t consider the indoor users. 2\) The authors didn’t consider different 5G [   ]{} frequency bands.[                              ]{} 3\) The locations of outdoor users are 2D.[      ]{}  [@series2009guidelines] 2 GHz to 6 GHz LAP Indoor users Downlink Real experiments 1\) The authors didn’t consider the different [    ]{} types of the building structures.[          ]{} 2\) The authors didn’t consider different 5G [  ]{} frequency bands.[                            ]{}  [@al2017modeling] 850 MHz LAP Outdoor cellular Uplink Real experiments 1\) The authors didn’t investigate denser urban base station environments.[                             ]{} 2\) The authors didn’t consider different 5G [ ]{} frequency bands such as mmWave.[   ]{}  [@holis2008elevation] 2 GHz to 6 GHz HAP Outdoor-ground users Downlink Simulations 1\) The authors didn’t consider the indoor users[ ]{} 2\) The authors didn’t consider different 5G [  ]{} frequency bands.[                             ]{} 3\) The locations of outdoor users are 2D.[      ]{} Free space All frequency LAP, UAV to UAV Uplink, Real experiments 1\) High Doppler shift.[                            ]{} bands HAP channels downlink UAV Deployment Strategies ------------------------- UAVs deployment problem is gaining significant importance in UAV-based wireless communications where the performance of the aerial wireless network depends on the deployment strategy and the 3D placements of UAVs. In this section, we classify the UAV deployment strategies based on the objective functions as shown in Figure \[haz5\]. ![Cellular-to-UAV Path Loss Parameters.[]{data-label="haz4"}](srv_WC_f5.eps) ### [Deployment Strategies for Minimizing the Transmit Power of UAVs]{} The authors in [@mozaffari2016optimal] propose an efficient deployment framework for deploying the aerial base stations, where the goal is to minimize the total required transmit power of UAVs while satisfying the users rate requirements. They apply the optimal transport theory to obtain the optimal cell association and derive the optimal UAV’s locations using the facility location framework. The authors in [@mozaffari2015drone] investigate the downlink coverage performance of a UAV, where the objective is to find the optimal UAV altitude which leads to the maximum ground coverage and the minimum transmit power. The authors in [@haz2017efficient] propose using a single UAV to provide wireless coverage for indoor users inside a high-rise building under disaster situations. They study the problem of efficient UAV placement, where the objective is to minimize the total transmit power required to cover the entire high-rise building. In [@shakhatreh2017efficient], the authors propose a particle swarm optimization algorithm to find an efficient 3D placement of a UAV that minimizes the total transmit power required to cover the indoor users. The authors in [@MICC17] utilize UAVs to provide wireless coverage for indoor and outdoor users in massively crowded events, where the objective is to find the optimal UAV placement which lead to the minimum transmit power. In [@alzenad20173d], the authors propose an optimal placement algorithm for UAV that maximizes the number of covered users using the minimum transmit power. The algorithm decouple the UAV deployment problem in the vertical and horizontal dimensions without any loss of optimality. The authors in [@mozaffari2016unmanned] consider two types of users in the network: the downlink users served by the UAV and device-to-device users that communicate directly with one another. In the mobile UAV scenario, using the disk covering problem, the entire target geographical area can be completely covered by the UAV in a shortest time with a minimum required transmit power. They also derive the overall outage probability for device-to-device users, and show that the outage probability increases as the number of stop points that the UAV needs to completely cover the area increases. ### [Deployment Strategies for Maximizing the Wireless Coverage of UAVs]{} In [@bor2016efficient], the authors highlight the properties of the UAV placement problem, and formulate it as a 3D placement problem with the objective of maximizing the revenue of the network, which is proportional to the number of users covered by a UAV. They formulate an equivalent problem which can be solved efficiently to find the size of the coverage region and the altitude of a UAV. The authors in [@mozaffari2016efficient] study the optimal deployment of UAVs equipped with directional antennas, using circle packing theory. The 3D locations of the UAVs are determined in a way that the total coverage area is maximized. In [@kalantari2017backhaul], the authors introduce network-centric and user-centric approaches, the optimal 3D backhaul-aware placement of a UAV is found for each approach. In the network-centric approach, the network tries to serve as many users as possible, regardless of their rate requirements. In the user-centric approach, the users are determined based on the priority. The total number of served users and sum-rates are maximized in the network-centric and user-centric frameworks. The authors in [@alzenad20173da] study an efficient 3D UAV placement that maximizes the number of covered users with different Quality-of-Service requirements. They model the placement problem as a multiple circles placement problem and propose an optimal placement algorithm that utilizes an exhaustive search over a one-dimensional parameter in a closed region. They propose a low-complexity algorithm, maximal weighted area algorithm, to tackle the placement problem. In [@shakhatreh2017micc], the authors aim to maximize the indoor wireless coverage using UAVs equipped with directional antennas. They present two methods to place the UAVs; providing wireless coverage from one building side and from two building sides. The authors in [@shah2017distributed] utilize UAVs-hubs to provide connectivity to small-cell base stations with the core network. The goal is to find the best possible association of the small cell base stations with the UAVs-hubs such that the sum-rate of the overall system is maximized depending on a number of factors including maximum backhaul data rate of the link between the core network and mother-UAV-hub, maximum bandwidth of each UAV-hub available for small-cell base stations, maximum number of links that every UAV-hub can support and minimum signal-to-interference-plus-noise ratio. They present an efficient and distributed solution of the designed problem, which performs a greedy search to maximize the sum rate of the overall network. In [@mozaffari2017wireless], the authors propose an efficient framework for optimizing the performance of UAV-based wireless systems in terms of the average number of bits transmitted to users and UAVs’s hovering duration. They investigate two scenarios: UAV communication under hover time constraints and UAV communication under load constraints. In the first scenario, given the maximum possible hover time of each UAV, the total data service under user fairness considerations is maximized. They utilize the framework of optimal transport theory and prove that the cell partitioning problem is equivalent to a convex optimization problem. Then, they propose a gradient-based algorithm for optimally partitioning the geographical area based on the users’s distribution, hover times, and locations of the UAVs. In the second scenario, given the load requirement of each user at a given location, they minimize the average hover time needed for completely serving the ground users by exploiting the optimal transport theory. (b1) [Deployment strategies for minimizing the transmit power of UAVs ]{}; (b2) at (b1.east) [Deployment strategies for maximizing the wireless coverage of UAVs ]{}; (b3) at (b2.east) Deployment strategies for minimizing the number of UAVs required to perform task ; (b4) at (b3.east) Deployment strategies to collect data using UAVs ; (top) at ($(b2.north)!.5!(b3.north)$) ; (atop) at ($(top.south) + (0,-5pt)$); (btop) at ($(b3.south) + (0,-5pt)$); (top.south) – (atop) (b1.north) |- (atop) -| (b4.north) (b2.north) |- (atop) -| (b3.north); Reference Number of UAVs Type of environment Type of antenna Operating frequency Objective function Deployment strategy ------------------------------ ---------------- --------------------- ------------------ --------------------- ---------------------------------- --------------------------  [@mozaffari2016optimal] Multiple UAVs Outdoor Omni-directional 2 GHz Minimizing the transmit power Optimal transport theory of UAVs  [@mozaffari2015drone] Single UAV Outdoor Directional 2 GHz Minimizing the transmit power Closed-form expression of UAV for the UAV placement  [@haz2017efficient] Single UAV Indoor Omni-directional 2 GHz Minimizing the transmit power Gradient descent of UAV algorithm  [@shakhatreh2017efficient] Single UAV Indoor Omni-directional 2 GHz Minimizing the transmit power Particle swarm of UAV optimization  [@MICC17] Single UAV Outdoor-Indoor Omni-directional 2 GHz Minimizing the transmit power Particle swarm of UAV optimization, K-means with ternary search algorithms  [@alzenad20173d] Single UAV Outdoor Directional 2 GHz Minimizing the transmit power Optimal 3D placement of UAV algorithm  [@mozaffari2016unmanned] Single UAV Outdoor Directional 2 GHz Minimizing the transmit power Disk covering problem of UAV  [@bor2016efficient] Single UAV Outdoor Directional 2.5 GHz Maximizing the wireless coverage Bisection search of UAV algorithm  [@mozaffari2016efficient] Multiple UAVs Outdoor Directional 2 GHz Maximizing the wireless coverage Circle packing of UAVs theory  [@kalantari2017backhaul] Single UAV Outdoor Directional 2 GHz Maximizing the wireless coverage branch and of UAV bound algorithm  [@alzenad20173da] Single UAV Outdoor Directional 2 GHz Maximizing the wireless coverage exhaustive search of UAV algorithm  [@shakhatreh2017micc] Multiple UAVs Indoor Directional 2 GHz Maximizing the wireless coverage efficient algorithm of UAVs  [@shah2017distributed] Multiple UAVs Outdoor Directional 2 GHz Maximizing the wireless coverage branch and of UAVs bound algorithm  [@mozaffari2017wireless] Multiple UAVs Outdoor Directional 2 GHz Maximizing the wireless coverage Optimal transport of UAVs theory, Gradient algorithm  [@kalantari2016number] Multiple UAVs Outdoor Directional 2 GHz Minimizing the number Particle swarm of UAVs optimization  [@shakhatreh2016continuous] Multiple UAVs Outdoor Directional 2 GHz Minimizing the number The cycles with limited of UAVs energy algorithm  [@shakhatreh2017indoor] Multiple UAVs Indoor Omni-directional 2 GHz Minimizing the number Particle swarm of UAVs optimization, K-means algorithms  [@zhu2014using] Multiple UAVs Outdoor Directional 2 GHz Minimizing the number polynomial time of UAVs approximate algorithm  [@lyu2017placement] Multiple UAVs Outdoor Directional 2 GHz Minimizing the number Spiral UAVs of UAVs placement algorithm  [@mozaffari2016mobile] Multiple UAVs Outdoor Directional 2 GHz Collecting data using Optimal transport UAVs theory  [@yang2017energy] Single UAV Outdoor Directional 2 GHz Collecting data using Two practical UAV UAV trajectories: circular and straight flights  [@alejo2015efficient] Multiple UAVs Outdoor Directional 2 GHz Collecting data using Genetics UAVs algorithm  [@wang2015efficient] Multiple UAVs Outdoor directional 2 GHz Collecting data using efficient algorithm UAVs for path planning  [@zhan2017energy] Single UAV Outdoor Directional 2 GHz Collecting data using efficient iterative UAVs algorithm ### [Deployment Strategies for Minimizing the Number of UAVs Required to Perform Task]{} The authors in [@kalantari2016number] propose a method to find the placements of UAVs in an area with different user densities using the particle swarm optimization. The goal is to find the minimum number of UAVs and their 3D placements so that all the users are served. In [@shakhatreh2016continuous], the authors study the problem of minimizing the number of UAVs required for a continuous coverage of a given area, given the recharging requirement. They prove that this problem is NP-complete. Due to its intractability, they study partitioning the coverage graph into cycles that start at the charging station. Based on this analysis, they then develop an efficient algorithm, the cycles with limited energy algorithm, that minimizes the number of UAVs required for a continuous coverage. The authors in [@shakhatreh2017indoor] study the problem of minimizing the number of UAVs required to provide wireless coverage to indoor users and prove that this problem is NP-complete. Due to the intractability of the problem, they use clustering to minimize the number of UAVs required to cover the indoor users. They assume that each cluster will be covered by only one UAV and apply the particle swarm optimization to find the UAV 3D location and UAV transmit power needed to cover each cluster. In [@zhu2014using], the authors study the problem of deploying minimum number of UAVs to maintain the connectivity of ground MANETs under the condition that some UAVs have already been deployed in the field. They formulate this problem as a minimum steiner tree problem with existing mobile steiner points under edge length bound constraints and prove that the problem is NP-Complete. They propose an existing UAVs aware polynomial time approximate algorithm to solve the problem that uses a maximum match heuristic to compute new positions for existing UAVs. The authors in [@lyu2017placement] aim to minimize the number of UAVs required to provide wireless coverage for a group of distributed ground terminals, ensuring that each ground terminal is within the communication range of at least one UAV. They propose a polynomial-time algorithm with successive UAV placement, where the UAVs are placed sequentially starting from the area perimeter of the uncovered ground terminals along a spiral path towards the center, until all ground terminals are covered. ![Interference Mitigation Techniques and Challenges.[]{data-label="haz8"}](srv_WC_f6.png) ### [Deployment Strategies to Collect Data Using UAVs]{} The authors in [@mozaffari2016mobile] propose an efficient framework for deploying and moving UAVs to collect data from ground Internet of Things devices. They minimize the total transmit power of the devices by properly clustering the devices where each cluster being served by one UAV. The optimal trajectories of the UAVs are determined by exploiting the framework of optimal transport theory. In [@yang2017energy], the authors present a UAV enabled data collection system, where a UAV is dispatched to collect a given amount of data from ground terminals at fixed location. They aim to find the optimal ground terminal transmit power and UAV trajectory that achieve different Pareto optimal energy trade-offs between the ground terminal and the UAV. The authors in [@alejo2015efficient] study the problem of trajectory planning for wireless sensor network data collecting deployed in remote areas with a cooperative system of UAVs. The missions are given by a set of ground points which define wireless sensor network gathering zones and each UAV should pass through them to gather the data while avoiding passing over forbidden areas and collisions between UAVs. The proposed UAV trajectory planners are based on Genetics Algorithm, Rapidly-exploring Random Trees and Optimal Rapidly-exploring Random Trees. The authors in [@wang2015efficient] design a basic framework for aerial data collection, which includes the following five components: deployment of networks, nodes positioning, anchor points searching, fast path planning for UAV, and data collection from network. They identify the key challenges in each component and propose efficient solutions. They propose a Fast Path Planning with Rules algorithm based on grid division, to increase the efficiency of path planning, while guaranteeing the length of the path to be relatively short. In [@zhan2017energy], the authors jointly optimize the sensor nodes wake-up schedule and UAV’s trajectory to minimize the maximum energy consumption of all sensor nodes, while ensuring that the required amount of data is collected reliably from each sensor node. They formulate a mixed-integer non-convex optimization problem and apply the successive convex optimization technique, an efficient iterative algorithm is proposed to find a sub-optimal solution. UAVs can be classified into two types: fixed wing and rotary wing, each with its own strengths and weaknesses. Fixed-wing UAV usually has high speed and payload, but they need to maintain a continuous forward motion to remain aloft, thus are not suitable for stationary uses. In contrast, rotary-wing UAV such as quadcopter, usually has limited mobility and payload, but they are able to move in any direction as well as to stay stationary in the air. Thus, the choice of UAVs critically depends on the uses [@zeng2016wireless]. In Table \[tablehaz2\], we make a comparison among the research papers related to UAV deployment strategies based on the objective functions. Interference Mitigation ----------------------- One of the techniques to mitigate interference is the coordinated multipoint technique [@maattanen2012system]. In downlink coordinated multipoint technique, the transmission aerial base stations cooperate in scheduling and transmission in order to strength the received signal and mitigate inter-cell interference [@maattanen2012system]. On the other hand, the physical uplink shared channel (PUSCH) is received at aerial base stations in uplink coordinated multipoint technique. The scheduling decision is based on the coordination among UAVs [@sawahashi2010coordinated]. The new challenge here is that UAVs receive interfering signals from more ground terminals in the downlink and their uplink transmitted signals are visible to more ground terminals due to the high probability of line of sight links. Therefore, the coordinated multipoint techniques must be applied across a larger set of cells to mitigate the interference and hence the coordination complexity will increase [@lin2017sky]. We can also mitigate interference by utilizing receiver techniques such as interference rejection combining and network-assisted interference cancellation and suppression. Compared to smart phones, we can equip UAVs with more antennas, which can be used to mitigate interference from more ground base stations. With MIMO antennas, UAVs can use beamforming to enables directional signal transmission or reception to achieve spatial selectivity which is also an efficient interference mitigation technique [@lin2017sky]. Another interference mitigation technique is to partition radio resources so that ground traffic and aerial traffic are served with orthogonal radio resources. This simple technique may not be efficient since the reserved radio resources for aerial traffic may be not fully utilized. Therefore, UAV operators need to provide more information about the trajectories and 3D placements of UAVs to construct more dynamic radio resource management [@lin2017sky]. A powerful interference mitigation technique is the power control technique in which an optimized setting of power control parameters can be applied to reduce the interference generated by UAVs. This technique can minimize interference, increase spectral efficiency, and benefit UAVs as well as ground terminals [@lin2017sky]. Dedicated cells for the UAVs is another option to mitigate interference, where the directional antennas are pointed towards the sky instead of down-tilted. These dedicated cells will be a practical solution especially in UAV hotspots where frequent and dense UAV takeoffs and landings occur [@lin2017sky]. In Figure \[haz8\], we summarize the interference mitigation techniques and challenges. Category Learning techniques Key characteristics Application --------------- --------------------------------------- ----------------------------------------------- -------------------------------------------------------------- Regression models - Estimate the variables’ relationships      - Energy learning [@donohoo2014context]                - Linear and logistics regression            K-nearest neighbor - Majority vote of neighbors                - Energy learning [@donohoo2014context]                 Supervised Support vector machines - Non-linear mapping to high dimension - MIMO channel learning [@feng2012determination]       learning - Separate hyperplane classification        Bayesian learning - A posteriori distribution calculation      - Massive MIMO learning [@wen2015channel]      - Gaussians mixture, expectation max     - Cognitive spectrum learning       and hidden Markov models              [@choi2013estimation; @assra2016approach; @yu2010cognitive] Unsupervised K-means clustering - K partition clustering                      - Heterogeneous networks [@xia2012optical]      - Iterative updating algorithm               learning Independent component analysis - Reveal hidden independent factors      - Spectrum learning in [@nguyen2013binary]        cognitive radio                    Markov decision processes/A partially - Bellman equation maximization         - Energy harvesting [@aprem2013transmit]             observable Markov decision process - Value iteration algorithm                 Reinforcement Q-learning - Unknown system transition model      - Femto and small cells             learning - Q-function maximization                 [@alnwaimi2015dynamic]-[@onireti2016cell] Multi-armed bandit - Exploration vs. exploitation             - Device-to-device networks [@maghsudi2015channel]   - Multi-armed bandit game               Research Trends and Future Insights ----------------------------------- ### [Cloud and Big Data]{} A cloud for UAVs contains data storage and high-performance computing, combined with big data analysis tools [@bor2016new]. It can provide an economic and efficient use of centralized resources for decision making and network-wide monitoring [@bor2016new; @bradai2015cellular; @zhou2014toward]. If UAVs are utilized by a traditional cellular network operator (CNO), the cloud is just the data center of the CNO (similar to a private cloud), where the CNO can choose to share its knowledge with some other CNOs or utilize it for its own business uses. On the other hand, if the UAVs are utilized by an infrastructure provider, the infrastructure provider can utilize the cloud to gather information from mobile virtual network operators and service providers. Under such scenario, it is important to guarantee security, privacy, and latency. To better exploit the benefit of the cloud, we can use a programmable network allowing dynamic updates based on big data processing, for which network functions virtualization and software defined networking can be research trends  [@bor2016new]. ### [Machine Learning]{} Next-generation aerial base stations are expected to learn the diverse characteristics of users’ behavior, in order to autonomously find the optimal system settings. These intelligent UAVs have to use sophisticated learning and decision-making, one promising solution is to utilize machine learning. Machine learning algorithms can be simply classified as supervised, unsupervised and reinforcement learning as shown in Table \[table4\]. The family of supervised learning algorithms utilizes known models and labels to enable the estimation of unknown parameters. They can be used for spectrum sensing and white space detection in cognitive radio, massive MIMO channel estimation and data detection, as well as for adaptive filtering in signal processing for 5G communications. They can also be utilized in higher-layer applications, such as estimating the mobile users’ locations and behaviors, which can help the UAV operators to enhance the quality of their services. The family of unsupervised learning algorithms utilizes the input data itself in a heuristic manner. They can be used for heterogeneous base station clustering in wireless networks, for access point association in ubiquitous WiFi networks, for cell clustering in cooperative ultra-dense small-cell networks, and for load-balancing in heterogeneous networks. They can also be utilized in fault/intrusion detections and for the users’ behavior-classification. The family of reinforcement learning algorithms utilizes dynamic iterative learning and decision-making process. They can be used for estimating the mobile users’ decision making under unknown scenarios, such as channel access under unknown channel availability conditions in spectrum sharing, base station association under the unknown energy status of the base stations in energy harvesting networks, and distributed resource allocation under unknown resource quality conditions in femto/small-cell networks [@jiang2017machine]. ### [Network Functions Virtualization]{} Network functions virtualization (NFV) reduces the need of deploying specific network devices for the integration of UAVs [@bor2016new; @bradai2015cellular]. NFV allows a programmable network structure by virtualizing the network functions on storage devices, servers, and switches, which is practical for UAVs requiring seamless integration to the existing network. Moreover, virtualization of UAVs as shared resources among cellular virtual network operators can decrease OPEX for each party [@liang2015wireless]. Here, the SDN can be useful for the complicated control and interconnection of virtual network functions (VNFs) [@bor2016new; @bradai2015cellular]. ### [Software Defined Networking]{} For mobile networks, a centralized SDN controller can make a more efficient allocation of radio resources, which is particularly important to exploit UAVs [@bor2016new; @bradai2015cellular]. For instance, SDN-based load balancing can be useful for multi-tier UAV networks, such that the load of each aerial base station and terrestrial base station is optimized precisely. A SDN controller can also update routing such that part of traffic from the UAVs is carried through the network without any network congestions [@bor2016new; @bradai2015cellular; @zhou2014toward]. For further exploitation of the new degree of freedom provided by the mobility of UAVs, the 3D placements of UAVs can be adjusted to optimize paging and polling, and location management parameters can be updated dynamically via the unified protocols of SDN [@bor2016new]. ### [Millimeter-Wave]{} Millimeter-wave (mmWave) technology can be utilized to provide high data rate wireless communications for UAV networks [@xiao2016enabling]. The main difference between utilizing mmWave in UAV aerial networks and utilizing mmWave in terrestrial cellular networks is that a UAV aerial base station may move around. Hence, the challenges of mmWave in terrestrial cellular networks apply to the mmWave UAV cellular network as well, including rapid channel variation, blockage, range and directional communications, multi-user access, and others [@xiao2016enabling; @rangan2014millimeter]. Compared to mmWave communications for static stations, the time constraint for beamforming training is more critical due to UAV mobility. For fast beamforming training and tracking in mmWave UAV cellular networks, the hierarchical beamforming codebook is able to create highly directional beam patterns, and achieves excellent beam detection performance [@xiao2016enabling]. Although the UAV wireless channels have high Doppler frequencies due to the continuous movements of UAVs with different velocities, the major multipath components are only affected by slow variations due to high gain directional transmissions. In beam division multiple access (BDMA), multiple users with different beams may access the channel at the same time due to the highly directional transmissions of mmWave [@xiao2016enabling; @rangan2014millimeter; @sun2015beam]. This technique improves the capacity significantly, due to the large bandwidth of mmWave technology and the use of BDMA in the spatial domain. The blockage problem can be mitigated by utilizing intelligent cruising algorithms that enable UAVs to fly out of a blockage zone and enhance the probability of line of sight links [@xiao2016enabling]. ### [Free Space Optical]{} Free space optical (FSO) technology can be used to provide wireless connectivity to remote places by utilizing UAVs, where physical access to 3G or 4G network is either minimal or never present [@kaushal2017optical]. It can be involved in the integration of ground and aerial networks with the help of UAVs by providing last mile wireless coverage to sensitive areas (e.g., battlefields, disaster relief, etc.) where high bandwidth and accessibility are required. For instance, Facebook will provide wireless connectivity via FSO links to remote areas by utilizing solar-powered high altitude UAVs [@kaushal2017optical; @adweek; @mobileeurope]. For areas where deployment of UAVs is impractical or uneconomical, geostationary earth orbit and low earth orbit satellites can be utilized to provide wireless connectivity to the ground users using the FSO links. The main challenge facing UAV-free space optics technology is the high blockage probability of the vertical FSO link due to weather conditions. The authors in [@alzenad2016fso] propose some methods to tackle this problem. The first method is to use an adaptive algorithm that controls the transmit power according to weather conditions and it may also adjust other system parameters such as incident angle of the FSO transceiver to compensate the link degradation, e.g., under bad weather conditions, high power vertical FSO beams should be used while low power beams could be used under good weather conditions. The second method is to use a system optimization algorithm that can optimize the UAV placement, e.g., hovering below clouds over negligible turbulence geographical area. ### Future Insights {#future-insights-5} Some of the future possible directions for this research are: - The majority of literature focuses on providing the wireless coverage for outdoor ground users, although 90% of the time people are indoor and 80% of the mobile Internet access traffic also happens indoors [@ericsson; @alcatel; @cisco]. Therefore, it is important to study the indoor wireless coverage problems by utilizing UAVs. - The majority of literature does not consider the limited energy capacity of UAV as a constraint when they study the wireless coverage of UAVs, where the energy consumption during data transmission and reception is much smaller than the energy consumption during the UAV hovering. It only constitutes 10%-20% of the UAV energy capacity [@gupta2016survey]. - Some path loss models are based on simulations softwares such as Air-to-Ground path loss for low altitude platforms and Air-to-Ground path loss for high altitude platforms, therefore it is necessary to do real experiments to model the statistical behavior of the path loss. - To the best of our knowledge, no studies present the path loss models for uplink scenario and mmWave bands. - There are challenges facing the new technologies such as mmwave. The challenges facing UAV-mmWave technology are fast beamforming training and tracking requirement, rapid channel variation, directional and range communications, blockage and multi-user access [@xiao2016enabling]. - We need more studies about the topology of UAV wireless networks where this topology remains fluid during: a) Changing the number of UAVs; b) Changing the number of channels; c) The relative placements of the UAVs altering [@gupta2016survey]. - There is a need to study UAV routing protocols where it is difficult to construct a simple implementation for proactive or reactive schemes [@gupta2016survey]. - There is a need for a seamless mechanism to transfer the users during the handoff process between UAVs, when out of service UAVs are replaced by active ones [@gupta2016survey]. - Further research is required to determine which technology is right for UAV applications, where the common technologies utilized in UAV applications are the IEEE 802.11 due to wide availability in current wireless devices and their appropriateness for small-scale UAVs [@hayat2016survey]. - D2D communications is an efficient technique to improve the capacity in ground communications systems [@asadi2014survey]. An important problem for future research is the joint optimization of the UAV path planning, coding, node clustering, as well as D2D file sharing [@zeng2016wireless]. - Utilizing UAVs in public safety communications needs further research, where topology formation, cooperation between UAVs in a multi-UAV network, energy constraints, and mobility model are the challenges that are facing UAVs in public safety communications [@kumbhar2017survey]. - In UAVs-based IoT services, it is difficult to control and manage a high number of UAVs. The reason is that each UAV may host more than one IoT device, such as different types of cameras and aerial sensors. Moreover, conflict of interest between different devices likely to happen many times, e.g., taking two videos from two different angles from one fixed placement. For future research, we need to propose efficient algorithms to solve the conflict of interest among Internet of things devices on-board [@motlagh2016low]. - For future research, it is important to propose efficient techniques that manage and control the power consumption of Internet of things devices on-board [@motlagh2016low]. - The security is one of the most critical issues to think about in UAVs-based Internet of things services, methods to avoid the aerial jammer on the communications are needed [@motlagh2016low]. PART III: Key Challenges and Conclusion {#part-iii-key-challenges-and-conclusion .unnumbered} ======================================= Key Challenges ============== Charging Challenges ------------------- UAV missions necessitate an effective energy management for battery-powered UAVs. Reliable, continuous, and smart management can help UAVs to achieve their missions and prevent loss and damage. The UAV’s battery capacity is a key factor for enabling persistent missions. But as the battery capacity increases, its weight increases, which cause the UAV to consume more energy for a given mission. The main directions in the literature to mitigate the limitations in UAV’s batteries are: (1) UAV battery management, (2) Wireless charging for UAVs, (3) Solar powered UAVs, and (4) Machine learning and communications techniques. ### Battery Management Battery management research in UAVs includes planning, scheduling, and replacement of battery so UAVs can accomplish their flight missions. This has been studied in the literature from different perspectives. Saha et al. in [@Saha2011] build a model to predict the end of battery charge for UAVs based on particle filter algorithm. They utilize a discharge curve for UAV Li-Po battery to tune the particle filter. They have shown that the depletion of the battery is not only related to the initial Start of Charge (SOC) but also, load profile and battery health conditions can be crucial factors. Park et al. in [@Park2017] investigate the battery assignment and scheduling for UAVs used in delivery business. Their objective is to minimize the deterioration in battery health. After splitting the problem into two parts; one for battery assignment and the other for battery scheduling. Heuristic algorithm and integer linear programming are used to solve the assignment and scheduling problems, respectively. The idle time between two successive charging cycles and the depth of the discharge are the main factors that control these algorithms. The use of UAVs in long time and enduring missions makes UAV battery swapping solutions necessary to accomplish these missions. An autonomous battery swapping system for UAVs is first introduced in [@Swieringa2010]. The battery swapping consists of landing platform, battery charger, battery storage compartment, and micro-controller. Similar concept to this autonomous battery swapping system was adopted and improved by different researchers in [@Michini2011; @Suzuki2012; @Lee2015; @Ure2015]. The hot swap term is adopted to refer to the continuous powering for the UAV during battery swapping. Figure \[eyad4\] shows an illustration of hot swapping systems. First, the UAV is connected to an external power supply during the swapping process. This will prevent data loss during swapping as it usually happens in cold swapping. Second, the drained battery is removed and stored in multi-battery compartment and charging station to be recharged. Third, a charged battery is installed in the UAV, and finally, the external power supply is disconnected. Another important aspect of the improvements presented in [@Michini2011; @Suzuki2012; @Lee2015; @Ure2015] is the several designs for the landing platform, which is an important part of the swapping system because it can compensate the error in the UAV positioning on the landing point. Table \[eyadt3\] shows the classification of autonomous battery swapping systems. It can be seen from this table that the swapping time for most of these systems is around one minute. This is a short period comparing to the battery charging average time, which is between 45-60 minutes. ![Illustration of Battery Hot Swapping System.[]{data-label="eyad4"}](srv_key-ch_f1){width="6cm" height="7cm"} **Ref \# /Year** **UAV Type** **Battery Type** **[Positioning System]{}** **Swapping time** **Hot vs Cold Swapping** --------------------------- --------------- ------------------ ------------------------------------------------ ------------------- -------------------------- [[@Swieringa2010]/2010]{} [Quadrotor]{} [Li-Po]{} [V-Shape Basket]{} [2 min.]{} [Cold swap]{} [[@Michini2011]/2011]{} [Quadrotor]{} [Li-Po]{} [Sloped landing pad]{} [11.8±3 sec.]{} [Hot swaps]{} [[@Suzuki2012]/2012]{} [Quadrotor]{} [Li-Po]{} [Dynamic forced positioning (Arm Actuation)]{} [47-60 sec.]{} [Cold swap]{} [[@Ure2015]/2015]{} [Quadrotor]{} [Li-Po]{} [Sloped landing pad]{} [60-70 sec.]{} [Hot swaps]{} [[@Lee2015]/2015]{} [Quadrotor]{} [Li-Po]{} [Rack with landing guide]{} [60 sec.]{} [Hot swaps]{} ### Wireless Charging For UAV Simic et. Al in [@Simic2015] study the feasibility of recharging the UAV from power lines while inspecting these power lines. Experimental tests show the possibility of energy harvesting from power lines. The authors design a circular antenna to harvest the required energy from power lines. Wang et al. in [@Wang2016] propose an automatic charging system for the UAV. The system uses charging stations allocated along the path of UAV mission. Each charging station consists of wireless charging pad, solar panel, battery, and power converter. All these components are mounted on a pole. The UAV employs GPS module and wireless network module to navigate to the autonomous charging station. When the power level in the UAV’s battery drops to a predefined level. The UAV will communicate with the central control room, which will direct the UAV to the closest charging station. The GPS module will help the UAV to navigate to the assigned station. Different wireless power transfer technologies were used in literature to implement the automatic charging stations. In [@Junaid2016; @Choi2017; @Aldhaher2017], the authors utilize the magnetic resonance coupling technology to implement an automatic drone charging station. Dunbar et al. in [@Dunbar2015] use RF Far field Wireless Power Transfer (WPT) to recharge micro UAV after landing. A flexible rectenna was mounted on the body of the UAV to receive RF signals from power transmitter. Mostafa et al. in [@Mostafa2017] develop a WPT charging system based on the capacitive coupling technology. One of the major challenges that face the deployment of autonomous wireless charging stations is the precise landing of the UAV on the charging pad. The imprecise landing will lead to imperfect alignment between the power transmitter and the power receiver, which means less efficiency and long charging time. Table \[eyadt4\] shows precise methodologies adopted in the literature. In [@Wang2016], the authors use GPS to land the UAV on the charging pad. In [@Junaid2017], the authors corporate the GPS system with camera and image processing system to detect the charging station. The authors in [@Choi2017] allow for imprecise landing on wide frame landing pad. Then a wireless charging transmitting coil is stationed perfectly in the bottom of the UAV by using a stepper motor and two laser distance sensors. [Ref \# /Year]{} [Precise Landing Methodology]{} ------------------------ -------------------------------------------------- [[@Wang2016]/2016]{} [GPS system]{} [[@Choi2017]/2017]{} [Landing frame with moving transmitting coil]{} [[@Junaid2017]/2017]{} [GPS system and vision-based target detection]{} : []{data-label="eyadt4"} ### Solar Powered UAVs For long endurance and high altitude flights, solar-powered UAVs (SUAVs) can be a great choice. These UAVs use solar power as a primary source for propulsion and battery as a secondary source to be used during night and sun absence conditions. The literature in flight endurance and persistent missions for SUAVs can be classified into two main directions: path planning and optimization and hybrid Models. Path planning and optimization includes: (i) gravitational potential energy planning. This scheme optimizes the path of the UAV over three stages, ascending and battery charging stage during the day using solar power, descending stage during the night using gravitational potential energy, and level flight stage using battery [@Hosseini2016; @Lee2017; @Gao2013], (ii) optimal path planing while tracking ground object [@Huang2016; @Spangelo2013], (iii) optimal path planning based on UAV kinematics and its energy loss model [@Klesh2007], and (iv) optimal path planning utilizing wind and metrological data [@Chakrabarty2011; @Oettershagen2017]. On the other hand, Hybrid models include: (i) UAVs with hybrid power sources such as solar, battery, and fuel cells [@Lee2014] and (ii) UAVs with a hybrid propulsion system that can be transformed from fixed wing to quadcopter UAV [@DSa2016; @DSa2017]. ### Machine Learning and Communications Techniques Communication equipment and their status (transmit, receive, sleep, or idle) can affect drone flight time. Utilizing the state-of-the-art machine learning techniques can result in smarter energy management. These technologies and their role in UAV persistent missions are covered in the literature under the following themes: - Energy efficient UAV networks: This area covers maximizing energy efficiency in the network layer, data link layer, physical layer, and cross-layer protocols in a bid to help UAVs to perform long missions. Gupta et al. in [@Gupta2016] list and compare several algorithms used in each layer. - Energy-based UAV fleet management: This includes modifying the mobility model of UAV fleet to incorporate energy as decision criterion as in [@Messous2016] to determine the next move of each UAV inside a fleet. - Machine learning: This area covers research that utilizes machine learning techniques for path planning and optimization applications, taking into consideration energy limitations. Zhang et al. in [@Zhang2017] use deep reinforcement learning to determine the fastest path to a charging station. While Choi et al. in [@Choi2017a] use a density-based spatial clustering algorithm to build a two-layer obstacle collision avoidance system. Several challenges still exist in the area of battery recharging and need to be appropriately addressed, including: - Advancements in wireless charging techniques such as magnetic resonant coupling and inductive coupling techniques are popular and suitable to be used because both of them have acceptable efficiency for small to mid-range distances. The advancements in this technology are expected to have a positive impact on the use of UAVs. - Development of battery technologies will allow UAVs to extend flight ranges. While Lithium-Ion batteries still dominant, PEM Fuel cells can be more convenient in UAVs because of their higher power density. - Motion planning of UAVs is an extremely important factor since it affects the wireless charging efficiency and the average power delivered . - The vast majority of the research, concerning UAV power management, focuses on multi-rotor UAVs. Other types of UAVs, such as fixed-wing UAVs, need more attention in future research. - Artificial intelligence, especially deep learning, can promote more advances in the field of UAV power management. Many research frontiers still need to be explored in this field such as; Deep reinforcement learning in the path planning and battery scheduling, convolutional neural networks in identifying charge stations and precise landing in a charging station, and recurrent neural networks in developing discharge models and precisely predicting the end of the charge. - Image processing techniques and smart sensors can also play an important role in identifying charging stations and facilitating precise positioning of UAVs on them. Collision Avoidance and Swarming Challenges ------------------------------------------- One of the challenges facing UAVs is collision avoidance. UAV can collide with obstacles, which can either be moving or stationary objects in either indoor or outdoor environment. During UAV flights, it is important to avoid accidents with these obstacles. Therefore, the development of UAV collision avoidance techniques has gained research interest [@pham2015survey; @lacher2007unmanned]. In this section, we first present the major categories of collision avoidance methods. Then, the challenges associated with UAVs collision avoidance are presented. Moreover, we provide research trends and some future insights. ### Collision Avoidance Approaches Many collision avoidance approaches have been proposed in order to avoid potential collisions by UAVs. The authors in [@pham2015survey] presented major categories of collision avoidance methods. These methods can be summarized as geometric approaches [@park2008uav], path planning approaches [@alexopoulos2013comparative], potential field approaches [@mujumdar2009nonlinear; @khatib1986real], and vision-based approaches [@saunders2009obstacle]. Several collision avoidance techniques can be utilized for an indoor environment such as vision based methods (using cameras and optical sensors), on-board sensors based methods (using IR, ultrasonic, laser scanner, etc.) and vision based combined with sensor based methods [@luo2013uav]. The collision avoidance approaches are discussed in the following: - Geometric Approach is a method that utilizes the geometric analysis to avoid the collision. In [@park2008uav], the geometric approach was utilized to ensure that the predefined minimum separation distance was not violated. This was done by computing the distance between two UAVs and the time required for the collision to occur. Moreover, it was used for path planning to avoid UAV collisions with obstacles. Several different approaches utilize this method to avoid collision such as Point of Closest Approach [@park2008uav], Collision Cone Approach [@chakravarthy1998obstacle; @mujumdar2009nonlinear], and Dubins Paths Approach [@dubins1957curves; @shanmugavel2010co]. - Path Planning Approach, which is also referred to as the optimized trajectory approach [@jun2003path], is a grid based method that utilizes the path re-planning algorithm with graph search algorithm to find a collision free trajectory during the flight. It uses geometric techniques to find an efficient and collision free trajectory, so it shares some similarities with the geometric approach. This method divides the map into a grid and represents the grid as a weighted graph [@alexopoulos2013comparative]. The grid with graph search algorithm are used to find collision free trajectory towards the desired target. - Potential Field Approach is proposed by [@khatib1986real] to be used as a collision avoidance method for ground robots. It has also been utilized for collision avoidance among UAVs and obstacles. This method uses the repulsive force fields which cause the UAV to be repelled by obstacles. The potential function is divided into attracting force field which pulls the UAV towards the goal, and repulsive force field which is assigned with the obstacle[@mujumdar2009nonlinear]. - Vision-based obstacle detection approaches utilize images from small cameras mounted on UAVs to tackle collision challenges. Advances in integrated circuits technology have enabled the design of small and low power sensors and cameras. Combined with advances in computer vision methods, such camera can be used in effective obstacle detection and collision avoidance [@saunders2009obstacle]. Moreover, this approach can be used efficiently for collision avoidance in the indoor environment. Such cameras provide real-time information about walls and obstacles in this environment. Many researchers utilized this method to avoid a collision and to provide fully autonomous UAV flights [@schmid2013stereo; @alvarez2016collision; @schmid2014autonomous; @mustafah2012indoor]. The researchers in [@alvarez2016collision] use a monocular camera with forward facing to generate collision-free trajectory. In this method, all computations were performed off-board the UAV. To solve the off-board image processing problem, the authors in [@roelofsen2015reciprocal] proposed a collision avoidance system for UAVs with visual-based detection. The image processing operation was performed using two cameras and a small on-board computation unit. ### Challenges There are several challenges in the area of collision avoidance approaches and need to be appropriately addressed, including: - The geometric approach utilizes the information such as location and velocity, which can be obtained using Automatic Dependent Surveillance Broadcast (ADS-B) sensing method. Thus, it is not applicable to non-aircraft obstacles. Furthermore, input data from ADS-B is sensitive to noise which hinders the exact calculation requirement of this approach. Moreover, ADS-B requires cooperation from another aircraft, which can be an intruder, which is referred to as cooperative sensing. - In non-cooperative sensing, the geometric approach requires UAVs to sense and extract information about the environment and obstacles such as position, speed and size of the obstacles. One of the possible solutions is to combine with a vision-based approach that uses a passive device to detect obstacles. However, this approach requires significant data processing and only can be used when the objects are close enough. Therefore, hardware limitation for on-board processing should be taken into consideration. - The diverse robotic control problem that allows UAV to perform complex maneuvers without collision can be solved using the standard control theory. However, each solution is limited to a specific case and not able to adapt to changes in the environments. This limitation can be overcome by learning from experience. This approach can be achieved using deep learning technique. More specifically, deep learning allows inferring complex behaviors from raw observation data. However, this approach has issues with samples efficiency. For real time application, deep learning requires the usage of on-board GPUs. In [@carrio2017review], a review of deep learning methods and applications for UAVs were presented. - In the context of multi-UAV systems, the collision avoidance among UAVs is a very complicated task, a technique known as formation control can be used for obstacle avoidance. More specifically, cooperative formation control algorithms were developed as a collision-avoidance strategy for the multiple UAVs [@6858777; @7331006]. A recent advancement in this area employs Model Predictive Controllers (MPC) in order to reduce computation time for the optimization of UAV’s trajectory. - The limited available power and payload of UAVs are challenging issues, which restrict on-board sensors requirements such as sensor weight, size and required power. These sensors such as IR, Ultrasonic, laser scanner, LADAR and RADAR are typically heavy and large for sUAV [@griffiths2007obstacle]. - The high speed of UAVs is another challenge. With speed ranges between 35 and 70 Km/h, obstacle avoidance approach must be executed quickly to avoid the collision [@griffiths2007obstacle]. - The use of UAVs in an indoor environment is a challenging task and it needs higher requirements than outdoor environment. It is very difficult to use GPS for avoiding collisions in an indoor environment, *usually indoor is a GPS denied environment*. Moreover, RF signals cannot be used in this environment, RF signals could be reflected, and degraded by the indoor obstacles and walls. - Vision-based collision avoidance methods using cameras suffer from heavy computational operations for image processing. Moreover, these cameras and optical sensors are light sensitive (need sufficient lighting to work properly), so steam and smoke could cause the collision avoidance system to fail. Therefore, sensor based collision avoidance methods can be used to tackle these problems [@chee2013control; @gageik2012obstacle]. - Some of the vision based collision avoidance approaches use a monocular camera with forward facing to generate collision free trajectory. In this method, all computations are performed off-board, with 30 frames per second which are sent in real time over Wi-Fi network to perform image processing at a remote base station. This makes UAV sensing and avoiding obstacles a challenging task [@alvarez2016collision] ### [Future Insights]{} {#future-insights-6} Based on the reviewed literature of collision avoidance challenges, we suggest these possible future directions: - UAVs could be integrated with vision sensors, laser range finder sensors, IR, and/or ultrasound sensors to avoid collisions in all directions [@chee2013control]. - UAV control algorithms can be developed for autonomous hovering without collision, and to ensure the completion of their mission successfully. - Standardization rules and regulations for UAVs are highly needed around the world to regulate their operations, to reduce the likelihood of collision among UAVs, and to guarantee safe hovering [@clarke2014regulation; @archick2005european]. - Under the deep learning techniques with Model Predictive Controllers (MPC), further work can be done on developing methods to generate guiding samples for more superior obstacle avoidance. The main concern in the deploying deep learning models is the processing requirement. Therefore, hardware limitation for on-board processing should be taken into consideration. - More studies are required to improve indoor and outdoor collision avoidance algorithms in order to compute smoother collision free paths, and to evaluate optimized trajectory in terms of aspects such as energy consumption [@roelofsen2015reciprocal]. - On-board processing is required for many UAV operations, such as dynamic sense and avoid algorithms, path re-planning algorithms and image processing. Design on-board powerful processor devices with low power consumption is an active area for future researches [@carrio2017review]. Networking Challenges {#FANET} --------------------- Fluid topology and rapid changes of links among UAVs are the main characteristics of multi-UAV networks or FANET [@gupta2016survey]. A UAV in FANET is a node flying in the sky with 3D mobility and speed between 30 to 460 km/h. This high speed causes a rapid change in the link quality between a UAV and other nodes, which introduces different design challenges to the communications protocols. Therefore, FANET needs new communication protocols to fulfill the communication requirements of Multi-UAV systems [@tareque2015routing]. ### FANET Challenges - One of the challenging issues in FANET is to provide wireless communications for UAVs, (wireless communications between UAV-GCS, UAV-satellite, and UAV-UAV). More specifically, coordination, cooperation, routing and communication protocols for the UAVs are challenging tasks due to the frequent connections interruption, fluid network topology, and limited energy resources of UAVs. Therefore, FANET requires some special hardware and a new networking model to address these challenges [@sahingoz2014networking; @motlagh2016low]. - The UAV power constraints limit the computation, communication, and endurance capabilities of UAVs. Energy-aware deployment of UAVs with Low power and Lossy Networks (LLT) approach can be efficiently used to handle this challenge [@zeng2016wireless; @winter2012rpl]. - FANET is a challenging environment for resources management in view of the special characteristics of this network. The challenge is the complexity of network management, such as the configuration difficulties of the hardware components of this network [@kirichek2016model]. Software-Defined Networking (SDN) and Network Function Virtualization (NFV) are useful approaches to tackle this challenge [@white2017programmable]. - Setting up an ad-hoc network among nodes of UAVs is a challenging task for FANET, due to the high node mobility, large distance between nodes, fluid network topology, link delays and high channel error rates. Therefore, transmitted data across these channels could be lost or delayed. Here the loss is usually caused by disconnections and rapid changes in links among UAVs. Delay-Tolerant Networking (DTN) architecture was introduced to be used in FANET to address this challenge [@burleigh2003delay; @loo2016mobile]. ### UAV New Networking Trends ### $a$) Delay-Tolerant Networking (DTN) {#a-delay-tolerant-networking-dtn .unnumbered} The DTN architecture was designed to handle challenges facing the dynamic environments as in FANET. [@warthman2012delay]. In FANET, DTN approach based on store-carry-forward model can be utilized to tackle the long delay for packets delivery. In this model, a UAV can store, carry and forward messages from source to destination with long term data storage and forwarding functions in order to compensate intermittent connectivity of links [@fall2007delay; @le2006uav]. DTN can be used with a set of protocols operating at MAC, transport and application layers to provide reliable data transport functions, such as Bundle Protocol (BP) [@scott2007bundle], Licklider Transmission Protocol (LTP) [@burleigh2008licklider] and Consultative Committee for Space Data Systems (CCSD) File Delivery Protocol (CFDP) [@protocol2007part]. These protocols use store, carry and forward model, so UAV node keeps a copy for each sent packet until it receives acknowledgment from the next node to confirm that the packet has been received successfully. BP and LTP are protocols developed to cope with FANET challenges and solve the performance problems for FANET. The BP, forms a store, carry and forward overlay networks, to handle message transmissions, receptions and re-transmissions using a convergence layer protocol (CLP) services [@scott2007bundle; @yang2014analytical] with the underlying transport protocols such as TCP-based [@demmer2014delay], UDP-based [@kruseudp] or LTP-based [@burleigh2008licklider]. The CFDP was designed for file transfer from source to destination based on store, carry and forward approach for DTN paradigm. It can be run over either reliable or unreliable service mode using transport layer protocols (TCP or UDP) [@wang2009protocols]. A routing strategy that combines DTN routing protocols in the sky and the existing Ad-hoc On-demand Distance Vector (AODV) on the ground for FANETs was proposed in [@le2006uav]. In this work, they implement a DTN routing protocol on the top of traditional and unmodified AODV. They also use UAV to store, carry and forward the messages from source to destination. ### $b$) Network Function Virtualization (VFV) {#b-network-function-virtualization-vfv .unnumbered} NFV is a new networking architecture concept for network softwarization and it is used to enable the networking infrastructure to be virtualized. More specifically, an important change in the network functions provisioning approach has been introduced using NFV, by leveraging existing IT virtualization technologies, therefore, network devices, hardware and underlying functions can be replaced with virtual appliances. NFV can provide programming capabilities in FANETs and reduces the network management complexity challenge [@han2015network; @jain2013network]. The research in [@bor2016new], proposed drone-cell management framework (DMF) using UAVs act as aerial base stations with a multi-tier drone-cell network to complement the terrestrial cellular network based on NFV and SDN paradigms. In [@rametta2017designing], the authors proposed a video monitoring platform as a service (VMPaaS) using swarm of UAVs that form a FANET in rural areas. This platform utilizes the recent NFV and SDN paradigms. Due to the complexity of the interconnections and controls in NFV, SDN can be consolidated with NFV as a useful approach to address this challenge. ### $c$) Software-Defined Networking (SDN) {#c-software-defined-networking-sdn .unnumbered} SDN is a promising network architecture, which provides a separation between control plane and data plane. It can also provide a centralized network programmability with global view to control network. Benefiting from the centralized controller in SDN, the original network switches could be replaced by uniform SDN switches [@zhao2014leveraging]. Therefore, the deployment and management of new applications and services become much easier. Moreover, the network management, programmability and reconfiguration can be greatly simplified [@zhang2017software]. FANET can utilize SDN to address its environment’s challenges and performance issues, such as dynamic and rapid topology changes, link intermittent between nodes; when UAV goes out of service due to coverage problems or for battery recharging. SDN also can help to address the complexity of network management[@gupta2016survey]. OpenFlow is one of the most common SDN protocols, used to implement SDN architecture and it separates the network control and data planes functionalities. OpenFlow switch consists of flow Table, OpenFlow Protocol and a Secure Channel [@mckeown2008openflow]. UAVs in FANETs can carry OpenFlow Switches. The SDN control plane could be centralized (one centralized SDN controller), decentralized (the SDN controller is distributed over all UAV nodes), or hybrid in which the processing control of the forwarded packets can be performed locally on each UAV node and control traffic also exists between the centralized SDN controller and all other SDN elements [@gupta2016survey]. The controller collects network statistics from OpenFlow switches, and it also needs to know the latest topology of the UAVs network. Therefore, it is important to maintain the connectivity of the SDN controller with the UAV nodes. The centralized SDN controller has a global view of the network. OpenFlow switches contain software flow tables and protocols to provide a communication between network and control planes, then the controller determines the path and tells network elements where to forward the packets [@mckeown2008openflow]. Figure \[fig:ahmad14\] shows separation of control and data planes in SDN platform with OpenFlow interface. ![Control and Data Planes in SDN Platform. []{data-label="fig:ahmad14"}](srv_key-ch_f2.png) ### $d$) Low power and Lossy Networks (LLT) {#d-low-power-and-lossy-networks-llt .unnumbered} In FANETs, UAV nodes are typically characterized by limited resources, such as battery, memory and computational resources. LLN composed of many embedded devices such as UAVs with limited power, and it can be considered as a promising approach to be used in FANET to handle power challenge in UAVs [@winter2012rpl; @rault2013multi]. The Internet Engineering Task Force (IETF) has defined routing protocol for LLN known as routing for low-power and lossy network (RPL) [@winter2012rpl], this protocol can utilize different low power communication technologies, such as low power WiFi, IEEE 802.15.4 [@domingo2012overview], which is a standard for low-power and low rate wireless communications and can be used for UAV-UAV communications in FANET [@zafar2017reliable]. The IPv6 over Low Power Wireless Personal Area Networks (6LoWPAN) defines a set of protocols that can be used to integrate nodes with IPv6 datagrams over constrained networks such as FANET [@atzori2010internet]. ### [Future Insights]{} {#future-insights-7} Based on the reviewed literature of the networking challenges and new networking trends of FANET, we suggest these future insights: - SDN services can be developed to support FANET functions, such as surveillance, safety and security services. - Further studies are needed to study the SDN deployment in FANET with high reliability, reachability and fast mobility of UAV nodes [@xia2015survey] - More research is needed to develope specific security protocols for FANETs based on the SDN paradigm. - More studies are needed to replace the conventional network devices by fully decoupled SDN switching devices, in which the control plane is completely decoupled from data plane and the routing protocols can be executed on-board from SDN switching devices [@xia2015survey]. - For NFV, more research is needed on optimal virtual network topology, customized end-to-end protocols and dynamic network embedding [@zhang2017software]. - New routing protocols need to be tailored to FANETs to conserve energy, satisfy bandwidth requirement, and ensure the quality of service [@sahingoz2014networking]. - Most of the existing MANET routing protocols partially fail to provide reliable communications among UAVs. So, there is a need to design and implement new routing protocols and networking models for FANETs [@bekmezci2013flying]. - FANETs share the same wireless communications bands with other applications such as satellite communications and GSM networks. This leads to frequency congestion problems. Therefore, there is a need to standardize FANET communication bands to mitigate this problem [@gupta2016survey]. - FANETs lack security support. Each UAV in FANET is required to exchange messages and routing information through wireless links. Therefore, FANETs are vulnerable to attacks. As a result, it is important to design and implement secure FANET routing protocols. - The design of congestion control algorithms become an important issue for FANETs. Current research efforts focus on modifying and improving protocols instead of developing new transport protocols that better suite FANETs. [@ivancic2012evaluation]. Security Challenges ------------------- Figure \[fig1\] illustrates the general architecture of UAV systems [@javaid2012cyber], which includes: (1) UAV units; (2) GCSs; (3) satellite (if necessary); and (4) communications links. Various components of the UAV systems provide a large attack surface for malicious intruders which brings huge cyber security challenges to the UAV systems. In this section, to present a comprehensive view of the cyber security challenges of UAV systems, we first summarize the attack vectors of general UAV systems. Then, based on the attack vectors and the potential capabilities of attackers, we classify the cyber attacks/challenges against UAV systems into different categories. More importantly, we present a comprehensive literature review of the state-of-the-art countermeasures for these security challenges. Finally, based on the reviewed literature, we summarize the security challenges and provide high level insights on how to approach them. ### Attack Vectors in UAV Systems Based on the architecture illustrated in Figure \[fig1\], we identify four attack vectors: - Communications links: various attacks could be applied to the communications links between different entities of UAV systems, such as eavesdropping and session hijacking. - UAVs themselves: direct attacks on UAVs can cause serious damage to the system. Examples of such attacks are signal spoofing and identity hacking. - GCSs: attacks on GCSs are more fatal than others because they issue commands to actual devices and collect all the data from the UAVs they control. Such attacks normally involve malwares, viruses and/or key loggers. - Humans: indirect attacks on human operators can force wrong/malicious operating commands to be issued to the system. This type of attacks is usually enabled with social engineering techniques. ### Taxonomy of Cyber Security Attacks/Challenges Against UAV Systems Based on the attack vectors and capabilities of attackers, as well as the work in [@javaid2012cyber], we classify the possible attacks against UAV systems into different categories. Figure \[fig2\] presents a graph that summarizes the proposed attack taxonomy. This attack model defines three general cyber security challenges for the UAV systems, namely, confidentiality challenges, integrity challenges, and availability challenges. Confidentiality refers to protecting information from being accessed by unauthorized parties. In other words, only the people who are authorized to do so can gain access to data. Attackers could compromise the confidentiality of UAV systems by various approaches (e.g., malware, hijacking, social engineering, etc.) utilizing different attack vector. Integrity ensures the authenticity of information. Attackers could modify or fabricate information of UAV systems (e.g., data collected, commands issued, etc.) through communications links, GCSs or compromised UAVs. For example, GPS signal spoofing attack. Availability ensures that the services (and the relevant data) that UAV systems carry are running as expected and are accessible to authorized users. Attackers could perform DoS (Deny of Service) attacks on UAV systems by, for example, flooding the communications links, overloading the processing units, or depleting the batteries. ![General Architecture of UAV Systems. Reprinted from [@javaid2012cyber].[]{data-label="fig1"}](srv_key-ch_f3) for tree=[ align=center, parent anchor=south, child anchor=north, font=, edge=[thick, -[Stealth\[\]]{}]{}, l sep+=10pt, edge path=[ (!u.parent anchor) – +(0,-10pt) -| (.child anchor); ]{}, if level=0[ inner xsep=0pt, tikz=[(.south east) – (.south west);]{} ]{} ]{} \[ [Cyber security challenges of UAV applications]{} \[Confidentiality \[UAVs \[[Key-logger,\ malware, etc.]{}\] \] \[GCSs \[ [ Hijacking,\ etc.]{}\] \] \[Humans \[[Social\ engineering]{}\] \] \[[Comms. links]{} \[[Eavesdrop,\ Hijacking, etc.]{}\] \] \] \[Availability \[DOS/DDOS \[[Buffer overflow,\ flooding, etc.]{}\] \] \] \[Integrity \[Fabrication Modification \[[Signal spoofing,\ MitM, etc.]{}\] \] \] \] \] ### Literature Review of the State-of-the-art Security Attacks/Challenges and Countermeasures {#review_security} The UAV related cyber security research can be classified into three main categories as shown in Figure \[fig3\], namely: - A: Specific attack discussion: This line of research focuses on one specific type of attack (e.g., signal spoofing) and proposes corresponding analysis or countermeasures. - B: General security analysis: This line of research presents the high level analysis, discussion or modeling of various attacks that exist in current UAV systems. - C: Security framework development: This line of research introduces new monitoring systems, simulation test beds or anomaly detection frameworks for the state-of-the-art UAV applications. Table \[t3\] summarizes the 15 relevant papers that we have reviewed including their attack vectors and categories. In addition, for the papers that discuss specific attacks/challenges, we summarize the proposed countermeasures, their limitations and propose high level countermeasures as guidelines for future enhancements. for tree=[ align=center, parent anchor=south, child anchor=north, font=, edge=[thick, -[Stealth\[\]]{}]{}, l sep+=10pt, edge path=[ (!u.parent anchor) – +(0,-10pt) -| (.child anchor); ]{}, if level=0[ inner xsep=0pt, tikz=[(.south east) – (.south west);]{} ]{} ]{} \[ Security works \[A: Specific attacks \] \[B: Security framework\ development \[[B-1: Monitoring\ systems]{} \] \[[B-2: Simulation\ test beds]{} \] \[[B-3: Anomaly\ detection frameworks]{} \] \] \[C: General security\ discussion \] \] [|l|l|l|l|l|l|l|]{} & & & & & &\ [@javaid2012cyber] & 2012 & Communication Links, UAV, GCSs & C & N/A & N/A & N/A\ [@giray2013anatomy] & 2013 & Communication Links & --------------------- A: Signal integrity (GPS spoofing) --------------------- & ------------------------------- 1\. Cryptography based signal authentication; 2\. Multiple receiver design. ------------------------------- & ------------------------------------ 1\. Lack of details; 2\. Require hardware changes (i.e., complexity and cost issue). ------------------------------------ & ---------------------------------- 1\. Strong authentication (e.g., trust platform module, Kerberos, etc.); 2\. Signal distortion detection; 3\. Direction-of-arrival sensing (i.e., transmitter antenna direction detection). ---------------------------------- \ [@javaid2013uavsim] & 2013 & Communication Links, UAV, GCSs & B-1 & N/A & N/A & N/A\ [@goppert2014software] & 2014 & Communication Links, UAV, GCSs & C & N/A & N/A & N/A\ [@maxa2015secure] & 2015 & Communication Links, UAVs & ---------------------------- A: Routing attack (eavesdropping, DOS, etc.) ---------------------------- & --------------------------- A secure routing protocol and its modeling process. --------------------------- & --------------------- Lack of evaluations or security proof. --------------------- & --------------------------- Perform evaluations under various attack scenarios or provide theoretical security proof. --------------------------- \ [@schumann2015r2u2] & 2015 & UAVs & B-2 & N/A & N/A & N/A\ [@birnbaum2015unmanned] & 2015 & UAVs & B-1 & N/A & &\ [@muzzi2015using] & 2015 & Communication Links, UAVs & A: DDoS & --------------------- Botnet platform (as a future work). --------------------- & ---------------------- There is no concrete solution provided. ---------------------- & ------------------------------ 1\. Delivers behavior-based anomaly detection (e.g., packet analysis about its type, rate, volume, source IP, etc.); 2\. Enables immediate response to DDoS attacks; ------------------------------ \ [@vattapparamban2016drones] & 2016 & Communication Links, UAV, GCSs & C & N/A & N/A & N/A\ [@sedjelmaci2016detect] & 2016 & Communication Links, UAV, GCSs & B-3 & N/A & N/A & N/A\ [@davidson2016controlling] & 2016 & Communication Links, UAVs & ------------------------- A: Signal integrity (sensor input spoofing) ------------------------- & ------------------------ Optical flow analysis (RANSAC algorithm [@fischler1981random]) ------------------------ & ---------------------------------- The attack model is too demanding (i.e., it is difficult to perform this attack. ). ---------------------------------- & --------------------------------------- 1\. Strong firewall/authentication schemes to prevent hijacking attacks; 2\. Delivers behavior-based anomaly detection. --------------------------------------- \ [@mcneely2016detection] & 2016 & UAVs & A: UAV hijacking & ------------------------ Profiling of in-flight statistics. ------------------------ & ---------------------------------- Temporary control instability (e.g., short time decreasing of amplitude) caused by the attacker cannot be detected. ---------------------------------- & --------------------------------------- 1\. Decreasing the detection interval; 2\. Strong firewall/authentication schemes to prevent hijacking attacks; 3\. Packets analysis of malicious injected commands. --------------------------------------- \ [@hagerman2016security] & 2016 & N/A & B-2 & N/A & N/A & N/A\ [@rodday2016exploring] & 2016 & Communication Links & --------------- A: Hijacking (MitM attack) --------------- & ------------------------- 1\. XBee 868LP on-board encryption; 2\. Dedicated hardware encryption; 3\. Application layer encryption. ------------------------- & No details provided. & ------------------------------------------------ 1\. Integrate detection and immediate response mechanisms. 2\. Strong authentication (e.g., trust platform module, Kerberos, etc.); ------------------------------------------------ \ [@wang2016authentication] & 2017 & Communication Links & ----------------------------- A: Authentication/key management targeted attacks ----------------------------- & ------------------------ A set of modified authentication and key management protocols. ------------------------ & --------------------- Lack of evaluations or security proof. --------------------- & --------------------------- Perform evaluations under various attack scenarios or provide theoretical security proof. --------------------------- \ [@podhradsky2017improving] & 2017 & Communication Links, GCS, UAV & C & N/A & N/A & N/A\ - Challenge of specific attack - GPS spoofing: GPS spoofing attack refers to the malicious attempt to manipulate the GPS signals in order to achieve the benefits of the attackers. In [@giray2013anatomy], the authors provide an analysis of UAV hijacking attack (e.g., GPS signal spoofing) with an anatomical approach and outline a model to illustrate: (1) that such attacks can be observed to reveal their vulnerabilities; (2) how to exploit such attacks; (3) how to provide countermeasures and risk mitigation; and (4) details of the impact of such attacks. The use of anatomical investigation of a UAV hijacking aims to provide insights that help all the practitioners of the UAV systems. The article mainly focuses on the analysis of GPS signal spoofing attacks. However, the proposed countermeasures (i.e., cryptography based signal authentication and multiple receiver design) are not novel and are only described at high level without sufficient details. - Challenge of specific attack - DDoS (Distributed Deny of Service) attack: DDoS attack is an attempt to make the UAV systems unavailable/unreachable by overwhelming it with traffic from multiple sources. In [@muzzi2015using], the authors present a platform to conduct a DDoS attack against UAV systems and layout the general mitigation method. Firstly, the authors present a DDoS attack modeling and specification. Then, they analyze the effects caused by this kind of attack. However, the proposed attack scenario is too simple (i.e., UDP flooding) and only a high level solution is said to be included in the future work. - Challenge of specific attack - sensor input spoofing attack: In [@davidson2016controlling], the authors introduce a new attack against UAV systems, named sensor input spoofing attack. If the attacker knows exactly how the sensor algorithms work, he can manipulate the victim’s environment to create a new control channel such that the entire UAV system is in dangerous. In addition, they provide suitable mitigation for the introduced attack. To perform the proposed sensor input spoofing attack, the attacker must meet three requirements: (1) environment influence requirement: the attacker must be able to modify the physical phenomenon that the sensor measures; (2) plausible input requirement: the attacker must be able to generate valid input that the sensor system can read; and (3) meaningful response requirement: the attacker must be able to generate meaningful UAV behavior corresponding to the spoofed input. This attack model is too demanding to be practical for many UAV applications and no convincing justifications were provided. - Challenge of specific attack - hijacking attacks: hijacking attack is a type of attack in which the attacker takes control of a communication between two parties and masquerades as one of them (e.g., inject extra commands). In [@rodday2016exploring], the authors demonstrate the capabilities of the attackers who target UAV systems to perform Man-in-the-Middle attacks (MitM) and control command injection attacks. They also propose corresponding countermeasures to help address these vulnerabilities. However, the introduced MitM attack requires fully compromised UAV system via reverse-engineering techniques which largely limits the feasibility/practicability of this type of attacks. In addition, the proposed encryption scheme lacks the necessary details that warrant credible evaluation of robustness and security. The article of [@mcneely2016detection] proposes an anomaly detection scheme based on flight patterns fingerprint via statistical measurement of flight data. Firstly, a baseline flight profile is generated. Then, simulated hijacking scenarios are compared to the baseline profile to determine the detection result. The proposed scheme is able to detect all direct hijacking scenarios (i.e., assume a fully compromised UAV and the flight plan has been altered to some random places.). However, temporary control instability (e.g., short time decreasing of amplitude) caused by the attacker cannot be detected. - Challenge of specific attack - Authentication/key management targeted attacks: The authors in [@wang2016authentication] first propose a communication architecture to integrate LTE technology into integrated CNPC (Control and Non-Payload Communication) networks. Then, they define several security requirements of the proposed architecture. Also, they modify the authentication, the key agreement and the handover key management protocols to make them suitable to the integrated architecture. The proposed modified protocol is proved to outperform the LTE counterpart protocols via a comparative analysis. In addition, it introduces almost the same amount of communications overhead. - Challenge of security framework development: In [@javaid2013uavsim], the authors introduce UAVSim, a simulation testbed for Unmanned Aerial Vehicle Networks cyber security analysis. The proposed test bed can be used to perform various experiments by adjusting different parameters of the networks, hosts and attacks. The UAVSim consists of five modules: (1) Attack library: generating various attacks (i.e., jamming attacks and DoS attacks); (2) UAV network module: forming different UAV networks; (3) UAV model library: providing different host functions (e.g., attack hosts or UAV hosts); (4) Graphical user interface; (5) Result analysis module; and (6) UAV Module browser. The proposed test bed is helpful in analyzing existing attacks against UAV systems by adjusting different parameters. However, the proposed attack library only contains two types of attacks for now (i.e., jamming attacks and DoS attacks). Therefore, UAVSim must be enriched with other types of attacks to improve its applicability and usability. The authors in [@schumann2015r2u2] present R2U2, a novel framework for run-time monitoring of security threats and system behaviors for Unmanned Aerial Systems (UAS). Specifically, they extend previous R2U2 from only monitoring of hardware components to both hardware and software configuration monitoring to achieve security threats diagnostic. Also, the extended version of R2U2 provides detection of attack patterns (i.e., ill-formatted and illegal commands, dangerous commands, nonsensical or repeated navigation commands and transients in GPS signals) rather than component failures. In addition, the FPGA implementation ensures independent monitoring and achieves software re-configurable feature. The extended version of R2U2 is more advanced than the original prototype in terms of attack pattern recognition and secure & independent monitoring. However, the independent monitoring system introduces a new security hole (i.e., the monitoring system itself) without introducing appropriate security mechanism to mitigate it. In the work of [@birnbaum2015unmanned], the authors present a UAV monitoring system that captures flight data to perform real-time abnormal behavior detection. If an abnormal behavior is detected, the system will raise an alert. The proposed monitoring system includes the following features: - A specification language for UAV in-flight behavior modeling; - An algorithm to covert a given flight plan into a behavioral flight profile; - A decision algorithm to determine if an actual UAV flight behavior is normal or not based on the behavioral flight profile; - A visualization system for the operator to monitor multiple UAVs. The proposed monitoring system can detect abnormal behaviors based on the in flight data. However, attacks which do not alter in-flight data are not detectable (e.g., attacks that only collect flight data). The authors of [@sedjelmaci2016detect] propose and implement a cyber security system to protect UAVs from several dangerous attacks, such as attacks that target data integrity and network availability. The detection of data integrity attacks uses Mahalanobis distance method to recognize the malicious UAV that forwards erroneous data to the base station. For the availability attacks (i.e., wormhole attack in this paper), the detection process is summarized as follows: - The UAV relays a packet when it becomes within the range of the base station. The packet includes: (1) node’s type (source, relay or destination); and (2) its next (and previous) hops. - The base station collects the forwarded packets from the UAVs, verifies whether the relay node forwards a packet or not and computes the Message Dropping Rate (MDR). - The base station will raise an alert if the MDR is higher than a false MDR assigned to a normal UAV. The proposed detection scheme for data integrity attacks and network availability attacks is proved to achieve high detection accuracy via simulations. The attack model assumes that every UAV node has the detection module and the node which is determined as malicious node would lose the ability to invoke the detection module. However, there is no co-operation detection algorithm provided and only single node detection procedure is introduced. In the work of [@hagerman2016security], the authors present an improved mechanism for UAV security testing. The proposed approach uses a behavioral model, an attack model, and a mitigation model to build a security test suite. The proposed security testing goes as follows: - Model system behavior: associates behavior criteria (BC) with a behavioral model (BM) and build a behavioral test (BT). - Attack type definition: defines attack types (A) with attack criteria (AC) and determine attack applicability matrix (AM). - Determine security test requirements: determines at which point during the operation of a behavioral test an attack will occur. - Test generation: a required mitigation process is injected. This work provides a systematic approach to identify vulnerabilities in UAV systems and provides corresponding mitigation. However, the proposed mitigation methods are limited to state roll back or re-executing. In other words, other prevention or detection methods (e.g., encryption or anomaly detection) cannot be included in the proposed scheme. - Challenges of security analysis works: In the work of [@javaid2012cyber], the authors perform an overall security threat analysis of a UAV system as described in Figure \[fig1\]. A cyber security threat model is proposed and analyzed to show existing or possible attacks against UAV systems. This model helps designers and users of UAV systems across different aspects, including: (1) understanding threat profiles of different UAV systems; (2) addressing various system vulnerabilities; (3) identifying high priority threats; and (4) selecting appropriate mitigation techniques. In addition, a risk evaluation mechanism (i.e., a standard risk evaluation grid included in the ETSI threat assessment methodology ) is used to assess the risk of different threats in UAV systems. Even though this paper provides a comprehensive profile of existing or potential attacks for UAV systems, the proposed attacks are general/high level threats that did not take into consideration hardware and software differences among different UAV systems. The survey paper of [@vattapparamban2016drones] reviews various aspects of drones (UAVs) in future smart cities, relating to cyber security, privacy, and public safety. In addition, it also provides representative results on cyber attacks using UAVs. The studied cyber attacks include de-authentication attack and GPS spoofing attack. However, no countermeasures are discussed. The authors in [@podhradsky2017improving] introduce an implementation of an encrypted Radio Control (RC) link that can be used with a number of popular RC transmitters. The proposed design uses Galois Embedded Crypto library together with openLRSng open-source radio project. The key exchange algorithm is described below: - $TX$ first generates the ephemeral key $K_e$ and , then encrypts $K_e$ using permanent key $K_p$ and $IV_{rand}$. The encrypted result is $m_1^{TX}$ and sent to $RX$. - $RX$ decrypts $m_1^{TX}$ with $K_p$ to obtain $K_e$, then generate $K_e^{'}$ and encrypts $K_e^{'}$ with $K_e$ and $IV_0=0$. The encrypted result is $m_2^{RX}$ and sent to $TX$. - $TX$ decrypts $m_2^{RX}$ with $K_e$ and $IV_0=0$ to verify that $RX$ has a copy of $K_e$. Then $TX$ sends $ACK$ message encrypted with $K_e^{'}$ and $IV_0$ to $RX$ as a confirmation (denoted as $m_3^{TX}$ ). - $RX$ receives $m_3^{TX}$it to verify that $TX$ has a copy of $K_e^{'}$. Key exchange is then successful. The proposed scheme achieves secure communication link for open source UAV systems. However, the symmetric key is assumed to be generated in a third-party trusted computer and hard-coded in the source code, which degrades the security guarantees and limits the feasibility of the scheme. ### Summarizing of Cyber Security Challenges for UAV Systems Based on the reviewed literature, we summarize four key findings of the cyber security challenges on UAV applications: - DoS attacks and hijacking attacks (i.e., for signal spoofing) are the most prevailing threats in the UAV systems: Various DoS attacks have been found and proved to cause serious availability issues in UAV systems. In addition, signal spoofing via hijacking attacks is able to severely damage the behaviors of certain UAV systems. *Possible solution*: (1) Strong authentication (e.g., trust platform module, Kerberos, etc.); (2) Signal distortion detection; and (3) Direction-of-arrival sensing (i.e., transmitter antenna direction detection). - Existing countermeasure algorithms are limited to single UAV systems: most of the existing schemes are designed only for single UAV systems. A few of them discuss multiple UAV scenarios without providing any concrete solutions. *Possible solution*: develop co-operation countermeasure algorithms for multiple UAVs. For example, modify and adopt existing distributed security frameworks (e.g., Kerberos) to multiple UAVs systems; (2) - Most of the current security analysis of UAV systems overlook the hardware/software differences (e.g., different hardware platform, different communication protocols, etc.) among various UAV systems: On one hand, some of the attacks only exist in specific hardware or software configuration. On the other hand, most of the proposed countermeasures are well studied solutions in other communication systems and there could be many deployment difficulties when applying them on different UAV systems. *Possible solution*: design unified/standard deployment interface or language for the diverse UAV systems. - Current UAV simulation test beds are still far from mature: Existing emulators for UAV security analysis are limited to few attack scenarios and specific hardware/software configurations. *Possible solution*: leveraging powerful simulation tools (e.g., Labview) or design customized simulation environments. Conclusion ========== The use of UAVs has become ubiquitous in many civil applications. From rush hour delivery services to scanning inaccessible areas, UAVs are proving to be critical in situations where humans are unable to reach or cannot perform dangerous/risky tasks in a timely and efficient manner. In this survey, we review UAV civil applications and their challenges. We also discuss current research trends and provide future insights for potential UAV uses. In SAR operations, UAVs can provide timely disaster warnings and assist in speeding up rescue and recovery operations. They can also carry medical supplies to areas that are classified as inaccessible. Moreover, they can quickly provide coverage of a large area without ever risking the security or safety of the personnel involved. Using UAVs in SAR operations reduces costs and human lives. In remote sensing, UAVs equipped with sensors can be used as an aerial sensor network for environmental monitoring and disaster management. They can provide numerous datasets to support research teams, serving a broad range of applications such as drought monitoring, water quality monitoring, tree species, disease detection, etc. In risk management, insurance companies can utilize UAVs to generate NDVI maps in order to have an overview of the hail damage to crops, for instance. Civil infrastructure is expected to dominate the addressable market value of UAV that is forecast to reach $\$45$ Billion in the next few years. In construction and infrastructure inspection applications, UAVs can be used to monitor real-time construction project sites. They can also be utilized in power line and gas pipeline inspections. Using UAVs in civil infrastructure applications can reduce work-injuries, high inspection costs and time involved with conventional inspection methods. In agriculture, UAVs can be efficiently used in irrigation scheduling, plant disease detection, soil texture mapping, residue cover and tillage mapping, field tile mapping, crop maturity mapping and crop yield mapping. The next generation of UAV sensors can provide on-board image processing and in-field analytic capabilities, which can give farmers instant insights in the field, without the need for cellular connectivity and cloud connection. With the rapid demise of snail mail and the massive growth of e-Commerce, postal companies have been forced to find new methods to expand beyond their traditional mail delivery business models. Different postal companies have undertaken various UAV trials to test the feasibility and profitability of UAV delivery services. To make UAV delivery practical, more research is required on UAVs design. UAVs design should cover creating aerial vehicles that can be used in a wide range of conditions and whose capability rivals that of commercial airliners. UAVs have been considered as a novel traffic monitoring technology to collect information about real-time road traffic conditions. Compared to the traditional monitoring devices, UAVs are cost-effective and can monitor large continuous road segments or focus on a specific road segment. However, UAVs have slower speeds compared to vehicles driving on highways. A possible solution might entail changing the regulations to allow UAVs to fly at higher altitudes. Such regulations would allow UAVs to benefit from high views to compensate the limitation in their speed. One potential benefit of UAVs is the capability to fill the gaps in current border surveillance by improving coverage along remote sections of borders. Multi-UAV cooperation brings more benefits than single UAV surveillance, such as wider surveillance scope, higher error tolerance, and faster task completion time. However, multi-UAV surveillance requires more advanced data collection, sharing, and processing algorithms. To develop more efficient and accurate multi-UAV cooperation algorithms, advanced machine learning algorithms could be utilized to achieve better performance and faster response. The use of UAVs is rapidly growing in a wide range of wireless networking applications. UAVs can be used to provide wireless coverage during emergency cases where each UAV serves as an aerial wireless base station when the cellular network goes down. They can also be used to supplement the ground base station in order to provide better coverage and higher data rates. Utilizing UAVs in wireless networks needs further research, where topology formation, cooperation between UAVs in a multi-UAV network, energy constraints, and mobility modelس are challenges facing UAVs in wireless networks. Throughout the survey, we discuss the new technology trends in UAV applications such as mmWave, SDN, NFV, cloud computing and image processing. We thoroughly identify the key challenges for UAV civil applications such as charging challenges, collision avoidance and swarming challenges, and networking and security related challenges. In conclusion, a complete legal framework and institutions regulating the civil uses of UAVs are needed to spread UAV services globally. We hope that the key research challenges and opportunities described in this survey will help pave the way for researchers to improve UAV civil applications in the future. ACRONYMS ======== The acronyms and abbreviations used throughout the survey and their definitions are listed bellow: IPv6 over Low Power Wireless Personal Area Networks. Attack Criteria. Ad-hoc On-demand Distance Vector. Air-to-Ground. Behavior Criteria. Beam Division Multiple Access. Behavioral Model. Bundle Protocol. Behavioral Test. Capacitive Coupling. Consultative Committee for Space Data Systems. CCSD File Delivery Protocol. Convergence Layer Protocol. Convolutional Neural Network. Cellular Network Operator. Control and Non-Payload Communication. Certificate of Authorization. Distributed Deny of Service. Drone-cell management frame-work. Deny of Service. Disruption Tolerant Networking. Enhanced Vegetation Index. Federal Aviation Administration. Flying Ad-Hoc Networks. Forward Looking Infrared. Fourier Mellin Invariant. Free Space Optical. Ground Control Station. Geographic Information System. Global Position System. Green Normalized Difference Vegetation index. Green Vegetation Index. High Altitude Platform. Internet Engineering Task Force. Inertial Sensor. Internet of Things. Infra Red. International Telecommunication Union. Kansas Department of Transportation. Low Altitude Platform. Low power and Lossy Networks. Line of Sight. Laser Range Finder. Licklider Transmission Protocol. Medium Access Control. Mobile Ad-hoc Networks. Multi-scale Block Local Binary Patterns. Message Dropping Rate. Mobile Edge Computing. Man-in-the-Middle attacks. Millimeter-Wave. Model Predictive Controller. Magnetic Resonance Coupling. Normalized difference vegetation index. Non-Line of Sight. Object-Based Image Analysis. Origin-Destination. Precision Agriculture. Performance-Based Regulations. Point of closest Approach. Pacific Gas and Electric Company. Physical Uplink Shared Channel. Perpendicular Vegetation Index. Radio Control. Routing for low-power and Lossy network. Roadside Unit. Special Airworthiness Certificate. Search and Rescue. Soil Adjusted Vegetation Index. Software-Defined Networking. Structural Health Monitoring. Start of Charge. Solar-Powered UAVs. Small Unmanned Aerial Vehicle. Support Vector Machine. Thermal Infrared. Unmanned Aircraft Systems. Unmanned Aerial Vehicle. Vehicular Ad-hoc Networks. Network Function Virtualization. Vegetation Indices. Video Monitoring Platform as a Service. Vertical Take-Off and Landing. Wireless Power Transfer. Wireless Sensor Networks. [Hazim Shakhatreh]{} is a Ph.D. student at the ECE department of New Jersey Institute of Technology. He received the B.S. degree and M.S. degree in wireless communications engineering from Yarmouk University, Jordan, in 2008 and 2012, respectively. His research interests include wireless communications and emerging technologies with focus on Unmanned Aerial Vehicle (UAV) networks. [Ahmad H. Sawalmeh]{} is a PhD student at the Engineering College of Universiti Tenaga Nasional (UNITEN), Malaysia. He received his M.S. degree in Computer Engineering from Jordan University of Science and Technology in 2005. He also received his B.S. degree from Jordan University of Science and Technology in 2003. He worked as a lecturer in the Computer Department at the Technical Vocational Training Corporation (TVTC), Kingdom of Saudi Arabia from 2006 – 2016. His research interests include wireless communications, UAVs networks and Flying Ad-Hoc Networks (FANETs). [Ala Al-Fuqaha]{} (S’00-M’04-SM’09) received his M.S. and Ph.D. degrees in Electrical and Computer Engineering from the University of Missouri-Columbia and the University of Missouri-Kansas City, in 1999 and 2004, respectively. Currently, he is Professor and director of NEST Research Lab at the Computer Science Department of Western Michigan University. His research interests include Wireless Vehicular Networks (VANETs), cooperation and spectrum access etiquettes in cognitive radio networks, smart services in support of the Internet of Things, management and planning of software defined networks (SDN) and performance analysis and evaluation of high-speed computer and telecommunications networks. In 2014, he was the recipient of the outstanding researcher award at the college of Engineering and Applied Sciences of Western Michigan University. He is currently serving on the editorial board for John Wiley’s Security and Communication Networks Journal, John Wiley’s Wireless Communications and Mobile Computing Journal, EAI Transactions on Industrial Networks and Intelligent Systems, and International Journal of Computing and Digital Systems. He is a senior member of the IEEE and has served as Technical Program Committee member and reviewer of many international conferences and journals. \[[{width="1in" height="1.25in"}]{}\][Zuochao Dou]{} received his B.S. degree in Electronics in 2009 at Beijing University of Technology. From 2009 to 2011, he studied at University of Southern Denmark, concentrating on embedded control systems for his M.S degree. Then, he received his second M.S. degree at University of Rochester in 2013 majoring in communications and signal processing. He is currently working towards his Ph.D. degree in the area of cloud computing security and network security with the guidance of Dr. Abdallah Khreishah and Dr. Issa Khalil. \[[{width="1in" height="1.25in"}]{}\][Eyad Almaita]{} received the B.Sc. from the Al-Balqa Applied University, Jordan in 2000, M.Sc. from Al-Yarmouk University, Jordan in 2006, Ph.D. from Western Michigan University, USA in 2012. He is now serving as associate professor at Mechatronics and power engineering dept. in Tafila Technical University, Jordan. His research interests include energy efficient systems, smart systems, power quality, and artificial intelligence. \[[{width="1in" height="1.25in"}]{}\][Issa Khalil]{} received PhD degree in Computer Engineering from Purdue University, USA in 2007. Immediately thereafter he joined the College of Information Technology (CIT) of the United Arab Emirates University (UAEU) where he served as an associate professor and department head of the Information Security Department. In 2013, Khalil joined the Cyber Security Group in the Qatar Computing Research Institute (QCRI), a member of Qatar Foundation, as a Senior Scientist, and a Principal Scientist since 2016. Khalil’s research interests span the areas of wireless and wireline network security and privacy. He is especially interested in security data analytics, network security, and private data sharing. His novel technique to discover malicious domains following the guilt-by-association social principle attracts the attention of local media and stakeholders, and received the best paper award in CODASPY 2018. Dr. Khalil served as organizer, technical program committee member and reviewer for many international conferences and journals. He is a senior member of IEEE and member of ACM and delivers invited talks and keynotes in many local and international forums. In June 2011, Khalil was granted the CIT outstanding professor award for outstanding performance in research, teaching, and service. \[[{width="1in" height="1.25in"}]{}\][Noor Shamsiah Othman]{} received the B.Eng. degree in electronic and electrical engineering and the M.Sc. degree in microwave and optoelectronics from the University College London, London, U.K., in 1998 and 2000, respectively, and and the Ph.D. degree in wireless communications from the University of Southampton, U.K., in 2008. She is currently with Universiti Tenaga Nasional, Malaysia, as Senior Lecturer. Her research interest include audio and speech coding, joint source/channel coding, iterative decoding, unmanned aerial vehicle communications and SNR estimator. [Abdallah Khreishah]{} is an associate professor in the Department of Electrical and Computer Engineering at New Jersey Institute of Technology. His research interests fall in the areas of visible-light communication, green networking, network coding, wireless networks, and network security. Dr. Khreishah received his BS degree in computer engineering from Jordan University of Science and Technology in 2004, and his MS and PhD degrees in electrical & computer engineering from Purdue University in 2006 and 2010. While pursuing his PhD studies, he worked with NEESCOM. He is a senior member of the IEEE and the chair of North Jersey IEEE EMBS chapter. [Mohsen Guizani]{} (S’85, M’89, SM’99, F’09) received his B.S. (with distinction) and M.S. degrees in electrical engineering, and M.S. and Ph.D. degrees in computer engineering from Syracuse University, New York, in 1984, 1986, 1987, and 1990, respectively. He is currently a professor and ECE Department Chair at the University of Idaho. Previously, he served as the associate vice president of Graduate Studies and Research, Qatar University, chair of the Computer Science Department, Western Michigan University, and chair of the Computer Science Department, University of West Florida. He also served in academic positions at the University of Missouri-Kansas City, University of Colorado-Boulder, Syracuse University, and Kuwait University. His research interests include wireless communications and mobile computing, computer networks, mobile cloud computing, security, and smart grid. He currently serves on the Editorial Boards of several international technical journals, and is the Founder and Editor-in-Chief of Wireless Communications and Mobile Computing (Wiley). He is the author of nine books and more than 400 publications in refereed journals and conferences. He has guest edited a number of special issues in IEEE journals and magazines. He has also served as a member, Chair, and the General Chair of a number of international conferences. He was selected as the Best Teaching Assistant for two consecutive years at Syracuse University. He was Chair of the IEEE Communications Society Wireless Technical Committee and TAOS Technical Committee. He served as an IEEE Computer Society Distinguished Speaker from 2003 to 2005. [^1]: Hazim Shakhatreh, Zuochao Dou, and Abdallah Khreishah are with the Department of Electrical and Computer Engineering, New Jersey Institute of Technology. (e-mail: {hms35,zd36,abdallah}@njit.edu) [^2]: Ahmad Sawalmeh and Noor Shamsiah Othman are with the Department of Electronics and Communication Engineering, Universiti Tenaga Nasional. (e-mail: {[email protected],[email protected]}) [^3]: Ala Al-Fuqaha is with the Department of Computer Science, Western Michigan University, Kalamazoo, MI, 49008, USA. (e-mail: ([email protected]) [^4]: Eyad Almaita is with the Department of Power and Mechatronics Engineering, Tafila Technical University. (e-mail: ([email protected]) [^5]: Issa Khalil is with Qatar Computing Research Institute (QCRI), HBKU, Doha, Qatar. (e-mail: [email protected]) [^6]: Mohsen Guizani is with the University of Idaho, Moscow, ID, 83844, USA. (e-mail: [email protected])

--- author: - | Drew A. Hudson\ Department of Computer Science\ Stanford University\ `[email protected]`\ Christopher D. Manning\ Department of Computer Science\ Stanford University\ `[email protected]`\ bibliography: - 'iclr2018\_conference.bib' title: | Compositional Attention Networks\ for Machine Reasoning --- abstract introduction The MAC Network {#sec:model} =============== ![**The MAC cell architecture.** The MAC recurrent cell consists of a control unit, read unit, and write unit, that operate over dual ****** and ****** hidden states. The **control unit** successively attends to different parts of the task description (question), updating the control state to represent at each timestep the reasoning operation the cell intends to perform. The **read unit** extracts information out of a knowledge base (here, image), guided by the control state. The **write unit** integrates the retrieved information into the memory state, yielding the new intermediate result that follows from applying the current reasoning operation.[]{data-label="fig:mac"}](mac.pdf){width="0.4\linewidth"} A MAC network is an end-to-end differentiable architecture primed to perform an explicit multi-step reasoning process, by stringing together $p$ recurrent MAC cells, each responsible for performing one reasoning step. Given a knowledge base $K$ (for VQA, an image) and a task description $q$ (for VQA, a question), the model infers a decomposition into a series of $p$ reasoning operations that interact with the knowledge base, iteratively aggregating and manipulating information to perform the task at hand. It consists of three components: (1) an input unit, (2) the core recurrent network, composed out of $p$ MAC cells, and (3) an output unit, all described below. The Input Unit -------------- The input unit transforms the raw inputs given to the model into distributed vector representations. Naturally, this unit is tied to the specifics of the task we seek to perform. For the particular case of VQA, it receives a question and an image and processes each of them respectively: **The question** string, of length $S$, is converted into a sequence of learned word embeddings that is further processed by a $d$-dimensional biLSTM yielding: (1) *contextual words*: a series of output states ${{\boldsymbol{cw}_1},\ldots,{\boldsymbol{cw}_S}}$ that represent each word in the context of the question, and (2) the question representation: ${q} = \left[ {\overleftarrow {\boldsymbol{cw}_1}} ,\overrightarrow {\boldsymbol{cw}_S} \right] $, the concatenation of the final hidden states from the backward and forward LSTM passes. Subsequently, for each step $i=1,\ldots,p$, the question ${q}$ is transformed through a learned linear transformation into a position-aware vector $\boldsymbol{q_i} ={{W}_{i}^{d\times 2d}}{q} + {{b}_{i}^d} $, representing the aspects of the question that are relevant to the $i^{th}$ reasoning step. **The image** is first processed by a fixed feature extractor pre-trained on ImageNet [@imagenet] that outputs *conv4* features from ResNet101 [@imp1], matching prior work for CLEVR [@nmn3; @rn; @film]. The resulting tensor is then passed through two CNN layers with $d$ output channels to obtain a final image representation, the *knowledge base* $\boldsymbol{K}^{H{\times}W{\times}d}={\{\boldsymbol{k_{h,w}^d}|_{h,w=1,1}^{H,W}\}}$, where $H = W = 14 $ are the height and width of the processed image, corresponding to each of its regions. The MAC cell {#sec:MACcell} ------------ The MAC cell is a recurrent cell designed to capture the notion of an atomic and universal reasoning operation and formulate its mechanics. For each step $i=1,\ldots,p$ in the reasoning process, the $i^{th}$ cell maintains dual hidden states: control $\boldsymbol{c_i}$ and memory $\boldsymbol{m_i}$, of dimension $d$, initialized to learned parameters $\boldsymbol{m_0}$ and $\boldsymbol{c_0}$, respectively. **The control** $\boldsymbol{c_i}$ represents the *reasoning operation* the cell should accomplish in the $i^{th}$ step, selectively focusing on some aspect of the question. Concretely, it is represented by a soft attention-based weighted average of the question words ${\boldsymbol{cw}_s}; s=1,\ldots,S$. **The memory** $\boldsymbol{m_i}$ holds the *intermediate result* obtained from the reasoning process up to the $i^{th}$ step, computed recurrently by integrating the preceding hidden state $\boldsymbol{m_{i-1}}$ with new information $\boldsymbol{r_{i}}$ retrieved from the image, performing the $i^{th}$ reasoning operation $\boldsymbol{c_i}$. Analogously to the control, $\boldsymbol{r_{i}}$ is a weighted average over its regions $\{\boldsymbol{k_{h,w}}|_{h,w=1,1}^{H,W}\}$. Building on the design principles of computer organization, the MAC cell consists of three operational units: control unit CU, read unit RU and write unit WU, that work together to accomplish tasks by performing an iterative reasoning process: The control unit identifies a series of operations, represented by a recurrent control state; the read unit extracts relevant information from a given knowledge base to perform each operation, and the write unit iteratively integrates the information into the cell’s memory state, producing a new intermediate result. Through their operation, the three units together impose an interface that regulates the interaction between the control and memory states. Specifically, the control state, which is a function of the question, guides the integration of content from the image into the memory state only through indirect means: soft-attention maps and sigmoidal gating mechanisms. Consequently, the interaction between these two modalities – visual and textual, or knowledge base and query – is mediated through probability distributions only. This stands in stark contrast to common approaches that fuse the question and image together into the same vector space through linear combinations, multiplication, or concatenation. As we will see in [section \[sec:experiments\]]{}, maintaining a strict separation between the representational spaces of question and image, which can interact only through interpretable discrete distributions, greatly enhances the generalizability of the network and improves its transparency. In the following, we describe the cell’s three components: control, read and write units, and detail their formal specification. Unless otherwise stated, all the vectors are of dimension $d$. ### The Control Unit {#sec:CU} $$\begin{gathered} {\mathit{cq}_i} = {W}^{d\times 2d}\left[\boldsymbol{c_{i-1}},\boldsymbol{q_i}\right] + {b}^d\tag{c1} \\[2pt] {\mathit{ca}_{i,s}} = W^{1\times d}({\mathit{cq}_i} \odot \boldsymbol{\mathit{cw}_s}) + b^1\tag{c2.1} \\[5pt] {\mathit{cv}_{i,s}} = \textrm{softmax}({{ca}_{i,s}})\tag{c2.2} \\[-0.5pt] \boldsymbol{c_{i}} = \sum\limits_{s = 1}^S {\mathit{cv}_{i,s}} \cdot {\boldsymbol{\mathit{cw}}_s}\tag{c2.3}\end{gathered}$$ The control unit (see [figure \[fig:control\]]{}) determines the reasoning operation that should be performed at each step $i$, attending to some part of the question and updating the control state $\boldsymbol{c_i}$ accordingly. It receives the contextual question words ${\boldsymbol{\mathit{cw}}_1,\ldots,\boldsymbol{\mathit{cw}}_S}$, the question position-aware representation $\boldsymbol{q_i}$, and the control state from the preceding step $\boldsymbol{c_{i-1}}$ and consists of two stages: 1. First, we combine $\boldsymbol{q_i}$ and $\boldsymbol{c_{i-1}}$ through a linear transformation into ${{cq}_i}$, taking into account both the overall question representation $\boldsymbol{q_i}$, biased towards the $i^{th}$ reasoning step, as well as the preceding reasoning operation $\boldsymbol{c_{i-1}}$. This allows the cell to base its decision for the $i^{th}$ reasoning operation $\boldsymbol{c_i}$ on the previously performed operation $\boldsymbol{c_{i-1}}$. 2. Subsequently, we *cast* ${\mathit{cq}_i}$ onto the space of the question words. Specifically, this is achieved by measuring the similarity between ${{cq}_i}$ and each question word ${\boldsymbol{\mathit{cw}}_s}$ and passing the result through a softmax layer, yielding an attention distribution over the question words ${\boldsymbol{\mathit{cw}}_1,\ldots,\boldsymbol{\mathit{cw}}_S}$. Finally, we sum the words according to this distribution to produce the reasoning operation $\boldsymbol{c_i}$, represented *in terms of* the question words. The casting of ${\mathit{cq}_i}$ onto question words serves as a form of regularization that restricts the space of the valid reasoning operations by anchoring them back in the original question words, and due to the use of soft attention may also improve the MAC cell transparency, since we can interpret the control state content and the cell’s consequent behavior based on the words it attends to. ### The Read Unit {#sec:RU} ![**The Read Unit (RU) architecture.** The read unit retrieves information from the knowledge base that is necessary for performing the current reasoning operation (control) and potentially related to previously obtained intermediate results (memory). It extracts the information by performing a two-stage attention process over the knowledge base elements. See [section \[sec:RU\]]{} for details.[]{data-label="fig:read"}](ruh.pdf){width="1.0\linewidth"} $$\begin{gathered} \begin{split} I_{i,h,w} = & [W_m^{d\times d}\boldsymbol{m_{i-1}} + b_m^d] \; \odot \\[2pt] & [W_k^{d\times d}{{\boldsymbol{k}_{h,w}}} + b_k^d] \end{split}\tag{r1} \\[5pt] {{{I'_{i,h,w}}}} = W^{d\times 2d}\left[ {{I_{i,h,w}}},{\boldsymbol{k}_{h,w}} \right] + b^d\tag{r2} \\[3pt] {{\mathit{ra}_{i,h,w}}} = W^{d\times d}({\boldsymbol{c_i} \odot {{{I'_{i,h,w}}}}}) + b^d\tag{r3.1} \\[5pt] {{\mathit{rv}_{i,h,w}}} = \textrm{softmax}({{\mathit{ra}_{i,h,w}}})\tag{r3.2} \\[-0.5pt] \boldsymbol{r_i} = \sum\limits_{h,w = 1,1}^{H,W} {\mathit{rv}_{i,h,w}}\tag{r3.3} \cdot {\boldsymbol{k}_{h,w}} \end{gathered}$$ For the $i^{th}$ step, the read unit (see [figure \[fig:read\]]{}) inspects the knowledge base (the image) and retrieves the information $\boldsymbol{r_i}$ that is required for performing the $i^{th}$ reasoning operation $\boldsymbol{c_i}$. The content’s relevance is measured by an attention distribution $\boldsymbol{\mathit{rv}_i}$ that assigns a probability to each element in the knowledge base ${\boldsymbol{k}_{h,w}^d}$, taking into account the current reasoning operation $\boldsymbol{c_i}$ and the prior memory $\boldsymbol{m_{i-1}}$, the intermediate result produced by the preceding reasoning step. The attention distribution is computed in several stages: 1. First, we compute the direct interaction between the knowledge-base element ${\boldsymbol{k}_{h,w}}$ and the memory $\boldsymbol{m_{i-1}}$, resulting in $I_{i,h,w}$. This term measures the relevance of the element to the preceding intermediate result, allowing the model to perform transitive reasoning by considering content that now seems important in light of information obtained from the prior computation step. 2. Then, we concatenate the element ${\boldsymbol{k}_{h,w}}$ to $I_{i,h,w}$ and pass the result through a linear transformation, yielding $I'_{i,h,w}$. This allows us to also consider new information that is *not* directly related to the prior intermediate result, as sometimes a cogent reasoning process has to combine together *independent* facts to arrive at the answer (e.g., for a logical OR operation, set union and counting). 3. Finally, aiming to retrieve information that is relevant for the reasoning operation $\boldsymbol{c_i}$, we measure its similarity to each of the interactions $I_{i,h,w}$ and pass the result through a softmax layer. This produces an attention distribution over the knowledge base elements, which we then use to compute a weighted average over them – $\boldsymbol{r_i}$. [r]{}[0.32]{} To give an example of the read unit operation, consider the question in [figure \[fig:rexample\]]{}, which refers to the purple cylinder in the image. Initially, no cue is provided to the model to attend to the cylinder, since no direct mention of it is given in the question. Instead, the model approaches the question in steps: in the first iteration it attends to the , updating $\boldsymbol{m_1}$ accordingly to the visual representation of the block. At the following step, the control unit realizes it should now look for of the block, storing that in $\boldsymbol{c_2}$. Then, when considering *both* $\boldsymbol{m_1}$ and $\boldsymbol{c_2}$, the read unit realizes it should look for “the sphere in front" ($\boldsymbol{c_2}$) of the blue block (stored in $\boldsymbol{m_1}$), thus finding the cyan sphere and updating $\boldsymbol{m_2}$. Finally, a similar process repeats in the next iteration, allowing the model to traverse from the cyan ball to the final objective – , and answer the question correctly. ### The Write Unit {#sec:WU} The write unit (see [figure \[fig:write\]]{}) is responsible for computing the $i^{th}$ intermediate result of the reasoning process and storing it in the memory state $\boldsymbol{m_i}$. Specifically, it integrates the information retrieved from the read unit $\boldsymbol{r_i}$ with the preceding intermediate result $\boldsymbol{m_{i-1}}$, guided by the $i^{th}$ reasoning operation $\boldsymbol{c_i}$. The integration proceeds in three steps, the first mandatory while the others are optional[^1]: 1. First, we combine the new information $\boldsymbol{r_i}$ with the prior intermediate result $\boldsymbol{m_{i-1}}$ by a linear transformation, resulting in ${m_{i}^{info}}$. 2. **Self-Attention** (Optional). To support non-sequential reasoning processes, such as trees or graphs, we allow each cell to consider all previous intermediate results, rather than just the preceding one $\boldsymbol{m_{i-1}}$: We compute the similarity between the $i^{th}$ operation $\boldsymbol{c_i}$ and the previous ones ${\boldsymbol{c}_1,\ldots,\boldsymbol{c}_{i-1}}$ and use it to derive an attention distribution over the prior reasoning steps ${\mathit{sa}_{i,j}}$ for $j = 0,\ldots,{i-1}$. The distribution represents the relevance of each previous step $j$ to the current one $i$, and is used to compute a weighted average of the memory states, yielding ${m_{i}^{sa}}$, which is then combined with ${m_{i}^{info}}$ to produce ${m'_{i}}$. Note that while we compute the attention based on the *control states*, we use it to average over the *memory states*, in a way that resembles Key-Value Memory Networks [@kvmn]. 3. **Memory Gate** (Optional). Not all questions are equally complex – some are simpler while others are more difficult. To allow the model to dynamically adjust the reasoning process length to the given question, we add a sigmoidal gate over the memory state that interpolates between the previous memory state $\boldsymbol{m_{i-1}}$ and the new candidate ${m'_{i}}$, *conditioned on the reasoning operation $\boldsymbol{c_{i}}$*. The gate allows the cell to skip a reasoning step if necessary, passing the previous memory value further along the network, dynamically reducing the effective length of the reasoning process as demanded by the question. ![**The Write Unit (WU) architecture**. The write unit integrates the information retrieved from the knowledge base into the recurrent memory state, producing a new intermediate result $\boldsymbol{m_i}$ that corresponds to the reasoning operation $\boldsymbol{c_i}$. See [section \[sec:WU\]]{} for details. []{data-label="fig:write"}](wun.pdf){width="1.0\linewidth"} $$\begin{gathered} m_i^{info} = {W}^{d\times 2d}[\boldsymbol{r_{i}},\boldsymbol{m_{i-1}}] + {b}^d\tag{w1} \\[1pt] {\mathit{sa}_{ij}} = \textrm{softmax}\left({W}^{1\times d}{\left( \boldsymbol{c_i} \odot \boldsymbol{c_j} \right)} + {b^1} \right)\tag{w2.1} \\[-3pt] {m_i^{sa}} = \sum\limits_{j = 1}^{i - 1} {\mathit{sa}_{ij}} \cdot \boldsymbol{{m}_{j}}\tag{w2.2} \\[1pt] {m'_i} = {W_s^{d\times d}}{m_i^{sa}} + {W_p^{d\times d}} m_i^{info} + b^d\tag{w2.3} \\[4pt] {c'_i} = W^{1\times d}\boldsymbol{c_i} + b^1\tag{w3.1} \\[2pt] \boldsymbol{m_i} = \sigma \left( {{c'_i}} \right) \boldsymbol{m_{i - 1}} + \left( {1 - \sigma \left( {{c'_i}} \right)} \right){m'_i}\tag{w3.2} \end{gathered}$$ The Output Unit {#sec:output} --------------- [r]{}[0.36]{} {width="1.0\linewidth"} The output unit predicts the final answer to the question based on the question representation $q$ and the final memory $\boldsymbol{m_p}$, which represents the final intermediate result of the reasoning process, holding relevant information from the knowledge base.[^2] For CLEVR, where there is a fixed set of possible answers, the unit processes the concatenation of $q$ and $\boldsymbol{m_p}$ through a 2-layer fully-connected softmax classifier that produces a distribution over the candidate answers. experiments Conclusion ========== We have introduced the Memory, Attention and Composition (MAC) network, an end-to-end differentiable architecture for machine reasoning. The model solves problems by decomposing them into a series of inferred reasoning steps that are performed successively to accomplish the task at hand. It uses a novel recurrent MAC cell that aims to formulate the inner workings of a single universal reasoning operation by maintaining a separation between memory and control. These MAC cells are chained together to produce explicit and structured multi-step reasoning processes. We demonstrate the versatility, robustness and transparency of the model through quantitative and qualitative studies, achieving state-of-the-art results on the CLEVR task for visual reasoning, and generalizing well even from a 10% subset of the data. The experimental results further show that the model can adapt to novel situations and diverse language, and generate interpretable attention-based rationales that reveal the underlying reasoning it performs. While CLEVR provides a natural testbed for our approach, we believe that the architecture will prove beneficial for other multi-step reasoning and inference tasks, including reading comprehension, textual question answering, and real-world VQA. Acknowledgments =============== We wish to thank Justin Johnson, Aaron Courville, Ethan Perez, Harm de Vries, Mateusz Malinowski, Jacob Andreas, and the anonymous reviewers for the helpful suggestions, comments and discussions. Stanford University gratefully acknowledges the support of the Defense Advanced Research Projects Agency (DARPA) Communicating with Computers (CwC) program under ARO prime contract no. W911NF15-1-0462 for supporting this work. supplement [^1]: Both self-attention connections as well as the memory gate serve to reduce long-term dependencies. However, note that for the CLEVR dataset we were able to maintain almost the same performance with the first step only, and so we propose the second and third ones as optional extensions of the basic write unit, and explore their impact on the model’s performance in [section \[sec:ablations\]]{}. [^2]: Note that some questions refer to important aspects that do not have counterpart information in the knowledge base, and thus considering both the question and the memory is critical to answer them.

--- author: - 'Philip H. Handle' - Francesco Sciortino - Nicolas Giovambattista bibliography: - 'tip4p2005-polyam.bib' title: 'Supplementary Material: “Glass Polymorphism in TIP4P/2005 Water: A Description Based on the Potential Energy Landscape Formalism"\' --- In this supplementary material we compare the PEL regions sampled by the system in the (stable/metastable) equilibrium liquid state and during the pressure-induced LDA$\rightleftarrows$HDA transformations. The data for the equilibrium liquid stem from Ref. . We follow the structure of the main document and discuss, separately, the influence of temperature $T$, cooling rate $q_\text{c}$ and compression rate $q_\text{P}$. Temperature Dependence ====================== ![ (a) IS Pressure, $P_\text{IS}$, and (b) shape function, $\mathcal S$, as a function of the IS energy sampled by the system during the pressure-induced LDA$\rightleftarrows$HDA transformation (solid lines). Sharp and light colors correspond, respectively, to the LDA$\rightarrow$HDA and HDA$\rightarrow$LDA transformations. Data is taken from Fig. 6 of the main manuscript. LDA is prepared by isobaric cooling at $P=0.1$ MPa using a cooling rate $q_\text{c}=30$ K/ns. The compression/decompression rate is $q_\text{P}=300$ MPa/ns. The circles represent the $P_\text{IS}(e_\text{IS})$ and $\mathcal{S}(e_\text{IS})$ for the stable/metastable equilibrium liquid from Ref.  with densities in the range $0.90$–$1.42$ g/cm$^3$ (bottom to top lines, in steps of 0.04 g/cm$^3$) and temperatures from $T\geq200$ K (blues circles) to $T=270$K (magenta circles). At low temperatures, the PEL regions sampled by the equilibrium liquid and the amorphous ices are different. []{data-label="fig:delta-tdep"}](pis-vgl-tdep.pdf "fig:"){width="49.00000%"} ![ (a) IS Pressure, $P_\text{IS}$, and (b) shape function, $\mathcal S$, as a function of the IS energy sampled by the system during the pressure-induced LDA$\rightleftarrows$HDA transformation (solid lines). Sharp and light colors correspond, respectively, to the LDA$\rightarrow$HDA and HDA$\rightarrow$LDA transformations. Data is taken from Fig. 6 of the main manuscript. LDA is prepared by isobaric cooling at $P=0.1$ MPa using a cooling rate $q_\text{c}=30$ K/ns. The compression/decompression rate is $q_\text{P}=300$ MPa/ns. The circles represent the $P_\text{IS}(e_\text{IS})$ and $\mathcal{S}(e_\text{IS})$ for the stable/metastable equilibrium liquid from Ref.  with densities in the range $0.90$–$1.42$ g/cm$^3$ (bottom to top lines, in steps of 0.04 g/cm$^3$) and temperatures from $T\geq200$ K (blues circles) to $T=270$K (magenta circles). At low temperatures, the PEL regions sampled by the equilibrium liquid and the amorphous ices are different. []{data-label="fig:delta-tdep"}](shape-vgl-tdep.pdf "fig:"){width="49.00000%"} Fig. \[fig:delta-tdep\](a) shows the pressure of the inherent structures (IS) sampled by the system during the LDA$\rightleftarrows$HDA transformation as a function of the corresponding IS energy, $P_\text{IS}(e_\text{IS})$. For comparison, also included is the $P_\text{IS}(e_\text{IS})$ of the liquid in (stable and metastable) equilibrium at $200\leq T \leq270$ K and $0.90\leq\rho\leq1.42$ g/cm$^3$ taken from Ref.  (circles). Similarly, Fig. \[fig:delta-tdep\](b) shows the shape function as function of the IS energy, $\mathcal{S}(e_\text{IS})$, sampled by the system in liquid state and during the LDA$\rightleftarrows$HDA transformation. It follows from Fig. \[fig:delta-tdep\] that the system samples different regions of the PEL during compression (sharp color lines) and decompression (light color lines). At high compression/decompression temperatures, within the liquid state, the system samples the same IS accessed by the liquid in equilibrium. For example, the values of $P_\text{IS}(e_\text{IS})$ and $\mathcal{S}(e_\text{IS})$ for the equilibrium liquid at $T=270$ K (magenta circles) practically coincide with the corresponding values obtained here during compression/decompression at $T=280$  K (violet lines). However, at low temperatures, the amorphous ices at $T=20$–80 K (orange and red lines) can access IS with $e_\text{IS}<-60$ kJ/mol and $P_\text{IS}>0.5$ GPa while IS sampled by the liquid with $e_\text{IS}<-60$ kJ/mol are characterized by $P_\text{IS}<0$. Similar conclusions follow from Fig. \[fig:delta-tdep\](b) regarding $\mathcal{S}(e_\text{IS})$. Cooling Rate Dependence {#sec:qdep} ======================= Figs. \[fig:delta-qdep\](a) and \[fig:delta-qdep\](b) show $P_\text{IS}(e_\text{IS})$ and $\mathcal{S}(e_\text{IS})$ during the pressure-induced LDA$\rightleftarrows$HDA transformation at $T=80$ K and for different cooling rates $q_\text{c}$. The values of $P_\text{IS}(e_\text{IS})$ and $\mathcal{S}(e_\text{IS})$ sampled by the liquid, shown in Fig. \[fig:delta-tdep\], are also included. We again find that the PEL regions sampled by the liquid and the amorphous ices are, in general, different. Specifically, the IS sampled by the system during the compression-induced LDA$\rightarrow$HDA transformation (sharp color lines) are never sampled by the liquid (circles). In particular, the regions of the PEL sampled by the liquid and the amorphous ices during compression become increasingly different as $q_\text{c}$ decreases. During decompression (light color lines), the PEL regions accessible to the amorphous ices and the liquid also differ. We note that, at low pressures (low density), the amorphous ices seem to explore the same IS accessed by the liquid in equilibrium \[i.e., the light color lines overlap with the circles in Figs.  \[fig:delta-qdep\](a) and (b)\]. However, when this occurs, the IS of the amorphous ice and the liquid have indeed different densities and hence, are different (cf. Refs. ). We also note that, as observed in Fig. \[fig:delta-tdep\], the regions explored by the system during the LDA$\rightarrow$HDA and HDA$\rightarrow$LDA transformations do not necessarily coincide at the temperature and rates explored, i.e., the observed LDA$\rightarrow$HDA transformation is not reversible. ![ (a) IS Pressure, $P_\text{IS}$, and (b) shape function, $\mathcal S$, as a function of the IS energy sampled by the system during the pressure-induced LDA$\rightleftarrows$HDA transformation (solid lines) at 80 K. Sharp and light colors correspond, respectively, to the LDA$\rightarrow$HDA and HDA$\rightarrow$LDA transformations. Data is taken from Fig. 10 of the main manuscript. LDA is prepared by isobaric cooling at $P=0.1$ MPa using using different cooling rates $q_\text{c}=0.01$–100 K/ns. The compression/decompression rate is $q_\text{P}=10$ MPa/ns. The circles represent the $P_\text{IS}(e_\text{IS})$ and $\mathcal{S}(e_\text{IS})$ for the stable/metastable equilibrium liquid (see Fig. \[fig:delta-tdep\]). The PEL regions sampled by the equilibrium liquid and the amorphous ices are, in general, different. []{data-label="fig:delta-qdep"}](pis-vgl-qdep.pdf "fig:"){width="49.00000%"} ![ (a) IS Pressure, $P_\text{IS}$, and (b) shape function, $\mathcal S$, as a function of the IS energy sampled by the system during the pressure-induced LDA$\rightleftarrows$HDA transformation (solid lines) at 80 K. Sharp and light colors correspond, respectively, to the LDA$\rightarrow$HDA and HDA$\rightarrow$LDA transformations. Data is taken from Fig. 10 of the main manuscript. LDA is prepared by isobaric cooling at $P=0.1$ MPa using using different cooling rates $q_\text{c}=0.01$–100 K/ns. The compression/decompression rate is $q_\text{P}=10$ MPa/ns. The circles represent the $P_\text{IS}(e_\text{IS})$ and $\mathcal{S}(e_\text{IS})$ for the stable/metastable equilibrium liquid (see Fig. \[fig:delta-tdep\]). The PEL regions sampled by the equilibrium liquid and the amorphous ices are, in general, different. []{data-label="fig:delta-qdep"}](shape-vgl-qdep.pdf "fig:"){width="49.00000%"} Compression Rate Dependence =========================== Figs. \[fig:delta-pidep\](a) and \[fig:delta-pidep\](b) show show $P_\text{IS}(e_\text{IS})$ and $\mathcal{S}(e_\text{IS})$ during the LDA$\rightleftarrows$HDA transformations at $T=80$ K and for different compression/decompression rates $q_\text{p}=0.1$–1000 MPa/ns. The results are similar to those presented in Sec. \[sec:qdep\]. Briefly, the IS sampled by the liquid and the amorphous ices differ, and the transformation is not reversible, i.e., the IS sampled by the system during the LDA$\rightarrow$HDA and HDA$\rightarrow$LDA transformation are also different. ![ (a) IS Pressure, $P_\text{IS}$, and (b) shape function, $\mathcal S$, as a function of the IS energy sampled by the system during the pressure-induced LDA$\rightleftarrows$HDA transformation (solid lines) at 80 K. Sharp and light colors correspond, respectively, to the LDA$\rightarrow$HDA and HDA$\rightarrow$LDA transformations. Data is taken from Fig. 14 of the main manuscript. LDA is prepared by isobaric cooling at $P=0.1$ MPa using using a cooling rate $q_\text{c}=0.1$ K/ns. Different compression/decompression rates $q_\text{P}=0.1$–$1000$ MPa/ns are used. The circles represent the $P_\text{IS}(e_\text{IS})$ and $\mathcal{S}(e_\text{IS})$ for the stable/metastable equilibrium liquid (see Fig. \[fig:delta-tdep\]). The PEL regions sampled by the equilibrium liquid and the amorphous ices are, in general, different. []{data-label="fig:delta-pidep"}](pis-vgl-pidep.pdf "fig:"){width="49.00000%"} ![ (a) IS Pressure, $P_\text{IS}$, and (b) shape function, $\mathcal S$, as a function of the IS energy sampled by the system during the pressure-induced LDA$\rightleftarrows$HDA transformation (solid lines) at 80 K. Sharp and light colors correspond, respectively, to the LDA$\rightarrow$HDA and HDA$\rightarrow$LDA transformations. Data is taken from Fig. 14 of the main manuscript. LDA is prepared by isobaric cooling at $P=0.1$ MPa using using a cooling rate $q_\text{c}=0.1$ K/ns. Different compression/decompression rates $q_\text{P}=0.1$–$1000$ MPa/ns are used. The circles represent the $P_\text{IS}(e_\text{IS})$ and $\mathcal{S}(e_\text{IS})$ for the stable/metastable equilibrium liquid (see Fig. \[fig:delta-tdep\]). The PEL regions sampled by the equilibrium liquid and the amorphous ices are, in general, different. []{data-label="fig:delta-pidep"}](shape-vgl-pidep.pdf "fig:"){width="49.00000%"}

[**Noncommutative $SU(N)$ theories, the axial anomaly, Fujikawa’s method and the Atiyah-Singer index.**]{}\ 1.2 true cm [C.P. Martín]{}[^1] and C. Tamarit[^2] 0.3 true cm [*Departamento de Física Teórica I, Facultad de Ciencias Físicas\ Universidad Complutense de Madrid, 28040 Madrid, Spain*]{}\ 0.75 true cm 0.25 true cm Fujikawa’s method is employed to compute at first order in the noncommutative parameter the $U(1)_A$ anomaly for noncommutative $SU(N)$. We consider the most general Seiberg-Witten map which commutes with hermiticity and complex conjugation and a noncommutative matrix parameter, $\theta^{\mu\nu}$, which is of “magnetic” type. Our results for $SU(N)$ can be readily generalized to cover the case of general nonsemisimple gauge groups when the symmetric Seiberg-Witten map is used. Connection with the Atiyah-Singer index theorem is also made. [*PACS:*]{} 11.15.-q; 11.30.Rd; 12.10.Dm\ [*Keywords:*]{} $U(1)_A$ anomaly, Seiberg-Witten map, noncommutative gauge theories. It is difficult to overstate the importance of the abelian chiral anomaly in Physics [@Frohlich:2000en; @Adler:2004ih]. A most beautiful explanation of the existence of this anomaly was supplied by Fujikawa [@Fujikawa:1980eg], who showed that it comes from the lack of invariance of the fermionic measure under chiral transformations. Fujikawa’s method of computing anomalies also provides a way of easily exhibiting the relationship between the abelian chiral anomaly and the Atiyah-Singer index theorem [@Fujikawa:1980eg; @Alvarez-Gaume:1985ex]. The method in question is called a nonperturbative method since no expansion in the coupling constant is carried out. The purpose of this note is to use Fujikawa’s method to work out the abelian chiral anomaly for noncommutative $SU(N)$ gauge theories with Dirac fermions [@Jurco:2001rq] up to first order in the noncommutative matrix parameter $\theta^{\mu\nu}$ and for the most general Seiberg-Witten map which is local at each order in $\theta^{\mu\nu}$ and commutes with hermiticity and complex conjugation. The case of noncommutive gauge theories with Dirac fermions and with a nonsemisiple gauge group is also analysed when the theory is defined by means of the symmetric Seiberg-Witten map [@Calmet:2001na; @Aschieri:2002mc]. Let $a_{\mu}$ be an ordinary $SU(N)$ gauge field. Let $\psi$ denote an ordinary massive Dirac fermion carrying a given representation of $SU(N)$. Following ref. [@Jurco:2001rq], we construct the noncommutative fields $A_{\mu}$ –the gauge field– and $\Psi$ –the Dirac fermion– by applying the Seiberg-Witten map to their ordinary counterparts. As in ref. [@Aschieri:2002mc; @Aschieri:2004ka], we shall assume that $\psi$ does not enter the Seiberg-Witten map that yields $A_{\mu}$, that this map renders $A_{\mu}$ hermitian and that it commutes, the Seiberg-Witten map, with complex conjugation when acting on fermion fields. We shall also assume that at each order in the noncommutative matrix parameter $\theta^{\mu\nu}$ the Seiberg-Witten map is local, i.e., that it is a polynomial of the fields and their derivatives with dimensionless coefficients other than $\theta^{\mu\nu}$. Note that if, barring $\theta^{\mu\nu}$, we would allow for dimensionful coefficients, such as masses, [*etc ...*]{}, then, the Seiberg-Witten map would have an infinite number of terms at each order in $\theta^{\mu\nu}$ and the theory would not be local at each order in $\theta^{\mu\nu}$. It is not difficult to show that at first order in $\theta^{\mu\nu}$ the most general Seiberg-Witten map that fulfils the previous requirements reads $$\begin{array}{l} {A_\mu=a_\mu-\frac{1}{4}\{\aa,\Db\am+f_{\b\mu}\}+i\left(\kappa_2-\frac{\kappa_1}{2}\right)\,\theta^{\alpha\beta}\,\Dcal_\mu[\aa,\ab]+\kappa_3\,\theta^{\a\b}\,\Dcal_\mu f_{\a\b}+\kappa_4\,{\theta_\mu}^\b \Dcal^\nu f_{\nu\beta},}\\ {\Psi=\psi+\Big[-\frac{1}{2}\theta^{\alpha\beta}a_\a\Db+\frac{i}{4}\theta^{\a\b} \aa\,\ab+i\kappa_3\,\theta^{\a\b}\,f_{\a\b}-\left(\kappa_2-\frac{\kappa_1}{2}\right)\,\theta^{\a\b}\,[\aa,\ab]+z_1\,\theta^{\a\b}\,f_{\a\b}}\\ {\phantom{\Psi=\psi}+\frac{i}{2}\,z_2\,\theta^{\a\b}[\g_\b,\g_\r]D_\a D^\r-\frac{z_3}{2}\,\theta^{\a\b}\,[\g_\a,\g_\r]{f^\rho}_\b +{i}z_4\,\theta^{\a\b}\,\g_\a\g_\b D^2-z_5\,\theta^{\a\b}\,\g_\a\g_\b\g_\mu\g_\nu f^{\mu\nu}\Big]\psi.}\\ \end{array} \label{SW}$$ Hermiticity of $A_\mu$ demands $\kappa_{i}$, $i=1,...,4$, to be real numbers. That the Seiberg-Witten map commutes with complex conjugation –i.e., $\Psi[\psi,a_{\mu},\theta^{\mu\nu}]= \Psi[\psi^{*},-a^{*}_{\mu},-\theta^{\mu\nu}]$, see refs. [@Aschieri:2002mc; @Aschieri:2004ka]– leads to $z_2=z_3=z_4=z_5=0$ and restricts $z_1$ to be a real number. Notice that the terms in the Seiberg-Witten map that go with $\kappa_4$ and $z_1$ correspond, respectively, to field redefinitions of $a^{\mu}$ and $\psi$, so that their actual values have no effect on physical quantities. However, we shall keep these parameters arbitrary and see whether they can be used to simplify the values of the (non-physical) Green functions of the fields we shall compute. The action of the noncommutative $SU(N)$ theory we shall study is given by $$S\,=\,\idx -\frac{1}{4g^2}\mathrm{Tr}\,F^{\mu\nu}{\star}F_{\mu\nu}\,+\, \Psib{\star}(i\Dirac_{{\star}}-m)\Psi.$$ $\mathrm{Tr}$ denotes the trace operation on the matrix representation of $SU(N)$ carried by $\psi$. In the previous equation, $F^{\mu\nu}=\partial_{\mu}A_\nu-\partial_{\nu}A_\mu-i[A_\mu,A_\nu]_{{\star}}$, $\Dirac_{{\star}}=\prslash-i\Aslash{\star}$ and $A_{\mu}$ and $\Psi$ are given by the Seiberg-Witten map above. ${\star}$ stands for the Moyal product of functions:$(f{\star}g)(x)=f(x)\exp(\frac{i}{2}\theta^{\alpha\beta} \overleftarrow{\partial_{\alpha}}\overrightarrow{\partial_{\beta}})g(x)$. Since we shall use Fujikawa’s method to compute the abelian anomaly, we must define the theory for the Euclidean signature of space-time. Upon Wick rotation –we shall play it safe [@Gomis:2000zz] and consider $\theta^{\mu\nu}$ to be of “magnetic” type: $\theta^{0 i}=0$–, we obtain a theory whose action, $S_{E}$, at first order in $\theta^{\mu\nu}$ reads: $$S_{E}=S_{YM}-\idx\;\psib\,({\cal K}+i{\cal M}(x))\,\psi. \label{SEaction}$$ $S_{YM}$ is the contribution coming from the pure noncommutative Yang-Mills action –whose actual value will be irrelevant to us. The differential operator ${\cal K}$ and the function ${\cal M}(x)$ are given by $$\begin{array}{l} {{\cal K}\,=\,i\Dirac\,+\,i \Rs}\\ {\Rs\,=\,(-\frac{1}{4}+2z_1) \,\theta^{\a\b}\,f_{\a\b} \g^\mu D_\mu -\frac{1}{2}\,\theta^{\a\b}\,\g^\rho f_{\r\a} D_\b + z_1\,\theta^{\a\b}\,\g^\mu\,\Dcal_\mu f_{\a\b}- i\kappa_4\,{\theta_\mu}^\beta\,\Dcal^\nu f_{\nu\b}\, \g^\mu,}\\ {{\cal M}(x)\,=\,m\,[\,1+(-\frac{1}{4}+2z_1) \,\theta^{\a\b}\,f_{\a\b}(x)\,].}\\ \end{array} \label{Kop}$$ The operator $i \Rs$ is gauge covariant and formally self-adjoint and, in ${\cal K}$, it should be understood as a perturbation of the ordinary Dirac operator $i\Dirac$. Note that this perturbation does not destroy the pairing between positive and negative eigenvalues that occurs in the spectrum of $i\Dirac$. We shall assume that the ordinary Dirac operator has a discrete spectrum. The latter is achieved by imposing on the fields boundary conditions that allow, by means of the stereographic projection, for the compactification of ordinary 4-dimensional Euclidean space to a 4-dimensional unit sphere [@Jackiw:1976dw; @Chadha:1977mh]. In particular, we shall assume that the ordinary gauge fields satisfy the standard boundary condition: $a_{\mu}(x)\rightarrow i g(x)\partial_{\mu}g^{-1}(x)+O(1/|x|^2)$ as $x\rightarrow\infty$. In keeping with the philosophy adopted in this paper, we shall take for granted that the eigenvalues and eigenfunctions of $K$ can be computed by employing standard perturbation theory, using $i \Rs$ as a perturbation. Thus, following Fujikawa [@Fujikawa:1980eg], we shall use the eigenfunctions of ${\cal K}$ to define the fermionic measure of the path integral. One expands first the fermion fields $\psi(x)=\sum_{n}\,a_n \varphi_{n}(x)$, $\psib(x)=\sum_{n}\,\bar{b}_n \varphi^{\dagger}_{n}(x)$, in terms of the of a orthonormal set of eigenfunctions of ${\cal K}$, say $\{\varphi_{n}(x)\}_n$. Recall that $a_n$ and $\bar{b}_n$ are Grassmann variables. Then, the fermionic measure is defined as follows $d\psi d\psib = \prod_{n}da_n d\bar{b}_n$. The generating functional, $Z[J^{a\mu},\omega,\bar{\omega}]$, of the complete Green functions of our theory is defined by the following path integral $$Z[J^a_{\mu},\omega,\bar{\omega}]=\frac{1}{N}\int\,d\mu\quad e^{-S_{E}+\idx\;[J^{a\mu}(x)a^a_{\mu}(x)+\bar{\omega}(x)\psi(x)+\psib(x) \omega(x)]}, \label{pathintegral}$$ where $S_{E}$ is defined by eqs. (\[SEaction\]) and (\[Kop\]), and the path integral measure $d\mu$ is equal to $[da^a_{\mu}]\prod_{n}da_n d\bar{b}_n$. $[da^a_{\mu}]$ is the measure over the space of gauge fields and contains the Faddeev-Popov factor. In the massless limit $S_{E}$ in eq. (\[SEaction\]) is invariant under the following infinitesimal $U(1)$ rigid chiral transformations $\delta\psi(x)=i\alpha\gamma_{5}\psi(x)$, $\delta\psib(x)=i\alpha\psib(x)\gamma_{5}$. Hence, under the infinitesimal local abelian chiral transformations $\delta\psi(x)=i\alpha(x)\gamma_{5}\psi(x)$, $\delta\psib(x)=i\alpha(x)\psib(x)\gamma_5$, the action $S_{E}$ undergoes the change $$\delta S_{E}\,=-\,\idx\; [\alpha(x)\partial_{\mu}j_{5}^{\mu}(x) -2\alpha(x)\psib(x){\cal M}(x)\gamma_5\psi(x)].$$ The current $j_{5}^{\mu}(x)$ is the $U(1)_A$ current, which is classically conserved and is given by $$j_{5}^{\mu}(x)\,=\,\psib(x)\left[\gamma^{\mu} -\left(\frac{1}{4}-2z_1\right) \, \theta^{\a\b}\,f_{\a\b} \g^\mu -\frac{1}{2}\,\theta^{\a\mu}\,\g^\rho f_{\r\a}\right]\gamma_5\psi(x). \label{current}$$ The measure of the path integral above also changes under the previous local chiral transformations: $d\mu\rightarrow d\mu\,[1-\idx \alpha(x) {\cal A}(x)]$. The symbol ${\cal A}(x)$ denotes the following formal expression $${\cal A}(x)\,=\, 2i \sum_{n}\varphi_{n}^{\dagger}(x)\gamma_5 \varphi_{n}(x). \label{formalanom}$$ These results and the fact that the path integral in eq. (\[pathintegral\]) does not change under changes of $\psi$ and $\psib$, leads to the following anomalous Ward identity $$\ll[\partial_{\mu}j_{5}^{\mu}(x) -2\psib(x){\cal M}(x)\gamma_5\psi(x)+i\bar{\omega}(x) \gamma_5\psi(x)+i\psib(x)\gamma_5\omega(x)]\gg= \ll{\cal A}(x)\gg,$$ where $\ll\cdots\gg=\frac{1}{N}\int\,d\mu\;\cdots\; e^{-S_{E}+\idx\;[J^{a\mu}(x)a^a_{\mu}(x)+\bar{\omega}(x)\psi(x)+\psib(x) \omega(x)]}$. As it stands in eq. (\[formalanom\]), ${\cal A}(x)$ is a formal object that is in demand of a proper definition. The latter is achieved as follows $${\cal A}(x)\,=\, 2i \lim_{\Lambda\rightarrow \infty}\; \sum_{n}\varphi_{n}^{\dagger}(x)\gamma_5 e^{-\frac{\lambda_n^2}{\Lambda^2}} \varphi_{n}(x)=2i \lim_{\Lambda\rightarrow \infty}\; \sum_{n}\varphi_{n}^{\dagger}(x)\gamma_5 e^{-\frac{{\cal K}^2}{\Lambda^2}} \varphi_{n}(x). \label{reganom}$$ $\lambda_n$ denotes a generic eigenvalue of ${\cal K}$, ${\cal K}$ being defined in eq. (\[Kop\]). The previous equation provides a gauge invariant definition of ${\cal A}(x)$ obtained by using the operator that gives the dynamics of fermions in the chiral limit. Besides, the spectrum of the operator $K$ has in common with the spectrum of $i\Dirac$ the following paring property of the nonvanishing eigenvalues: for each nonvanishing eigenvalue $\lambda_{n}$ with eigenfunction, say, $\varphi_{n}(x)$, there exists an eigenvalue $-\lambda_{n}$ with eigenfunction $\gamma_5\varphi_{n}(x)$. That this pairing property holds is necessary to establish a connection between of the value of ${\cal A}(x)$ and the index of the operator ${\cal K}(1+\gamma_5)/2$. We shall come back to this issue at the end of this paper. By going over to a plane wave basis, expanding the exponential $e^{-\frac{{\cal K}^2}{\Lambda^2}}$, dropping all contributions with more that one $\theta^{\mu\nu}$ and ignoring terms that yield traces of the type $\tr\,\g_5=\tr\,\g_5\g^\mu\g^\nu=0$, one obtains the following expression for the far r.h.s of eq. (\[reganom\]): $$\begin{array}{l} {\mathcal{A}(x)=\mathcal{A}_{\rm ordinary}(x)+\mathcal{A}_{\theta}(x)\,+\,O(\theta^2),}\\ {\mathcal{A}_{\rm ordinary}(x)=\sum_{k=2}^{\infty} \lim_{\Lambda\rightarrow\infty} 2i\idq e^{-q^2}\frac{\Lambda^{2(2-k)}}{k!}{\mathbf{Tr}}\g_5 \Dp^{2k}(\Lambda q)\id,}\\ {\mathcal{A}_{\theta}(x)\!=\!\sum_{k=2}^{\infty}\sum_{l=0}^{k-1}\lim_{\Lambda\rightarrow\infty}2i\idq\! e^{-q^2}\frac{\Lambda^{2(2-k)}}{k!}{\mathbf{Tr}}\g_5 \Dp^{2l}(\Lambda q)\left\{\Dirac(\Lambda q),\Rs(\Lambda q)\right\}\Dp^{2(k-1-l)}(\Lambda q)\id.}\\ \end{array} \label{A1A2}$$ $\id$ denotes the identity function on $\rm{I\!R}^4$. Notice that ${\mathbf{Tr}}$ also denotes trace over $\gamma$ matrices, when there occur such matrices in the expression affected by ${\mathbf{Tr}}$. The symbols $\Dirac(\Lambda q)$, $\Dp^{2}(\Lambda q)$ and $\Rs(\Lambda q)$ are defined, respectively, by the following equalities: $$\begin{array}{l} {\Dirac(\Lambda q)=\Dirac+i\Lambda\qslash ,\quad \Dp^{2}(\Lambda q)= D^2+2i\Lambda q\cdot D -\frac{i}{2} f_{\mu\nu}\gamma^{\mu} \gamma^{\nu},}\\ {\Rs(\Lambda q)=\Rs+i\left[(-\frac{1}{4}+2z_1)\,\theta^{\a\b}\,f_{\alpha\beta}\Lambda\qslash-\frac{1}{2}\theta^{\a\b}\gamma^\rho\,f_ {\rho\alpha}\Lambda q_\beta\right].}\\ \end{array}$$ $\Dirac$ and $\Rs$ are given in eq. (\[Kop\]). $\mathcal{A}_{\rm ordinary}(x)$ gives, of course, the abelian anomaly in ordinary 4-dimensional Euclidean space: $$\mathcal{A}_{\rm ordinary}(x)= \frac{i}{(4\pi)^2}\,\epsilon^{\mu\nu\rho\sigma} {\mathbf{Tr}}\, f_{\mu\nu}f_{\rho\sigma}. \label{ordinary}$$ Let us show next that the terms with $k$ such that $k\geq 5$ yield a vanishing contribution to $\mathcal{A}_{\theta}(x)$ in eq. (\[A1A2\]). Let us consider a term coming from the expansion of $$\begin{array}{l} {\Dp^{2l}(\Lambda q)\left\{\Dirac(\Lambda q), \Rs(\Lambda q)\right\}\Dp^{2(k-1-l)}(\Lambda q)}=\\ {\left[D^2-\frac{i}{2}\gamma^\mu\gamma^\nu f_{\mu\nu}+2i\Lambda q\cdot D\right]^{l}\left\{\Dirac(\Lambda q), \Rs(\Lambda q)\right\}\left[D^2-\frac{i}{2}\gamma^\mu\gamma^\nu f_{\mu\nu}+2i\Lambda q\cdot D\right]^{k-1-l}}, \end{array}$$ which contains $a$, $b$ and $c$ factors of type $D^2$, $\gamma^\mu\gamma^\nu f_{\mu\nu}$ and $2i\Lambda q\cdot D$, respectively. Since $\left\{\Dirac(\Lambda q),\Rs(\Lambda q)\right\}$ supplies two $\gamma$ matrices to the term in question, we conclude that the trace over the Dirac matrices will vanish unless $2b+2 \geq 4$, i.e., unless $b\geq 1$. Now, notice that $a+b+c=k-1$, so that $c$ is bounded from above as follows: $c\leq c_{\max}=k-1-b$. Hence, the highest power of $\Lambda$ that occurs in the term that we are analysing is $c_{max}+2=k-b+1$. Next, this term is to be multiplied by $\Lambda^{2(2-k)}$, so, for $k>2$, it will not survive in the large $\Lambda$ limit if $k-b+1< 2(k-2)$. This inequality and the constraint $b\geq 1$, leads to $k>4$. We thus conclude that $$\mathcal{A}_{\theta}= T_1+T_2+T_3, \label{ateta}$$ where $T_1$, $T_2$ and $T_3$ correspond, respectively, to the contributions to $\mathcal{A}_{\theta}(x)$ –see eq. (\[A1A2\])– with $k=2$, $3$ and $4$: $$\begin{array}{l} {T_1=\sum_{l=0}^{1}\limit 2i\idq e^{-q^2}\frac{1}{2}{\mathbf{Tr}}\g_5 \Dp^{2l}(\Lambda q)\left\{\Dirac(\Lambda q),\Rs(\Lambda q)\right\}\Dp^{2(1-l)}(\Lambda q)\id,}\\ { T_2=\sum_{l=0}^{2}\limit 2i\idq e^{-q^2}\frac{1}{3!\Lambda^2}{\mathbf{Tr}}\g_5 \Dp^{2l}(\Lambda q)\left\{\Dirac(\Lambda q),\Rs(\Lambda q)\right\}\Dp^{2(2-l)}(\Lambda q)\id,}\\ {T_3=\sum_{l=0}^{3}\limit 2i\idq e^{-q^2}\frac{1}{4!\Lambda^4}{\mathbf{Tr}}\g_5 \Dp^{2l}(\Lambda q)\left\{\Dirac(\Lambda q),\Rs(\Lambda q)\right\}\Dp^{2(3-l)}(\Lambda q)\id.} \end{array} \label{t1t2t3}$$ To carry out the computation of $T_1$, $T_2$ and $T_3$, we shall need the expansion of $\left\{\Dirac(\Lambda q),\Rs(\Lambda q)\right\}$ in powers of $\Lambda$: $$\begin{array}{l} {\left\{\Dirac(\Lambda q),\Rs(\Lambda q)\right\}=\g^\mu\g^\nu S_{\mu\nu}= \g^\mu\g^\nu(S_{\mu\nu}|_{\Lambda^0}+S_{\mu\nu}|_{\Lambda^1}+S_{\mu\nu}|_{\Lambda^2}),}\\ {S_{\mu\nu}|_{\Lambda^0}=d_1\theta^{\a\b}\left(\Dcal_\mu f_{\a\b}D_\nu+ 2 f_{\a\b}D_\mu D_\nu\right)-\frac{1}{2}\,\theta^{\a\b} \left(\Dcal_\mu f_{\nu\a} D_\beta+f_{\nu\a}D_\mu D_\b+f_{\mu\a}D_\b D_\nu\right),}\\ {\phantom{S_{\mu\nu}|_{\Lambda^0}}+z_1\,\theta^{\a\b} (\Dcal_\mu\Dcal_\nu f_{\a\b}+\Dcal_{\{\nu} f_{\a\b}D_{\mu\}^{o}}) -i\kappa_4\,({\theta_\nu}^\b\Dcal_\mu\Dcal^\a f_{\a\b}+ {\theta_{\{\nu}}^\b\Dcal^\a f_{\a\b}D_{\mu\}^{o}}),}\\ {S_{\mu\nu}|_{\Lambda^1}=i\Lambda (q_{\{\mu} R_{\nu\}^{o}})+ i d_1 \Lambda\,\theta^{\a\b}(q_\nu\Dcal_\mu f_{\a\b}+f_{\a\b}q_{\{\mu}D_{\nu\}^{o}}) -\frac{i}{2}\Lambda\,\theta^{\a\b}q_\beta(\Dcal_\mu f_{\nu\a}+ f_{\{\mu\a}D_{\nu\}^{o}})}\\ {S_{\mu\nu}|_{\Lambda^2}=-\Lambda^2\,\theta^{\a\b}\, q_{\{\mu}\left(d_1 f_{\a\b}q_{\nu\}^{o}}-\frac{1}{2} f_{\nu\}^{o}\a}q_\b\right).} \end{array} \label{DR+RD}$$ $\{\}^{o}$ indicates that only the indices $\mu$ and $\nu$ are symmetrized. $d_1=-1/4+z_1$. Let us work out $T_1$ in eq. (\[t1t2t3\]). Using the fact that $\tr\, \gamma_5=\tr\, \gamma_5\gamma_{\mu}\gamma_{\nu}=0$, one concludes that $$T_1=\limit\!\! 2i\!\idq\! e^{-q^2}\frac{1}{2}{\mathbf{Tr}}\g_5 \left\{-\frac{i}{2}\g^\mu\g^\nu f_{\mu\nu}\left\{\Dirac(\Lambda q), \Rs(\Lambda q)\right\}{\id}-\frac{i}{2}\left\{\Dirac(\Lambda q), \Rs(\Lambda q)\right\}\g^\mu\g^\nu f_{\mu\nu}\right\}.$$ Substituting in the previous equation the results in eq. (\[DR+RD\]), one shows that the contribution coming from $S_{\mu\nu}|_{\Lambda^1}$ vanishes upon integration over $q$ and that $S_{\mu\nu}|_{\Lambda^2}$ yields a vanishing contribution since $S_{\mu\nu}|_{\Lambda^2}$ is symmetric in $\mu$ and $\nu$. Then, the computation of the integral and traces on the r.h.s. of the previous equation leads to $$\begin{array}{l} {T_1=-\frac{2}{(4\pi)^2}\,\epsilon^{\mi\mii\miii\miv}{\mathbf{Tr}}(f_{\mi\mii} S_{\miii\miv}|_{\Lambda^0}+S_{\mi\mii}|_{\Lambda^0}f_{\miii\miv})}\\ {\phantom{T_1}=\frac{i}{8\pi^2}\,\theta^{\a\b}\,\epsilon^{\mi\mii\miii\miv} \,{\mathbf{Tr}}\,(2 d_1 f_{\a\b}f_{\mi\mii}f_{\miii\miv}-f_{\mii\a} f_{\mi\b}f_{\miii\miv})}\\ {\phantom{T_1=}+\frac{1}{16\pi^2}\,\theta^{\a\b}\, \epsilon^{\mi\mii\miii\miv}\,{\mathbf{Tr}}\,\big[f_{\mi\mii}(\Dcal_\miii f_{\miv\a} D_\b \id-2 d_1 \Dcal_\miii f_{\a\b} D_{\mu_4} \id)+\Dcal_\mi f_{\mii\a} f_{\miii\miv}D_\b \id}\\ {\phantom{T_1=}-2 d_1 \Dcal_\mi f_{\a\b}f_{\miii\miv} D_\mii \id\big]+\frac{i\kappa_4}{4\pi^2}\,{\theta_\miv}^\b\, \epsilon^{\mi\mii\miii\miv}\,{\mathbf{Tr}}\, \Dcal_\miii(f_{\mi\mii} \Dcal^\nu f_{\nu\b}),} \end{array} \label{T1}$$ where $\id$ is the unit function on $\rm{I\!R}^4$ and $d_1=-1/4+2z_1$. To calculate $T_2$ in eq. (\[t1t2t3\]) will be shall express it as the sum of two terms, say, $T_2^{(6\gamma)}$ and $T_2^{(4\gamma)}$, which involve the computation of the trace over six and four $\gamma$ matrices, respectively: $$\begin{array}{l} {T_2\,=\,T_2^{(6\gamma)}\,+\,T_2^{(4\gamma)},}\\ {T_2^{(6\gamma)}=\sum_{l=0}^{2}\limit \left(-\frac{i}{2}\right)\idq e^{-q^2}\frac{1}{(3)!\Lambda^2}{\mathbf{Tr}}\g_5 [(\g^\r\g^\s f_{\r\s})^l\g^\mu\g^\nu S_{\mu\nu}(\g^\kappa\g^\tau f_{\kappa\tau})^{(2-l)}]\id}\\ { T_2^{(4\gamma)}=\limit\idq e^{-q^2}\frac{1}{(3)!\Lambda^2}{\mathbf{Tr}}\g_5 [\{D^2\g^\r\g^\s f_{\r\s}+\g^\r\g^\s f_{\r\s}D^2,\g^\mu\g^\nu S_{\mu\nu}\}}\\ {\phantom{T_2^{(4\gamma)}=}+D^2\g^\mu\g^\nu S_{\mu\nu}\g^\r\g^\s f_{\r\s}+\g^\r\g^\s f_{\r\s}\g^\mu\g^\nu S_{\mu\nu}D^2]\id}\\ {\phantom{T_2^{(ii)}=}+\limit\idq e^{-q^2}\frac{2i}{(3)!\Lambda}{\mathbf{Tr}}\g_5 [\{q\cdot D\g^\r\g^\s f_{\r\s}+\g^\r\g^\s f_{\r\s}q\cdot D,\g^\mu\g^\nu S_{\mu\nu}\}}\\ {\phantom{T_2^{(ii)}=}+q\cdot D\g^\mu\g^\nu S_{\mu\nu}\g^\r\g^\s f_{\r\s}+\g^\r\g^\s f_{\r\s}\g^\mu\g^\nu S_{\mu\nu}q\cdot D\,]\id}.\\ \end{array} \label{T26g-4g}$$ In the limit $\Lambda\rightarrow\infty$, only the piece $S_{\mu\nu}|_{\Lambda^2}$ of $S_{\mu\nu}$ –see eq. (\[DR+RD\])– contributes to $T_2^{(6\gamma)}$. The computation of the corresponding integrals and some algebra yields $$T_2^{(6\gamma)}=\frac{i}{2(4\pi)^2}\,\theta^{\a\b}\,(-8d_1-1)\,\epsilon^{\mi\mii\miii\miv}\,{\mathbf{Tr}}\,f_{\a\b}f_{\mi\mii}f_{\miii\miv}. \label{T26g}$$ Since $\tr\, \gamma_5\gamma^{\rho}\gamma^{\sigma}\gamma^{\mu}\gamma^{\nu}\sim \epsilon^{\rho\sigma\mu\nu}$, one concludes that only the antisymmetric part of $S_{\mu\nu}$ in eq. (\[DR+RD\]) is relevant to the computation of $T_2^{(4\gamma)}$ in eq. (\[T26g-4g\]). Then, in the large $\Lambda$ limit we have $$\begin{array}{l} {T_2^{(4\gamma)}=\idq e^{-q^2}\frac{i}{(3)!}{\mathbf{Tr}}\g_5 [\{q\cdot D\g^\r\g^\s f_{\r\s}+\g^\r\g^\s f_{\r\s}q\cdot D, \g^\mu\g^\nu S_{[\mu\nu]}|_{\Lambda^1}\}}\\ {\phantom{T_2^{(i)}=}+q\cdot D\g^\mu\g^\nu S_{\mu\nu}|_{\Lambda^1} \g^\r\g^\s f_{\r\s}+\g^\r\g^\s f_{\r\s}\g^\mu\g^\nu S_{[\mu\nu]}|_{\Lambda^1}q \cdot D]\id.}\\ \end{array}$$ $S_{[\mu\nu]}|_{\Lambda^1}$ is the antisymmetric part of $S_{\mu\nu}|_{\Lambda^1}$ in eq. (\[DR+RD\]). By substituting in the previous equation the necessary integrals, and after some algebra, one obtains the following result: $$\begin{array}{l} {T_2^{(4\gamma)}=-\frac{1}{16\pi^2}\theta^{\a\b}\,\epsilon^{\mi\mii\miii\miv}\,{\mathbf{Tr}}\, [f_{\mi\mii}\Dcal_\miii f_{\miv\a}D_\b \id+\Dcal_\mi f_{\mii\a}f_{\miii\miv}D_\b \id-2d_1 (f_{\mi\mii}\Dcal_\miii f_{\a\b}D_\miv\id}\\ {\phantom{T_2^{(4\gamma)}=-\frac{1}{16\pi^2}\theta^{\a\b}\,\epsilon^{\mi\mii\miii\miv}\,{\mathbf{Tr}}\, [}+\Dcal_\mi f_{\a\b}f_{\miii\miv}D_\mii\id)].} \end{array} \label{T24g}$$ From the definition of $T_3$ in eq. (\[t1t2t3\]), one readily learns that there are contributions to it involving $8$, $6$ and $4$ $\gamma^{\mu}$ matrices. The contributions with $8$ and $6$ $\gamma^{\mu}$ matrices vanish in the large $\Lambda$ limit as $\Lambda^{-2}$ and $\Lambda^{-1}$, respectively. The contributions with $4$ $\gamma^{\mu}$ matrices also go away as $\Lambda\rightarrow\infty$, since in this limit they are proportional $1/\Lambda^2\,\epsilon^{\rho\sigma\mu\nu}S_{\mu\nu}|_{\Lambda^2}$ and $S_{\mu\nu}|_{\Lambda^2}$ is symmetric in its indices $\mu$ and $\nu$. In conclusion: $$T_3\,=\,0.$$ Substituting the previous equation and eq. (\[T24g\]), (\[T26g\]) and (\[T1\]) in eq. (\[ateta\]), one obtains the following result: $$\begin{array}{l} {{\cal A}_{\theta}(x)=T_1+T_2^{(6\gamma)}+T_2^{(4\gamma)}+T_3 =-\frac{i}{32\pi^2}\,\theta^{\a\b}\,\epsilon^{\mi\mii\miii\miv}\, {\mathbf{Tr}}\,[f_{\a\b}f_{\mi\mii}f_{\miii\miv}+4 f_{\a\miii}f_{\miv\b}f_{\mi\mii}]}\\ {\phantom{{\cal A}_{\theta}(x)=T_1+T_2^{(6\gamma)}+T_2^{(4\gamma)}+T_3=}+\frac{i\kappa_4}{4\pi^2}\,{\theta_\miv}^\beta\,\epsilon^{\mi\mii\miii\miv}\,{\mathbf{Tr}}\,\Dcal_\miii(f_{\mi\mii}\Dcal^\nu f_{\nu\b})}\\ {\phantom{{\cal A}_{\theta}(x)=T_1+T_2^{(6\gamma)}+T_2^{(4\gamma)}+T_3}= \partial_{\mu}\big( \frac{i\kappa_4}{4\pi^2}\,{\theta_\miii}^\beta\,\epsilon^{\mu\mi\mii\miii}\,{\mathbf{Tr}}\,f_{\mi\mii}\Dcal^\nu f_{\nu\b}\big).}\\ \end{array} \label{atetares}$$ The identity ${\mathbf{Tr}}\,\theta^{\a\b}\,\epsilon^{\mi\mii\miii\miv}[f_{\a\b}f_{\mi\mii}f_{\miii\miv}+4 f_{\a\miii}f_{\miv\b}f_{\mi\mii}]=0$ has been used to get the second equality in the previous expression. In view of eq. (\[atetares\]), one concludes that there is no anomalous contribution at first order in $\theta^{\mu\nu}$. Notice that one can always set $\kappa_4=0$ and that even in the event that one insisted in having a nonvanishing $\kappa_4$, the contribution to ${\cal A}_{\theta}$ can be absorbed by performing the following finite and gauge invariant renormalization of the current $j^{\mu}_5$ in eq. (\[current\]): $j^{\mu}_{5\,\rm{ren}}=j^{\mu}_5 -\frac{i\kappa_4}{4\pi^2}\,{\theta_\miii}^\beta\,\epsilon^{\mu\mi\mii\miii}\,{\mathbf{Tr}}\,f_{\mi\mii}\Dcal^\nu f_{\nu\b}.$ Let us choose $SU(N)$, with $N>2$, as our ordinary gauge group. That ${\cal A}(x)$ in eq. (\[A1A2\]) be equal to ${\cal A}(x)_{\rm ordinary}$ up to first order in $\theta^{\mu\nu}$ is a highly nontrivial result. Indeed, ${\cal A}_{\theta}(x)$ being proportional to a truly anomalous term like ${\mathbf{Tr}}\,\theta^{\a\b}\,\epsilon^{\mi\mii\miii\miv}[f_{\a\b}f_{\mi\mii}f_{\miii\miv}]$ is consistent with power counting and gauge invariance. And yet, as shown in eq. (\[atetares\]) all contributions of this type cancel each other. Why? One may answer this question by establishing the connection between the abelian anomaly ${\cal A}(x)$ and the index of ${\cal K}(1+\gamma_5)/2$, ${\cal K}$ being defined in eq. (\[Kop\]). But first let us exhibit some properties of ${\cal P}(x)={\mathbf{Tr}}\,\theta^{\a\b}\,\epsilon^{\mi\mii\miii\miv}[f_{\a\b}f_{\mi\mii}f_{\miii\miv}(x)]$. The first property we want to display is that for $SU(2)$, ${\cal P}(x)=0$. The second property is that for $SU(N)$, with $N>2$, ${\cal P}(x)$ cannot be expressed as $\partial_{\mu}\,X^{\mu}$, $X^{\mu}$ being a gauge invariant polynomial of the gauge field and its derivatives. This is why we called ${\cal P}(x)$ a truly anomalous contribution for $SU(N)$, $N>2$. That ${\cal P}(x)$ possesses this property can be shown as follows. If there exist such an $X^{\mu}$, it would be a polynomial on the field $a^{a}_{\mu}$ and its derivatives such that $s_0\,X^{\mu}|_{aaa}=0$. $s_0$ is the free BRS operator –$s_0\, a^{a}_{\mu}=\partial_{\mu}c^a$– and $X^{\mu}|_{aaa}$ is the contribution to $X^{\mu}$ which has 3 fields $a^{a}_{\mu}$ and 2 partial derivatives. Now, it has been shown in ref. [@Brandt:1989rd] that the cohomology of $s_0$ over the space of polynomials of $a^{a}_{\mu}$ and its derivatives is constituted by polynomials of $f_{0\,\mu\nu}^{a}= \partial_{\mu}a^{a}_{\nu}-\partial_{\nu}a^{a}_{\mu}$ and its derivatives. Hence, $X^{\mu}|_{aaa}=0$, for it cannot expressed as a polynomial of $f_{0\,\mu\nu}^{a}$ and its derivatives: we are one derivative short in $X^{\mu}|_{aaa}$. $X^{\mu}|_{aaa}=0$ implies that $X^{\mu}$ does not exist. The third property is that $\idx\; {\cal P}(x)$ does not necessarily vanish for fields with well-defined Pontrjagin number. For instance, in the $SU(3)$ case, $a^{a}_{\mu}=a^{({\rm BPST})\,a}_{\mu}+\delta^{a8}\,b_{\mu}$, with $a^{({\rm BPST})\,a}_{\mu}$ being standard embedding of the BPST $SU(2)$ instanton into $SU(3)$ and $b_{\mu}$ being a 4-dimensional vector field with components $b_\mu=\frac{\omega_{\mu\nu}x_\nu \rho^{2(n-1)}}{(r^2+\rho^2)^n},\,\, n>1,\,\omega_{\mu\nu}=-\omega_{\nu\mu},\,\,\omega_{\mu\nu}=\text{Sign}(\nu-\mu)(\mu\cdot\nu) \,\text{ for $\mu\neq\nu$}$, yields $\idx\;{\cal P}(x)\neq 0$ and has Pontrjagin number equal to 1. To symplify the computation choose a $\theta^{\mu\nu}$ with $\theta^{12}=-\theta^{21}$ as its only nonvanishing components. Let us now establish the connection between the abelian axial anomaly and the index of ${\cal K}(1+\gamma_5)/2$. Using eq. (\[reganom\]), one readily shows that $$-\frac{i}{2}\,\idx\;{\cal A}(x)=\idx\sum_{n=1}^{n_+}\idx\, \varphi_{n\,+}^{\dagger}(x)\varphi_{n\,+}(x)- \idx\sum_{n=1}^{n_-}\idx\, \varphi_{n\,-}^{\dagger}(x)\varphi_{n\,-}(x)=n_{+}-n_{-}.$$ $n_{+}$ and $n_{-}$ are respectively the number of positive and negative chirality zero modes of ${\cal K}$ in eq. (\[Kop\]). Of course, $n_{+}=\text{ dim Ker\;}{\cal K}(1+\gamma_5)/2$ and $n_{-}=\text{ dim Ker\;}{\cal K}^{\dagger}(1-\gamma_5)/2$. Hence, the abelian anomaly is given by the index of ${\cal K}(1+\gamma_5)/2$: $$-\frac{i}{2}\,\idx\;{\cal A}(x)\,=\,\text{ index }{\cal K}(1+\gamma_5)/2. \label{anomindex}$$ Now, we have assumed that the operator ${\cal K}$ differs from the Dirac operator $i\Dirac$ in a “infinitesimally small” –otherwise our expansions in $\theta^{\mu\nu}$ would not make much sense– operator $i\Rs$ that is hermitian and such that $\gamma_5\,{\cal K}=-{\cal K}\gamma_5$. Then, one would hope [@Green:1987mn] that $$\text{ index }{\cal K}(1+\gamma_5)/2\,=\,\text{ index }i\Dirac(1+\gamma_5)/2 \,=\,\frac{1}{32\pi^2}\,\idx\;\epsilon^{\mu\nu\rho\sigma}\,{\mathbf{Tr}}\; f_{\mu\nu}(x)f_{\rho\sigma}(x). \label{ASindex}$$ A by-product of our calculations is that the previous equation indeed holds as far as we have computed. Notice that if ${\cal A}_{\theta}$ in eq. (\[A1A2\]) had received a contribution like ${\cal P}(x)={\mathbf{Tr}}\,\theta^{\a\b}\,\epsilon^{\mi\mii\miii\miv}[f_{\a\b}f_{\mi\mii}f_{\miii\miv}(x)]$, then, in view of the discussion in the previous paragraph and eq. (\[anomindex\]), we would have concluded that the first equality in eq. (\[ASindex\]) would not be correct. Obviously, this analysis can be extended and conjecture that at any order in $\theta$ the abelian anomaly for noncommutative $SU(N)$ is saturated by the ordinary abelian anomaly. This conjecture is further supported by the second order in $\theta^{\mu\nu}$ Feynman diagram calculations carried out in ref. [@Martin:2005gt]. Finally, our results can be readily extended to the case of noncommutative gauge theories with a nonsemisimple gauge group, when the noncommutative theory is constructed by using the symmetric form of the Seiberg-Witten map as defined in ref. [@Aschieri:2002mc]. In this case the Seiberg-Witten map is the same as the map displayed in eq. (\[SW\]) by now $a_{\mu}$ is given by $ a_\m = \sum_{k=1}^s g_k\, (a_\mu^k)^a\,(T^k)^a + \sum_{l=s+1}^N g_l\,a_\mu^l\,T^l$ and the spinor $\psi$ denotes a hypermultiplet carrying a given representation of the nonsemisimple gauge group. $a_\m^k$, $g_k$ and $a_\mu^l$, $g_l$ are the ordinary gauge field and coupling constants associated, respectively, to each simple and $U(1)$ factor of the nonsemisimple group. The reader is referred to ref. [@Brandt:2003fx] for further details on the notation. It is clear that eqs. (\[ordinary\]) and (\[atetares\]) will also be valid in the nonsemisimple case provided $a_\m$ is defined as in the previous equation. It is a very interesting and open question to obtain the results presented in this paper by using the heat kernel expansion [@Vassilevich:2003xt] due to its relevance in the mathematically rigorous proof of index theorems. Acknowledgments {#acknowledgments .unnumbered} =============== This work has been financially supported in part by MEC through grant BFM2002-00950. The work of C. Tamarit has also received financial support from MEC trough FPU grant AP2003-4034. [99]{} J. Frohlich and B. Pedrini, arXiv:hep-th/0002195. S. L. Adler, arXiv:hep-th/0411038. K. Fujikawa, Phys. Rev. D [**21**]{} (1980) 2848 \[Erratum-ibid. D [**22**]{} (1980) 1499\]. L. Alvarez-Gaume, HUTP-85/A092 [*Lectures given at Int. School on Mathematical Physics, Erice, Italy, Jul 1-14, 1985*]{} B. Jurco, L. Moller, S. Schraml, P. Schupp and J. Wess, Eur. Phys. J. C [**21**]{} (2001) 383 \[arXiv:hep-th/0104153\]. X. Calmet, B. Jurco, P. Schupp, J. Wess and M. Wohlgenannt, Eur. Phys. J. C [**23**]{} (2002) 363 \[arXiv:hep-ph/0111115\]. P. Aschieri, B. Jurco, P. Schupp and J. Wess, Nucl. Phys. B [**651**]{} (2003) 45 \[arXiv:hep-th/0205214\]. P. Aschieri, arXiv:hep-th/0404054. J. Gomis and T. Mehen, Nucl. Phys. B [**591**]{} (2000) 265 \[arXiv:hep-th/0005129\]. R. Jackiw and C. Rebbi, Phys. Rev. D [**14**]{} (1976) 517. S. Chadha, A. D’Adda, P. Di Vecchia and F. Nicodemi, Phys. Lett. B [**67**]{} (1977) 103. F. Brandt, N. Dragon and M. Kreuzer, Phys. Lett. B [**231**]{} (1989) 263. M. B. Green, J. H. Schwarz and E. Witten, [*Superstring Theory. Vol. 2: Loop Amplitudes, Anomalies And Phenomenology*]{}, Cambridge Universty Press, 1987. C. P. Martin and C. Tamarit, arXiv:hep-th/0503139. F. Brandt, C. P. Martin and F. R. Ruiz, JHEP [**0307**]{} (2003) 068 \[arXiv:hep-th/0307292\]. D. V. Vassilevich, Phys. Rept.  [**388**]{} (2003) 279 \[arXiv:hep-th/0306138\]. [^1]: E-mail: [email protected] [^2]: E-mail: [email protected]

--- abstract: | We consider a quasilinear parabolic differential equation associated with the renormalization group transformation of the two–dimensional hierarchical Coulomb system in the limit as the size of the block $L\downarrow 1$. We show that the initial value problem is well defined in a suitable function space and the solution converges, as $t\rightarrow \infty$, to one of the countably infinite equilibrium solutions. The $j$–th nontrivial equilibrium solution bifurcates from the trivial one at $\beta _{j}=8\pi /j^{2}$, $j=1,2,\ldots $. These solutions are fully described and we provide a complete analysis of their local and global stability for all values of inverse temperature $\beta >0$. Gallavotti and Nicoló’s conjecture on infinite sequence of “phases transitions” is also addressed. Our results rule out an intermediate phase between the plasma and the Kosterlitz–Thouless phases, at least in the hierarchical model we consider. author: - | **Leonardo F. Guidi**[^1] &**Domingos H. U. Marchetti** [^2]\ Instituto de Física\ Universidade de São Paulo\ Caixa Postal 66318\ 05315 São Paulo, SP, Brasil title: | Renormalization Group Flow\ of the Two-Dimensional Hierarchical Coulomb Gas --- Introduction ============ We consider, for each $\beta >0$, the partial differential equation $$u_{t}-\frac{\beta }{4\pi }(u_{xx}-u_{x}^{2})-2u=0\, \label{pde}$$ on $\mathbb{R}_{+}\times \left( -\pi ,\pi \right) $ with periodic boundary condition, $u(t,-\pi )=u(t,\pi )$ and $u_{x}(t,-\pi )=u_{x}(t,\pi )$, in the space of even functions, satisfying an additional condition $u(t,0)=0$[^3]. We show that the initial value problem is well defined in an appropriate function space $\mathcal{B}$ and the solution exists and is unique for all $ t>0$. Furthermore, as $t\rightarrow \infty $, the solution converges in $ \mathcal{B} $ to one of the (equilibrium) solutions $\phi $ of $$\frac{\beta }{4\pi }\left( \phi ^{\prime \prime }-(\phi ^{\prime })^{2}\right) +2\phi =0\,, \label{stateq}$$ with $\phi (-\pi )=\phi (\pi )$ and $\phi ^{\prime }(-\pi )=\phi ^{\prime }(\pi )$. For $\beta >8\pi $, $\phi _{0}\equiv 0$ is the (globally) asymptotically stable solution of (\[pde\]). For $\beta <8\pi $ such that $ 8\pi /\left( k+1\right) ^{2}\leq \beta <8\pi /k^{2}$ holds for some $k\in \mathbb{N}_{+}$, $\phi _{0}$ is unstable and there exist $2k$ non–trivial equilibria solutions $\phi _{1}^{\pm },\ldots ,\phi _{k}^{\pm }$ of (\[stateq\]) among which $\phi _{1}^{\pm }$ are the only asymptotically stable ones. The aim of the present work is to show that, for $j\geq 1$, $\phi _{j}^{\pm } $ have a $\left( j-1\right) $–dimensional unstable manifold $\mathcal{M} _{j}\subset \mathcal{B}$ so $\phi _{j}^{\pm }$ are more stable than $\phi _{j^{\prime }}^{\pm }$ if $j<j^{\prime }$. As a consequence, there exists a dense open set of initial conditions in $\mathcal{B}$ such that $\phi _{1}^{+}$ ($\phi _{1}^{-}$ is not physically admissible) is the non–trivial stable solution for all $\beta <8\pi $. Our description of equation (\[pde\]) is motivated by two distinct goals. Firstly, it provides a new example of nonlinear parabolic differential equation by which a geometric theory can be carried out (see e.g. Henry [@H]). According to this theory, the above scenario can be stated as follows: there exist a sufficient large ball $\mathcal{B}_{0}\subset \mathcal{B}$ about the origin such that, if $u(t,\mathcal{B}_{0})$ denotes the set of points reached at time $t$ starting from any initial function in $\mathcal{B} _{0}$, then the invariant set $\bigcap_{t\geq 0}u(t,\mathcal{B}_{0})$ coincides with the $k$–dimensional unstable manifold $\mathcal{K}_{k}=\bigcup_{0\leq j\leq k}\mathcal{M}_{j}=\overline{\mathcal{M}_{0}}$ provided $8\pi /(k+1)^{2}\leq \beta <8\pi /k^{2}$. Secondly, the solution of the initial value problem (\[pde\]) describes the renormalization group (RG) flow of the effective potential in the two–dimensional hierarchical Coulomb system and the stationary solutions $ \left\{ \phi _{j}^{+}\right\} $, the fixed points of RG, contain informations on its critical phenomena. The analysis of equation (\[pde\]) presented here can hopefully bring some light to a question raised by Gallavotti and Nicoló [@GN] on the “screening phase transitions” in two–dimensional Coulomb systems. The existence of infinitely many thresholds of “instabilities” found in the Mayer series at inverse temperature $\beta _{n}=8\pi (1-1/(2n))$, $n\in \mathbb{N}_{+}$, indicates, according to the authors, a sequence of “intermediate” phase transitions from the plasma phase ($\beta \leq \beta _{1}=4\pi )$ to the multipole phase ($\beta \geq \beta _{\infty }=8\pi $). They conjectured that some partial screening takes place when the inverse temperature decreases from $8\pi $ to $4\pi $, which prevents the formation of neutral multipole of order larger than $2n$ where $n$ is the integer part of $1/(2-\beta /4\pi )$ (dipoles are the last to be prevented at $4\pi $). The Kosterlitz–Thouless phase (multipole phase) was established by Fröhlich–Spencer [@FS] and extended up to $8\pi $ by one of the present authors and A. Klein [@MK]. Debye screening (plasma phase) was only proved for sufficiently small $\beta <<4\pi $ [@BF]. Study of the region $[4\pi ,8\pi ]$ began with the work by Benfatto, Gallavotti and Nicoló [@BGN] on the ultraviolet collapses of neutral clusters in the Yukawa gas which served as a base for the results in [@GN]. It seems improbable, on the light of the present knowledge, that a conclusive answer to the Gallavotti–Nicoló conjecture will come up soon. It may be noted, however, that the scenario of an intermediate phase, which has challenged the conventional picture due to Jose *et al* [@JKKN], has been contested by Fisher *et al* [@FLL] based on Debye–Hückel–Bjerrum theory and by Dimock and Hurd [@DH] who have reinterpreted the ultraviolet collapses in the Yukawa gas. The Kosterlitz–Thouless phase is manifested in the hierarchical model as a bifurcation from the trivial solution [@MP]. Our results rule out the existence of further phase transitions since no other bifurcation arises from the stable solution (see Theorem \[lstb\] on the stability of $\phi _{1}^{+}$). Even though the existence of the invariant unstable manifold $\mathcal{K} _{k} $ may provide a suitable explanation to the appearance of Gallavotti–Nicoló’s thresholds, the nature (and location) of the instabilities in the hierarchical Coulomb gas differs substantially from the one we have just described, because neutral multipoles cannot be formed in the hierarchical model. We believe, however, our investigation may be helpful for the plasma phase. Numerical analysis shows the stable solution $ \phi _{1}^{+}$ looks like the Debye–Hückel potential $\phi _{DH}=(2\pi /\beta )\,x^{2}$ in $(-\pi ,\pi )$ right after the transition takes place (see Remark \[sinegordon\]). As in [@F], the renormalization group (GR) flow (\[pde\]) may be derived from the block–spin RG transformation of a two–dimensional hierarchical Coulomb system in the limit as the block size $L\downarrow 1$. This procedure, called *local potential approximation*, has been discussed by Felder [@F] in the context of Dyson’s hierarchical model, whose partial differential equation, $$u_{t}-\frac{1}{2}u_{xx}+\frac{d-2}{2}x\,u_{x}-d\,u+\frac{1}{2}u_{x}^{2}=0\,, \label{pde-f}$$ coincides with (\[pde\]) when his dimensional parameter $d=2$ if $\beta $ is equal to $2\pi $ (without boundary conditions). Felder showed that (\[pde-f\]) has global stationary solutions $u_{2n}^{\ast }$ on $\mathbb{R}$ for $2<d<d_{n}$ with $u_{2n}^{\ast }(x)\rightarrow 0$ as $d\uparrow d_{n}$ and calculated their profile. Here, $d_{n}=2+2/(n-1)$, $n=2,3,\ldots $, is the sequence of thresholds where nontrivial fixed points are expected to appear as a bifurcation from the trivial solution. We mean by global solution one which doesn’t blow up at finite $x$. The present paper begins with a derivation of equation (\[pde\]) in Section \[rgflow\]. The existence, uniqueness and continuous dependence on the initial value are presented in Section \[EU\] and the precise statements are given in Theorems \[thivp\] and \[cdi\]. We describe all global solutions of (\[stateq\]) completely in Section \[SS\]. Due to smoothness and the periodic condition, blow–up of an admissible stationary solution is impossible. We show that the non–trivial stationary solution for $\beta <8\pi $ is *unique* modulo solutions with period $2\pi /j$, $ j=2,3,\ldots $, which are responsible for the existence of the unstable manifold (see Theorem \[existence\]). Finally, we analyze in detail the local and global stability of equilibrium solutions of (\[pde\]) in Section \[Stability\]. The main results are stated in Theorems \[lstb\] and \[gstb\]. The Flow Equation \[rgflow\] ============================ This section is devoted to the derivation of (\[pde\]) from the RG transformation of two–dimensional hierarchical Coulomb system. We begin with a brief review of this model. A Coulomb system is an ensemble of two species (for simplicity) of charged particles, interacting via a two–body Coulomb potential $V$. In the grand canonical ensemble the total number of particles fluctuates around a mean value determined by the particle activity $z$. It will become clear that the charge ensemble, rather than the particle ensemble, is more appropriate for RG transformation. A configuration $q$ of this system is a function $q:\Lambda \subset \mathbb{Z }^{2}\longrightarrow \mathbb{Z}$ which associates to each site $x$ of the lattice $\Lambda $ the total charge $q(x)$ at this position. To each configuration we introduce two functionals: the total energy $E: \mathbb{Z}^{\Lambda }\longrightarrow \mathbb{R}_{+}$, $$E(q)=\frac{1}{2}\sum_{x,y\in \Lambda }q(x)\,V(x,y)\,q(y) \label{E}$$ (self–energy is included) and an “a priori” weight $F:\mathbb{Z}^{\Lambda }\longrightarrow \mathbb{R}_{+}$, $$F(q)=\prod_{x\in \Lambda }\lambda (q(x)) \label{F}$$ defined for positive real valued functions $\lambda $. The equilibrium Gibbs measure $\mu _{\Lambda }:\mathbb{Z}^{\Lambda }\longrightarrow \mathbb{R}_{+}$ is thus given by $$\mu _{\Lambda }(q):=\frac{1}{\Xi _{\Lambda }}F(q)\,e^{-\beta \,E(q)} \label{mu}$$ where $\beta $ is the inverse temperature and $$\Xi _{\Lambda }=\sum_{q\in \mathbb{Z}^{\Lambda }}F(q)\,e^{-\beta \,E(q)} \label{Xi}$$ is the grand partition function. It has been shown (see e.g. [@FS]) that the standard Coulomb system in the grand canonical ensemble with particle activity $z$ has charge activity given by $\lambda (q)=I_{q}(2z)$, where $I_{q}$ is the $q$–th modified Bessel function. If $\lambda (q)=\delta _{q,0}+z\left( \delta _{q,1}+\delta _{q,-1}\right) $, $\Xi _{\Lambda }$ is the grand canonical ensemble of charged particles with hard core. Let us introduce our hierarchical model as proposed in ref. [@MP]. The potential $V$ in (\[E\]) is replaced by a function $$V_{h}(x,y)=-\frac{1}{2\pi }\ln d_{h}(x,y)\,,$$ given by the asymptotic behavior of  the two–dimensional Coulomb potential with the Euclidean distance $\left| x-y\right| $ replaced by hierarchical distance $$d_{h}(x,y):=L^{N(x,y)}\,, \label{dh}$$ defined for an integer $L>1$, where $$N(x,y):=\inf \left\{ N\in \mathbb{N}_{+}:\left[ \frac{x}{L^{N}}\right] = \left[ \frac{y}{L^{N}}\right] \right\} \label{N}$$ and $[z]\in \mathbb{Z}^{2}$ has components the integer part of the components of $z\in \mathbb{R}^{2}$. Notice that $d_{h}$ is not invariant by translations. Now, given an integer number $N>1$ , let $\Lambda =\Lambda _{N}=[-L^{N},L^{N}-L^{N-1}]^{2}\cap \mathbb{Z}^{2}$ and define, for each configuration $q\in \mathbb{Z}^{\Lambda }$, the block configuration $q^{1}:\Lambda _{N-1}\longrightarrow \mathbb{Z}$, $$q^{1}(x)=\sum_{\underset{i=1,2}{0\leq y_{i}<L}}q(Lx+y)\,. \label{q1}$$ The renormalization group transformation $\mathcal{R}$ acting on the space of Gibbs measures (\[mu\]), $$\mu _{\Lambda _{N-1}}^{1}(q^{1})=[\mathcal{R}\mu _{\Lambda _{N}}](q^{1}):=\sum_{\underset{q^{1}\mathrm{fixed}}{q\in \mathbb{Z}^{\Lambda _{N}}:}}\mu _{\Lambda _{N}}(q)\,, \label{mu1}$$ involves an integration over the fluctuations about $q^{1}$ following by a rescaling back to the original lattice. As it has been shown in [@MP], the RG transformation $\mathcal{R}$ preserves the form of the Gibbs measure in the grand canonical ensemble of charges. The measure $\mu _{\Lambda _{N-1}}^{1}$ is thus given by (\[mu\]) with the “a priori weight” $F$ replaced by $$F^{1}(q^{1})=\prod_{x\in \Lambda _{N-1}}\lambda ^{1}(q^{1}(x)) \label{F1}$$ where $$\lambda ^{1}(p)=L^{-\beta p^{2}/(4\pi )}\underset{L^{2}-\mathrm{times}}{(\underbrace{\lambda \star \lambda \star \cdots \star \lambda })}(p) \label{lambda}$$ with $(\lambda \star \varrho )(p)=\sum\limits_{q\in \mathbb{Z}}\lambda (p-q)\,\varrho (q)$. Note that $\Xi _{\Lambda _{N}}(\lambda )=\Xi _{\Lambda _{N-1}}(\lambda ^{1})$. A peculiar feature of hierarchical models is the reduction of the measure space where $\mathcal{R}$ acts to local functions. The RG transformation (\[mu1\]) induces a transformation $\lambda ^{1}=r\lambda $ given by (\[lambda\]) on the space of infinite sequences. Note that the space $\ell _{1}(\mathbb{Z})$ of summable sequences is closed by the $r$ transformation: $\,(\lambda \star \lambda )\in \ell _{1}(\mathbb{Z})$ if $\lambda \in \ell _{1}(\mathbb{Z})$ by the Hausdorff-Young inequality. In order to take $L\downarrow 1$ limit of the RG transformation $r$ it is convenient to write the system in the *sine–Gordon representation.* Fourier transforming (\[lambda\]), $$\widehat{\lambda }(\varphi )=\sum_{q\in \mathbb{Z}}\,\lambda (q)\,e^{iq\varphi }\,,$$ and using the convolution theorem, yields $$\widehat{\lambda ^{1}}(\varphi )=\widehat{r\lambda }(\varphi )=\frac{1}{2\pi }\int_{-\pi }^{\pi }\vartheta (\varphi -\tau )\,\,\widehat{\lambda }^{L^{2}}(\tau )\,d\tau \label{l-tilde}$$ where $$\begin{aligned} \vartheta (\varphi ) &=&\sum_{q\in \mathbb{Z}}\,L^{-\beta q^{2}/(4\pi )}\,e^{iq\varphi } \notag \\ &=&\frac{1}{(\beta \ln L)^{1/2}}\sum_{n\in \mathbb{Z}}e^{-\pi (\varphi +2\pi n)^{2}/(\beta \ln L)} \label{nu}\end{aligned}$$ by the Poisson formula. Plugging (\[nu\]) into (\[l-tilde\]) and changing the variable $\zeta =\tau +2\pi n$, equation (\[l-tilde\]) can be written as $$\widehat{r\lambda }(\varphi )=\left( \nu \ast \widehat{\lambda }^{L^{2}}\right) (\varphi ) \label{l-t1}$$ where $\nu \ast $ means convolution by a Gaussian measure with mean zero and variance $\beta \ln L/(2\pi )$: $$\begin{aligned} (\nu \ast f)(\varphi ) &=&\left( \beta \ln L\right) ^{-1/2}\int_{-\infty }^{\infty }d\zeta \,\,e^{-\pi \left( \varphi -\zeta \right) ^{2}/(\beta \ln L)}\,f(\zeta ) \label{gauss} \\ &=&e^{\left( \beta \ln L/4\pi \right) \left( d^{2}/d\varphi ^{2}\right) }\,f(\varphi )\,\,, \notag\end{aligned}$$ where in the second form of the Gaussian convolution we have used Wick’s theorem. Note that (\[l-t1\]) is precisely the RG transformation derived by Gallavotti who has started directly from the sine-Gordon representation. In order to let the block size $L$ to $1$, we introduce a variable $t:=n\,\ln L$ which keeps track of the number of times the RG transformation (\[mu1\]) has to be iterated in order to bring two sites at hierarchical distance $L^{n}$ to $\mathcal{O}(1)$ distance. We shall take the limit $L\downarrow 1$ together with $n\rightarrow \infty $ maintaining $t$ fixed. Define $$u(t,x)=-\ln \widehat{\lambda ^{n}}(x) \label{utx}$$ where $\widehat{\lambda ^{n}}=\widehat{r^{n}\lambda }$ denotes the $n$ –th iteration of the transformation (\[l-t1\]). If one writes $t^{\prime }=(n+1)\ln L$ then, by taking the logarithm and using (\[utx\]), equation (\[l-t1\]) reads $$\begin{aligned} u(t^{\prime },x) &=&-\ln \left\{ \exp \left( \frac{\beta t}{4\pi n}\frac{d^{2}}{dx^{2}}\right) \exp \left( -e^{2t/n}u(t,x)\right) \right\} \notag \\ &=&u(t,x)-\ln \left\{ 1+\frac{t}{n}\left( \frac{\beta }{4\pi }\left( u_{x}^{2}(t,x)-u_{xx}(t,x)\right) -2u(t,x)\right) +\mathcal{O}\left( \frac{1 }{n^{2}}\right) \right\} \label{uutx} \\ &=&u(t,x)+\frac{t}{n}\left( \frac{\beta }{4\pi }\left( u_{xx}(t,x)-u_{x}^{2}(t,x)\right) +2u(t,x)\right) +\mathcal{O}\left( \frac{1 }{n^{2}}\right) \notag\end{aligned}$$ which, combined with $$\begin{aligned} u_{t}(t,x) &=&\lim_{t^{\prime }\downarrow t}\frac{u(t^{\prime },x)-u(t,x)}{ t^{\prime }-t} \notag \\ &=&\lim_{n\rightarrow \infty }\frac{n}{t}\left( u(t^{\prime },x)-u(t,x)\right) \,, \label{utxutx}\end{aligned}$$ yields equation (\[pde\]). Existence, Uniqueness and Continuous Dependence \[EU\] ====================================================== In this section the existence, uniqueness and continuous dependence on the initial value of equation (\[pde\]) will be established by Picard’s theorem for Banach spaces. To avoid the appearance of zero modes upon linearization, we differentiate (\[pde\]) with respect to $x$ and consider the equation for $v=u_{x}$, $$v_{t}-\frac{\beta }{4\pi }\left( v_{xx}-2v\,v_{x}\right) -2v=0\,, \label{dpde}$$ with $v\left( t,-\pi \right) =v\left( t,\pi \right) $ and $v_{x}\left( t,-\pi \right) =v_{x}\left( t,\pi \right) $, in the subspace of odd functions and initial value $v(0,\cdot )=v_{0}$. Note that the operator defined by the l. h. s. of (\[dpde\]) preserves this subspace. Before we proceed, we have the following \[Lagrange\]The “a priori weight” $\lambda (t,q):=\lambda ^{n}(q)\,$at scale $t=n\ln L$, is a positive **symmetric**, $\lambda (t,q)=\lambda (t,-q)$, sequence of real numbers and has to be normalized at all scales. In [@MP] equation (\[lambda\]) was redefined so that $\lambda ^{n}(0)=1$ holds for all $n$. Here, the appropriated normalizationis given by $$\sum_{q\in \mathbb{Z}}\lambda (t,q)=1\,,$$ since, in view of equation (\[utx\]), this leads to the condition $ \widetilde{u}\left( t,0\right) =0$, which is already imposed for all $t$ if $$\widetilde{u}(t,x)=\int_{0}^{x}v(t,y)\,dy \label{utilde}$$ with $v(s,x)$ an odd solution of (\[dpde\]). From (\[utilde\]), we have $$\begin{aligned} \widetilde{u}_{t} &=&\int_{0}^{x}v_{t}(t,y)\,dy \notag \\ &=&\int_{0}^{x}\left[ \alpha \left( \widetilde{u}_{xx}-\widetilde{u} _{x}^{2}\right) +2\widetilde{u}\right] _{x}\,dy \notag \\ &=&\alpha \left( \widetilde{u}_{xx}-\widetilde{u}_{x}^{2}\right) +2 \widetilde{u}-\alpha \widetilde{u}_{xx}(t,0) \label{utildeq}\end{aligned}$$ where $\widetilde{u}_{x}(t,0)=v(t,0)=0$ by parity. Note that $\widetilde{u} (t,x)=-\ln \widetilde{\lambda ^{n}}(x)+\ln \widetilde{\lambda ^{n}}(0)$ also satisfies (\[utildeq\]) by equations (\[uutx\]) and (\[utxutx\]). Moreover, note that there is a one–to–one correspondence between the solution of (\[pde\]) and the solution of (\[utildeq\]), with the same initial value $u_{0}$, given by $$\widetilde{u}(t,x)=u(t,x)-u(t,0) \label{utildeu}$$ and $$u(t,x)=\widetilde{u}(t,x)+\alpha \int_{0}^{t}e^{2(t-s)}\widetilde{u} _{xx}(s,0)\,ds\,\,, \label{uutilde}$$ where $\alpha \widetilde{u}_{xx}(t,0)$ is the required Lagrange multiplier introduced in (\[utildeq\]) to assure that $\widetilde{u}(t,0)=0$ (see comments after equation (*1.1*$^{\prime }$) in ref. [@F]). This correspondence will be useful in Section \[Stability\]. Because the standard initial condition $u_{0}(x)=z\left( 1-\cos x\right) $ satisfies $\ u_{0}^{\prime }(0)=u_{0}^{\prime }(\pi )=0$, equation (\[dpde\]) may equivalently be considered on $\left( 0,\pi \right) $ with Dirichlet boundary conditions $v\left( t,0\right) =v\left( t,\pi \right) =0$. Another reason for considering (\[dpde\]) instead of (\[pde\]) is the fact that the nonlinearity $2v\,v_{x}$ is more suitable than $u_{x}^{2}$ for the analysis of equilibrium solutions and corresponding stabilities given in the next sections. The boundary and initial value problem (\[dpde\]) may be written as an ordinary differential equation $$\frac{dz}{dt}+Az=F(z) \label{ode}$$ in a conveniently defined Banach space $\mathcal{B}$ where $$Az=-\alpha z\,^{\prime \prime }-2z\qquad \mathrm{and}\qquad F(z)=-2\alpha z^{\prime }z\,, \label{A}$$ with $\alpha =\beta /(4\pi )$ and initial value $z(0)=z_{0}$. The linear operator $A$ is defined on the space $C_{\mathrm{o,p}}^{2}$ of smooth odd and periodic real–valued functions in $[-\pi ,\pi ]$,[^4] with inner product $\left( f,g\right) :=\displaystyle\int_{-\pi }^{\pi }f(x)\,g(x)\,dx$. Because of $\left( f,Ag\right) =\left( Af,g\right) \,$, $A$ may be extended to a self–adjoint operator in $L_{\mathrm{o,p} }^{2}\left( -\pi ,\pi \right) $. The domain $D(A)$ of $A$ is $$D(A)=\left\{ f\in L_{\mathrm{o,p}}^{2}\left( -\pi ,\pi \right) :Af\in L_{ \mathrm{o,p}}^{2}\left( -\pi ,\pi \right) \right\}$$ and the spectrum of $A$, $$\sigma (A)=\left\{ \lambda _{n}=\alpha n^{2}-2,\,n\in \mathbb{N}_{+}\right\} \,, \label{spectrum}$$ consists of simple eigenvalues with corresponding eigenfunctions $\phi _{n}(x)=\left( 1/\pi \right) ^{1/2}\sin \,nx\,$. Let $A_{1}$ denote a positive definite linear operator given by $A$ if $ \alpha >2$ and $A+aI$ for some $a>2-\alpha $, otherwise. The following properties also hold for $A$ given by the closure in $L_{\mathrm{o,p} }^{q}\left( -\pi ,\pi \right) $, $1\leq q<\infty $, of the operator $\left. \left( -\alpha \,d^{2}/dx^{2}-2\right) \right| _{\mathcal{C}_{\mathrm{o,p} }^{2}}$. 1. The operator $A$ generates an analytic semi–group $T(t)=e^{-tA}$ given by the formula $$T(t)=\frac{1}{2\pi i}\int_{\Gamma }\frac{1}{\lambda +A}\,e^{\lambda t}\,d\lambda$$ where $\Gamma $ is a contour in the resolvent set of $A$ with $\arg \lambda \longrightarrow \pm \theta $, $\pi /2<\theta <\pi $, as $\left| \lambda \right| \rightarrow \infty $. From this, we have $$\left\| e^{-tA}\right\| \leq C\,e^{-ct}\hspace{0.5in}\mathrm{and}\hspace{ 0.5in}\left\| Ae^{-tA}\right\| \leq \frac{C}{t}\,e^{-ct} \label{bounds}$$ for $t>0$, $c<\inf_{\lambda }\sigma \left( A\right) $ and $C<\infty $. 2. Given $\gamma \geq 0$, let the fractional power of $A_{1}$ be given by $$A_{1}^{-\gamma }=\frac{1}{\Gamma (\gamma )}\int_{0}^{\infty }\,t^{\gamma -1}\,e^{-A_{1}t}\,dt$$ and define $A_{1}^{\gamma }=\left( A_{1}^{-\gamma }\right) ^{-1}$. $ A_{1}^{-\gamma }$ is a bounded operator (compact if $\gamma >0$) with $ A_{1}^{-1/2}\left( d/dx\right) $ and $\left( d/dx\right) A_{1}^{-1/2}$ bounded in the $L_{\mathrm{o,p}}^{2}\left( -\pi ,\pi \right) $ norm. In addition, for $\gamma >0$, $A_{1}^{\gamma }$ is closely defined with the inclusion $D(A_{1}^{\gamma })\subset D(A_{1}^{\tau })$ if $\gamma >\tau $. It thus follows from $1.$ and $2.$ (see e.g. [@H]) $$\left\| A_{1}^{\gamma }e^{-tA_{1}}\right\| \leq \frac{C_{\gamma }}{t^{\gamma }}\,e^{-ct} \label{b1}$$ holds for $0<\gamma <1$, $t>0$. Here $C_{\gamma }$ is bounded in any compact interval of $\left( 0,1\right) $ and also bounded as $\gamma \searrow 0$. Note that, if the operator norm is induced by the $L^{2}$–norm, equation (\[b1\]) hold with $$C_{\gamma }=\sup_{n\in \mathbb{N}_{+}}\left| \left( t\lambda _{n}\right) ^{\gamma }\,e^{-t\lambda _{n}}\right| \leq \sup_{r>tc}\left| r^{\gamma }\,e^{-r}\right| \leq \left( \frac{\gamma }{e}\right) ^{\gamma }\,, \label{Cgamma}$$ uniformly in $\gamma ,t\geq 0$. Following Picard’s method, let us replace $F$ in (\[ode\]) by a locally Hölder continuous function $f:[0,T]\longrightarrow \mathcal{B}$: $$\left\| f(r)-f(s)\right\| \leq C\left| r-s\right| ^{\theta }$$ for $0\leq r\leq s<T$ and $\theta >0$. In this case, a solution to (\[ode\]) is given by the variation of constants formula $$z(t)=e^{-tA}z_{0}+\int_{0}^{t}\,e^{-\left( t-s\right) A}\,f(s)\,ds\,. \label{cvm}$$ Note that $z:[0,T)\longrightarrow \mathcal{B}$ is continuously differentiable with $z\in D(A)$ satisfying the differential equation (\[ode\]). Moreover, $z (t)$ is the unique solution with $z(0)=z_{0}$ provided $ f$ is such that $\displaystyle\lim _{\rho \to 0} \int _{0}^{\rho } \left\| f (s)\right\| ds =0 $. Now, substituting $f(s)=F\left( z(s)\right) $ into (\[cvm\]) leads to an integral equation $$z(t)=e^{-tA}z_{0}+\int_{0}^{t}\,e^{-(t-s)A}\,F\left( z(s)\right) \,ds \label{integral}$$ whose solution, whether it exists, also solves the initial value problem (\[dpde\]) provided $F\left( z(s)\right) $ is shown to be locally Hölder continuous on the interval $0\leq t<T$. To formulate the necessary condition on $F$ and state our results, let $ \mathcal{B}^{\gamma }=D(A^{\gamma })$, $\gamma \geq 0$, denote the Banach space with the graph norm $$\left\| f\right\| _{\gamma }:=\left\| A^{\gamma }f\right\| .$$ $F:\mathcal{B}^{\gamma }\longrightarrow L_{\mathrm{p,o}}^{2}\left( -\pi ,\pi \right) $ is said to be locally Lipschtzian if there exist $U\subset \mathcal{B} ^{\gamma }$ and a finite constant $L$ such that $$\left\| F(z_{1})-F(z_{2})\right\| \leq L\left\| z_{1}-z_{2}\right\| _{\gamma }\, \label{F-F}$$ holds for any $z_{1},\,z_{2}\in U$. \[thivp\] The initial value problem (\[ode\]) has a unique solution $ z(t)$ for all $t\in \mathbb{R}_{+}$ with $z(0)=z_{0}\in \mathcal{B}^{1/2}$. In addition, if $\left\| z(t)\right\| _{1/2}$ is bounded as $t\rightarrow \infty $, the trajectories $\left\{ z(t)\right\} _{t\geq 0}$ lie on a compact set in $\mathcal{B}^{1/2}$. **Proof.** The proof of Theorem \[thivp\] will be divided into four parts. Firstly, $F(z(t))$ will be shown to be Hölder continuous under the Lipschtzian condition (\[F-F\]), which establishes the equivalence between the integral equation (\[integral\]) and the initial problem (\[ode\]). Secondly, the Banach fixed point theorem will be used to show the existence of a unique solution $z(t)$ of (\[integral\]) for $0\leq t\leq T$. Hence, by a compactness argument, the solution $z(t)$ will be extended to all $t\in \mathbb{R}_{+}$. Finally, assuming that $ \left\| z(t)\right\| _{1/2}$ stays bounded for all $t>0$, we conclude the proof. We have to wait till Section \[Stability\] for the boundedness hypothesis to be established. *Part I: Continuity.* Let us show that $F:D(A^{1/2}) \longrightarrow L_{\mathrm{o,p}}^{2}\left( -\pi ,\pi \right) $ given by $ F(z)=-2\alpha z\,z^{\prime }$ is locally Lipschitz. We note that $ D(A^{1/2})=H_{\mathrm{o,p}}^{1}\left( -\pi ,\pi \right) $ where $H_{\mathrm{ o,p}}^{k}\left( -\pi ,\pi \right) $ is the Sobolev space of odd periodic functions which have distributional derivatives up to order $k$. It thus follows that, if $z\in H_{\mathrm{o,p}}^{1}$, then $z(x)=\int_{0}^{x}z^{ \prime }(\xi )\,d\xi $ is absolutely continuous with $$\sup_{x\in \lbrack -\pi ,\pi ]}\left| z(x)\right| \leq \sqrt{2\pi }\left\| z\right\| _{1/2}\,,$$ by the Schwarz inequality. Moreover, using (\[b1\]), we have $$\begin{aligned} \left\| F(z_{1})-F(z_{2})\right\| &\leq &2\alpha \left\{ \left\| z_{1}(z_{1}^{\prime }-z_{2}^{\prime })\right\| +\left\| (z_{1}-z_{2})z_{2}^{\prime }\right\| \right\} \label{lipsch} \\ &\leq &2\alpha \sqrt{2\pi }\left\{ \left\| z_{1}\right\| \left\| z_{1}-z_{2}\right\| _{1/2}+\left\| z_{1}-z_{2}\right\| \left\| z_{2}\right\| _{1/2}\right\} \notag\end{aligned}$$ which satisfies (\[F-F\]) with $\gamma =1/2$ and $L=2\alpha \sqrt{2\pi } \left( \left\| z_{1}\right\| _{1/2}+\left\| z_{2}\right\| _{1/2}\right) $. Suppose that $z:(0,T)\longrightarrow \mathcal{B}^{1/2}$ is a continuous solution of (\[integral\]). From the estimate (\[b1\]), we have $$\begin{aligned} \left\| \left( e^{-hA}-I\right) e^{-\tau A}w\right\| _{1/2} &\leq &\int_{0}^{h}\left\| A\,e^{-(s+\tau )A}w\right\| _{1/2}\,\,ds \notag \\ &=&\int_{0}^{h}\left\| A^{1-\delta }\,e^{-sA}\right\| ds\,\left\| A^{\delta }e^{-\tau A}w\right\| _{1/2} \label{contbound} \\ &\leq &C_{1-\delta }\int_{0}^{h}\frac{1}{s^{1-\delta }}ds\,\left\| A^{\delta }e^{-\tau A}w\right\| _{1/2} \notag \\ &\leq &\frac{C_{1-\delta }}{\delta \,}h^{\delta }C_{\delta +1/2}\,\frac{ e^{-c\tau }}{\tau ^{\delta +1/2}}\,\left\| w\right\| \notag\end{aligned}$$ for $0<\delta <1/2$ which can be used in the equation (\[integral\]) along with (\[F-F\]), to get $$\begin{array}{ccc} \left\| z(t+h)-z(t)\right\| _{1/2} & \leq & \left\| \left( e^{-hA}-I\right) e^{-tA}z_{0}\right\| _{1/2}+\int_{0}^{t}\left\| \left( e^{-hA}-I\right) e^{-\left( t-s\right) A}F(z(s))\right\| _{1/2}\,ds \\ & & \\ & & +\int_{t}^{t+h}\left\| e^{-\left( t+h-s\right) A}F(z(s))\right\| _{1/2}\,ds\leq K\,h^{\delta } \end{array} \label{continuous}$$ for some constant $K<\infty $ in the open interval $\left( 0,T\right) $. Combined with (\[F-F\]), this implies the Hölder continuity of $ f(t)=F\left( z(t)\right) $ and the equivalence between the equations (\[ode\]) and (\[integral\]). *Part II: Local existence.* Let $V=\left\{ z\in \mathcal{B} ^{1/2}:\left\| z-z_{0}\right\| \leq \varepsilon \right\} $ be an $\varepsilon $–neighborhood and let $L$ be the Lipschitz constant of $F$ on $ V$. We set $B=\left\| F(z_{0})\right\| $ and let $T$ be a positive number such that $$\left\| \left( e^{-hA}-I\right) \,z_{0}\right\| _{1/2}\leq \frac{\varepsilon }{2} \label{H1}$$ with $0\leq h\leq T$ and $$C_{1/2}\left( B+L\varepsilon \right) \int_{0}^{T}s^{-1/2}\,e^{-cs}\,ds\leq \frac{\varepsilon }{2} \label{H2}$$ hold. Let $\mathcal{S}$ denote the set of continuous functions $y:[t_{0},t_{0}+T]\longrightarrow \mathcal{B}^{1/2}$ such that $\left\| y(t)-z_{0}\right\| \leq \varepsilon $. Equipped with the sup–norm $$\left\| y\right\| _{T}:=\sup_{t_{0}\leq t\leq t_{0}+T}\left\| y(t)\right\| _{1/2}$$ $\mathcal{S}$ is a complete metric space. Defining $\Phi \lbrack y]:[t_{0},t_{0}+T]\longrightarrow \mathcal{B}^{1/2}$ for each $y\in \mathcal{S}$ by $$\Phi \lbrack y](t)=e^{-(t-t_{0})A}z_{0}+\int_{t_{0}}^{t}\,e^{-\left( t-s\right) A}\,F\left( y(s\right) )\,\,ds\,,$$ we now show that, under the conditions (\[H1\]) and (\[H2\]), $\Phi : \mathcal{S}\longrightarrow \mathcal{S}$ is a strict contraction. Using $$\left\| F(y(t))\right\| \leq \left\| F(y(t))-F(z_{0})\right\| +\left\| F(z_{0})\right\| \leq L\left\| y(t)-z_{0}\right\| _{1/2}+B\leq L\varepsilon +B$$ and (\[b1\]), we have $$\begin{aligned} \left\| \Phi \lbrack y](t)-z_{0}\right\| _{1/2} &\leq &\left\| \left( e^{-(t-t_{0})A}-I\right) e^{-tA}z_{0}\right\| _{1/2}+\int_{t_{0}}^{t_{0}+T}\left\| A^{1/2}e^{-\left( t-s\right) A}\right\| \,\left\| F(y(s))\right\| \,ds \\ &\leq &\frac{\varepsilon }{2}+C_{1/2}\left( B+L\varepsilon \right) \int_{0}^{T}s^{-1/2}\,e^{-cs}\,ds\leq \varepsilon\end{aligned}$$ and since $\Phi \lbrack y]$ is continuous by an estimate analogous to (\[continuous\]), $\Phi \lbrack y]\in \mathcal{S}$. Analogously, from (\[F-F\]) and (\[H2\]), for any $y,w\in \mathcal{S}$ $$\begin{aligned} \left\| \Phi \lbrack y](t)-\Phi \lbrack w](t)\right\| _{1/2} &\leq &\int_{t_{0}}^{t_{0}+T}\left\| A^{1/2}e^{-\left( t-s\right) A}\right\| \,\left\| F(y(s))-F(w(s))\right\| \,ds \\ &\leq &C_{1/2}L\int_{0}^{T}s^{-1/2}\,e^{-cs}\,ds\,\left\| y-w\right\| _{T}\leq \frac{1}{2}\left\| y-w\right\| _{T}\end{aligned}$$ holds uniformly in $t\in \lbrack t_{0},t_{0}+T]$ concluding our claim. By the contraction mapping theorem, $\Phi $ has a **unique** fixed point $z $ in $\mathcal{S}$ which is the continuous solution of the integral equation (\[integral\]) on $(t_{0},t_{0}+T)$ and, by *Part I*, is the solution of (\[ode\]) in the same interval with $z(t_{0})=z_{0}\in \mathcal{B}^{1/2}$. *Part III: Global existence.* As the set $U$ where (\[F-F\]) holds is compact, the same $T$ can be chosen in *Part II* for any initial condition $z_{0}\in U$. Moreover, if $I_{1}=(t_{1},t_{1}+T)$ and $I_{2}=(t_{2},t_{2}+T)$ are two intervals containing $t_{0}$, then there exist $z_{0,1},z_{0,2}\,\in U$ such that the two solutions $z_{1}(t)$ and $ z_{2}(t)$ of equation (\[ode\]) on $I_{1}$ with $z_{1}(t_{1})=z_{0,1}$ and on $I_{2}$ with $z_{2}(t_{2})=z_{0,2}$, respectively, coincide in the open interval $I_{1}\cap I_{2}$. As a consequence, one can define an **open maximal interval** $I_{\mathrm{\max }}=(t_{-},t_{+})$ (containing the origin), where the solution $z(t)$ of (\[ode\]) is uniquely given by patching together the solutions $z_{j}(t)$ on intervals $I_{j}$ with $ z_{j}(t_{j})=z_{0,j}$. By construction, there is no solution to (\[ode\]) on $(t_{0},t^{\prime })$ if $t^{\prime }>t_{+}$. Therefore, either $ t_{+}=\infty $, or else there exist a sequence $\left\{ t_{n}\right\} _{n\in \mathbb{N}_{+}}$, with $t_{n}\rightarrow t_{+}$ as $n\rightarrow \infty $ such that $z(t_{n})$ tend to the boundary $\partial U$ of the compact set $U$ . It thus follows that, if $t_{+}$ is finite, the solution $z(t)$ blows–up at finite time. In what follows we show that $\left\| z(t)\right\| _{1/2}$ remains finite for all $t>t_{0}$ and this implies global existence of $z(t)$ . Let us start with the following generalization of the Gronwall inequality. \[gronwall\]Let $\xi $ and $\gamma $ be numbers and let $\theta $ and $ \zeta $ be non–negative continuous functions defined in a interval $ I=\left( 0,T\right) $ such that $\xi \geq 0$, $\gamma >0$ and $$\zeta (t)\leq \theta (t)+\xi \int_{0}^{t}\left( t-\tau \right) ^{\gamma -1}\,\zeta (\tau )\,d\tau \,. \label{zeta}$$ Then $$\zeta (t)\leq \theta (t)+\int_{0}^{t}E_{\gamma }^{\prime }(t-\tau )\,\theta (\tau )\,d\tau \label{zeta1}$$ holds for $t\in I$, where $E_{\gamma }^{\prime }=dE_{\gamma }/dt$, $$E_{\gamma }(t)=\sum_{n=0}^{\infty }\frac{1}{\Gamma \left( n\gamma +1\right) } \left( \xi \Gamma (\gamma )\,t^{\gamma }\right) ^{n} \label{Egamma}$$ and $\Gamma (z)=\displaystyle\int_{0}^{\infty }t^{z-1}e^{-t}dt$ is the gamma function. In addition, if $\theta (t)\leq K$ for all $t\in I$, then $$\zeta (t)\leq K\,E_{\gamma }(t)\,\leq K^{\prime }\,e^{\xi \Gamma (\gamma )T} \label{zeta2}$$ holds for some finite constant $K^{\prime }$. **Proof.** If $\mathcal{T}$  is an integral operator given by the convolution $$\mathcal{T}\zeta (t)=\xi \int_{0}^{t}\left( t-\tau \right) ^{\gamma -1}\zeta (\tau )\,d\tau \,, \label{tau}$$ then the inequality (\[zeta\]) can be formally solved by $$\zeta (t)=\theta (t)+\sum_{n=1}^{\infty }\mathcal{T}^{n}\theta (t)$$ where $\mathcal{T}^{n}$ is also an convolution integral operator which can be explicitly evaluated by the Laplace transform, $$\begin{aligned} \mathcal{T}^{n}\theta (t) &=&\frac{1}{\Gamma \left( n\gamma \right) }\left( \xi \Gamma (\gamma )\right) ^{n}\int_{0}^{t}\left( t-\tau \right) ^{n\gamma -1}\theta (\tau )\,d\tau \\ &=&\frac{1}{\Gamma \left( n\gamma +1\right) }\left( \xi \Gamma (\gamma )\right) ^{n}\int_{0}^{t}\frac{d}{dt}\left( t-\tau \right) ^{n\gamma }\theta (\tau )\,d\tau \equiv \left( f_{n}^{\prime }\ast \theta \right) (t)\,,\end{aligned}$$ with $f_{n}(t)=\left( \xi \Gamma (\gamma )\,t^{\gamma }\right) ^{n}/\Gamma \left( n\gamma +1\right) $. Equation (\[zeta1\]) (and (\[zeta2\]) by the fundamental theorem of calculus) thus follows by setting $E_{\gamma }(t)=\sum_{n\in \mathbb{N} }f_{n}(t)$. Note that this series is absolutely and uniformly convergent in $ t\in I$, with $E_{\gamma }(0)=1$, and it cannot grow faster than exponential $$E_{\gamma }(T)\sim \frac{1}{\gamma }e^{\xi \Gamma (\gamma )T} \label{exp}$$ as $T\rightarrow \infty $ (see Lemma $7.1.1$ in [@H]). This concludes the proof of Lemma \[gronwall\]. $\Box $ Taking the graph norm of (\[integral\]), we have in view of (\[bounds\]), (\[b1\]) and (\[exp\]) $$\begin{aligned} \left\| z(t)\right\| _{1/2} &\leq &\left\| e^{-(t-t_{0})A}z_{0}\right\| _{1/2}+L\int_{t_{0}}^{t}\,\left\| A^{1/2}e^{-(t-s)A}\right\| \,\left\| \,z(s)\right\| _{1/2}\,\,ds \notag \\ &\leq &C\left\| z_{0}\right\| _{1/2}+L\int_{t_{0}}^{t}\,\left( t-s\right) ^{-1/2}\,\left\| \,z(s)\right\| _{1/2}\,\,ds \label{b2} \\ &\leq &C\,\exp \left( LC_{1/2}\sqrt{\pi }t\right) \,\left\| z_{0}\right\| _{1/2}\,, \notag\end{aligned}$$ which is finite for any $t\in \mathbb{R}_{+}$. *Part IV: Compact trajectories.* Since $\mathcal{B}^{\gamma }\subset \mathcal{B }^{1/2}$ has compact inclusion if $1/2<\gamma <1$ [@H], it suffices to show that $\left\| z(t)\right\| _{\gamma }$ remains bounded as $ t\rightarrow \infty $. The hypothesis $\left\| z(t)\right\| _{1/2}<\infty $ combined with (\[lipsch\]) implies the existence of $C^{\prime }<\infty $ such that, analogously as in (\[b2\]), $$\begin{aligned} \left\| z(t)\right\| _{\gamma } &\leq &\left\| e^{-tA}z_{0}\right\| _{\gamma }+\int_{0}^{t}\,\left\| A^{\gamma }e^{-(t-s)A}\right\| \,\left\| F\left( \,z(s)\right) \right\| \,\,ds \\ &\leq &C_{\gamma -1/2}\,t^{1/2-\gamma }\,e^{-ct}\left\| z_{0}\right\| _{1/2}+C^{\prime }\,C_{\gamma }\int_{0}^{t}\,\left( t-s\right) ^{-\gamma }\,e^{-c(t-s)}\,\,ds\,,\end{aligned}$$ which is bounded for $t>0$ provided $c>0$ (i.e. $\inf_{\lambda }\sigma (A)>0$ ). Although the spectrum of $A$ is not positive if $\beta \leq 8\pi ,$ we shall see in Section \[Stability\] that $A$ in the integral equation (\[integral\]) can be replaced by a positive linear operator $L$ (see Theorems \[asym\] and \[positive\]). This concludes the proof of Theorem \[thivp\]. $\Box $ It follows by analogous procedure that if $z_{1}$ and $z_{2}$ are solutions of (\[ode\]) differing by their initial value in $\mathcal{B}^{1/2}$, then $$\begin{aligned} \left\| z_{1}(t)-z_{2}(t)\right\| _{1/2} &\leq &\left\| e^{-tA}\left( z_{0,1}-z_{0,2}\right) \right\| _{1/2}+\int_{0}^{t}\left\| A^{1/2}e^{-\left( t-s\right) A}\right\| \,\left\| F(z_{1}(s))-F(z_{2}(s))\right\| \,ds \\ &\leq &\left\| e^{-tA}\left( z_{0,1}-z_{0,2}\right) \right\| _{1/2}+C_{1/2}L\int_{0}^{t}\left( t-s\right) ^{-1/2}\,e^{-cs}\,ds\,\left\| z_{1}(s)-z_{2}(s)\right\| _{1/2}\end{aligned}$$ which implies, by the Gronwall inequality, the continuous dependence of $ z(t) $ with respect to its initial condition. We may also consider the dependence of $z$ with respect to the parameter $ \alpha =\beta /(4\pi )$. The next statement is a corollary of the above analysis. \[cdi\] The solution $z(t):\mathbb{R}_{+}\times \mathcal{B}^{1/2}\longrightarrow \mathcal{B}^{1/2}$ to the initial value problem (\[ode\]) as a function of the bifurcation parameter $\alpha $ and the initial value $z_{0}$ is continuous. \[DA\] It can be shown (see [@H]) that for any initial value $ z_{0}\in \mathcal{B}^{\gamma }$, $0<\gamma <1$, the solution is actually in $ D(A)$ at any later time. Moreover, since $F:\mathcal{B}^{1/2}\longrightarrow L_{\mathrm{o,p}}^{2}\left( -\pi ,\pi \right) $ is $\mathcal{C}^{\infty }$ (has Fréchet derivatives of all orders), it can also be shown that $ \left( \alpha ,z_{0}\right) \in \mathbb{R}_{+}\times \mathcal{B} ^{1/2}\longrightarrow z(t;\alpha ,z_{0})$ is $\mathcal{C}^{\infty }$  for all $t>0$. \[sobolev\] Under minor modifications, one can show existence, uniqueness and continuous dependence of (\[dpde\]) in Sobolev space $H_{ \mathrm{o,p}}^{1}\left( -\pi ,\pi \right) $ with norm $\left\| z\right\| _{1}=\left\| z^{\prime }\right\| _{L_{\mathrm{o,p}}^{2}}$ (just include the linear term of (\[dpde\]) in the definition of $F$). The same results hold for equation (\[pde\]) in the Sobolev space of even and periodic function $ H_{\mathrm{e,p}}^{1}\left( -\pi ,\pi \right) $ with both norms $\left\| \cdot \right\| _{1}$ and $\left\| \cdot \right\| _{1/2}$. Note from item $2.$ after (\[bounds\]) and (\[A\]) that $\alpha \left\| z\right\| _{1}=\left\| z\right\| _{1/2}+2\left\| z\right\| _{L_{\mathrm{o,p}}^{2}}$ so, both norms are equivalent. Equilibrium Solutions \[SS\] ============================ Time independent (equilibrium) solutions of (\[dpde\]) are odd solutions of the ordinary differential equation $$\alpha \left( \psi ^{\prime \prime }-2\psi \psi ^{\prime }\right) +2\psi =0\,, \label{stationary}$$ with periodic conditions $\psi (-\pi )=\psi (\pi )$ and $\psi ^{\prime }(-\pi )=\psi ^{\prime }(\pi )$, $\alpha =\beta /\left( 4\pi \right) \geq 0$ , which can be written as $$\left\{ \begin{array}{lll} w^{\prime } & = & 2p\left( w-\alpha ^{-1}\right) \\ & & \\ p^{\prime } & = & w\,, \end{array} \right. \label{syst}$$ by setting $p=\psi $ and $w=\psi ^{\prime }$. In this section we give a qualitative and quantitative description of the solutions of (\[syst\]) in the phase space $\mathbb{R}^{2}$ and study their implications for the equilibrium solutions of (\[dpde\]). Our results are summarized as follows. \[existence\] The stationary equation (\[stationary\]) has two distinct regimes separated by $\alpha =2$ ($\beta =8\pi $). For $\alpha \geq 2$, $\psi _{0}\equiv 0$ is the unique solution. For $\alpha <2$ such that $ 2/\left( k+1\right) ^{2}\leq \alpha <2/k^{2}$ holds for some $k\in \mathbb{N} _{+}$, there exist $2k$ non–trivial solutions $\psi _{j}^{+},\psi _{j}^{-}$ , $j=1,\ldots ,k$, with fundamental period $2\pi /j$, $\psi _{j}^{\pm }(-x)=-\psi _{j}^{\pm }(x)$ and $\psi _{j}^{-}(x)=\psi _{j}^{+}(x+\pi )$. Moreover, each pair of non–trivial solutions bifurcate from the trivial solution $\psi _{0}$ at $\alpha _{j}=2/j^{2}$ ($\beta _{j}=8\pi /j^{2}$) with $\lim\limits_{\alpha \uparrow \alpha _{j}}\psi _{j}^{\pm }=0$. In the phase space, these solutions $\left( \psi _{j}^{\prime },\psi _{j}\right) $, are closed orbits around $\left( 0,0\right) $ whose distance from the origin increases monotonically as $\alpha $ decreases. Numerical computations indicate that these orbits approach rapidly to the open orbit $ \left\{ \left( \alpha ^{-1},\alpha ^{-1}x\right) ,\,x\in \mathbb{R}\right\} $ from the left as $\alpha \rightarrow 0$. Let us begin by stating the general properties derived by the same tools used in the analysis performed in Section \[EU\]. The vector field $f:\mathbb{R}^{2}\longrightarrow \mathbb{R}^{2}$, $$(w,p)\longrightarrow f(w,p)=\left( 2p(w-\alpha ^{-1}),w\right) \,,$$ in the right hand side of (\[syst\]), defines a smooth autonomous dynamical system. It thus follows from Piccard’s theorem (see e.g. [@CL]) that there exist a unique solution $(w(x),p(x))$ of this system, globally defined in $\mathbb{R}^{2}$, with $(w(0),p(0))=(w_{0},p_{0})$. As we have seen in Section \[EU\], the existence of a global solution and its continuous dependence on the value $(w_{0},p_{0})$, and on the parameter $ \alpha $, follow from Gronwall’s lemma, which holds here in its standard form. As a consequence, the phase space $\mathbb{R}^{2}$ is foliated by non–overlapping orbits $$\gamma _{P}=\left\{ (w(x),h(x)):x\in \mathbb{R\;}\mathrm{and}\;P=(w(0),p(0)) \right\}$$ which passes by $P=(w_{0},p_{0})\in \mathbb{R}^{2}$ at $x=0$. Note that, by varying continuously $P$ and $\alpha $, the orbit $\gamma _{P}$ varies continuously in the phase space. We shall now determine the values $(P,\alpha )$ by which the solution of (\[syst\]) defines closed orbits. Note that the orbits are symmetric with respect to the $w$–axis, $L=\{(w,0):w\in \mathbb{R}\}$, since the system of equations (\[syst\]) remains invariant if the sign of both, $x$ and $p$, are reversed. As we shall see, there is no loss of generality if the initial value $(w(0),p(0))=P$ belongs to $L$. We write $\gamma _{P}=\gamma _{w_{0}}$. \[orbit\] Every orbit $\gamma _{P}$ is determined by a single value $P$ in the positive semi–axis $L^{+}=\{(w_{0},0):w_{0}\geq 0\}$. For $w_{0}>0$, the orbit $\gamma _{w_{0}}$ is either closed or unbounded depending on whether $\alpha \,w_{0}<1$ or $\alpha \,w_{0}\geq 1$, respectively. The orbit $\gamma _{\alpha ^{-1}}=\{(\alpha ^{-1},\alpha ^{-1}x):x\in \mathbb{R} \}$ separates the phase space $\mathbb{R}^{2}$ in such way that $\gamma _{P}$ is closed if $P$ is on the left of $\gamma _{\alpha ^{-1}}$ and unbounded otherwise. In addition, if $w_{0}=0$, then $\gamma _{0}=\{(0,0)\}$, and the origin is enclosed by every closed orbit. **Proof.** The proof of Proposition \[orbit\] follows from an explicit computation. By the chain rule, equation (\[syst\]) can be written as $$\frac{dp}{dw}=\frac{w}{2p\left( w-\alpha ^{-1}\right) } \label{dh/dw}$$ provided $\alpha w\neq 1$. The trajectories $\gamma _{w_{0}}$, obtained by integrating $2p\,dp=w\,dw/\left( w-\alpha ^{-1}\right) $ with initial point $ P=(w_{0},0)$, $$p^{2}=w-w_{0}+\alpha ^{-1}\ln \left( \frac{1-\alpha w}{1-\alpha w_{0}} \right) \,, \label{h2}$$ are portrayed in Figure $1$. We note that $P=(0,0)$ is the only critical point of (\[syst\]) which is a center for all $\alpha >0$ since, by linearizing $f(w,p)$ around $P=(0,0)$ gives a matrix whose eigenvalues are $\lambda _{\pm }=\pm i\sqrt{2\alpha ^{-1}}$. This implies that $\gamma _{0}=\{(0,0)\}$ and the orbits $\gamma _{w_{0}}$ with $w_{0}$ sufficiently closed to $0$ are, in view of (\[h2\]), ellipses defined by the equation $2\alpha ^{-1}p^{2}+w^{2}=C$. When $\alpha w_{0}=1$, using mathematical induction and equations (\[syst\]) with $(w(0),p(0))=(w_{0},0)$, we have $$\frac{d^{n}w}{dx^{n}}(0)=0\,,$$ for all $n\geq 1$, which leads $$\gamma _{\alpha ^{-1}}=\left\{ \left( \alpha ^{-1},\alpha ^{-1}x\right) :x\in \mathbb{R}\right\} \,.$$ Hence, if $\omega =\omega (P)$ denotes the set of limit points (the $\omega $ –limit set) given by $$\omega (P)=\left\{ (w^{\ast },h^{\ast })\in \mathbb{R}^{2}:\lim_{n\rightarrow \infty }\left( w(x_{n}),h(x_{n})\right) =(w^{\ast },h^{\ast })\right\} \label{omega}$$ for some sequence of points $\{x_{n}\}$ such that $x_{n}\rightarrow \infty $ , as $n\rightarrow \infty $, $\gamma _{\alpha ^{-1}}$ separates two different type of orbits: $\omega (P)=\gamma _{P}$ or $\omega (P)=\{\infty \} $ depending on whether the point $P$ is at the left or at the right of $ \gamma _{\alpha ^{-1}}$. $\Box $ **Proof of Theorem \[existence\].** The stationary solutions satisfy (\[syst\]) with periodic  conditions $w(0)=w(2\pi )$ and $p(0)=p(2\pi )$. By fixing the period $T$ of an orbit $\gamma _{w_{0}}$ in $2\pi $, the label $w_{0}$ becomes implicitly dependent on the parameter $ \alpha $. In view of Proposition \[orbit\], Theorem \[existence\] follows if for $\alpha \geq 2$, except by the orbit $\gamma _{0}=\{(0,0)\}$, no (non–trivial) solution has period $T=2\pi $ and for $\alpha <2$ there is a one–to–one correspondence between $w_{0}$ and $\alpha $ for $T$ fixed at any value $2\pi /k$, $k=1,\ldots ,\left[ \sqrt{2/\alpha }\right] $. More precisely, let $T=T(\alpha ,w_{0})$ denote the period of the dynamical system (\[syst\]) with initial value $(w(0),p(0))=\left( w_{0},0\right) $: $$T=\int_{\gamma _{w_{0}}}dx=2\int \frac{dp}{w}\,, \label{T}$$ where, by symmetry, the second integration is over the semi–orbit above the $w$–axis. For $\mathcal{D}=\left\{ \left( \alpha ,w_{0}\right) \in \mathbb{R}_{+}\times \mathbb{R}_{+}:\alpha w_{0}\leq 1\right\} $, we set $$G_{j}=T-\frac{2\pi }{j}$$ and note that $G_{j}:\mathcal{D}\longrightarrow \mathbb{R}$ is a continuous function of both variables satisfying $$G_{j}\left( 2/j^{2},0\right) =0\text{.} \label{Gj}$$ To see (\[Gj\]), we compute the period $T_{L}$ of an elliptic orbit, e.g. $ \left\{ \left( 2/\alpha \right) p^{2}+w^{2}=1\right\} $, of (\[syst\]) linearized at the origin ($f(w,p)$ replaced by $\left( 2\alpha ^{-1}p,w\right) $), $$T_{L}=4\int_{0}^{\sqrt{\alpha /2}}\frac{dp}{\sqrt{1-\left( 2/\alpha \right) p^{2}}}=2\pi \sqrt{\frac{\alpha }{2}\,}\,, \label{TL}$$ and note that $\lim_{w_{0}\rightarrow 0}T(\alpha ,w_{0})=T_{L}$. Continuity follows from the general properties stated previously. Hence, provided $$\frac{\partial T}{\partial w_{0}}>0 \label{dtdw0}$$ holds for all $\left( \alpha ,w_{0}\right) \in \mathcal{D}$, by the implicit function theorem, there exists a **unique** (strictly) monotone decreasing function $\widehat{w}_{j}:\left[ 0,2/j^{2}\right] \longrightarrow \mathbb{R}_{+}$ with $\widehat{w}_{j}(2/j^{2})=0$ such that $G_{j}(\alpha , \widehat{w}_{j}(\alpha ))=0$. Note that (\[dtdw0\]) and $$T(\alpha ,w_{0})=\sqrt{\alpha }T(1,\alpha w_{0})\, \label{TT}$$ imply that $T$ is an increasing function of both $\alpha $ and $w_{0}$, independently. This fact, which can be seen by rescaling (\[syst\]) by $ x\rightarrow \overline{x}=x/\sqrt{\alpha }$, $w\rightarrow \overline{w} =\alpha w$ and $p\rightarrow \overline{p}=\sqrt{\alpha }p$, explains the monotone behavior of $\widehat{w}_{j}$. It thus follows that, if $\alpha <2$, for each $j=1,\ldots ,k$ such that $ 2/\left( k+1\right) ^{2}\leq \alpha <2/k^{2}$ holds, a unique function $ \widehat{w}_{j}$ such that $\widehat{w}_{j}(2/j^{2})=0$ exists. The non–trivial solutions $\psi _{1}^{\pm },\ldots ,\psi _{k}^{\pm }$ of (\[stationary\]) are the $p$–component of $\gamma _{\widehat{w}_{j}}$, $ j=1,\ldots ,k$, which winds around the origin $j$–times: $\psi _{j}^{+}$ is $2\pi $–periodic odd function with fundamental period $2\pi /j$, $\left( \psi _{j}^{+}\right) ^{\prime }(0)>0$ and satisfies $\psi _{j}^{+}(x+\pi )=\psi _{j}^{-}(x)$. If $\alpha \geq 2$, because $T(\alpha ,w_{0})$ is a strictly increasing function of $w_{0}$ and $T(\alpha ,0)\geq 2\pi $ (see eq. (\[TL\])), there is no solution of $G_{j}(\alpha ,w_{0})=0$ besides $ \widehat{w}_{j}(\alpha )=0$ for $j=1$. This reduces the proof of Theorem \[existence\] to the proof of inequality (\[dtdw0\]). To prove (\[dtdw0\]), it is convenient to change variables. Let $$q=\ln \left( 1-\alpha \,w\right) \, \label{phi}$$ be defined for $\alpha w<1$. From (\[TT\]), there is no loss of generality in taking $\alpha =1$. The system of equations (\[syst\]) under this condition is thus equivalent to the following Hamiltonian system[^5] $$\left\{ \begin{array}{lll} q^{\prime } & = & 2p \\ & & \\ p^{\prime } & = & 1-e^{q}, \end{array} \right. \label{syst1}$$ whose energy function is given by $$H(q,p)=p^{2}+e^{q}-q-1\,. \label{hamiltonian}$$ The trajectory equation (\[h2\]), when written in terms of the $q$–variable,  gives exactly the energy level equation $H(q,p)=E$ with $$E=-w_{0}-\ln \left( 1-w_{0}\right) \,. \label{Ew0}$$ We denote by $\gamma _{E}$ the orbits of (\[syst1\]) and note that, in view of the fact $$\frac{dE}{dw_{0}}=\frac{w_{0}}{1-w_{0}}\,>0,$$ there is a one–to–one correspondence between the two families of closed orbits $\left\{ \gamma _{w_{0}},\,0\leq w_{0}<1\right\} $ and $\left\{ \gamma _{E},\,0\leq E<\infty \right\} $. Now, let $\widetilde{T}=\widetilde{T}(E)$ be the period of an orbit $\gamma _{E}$, $$\widetilde{T}=\int_{\gamma _{E}}dx=\int_{q_{-}}^{q_{+}}\frac{dq}{p}\,. \label{Ttilde}$$ Using the energy conservation law, we have $$p=p(q,E)=\sqrt{E-v(q)}\,, \label{p}$$ where the potential energy is given by $$v(q)=e^{q}-q-1\,,\, \label{v}$$ and $q_{\pm }=q_{\pm }(E)$ are the positive and negative roots of equation $v(q)=E$. Equation (\[dtdw0\]) holds if and only if $\dfrac{d\widetilde{T}}{dE}>0$ holds uniformly in $E\in \mathbb{R}_{+}$. But this follows from the monotonicity criterion given by C. Chicone [@C] (see also [@CG]): \[monotone\]Let $v\in \mathcal{C}^{3}(\mathbb{R})$ be a three–times differentiable function and let $f(q)=-v^{\prime }(q)$ be the force acting at $q$. If $v/f^{2}$ is a convex function with $$\left( \frac{v}{f^{2}}\right) ^{\prime \prime }=\frac{6v\left( v^{\prime \prime }\right) ^{2}-3\left( v^{\prime }\right) ^{2}v^{\prime \prime }-2vv^{\prime }v^{\prime \prime \prime }}{\left( v^{\prime }\right) ^{4}} >0\,,\qquad q\neq 0\,, \label{convex}$$ then the period $\widetilde{T}$ is a monotone (strictly) increasing function of $E$. **Proof.** It follows from (\[p\]) two basic facts: $$\frac{\partial p}{\partial q}=\frac{f}{2p}\hspace{1in}\mathrm{and\hspace{1in}} p(q_{\pm },E)=0\,. \label{dphi}$$ These will be used for deriving an appropriated integral representation of $d \widetilde{T}/dE$. Let $$K:=\frac{1}{3}\int_{q_{-}}^{q_{+}}p^{3}\left( \frac{v}{f^{2}}\right) ^{\prime \prime }\,dq\,. \label{K}$$ Integrating twice by parts, gives $$\begin{aligned} K &=&\left. \frac{p^{3}}{3}\,\left( \frac{v}{f^{2}}\right) ^{\prime }\right| _{q_{-}}^{q_{+}}-\left. \frac{pv}{2f}\right| _{q_{-}}^{q_{+}}+\int_{q_{-}}^{q_{+}}\left( pf\right) ^{\prime }\,\frac{v}{ f^{2}}\,dq \\ &=&\frac{1}{2}\int_{q_{-}}^{q_{+}}\left( \frac{v}{2p}+vp\frac{f^{\prime }}{ f^{2}}\right) \,dq\end{aligned}$$ in view of (\[dphi\]). Note that $f(q_{\pm })\not=0$ since $$v^{\prime }(q_{\pm })=v(q_{\pm })-q_{\pm }=E-q_{\pm }$$ vanishes only at $E=0$. This follows from the fact that $v$ is a convex positive function with $v(0)=0$ and asymptotic behavior $v(q)\sim q-1$ and $ \sim e^{\alpha q}$, as $q$ goes to $-\infty $ and $\infty $. Now, using $(v/f)^{\prime }=v^{\prime }/f-vf^{\prime }/f^{2}=-1-vf^{\prime }/f^{2}$, and integrating by parts, we continue $$\begin{aligned} K &=&\frac{1}{2}\int_{q_{-}}^{q_{+}}\left( \frac{v}{2p}-p\left( \frac{v}{f} \right) ^{\prime }-p\right) \,dq \notag \\ &=&\frac{1}{2}\int_{q_{-}}^{q_{+}}\left( \frac{v}{p}-p\right) \,dq-\left. \frac{1}{2}p\,\left( \frac{v}{f}\right) \right| _{q_{-}}^{q_{+}} \label{KK} \\ &=&\frac{1}{2}\int_{q_{-}}^{q_{+}}\left( \frac{E}{p}-2p\right) \,dq \notag\end{aligned}$$ where in the last equation we have used $v=E-p^{2}$. From (\[Ttilde\]), (\[K\]) and (\[KK\]), we have $$E\widetilde{T}=2\int_{q_{-}}^{q_{+}}p\,dq+\frac{2}{3}\int_{q_{-}}^{q_{+}}p^{3}\,\left( \frac{v}{f^{2}}\right) ^{\prime \prime }\,dq\,.$$ Differentiating this with respect to $E$ and using (\[dphi\]), gives $$\widetilde{T}+E\,\frac{d\widetilde{T}}{dE}=\int_{q_{-}}^{q_{+}}\frac{dq}{p} +\int_{q_{-}}^{q_{+}}p\,\left( \frac{v}{f^{2}}\right) ^{\prime \prime }\,dq$$ which, in view of (\[Ttilde\]) and the assumption of Lemma \[convex\], implies $$\frac{d\widetilde{T}}{dE}=\frac{1}{E}\int_{q_{-}}^{q_{+}}p\,\left( \frac{v}{ f^{2}}\right) ^{\prime \prime }\,dq>0\,.$$ $\Box $ It remains to verify (\[convex\]) for $v$ given by (\[v\]). By an explicit computation (see Chicone [@C]) $$\left( \frac{v}{f^{2}}\right) ^{\prime \prime }\left( v^{\prime }\right) ^{4}=e^{q}\,g(q)$$ where $$g(q):=e^{2q}+4\left( 1-q\right) e^{q}-2q-5$$ is such that $g(0)=g^{\prime }(0)=0$ and $g^{\prime \prime }(q)=4e^{q}v(q)\geq 0$. This implies $g(q)\geq 0$ ($g(q)=0$ only if $q=0$), the hypothesis of Lemma \[monotone\] and concludes the proof of Theorem \[existence\]. $\Box $ Turning back to the Coulomb system problem, some remarks are now in order. Recalling $v(t,x)=u_{x}(t,x)$ and denoting $\lambda ^{\ast }=\lim\limits_{n\rightarrow \infty }\lambda ^{n}$ the charge activity at the fixed point, we have from (\[utx\]) $$\psi (0)=-i\sum_{q\in \mathbb{Z}}q\,\lambda ^{\ast }(q)\left/ \sum_{q\in \mathbb{Z}}\,\lambda ^{\ast }(q)\right. =0$$ and $$\psi ^{\prime }(0)=\sum_{q\in \mathbb{Z}}q^{2}\,\lambda ^{\ast }(q)\left/ \sum_{q\in \mathbb{Z}}\,\lambda ^{\ast }(q)\right. \geq 0\,.$$ These boundary conditions select $\psi _{j}^{+}$, $j=1,\ldots ,k$, as being the only physically meaningful stationary solutions and implies $\phi ^{+}(x)=\displaystyle\int_{0}^{x}\psi ^{+}(y)\,dy\geq 0$ on $\left( -\pi ,\pi \right) $. The value $\alpha =2$ is a bifurcation point as one can see by linearizing (\[stationary\]) about $\psi \equiv 0$. The linear operator $L[0]=A$ given by (\[A\]) in the subspace of odd $2\pi $–periodic functions has eigenvalues and associate eigenfunctions given by (\[spectrum\]). Hence, if $\alpha >2$, the eigenvalues are all positive and $\psi \equiv 0$ is locally stable. When $\alpha <2$ (but close to $2$) a single eigenvalue becomes negative and one can apply the Crandall–Rabinowitz bifurcation theory [@C] to locally describe the stable solution which bifurcates from the trivial one. Note that Crandall–Rabinowitz theory can also be applied in the neighborhood of $\alpha _{j}=2/j^{2}$, $j>1$, in the orthogonal complement of the span $\left\{ \pi ^{-1/2}\sin mx,\,m=1,...,j-1\right\} $ corresponding to the odd functions with fundamental period $T=2\pi /j$. These points were referred to in the introduction as a sequence of instability thresholds. In Theorem \[existence\] we have given a global characterization of the non–trivial stationary solutions. \[sinegordon\] In the sine–Gordon representation, the effective potential $\phi (x)=\int_{0}^{x}\psi (y)\,dy=x^{2}/\left( 2\alpha \right) $ at $\gamma _{\alpha ^{-1}}$ corresponds the Debye–Hückel regime with Debye length $\alpha $. Although this regime is not reached for all $\beta >0 $, it gets closed quite fast as $\beta =4\pi \alpha $ approaches $0$. Numerical calculation is shown in Figure 2. Note that at $\alpha =1$ ($\beta =4\pi $), $\widehat{w}_{1}$ cannot be distinguished from $\alpha ^{-1}$ (numerical error is in the sixth decimal order). The derivative of (\[T\]) with respect to $w_{0}$, computed from equation (\[h2\]), $$\frac{\partial T}{\partial w_{0}}=\frac{2\alpha w_{0}}{1-\alpha w_{0}}\int \mathrm{sign}\left( w\right) \frac{2\left( 1-\alpha w\right) p^{2}/\alpha }{ \left( 1+2\left( 1-\alpha w\right) p^{2}/\alpha \right) }dp\,,$$ indicates that an estimate from below can be very delicate to obtain. Note $ \mathrm{sign}\left( w\right) $ changes along the orbit $\gamma _{w_{0}}$. This shows how amusing Chicone’s monotonicity result is for the problem at hand. Stability \[Stability\] ======================= Let $z(t;z_{0})$ denote the solution of  the initial value problem (\[ode\]) – (\[A\]). It follows from the analysis in Section \[EU\] that $$S(t)z_{0}=z(t;z_{0}) \label{dyn}$$ defines a dynamical system on a closed subset $\mathcal{V}\subset D\left( A\right) $ of $\mathcal{B}^{1/2}$ with the topology induced by the graph norm $\left\| \cdot \right\| _{1/2}$. Note that $z(t;z_{0})$ is continuous in both $t$ and $z_{0}$ with $z(0;z_{0})=z_{0}$ and satisfies the (nonlinear) semi–group property $S(t+\tau )z_{0}=z(t;z(\tau ;z_{0}))=S(t)S(\tau )z_{0}$. This section is devoted to the stability analysis of the equilibrium solutions described in Section \[SS\]. By local stability it is meant that $z(t;z_{0})$ is uniformly continuous in $\mathcal{V}$ for all $t\geq 0$: given $\varepsilon >0$, $\left\| z(t;z_{0})-z(t;z_{1})\right\| _{1/2}<\varepsilon $ for all $t\geq 0$ and $z_{1}\in \mathcal{V}$ such $ \left\| z_{1}-z_{0}\right\| _{1/2}<\delta $ for some $\delta =\delta (\varepsilon )>0$. It is uniformly asymptotically stable if, in addition, $ \lim\limits_{t\rightarrow \infty }\left\| z(t;z_{0})-z(t;z_{1})\right\| _{1/2}=0$. The Liapunov (global) stability analysis as developed by LaSalle and applied to semilinear parabolic differential equations by Chafee and Infante [@CI] (see also [@H]) will also be discussed and extended in this section. Let us begin with the local analysis. \[lstb\] There exists a neighborhood $\mathcal{U}\subset \mathcal{B} ^{1/2}$ of the origin such that, if $\alpha >2$ and $z_{0}$ in $\mathcal{U}$ , then $\psi _{0}\equiv 0$ is stable, i.e., $\lim\limits_{t\rightarrow \infty }\left\| z(t;z_{0})\right\| _{1/2}=0$. If $\alpha <2$ is such that $ 2/\left( k+1\right) ^{2}\leq \alpha <2/k^{2}$ holds, among all equilibrium solutions of (\[stationary\]), $\psi _{0},\psi _{j}^{\pm }$, $j=1,\ldots ,k $, $\psi _{1}^{\pm }$ are the only asymptotically stables. So, there exists $ \rho >0$ such that if $\left\| z_{0}-\psi \right\| _{1/2}\leq \rho $, then $ \lim\limits_{t\rightarrow \infty }\left\| z(t;z_{0})-\psi \right\| _{1/2}=0$ for $\psi =\psi _{1}^{\pm }$ and, for any sequence $\left\{ z_{n}\right\} _{n\geq 1}$ with $\lim\limits_{n\rightarrow \infty }\left\| z_{n}-\psi \right\| =0$, we have $\sup\limits_{t>0}\left\| z(t;z_{n})-\psi \right\| _{1/2}\geq \varepsilon >0$ for all $n$ and $\psi =\psi _{j}^{\pm }$, $j\neq 1 $. It is convenient to consider the equation $$\frac{d\zeta }{dt}+L\zeta =F\left( \zeta \right) \label{ode1}$$ for $\zeta =z-\psi $ where $\psi $ is a solution of (\[stationary\]). Here $$L\zeta =L\left[ \psi \right] \zeta =-\alpha \zeta ^{\prime \prime }+2\alpha \psi \zeta ^{\prime }-2\left( 1-\alpha \psi ^{\prime }\right) \zeta \label{L}$$ is the linearization of the differential operator (\[dpde\]) around $\psi $ and $F$ is as in (\[A\]). Note $L=A$ and (\[ode1\]) reduces to (\[ode\]) if $\psi =\psi _{0}=0$. **Proof.** The proof of the Theorem \[lstb\] follows from the next two theorems. \[asym\] If the spectrum $\sigma (L)$ of (\[L\]) lies in $\left\{ \lambda \in \mathbb{R}:\lambda \geq c\right\} $ for some $c>0$, then $\zeta =0$ is the unique uniformly asymptotically stable solution of (\[ode1\]). On the other hand, if $\sigma (L)\cap \left\{ \lambda \in \mathbb{R}:\lambda <0\right\} \neq \emptyset $, then $\zeta =0$ is unstable. \[positive\] Let $L=L[\psi ]$ be given by (\[L\]). Then $\sigma (L)>0$ whenever $\psi =\psi _{0}$ and $\alpha >2$ or $\psi =\psi _{1}^{\pm }$ and $ \alpha <2$. If $\alpha $ is such that $2/\left( k+1\right) ^{2}\leq \alpha <2/k^{2}$ holds for some $k\in \mathbb{N}_{+}$, then $\sigma (L)\cap \left\{ \lambda \in \mathbb{R}:\lambda <0\right\} \neq \emptyset $ for $\psi =\psi _{0}$ and $\psi =\psi _{j}^{\pm },\;j=2,\ldots ,k$. **Proof of Theorem \[asym\].** We shall prove only the first part of Theorem \[asym\] and refer to Theorem 5.1.3 of Henry’s book [@H] for the instability part. It follows from (\[integral\]), (\[b1\]), (\[lipsch\]) and the hypothesis on $\sigma (L)$ that $$\left\| \zeta (t)\right\| _{1/2}\leq C_{1/2}\,e^{-ct}\left\| \zeta _{0}\right\| _{1/2}+\xi \int_{0}^{t}\,\left( t-s\right) ^{-1/2}\,e^{-c(t-s)}\left\| \,\zeta (s)\right\| _{1/2}^{2}\,\,ds\,, \label{zt}$$ with $c>0$, $C_{1/2}=1/\sqrt{2e}$ and $\xi =2\sqrt{2\pi }\alpha $.   Let us assume that $\left\| \zeta (s)\right\| _{1/2}\leq \rho $ on a interval $\left( 0,t\right) $ for some $\rho $ satisfying $$\xi \int_{0}^{\infty }t^{-1/2}\,e^{-ct}\,dt=\xi \sqrt{\frac{\pi }{c}}<\frac{ 1 }{2\rho }, \label{rho}$$ i. e., $\rho <\dfrac{1}{4\pi \alpha }\sqrt{\dfrac{c}{2}}$. If $\left\| \zeta _{0}\right\| _{1/2}\leq \rho \sqrt{\dfrac{e}{2}}$, then equation (\[zt\]) can be bounded as $$\left\| \zeta (t)\right\| _{1/2}\leq \frac{\rho }{2}+\rho ^{2}\xi \int_{0}^{t}\,\left( t-s\right) ^{-1/2}\,e^{-c(t-s)}<\rho \, \label{zt1}$$ and this implies the existence of a unique solution of (\[ode1\]) with $ \left\| \zeta (t)\right\| _{1/2}\leq \rho $ for all $t>0$. Note that $ \left\| \zeta _{0}\right\| _{1/2}<\rho $ and if $t_{1}$ is the maximum value under which $\left\| \zeta (t)\right\| _{1/2}<\rho $ for all $0<t<t_{1}$, then either $\left\| \zeta (t_{1})\right\| _{1/2}=\rho $ or $t_{1}=\infty $. But the first case is impossible by (\[zt1\]). Going back to (\[zt\]), using $\left\| \,\zeta (s)\right\| _{1/2}<\rho $ and a slightly modification of Gronwall inequality (\[gronwall\]) with $ E_{1/2}(t)=$ $\displaystyle\sum\limits_{n=0}^{\infty }\left( \rho \xi \sqrt{ \pi }t^{1/2}\right) ^{n}\left/ \Gamma (n/2+1)\right. $, we have $$\begin{aligned} \left\| \zeta (t)\right\| _{1/2} &\leq &C_{1/2}\left\| \zeta _{0}\right\| _{1/2}E_{1/2}(t)\,\,e^{-ct} \\ &\leq &C_{1/2}\left\| \zeta _{0}\right\| _{1/2}\left( 1+\rho \xi t^{1/2}\right) e^{-\left( c-\rho ^{2}\xi ^{2}\pi \right) t} \\ &\leq &\frac{1}{2e}\left\| \zeta _{0}\right\| _{1/2}\left( 1+\frac{1}{2} \sqrt{\frac{ct}{\pi }}\right) \,e^{-3ct/4}\,,\end{aligned}$$ in view of (\[rho\]). This proves the stability statement of Theorem \[asym\], since (\[ode1\]) defines a dynamical system in a closed subset $\mathcal{V}_{\rho }=\left\{ \zeta \in \mathcal{B}^{1/2}:\left\| \zeta \right\| _{1/2}\leq \rho \right\} $ with $\lim\limits_{t\rightarrow \infty }\left\| \zeta (t)\right\| _{1/2}=0$ if $\left\| \zeta _{0}\right\| _{1/2}=\left\| z_{0}-\psi \right\| _{1/2}\leq \rho \sqrt{\dfrac{e}{2}}$. $\Box $ One can actually show that if $c=\inf_{\lambda }\sigma (L)$ then $\zeta (t;\zeta _{0})=z(t;z_{0})-\psi $ decays exponentially fast to $0$ as $$\zeta (t;\zeta _{0})=\kappa (\zeta _{0})\,e^{-ct}+\varepsilon (t;\zeta _{0})$$ where $\left\| \varepsilon (t;\zeta _{0})\right\| _{1/2}\leq C\left\| \zeta _{0}\right\| _{1/2}e^{-c^{\prime }t}$ with $0<c<c^{\prime }$ and $\kappa : \mathcal{V}_{\rho }\longrightarrow \mathcal{N}\left( L-cI\right) $ is continuous and such that $\kappa (0)=0$, where $\mathcal{N}\left( L-cI\right) $ is the one–dimensional span of the eigenfunction of $L$ associated to $c$. **Proof of Theorem \[positive\].** Since $L[\psi _{0}]=A$, Theorem \[positive\] for $\psi =\psi _{0}$ with $\alpha \geq 0\ $follows from the spectral computation in (\[spectrum\]). Now, let $\psi $ be a nontrivial solution of the equilibrium equation (\[stationary\]) and note that $\psi (0)=\psi (\pi )=0$ by parity. According to Theorem \[asym\], $\psi $ is asymptotically stable if $\sigma (L)>0$ and unstable if $\sigma (L)\cap \left\{ \lambda <0\right\} \neq \emptyset $. Let $\varphi $ be the solution of $$L[\psi ]\varphi =0 \label{Lphi}$$ in the domain $0<x<\pi $ satisfying $$\varphi (0)=0\qquad \mathrm{and}\qquad \varphi ^{\prime }\left( 0\right) =1. \label{phiphi}$$ As in [@H], we shall use the comparison theorem to establish that $\psi $ is asymptotically stable if $\varphi (x)>0$ on $0<x\leq \pi $ and unstable if $\varphi (x)<0$ somewhere in $0<x<\pi $.   To apply the comparison theorem and complete the proof of Theorem \[positive\], let $$p(x):=e^{-2\int_{0}^{x}\psi (y)\,dy} \label{p(x)}$$ be the weight which makes $L$ a self–adjoint operator: $$p\,L[\psi ]\zeta =-\alpha \left( p\,\zeta ^{\prime }\right) ^{\prime }-2p\left( 1-\alpha \psi ^{\prime }\right) \zeta \,. \label{pL}$$ Note that $\left( L\zeta ,\eta \right) _{p}=\left( \zeta ,L\eta \right) _{p}$ for any odd periodic functions $\zeta $ and $\eta $ of period $2\pi $ were $ \left( f,g\right) _{p}:=\displaystyle\int_{-\pi }^{\pi}f(x)\,g(x)\,p(x)\,dx$. \[comp\] Suppose $\zeta _{1}$ and $\zeta _{2}$ are two real solutions on the domain $\left( 0,\pi \right) $ of $$p\,L[\psi ]\zeta =f_{i}\,,\qquad i=1,2\,,$$ respectively, with $\zeta _{1}(0)=\zeta _{1}(\pi )=0$, $\zeta _{1}^{\prime }(0)>0$ and $\zeta _{2}(0)=0$, $\zeta _{2}^{\prime }(0)>0$. If $\zeta _{1}>0$ and $f_{i}=f_{i}(\zeta ;x)$ is such that $$f_{2}>f_{1} \label{f2f1}$$ on $\left( 0,\pi \right) $, then $\zeta _{2}$ must vanish at some point of this domain. **Proof.** Let assume that $\zeta _{2}>0$ on $\left( 0,\pi \right) $. Then, from (\[pL\]) and the hypotheses of Theorem \[comp\], we have $$\begin{aligned} 2\int_{0}^{\pi }\,\left( f_{2}-f_{1}\right) \,dx &=&\left( \zeta _{1},L\zeta _{2}\right) _{p}-\left( L\zeta _{1},\zeta _{2}\right) _{p} \\ &=&2\alpha \int_{0}^{\pi }\left[ \left( p\,\zeta _{1}^{\prime }\right) ^{\prime }\zeta _{2}-\zeta _{1}\left( p\,\zeta _{2}^{\prime }\right) ^{\prime }\right] \,dx \\ &=&2\alpha \int_{0}^{\pi }\left[ p\,\left( \zeta _{1}^{\prime }\zeta _{2}-\zeta _{1}\,\zeta _{2}^{\prime }\right) \right] ^{\prime }\,dx\end{aligned}$$ which, in view of the boundary conditions and (\[f2f1\]), implies a contradiction $$p(\pi )\,\zeta _{1}^{\prime }(\pi )\,\zeta _{2}(\pi )>0\,.$$ Note that $\zeta _{1}^{\prime }(\pi )<0$ since $\zeta _{1}>0$ on $\left( 0,\pi \right) $ and $\zeta _{1}(\pi )=0$. So, there must exist $\overline{x} \in \left( 0,\pi \right) $ such that $\zeta _{2}(\overline{x})=0$. $\Box $ If we consider the eigenvalue equation $$L[\psi ]\theta =\lambda \theta \label{L-l}$$ on $\left( 0,\pi \right) $ for the smallest eigenvalue $\lambda $ in the space of odd periodic function, $\theta $ satisfies the conditions of $\zeta _{1}$ in Theorem \[comp\] with $f_{1}=\lambda p\zeta $. Note the eigenfunction associated to the smallest eigenvalue may be chosen to be positive in the domain $\left( 0,\pi \right) $. Applying Theorem \[comp\] for (\[Lphi\]) and (\[L-l\]) we arrive to the following stability criterium: \[critm\]The smallest eigenvalue $\lambda $ of $L[\psi ]$ is positive if $\varphi >0$ on $\left( 0,\pi \right) $ and negative if there exist $ \overline{x}\in \left( 0,\pi \right) $ such that $\varphi (\overline{x})=0$, where $\varphi $ is the solution of equations (\[Lphi\]) and (\[phiphi\]). Now, for a given non–trivial stationary solution $\psi $ let $$\chi =c\left( -\alpha \psi ^{\prime \prime }+4\psi \right) \,, \label{chi}$$ where $c>0$ is chosen so that $\chi ^{\prime }(0)=1$. It follows from the equation $-\alpha \psi ^{\prime \prime }=2\left( 1-\alpha \psi ^{\prime }\right) \psi \,$ (see (\[stationary\])), that $$\chi (0)=0\qquad \qquad \mathrm{and}\qquad \qquad \chi >0$$ whenever $\psi >0$ (recall $\psi (0)=0$ and $1-\alpha \psi ^{\prime }>0$ for all closed orbits). Moreover, we have \[equal\] $$L[\psi ]\chi =8c\alpha ^{2}\psi \left( \psi ^{\prime }\right) ^{2}>0 \label{Lchi}$$ on the same domain $\left( 0,\overline{x}\right) $ that $\psi >0$. **Proof.** Differentiating (\[stationary\]) twice, $$-\alpha \left( \psi ^{\prime \prime }\right) ^{\prime \prime }=-2\alpha \psi \left( \psi ^{\prime \prime }\right) ^{\prime }+2\left( 1-3\alpha \psi ^{\prime }\right) \psi ^{\prime \prime }\,,$$ and using (\[stationary\]) again, gives $$\begin{aligned} L[\psi ]\psi ^{\prime \prime } &=&-\alpha \left( \psi ^{\prime \prime }\right) ^{\prime \prime }+2\alpha \psi \left( \psi ^{\prime \prime }\right) ^{\prime }-2\left( 1-\alpha \psi ^{\prime }\right) \psi ^{\prime \prime } \\ &=&-4\alpha \psi ^{\prime }\psi ^{\prime \prime } \\ &=&8\left( 1-\alpha \psi ^{\prime }\right) \psi \,\psi ^{\prime }\end{aligned}$$ In addition, we have $$\begin{aligned} L[\psi ]\psi &=&-\alpha \psi ^{\prime \prime }+2\alpha \psi \psi ^{\prime }-2\left( 1-\alpha \psi ^{\prime }\right) \psi \\ &=&2\alpha \psi \psi ^{\prime }\end{aligned}$$ which combined with the above equation, gives the equality in Proposition \[equal\]. $\Box $ **Completion of the proof of Theorem \[positive\].** We are in position to prove Theorem \[positive\] for non–trivial equilibrium solutions. Let $\chi $ be given by (\[chi\]) with $\psi =\psi _{1}^{+}$. Then $\chi >0$ on $\left( 0,\pi \right) $ and Theorem \[comp\] can be used to compare equation (\[Lchi\]) with (\[Lphi\]). This yields $\varphi >\chi \geq 0$ on $(0,\pi ]$ which implies the stability of $\psi _{1}^{+}$ by Criterium \[critm\]. For instability, we observe that $\psi ^{\prime }$ satisfies $$\begin{aligned} L[\psi ]\psi ^{\prime } &=&-\alpha \psi ^{\prime \prime \prime }+2\alpha \psi \psi ^{\prime \prime }-2\left( 1-\alpha \psi ^{\prime }\right) \psi ^{\prime } \\ &=&\left( -\alpha \psi ^{\prime \prime }+2\alpha \psi \psi ^{\prime }-2\psi \right) ^{\prime }=0\,,\end{aligned}$$ in view of equation (\[stationary\]). Recall that $\psi =\psi _{j}^{+}$ with $j\geq 2$, has fundamental period $2\pi /j$ and satisfies $\psi (\pi /j)=\psi ^{\prime \prime }(\pi /j)=0$ by the odd parity and equation (\[stationary\]). Since $\psi ^{\prime }(0)>0$, this implies $\psi <0$ on $ \left( \pi /j,2\pi /j\right) $ and the minimum of $\psi $ is attained at $ \underline{x}=\dfrac{3\pi }{2j}$. Since $\psi ^{\prime }$ and $\varphi $ satisfies the same self–adjoint equation $pL[\psi ]\zeta =0$, their Wronskian $$\begin{aligned} W\left( \varphi ,\psi ^{\prime };x\right) &=&\left| \begin{array}{cc} \varphi & \psi ^{\prime } \\ -\alpha p\varphi ^{\prime } & -\alpha p\psi ^{\prime \prime } \end{array} \right| \\ &=&\alpha p\left( \varphi ^{\prime }\psi ^{\prime }-\varphi \psi ^{\prime \prime }\right) =\alpha \psi ^{\prime }(0)>0\end{aligned}$$ is a non–vanishing constant (recall $p(0)=1$, $\varphi (0)=0$ and $\left( \psi _{j}^{+}\right) ^{\prime }(0)>0$). As a consequence $$W\left( \varphi ,\psi ^{\prime };\pi /j\right) =-\alpha p(\underline{x} )\varphi \left( \underline{x}\right) \psi ^{\prime \prime }\left( \underline{ x}\right) >0$$ implies $\varphi \left( \underline{x}\right) <0$ because $\psi ^{\prime \prime }\left( \underline{x}\right) >0$. It thus follows from the stability criterium that $\psi _{j}^{+},\,j=2,\ldots ,k$, are unstable since $ \underline{x}\in \left( 0,\pi \right) $ provided $j\geq 2$ and there exist $ \overline{x}\in \left( 0,\pi \right) $, $\overline{x}<\underline{x}$, such that $\varphi (\overline{x})=0$. By a slight modification of these arguments, one may conclude the stability of $\psi _{1}^{-}$and instability of $\psi _{j}^{-},\,j=2,\ldots ,k$, as well. This concludes the proof of Theorem \[positive\] and, consequently, the proof of Theorem \[lstb\]. $\Box $ Now, we turn to the Liapunov stability analysis with a proof of global stability of the trivial solution $\phi _{0}\equiv 0$. Let $V$ be a real–valued functional on the subspace of absolutely continuous function of $D(A)$ given by $$V\left( v\right) =\int_{-\pi }^{\pi }\left\{ \left( \alpha ^{-1}-\,v^{\prime }\right) \ln \left( 1-\alpha v^{\prime }\right) +v^{\prime }-v^{2}\right\} \,dx \label{V}$$ and notice that $V(0)=0$ and $V\left( \eta \right) =W(\eta )+o\left( \left\| \eta \right\| ^{2}\right) $, as $\left\| \eta \right\| \rightarrow 0$, where $$W\left( v\right) =\frac{1}{2}\int_{-\pi }^{\pi }\left( \alpha \,v^{\prime 2}-2v^{2}\,\right) \,dx\,,$$ by Taylor expanding $g(w)=\left( \alpha ^{-1}-w\right) \ln \left( 1-\alpha w\right) -w$ around $w=0$. Observe that $W\left( v\right) =\left( 1/2\right) \left\| v\right\| _{1/2}^{2}$ if $\alpha >2$ and since $g(w)-(\alpha /2)w^{2}\geq 0$ if $\alpha w<1$, $V\left( v\right) \leq W\left( v\right) $ holds on the space $$\mathcal{V}=\left\{ v\in H_{\mathrm{o,p}}^{1}\cap H_{\mathrm{o,p} }^{2}:\alpha v^{\prime }<1\right\} \,\,,$$ of odd, positive and $2\pi $–periodic functions with distributional derivative up to second order. A Liapunov function $V$ of a dynamical system $\left\{ S(t),t\geq 0\right\} $ satisfies $$\overset{\cdot }{V}(v)=\overline{\lim\limits_{t\downarrow 0}}\frac{1}{t} \left( V\left( S(t)v\right) -V(v)\right) \leq 0 \label{Vdot}$$ for all $v\in \mathcal{V}$. We now show that (\[Vdot\]) holds if $S(t)$ is given by equation (\[dpde\]). More precisely, Let $S(t)v_{0}=v(t;v_{0})$ be the dynamical system in $\mathcal{V}$ given by (\[dyn\]). Then, the pair of functions $$\rho (w)=\frac{1}{1-\alpha w} \label{rhow}$$ and $$\Phi (p,w)=\left( \alpha ^{-1}-\,w\right) \ln \left( 1-\alpha w\right) +w-p^{2} \label{Ppw}$$ generate the Liapunov function given by (\[V\]): $$V(v)=\int_{0}^{\pi }\Phi (v,v_{x})\,dx\qquad \mathrm{with}\qquad \overset{ \cdot }{V}(v)=\int_{0}^{\pi }\rho (v_{x})v_{t}^{2}\,dx\,. \label{VVv}$$ **Proof.** Note that, from the parity of $v$ the integral in (\[V\]) can be made over $[0,\pi ]$. By the calculus of variations and equations (\[rhow\]) and (\[Ppw\]) we have $$\begin{aligned} \dot{V}\left( v\left( t,\cdot \right) \right) &=&-\int_{0}^{\pi }\left( \frac{d}{dx}\frac{\partial \Phi }{\partial v_{x}}-\frac{\partial \Phi }{ \partial v}\right) v_{t}\,dx+\left. \frac{\partial \Phi }{\partial v_{x}} v_{t}\right| _{0}^{\pi } \notag \\ &=&-\int_{0}^{\pi }\left( -\frac{d}{dx}\ln (1-\alpha v_{x})+2v\right) v_{t}\,dx \notag \\ &=&-\int_{0}^{\pi }\rho (v_{x})\,\left( \alpha v_{xx}-2\alpha vv_{x}+2v\right) v_{t}\,dx\,, \label{Vv}\end{aligned}$$ where $v_{t}(t,0)=v_{t}(t,\pi )=0$, $t\geq 0$, in view of the boundary conditions on $\mathcal{V}$. Since $\rho (w)\geq 0$ for $\alpha w<1$, this with (\[dpde\]) concludes the proof of Proposition. $\Box $ We have used the construction method based in the Euler–Lagrange equation to find this Liapunov function (see e.g. Chap. 2 of Zelenyak, Lavrentiev and Vishnevskii [@ZLV]). A sufficient condition for (\[VVv\]) hold leads to a first order partial differential equation for $\rho $ $$w\rho _{p}-\frac{2}{\alpha }(1-\alpha w)p\rho _{w}=-2p\rho$$ whose characteristics are given by the orbits $\gamma _{w_{0}}$ described in Section \[SS\] in the study of the equilibrium solutions of (\[dpde\]). Note that equation (\[Ppw\]) is the Lagrangian associated with the Hamiltonian (\[hamiltonian\]) (with $q$ defined by (\[phi\])). Due to the requirement $\alpha w<1$, our particular solution takes into account only the closed orbits. There may be other suitable choices which includes all orbits. The proof of global stability of $\phi _{0}$ requires that a subspace of $ \mathcal{V}$ be invariant under the flow equation (\[dpde\]). This is shown in the following by using the maximum principle. \[cone\] If $v(t,x)$ is a classical solution of equation (\[dpde\]) with initial condition $v(0,x)=v_{0}(x)\in \mathcal{V}$, then $\alpha v_{x}(t,x)<1$ and $$\alpha ^{-1}\left( x-\pi \right) <v(t,x)<\alpha ^{-1}x\,, \label{tud}$$ hold for all $t\geq 0$ and $0\leq x\leq \pi $. **Proof.** Denoting $$L[v]:=F(v_{xx},v_{x},v)-v_{t}, \label{Fut}$$ where $F(a_{1},a_{2},a_{3})=\alpha \left( a_{1}-2a_{2}\,a_{3}\right) +2a_{3}\ $is a continuous and differentiable function of its variables, the differential equation (\[dpde\]) can be written as $$L[v]=0. \label{Lv}$$ For $v$ satisfying (\[Lv\]) with $v(t,0)=v(t,\pi )=0\,$, $0\leq t\leq \tau $, and initial data $v(0,\cdot )=v_{0}$, let us suppose $z=z(t,x)$ and $ Z=Z(t,x)$ are such that $$L[Z]\leq 0\leq L[z] \label{LZLz}$$ for all $\left( t,x\right) $ in $D=\left( 0,\tau \right) \times \left( 0,\pi \right) $ with $z(t,y)\leq 0\leq Z(t,y)$, $y=0,\pi $ and $0\leq t\leq \tau $ , and $$z(0,x)\leq v_{0}(x)\leq Z(0,x)\,,$$ for $0\leq x\leq \pi $. Then, by the maximum principle (see [@PW], Theorem $12$ in Chap. $3$), $$z(t,x)\leq v(t,x)\leq Z(t,x)\, \label{zuZ}$$ in $\overline{D}=[0,\pi ]\times \lbrack 0,\tau ]$. The lower limit function $z$ is given by $$z(x)=\theta \left( x-\pi \right) \,, \label{z}$$ with $\theta \geq 0$. From (\[Lv\]), $$L[z]=2\theta (\alpha \theta -1)(\pi -x)\}\,,$$ is always positive provided $\alpha \theta \geq 1$. Analogously, the upper limit function $Z$ is given by $$Z(x)=\delta x\,, \label{Z}$$ from which $$L[Z]=-2\delta (\alpha \delta -1)x\,$$ is always negative provided $\alpha \delta \geq 1$. Since (\[LZLz\]) holds uniformly in $\tau $, equation (\[zuZ\]) holds for all $(t,x)$ in $\mathbb{R}_{+}\times \lbrack 0,\pi ]$. Note that $v$ remains bounded irrespective of $\alpha v_{x}<1$. However, if this condition holds for $t=0$, it remains for all $t>0$. To see this, observe from the equation $v_{t}=v_{xx}+(1-\alpha v_{x})v$ with $v_{xx}=0$, that the rate by which $|v|$ increases tends to zero when the inequality saturates. The inclusion of the Laplacian only smooths $v$ and prevents, even more, $v_{x}$ to increase beyond the threshold. The same argument justify the strict inequality (\[tud\]). This concludes proof of Theorem \[cone\]. $\Box $ We pause to discuss some properties of the classical solutions of equations (\[pde\]) and (\[utildeq\]). Recall that $\widetilde{u} (t,x)= \displaystyle\int_{0}^{x}v(t,y)\,dy$ with $v$ satisfying (\[dpde\]). Note that the cone $\mathcal{C}=\left\{ u\in H_{\mathrm{e,p}}^{1}\cap H_{ \mathrm{e,p}}^{2}:u\geq 0,\,\alpha u_{xx}<1\right\} $ is invariant under the unnormalized evolution (\[pde\]). For this, let $$M[u]:=\alpha (u_{xx}-u_{x}^{2})+2u.$$ If $u(t,x)$ is a classical solution of (\[pde\]) with initial value $ u_{0}\in \mathcal{C}$, since $M[u]=0$ for $u\equiv 0$, we have by Theorem $7$ in Chap. $3$ of [@PW] (see also Remark (ii) after this) that $u(t,x)\geq 0$ for all $t>0$. This, however, does not imply that $\widetilde{u}(t,x)$ remains positive (recall (\[uutilde\])). A proof of this assertion goes as follows. Theorem \[cone\] implies $\widetilde{u}(t,x)$ remains bounded, and $ \widetilde{u}_{xx}(t,0)$ bounded from above, if $\widetilde{u}$ satisfies (\[utildeq\]) with initial condition $u_{0}$ satisfying $\alpha ^{-1}[(x-\pi )^{2}/2-\pi ^{2}/2]<u_{0}<\alpha ^{-1}x^{2}/2$ (by integrating (\[tud\])). The comparison principle applied directly to equation (\[utildeq\]) leads to (\[LZLz\]) with $L$ replaced by $M$ and $0$ replaced by $\alpha \widetilde{u}_{xx}(t,0)$. An upper and lower solutions, $z$ and $Z $, can be obtained from the solution of the equilibrium initial value problem (\[syst\]): $$\Phi ^{\pm }(\alpha ,w_{0};x)=\int_{0}^{x}\Psi ^{\pm }(\alpha ,w_{0};y)\,dy\,,$$ where $\Psi ^{\pm }(\alpha ,w_{0};x)$ is the $p$–component of the closed orbit $\gamma _{\pm w_{0}}$ starting at $(\pm w_{0},0)$. Note that $\Phi ^{\pm }$ is an even periodic function of period $T=T(\alpha ,w_{0})$ given by a monotonically increasing function of both $w_{0}$ and $\alpha $ with $ T\rightarrow \infty $ as $\alpha w_{0}\uparrow 1$ for $\Phi ^{+}$ and as $ w_{0}\rightarrow \infty $ for $\Phi ^{-}$ (see proof of Theorem \[existence\] for details). $\Phi ^{+(-)}$ is also a monotone increasing (decreasing) function of $x$ in $[0,T/2]$ and satisfies $$M[\Phi ^{\pm }]=\pm \alpha w_{0}\,,$$ with $\Phi ^{\pm }(0)=\Phi ^{\pm \prime }(0)=\Phi ^{\pm \prime }(T/2)=0$ and $\Phi ^{\pm \prime \prime }(0)=w_{0}$. The lower limit function $z$ is given by $$z(x)=\theta \left( \Phi ^{-}(\alpha ,w_{0};x+\widetilde{x})+\frac{\alpha }{2} \widetilde{w}_{0}\right) \,, \label{zz}$$ where $\theta <1$, $\widetilde{w}_{0}\geq w_{0}$, $\widetilde{x}=\widetilde{x }(\alpha ,w_{0},\widetilde{w}_{0})$ is such that $z(0)=0$, with $w_{0}$ and $ \widetilde{w}_{0}$ so that $T$ is very large and $z^{\prime }(\pi )=0$ which can always be done in view of the properties of $\Phi ^{-}$. The upper limit function $Z$ can be written also as (\[zz\]) with $\Phi ^{-}$ replaced by $ \Phi ^{+}$ and the second term with minus sign. We have $$M[W^{\pm }]=\alpha \theta \{(1-\theta )(\Phi ^{\pm \prime })^{2}\mp ( \widetilde{w}_{0}-w_{0})\}\,,$$ with $W^{+}=Z$ and $W^{-}=z$. In order inequality (\[zuZ\]) holds uniformly in $\tau $, $\widetilde{u}_{xx}(t,0)$ has to remain bounded from above and below. Since $\widetilde{u}_{xx}(t,0)<\alpha ^{-1}$ by Theorem \[cone\], one may choose $\theta $ arbitrarily small in (\[zz\]) and take $\widetilde{w}_{0}$ and $w_{0}$ so large that $\theta (\widetilde{w} _{0}-w_{0})>\alpha ^{-1}$. In the limit as $\theta \rightarrow 0$ we have $ \widetilde{u}(t,x)\geq 0$ and $0\leq \widetilde{u}_{xx}(t,0)<\alpha ^{-1}$ for $0\leq t\leq \tau $, uniformly in $\tau $, implying $\widetilde{u} (t,x)\geq 0$ for all $t\geq 0$. LaSalle’s invariance principle allows us to apply Liapunov function techniques under milder assumptions. A subset $\mathcal{K}\subset \mathcal{V} $ of a complete metric space $\mathcal{V}$ is said to be *invariant* (positive invariant) if, for any $v_{0}\in \mathcal{K}$, there exist a continuous curve $v:\mathbb{R}\longrightarrow \mathcal{K}$ with $v(0)=v_{0}$ and $S(t)v(\tau )=v(t+\tau )$ for all $t\geq 0$ and $\tau \in \mathbb{R}$ ($ \mathbb{R}_{+}$). The following two theorems express the content of this principle. \[limitset\] Suppose $v_{0}\in \mathcal{V}$ is such that the orbit $ \gamma (v_{0})=\left\{ S(t)v_{0},t\geq 0\right\} $ through $v_{0}$ lies in a compact set in $\mathcal{V}$ and let $\omega (v_{0})$ denote its $\omega $ –limit set, i.e., $$\omega (v_{0})=\bigcap_{\tau \geq 0}\overline{\gamma \left( S(\tau )v_{0}\right) }\,$$ (see (\[omega\]) for alternative definition). Then $\omega (v_{0})$ is nonempty, compact, invariant, connected and $\mathrm{dist}\left( S(\tau )u_{0},\omega (v_{0})\right) \longrightarrow 0$ as $t\rightarrow \infty $. **Proof.** We refer to Theorem $4.3.3$ of [@H] for details. Note that $\omega (v_{0})$ is the intersection of a decreasing collection of nonempty compact sets. Note, in addition, that $\omega (v_{0})$ is positive invariant by definition and is invariant by compactness argument. $\Box $ \[invariant\] Let $V$ be a Liapunov function for $t\geq 0$ and, for $$\mathcal{E}:=\left\{ v\in \mathcal{V}:\,\overset{\cdot }{V}(v)=0\right\} , \label{EV}$$ let $\mathcal{K}$ be the maximal invariant set in $\mathcal{E}$.  If the orbit $\gamma (v_{0})$ lies in a compact set in $\mathcal{V}$, then $S(t)v_{0}\longrightarrow \mathcal{K}$ as $\ t\rightarrow \infty $. **Proof.**  By definition, $V\left( S(t)v_{0}\right) $ is a nonincreasing function of $t$ and bounded from below, by hypothesis. So, $ \lim\limits_{t\rightarrow \infty }V\left( S(t)v_{0}\right) =\upsilon $ exists. If $y\in \omega (v_{0})$, then $V(y)=\upsilon $ and, in view of the fact that $S(t)y=y$, we have $V\left( S(t)y\right) =\upsilon $ which implies $\overset{\cdot }{V}(t)=0$ and $\omega (v_{0})\in \mathcal{K}$. $\Box $ Now, we apply the invariance principle to the problem at our hand. As we will see, if $\mathcal{B}_{0}$ is a sufficient large ball around $\phi _{0}=0 $ in the cone $\mathcal{C}$ (with the induced topology of $H_{\mathrm{e,p} }^{1}$), the invariant set $\mathcal{K}_{k}=\left\{ \omega (u_{0}),\,u_{0}\in \mathcal{B}_{0}\right\} \subset \mathcal{E}$ consists of the union of unstable manifolds for the equilibrium points $\phi _{0},\phi _{1},\ldots ,\phi _{k}$, with $\phi _{j}(x)=\displaystyle\int_{0}^{x}\psi _{j}^{+}(y)\,dy$, provided $\alpha $ is such that $2/\left( k+1\right) ^{2}\leq \alpha <2/k^{2}$ holds for some $k\in \mathbb{N}$. Note that the hypotheses of Theorems \[limitset\] and \[invariant\] hold since the orbits of $S(t)v_{0}$ are bounded in $H_{\mathrm{o,p}}^{1}$ by Theorem \[cone\] and remain in a compact set of $H_{\mathrm{o,p}}^{1}$ in view of Theorem \[thivp\]. For this the Sobolev embedding theorem is evoked: $ W^{2,2}\left( -\pi ,\pi \right) \subset C^{1+a}\left( -\pi ,\pi \right) $ with continuous inclusion, so $v$ has a continuous representative in $C_{ \mathrm{o,p}}^{1+a}$ which belongs to $C_{\mathrm{o,p}}^{2+a}$ by Schauder estimates (see e.g [@S] and references therein). Therefore, any solutions $\widetilde{u}(t,x)=\int_{0}^{x}v(t,y)\,dy$ of (\[utildeq\]) in $ \mathcal{C}$ has a continuously three–times differentiable representative. We thus have \[gstb\] If $\alpha >2$, $\phi _{0}=0$ is globally asymptotically stable solution of (\[utildeq\]) in $$\widetilde{\mathcal{C}}=\left\{ u\in H_{\mathrm{e,p}}^{1}\cap H_{\mathrm{e,p} }^{2}:u(0)=0,\,u\geq 0\,\,\mathrm{and}\,\,\alpha u_{xx}<1\right\} \,.$$ If $\alpha <2$, the origin is unstable in $\widetilde{\mathcal{C}}$ and there exits an open dense set $\mathcal{U}\subset \widetilde{\mathcal{C}}$ of initial conditions such that $\lim\limits_{t\rightarrow \infty } \widetilde{u}(t;u_{0})\longrightarrow \phi _{1}^{+}$ for all $u_{0}\in \mathcal{U}$. **Proof.** It follows from Theorem \[invariant\] $v(t;\cdot )\longrightarrow \omega \left( v(0;\cdot )\right) \subset \left\{ \psi : \overset{\cdot }{V}\left( \psi \right) =0\right\} $ in $\mathcal{V}$ as $ t\rightarrow \infty $. But, from (\[Vv\]), $\overset{\cdot }{V}\left( \psi \right) =0$ iff $$\alpha \psi ^{\prime \prime }-2\alpha \,\psi \psi ^{\prime }+2\psi =0\,, \label{phixx}$$ whose solutions are $\psi =\psi _{0}$ and $\psi _{j}^{+},\,j=1,\ldots ,k$, studied in Section \[SS\].  We note that $\phi _{j}^{+}(x)=\displaystyle \int_{0}^{x}\psi _{j}^{+}(y)\,dy\geq 0$ ($\phi _{j}^{-}(x)\leq 0$) for all $ x\in \lbrack -\pi ,\pi ]$ and $j\geq 1$, since $\left( \psi _{j}^{+}\right) ^{\prime }(0)>0$ ($\left( \psi _{j}^{-}\right) ^{\prime }(0)<0$). Multiplying (\[phixx\]) by $\psi $ and integrating over $\left( -\pi ,\pi \right) $, gives $$\int_{-\pi }^{\pi }\left( \alpha \psi ^{\prime \prime }+2\psi \right) \psi \,dx=-\left\| \psi \right\| _{1/2}^{2}\leq 0$$ if $\alpha >2$. The nonlinear term vanishes since, by integration by parts, $$\int_{-\pi }^{\pi }\psi ^{\prime }\psi ^{2}\,dx=-2\int_{-\pi }^{\pi }\psi ^{\prime }\psi ^{2}\,dx\,.$$ This implies $\psi \equiv 0$ and proves that $S(t)v_{0}\longrightarrow 0$ as $t\rightarrow \infty $ in $\mathcal{V}$. We quote Theorem $4.3.5$ in [@H] for the instability assertion. Since the spectrum $\sigma (L)$ of the linearized operator around the equilibrium points (see Theorem \[positive\]) lies on the real line, all equilibrium points are hyperbolic, $\mathcal{E}$ given in (\[EV\]) is a discrete and finite set and $$\mathcal{V=}\bigcup_{\psi \in \mathcal{E}}\mathcal{W}^{s}(\psi )$$ holds with $\mathcal{W}^{s}(\psi )=\left\{ u_{0}\in \mathcal{V}:S(t)v_{0} \longrightarrow \psi \;\mathrm{as}\;t\rightarrow \infty \right\} $. It is proven in [@H] that each stable manifold $\mathcal{W}^{s}(\psi )$ is a $ C^{2}$ embedded submanifold of $\mathcal{V}$ ($\mathcal{W}^{s}(\phi )$ is $ C^{3}$ submanifold of $\widetilde{\mathcal{C}}$) and, if $\psi $ is locally unstable, than $\mathcal{W}^{s}(\psi )$ has codimension larger than or equal to $1$. Therefore, $\mathcal{V}$, and consequently $\widetilde{\mathcal{C}}$ , can be written as a *finite* union of open connected sets together with a closed nowhere–dense remainder. $\Box $ Finally, we show that, for an open set $\mathcal{V}_{0}\subset \mathcal{V}$ given as before,  the maximal invariant set $$\mathcal{K}_{k}=\bigcup_{\psi \in \mathcal{E}}\mathcal{W}^{u}(\psi ) \label{Mk}$$ where $\mathcal{W}^{u}(\psi )=\left\{ v_{0}\in \mathcal{V}:S(t)v_{0}\longrightarrow \psi \;\mathrm{as}\;t\rightarrow -\infty \right\} $ is the unstable manifold of $\psi $. By Theorem \[invariant\], the orbit $ v(t;u_{0})=S(t)v_{0}$ exists and remains, by invariance, in $\mathcal{K}_{k}$ for all $t\in \mathbb{R}$. Therefore, $\lim\limits_{t\rightarrow \infty }v(t,u_{0})=\psi $ exists and $\psi \in \mathcal{K}_{k}$, so $\mathcal{K} _{k}\subset \bigcup_{\psi \in \mathcal{E}}\mathcal{W}^{u}(\psi )$. Since the converse is also true, the equality (\[Mk\]) thus holds. **Acknowledgments** We wish to thank J. Fernando Perez for posing to one of the authors (D.H.U.M.), together with C. Ragazzo, the problem of what is the effect of the sequence of thresholds on the stable branch. The authors have benefited from the discussions with C. Ragazzo, W. F. Wreszinski and J. C. A. Barata. [JKKN]{} D. Brydges and P. Federbush, *Debye Screening*. Commun. Math. Phys. **73**, 197-245 (80). G. Benfatto, G. Gallavotti and F. Nicoló, *On the massive sine–Gordon equation in the first few regions of collapse*. Commun. Math. Phys. **85**, 387-410 (82). Carmen Chicone, *The monotonicity of the period function for planar Hamiltonian vector fields*. J. Diff. Eqns. **69**, 310-321 (87). W. A. Coppel and L. Gavrilov, *The period function of a Hamiltonian quadratic system*. Diff. and Int. Eqns. **6**, 1357-1365 (93). N. Chafee and E. F. Infante, *A bifurcation problem for a nonlinear partial differential equation of parabolic type*. J. Aplicable Anal. **4**, 17-37 (74). Earl A. Coddington and N. Levinson, *Theory of ordinary differential equations*. MacGraw-Hill Book Company, 1955. J. Dimock and T. R. Hurd, *Construction of the two–dimensional sine–Gordon model for* $\beta <8\pi $. Commun. Math. Phys. **156**, 547-580 (93). Giovanni Felder, *Renormalization group in the local potential approximation*, Commun. Math. Phys. **111**, 101-121 (87). Michael E. Fisher, Xiao–jun Li and Yan Levin,*On the absence of intermediate phases in the two–dimensional Coulomb gas*. J. Stat. Phys. ** 79**, 1-11 (95). J. Fröhlich and T. Spencer, *The Kosterlitz–Thouless in two-dimensional abelian spin systems and Coulomb gas* . Commun. Math. Phys. **81**, 527-602 (81) G. Gallavotti and F. Nicoló, *The “The screening phase transitions” in the two–dimensional Coulomb gas*. J. Stat. Phys. **39**, 133-156 (85).* * Daniel Henry, *Geometric theory of semilinear parabolic equations*. Lecture Notes in Mathematics **840**, Springer-Verlag, 1981. J. V. José, L. P. Kadanoff, S. Kirkpatric and D. R. Nelson, *Renormalization, vortices, and symmetry–breaking perturbations in the two–dimensional planar model*. Phys. Rev. D **16** , 1217-1241 (77). D. H. U. Marchetti and A. Klein, *Power–law falloff in two–dimensional Coulomb gases at inverse temperature* $\beta >8\pi $. J. Stat. Phys. **64**, 135-162 (91). D. H. U. Marchetti and J. F. Perez, *The Kosterlitz–Thouless phase transition in two–dimensinal hierarchical Coulomb gases*. J. Stat. Phys. **55**, 141-156 (89). Murray H. Protter and Hans F. Weinberger, *Maximum principles in differential equations*. Prentice–Hall, Inc., New Jersey, 1964. D. H. Sattinger, *Monotone methods in nonlinear elliptic and parabolic boundary value problems*. Indiana Univ. Math. J. **21**, 979-1000 (72). T. I. Zelenyak, M. M. Lavrentiev Jr. and M. P. Vishnevskii, *Qualitative theory of parabolic equations*. VSP, Utrecht, 1997. [^1]: Supported by FAPESP under grant $\#98/10745-1$. E-mail: *[email protected]*.        [^2]: Partially supported by CNPq, FINEP and FAPESP. E-mail: *[email protected]* [^3]: This is assured by a Lagrange multiplier (see Remark \[Lagrange\]). [^4]: From here on, the subindexes in $C_{\mathrm{o,p}}^{2}$, $L_{\mathrm{o,p} }^{2} $, $L_{\mathrm{e,p}}^{2}$, $H_{\mathrm{e,p}}^{1}$ and *etc.*, indicate spaces of odd and periodic (o,p) or even and periodic (e,p) functions. [^5]: We thank G. Benfatto for explaining this tranformation and for pointing us equation (\[h2\]) in a footnote of [@F].

--- abstract: 'We show that a number of model liquids at fixed volume exhibit strong correlations between equilibrium fluctuations of the configurational parts of (instantaneous) pressure and energy. We present detailed results for thirteen systems, showing in which systems these correlations are significant. These include Lennard-Jones liquids (both single- and two-component) and several other simple liquids, but not hydrogen-bonding liquids like methanol and water, nor the Dzugutov liquid which has significant contributions to pressure at the second nearest neighbor distance. The pressure-energy correlations, which for the Lennard-Jones case are shown to also be present in the crystal and glass phases, reflect an effective inverse power-law potential dominating fluctuations, even at zero and slightly negative pressure. An exception to the inverse-power law explanation is a liquid with hard-sphere repulsion and a square-well attractive part, where a strong correlation is observed, but only after time-averaging. The companion paper \[arXiv:0807.0551\] gives a thorough analysis of the correlations, with a focus on the Lennard-Jones liquid, and a discussion of some experimental and theoretical consequences.' author: - 'Nicholas P. Bailey' - 'Ulf R. Pedersen' - Nicoletta Gnan - 'Thomas B. Schr[ø]{}der' - 'Jeppe C. Dyre' title: 'Pressure-energy correlations in liquids. I. Results from computer simulations' --- Introduction ============ ![(Color online) Equilibrium fluctuations of (a) pressure $p$ and energy $E$ and (b) virial $W$ and potential energy $U$, in a single-component Lennard-Jones system simulated in the NVT ensemble at $\rho=34.6$ mol/l and $T=80$K (Argon units). The time-averaged pressure was close to zero (1.5 MPa). The correlation coefficient $R$ between $W$ and $U$ is 0.94, whereas the correlation coefficient is only 0.70 between $p$ and $E$. Correlation coefficients were calculated over the total simulation time (10 ns).[]{data-label="EP_timeFluc"}](EtotPIntime.png "fig:"){width="8.5cm"} ![(Color online) Equilibrium fluctuations of (a) pressure $p$ and energy $E$ and (b) virial $W$ and potential energy $U$, in a single-component Lennard-Jones system simulated in the NVT ensemble at $\rho=34.6$ mol/l and $T=80$K (Argon units). The time-averaged pressure was close to zero (1.5 MPa). The correlation coefficient $R$ between $W$ and $U$ is 0.94, whereas the correlation coefficient is only 0.70 between $p$ and $E$. Correlation coefficients were calculated over the total simulation time (10 ns).[]{data-label="EP_timeFluc"}](EpotVirIntime.png "fig:"){width="8.5cm"} Physicists are familiar with the idea of thermal fluctuations in equilibrium. They also know how to extract useful information from them, using linear response theory.[@Landau/Lifshitz:1980; @Hansen/McDonald:1986; @Reichl:1998; @Allen/Tildesley:1987] These methods started with Einstein’s observation that the specific heat in the canonical ensemble is determined by the magnitude of energy fluctuations. In any thermodynamic system some variables are fixed, and some fluctuate. The magnitude of the variances of the latter, and of the their mutual covariance, determine the thermodynamic “response” parameters.[@Landau/Lifshitz:1980] For example, in the canonical (NVT) ensemble, pressure $p$ and energy $E$ fluctuate; the magnitude of pressure fluctuations is related to the isothermal bulk modulus $K_T\equiv - V\left(\frac{\partial p}{\partial V}\right)_T$, that of the energy fluctuations to the specific heat at constant volume $c_V\equiv T\left(\frac{\partial S}{\partial T}\right)_V$, while the covariance $\langle \Delta p\Delta E \rangle$ is related[@Allen/Tildesley:1987] to the thermal pressure coefficient $\beta_V \equiv \left(\frac{\partial p}{\partial T}\right)_V$. If the latter is non-zero, it implies a degree of correlation between pressure and energy fluctuations. There is no obvious reason to suspect any particularly strong correlation, and to the best of our knowledge none has ever been reported. But in the course of investigating the physics of highly viscous liquids by computer simulation, we noted strong correlations between pressure and energy equilibrium fluctuations in several model liquids, also in the high temperature, low-viscosity state. These included the most studied of all computer liquids, the Lennard-Jones system. Surprisingly, these strong correlations survive crystallization, and they are also present in the glass phase. “Strong” here and henceforth means a correlation coefficient of order 0.9 or larger. In this paper we examine several model liquids and detail which systems exhibit strong correlations and which do not. In the companion paper[@Bailey/others:2008c] (referred to as Paper II) we present a detailed analysis of the correlations for the single-component Lennard-Jones system, and discuss some consequences. ![\[WUscatterArgon0P80\] (Color online) (a) Scatter-plot of instantaneous virial $W$ and potential energy $U$ from the simulation of Fig. \[EP\_timeFluc\]. The dashed line is a guide to the eye, with a slope determined by the ratio of standard deviations of $W$ and $U$ (Eq. (\[slopeDefinition\])). (b) Example of a system with almost no correlation between $W$ and $U$: TIP5P water at T=12.5$^\circ$C, density 1007.58 kg/m$^3$ (NVT). This system has Coulomb, in addition to Lennard-Jones, interactions.](WUscatterArgon0P80.png "fig:"){width="8.5"} ![\[WUscatterArgon0P80\] (Color online) (a) Scatter-plot of instantaneous virial $W$ and potential energy $U$ from the simulation of Fig. \[EP\_timeFluc\]. The dashed line is a guide to the eye, with a slope determined by the ratio of standard deviations of $W$ and $U$ (Eq. (\[slopeDefinition\])). (b) Example of a system with almost no correlation between $W$ and $U$: TIP5P water at T=12.5$^\circ$C, density 1007.58 kg/m$^3$ (NVT). This system has Coulomb, in addition to Lennard-Jones, interactions.](WUscatterWater.png "fig:"){width="8.5"} Specifically, the fluctuations which are in many cases strongly correlated are those of the configurational parts of pressure and energy (the parts in addition to the ideal gas terms). The (instantaneous) pressure $p$ and energy $E$ have contributions both from particle momenta and positions: $$\begin{aligned} p &= Nk_BT({\mathbf{p}}_1,\ldots,{\mathbf{p}}_N)/V+W({\mathbf{r}}_1,\ldots,{\mathbf{r}}_N)/V \nonumber\\ E &= K({\mathbf{p}}_1,\ldots,{\mathbf{p}}_N) + U({\mathbf{r}}_1,\ldots,{\mathbf{r}}_N),\end{aligned}$$ where $K$ and $U$ and the kinetic and potential energies, respectively. Here $T({\mathbf{p}}_1,\ldots,{\mathbf{p}}_N)$ is the “kinetic temperature”,[@Allen/Tildesley:1987] proportional to the kinetic energy per particle. The configurational contribution to pressure is the virial $W$, which is defined[@Allen/Tildesley:1987] by $$\label{generalWformula} W = -\frac{1}{3} \sum_i {\mathbf{r}}_i \cdot {\mathbf{\nabla}}_{{\mathbf{r}}_i} U $$ where ${\mathbf{r}}_i$ is the position of the $i$th particle. Note that $W$ has dimension energy. For a pair interaction we have $$U_{\textrm{pair}} = \sum_{i<j} v(r_{ij})$$ where $r_{ij}$ is the distance between particles $i$ and $j$ and $v(r)$ is the pair potential. The expression for the virial (Eq. (\[generalWformula\])) becomes[@Allen/Tildesley:1987] $$W_{\textrm{pair}} = -\frac{1}{3}\sum_{i<j} r_{ij} v'(r_{ij}) = -\frac{1}{3} \sum_{i<j} w(r_{ij})$$ where for convenience we define $$\label{pairVirialDef} w(r)\equiv r v'(r).$$ Fig. \[EP\_timeFluc\] (a) shows normalized instantaneous values of $p$ and $E$, shifted and scaled to have zero mean and unit variance, as a function of time for the standard single-component Lennard-Jones (SCLJ) liquid, while Fig. \[EP\_timeFluc\] (b) shows the corresponding fluctuations of $W$ and $U$. We quantify the degree of correlation by the standard correlation coefficient $R$, defined by $$\label{correlationCoeff} R=\frac {\langle\Delta W \Delta U\rangle}{\sqrt{\langle(\Delta W)^2\rangle}\sqrt{\langle(\Delta U)^2\rangle}}.$$ Here angle brackets $\langle\rangle$ denote thermal averages while $\Delta$ denotes deviation from the average value of the given quantity. The correlation coefficient is ensemble-dependent, but our main focus—the $R \rightarrow1$ limit–is not. Most of the simulations reported below were carried out in the NVT ensemble. Another important characteristic quantity is the “slope” $\gamma$, which we define as the ratio of standard deviations: $$\label{slopeDefinition} \gamma \equiv \frac{\sqrt{(\Delta W)^2}} {\sqrt{(\Delta U)^2}}.$$ Considering the “total” quantities, $p$ and $E$, (Fig. \[EP\_timeFluc\](a)) there is some correlation; the correlation coefficient $0.70$. For the configurational parts, $W$ and $U$, on the other hand (Fig. \[EP\_timeFluc\](b)), the degree of correlation is much higher, $R=0.94$ in this case. Another way to exhibit the correlation is a scatter-plot of $W$ against $U$, as shown in Fig. \[WUscatterArgon0P80\](a). Is this correlation surprising? Actually, there are some interatomic potentials for which there is a 100% correlation between virial and potential energy. If we have a pair potential of the form $v(r) \propto r^{-n}$, an inverse power-law, then $w(r) = -n v(r)$ and $W_{\textrm{pair}} = (n/3) U_{\textrm{pair}}$ holds exactly. In this case the correlation is 100% and $\gamma=n/3$. Conversely, suppose a system is known to be governed by a pair potential, and that there is 100% correlation between $W$ and $U$. We can write both $U$ and $W$ at any given time $t$ as integrals over the instantaneous radial distribution function defined[@Allen/Tildesley:1987] as $$\label{timeDepRDF} g(r,t) \equiv \frac{2}{N \rho} \sum_{i<j} \delta(r-r_{ij}(t))/(4\pi r^2)$$ from which $$\label{U_from_RDF} U(t) = \frac{N}{2} \rho \int_0^{\infty} \! dr \, 4\pi r^2 g(r,t) v(r)$$ and $$\label{W_from_RDF} W(t) = -\frac{N}{6} \rho \int_0^{\infty} \! dr \, 4\pi r^2 g(r,t) w(r).$$ ![\[effectivePowerLawFitSimple\] (Color online) Illustration of the “effective inverse power-law” chosen in this case to match the Lennard-Jones potential and its first two derivatives at the point $r=\sigma$. The vertical line marks the division into the repulsive and attractive parts of the Lennard-Jones potential.](effectivePowerLawFitSimple.png){width="8"} Here the factor of ${\frac{1}{2}}$ is to avoid double-counting, and $\rho=N/V$ is the number density. 100% correlation means that $W(t) = \gamma U(t)$ holds for arbitrary $g(r,t)$ (a possible additive constant could be absorbed into the definition of $U$). In particular we could consider $g(r,t)=\delta(r-r_0)$.[^1] Substituting this into the above expressions, the integrals go away and we find $w(r_0)=-3\gamma v(r_0)$. Since $r_0$ was arbitrary, $v'(r) = -3\gamma v(r)/r$, which has the solution $v(r) \propto r^{-3\gamma}$. This connection between an inverse power-law potential and perfect correlations suggests that strong correlations can be attributed to an [*effective inverse power-law potential*]{}, with exponent given by three times the observed value of $\gamma$. This will be detailed in Paper II, which shows that while this explanation is basically correct, matters are somewhat more complicated than this. For instance, the fixed volume condition, under which the strong correlations are actually observed, imposes certain constraints on $g(r,t)$. The celebrated Lennard-Jones potential is given[@Lennard-Jones:1931] by $$\label{LJpotential} v_{LJ}(r) = 4\epsilon\left[ \left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^{6} \right].$$ One might think that in the case of the Lennard-Jones potential the fluctuations are dominated by the repulsive $r^{-12}$ term, but this naïve guess leads to a slope of four, rather than the 6.3 seen in Fig. \[WUscatterArgon0P80\] (a). Nevertheless the observed correlation, and the above mentioned association with inverse power-law potentials, suggest that an effective inverse power-law description (involving short distances), with a more careful identification of the exponent, may apply. In fact, the presence of the second, attractive, [*term*]{}, increases the effective steepness of the repulsive [*part*]{}, thus increasing the slope of the correlation, or equivalently the effective inverse power-law exponent (Fig. \[effectivePowerLawFitSimple\]). Note the distinction between repulsive [*term*]{} and repulsive [*part*]{} of the potential: the latter is the region where $v(r)$ has negative slope, thus the region $r<r_m$ ($r_m$ being the distance where the pair potential has its minimum, $2^{1/6}\sigma$ for $v_{LJ}$). This region involves both the repulsive and attractive terms (see Fig. \[effectivePowerLawFitSimple\], which also illustrates the approximation of the repulsive part by a power law with exponent 18). The same division was made by Weeks, Chandler, and Andersen (WCA) in their noted paper of 1971,[@Weeks/Chandler/Andersen:1971] in which they showed that the the thermodynamic and structural properties of the Lennard-Jones fluid were dominated by the repulsive part at high temperatures for all densities, and also at low temperatures for high densities. Ben-Amotz and Stell[@Ben-Amotz/Stell:2003] noted that the repulsive core of the Lennard-Jones potential may be approximated by an inverse power-law with $n\sim$18–20. The approximation by an inverse power-law may be directly checked by computing the potential and virial with an inverse power-law potential for configurations drawn from actual simulations using the Lennard-Jones potential. The agreement (apart from additive constants) is good, see Paper II. Consider now a system with different types of pair interactions, for example a binary Lennard-Jones system with AA, BB, and AB interactions, or a hydrogen-bonding system modelled via both Lennard-Jones and Coulomb interactions. We can write arbitrary deviations of $U$ and $W$ from their mean values, denoted $\Delta U$ and $\Delta W$, as a sum over types (indexed by $t$; sums over pairs of a given type are implicitly understood): $$\Delta U = \sum_t \Delta U_t;\ \Delta W = \sum_t \Delta W_t .$$ Now, supposing there is near-perfect correlation for the individual terms with corresponding slopes $\gamma_t$, we can rewrite $\Delta W$ as $$\Delta W = \sum_t \gamma_t \Delta U_t.$$ If the $\gamma_t$ are all more or less equal to a single value $\gamma$, then this can be factored out and we get $\Delta W \simeq \gamma \Delta U$. Thus the existence of different Lennard-Jones interactions in the same system does not destroy the correlation, since they have $\gamma_t\sim6$. On the other hand the slope for Coulomb interaction, which as an inverse power-law has perfect $W,U$ correlations, is $1/3$, so we cannot expect overall strong correlation in this case (Fig. \[WUscatterArgon0P80\] (b)). Indeed such reasoning also accounts for the reduction of correlation when the total pressure and energy are considered: $\Delta E=\Delta U+ \Delta K$, while (for a large atomic system) $V\Delta p=\gamma\Delta U + (2/3) \Delta K$. The fact that $\gamma$ is (for the Lennard-Jones potential) quite different from 2/3 implies that the $p,E$ correlation is significantly weaker that of $W,U$ (Fig. \[EP\_timeFluc\]). Even in cases of unequal slopes, however, there can be circumstances under which one kind of term, and therefore one slope, dominates the fluctuations. In this case strong correlations will be observed. Examples include the high-temperature limits of hydrogen-bonded liquids (section \[ResultsAllSystems\]) and the time-averaged (total) energy and pressure in viscous liquids (Paper II). Some of the results detailed below were published previously in Letter form;[@Pedersen/others:2008] the aim of the present contribution is to make a comprehensive report covering more systems, while Paper II contains a detailed analysis and discusses applications. In the following section, we describe the systems simulated. In section \[Results\] we present the results for all the systems investigated, in particular the degree of correlation (correlation coefficient $R$) and the slope. Section \[Summary\] gives a summary. \[Simulations\]Simulated systems ================================ A range of simulation methods, thermodynamic ensembles and computational codes were used. One reason for this was to eliminate the possibility that the strong correlations are an artifact of using a particular ensemble or code. In addition, no one code can simulate the full range of systems presented. Most of the data we present are from molecular dynamics (MD) simulations, although some are from Monte Carlo [@Landau/Binder:2005] (MC) and event-driven[@Zaccarelli/others:2002] (ED) simulations. Most of the MD simulations (and of course all MC simulations), had fixed temperature (NVT), while some had fixed total energy (NVE). Three MD codes were used: Gromacs (GRO),[@Berendsen/vanderSpoel/vanDrunen:1995; @Lindahl/Hess/vanderSpoel:2001] Asap (ASAP), [@Asap] and DigitalMaterial (DM). [@Bailey/others:2006] Home-made (HM) codes were used for the MC and ED simulations. We now list the thirteen systems studied, giving each a code-name for future reference. The systems include monatomic systems interacting with pair potentials, binary atomic systems interacting with pair potentials, molecular systems consisting of Lennard-Jones particles joined rigidly together in a fixed configuration (here the Lennard-Jones interaction models the van der Waals forces), molecular systems which have Coulomb as well as Lennard-Jones interactions, metallic systems with a many-body potential, and a binary system interacting with a discontinuous “square-well” potential. Included with each system is a list specifying which simulation method(s), which ensemble(s), and which code(s) were used (semi-colons separate the method(s) from the ensemble(s) and the ensemble(s) from the code(s)). Details of the potentials are given in Appendix \[potentialDetails\]. CU : Pure liquid Cu simulated using the many-body potential derived from effective medium theory (EMT); [@Jacobsen/Norskov/Puska:1987; @Jacobsen/Stoltze/Norskov:1996] (MD; NVE; ASAP) DB : Asymmetric “dumb-bell” molecules,[@Pedersen/others:2008a] consisting of two unlike Lennard-Jones spheres connected by a rigid bond; (MD; NVT; GRO) DZ : The potential introduced by Dzugutov[@Dzugutov:1992] as a candidate for a monatomic glass-forming system. Its distinguishing feature is a peak in $v(r)$ around 1.5$\sigma$, after which it decays exponentially to zero at a finite value of $r$; (MD; NVT, NVE; DM) EXP : A system interacting with a pair potential with exponential repulsion and a van der Waals attraction; (MC; NVT; HM) KABLJ : The Kob-Andersen binary Lennard-Jones liquid;[@Kob/Andersen:1994] (MD; NVT, NVE; GRO, DM) METH : The Gromos[@Scott/others:1999] 3-site model for methanol; (MD; NVT; GRO) MGCU : A model of the metallic alloy Mg$_{85}$Cu$_{15}$ using an EMT-based potential; [@Bailey/Schiotz/Jacobsen:2004a] (MD; NVE; ASAP) OTP : A three-site model of the fragile glass-former Ortho-terphenyl (OTP);[@Lewis/Wahnstrom:1994](MD; NVT; GRO) SCLJ : The standard single-component Lennard-Jones system with the interaction given in Eq. (\[LJpotential\]); (MD, MC; NVT, NVE; GRO, DM) SPC/E : The SPC/E model of water;[@Berendsen/Grigera/Straatsma:1987] (MD; NVT; GRO) SQW : A binary model with a pair interaction consisting of an infinitely hard core and an attractive square well; [@Zaccarelli/others:2002; @Zaccarelli/Sciortino/Tartaglia:2004](ED; NVE; HM) TIP5P : A five-site model for liquid water which reproduces the density anomaly.[@Mahoney/Jorgensen:2000] (MD; NVT; GRO) TOL : A 7-site united-atom model of toluene; (MD; NVT; GRO) The number of particles (atoms or molecules) was in the range 500–2000. Particular simulation parameters ($N$, $\rho$, $T$, duration of simulation) are given when appropriate in the results section. \[Results\]Results ================== \[resultsSCLJ\]The standard single-component Lennard-Jones system ----------------------------------------------------------------- $\rho$ (mol/l) $T$(K) p(MPa) phase $R$ $\gamma$ ---------------- -------- -------- --------- ------- ---------- 42.2 12 2.6 glass 0.905 6.02 39.8 50 -55.5 crystal 0.987 5.85 39.8 70 -0.5 crystal 0.989 5.73 39.8 90 54.4 crystal 0.990 5.66 39.8 110 206.2 liquid 0.986 5.47 39.8 150 309.5 liquid 0.988 5.34 37.4 60 -3.7 liquid 0.965 6.08 37.4 100 102.2 liquid 0.976 5.74 37.4 140 192.7 liquid 0.981 5.55 37.4 160 234.3 liquid 0.983 5.48 36.0 70 -0.7 liquid 0.954 6.17 36.0 110 90.3 liquid 0.969 5.82 36.0 150 169.5 liquid 0.977 5.63 36.0 190 241.4 liquid 0.981 5.49 36.0 210 275.2 liquid 0.982 5.44 34.6 60 -42.5 liquid 0.900 6.53 34.6 100 41.7 liquid 0.953 6.08 34.6 140 114.5 liquid 0.967 5.80 34.6 200 211.0 liquid 0.977 5.57 32.6 70 -35.6 liquid 0.825 6.66 32.6 90 -0.8 liquid 0.905 6.42 32.6 110 31.8 liquid 0.929 6.22 32.6 150 91.7 liquid 0.954 5.95 32.6 210 172.7 liquid 0.968 5.68 37.4 60 -3.7 liquid 0.965 6.08 36.0 70 -0.7 liquid 0.954 6.17 34.6 80 1.5 liquid 0.939 6.27 32.6 90 0.0 liquid 0.905 6.42 42.2 12 2.6 glass 0.905 6.02 : \[SCLJ\_correlationCoeffs\] Correlation coefficients $R$ and effective slopes $\gamma$ for the single-component Lennard-Jones system (SCLJ) for the state points in Fig. \[completeArgonTP\]. $p$ is the thermally averaged pressure. The last five states were chosen to approximately follow the isobar $p=0$. SCLJ is the system we have most completely investigated. $W,U$-plots are shown for a range of thermodynamic state points in Fig. \[completeArgonTP\]. Here the ensemble was NVT with N=864, and each simulation consisted of a 10 ns run taken after 10ns of equilibration; for all SCLJ results so-called “Argon” units are used ($\sigma=0.34$ nm, $\epsilon=0.997$ kJ/mol). Each elongated oval in Fig. \[completeArgonTP\] is a collection of $W,U$ pairs for a given state point. Varying temperature at fixed density moves the oval parallel to itself, following an almost straight line as indicated by the dashed lines. Different densities correspond to different lines, with almost the same slope. In a system with a pure inverse power-law interaction, the correlation would be exact, and moreover the data for all densities would fall on the same straight line (see the discussion immediately after Eq. (\[pairVirialDef\])). Our data, on the other hand, show a distinct dependence on volume, but for a given volume, because of the strong correlation, the variation in $W$ is almost completely determined by that of $U$. ![\[CC\_Slope\_vs\_T\] (Color online) Upper plot, correlation coefficient $R$ for the SCLJ system as a function of temperature for several densities (NVT). This figure makes clear the different effects of density and temperature on $R$. Lower plot, effective slope $\gamma$ as a function of $T$. Simulations at temperatures higher than those shown here indicate that the slope slowly approaches the value four as $T$ increases. This is to be expected because as collisions become harder, involving shorter distances, the effective inverse power-law exponent approaches the 12 from the repulsive term of the Lennard-Jones potential.](CC_Slope_vs_T.png){width="8.5cm"} Values of correlation coefficient $R$ for the state points of Fig. \[completeArgonTP\] are listed in Table \[SCLJ\_correlationCoeffs\], along with the slope $\gamma$. In Fig. \[CC\_Slope\_vs\_T\] we show the temperature dependence of both $R$ and $\gamma$ for different densities. Lines have been drawn to indicate isochores and one isobar ($p=0$). Note that when we talk of an isobar here, we mean a set of NVT ensembles with $V,T$ chosen so that the thermal average of $p$ takes on a given value, rather than fixed-pressure ensembles. This figure makes it clear that for fixed density, $R$ increases as $T$ increases, while it also increases with density for fixed temperature; the slope slowly decreases in these circumstances. In fact it eventually reaches four, the value expected for a pure $r^{-12}$ interaction (e.g., at $\rho=34.6$ mol/l, $T=1000$ K, $\gamma=4.61$, see Ref. ). This is consistent with the idea that the repulsive part, characterized by an effective inverse power-law, dominates the fluctuations: increasing either temperature or density increases the frequency of short-distance encounters while reducing the typical distances of such encounters. On the other hand, along an isobar, these two effects work against each other, since as $T$ increases, the density decreases. The density effect “wins” in this case, which is equivalent to a statement about the temperature and volume derivative of $R$: Our simulations imply that $$\left(\frac{\partial R}{\partial T}\right)_p = \left(\frac{\partial R}{\partial T}\right)_V + \left(\frac{\partial R}{\partial V}\right)_T \left(\frac{\partial V}{\partial T}\right)_p < 0$$ which is equivalent to $$\left(\frac{\partial R}{\partial T}\right)_V < -\left(\frac{\partial R}{\partial V}\right)_T V \alpha_p = \rho \left(\frac{\partial R}{\partial \rho}\right)_T \alpha_p$$ where $\alpha_p \equiv (\partial V/\partial T)_p/V$ is the thermal expansivity at constant pressure and $\rho$ is the particle density. This can be recast in terms of logarithmic derivatives (valid whenever $(\partial R/\partial \rho)_T>0$) as follows $$\label{rhoBeatsTempequiv} \frac{ \left( \frac{\partial R}{\partial \ln(T)} \right)_V } { \left( \frac{\partial R}{\partial \ln(\rho)}\right)_T } < T\alpha_p.$$ Thus what we observe in the simulations, namely that the correlation becomes stronger as temperature is reduced at fixed pressure, is a priori more to be expected when the thermal expansivity is large (since then the right hand side of Eq. (\[rhoBeatsTempequiv\]) is large). This has particular relevance in the context of supercooled liquids, which we discuss in paper II, because these are usually studied by lowering temperature at fixed pressure. On the other hand if the expansivity becomes small, as for example, when a liquid passes through the glass transition, the inequality (\[rhoBeatsTempequiv\]) is a priori less likely to be satisfied. We have in fact observed this in a simulation of OTP: upon cooling through the (computer) glass transition, the correlation became weaker with further lowering of temperature at constant pressure. Remarkably, the correlation persists when the system has crystallized, as seen in the data for the highest density—the occurrence of the first-order phase transition can be inferred from the gap between the data for 90K and 110K, but the data fall on the same line above and below the transition. One would not expect the dynamical fluctuations of a crystal, which are usually assumed to be well-described by a harmonic approximation, to resemble those of the high-temperature liquid. In fact for a one-dimensional crystal of particles interacting with a harmonic potential $v(r) = {\frac{1}{2}}k (r-r_m)^2$ it is easy to show (Paper II) that there is a negative correlation with slope equal to -2/3. To investigate whether the harmonic approximation ever becomes relevant for the correlations, we prepared a perfect FCC crystal of SCLJ particles at zero temperature and simulated it at increasing temperatures, from 0.02K to 90K in Argon units, along a constant density path. The results are shown in Fig. \[fccArgon\]. Clearly the correlation is maintained right down to zero temperature. The harmonic approximation is therefore useless for dealing with the pressure fluctuations even as $T\rightarrow0$, because the slope is far from -2/3. The reason for this is that the dominant contribution to the virial fluctuations comes from the third order term, as shown in Paper II. ![\[fccArgon\] (Color online) Scatter-plot of the $W,U$ correlations for a perfect face-centered cubic (FCC) crystal of Lennard-Jones atoms at temperatures 1K, 2K, 3K, 5K, 10K, 20K, 30K, 40K, 50K, 60K, 70K and 80K, as well as for defective crystals (i.e., crystallized from the liquid) at temperatures 50K, 70K and 90K (NVT). The dashed line gives the best fit to the (barely visible) lowest-temperature data ($T=1$ K). The inset shows the temperature dependence of $R$ at very low temperatures. The crystalline case is examined in detail in Paper II, where we find that $R$ does not converge to unity at $T=0$, but rather to a value very close to unity.](fccArgon.png){width="8.5cm"} \[Dzugutov\]A case with little correlation: the Dzugutov system --------------------------------------------------------------- ![\[DZpotential\] (Color online) A plot of the Dzugutov pair potential, with the Lennard-Jones potential (shifted by a constant) shown for comparison.](DZpotential.png){width="8.5cm"} ![\[DzugutovVPE\] (Color online) Scatter-plot of $W,U$ correlations for the Dzugutov system at density 0.88 and temperatures 0.65 and 0.70 (NVE). The dashed lines indicate the best-fit line using linear regression. These are consistent with the temperature dependence of the mean values of $\langle W\rangle$ and $\langle U\rangle$, as they should be (see appendix \[FlucThermDerivs\]), but they clearly do not represent the direction of greatest variance. The full lines have slopes equal to the ratio of standard deviations of the two quantities (Eq. (\[slopeDefinition\])). The correlation coefficient is 0.585 and 0.604 for $T=0.65$ and $T=0.70$, respectively.](DzugutovVPE.png){width="8.5cm"} Before presenting data for all the systems studied, it is useful to see what it means for the correlation not to hold. In this subsection we consider the Dzugutov system,[@Dzugutov:1992] whose potential contains a peak at the second-neighbor distance, (Fig. \[DZpotential\], see appendix \[potentialDetails\] for details) whose presence might be expected to interfere with the effectiveness of an inverse power-law description. In the next subsection we show how in a specific model of water the lack of correlation can be explicitly seen to be the result of competing interactions. Fig. \[DzugutovVPE\] shows $W,U$ plots for the Dzugutov system for two nearby temperatures at the same density. The ovals are much less elongated than was the case for SCLJ, indicating a significantly weaker correlation—the correlation coefficients here are 0.585 and 0.604, respectively. In paper II it is shown explicitly that the weak correlation is due to contributions arising from the second peak. Note that the major axes of the ovals are not aligned with the line joining the state points, given by the mean values of $W$ and $U$, here identifiable as the intersection of the dashed and straight lines. On the other hand, the lines of best fit from linear regression, indicated by the dashed lines in each case, [*do*]{} coincide with the line connecting state points. This holds generally, a fact which follows from statistical mechanics (appendix \[FlucThermDerivs\]). The interesting thing is rather that the major axes point in different directions, whereas in the SCLJ case they are also aligned with the state-point line. The linear-regression slope, being equal to $\langle \Delta U \Delta W\rangle/ \langle (\Delta U)^2\rangle$, treats $W$ and $U$ in an asymmetric manner by involving $\langle (\Delta U)^2\rangle$, but not $\langle (\Delta W)^2\rangle$. This is because a particular choice of independent and dependent variables is made. If instead we plotted $U$ against $W$, we would expect the slope to be simply the inverse of the slope in the $W,U$ plot, but in fact the new slope is $\langle \Delta U \Delta W\rangle/\langle (\Delta W)^2\rangle$. This equals the inverse of the original slope only in the case of perfect correlation, where $\langle \Delta U \Delta W\rangle^2 = \langle (\Delta W)^2\rangle \langle (\Delta U)^2\rangle$. For our purposes a more symmetric estimate of the slope is desired, one which agrees with the linear regression slope in the limit of perfect correlation. We use simply the ratio of standard deviations $\sqrt{\langle(\Delta W)^2\rangle} /\sqrt{\langle(\Delta U)^2\rangle}$ (Eq. (\[slopeDefinition\])). This slope was used to plot the dashed line in Fig. \[WUscatterArgon0P80\](a) and the full lines in Fig. \[DzugutovVPE\], where it clearly represents the orientation of the data better.[^2] \[TIP5Pwater\]When competition between van der Waals and Coulomb interactions kills the correlation: TIP5P water ---------------------------------------------------------------------------------------------------------------- As we shall see in the next section, the systems which show little correlation include several which involve both van der Waals and Hydrogen bonding, modeled by Lennard-Jones and Coulomb interactions respectively. As noted already, the latter, being a pure inverse power-law ($n=1$), by itself exhibits perfect correlation with slope $\gamma=1/3$, while the Lennard-Jones part has near perfect correlation. But the significant difference in slopes means that no strong correlation is seen for the full interaction. To check explicitly that this is the reason the correlation is destroyed we have calculated the correlation coefficients for the Lennard-Jones and Coulomb parts separately in a model of water. Water is chosen because the density of Hydrogen bonds is quite high. Simulations were done with the TIP5P model of water [@Mahoney/Jorgensen:2000] which has the feature that the density maximum is reasonably well reproduced. This existence of the density maximum is in fact related to pressure and energy becoming uncorrelated, as we shall see. ![\[TIP5PwaterCorrCoeff\]Plot of $R$ for TIP5P water in NVT simulations with densities chosen to give an average pressure of one atmosphere. Not only is the magnitude of $R$ low (less than 0.2) in the temperature range shown, but it changes sign around the density maximum. The vertical arrow indicates the state point used for Fig. \[WUscatterArgon0P80\] (b). ](TIP5PwaterCorrCoeff.png){width="8.5cm"} Figure \[TIP5PwaterCorrCoeff\] shows the correlation coefficients and slopes for a range of temperatures; the correlation is almost non-existent, passing through zero around where the density attains its maximum value. We have separately determined the correlation coefficient of the Lennard-Jones part of the interaction; it ranges from 0.9992 at -25 $^\circ$C to 0.9977 at 75 $^\circ$C, even larger than we have seen in the SCLJ system. The reason for this is that the (attractive) Coulomb interaction forces the centers of the Lennard-Jones interaction closer together than they would be otherwise, thus the relevant fluctuations are occurring higher up the repulsive part of the Lennard-Jones pair potential. Correspondingly the slope from this interaction ranges between 4.45 and 4.54, closer to the high-$T$, high density limit of 4 than was the case for the SCLJ system. This is confirmed by inspection of the oxygen-oxygen radial distribution function in Ref.  where it can be seen that the first peak lies entirely to the left of the $v_{LJ}=0$ distance $\sigma=0.312$ nm. Finally note that the near coincidence between the vanishing of the correlation coefficient and the density maximum, which is close to the experimental value of 4$^\circ$C, is not accidental: The correlation coefficient is proportional to the configurational part of the thermal pressure coefficient $\beta_V$ (Paper II). The full $\beta_V$ vanishes exactly at the density maximum (4$^\circ$C), but the absence of the kinetic term means that the correlation coefficient vanishes at a slightly higher temperature ($\sim12^\circ$C). \[ResultsAllSystems\]Results for all systems -------------------------------------------- {width="15.5cm"} In Fig. \[correlationPlotAllLiquids\] we summarize the results for the various systems. Here we plot the numerator of Eq. (\[correlationCoeff\]) against the denominator, including factors of $1/(k_BTV)$ in both cases to make an intensive quantity with units of pressure. Since $R$ cannot be greater than unity, no points can appear above the diagonal. Being exactly on the diagonal indicates perfect correlation ($R=1$), while being significantly below it indicates poor correlation. Different types of symbols indicate different systems, as well as different densities for the same system, while symbols of the same type correspond to different temperatures. system $\rho$(mol/L) $T$(K) phase $R$ $\gamma$ -------- --------------- -------- -------- -------- ---------- CU 125.8 1500 liquid 0.907 4.55 CU 125.8 2340 liquid 0.926 4.15 DB 11.0 130 liquid 0.964 6.77 DB 9.7 300 liquid 0.944 7.45 DZ 37.2 78 liquid 0.585 3.61 EXP 30.7 96 liquid 0.908 5.98 EXP 33.2 96 liquid 0.949 5.56 KABLJ 50.7 30 glass 0.858 6.93 KABLJ 50.7 70 liquid 0.946 5.75 KABLJ 50.7 240 liquid 0.995 5.10 METH 31.5 150 liquid 0.318 22.53 METH 31.5 600 liquid 0.541 6.88 METH 31.5 2000 liquid 0.861 5.51 MGCU 85.0 640 liquid 0.797 4.74 MGCU 75.6 465 liquid 0.622 6.73 OTP 4.65 300 liquid 0.913 8.33 OTP 4.08 500 liquid 0.884 8.78 OTP 3.95 500 liquid 0.910 7.70 SPC/E 50.0 200 liquid 0.016 208.2 SPC/E 55.5 300 liquid 0.065 48.6 SQW 60.8 48 liquid -0.763 -50.28 SQW 60.8 79 liquid -0.833 -49.11 SQW 60.8 120 liquid -0.938 -52.02 SQW 59.3 120 liquid -0.815 -30.07 TIP5P 55.92 273 liquid -0.051 -2.47 TIP5P 55.94 285.5 liquid 0.000 2.51 TOL 10.5 75 glass 0.877 7.59 TOL 10.5 300 liquid 0.961 8.27 : \[allSystems\_correlationCoeffs\]Correlation coefficients and effective slopes at selected state points for all investigated systems besides SCLJ. Argon units were used for DZ, EXP, KABLJ and SQW by choosing the length parameter (of the larger particle when there were two types) to be 0.34 nm and the energy parameter to be 0.997 kJ/mol. The phase is indicated as liquid or glass. SQW data involves time averaging over periods 3.0, 3.0, 8.0 and 9.0, respectively, for the four listed state points. A minus sign has been included with the slope when $R<0$; note that the $\gamma$ values only really make sense as slopes when $|R|$ is close to unity. The ensemble was NVT except for CU, DZ, MGCU, and SQW, where it was NVE. All of the simple liquids, SCLJ, KABLJ, EXP, DB, TOL, show strong correlations, while METH, SPC/E, and TIP5P show little correlation. Values of $R$ and $\gamma$ at selected state points for all systems are listed in Table \[allSystems\_correlationCoeffs\]. What determines the degree of correlation? The Dzugutov and TIP5P cases have already been discussed. The poor correlation for METH and SPC/E is (presumably) because these models, like TIP5P, involve both Lennard-Jones and Coulomb interactions. In systems with multiple Lennard-Jones species, but without any Coulomb interaction, there is overall a strong correlation because the slope is almost independent of the parameters for a given kind of pair. As the temperature is increased, the data for the most poorly correlated systems, which are all hydrogen-bonding organic molecules, slowly approach the perfect-correlation line. This is consistent with the idea that this correlation is observed when fluctuations of both $W$ and $U$ are dominated by close encounters of pairs of neighboring atoms; at higher temperature there are increasingly many such encounters, which therefore come to increasingly dominate the fluctuations. Also because the Lennard-Jones potential rises much more steeply than the Coulomb potential, the latter becomes less important as short distances dominate more. Although not obvious in the plot, we find that increasing the density at fixed temperature generally increases the degree of correlation, as found in the SCLJ case; this is also consistent with the increasing relevance of close encounters or collisions. ![\[SQW\_Illustration\]Illustration of the square-well potential, indicating the four microscopic processes which contribute to the virial.](SQW_Illustration.png){width="8.5"} ![\[SQW\_correlationFuncs\] (Color online) Energy-energy, virial-virial, and energy-virial correlation functions for SQW at packing fraction $\phi=0.595$ and temperature $T=1.0$ (normalized to unity at $t=0$). The cross correlation has been multiplied by -1. The arrow marks the time $t=8$, roughly 1/10 of the relaxation time (determined from the long-time part of the energy-virial cross-correlation function). This time was used for averaging.](SQW_correlationFuncs.png){width="8.5"} A system quite different from the others presented so far is the binary square-well system, SQW, with a discontinuous potential combining hard-core repulsion and a narrow attractive well (Fig. \[SQW\_Illustration\]; see appendix \[potentialDetails\] for details). Such a potential models attractive colloidal systems,[@Zaccarelli/others:2002] one of whose interesting features, predicted from simulations and theory, is the existence of two different glass phases, termed the “repulsive” and “attractive” glasses. [@Sciortino:2002] The discontinuous potential not only makes the simulations substantially different from a technical point of view, but there are also conceptual differences—in particular, the instantaneous virial is a sum of delta-functions in time. The dynamical algorithm employed in the simulations is “event-driven”, where events involve a change in the relative velocity of a pair of particles due to hitting the hard-core inner wall of the potential or crossing the potential-step. The algorithm must detect the next event, advance time by the appropriate amount, and adjust the velocities of all particles appropriately. There are four kinds of events (illustrated in Fig. \[SQW\_Illustration\]): (1) “collisions”, involving the inner repulsive core; (2) “bounces”, involving bouncing off the outer (attractive) wall of the potential; (3) “trapping”, involving the separation going below the range of the outer wall and (4) “escapes”, involving the separation increasing beyond the outer wall. To obtain meaningful values of the virial a certain amount of time-averaging must be done—we can no longer consider truly instantaneous quantities. As shown in appendix \[Virial\_SQW\] the time-averaged virial may be written as the following sum over all events which take place in the averaging interval $ t_{\textrm{av}}$: $$\label{SQWvirial} \bar{W} = \frac{1}{3 t_{\textrm{av}}} \sum_{\textrm{events}\, e} m_{r,e}{\mathbf{r}}_{e} \cdot \Delta {\mathbf{v}}_{e}$$ Here ${\mathbf{r}}_e$ and ${\mathbf{v}}_e$ refer to the relative position and velocity for the pair of particles participating in event $e$, while $\Delta$ indicates the change taking place in that event. Positive contributions to $\bar{W}$ come from collisions; the three other event types involve the outer wall, which, as is easy to see, always gives a negative contribution. The default $t_{\textrm{av}}$ in the simulation was 0.05. Strong correlations emerge only at longer averaging times, however. An appropriate $t_{\textrm{av}}$ may be chosen by considering the correlation functions (auto- and cross-) for virial and potential energy, plotted in Fig. \[SQW\_correlationFuncs\], where the “instantaneous” values $E(t)$ and $W(t)$ correspond to averaging over 0.05 time units. We choose $t_{\textrm{av}} \simeq \tau_\alpha/10$, where $\tau_\alpha$ is the relaxation time determined from the cross-correlation function $\langle\Delta U(0)\Delta W(t)\rangle$. Data for a few state points are shown in Table \[allSystems\_correlationCoeffs\]. Remarkably, this system, so different from the continuous potential systems, also exhibits strong $W,U$-correlations, with $R=0.94$ in the case $T=1.0, \phi=0.595$ (something already hinted at in Fig. \[SQW\_correlationFuncs\] in the fact that the curves coincide). There is a notable difference, however, compared to continuous systems: The correlation is negative. The reason for the negative correlation is that at high density, most of the contributions to the virial are from collisions: a particle will collide with neighbor 1, recoil and then collide with neighbor 2 before there is a chance to make a bounce event involving neighbor 1. The number of collisions that occur in a given time interval is proportional to the number of bound pairs which is exactly anti-correlated with the energy. The effective slope $\gamma$ has a large (negative) value of order -50, which does not seem to depend strongly on temperature. This example is interesting because it shows that strong pressure-energy correlations can appear in a wider range of systems that might first have been guessed. Note, however, that the ordinary hard-sphere system cannot display such correlations, since potential energy does not exist as a dynamical variable in this system, i.e., it is identically zero. The idea of correlations emerging when quantities are averaged over a suitable time interval is one we shall meet again in Paper II in the context of viscous liquids. \[Summary\]Summary ================== We have demonstrated a class of model liquids whose equilibrium thermal fluctuations of virial and potential energy are strongly correlated. We have presented detailed investigations of the presence or absence of such correlations in various liquids, with extra detail presented for the standard single-component Lennard-Jones case. One notable aspect is how widespread these correlations are, appearing not just in Lennard-Jones potentials or potentials which closely resemble the Lennard-Jones one, but also in systems involving many-body potentials (CU, MGCU) and discontinuous potentials (SQW). We have seen how the presence of different types of terms in the potential, such as Lennard-Jones and Coulomb interactions, spoil the correlations, even though each by itself would give rise to a strongly correlating system. Most simulations were carried out in the NVT ensemble; $R$ is ensemble dependent, but the $R\rightarrow 1$ limit is not. Several of the hydrocarbon liquids studied here were simulated using simplified “united-atom” models where groups such as methyl-groups or even benzene rings were represent by Lennard-Jones spheres. These could also be studied using more realistic “all-atom” models, where every atom (including hydrogen atoms) is included. It would be worth investigating whether the strength of the correlations is reduced by the associated Coulomb terms in such cases. In paper II we provide a detailed analysis for the single-component Lennard-Jones case, including consideration of contributions beyond the effective inverse power-law approximation and the $T\rightarrow0$ limit of the crystal. There we also discuss some consequences, including a direct experimental verification of the correlations for supercritical Argon and consequences of strong pressure-energy correlations in highly viscous liquids and biomembranes. Useful discussions with S[ø]{}ren Toxv[æ]{}rd are gratefully acknowledged. Center for viscous liquid dynamics “Glass and Time” is sponsored by The Danish National Research Foundation. \[potentialDetails\]Details of interatomic potentials ===================================================== Here we give more detailed information about the interatomic potentials used. These details have been published elsewhere as indicated, except for the case of EXP and TOL. CU : Pure liquid Cu simulated using the many-body potential derived from effective medium theory (EMT). [@Jacobsen/Norskov/Puska:1987; @Jacobsen/Stoltze/Norskov:1996] This is similar to the embedded atom method of Daw and Baskes,[@Daw/Baskes:1984] where the energy of a given atom $i$, $E_i$ is some nonlinear function (the “embedding function”) of the electron density due to the neighboring atoms. In the EMT, it is given as the energy of an atom in an equivalent reference system, the “effective medium”, plus a correction term, $E_i=E_{C,i}(n_i) +{\frac{1}{2}}\left[\sum_{j\neq i} v_{ij}(r_{ij}) - \sum_{j\neq i}^{\textrm{ref}} v_{ij}(r_{ij})\right]$. Specifically, the reference system is chosen as an FCC crystal of the given kind of atom, and “equivalent” means that the electron density is used to choose the lattice constant of the crystal. The correction term is an ordinary pair potential involving a simple exponential, but notice that the corresponding sum in the reference system is subtracted (guaranteeing that the correct reference energy is given when the configuration is fact, the reference configuration). The parameters were $E_0$=-3.510 eV; $s_0$=1.413Å; $V_0$=2.476 eV; $\eta_2$= 3.122Å$^{-1}$; $\kappa$=5.178; $\lambda$=3.602; $n_0$=0.0614 Å$^{-3}$. DB : Asymmetric “dumb-bell” molecules,[@Pedersen/others:2008a] consisting of two unlike Lennard-Jones spheres, labelled P and M, connected by a rigid bond. The parameters were $\epsilon_p$=5.726 kJ/mol, $\sigma_p=$0.4963 nm, $m_p$=77.106 u; $\epsilon_m$=0.66944 kJ/mol, $\sigma_m=$0.3910 nm, $m_m$=15.035 u; the bond length was $d$=0.29 nm. Cross interactions, $\epsilon_{pm}$ and $\sigma_{pm}$, were set equal to the geometric and arithmetic means of the $p$ and $m$ parameters, respectively (Lorentz-Berthelot mixing rule[@Rowlinson:1969]). DZ : A monatomic liquid introduced by Dzugutov as a candidate for a monatomic glass-forming system.[@Dzugutov:1992] The potential is a sum of two parts, $v(r)=v_1(r)+v_2(r)$, with $v_1(r) = A(r^{-m}-B)\exp(c/(r-a))$ for $r<a$ and zero otherwise, and $v_2(r)=B \exp(d/(r-b))$ for $r<b$, zero otherwise. The parameters are chosen to match the location and curvature of the Lennard-Jones potential: $m=16$, $A=5.82$, $c=1.1$, $a=1.87$, $B=1.28$, $d=0.27$, $b=1.94$. EXP : A system interacting with a pair potential with exponential repulsion $U(r) = \frac{\epsilon}{8} \left[6\exp(-14(r/\sigma - 1))-14(\sigma/r)^6\right]$. Note that the attractive term has the same form as the Lennard-Jones potential. KABLJ : The Kob-Andersen binary Lennard-Jones liquid[@Kob/Andersen:1994], a mixture of two kinds of particles A and B, with A making 80% of the total number. The energy and length parameters are $\epsilon_{AA}=1.0$, $\epsilon_{BB}=0.5$, $\epsilon_{AB}=1.5$, $\sigma_{AA}=1.0$, $\sigma_{BB}=0.88$, $\sigma_{AB}=0.8$. The masses are both equal to unity. We use the standard density $\rho=1.2\sigma_{AA}^{-3}$. METH : The Gromos 3-site model for methanol.[@vanGunsteren/others:1996] The sites represent the methyl (M$\equiv$CH$_3$) group ($m=15.035$ u), the O atom ($m=15.999$ u), and the O-bonded H atom ($m=1.008$ u). The charges for Coulomb interactions are respectively 0.176 e, -0.574 e, and 0.398 e. The M and O groups additionally interact via Lennard-Jones forces, with parameters $\epsilon_{MM} = 0.9444$ kJ/mol, $\epsilon_{OO} = 0.8496$ kJ/mol, $\epsilon_{MO} = 0.9770$ kJ/mol, $\sigma_{MM} = 0.3646$ nm, $\sigma_{OO} = 0.2955$ nm, and $\sigma_{MO} = 0.3235$ nm. Lennard-Jones interactions are smoothly cutoff between 0.9 nm and 1.1 nm. The M-O and O-H distance is fixed at 0.136 nm and 0.1 nm, respectively, while the M-O-H bond angle is fixed at 108.53$^\circ$. MGCU : A model of the metallic alloy Mg$_{85}$Cu$_{15}$, simulated by EMT with parameters as in Ref. . In this potential there are seven parameters for each element. However, some of the Cu parameters were allowed to vary from their original values in the process of optimizing the potential for the Mg-Cu system. The parameters for Cu were $E_0$=-3.510 eV; $s_0$=1.413Å; $V_0$=1.994 eV; $\eta_2$= 3.040Å$^{-1}$; $\kappa$=4.944; $\lambda$=3.694; $n_0$=0.0637 Å$^{-3}$, while those for Mg were $E_0$=-1.487 eV; $s_0$=1.766Å; $V_0$=2.230 eV; $\eta_2$= 2.541Å$^{-1}$; $\kappa$=4.435; $\lambda$=3.293; $n_0$=0.0355 Å$^{-3}$. It should be noted that there is an error in Ref. : The parameter $s_0$ for Cu is given in units of bohr instead of Å. OTP : The Lewis-Wahnstr[ö]{}m three-site model of orthoterphenyl[@Lewis/Wahnstrom:1994] consisting of three identical Lennard-Jones spheres located at the apices A B, and C of an isosceles triangle. Sides AB and BC are 0.4830 nm long, while the ABC angle is 75$^\circ$. The Lennard-Jones interaction parameters are $\epsilon=4.989$ kJ/mol, $\sigma=0.483$ nm, while the mass of each sphere, not specified in Ref. [@Lewis/Wahnstrom:1994], was taken as one third of the mass of an OTP molecule, $m=76.768$ u. SCLJ : The standard single-component Lennard-Jones system with potential given by Eq. (\[LJpotential\]). SPC/E : The SPC/E model of water,[@Berendsen/Grigera/Straatsma:1987] in which each molecule consists of three rigidly bonded point masses, with an OH distance of 0.1 nm and the HOH angle equal to the tetrahedral angle. Charges on O and each H are equal to -0.8476 e and +0.4238 e, respectively. O atoms interact with each other via a Lennard-Jones potential with $\epsilon$=2.601 kJ/mol and $\sigma$=0.3166 nm. SQW : A binary model with a pair interaction consisting of an infinitely hard core and an attractive square well: [@Zaccarelli/others:2002; @Zaccarelli/Sciortino/Tartaglia:2004] $v_{ij}(r) = \infty$, $r < \sigma_{ij}$, $v_{ij}(r)=-u_0$, $\sigma_{ij}<r<\sigma_{ij}(1+\epsilon)$, $v_{ij}(r)=0$, $r>\sigma_{ij}(1+\epsilon)$. The radius parameters are $\sigma_{AA}=1.2$, $\sigma_{BB}=1$, $\sigma_{AB}=1.1$, while $\epsilon=0.03$ and $u_0=1$. The composition was equimolar, and the masses of both particles were equal to unity. TIP5P : In this water model[@Mahoney/Jorgensen:2000] there are five sites associated with a single water molecule. One for the O atom, one for each H, and two to locate the centers of negative charge corresponding to the electron lone-pairs on the O. The OH bond length, and HOH bond angle are fixed at the gas-phase experimental values, $r_{OH}=0.09572$ nm and $\theta_{HOH}=104.52^\circ$. The negative charge sites are located symmetrically along the lone-pair directions at distance $r_{OL}=0.07$ nm and with an intervening angle $\theta_{LOL}=109.47^\circ$. A charge of +0.241 $e$ is located on each Hydrogen site, while charges of equal magnitude and opposite sign are placed on the lone-pair sites. O atoms on different molecules interact via the Lennard-Jones potential with $\sigma_O=0.312$ nm and $\epsilon_O=0.669$ kJ/mol. TOL : A 7-site united-atom model of toluene, consisting of six “ring” C atoms and a methyl group (H atoms are not explicitly represented). In order to handle the constraints more easily, only three mass points were used; one at the ring C attached to the methyl group ($m=40.065$ u), and one at each of the two “meta” C atoms ($m=26.038$) (note that with this mass distribution, the moment of inertia is not reproduced correctly). Parameters were derived from the information in Ref. : $\epsilon_{ring}$=0.4602 kJ/mol, $\epsilon_{methyl}$=0.6694 kJ/mol, $\sigma_{ring}$=0.375 nm, $\sigma_{methyl}$=0.391 nm. The Lorentz-Berthelot rule was used for cross-interactions.[@Rowlinson:1969] \[FlucThermDerivs\]Connecting fluctuations to thermodynamic derivatives ======================================================================= If $A$ is a dynamical quantity which depends only on the configurational degrees of freedom, then its average in the canonical ensemble (NVT) is given by (where, for convenience, we use a discrete-state notation, with $A_i$ referring to the value of $A$ in the $i$th micro-state, etc.) $$\langle A \rangle = \frac{\sum_i A_i \exp(-\beta U_i)} {\sum_i \exp(-\beta U_i)} = \frac{\sum_i A_i \exp(-\beta U_i)}{Q}$$ where $\beta=1/k_BT$ and $Q$ is the configurational partition function. Then the inverse temperature derivative of $\langle A \rangle$ can be written in terms of equilibrium fluctuations: $$\begin{aligned} \left(\frac{\partial\langle A \rangle}{\partial \beta} \right)_V = & -\frac{\sum_i A_i \exp(-\beta U_i) U_i}{Q} \\ & + \frac{\sum_i A_i \exp(-\beta U_i) \sum_j \exp(-\beta U_j) U_j}{Q^2} \nonumber \\ =& - \left( \langle A U \rangle - \langle A\rangle \langle U \rangle \right) \\ =& - \langle \Delta A \Delta U \rangle.\end{aligned}$$ Now taking $A=W$ and $A=U$ successively we find that $$\begin{aligned} \left(\frac{\partial\langle W \rangle}{\partial T} \right)_V / \left(\frac{\partial\langle U \rangle}{\partial T}\right)_V & = \left(\frac{\partial\langle W \rangle}{\partial \beta}\right)_V / \left(\frac{\partial\langle U \rangle}{\partial \beta}\right)_V \nonumber \\ & = \frac{\langle \Delta W \Delta U \rangle}{\langle (\Delta U)^2 \rangle}. \label{derivativeLinearRegression}\end{aligned}$$ This last expression is precisely the formula for the slope obtained by linear-regression when plotting $W$ against $U$. Consider now volume derivatives. Because volume dependence comes in through the micro-state values, $A_i$ and $U_i$, and the volume derivatives of these are not necessarily related in a simple way, the corresponding expression is not as simple: The derivative of ${\langle W \rangle}$ with respect to volume at fixed temperature is given by $$\begin{aligned} \left(\frac{\partial {\langle W \rangle}}{\partial V}\right)_T & = \frac{\partial}{\partial V}\Bigl({\langle p \rangle}V - Nk_BT\Bigr)_T \\ &= {\langle p \rangle} + V\left(\frac{\partial {\langle p \rangle}}{\partial V}\right)_T = {\langle p \rangle} - K_T,\end{aligned}$$ where $K_T$ is the isothermal bulk modulus. The derivative of $U$ can be obtained by writing pressure as the derivative of Helmholtz free energy $F$ ($K$ is kinetic energy): $$\begin{aligned} {\langle p \rangle} & = -\left(\frac{\partial F}{\partial V}\right)_T = -\left(\frac{\partial ({\langle U \rangle}+{\langle K \rangle}-TS)}{\partial V}\right)_T \\ &= -\left(\frac{\partial {\langle U \rangle}}{\partial V}\right)_T + T \left(\frac{\partial S}{\partial V}\right)_T.\end{aligned}$$ Then using the thermodynamic identity $\left(\frac{\partial S} {\partial V}\right)_T = \left(\frac{\partial {\langle p \rangle}}{\partial T}\right)_V\equiv \beta_V$, we obtain the ratio of volume derivatives of ${\langle W \rangle}$ and ${\langle U \rangle}$ $$\left(\frac{\partial {\langle W \rangle}}{\partial V}\right)_T / \left(\frac{\partial {\langle U \rangle}}{\partial V}\right)_T = -\frac{K_T-{\langle p \rangle}}{T\beta_V - {\langle p \rangle}} $$ which becomes $-K_T/(T\beta_V)$ when the pressure is small compared to the bulk modulus. As discussed in Paper II, $\beta_V$ can be expressed in terms of $\langle\Delta U\Delta W\rangle$ again, but the fluctuation expression for $K_T$ is more complicated. Thus we cannot simply identify the lines of constant $T$, varying $V$, on a ${\langle W \rangle},{\langle U \rangle}$ plot, as we could with lines of fixed $V$, varying $T$, by examining the fluctuations at fixed $V,T$. \[Virial\_SQW\]Virial for square-well system ============================================ Here we derive the expression for the time-averaged virial, Eq. (\[SQWvirial\]), as a convenience for the reader. The idea is to replace the step $u_0$ in the potential with a a finite slope $u_0/\delta$ over a range $\delta$, and take the limit $\delta \rightarrow 0$. We start by replacing a two-body interaction in three dimensions with the equivalent one-dimensional, one-body problem using the radial separation $r$ and the reduced mass $m_r$. Let the potential step be at $r=r_s$ and define $x=r-r_s$ (see Fig. \[illustrateSQWvirial\]). We consider an “escape event” over a positive step, so that an initial (relative) velocity $v_0$ becomes a final velocity $v_1$ and $r$ goes from a value less than $r_0$ to a value greater than $r_0+\delta$. The resulting formula also applies for the other kinds of events. ![\[illustrateSQWvirial\]Illustration of replacement of discontinuous step by a finite slope for the square well potential for the purpose of calculating the virial. The limit $\delta\rightarrow 0$ is taken at the end.](illustrateSQWvirial.png){width="8.5cm"} The contribution to the time integral of the virial from this event is given by $$\Delta = \int_0^{t_\delta} \frac{(r_0+x)}{3} F dt = -\int_0^{t_\delta} \frac{(r_0 + x)u_0}{3\delta} dt$$ where $F$ is the (constant) force in the region $0<x<\delta$ and $t_\delta$ is the time taken for the ’particle’ (the radial separation) to cross this region. The trajectory $x(t)$ is given by the standard formula for uniform acceleration $$\label{x_of_t} x(t) = v_0 t - {\frac{1}{2}}\frac{u_0}{\delta m_r} t^2,$$ which by setting $x(t_\delta)=\delta$ gives the following expression for $t_\delta$: $$\label{t_delta} t_\delta = \delta \left( \frac{m_r v_0}{u_0} - \sqrt{\left( \frac{m_r v_0} {u_0}\right)^2 - \frac{2m_r}{u_0}} \right);$$ here we have taken the negative root, appropriate for a positive $u_0$ (we want the smallest positive $t_\delta$). Returning to $\Delta$, it can be split into two parts as follows $$\Delta = -\int_0^{t_\delta} \frac{r_0 u_0}{3\delta} dt - \int_0^{t_\delta} \frac{x(t) u_0}{3\delta} dt$$ Consider the second term: using the expression for $x(t)$ from Eq. (\[x\_of\_t\]), we see that the result of the integral will involve a term proportional to $t_\delta^2$ and one proportional to $t_\delta^3$. Using Eq. (\[t\_delta\]) to replace $t_\delta\propto \delta$, and noting the $\delta$ in the denominator, the terms will have linear and quadratic dependence on $\delta$, respectively. Thus they will vanish in the limit $\delta\rightarrow 0$. On the other hand, the first term gives $$\begin{aligned} \Delta = -\frac{r_0 u_0}{3\delta} t_\delta &= -\frac{r_0 u_0}{3} \left( \frac{m_r v_0}{u_0} - \sqrt{\left( \frac{m_r v_0} {u_0}\right)^2 - \frac{2m_r}{u_0}} \right)\nonumber \\ &= \frac{r_0 m_r}{3} \left( \sqrt{v_0^2 - 2 u_0/m_r} - v_0 \right).\end{aligned}$$ This expression can be simplified by writing it in terms of the change of velocity $\Delta v\equiv v_1 - v_0$. In the one-body problem conservation of momentum does not hold, and $v_1$ is given by energy conservation: $${\frac{1}{2}}m_r v_0^2 = {\frac{1}{2}}m_r v_1^2 + u_0$$ from which $\Delta v$ is obtained as $$\Delta v \equiv v_1 - v_0 = \sqrt{{v_0}^2-2u_0/m_r} - v_0,$$ thus the expression for $\Delta $ becomes $$\Delta = \frac{r_0 m_r}{3} \Delta v = \frac { m_r}{3} {\mathbf{r}} \cdot \Delta {\mathbf{v}}$$ where in the last expression a switch to three-dimensional notation was made, recognizing that for central potentials $\Delta {\mathbf{v}}$ will be parallel to the displacement vector between the two particles. This expression, derived for escape events, must also hold for capture events since these are time-reverses of each other, and the virial is fundamentally a configurational quantity, independent of dynamics (the above expression is time-reversal invariant because the change in the radial component of velocity is the same either way, since although the “initial” and “final” velocities are swapped, they also have opposite sign). Bounce and collision events may be treated by dividing the event into two parts at the turning point (where the relative velocity is zero), noting that each may be treated exactly as above, then adding the results back together. If we now consider all events that take place during an averaging time $t_{\textrm{av}}$), we get the time-averaged virial as $$\bar{W} = \frac{1}{3 t_{\textrm{av}}} \sum_{\textrm{events}\, e} m_{r,e}{\mathbf{r}}_{e} \cdot \Delta {\mathbf{v}}_{e}$$ [32]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{} , ** (, ). , ** (, ), ed. , ** (, ), ed. , ** (, ). , , , , (), , ****, (). , , , ****, (). , ****, (). , , , , ****, (). , ** (, ), ed. , , , , , , ****, (). , , , ****, (). , , , ****, (). Asap, **, , <https://wiki.fysik.dtu.dk/asap>. , , , , , , , (), . , , , ****, (). , , , ****, (). , , , , ****, (). , ****, (). , ****, (). , , , , , , , , , , ****, (). , , , ****, (). , ****, (). , , , ****, (). , , , ****, (). , ****, (). , ****, (). , ****, (). , ** (, ). , , , , , , , , ** (, ). , , , ****, (). , , , , ** (, ), ed. [^1]: If this seems unphysical, the argument could be given in terms of arbitrary deviations from equilibrium, $\Delta g(r,t)\equiv g(r,t)-{\langle g(r) \rangle}$. See however, Paper II. [^2]: Choosing this measure of the slope is equivalent to diagonalizing the correlation matrix (the covariance matrix where the variables have been scaled to have unit variance) to identify the independently fluctuating variable. This is often done in multivariate analysis (see, e.g., Ref. ), rather than diagonalizing the covariance matrix, when different variables have widely differing variances.

--- abstract: 'Using a combination of density functional theory and dynamical mean field theory we show that electric polarization and magnetism are strongly intertwined in (TMTTF)$_2$-$X$ (X$=$PF$_6$, As$F_6$, and SbF$_6$) organic crystals and they originate from short-range Coulomb interactions. Electronic correlations induce a charge-ordered state which, combined with the molecular dimerization, gives rise to a finite electronic polarization and to a ferroelectric state. We predict that the value of the electronic polarization is enhanced by the onset of antiferromagnetism showing a sizable magnetoelectric leading to a multiferroic behavior of (TMTTF)$_2$-$X$ compounds.' author: - 'Gianluca Giovannetti$^{1}$, Reza Nourafkan$^{2}$, Gabriel Kotliar$^{2}$, and Massimo Capone$^{1}$' title: 'Correlation-driven electronic multiferroicity in (TMTTF)$_2$-$X$ organic crystals' --- In the last decade the study of ferroelectric and multiferroic materials is emerging as a novel frontier in condensed matter physics for their wide range of potential applications[@Cheong]. Ferroelectricity appears when ionic or molecular distortions break the inversion symmetry and the coherent orientation of dipole moments creates a net polarization. Despite the traditional mechanisms of magnetism and ferroelectricity are typically incompatible[@Hill], the simultaneous ordering of electrical and magnetic degrees of freedom is possible and defines multiferroic materials. We can classify multiferroics according to the microscopic mechanism that determines their properties[@Khomskii], and to the relation between the two orderings. In particular, we have materials in which ferroelectricity and magnetism have different origin, but also systems in which magnetism controls ferroelectricity or even causes it. These latter multiferroics are promising candidate for observing sizable room-temperature magneto-electric responses, which can pave the way to the development of magnetic devices which can be switched by applying an external voltage[@Vaz]. A further promising direction is the development of [*[electronic ferroelectrics]{}*]{}, in which the polarization has a predominant electronic contribution[@Kobayashi; @Portengen96; @Batista02]. An electronic mechanism compatible with both ferroelectric and magnetic orderings is based on charge ordering[@KhomskiiBrink]. In charge-ordered (CO) systems, the coexistence of inequivalent bonds and inequivalent sites with different carrier density leads to a ferroelectric state, which can be multiferroic if the CO state also supports magnetic ordering. The realization of a similar mechanism in the Fabre charge-transfer organic salts (TMTTF)$_2$-$X$ has been hinted[@KhomskiiBrink; @Giovannetti], but the role of strong electron-electron correlation in the complex interplay between electric polarization and magnetism has not been investigated and elucidated so far. As in other molecular solids, the screened Coulomb interaction is expected to play a major role because of the narrow bands arising from the overlap between molecular orbitals. Here we use a combination of density functional theory (DFT) [@DFT] and dynamical mean-field theory (DMFT) [@DMFT] to study the correlation-driven emergence of a ferroelectric state in (TMTTF)$_2$-$X$ and its interplay with the magnetic order evaluating the electronic contribution to the polarization with a recently introduced method based on the interacting Green’s functions[@RezaNourafkanPdmft]. We find that short-range correlations give rise to a characteristic coupling between magnetism and polarization and consequently to a multiferroic state. The family of (TMTTF)$_2$-$X$ organic salts displays indeed diverse electronic properties that can be controlled by substituting the counterion X or by applying pressure and their phase diagram include various electronic phases such as charge ordering, spin density wave and antiferromagnetism (AFM)[@Brazovskii; @Jerome]. In particular, at least three members of the family, $X=$PF$_6$, As$F_6$, and SbF$_6$ develop a low-temperature CO state with a 4K$_F$ modulation, with critical temperature 67, 102, 157 K respectively, which in turn coincides with the onset of the ferroelectric order [@Monceau; @Chow]. At very low temperatures T $\lesssim 17K$, a spin-Peierls state establishes for $X=$PF$_6$ and As$F_6$ while for $X=$SbF$_6$ the CO state in coexist with an AFM phase at T $\lesssim 8K$[@Matsunaga]. The magnetic response is not trivial also at higher temperatures, where the electron-spin resonance spectra can be described by a spin-$1/2$ antiferromagnetic Heisenberg chain for all the three compounds we consider[@Dumm]. The existence of localized spins above the ordering temperature is a signature of Mott localization, whose interplay with CO and magnetic ordering will be shown to be the key to the electronic multiferroic behavior. The crystal structure is characterized by TMTTF molecules stacked along the $a$ axis, separated by the X counterion, and weakly interacting along the $b$ axis, leading to a quasi-onedimensional bandstructure. In the following we will use the electronic structure of (TMTTF)$_2$-PF$_6$, for which accurate structural data are available, as a baseline for the analysis of the whole family. It is important to notice that in the CO state the crystals preserve the centrosymmetric structure[@Jacko] despite the molecular dimerization, suggesting a small ionic contribution to the polarization. We performed DFT calculations in the Perdew-Burke-Ernzerhof (PBE)[@PBE] scheme using Quantum Espresso[@QE]. A two-dimensional tight-binding representation of the bandstructure is built using Wannier90[@W90]. This mapping from DFT electronic structure to the localized Wannier molecular orbitals of X$=$PF$_6$ is representative of all the members of the charge-transfer (TMTTF)$_2$-$X$ as indeed changes in the hopping parameters along the series are rather small. The low-energy electronic structure of all the members of the family (TMTTF)$_2$-$X$ is characterized by two bands arising from TMTTF HOMO orbitals[@Jacko]. The conduction bands arise from the highest occupied molecular orbital s of the two inequivalent TMTTF molecules and they have a width $W \sim 1.0$ eV [@Giovannetti]. The ratio 2:1 between cations (TMTTF molecules) and anions (PF$_6$ group) in the unit cell leads to a commensurate band filling of $3/4$ which leads to a metallic state within PBE[@Jacko]. We then include the interactions as described by the Hamiltonian $$H = \sum_{\langle i,j\rangle \sigma} t_{ij}( {\xi}) c^{\dagger}_{i \sigma} c^{~}_{j \sigma} + U \sum_{i} n_{i \uparrow} n_{i \downarrow} + V \sum_{\langle i,j\rangle} n_{i} n_{j}, \label{Ham}$$ where $c_{i \sigma}$ and $c^{\dagger}_{i \sigma}$ are annihilation and creation operators for electrons at site $i$ with spin $\sigma$, $t_{ij}$ denotes the hopping parameters. The hopping is two-dimensional, while the non-local repulsion is restricted to sites along the same chain. For the sake of clarity, we parameterize the distortion in terms of the deviation of the hoppings along the one-dimensional chains with respect to their undistorted value of $t_0 =$ 0.21 eV. When we approach the actual dimerized structure, the two inequivalent hoppings become $t_{\pm} = t_0 \pm \xi$, with $\xi =$ 0.01 eV in the fully distorted structure. $U$ is the on-site Hubbard repulsion and $V$ is an inter-site repulsion between TMTTF sites along the stacking direction. We mention here that the inclusion of interchain hopping parameters does not affect the interplay between CO and magnetism, while it is important to determine the actual two-dimensional magnetic and charge pattern[@Yoshimi]. We perform DMFT calculations for different lattice distortions (or $\xi$) and we compute the Green’s function and the electronic polarization according to Ref. [@RezaNourafkanPdmft]. The interaction $V$ is treated in the Hartree approximation and the DMFT equations are solved with a zero-temperature exact-diagonalization impurity solver [@Caffarel; @Capone] using an Arnoldi algorithm. We allow for charge- and spin-ordering introducing four independent sites. For each of them we define an impurity model and we compute a local dynamical self-energy. In order to identify the relative role of the two interaction terms, we vary $U$ and $V$ respectively in a range around previous estimates of $U$=2.2 eV[@Giovannetti] and $V$=0.4 eV[@Mila]. ![(Color online) Spectral density of states calculated in the paramagnetic state for U$=$2.2 and V$=$0.325 within DFT+DMFT scheme. Insets: (Left) Charge arrangement of poor (P) and rich (R) sites in the CO paramagnetic state along the stacking direction $a$ at ${\xi}$=0.01 ; (Right) Charge gap ${\Delta}$ as function of V.[]{data-label="fig1"}](Fig1){width="1.0\columnwidth"} We start our investigation from the paramagnetic (PM) state, where we inhibit magnetic ordering. We find however a CO solution with a charge pattern characterized by alternate charge-poor (P) and charge-rich (R) TMTTF molecules along the stacking direction $a$ as shown in the inset of Fig. \[fig1\]. The sum of the occupation of two neighboring sites is always three, and the charge disproportionation $\delta = n_+ - n_-$ ($n_+$ and $n_-$ being the occupancy of the two non-equivalent sites) is such that the R sites approach $n_+=2$ and the P sites approach half-filling. In this light, the present CO state can be interpreted as a Mott-like state[@Amaricci]. In Fig. \[fig1\] we show the spectral function $A(\omega) = -1/\pi Im G(\omega)$ for the PM-CO phase in the distorted structure with $\xi = 0.01$ which shows an insulating behavior with a charge gap $\Delta$. For a fixed value of $U$, ${\Delta}$ increases as a function of $V$ (see inset of the Fig. \[fig1\]). For the smallest values of V for which the system is insulating, the theoretical value of $\Delta$ is close to the experimental value[@Pashkin]. We also performed Hartree-Fock calculations in which the Hubbard interaction is treated at mean-field level, and we find that the PM solution has metallic nature, clearly demonstrating the strongly correlated nature of the insulating state and the need to use DMFT to properly account for dynamical correlations. An insulating solution is found in Hartree-Fock only allowing for simultaneous AFM and CO broken symmetries. On the other hand, within DMFT we find a sharp first-order transition from a CO correlated metal to a Mott insulator with charge ordering by increasing V, as obtained for the quarter-filled model in Ref. [@Amaricci]. The value of the disproportionation parameter is immediately large ($\delta \simeq 0.6$) as soon as V is sufficient to enter the insulating state. These values are most likely overestimated by the mean-field treatment of the inter-site repulsion in our single-site DMFT approximation, where only the local dynamical fluctuations are treated exactly. A possible strategy to overcome this limitation would be to use cluster extensions of DMFT. In these methods, however, the mirror symmetry with respect to each site is broken even in the case where the hopping is uniform for even number of sites in the cluster. Therefore we limited to the single-site DMFT which does not introduce a bias in favor of distortion. The comparison with experiments confirms the expectations, as all the experimental estimates are smaller than our values. It should however be noticed that nuclear magnetic resonance (NMR) systematically predicts larger values with respect to infrared or Raman spectroscopies. More precisely, NMR gives $\delta = 0.28$ [@Nakamura_CO] for X=PF$_6$, 0.33 [@Fujiyama_CO] and 0.5 [@Zamborszky_CO] for X=AsF$_6$ and 0.5 [@Yu] or 0.55 [@Iwase_CO] for X=SbF$_6$, while IR gives 0.16, 0.21 and 0.29 respectively [@Knoblauch]. However, infrared and Raman estimates are based on several assumptions and they rely on single-molecule estimates of the vibrational frequency, whose accuracy for the present solids is questionable. In this view, the discrepancy between the NMR measurements and our calculations remains significant, but is is acceptable if one takes into account the limitations of single site DMFT. The paramagnetic solution we have found displays simultaneous CO and dimerization which break the inversion symmetry and induces an electric polarization in the $a$ lattice direction [@Monceau]. In Fig. \[fig2\]a and \[fig2\]b we show the computed charge disproportionation $\delta$ and electronic contribution to the polarization, $P$ as a function of $V$ ($P$ is plotted as a function of $\xi$, and different $V$ are compared). Interestingly, $P$ decreases when we increase $V$ thereby increasing $\delta$. This trend is related to the symmetry of the dimerized ferroelectric state which changes from a bond-centered ordering (favored by $\xi$) towards a site-centered ordering (favored by $V$). Note that $\xi=0$, in which site-centered ordering is the ground state has by definition $P=0$. On the other hand, both $\delta$ and $P$ are not sensitive to $U$ in the range considered in our calculations. It is worth mentioning that however $U$ cooperates with $V$ in opening the insulating gap, confirming the Mott nature of the insulator. ![(Color online) Spectral density of states calculated in the AFM state for $U=$2.2 and $V=$0.325 within DFT+DMFT. Insets: (Left) Charge and spin arrangements of poor (P) and rich (R) sites in the AFM CO state along the stacking direction $a$ at ${\xi}$=0.1 ; (Right) Charge gap ${\Delta}$ as function of V.[]{data-label="Fig3"}](Fig3){width=".925\columnwidth"} The charge localization introduced by electronic correlations leads to the formation of local magnetic moments which are typically expected to arrange in some ordered state to minimize the exchange energy. We therefore release the constraint to paramagnetic solutions to study the interplay between ferroelectric and magnetic orders. The spin pattern we obtain is depicted in the inset of Fig.\[Fig3\]. The charge-rich sites have an occupation which approaches two and a small magnetic moment, while the nearly half-filled charge-poor sites have a large momentum. Within each dimer the two spins are ferromagnetically aligned, while neighboring dimers display antiferromagnetic ordering of the spins. This magnetic structure corresponds to have one effective spin per dimer with an antiferromagnetic inter-dimer coupling and what it has been found in Ref. [@Matsunaga] for X=SbF$_6$. As a consequence of the exchange interactions the spin-minority states at the charge-poor sites can be occupied by minority electrons jumping from charge-rich site creating exchange striction of the same kind found in novel inorganic multiferroic materials. When $V$ is reduced the difference between the magnetization at charge-poor and rich sites decreases, thereby enhancing the magnetic striction and the electronic polarization (see Fig. \[fig4\]c). The spin and charge orders found in our calculations are consistent with the experimental evidence of charge[@Monceau] and spin [@Dumm; @Matsunaga] orderings in TMTTF salts. In Fig. \[Fig3\] we show the DMFT spectral function in the AFM-CO phase, which is naturally insulating, with a gap ${\Delta}$ again increasing upon increasing $V$, and very close to its paramagnetic value (see inset Fig. \[Fig3\]). Fig. \[fig4\]a) and \[fig4\]b) show that the disproportionation and the large magnetization $m_P = n_{\uparrow P} - n_{\downarrow P}$ of the charge-poor sites both increase as $V$ increases while the small magnetization $m_R$ of the charge-rich sites decreases. We can therefore conclude that the magnetic ordering is controlled by the charge disproportionation which underlies the ferroelectric behavior. Indeed, the AFM phase favors the formation of a mixed bond-centered/site-centered polar charge ordering. By comparing panel(a) of the Fig. \[fig2\] and Fig. \[fig4\], one can see that the charge disproportionation is smaller in the AFM phase than in the PM phase and the difference between $\delta$ in the two phases decreases upon increasing $V$ at fixed $U$. In the other words, the onset of the AFM phase shifts the center of negative charge toward the center of bonds with a larger value at smaller $V$ and leads to a larger electric polarization (see Fig. \[fig2\]b and \[fig4\]a). Similarly to the PM-CO state the electronic polarization decreases with the charge disproportionation ${\delta}$ at the molecular sites and it is linear in the dimerization (see Fig. \[fig4\]a and \[fig4\]c). As pointed out above, magnetism cooperates with the dimerization to go toward a bond-centered/site-centered charge density. Therefore, the fact that ferroelectric polarization increases almost by a factor $2$ in the magnetic state is a prediction of the magneto-electric coupling in (TMTTF)$_2$-$X$ crystals. Our results show that the magnetic state changes following the ferroelectric phase transition which can be varied by applying electric field and inversely, the ferroelectric polarization can be manipulated by applying high magnetic fields. This peculiar magnetoelectric effect in (TMTTF)$_2$-$X$ will be also combined with the magnetoelastic effect that modifies both the superexchange interaction and the molecular bonding. The enhancement of the electronic contribution to the ferroelectric polarization at the onset of antiferromagnetism appears therefore as a common feature of half-filled correlated insulator [@GiovannettiTTFCA; @RezaNourafkanPdmft]. Our calculations predict an electronic polarization $P \simeq$ $0.2$ $\mu$C/cm$^2$ for the actual structure of (TMTTF)$_2$-PF$_6$. The ionic contribution is expected to be smaller because of the centrosymmetric crystal structure. A quantitative calculation of the electronic polarization, which is still to be determined experimentally, potentially requires the inclusion of more molecular orbitals, non-local quantum fluctuations and a fully microscopic account of the structural distortions. However our conclusions about the magnetoelectric effects are expected to be robust, as they are based on a generic properties of correlated systems with charge and magnetic ordering. We believe that the mechanism we present to be relevant for a wider family of charge-ordered ferroelectric systems as k-(BEDT-TTF)$_2$-Cu\[N(CN)$_2$\]Cl [@Lunkenheimer] and ${\beta}$’-(BEDT-TTF)$_2$ICl$_2$ [@Iguchi]. Furthermore, the same magneto-electric coupling we proposed for (TMTTF)$_2$-$X$ may also describe materials in which a low-temperature multiferroic state is replaced at higher temperature by ferroelectric ordering, as observed in other Mn-based multiferroic material [@Sakai; @GiovannettiSBMO]. In conclusion, using DFT+DMFT and a recent method based on Green’s function formalism to calculate the electronic contribution to the polarization we investigate the ferroelectric and multiferroic phase of (TMTTF)$_2$-$X$ molecular crystals. We show that (TMTTF)$_2$-$X$ are strongly correlated insulators developing a charge ordered state which, combined with molecular dimerization, gives a ferroelectric state as experimentally observed. The same correlation effects also drive and antiferromagnetic ordering. By comparing the values of electronic polarization in the paramagnetic and antiferromagnetic magnetic structure we show that magnetic and ferroelectric orderings are strongly intertwined. These findings establish (TMTTF)$_2$-$X$ to belong to the class of electronic-driven multiferroic materials, and that strong electron-electron correlations is the driving force behind charge-ordering, polarization and magnetic ordering. This kind of Mott state which hosts both charge ordering and antiferromagnetism provides us with a mechanism to overcome the apparent incompatibility of ferroelectric and magnetic ordering. We are grateful to C. Batista for useful discussions. 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--- author: - 'B. Andrei Bernevig and Shou-Cheng Zhang' title: Quantum Spin Hall Effect --- 10000 Recently, the intrinsic spin Hall effect has been theoretically predicted for semiconductors with spin-orbit coupled band structures[@murakami2003; @sinova2004]. The spin Hall current is induced by the external electric field according to the equation $$j_j^i = \sigma_s \epsilon_{ijk} E_k \label{spin_response}$$ where $j_j^i$ is the spin current of the $i$-th component of the spin along the direction $j$, $E_k$ is the electric field and $\epsilon_{ijk}$ is the totally antisymmetric tensor in three dimensions. The spin Hall effect has recently been detected in two different experiments[@kato2004A; @wunderlich2004], and there is strong indication that at least one of them is in the intrinsic regime[@bernevig2004A]. Because both the electric field and the spin current are even under time reversal, the spin current could be dissipationless, and the value of $\sigma_s$ could be independent of the scattering rates. This is in sharp contrast with the extrinsic spin Hall effect, where the effect arises only from the Mott scattering from the impurity atoms[@mott1929]. The independence of the intrinsic spin Hall conductance $\sigma_s$ on the impurity scattering rate naturally raises the question whether it can be quantized under certain conditions, similar to the quantized charge Hall effect. We start off our analysis with a question: Can we have Landau Level (LL) -like behavior *in the absence* of a magnetic field [@haldane1988]? The quantum Landau levels arise physically from a [*velocity dependent force*]{}, namely the Lorentz force, which contributes a term proportional to $\vec{A}\cdot \vec{p}$ in the Hamiltonian. Here $\vec{p}$ is the particle momentum and $\vec{A}$ is the vector potential, which in the symmetric gauge is given by $\vec{A} = \frac{B}{2}(y, -x,0)$. In this case, the velocity dependent term in the Hamiltonian is proportional to $B(xp_y-yp_x)$. In condensed matter systems, the only other ubiquitous velocity dependent force besides the Lorentz force is the spin-orbit coupling force, which contributes a term proportional to $(\vec{p} \times \vec{E})\cdot \vec{\sigma}$ in the Hamiltonian. Here $\vec{E}$ is the electric field, and $\vec{\sigma}$ is the Pauli spin matrix. Unlike the magnetic field, the presence of an electric field does not break the time reversal symmetry. If we consider the particle momentum confined in a two dimensional geometry, say the $xy$ plane, and the electric field direction confined in the $xy$ plane as well, only the $z$ component of the spin enters the Hamiltonian. Furthermore, if the electric field $\vec{E}$ is not constant but is proportional to the radial coordinate $\vec{r}$, as it would be, for example, in the interior of a uniformly charged cylinder $\vec{E} \sim E(x,y,0)$, then the spin-orbit coupling term in the Hamiltonian takes the form $E \sigma_z (xp_y-yp_x)$. We see that this system behaves in such a way as if particles with opposite spins experience the opposite “effective" orbital magnetic fields, and a Landau level structure should appear for each spin orientations. However, such an electric field configuration is not easy to realize. Fortunately, the scenario previously described is realizable in zinc-blende semiconductors such as GaAs, where the shear strain gradients can play a similar role. Zinc-blende semiconductors have the point-group symmetry $T_d$ which is half of the cubic-symmetry group $O_h$, and does not contain inversion as one of its symmetries. Under the $T_d$ point group, the cubic harmonics $xyz$ transform like the identity, and off-diagonal symmetric tensors ($xy +yx$, etc.) transform in the same way as vectors on the other direction ($z$, etc), and represent basis functions for the $T_1$ representation of the group. Specifically, strain is a symmetric tensor $\epsilon_{ij} = \epsilon_{ji}$, and its off-diagonal (shear) components are, for the purpose of writing down a spin-orbit coupling Hamiltonian, equivalent to an electric field in the remaining direction: $$\label{strainelectricfieldanalogy} \epsilon_{xy} \leftrightarrow E_z;\;\;\; \epsilon_{xz} \leftrightarrow E_y;\;\;\;\epsilon_{yz} \leftrightarrow E_x$$ The Hamiltonian for the conduction band of bulk zinc-blende semiconductors under strain is hence the analogous to the spin-orbit coupling term $(\vec{v} \times \vec{E})\cdot \vec{\sigma}$. In addition, we have the usual kinetic $p^2$ term and a trace of the strain term $tr{\epsilon} = \epsilon_{xx} + \epsilon_{yy} +\epsilon_{zz}$, both of which transform as the identity under $T_d$: $$\begin{aligned} & H = \frac{p^2}{2m} + B tr{\epsilon} +\frac{1}{2} \frac{C_3}{\hbar} [ (\epsilon_{xy} p_y - \epsilon_{xz} p_z) \sigma_x + \nonumber \\ & +(\epsilon_{zy} p_z - \epsilon_{xy} p_x) \sigma_y + (\epsilon_{zx} p_x - \epsilon_{yz} p_y) \sigma_z ]\end{aligned}$$ For GaAs, the constant $\frac{C_3}{\hbar} = 8 \times 10^5 m/s$ [@dyakonov1986]. This Hamiltonian is not new and was previously written down in Ref. [@pikus1984; @howlett1977; @khaetskii2001; @bahder1990] but the analogy with the electric field and its derivation from a Lorentz force is suggestive enough to warrant repetition. There is yet another term allowed by group theory in the Hamiltonian [@bernevig2004], but this is higher order in perturbation theory and hence not of primary importance. Let us now presume a strain configuration in which $\epsilon_{xy} = 0$ but $\epsilon_{xz}$ has a constant gradient along the $y$ direction while $\epsilon_{yz}$ has a constant gradient along the $x$ direction. This case then mimics the situation of the electric field inside a uniformly charged cylinder discussed above, as $\epsilon_{xz} (\leftrightarrow E_y) =g y$ and $\epsilon_{yz} (\leftrightarrow E_x) = g x$, $g$ being the magnitude of the strain gradient. With this strain configuration and in a symmetric quantum well in the $xy$ plane, which we approximate as being parabolic, the above Hamiltonian becomes: $$H= \frac{p_x^2 + p_y^2}{2m} + \frac{1}{2} \frac{C_3}{\hbar}g (y p_x - x p_y)\sigma_z + D(x^2 + y^2)$$ We first solve this Hamiltonian and come back to the experimental realization of the strain architecture in the later stages of the paper. We make the coordinate change $x \rightarrow (2mD)^{-1/4} x$, $y \rightarrow (2mD)^{-1/4} y$ and $R = \frac{1}{2} \frac{C_3}{\hbar} \sqrt{\frac{2m}{D}} g$. $R=2$ or $D=D_0\equiv\frac{2mgC_3^2}{16\hbar^2}$ is a special point, where the Hamiltonian can be written as a complete square, namely $H = \frac{1}{2m} (\vec{p} - e \vec{A} \sigma_z)^2$ with ${\vec{A} = \frac{m C_3 g}{2 \hbar e}(y, -x, 0)}$. At this point, our Hamiltonian is exactly equivalent to the usual Hamiltonian of a charged particle in an uniform magnetic field, where the two different spin directions experience the opposite directions of the “effective" magnetic field. Any generic confining potential $V(x,y)$ can be written as $D_0(x^2 + y^2) + \Delta V(x,y)$, where the first term completes the square for the Hamiltonian, and the second term $\Delta V(x,y)=V(x,y)-D_0(x^2 + y^2)$ describes the additional static potential within the Landau levels. Since $[H, \sigma_z] =0$ we can use the spin on the $z$ direction to characterize the states. In the new coordinates, the Hamiltonian takes the form: $$\begin{aligned} & H = \left(\begin{array}{cc} H_{\uparrow} & 0 \\ 0 & H_{\downarrow} \\ \end{array}\right) \nonumber \\ & H_{ \downarrow , \uparrow} = \sqrt{\frac{D}{2m}} [p_x^2 + p_y^2 + x^2+ y^2 \pm R(x p_y - y p_x)]\end{aligned}$$ The $H_{ \downarrow, \uparrow}$ is the Hamiltonian for the down and up spin $\sigma_z$ respectively. Working in complex-coordinate formalism and choosing $z = x+ i y$ we obtain two sets of raising and lowering operators: $$\begin{aligned} & a = \partial_{z^\star} + \frac{z}{2}, \;\;\; a^\dagger = - \partial_z + \frac{z^\star}{2} \nonumber \\ & b = \partial_{z} + \frac{z^\star}{2}, \;\;\; b^\dagger = - \partial_{z^\star} + \frac{z}{2}\end{aligned}$$ in terms of which the Hamiltonian decouples: $$H_{ \downarrow, \uparrow} =2 \sqrt{\frac{D}{2m}} \left[(1 \mp \frac{R}{2} ) a a^\dagger + (1 \pm \frac{R}{2}) b b^\dagger + 1 \right]$$ The eigenstates of this system are harmonic oscillators $|m,n\rangle = (a^\dagger)^m (b^\dagger)^n |0,0\rangle$ of energy $E^{ \downarrow, \uparrow}_{m,n} = \frac{1}{2} \sqrt{\frac{D}{2m}} \left[(1 \mp \frac{R}{2} ) m + (1 \pm \frac{R}{2}) n + 1 \right]$. We shall focus on the case of $R=2$ where there is no additional static potential within the Landau level. For the spin up electron, the vicinity of $R \approx 2$ is characterized by the Hamiltonian $H_{\uparrow} = \frac{1}{2} \frac{C_3}{\hbar} g (2 a a^\dagger + 1) $ with the LLL wave function $\phi^\uparrow_n(z) =\frac{z^n}{\sqrt{\pi n ! } }\exp(\frac{-z z^\star}{2})$. $a$ is the operator moving between different Landau levels, while $b$ is the operator moving between different degenerate angular momentum states within the same LL: $L_z= b b^\dagger - a a^\dagger$, $L_z \phi^\uparrow_n(z) = n \phi^\uparrow_n(z)$. The wave function, besides the confining factor, is holomorphic in $z$, as expected. These up spin electrons are the chiral, and their charge conductance is quantized in units of $e^2/h$. For the spin down electron, the situation is exactly the opposite. The vicinity of $R \approx 2$ is characterized by the Hamiltonian $H_{\downarrow} = \frac{1}{2} \frac{C_3}{\hbar} g (2 b b^\dagger + 1) $ with the LLL wave function $\phi^\downarrow_m(z) =\frac{(z^\star)^m}{\sqrt{\pi m ! } }\exp(\frac{-z z^\star}{2})$. $b$ is the operator moving between different Landau levels, while $a$ is the operator between different degenerate angular momentum states within the same LL: $L_z= b b^\dagger - a a^\dagger$, $L_z \phi^\downarrow_m(z) = - m \phi^\downarrow_m(z)$. The wave function, besides the confining factor, is anti-holomorphic in $z$. These down spin electrons are anti-chiral, and their charge conductance is quantized in units of $-e^2/h$. ![ Spin up and down electrons have opposite chirality as they feel the opposite spin-orbit coupling force. Total charge conductance vanishes but spin conductance is quantized. The inset shows the lattice displacement leading to the strain configuration.[]{data-label="edgecurrent"}](edgecurrent.eps) The picture that now emerges is the following: our system is equivalent to a bilayer system; in one of the layers we have spin down electrons in the presence of a down-magnetic field whereas in the other layer we have spin up electrons in the presence of an up-magnetic field. These two layers are placed together. The spin up electrons have positive charge Hall conductance while the spin down electrons have negative charge Hall conductance. As such, the charge Hall conductance of the whole system vanishes. The time reversal symmetry reverses the directions of the “effective" orbital magnetic fields, but interchanges the layers at the same time. However, the spin Hall conductance remains finite, as the chiral states are spin-up while the anti-chiral states are spin-down, as shown in Figure\[\[edgecurrent\]\]. The spin Hall conductance is hence quantized in units of $2 \frac{e^2}{h} \frac{\hbar}{2e}=2\frac{e}{4\pi}$. Since an electron with charge $e$ also carries spin $\hbar/2$, a factor of $\frac{\hbar}{2e}$ is used to convert charge conductance into the spin conductance. While it is hard to experimentally measure the spin Hall effect, and supposedly even harder to measure the quantum version of it, the measurement of the charge quantum Hall effect has become relatively common. In our system, however, the charge Hall conductance $\sigma_{xy}$ vanishes by symmetry. However, we can use our physical analogy of the two layer system placed together. In each of the layers we have a charge Quantum Hall effect at the same time (since the filling is equal in both layers), but with opposite Hall conductance. However, when on the plateau, the longitudinal conductance $\sigma_{xx}$ also vanishes ($\sigma_{xx} =0$) separately for the spin-up and spin-down electrons, and hence vanishes for the whole system. Of course, between plateaus it will have non-zero spikes (narrow regions). This is the easiest detectable feature of the new state, as the measurement is entirely electric. Other experiments on the new state could involve the injection of spin-polarized edge states, which would acquire different chirality depending on the initial spin direction. We now discuss the realization of a strain gradient of the specific form proposed in this paper. The strain tensor is related to the displacement of lattice atoms from their equilibrium position $u_i$ in the familiar way $\epsilon_{ij} = (\partial u_i /\partial x_j + \partial u_j /\partial x_i)/2$. Our strain configuration is $\epsilon_{xx} =\epsilon_{yy} = \epsilon_{zz} = \epsilon_{xy}=0$ as well as the strain gradients $\epsilon_{zx} = g y$ and $\epsilon_{yx} = g x$. Having the diagonal strain components non-zero will not change the physics as they add only a chemical potential term to the Hamiltonian. The above strain configuration corresponds to a displacement of atoms from their equilibrium positions of the form $\vec{u} = (0,0, 2 g x y)$. This can be possibly realized by pulverizing GaAs on a substrate in MBE at a rate which is a function of the position of the pulverizing beam on the substrate. The GaAs pulverization rate should vary as $xy \sim r^2 \sin(2\phi)$, where $r$ is the distance from one of the corners of the sample where the GaAs depositing was started. Conversely, we can keep the pulverizing beam fixed at some $r$ and rotate the sample with an angle-dependent angular velocity of the form $\sin(2 \phi)$. We then move to the next incremental distance $r$, increase the beam rate as $r^2$ and again start rotating the substrate as the depositing procedure is underway. The strain architecture we have proposed to realize the Quantum Spin Hall effect is by no means unique. In the present case, we have re-created the so-called symmetric gauge in magnetic-field language, but, with different strain architectures, one can create the Landau gauge Hamiltonian and indeed many other gauges. The Landau gauge Hamiltonian is maybe the easiest to realize in an experimental situation, by growing the quantum well in the $[110]$ direction. This situation creates an off-diagonal strain $\epsilon_{xy} = \frac{1}{4} S_{44} T$, and $\epsilon_{xz} = \epsilon_{yz} =0$ where $T$ is the lattice mismatch (or impurity concentration), $s_{44}$ is a material constant and $x,y,z$ are the cubic axes. The spin-orbit part of the Hamiltonian is now $\frac{C_3}{\hbar} \epsilon_{xy} (p_x \sigma_y - p_y \sigma_x)$. However, since the growth direction of the well is $[110]$ we must make a coordinate transformation to the $x',y',z'$ coordinates of the quantum well ($x', y'$ are the new coordinates in the plane of the well, whereas $z'$ is the growth coordinate, perpendicular to the well and identical to the $[110]$ direction in cubic axes). The coordinate transformation reads: $x' =\frac{1}{\sqrt{2} }(x-y)$, $y' = -z$, $z' = \frac{1}{\sqrt{2}} (x+y)$, and the momentum along $z'$ is quantized. We now vary the impurity concentration $T$ (or vary the speed at which we deposit the layers) linearly on the $y'$ direction of the quantum well so that $\epsilon_{xy}= g y'$ where $g$ is strain gradient, linearly proportional to the gradient in $T$. In the new coordinates and for this strain geometry, the Hamiltonian reads: $$H = \frac{p^2}{2m} + \frac{C_3}{\hbar} g y' p_{x'} \sigma_{z'} + D y^2$$ where we have added a confining potential. At the suitable match between $D$ and $g$, this is the Landau-gauge Hamiltonian. One can also replace the soft-wall condition (the Harmonic potential) by hard-wall boundary condition. We now estimate the Landau Level gap and the strain gradient needed for such an effect, as well as the strength of the confining potential. In the case $R \approx 2$ the energy difference between Landau levels is $\Delta E_{Landau} = 2 \times \hbar \frac{1}{2} \frac{C_3}{\hbar} g = C_3 g$. For a gap of $1mK$, we hence need a strain gradient or $1 \%$ over $60 \mu m$. Such a strain gradient is easily realizable experimentally, but one would probably want to increase the gap to $10mK$ or more, for which a strain gradient of $1 \%$ over $6 \mu m$ or larger is desirable. Such strain gradients have been realized experimentally, however, not exactly in the configuration proposed here [@shen1996; @shen1997]. The strength of the confining potential is in this case $D = 10^{-15} N/m$ which corresponds roughly to an electric field of $1 V/m$ for a sample of $60 \mu m$. In systems with higher spin-orbit coupling, $C_3$ would be larger, and the strain gradient field would create a larger gap between the Landau levels. We now turn to the question of the many-body wave function in the presence of interactions. For our system this is very suggestive, as the wave function incorporates both holomorphic and anti-holomorphic coordinates, by contrast to the pure holomorphic Laughlin states. Let the up-spin coordinates be $z_i$ while the down-spin coordinates be the $w_i$. $z_i$ enter in holomorphic form in the wave function whereas $w_i$ enter anti-holomorphically. While if the spin-up and spin-down electrons would lie in separate bi-layers the many-body wave function would be just $\prod_{i<j} (z_i -z_j)^m \prod_{k<l} (w^\star_k - w^\star_l)^m e^{-\frac{1}{2} (\sum_{i}z_i z_i^\star + \sum_k w_k w_k^\star)}$, where $m$ is an odd integer. Since the particles in our state reside in the same quantum well and may possibly experience the additional interaction between the different spin states, a more appropriate wave function is: $$\begin{aligned} & \psi(z_i, w_i) = \prod_{i<j} (z_i -z_j)^m \prod_{k<l} (w^\star_k - w^\star_l)^m \nonumber \\ & \prod_{r,s} (z_r -w^\star_s)^n e^{-\frac{1}{2} (\sum_{i} z_i z_i^\star + \sum_k w_k w_k^\star)}\end{aligned}$$ The above wave function is symmetric upon the interchange $z \leftrightarrow w^\star$ reflecting the spin-$\uparrow$ chiral - spin-$\downarrow$ anti-chiral symmetry. This wave function is of course analogous to the Halperin’s wave function of two different spin states [@halperin1983]. The key difference is that the two different spin states here experience the opposite directions of magnetic fields. Many profound topological properties of the quantum Hall effect are captured by the Chern-Simons-Landau-Ginzburg theory[@zhang1989]. While the usual spin orbit coupling for spin-$1/2$ systems is $T$-invariant but $P$-breaking, our spin-orbit coupling is also $P$-invariant due to the strain gradient. The low energy field theory of the spin Hall liquid is hence a double Chern-Simons theory with the action: $$S = \frac{\nu}{4 \pi} \int \epsilon^{\mu \nu \rho} a_\mu \partial_\nu a_\rho - \frac{\nu}{4 \pi} \int \epsilon^{\mu \nu \rho} c_\mu \partial_\nu c_\rho$$ where the $a_\mu$ and $c_\mu$ fields are associated with the left and right movers of our theory while $\nu$ is the filling factor. The fractional charge and statistics of the quasi-particle follow easily from this Chern-Simons action. Essentially, the two Chern-Simons terms have the same filling factor $\nu$ and hence the Hilbert space is not the tensor product of any two algebras, but of two identical ones. This is a mathematical statement of the fact that one can insert an up or down electron in the system with the same probability. Such special theories avoids the chiral anomaly [@freedman2003] and their Berry phases have been recently proposed as preliminary examples of topological quantum computation [@freedman2003]. It is refreshing to see that such abstract mathematical models can be realized in conventional semiconductors. A similar situation of Landau levels without magnetic field arises in rotating BECs where the mean field Hamiltonian is similar to either $H_\uparrow$ or $H_\downarrow$. In the limit of rapid rotation, the condensate expands and becomes effectively two-dimensional. The $L_z$ term is induced by the rotation vector $\Omega$ [@ho2001]. The LLL behavior is achieved when the rotation frequency reaches a specific value analogous to the case $R\approx 2$ in our Hamiltonian. In the BEC literature this is the so called mean-field quantum Hall limit and the ground state wave function is Laughlin-type. In contrast to our case, the theory is still $T$ breaking, the magnetic field is replaced by a rotation axial vector, and the lowest Landau level is chiral. In conclusion we predict a new state of matter where a quantum spin Hall liquid is formed in conventional semiconductors with spin-orbit coupling. The quantum Landau levels are caused by the gradient of strain field, rather than the magnetic field. The new quantum spin Hall liquid state shares many emergent properties similar to the charge quantum Hall effect, however, unlike the charge quantum Hall effect, our system does not violate time reversal symmetry. We wish to thank S. Kivelson, E. Fradkin, J. Zaanen, D. Santiago and C. Wu for useful discussions. B.A.B. acknowledges support from the Stanford Graduate Fellowship Program. This work is supported by the NSF under grant numbers DMR-0342832 and the US Department of Energy, Office of Basic Energy Sciences under contract DE-AC03-76SF00515. [19]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{} , , , ****, (). , ****, (). , . , . , . , ****, (). , ****, (). , . , ****, (). , ****, (). , ****, (). , , ****, (). , ****, (). , ****, (). , , , ****, (). , . , ****, ().

--- abstract: 'A linear Boltzmann equation is derived in the Boltzmann-Grad scaling for the deterministic dynamics of many interacting particles with random initial data. We study a Rayleigh gas where a tagged particle is undergoing hard-sphere collisions with background particles, which do not interact among each other. In the Boltzmann-Grad scaling, we derive the validity of a linear Boltzmann equation for arbitrary long times under moderate assumptions on higher moments of the initial distributions of the tagged particle and the possibly non-equilibrium distribution of the background. The convergence of the empiric dynamics to the Boltzmann dynamics is shown using Kolmogorov equations for associated probability measures on collision histories.' address: - 'Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom' - 'Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom' - 'Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom' author: - Karsten Matthies - George Stone - Florian Theil bibliography: - 'bib.bib' title: The Derivation of the Linear Boltzmann Equation from a Rayleigh Gas Particle Model --- Introduction ============ The derivation of continuum equations from atomistic particle models is currently a major problem in mathematical physics with origins in Hilbert’s Sixth Problem. A particular interest in this area is the derivation of the Boltzmann equation from atomistic particle dynamics. The first major work in this area was by Lanford [@lanford75] which showed convergence from a hard-sphere particle model for short times by using the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy, see e.g. [@born46; @cercignani94; @uchiyama1988]. A recent major work by Gallagher, Saint-Raymond and Texier [@saintraymond13] continued the BBGKY development and proved convergence to the Boltzmann equation for short times for both hard-sphere and short-range potentials. The latter of which was improved by Pulvirenti [@pul13]. A particular difficulty has been proving convergence for arbitrarily long times, since the aforementioned results hold only for times up to a fraction of the mean free flight time. In [@bod15brown] Bodineau, Gallagher and Saint-Raymond were able to utilise the tools from [@saintraymond13] to prove convergence from a hard-sphere particle model to the linear Boltzmann equation for arbitrary long times in the case that the initial distribution of the background is near equilibrium. They were further able to use the linear Boltzmann equation as an intermediary step to prove convergence to Brownian motion. Further in a following paper [@bod15] the authors were able to consider weaker assumptions on the initial data and prove convergence to the Stokes-Fourier equations. For a general overview of the Boltzmann equation and the BBGKY hierarchy we refer to the books [@cerci88; @cercignani94; @spohn91]. A new method to tackle this problem has recently been developed in a series of papers [@matt08; @matthies10; @matt12]. This method employs semigroup techniques to study the evolution of collision trees rather than the BBGKY hierarchy. This comes from studying the distribution of the history of the particles up to a certain time rather than the distribution of the particles at a specific time. These papers have been able to prove convergence for arbitrary times but only for a simplified particle interaction system. This is the approach that we continue to develop in this paper. The BBGKY Hierarchy ------------------- The standard approach to tackling the derivation of the Boltzmann equation is via the BBGKY hierarchy. We refer specifically to [@bod15brown] for hard spheres. If $f_N(t)$ represents the $N$ particle distribution resulting from hard sphere dynamics at time $t$ then, away from collisions, $f_N$ satisfies the Liouville equation, $$\partial_t f_N(t) + v \cdot \partial_x f_N(t)= 0.$$ Considering this equation in weak form and integrating we have, away from collisions, $$\label{eq-bbgky} \partial_t f_N^{(s)}(t) + \sum_{1 \leq i \leq s } v_i \cdot \nabla _{x_i} f_N(t) = C_{s,s+1}f_N ^{(s+1)}(t),$$ for $s=1,\dots,N$ where $f_N^{(s)}$ denotes the $s$ particle marginal and where $C_{s,s+1}$ is the effect on the distribution of the first $s$ particles by a collision with another particle given explicitly in below. The system of $N$ equations is known as the BBGKY hierarchy. If the initial distribution of the $N$ particles is given by, $$f_N(0)=\frac{1}{\mathcal{Z}_N} \mathbbm{1}_{\mbox{no}} f_0^{\otimes N},$$ where $\mathbbm{1}_{\mbox{no}}$ conditions on no initial overlap and $\mathcal{Z}_N$ is a normalising constant, then the initial distribution of $f_N^{(s)}$ is given by, $$f_N^{s}(0,Z_s) = \int f_N(0,Z_n) \, \mathrm{d}z_{s+1} \dots \, \mathrm{d}z_{N}.$$ After successive time integration of one obtains a representation in the form of a finite sum $$\begin{aligned} \label{Duhamel} f^{(s)}_N(t) = &\sum_{k=0}^{N-s} \int_0^t\int_0^{t_1}\ldots \int_0^{t_{k-1}} {\bf T}_s(t-t_1){\mathcal C}_{s,s+1}{\bf T}_{s+1} (t_1-t_2){\mathcal C}_{s+1,s+2}\ldots\\ \nonumber &\hspace{5cm}\ldots {\bf T}_{s+k}(t_k)f_N^{(s+k)}(0) \, \dd t_k\ldots \dd t_1,\end{aligned}$$ where ${\bf T}_s$ is the flow map of $s$ hard spheres. A popular method to establish the convergence of solutions of is to first demonstrate that the mild form of the BBGKY hierarchy is given by a contracting operator if $t$ is sufficiently small. This step ensures that the sum converges absolutely, and in a second step one checks the convergence of the individual terms. It is noteworthy that the necessity to establish the contraction property of the mild BBGKY operator (and thereby restricting the analysis to small values of $t$) is due to the fact that the individual terms in are unsigned. The key achievement of this paper is to demonstrate that in a simpler setting a representation formula similar to for $f_N^{(s)}$ can be found so that the individual terms are non-negative and can be interpreted as probabilities. This representation offers two significant advantages: 1. It is possible to allow irregular initial data which may not be exponentially tight.\ 2. The need to establish a contraction property is replaced by a tightness bound, which is related to the properties of physically relevant objects such as collision histories. In particular, convergence can be established for all times. The Lorentz and Rayleigh Gas ---------------------------- Instead of considering a system of $N$ identical hard spheres evolving via elastic collisions one can consider a single tagged or tracer particle evolving among a system of fluid scatterers or background particles. If the background particles are fixed and of infinite relative mass to the tracer particle then one has a model known as the Lorentz gas first introduced by Lorentz in [@lor05] to study the motion of electrons in a metal. Much research has been done deriving the linear Boltzmann equation from a Lorentz gas with randomly placed scatterers, for example [@bold83; @gall99; @spohn78] and a large number of references found in Part I Chapter 8 of [@spohn91]. The linear Boltzmann equation can however fail to hold if we consider non-random periodic scatterers, as shown for example by Golse [@golse08] and Marklof [@marklof10]. The existence of a limiting stochastic process for the periodic Lorentz gas from the Boltzmann-Grad limit was shown by Marklof and Strömbergsson in [@mark11]. When a force field is present the convergence of the distribution of the tracer particle in an absorbing Lorentz gas to the solution of a gainless linear Boltzmann equation was proved in [@desvi07]. The authors also proved that if the scatterers move with a constant random initial velocity then the convergence can be proven with significantly weaker assumptions on the force field. Closely related to the Lorentz gas is the Rayleigh gas, where the background particles are no longer of infinite mass. Convergence to Brownian motion is discussed in Part I Chapter 8 of [@spohn91]. In [@lebo82] Lebowitz and Spohn proved the convergence of the momentum process for a test particle to a jump process associated to the linear Boltzmann equation. This was proved for arbitrarily long times, via the BBGKY hierarchy, when the initial distribution of the velocities is at equilibrium. This builds on their previous work [@lebo78; @lebo82b; @bei80]. In this paper we consider a Rayleigh gas where the background particles are of equal mass to the tagged particle and have no self interaction. The particles evolve via a simplified form of hard-sphere dynamics whereby the background particles do not change velocity. We consider non-equilibrium initial data but require that the background particles are spatially homogeneous. Convergence is proved for arbitrary times. Model and Main Result ===================== Our Rayleigh gas model in three dimensional space is now detailed. Define $U:=[0,1]^3$ with periodic boundary conditions. Here a tagged particle evolves via the hard sphere flow and the remaining $N$ particles do not interact, i.e. move along straight lines. The initial distribution of the tagged particle is $f_0 \in L^1 (U \times \mathbb{R}^3 )$. The $N$ background particles are independently distributed according to the law $g_0 \in L^1 ( \mathbb{R}^3 )$ in velocity space and uniform in $U$. The tagged particle and the background particles are spheres with diameter $\varepsilon >0$ which is related to $N$ via the Boltzmann-Grad scaling, $$\label{eq-boltzgrad} N\varepsilon^2 =1.$$ The background particles always travel in free flow with their velocities never changing from the initial value. The tagged particle travels in free flow whilst its centre remains at least $\varepsilon$ away from the centre of all the background particles. When the centre of the tagged particle comes within $\varepsilon$ of the centre of a background particle the tagged particle collides as if it was a Newtonian hard-sphere collision and changes velocity. Explicitly this is described as follows. Denote the position and velocity of background particle $1\leq j \leq N$ at time $t$ by $(x_j(t),v_j(t))$. Then for all $t\geq 0$, $$\begin{aligned} \frac{\mathrm{d}x_j(t)}{\mathrm{d}t} = v_j(t) \textrm{ and } \frac{\mathrm{d}v_j(t)}{\mathrm{d}t} = 0.\end{aligned}$$ Further denote the position and velocity of the tagged particle at time $t$ by $(x(t),v(t))$. Then for all $t\geq 0$, $$\frac{\mathrm{d}x(t)}{\mathrm{d}t} = v(t) .$$ If at time $t$ for all $1\leq j \leq N$, $|x(t)-x_j(t)|>\varepsilon$ then ${\mathrm{d}v(t)}/{\mathrm{d}t}=0$. Otherwise there exists a $1\leq j \leq N$ such that $|x(t)-x_j(t)|=\varepsilon$ and the tagged particle experiences an instantaneous change of velocity. Define the collision parameter $\nu \in \mathbb{S}^2$ by, $$\nu := \frac{x(t)-x_j(t)}{|x(t)-x_j(t)|} .$$ ![The collision parameter $\nu$[]{data-label="fig-nu"}](nu) Then the velocity of the tagged particle instantaneously after the collision, $v(t)$, is given by $$v(t):= v(t^-) - \nu \cdot (v(t^-) - v_j)\nu.$$ It is noted that in this model we do not have conservation of momentum. The background particles do not change velocity and the root particle does. \[prop:wellposedHS\] For $N \in \mathbb{N}$ and $T>0$ fixed these dynamics are well defined up to time $T$ for all initial configurations apart from a set of measure zero. The proof is given in section \[sec:aux\], which establishes that almost surely all collisions involve only pairs. We are interested in studying the distribution of a tagged particle among $N$ background particles, $\hat{f}^N_t$, under the above particle dynamics as $N$ tends to infinity or equivalently as $\varepsilon$ tends to zero. \[admissible\] Probability densities $f_0\in L^1(U\times \R^3)$, $g_0 \in L^1(\R^3)$ are admissible if $$\begin{aligned} \label{eq-f0assmp2} \int_{ U \times \mathbb{R}^3} f_0 (x,v)(1+|v|^2) \, \mathrm{d}x \, \mathrm{d}v < \infty,\\ \label{eq-g0l1assmp} \int_{ \mathbb{R}^3} g_0 (v)(1+|v|^2) \, \, \mathrm{d}v < \infty,\\ \label{eq-g0linf5} \operatorname*{ess\,sup}_{v\in\mathbb{R}^3} g_0(v)(1+|v|^4) < \infty. \end{aligned}$$ The distribution of the tagged particle is shown to converge to the solutions of a linear Boltzmann equation up to a finite arbitrary time $T$. We now state the main theorem of this paper. \[thm-main\] Let $0<T<\infty$ and $f_0,g_0$ be admissible. Then $\hat f_t^N$ converges to a time-dependent density $f_t$ in the TV sense. Moreover, the limit $f_t$ satisfies the linear Boltzmann equation $$\label{eq-linboltz} \begin{cases} \partial_t f_t (x,v) & = -v \cdot \partial_x f_t(x,v) + Q[f_t](x,v), \\ f_{t=0}(x,v) & = f_0(x,v), \end{cases}$$ where the collision operator $Q$ is defined by $Q:=Q^+-Q^-$ and $Q^+$ and $Q^-$ known respectively as the gain and loss term are given as follows, $$Q^+[f](x,v) = \int_{\mathbb{S}^2} \int_{\mathbb{R}^3} f(x,v')g_0(\bar{v}')[(v-\bar{v})\cdot \nu]_+ \, \mathrm{d}\bar{v} \, \mathrm{d}\nu,$$ where the pre-collision velocities, $v'$ and $\bar{v}'$, are given by $v' = v+ \nu \cdot (\bar{v}-v) \nu$ and $\bar{v}'=\bar{v} - \nu \cdot (\bar{v}-v)\nu$, and $$Q^-[f](x,v) = f(x,v)\int_{\mathbb{S}^2} \int_{\mathbb{R}^3} g_0(\bar{v})[(v-\bar{v})\cdot \nu]_+ \, \mathrm{d}\bar{v} \, \mathrm{d}\nu.$$ Remarks ------- 1. The reader is reminded that solutions of only conserve mass, but not energy. 2. The analysis of the Rayleigh gas can also be done using traditional BBGKY approach. Here one uses the collision operator in $\mathbb{R}^d$: $$\begin{aligned} \mathcal C^{\text{Rayl}}_{s,s+1}f^{(s+1)} (t,Z_s)&=(N-s) \varepsilon^{d-1} \int_{S^{d-1} \times \mathbb{R}^d} \nu \cdot (v_{s+1}-v_i)\\ & \qquad \times f_N^{(s+1)}(t,Z_s,x_1+ \varepsilon \nu, v_{s+1}) \, \mathrm{d}\nu \, \mathrm{d}v_{s+1}.\end{aligned}$$ The hard sphere collision operator is given by $$\begin{aligned} \label{eq-Chs} \mathcal C^{\text{hs}}_{s,s+1} f^{(s+1)}(t,Z_s)&:= (N-s) \varepsilon^{d-1} \sum_{i=1}^s \int_{S^{d-1} \times \mathbb{R}^d} \nu \cdot (v_{s+1}-v_i) \\ \nonumber & \qquad \times f_N^{(s+1)}(t,Z_s,x_i+ \varepsilon \nu, v_{s+1}) \, \mathrm{d}\nu \, \mathrm{d}v_{s+1}.\end{aligned}$$ The only difference between $\mathcal C^{\text{Rayl}}_{s,s+1}$ and $\mathcal C^{\text{hs}}_{s,s+1}$ is the fact that in the hard sphere case one sums over all indices $i=1,\ldots ,s$ and in the Rayleigh case only over $i=1$. This gives estimates on $\mathcal C^{\text{Rayl}}_{s,s+1}$, which are independent of $s$ in the contraction proof for the mild form of the associated BBGKY hierarchy. Using the function spaces $\bf{X}_{\varepsilon,\beta,\mu}$ with norm $ \|.\|_{\varepsilon, \beta,\mu}$ as in [@saintraymond13 Def 5.1.4] for measurable functions $G: t \in[0,T] \mapsto (t)=(g_s(t))_{s \geq 1} \in \bf{X}_{\varepsilon,\beta,\mu} $ one can introduce another time-dependent variant compared to [@saintraymond13 Def 5.1.4] $$\||G|\|_{\varepsilon, \beta,\mu,\lambda}:= \sup_{0 \leq t \leq T} \|G(t)\exp(-\lambda t |v_1|^2) \|_{\varepsilon, \beta,\mu}.$$ This will lead to a contraction for arbitrary large times $T$. For a slightly different approach assuming only finite moments see [@spohn80 Section II.B].\ 3. Our method can be used to derive quantitative error estimates at the expense of more complex notation and additional regularity requirements for $f_0$ and $g_0$. In particular, see lemmas \[lem-intg0bhtbound\] and \[lem-emp3\] for some quantitative expressions.\ 4. The result should also hold in the case $d=2$ or $d \geq 4$ up to a change in moment assumptions on the initial data and minor changes in estimates and calculations throughout the paper.\ 5. One could consider a spatially inhomogeneous initial distribution for the background particles $g_0=g_0(x,v)$. This adds a complication to the equations since for example the operator $Q$ in becomes time-dependent, i.e. $Q_t = Q_t^+-Q^-_t$ with $$Q_t^-[f](x,v) = f(x,v)\int_{\mathbb{S}^2} \int_{\mathbb{R}^3} g_0(x-t\bar{v},\bar{v})[(v-\bar{v})\cdot \nu]_+ \, \mathrm{d}\bar{v} \, \mathrm{d}\nu,$$ and $Q^+_t$ analogous. Since the operator now depends on the time $t$ this would require evolution semigroup results to echo the semigroup results in [@arlotti06].\ 6. One could also attempt to adapt these results to more complex and involved models. For example a model where each particle has an associated counter and a collision occurs between particle $i$ and $j$ if and only if both counters are less than $k$, in the hope of letting $k$ tend to infinity. The main difficulty here will be that one will need to keep track of the current distribution of the background $g_t$ in contrast to our model where the background has constant with time distribution $g_0$. Method of Proof --------------- We closely follow the method of [@matt12]. That is we study the probability distribution of finding a given history of collisions, a given tree, at time $t$. Firstly in section \[sec:ideal\] we prove the main result, theorem \[thm-ideq\], which shows that there exists a solution $P_t$ to a Kolmogorov differential equation and relate this solution to the solution of the linear Boltzmann equation. We show existence by explicitly building a solution on the most simple trees and using this to iteratively build a full solution. In section \[sec:emp\] we consider the distribution $\hat{P}_t$ of finding a given history of collisions from our particle dynamics and show by direct calculation that this solves a similar differential equation in theorem \[thm-emp\] for sufficiently well controlled (good) trees. Finally in section \[sec:conv\] we prove the main theorem of the paper, theorem \[thm-main\], by proving the convergence between $P_t$ and $\hat{P}_t$ in theorem \[thm-ptphatcomp\] and then relating this to $f_t$ and $\hat{f}^N_t$. Tree Set Up ----------- We construct trees in a similar way to [@matt12]. A tree represents a specific history of collisions. The nodes of the tree are denoted by $m$ and represent particles while the edges, denoted $E$, represent collisions. The root of the tree represents the tagged particle and is marked with the initial position of the tagged particle $(x_0,v_0)\in U \times \mathbb{R}^3$. The child nodes of the root represent background particles that the root collides with and are denoted $(t_j,\nu_j,v_j) \in (0,T]\times \mathbb{S}^2\times \mathbb{R}^3$, where $t_j$ represents the collision time, $\nu_j$ the collision parameter and $v_j$ the incoming velocity of the background particle. Since the background particles only collide with the root particle and not each other we only consider trees of height at most 1, so the trees simplify to the initial position of the tagged particle and a list of its $n\geq 0$ collisions. The graph structure is mainly suppressed. The set of collision trees $\mathcal{MT}$ is defined by, $$\mathcal{MT}:= \{ (x_0,v_0),(t_1,\nu_1,v_1),\dots,(t_{n},\nu_{n},v_{n}) : n \in \mathbb{N}\cup \{0\} \}.$$ For a tree $\Phi \in \mathcal{MT}$, $n(\Phi)$ is the number of collisions. The final collision in a tree $\Phi$ plays a significant role. Define the maximum collision time $\tau(\Phi) \in [0,T]$, $$\tau(\Phi):= \begin{cases} 0 & \textrm{ if } n(\Phi) =0, \\ \max_{1\leq j \leq n(\Phi) } t_j & \textrm{ else. } \end{cases}$$ Further for $n(\Phi) \geq 1$ the marker for the final collision is denoted by, $$(\tau, \nu,v') := (t_{n(\Phi) },\nu_{n(\Phi) },v_{n(\Phi) } ).$$ The realisation of a tree $\Phi$ at a time $t\in[0,T]$ for a particle diameter $\varepsilon>0$ uniquely defines the position and velocity of the tagged particle for all times up to $t$ since the initial position and the collisions the root experiences are known. Further it determines the initial positions of the $n(\Phi)$ background particles involved in the tree since we can work backwards from the collision and we know that their velocity does not change. Finally it also includes information about the other $N-n$ background particles, because it is known that they do not interfere with the root up to time $t$. If the root collides at the instant $t$ denote the pre-collisional velocity by $v(t^-)$ and the post-collisional velocity by $v(t)$. Throughout this paper the dependence on $\Phi$ is often dropped from these and other variables when the context is clear. For $n\geq 1$, define $\bar{\Phi}$ as being the pruned tree of ${\Phi}$ with the node representing the final collision, which occurs at time $\tau$, removed. For example if $\Phi = ((x_0,v_0),(t_1,\nu_1,v_1),(\tau,\nu,v'))$ then $\bar{\Phi} = ((x_0,v_0),(t_1,\nu_1,v_1))$. Define a metric $\mathrm{d}$ on $\mathcal{MT}$ as follows. For $\Phi,\Psi \in \mathcal{MT}$ with components $\Phi_j$ and $\Psi_j$ respectively. $$\textrm{d}(\Phi,\Psi) := \begin{cases} 1 & \textrm{ if } n(\Phi) \neq n(\Psi) \\ \min \left\{ 1,\max_{0\leq j \leq n} |\Phi_j - \Psi_j|_\infty \right\} & \textrm{ else.} \end{cases}$$ Further denote by $B_h(\Phi)$ the ball of radius $h/2$ around $\Phi \in \mathcal{MT}$, $$\label{eq-balldeff} B_h(\Phi) := \{ \Psi \in \mathcal{MT} : \mathrm{d}(\Phi,\Psi) < h/2 \}.$$ The standard Lebesgue measure on $\mathcal{MT}$ is denoted by $\mathrm{d}\lambda$. The Idealized Distribution {#sec:ideal} ========================== In this section we show that there exists a solution, denoted $P_t$, to the idealized equation, equation , and relate this solution to the solution of the linear Boltzmann equation. We prove existence by constructing a solution iteratively on different sized trees. In section \[sec:conv\] $P_t$ is compared to the solution of a similar evolution equation defined by the particle dynamics in order to show the required convergence. The idealized system plays the same role as the Boltzmann hierarchy in [@saintraymond13]. For a given tree $\Phi \in \mathcal{MT}$, $P_0(\Phi)$ is zero unless $\Phi$ involves no collisions, in which case $P_0(\Phi)$ is given in terms of the initial distribution $f_0$. $P_t(\Phi)$ remains zero until $t=\tau$ when there is an instantaneous increase to a positive value depending on $P_\tau(\bar{\Phi})$ and the final collision in $\Phi$. For $t>\tau$, $P_t(\Phi)$ decreases at a rate that is obtained by considering all possible collisions. The idealized equation is given by, $$\label{eq-id} \begin{cases} \partial _t P_t (\Phi) & = \mathcal{Q}_t[P_t](\Phi) = \mathcal{Q}_t^+ [P_t](\Phi) - \mathcal{Q}^{-}_t[P_t](\Phi), \\ P_0(\Phi) & = f_0 (x_0,v_0)\mathbbm{1}_{n(\Phi)=0}, \end{cases}$$ where, $$\mathcal{Q}_t^+ [P_t](\Phi): = \left\{\begin{array}{cc}\delta(t-\tau)P_t (\bar{\Phi})g_0(v')[(v(\tau^-)-v')\cdot \nu]_+&\mbox{if } n(\Phi)\geq 1, \\ 0&\mbox{if } n(\Phi)=0,\end{array}\right.$$ $$\mathcal{Q}^{-}_t[P_t](\Phi): = P_t(\Phi) \int_{\mathbb{S}^{2}} \int_{\mathbb{R}^3} g_0(\bar{v})[(v(\tau)-\bar{v})\cdot \nu ]_+ \, \mathrm{d}\bar{v} \, \mathrm{d}\nu.$$ \[thm-ideq\] Suppose that $f_0$ and $g_0$ are admissible (in the sense of Def. \[admissible\]). Then there exists a solution $P:[0,T] \to L^1(\mathcal{MT})$ to the idealized equation, . Moreover for any $t\in[0,T]$ and for any $\Omega \subset U \times \mathbb{R}^3 $ define $$S_t(\Omega):=\{ \Phi \in \mathcal{MT} : (x(t),v(t)) \in \Omega \}.$$ Then $$\int_\Omega f_t(x,v) \, \mathrm{d}x \, \mathrm{d}v = \int_{S_t(\Omega)} P_t(\Phi) \, \mathrm{d}\Phi,$$ where $f_t$ is the unique mild solution of the linear Boltzmann equation given in proposition \[prop-linboltz\]. Condition can be relaxed to $$\label{eq-g0l4eta} \operatorname*{ess\,sup}_{v\in\mathbb{R}^3} g_0(v)(1+|v|^{3+\eta}) < \infty$$ for some $\eta > 0$. From now on assume that $f_0$ and $g_0$ are admissible with the provision that either or holds. We prove the existence by construction, taking several steps to build a solution by solving on the most simple trees first and using this solution to iteratively build a full solution. We begin by solving the linear Boltzmann equation. We establish existence, uniqueness and regularity of solutions of by adapting methods from semigroup theory. The difficulty here is that after writing the linear Boltzmann equation as the sum of two unbounded operators we need to ensure that a honest semigroup is generated in order to prove existence and uniqueness. Next we adapt these semigroup techniques to define functions $P_t^{(j)}$ that describe the distribution of finding the tagged particle such that it has experienced $j$ collisions. This is key to connecting $P_t$ to the solution of the linear Boltzmann. The following notion of mild solution suitable for transport equations is used (c.f. [@arendt11 Def 3.1.1]) \[deff-mildsol\] Consider the following system, $$\begin{cases} \label{eq-mildsol} \partial_t u(t)& = Lu(t), \\ u(0)& =u_0. \end{cases}$$ Where $L: D(L) \subset L^1(U\times \mathbb{R}^3) \to L^1(U\times \mathbb{R}^3)$ is an operator and $u_0 \in L^1(U\times \mathbb{R}^3)$ is given. The function $u:[0,T]\to U \times \mathbb{R}^3$ is called a mild solution of if for all $t\geq 0$, $$\int_0^t u(\theta)\, \mathrm{d}\theta \in D(L) \, \textrm{ and } \, L\int_0^t u(\theta)\, \mathrm{d}\theta = u(t)-u_0.$$ We split the right hand side of into two operators, $A$ and $B$. These will appear in the construction of $P_t$. \[deff-aoperator\] Define $D(A), D(B)\subset L^1(U \times \mathbb{R}^3)$ by, $$\begin{aligned} D(A)&:=\{ f \in L^1(U \times \mathbb{R}^3) : v\cdot \partial_x f(x,v) + Q^-[f](x,v) \in L^1(U \times \mathbb{R}^3) \},\\ D(B)&:= \{ f \in L^1(U\times \mathbb{R}^3) : Q^+[f] \in L^1(U\times \mathbb{R}^3) \}.\end{aligned}$$ Then define $A:D(A) \to L^1(U \times \mathbb{R}^3)$ and $B:D(B) \to L^1(U \times \mathbb{R}^3)$ by, $$\begin{aligned} \label{eq-adeff} (Af)(x,v) &:= -v\cdot \partial_x f(x,v) - Q^-[f](x,v),\\ \label{eq-bdeff} (Bf)(x,v) &:= Q^+[f](x,v).\end{aligned}$$ \[prop-linboltz\] Suppose that the assumptions in theorem \[thm-ideq\] hold. Then there exists a unique mild solution $f:[0,T] \to L^1(U \times \mathbb{R}^3)$ to . Furthermore $f_t$ remains non-negative and of mass $1$, and $$\begin{aligned} \label{lem-ftmom} \int_{U\times \mathbb{R}^3} f_t(x,v)(1+|v|) \, \mathrm{d}x \, \mathrm{d}v < \infty,\\ f_t\in D(B) \label{lem-ftindb} \end{aligned}$$ hold for all $t\in [0,T]$. See section \[sec:aux\]. \[prop-p0\] There exists a unique mild solution, $P^{(0)}:[0,T] \to L^1( U \times \mathbb{R}^3)$, to the following evolution equation, $$\label{eq-p0} \begin{cases} \partial_t P^{(0)}_t (x,v) & = (AP^{(0)}_t)(x,v), \\ P^{(0)}_0 (x,v) & = f_0(x,v). \end{cases}$$ Where $A$ is as in . The distribution $P^{(0)}_t(x,v)$ can be thought of as the probability of finding the tagged particle at $(x,v)$ at time $t$ such that it has not yet experienced any collisions. By lemma \[lem-tsemi\] $A$ generates the substochastic $C_0$ semigroup $T(t)$ given in . By the Hille-Yoshida theorem, [@pazy83 Thm 1.3.1] $A$ is closed. By [@arendt11 Thm 3.1.12] has a unique mild solution given by $$\label{eq-pt0ttf0} P_t^{(0)}=T(t)f_0.$$ \[lem-p0leqft\] For all $t \in [0,T]$, $P_t^{(0)} \leq f_t$ pointwise. This lemma is entirely expected. The probability of finding the tagged particle at $(x,v)$ at time $t$ is given by $f_t(x,v)$ and the probability of finding it at $(x,v)$ at time $t$ such that it has not experienced any collisions up to time $t$ is given by $P_t^{(0)}(x,v)$ so one expects $P_t^{(0)} \leq f_t$. For $t \in [0,T]$ define $F_t^{(0)}:=f_t-P_t^{(0)}$. Then since $f_t$ and $P_t^{(0)}$ are mild solutions of and respectively, $F_t^{(0)}$ is a mild solution of $$\begin{cases} \partial_t F_t^{(0)}(x,v) & = AF_t^{(0)} + Bf_t(x,v) \\ F_0^{(0)}(x,v) & = 0. \end{cases}$$ By and [@arendt11 Prop. 3.1.16] $F_t^{(0)}$ is given by, $$F_t^{(0)} = \int_0^t T(t-\theta)Bf_\theta \, \mathrm{d}\theta.$$ Now noting that $f_\theta$ is non-negative it follows that $Bf_\theta$ and hence $T(t-\theta)Bf_\theta$ are non-negative also. Hence $F_t^{(0)} \geq 0$ which implies $P_t^{(0)} \leq f_t$. \[deff-ptfull\] For $j \in \mathbb{N} \cup \{0\}$ denote by $\mathcal{T}_j$ the set of all trees with exactly $j$ collisions. Explicitly, $$\label{eq-tjdeff} \mathcal{T}_j := \{ \Phi \in \mathcal{MT} : n(\Phi) = j \}.$$ The required solution $P_t$ can now be defined iteratively on the space $\mathcal{MT}$. For $\Phi \in \mathcal{T}_0$ define $$\label{eq-ptt0} P_t(\Phi) := P^{(0)}_t (x(t),v(t)).$$ Else define, $$\begin{aligned} \label{eq-ptfull} P_t(\Phi): = \mathbbm{1}_{t \geq \tau} \exp \left( -(t-\tau) \int_{\mathbb{S}^2} \int_{\mathbb{R}^3} g_0(\bar{v}) [(v(\tau)-\bar{v})\cdot \nu']_+ \, \mathrm{d}\bar{v} \, \mathrm{d}\nu' \right) \nonumber \\ P_\tau (\bar{\Phi})g_0(v')[(v(\tau^-)-v')\cdot \nu]_+.\end{aligned}$$ The right hand side of this equation depends on $P_\tau (\bar{\Phi})$ but since $\bar{\Phi}$ has degree exactly one less than $\Phi$ the equation is well defined. The proof that $P_t$ has the required properties of theorem \[thm-ideq\] is given shortly. We first define the function $P^{(j)}_t$ which is thought of, in parallel to $P^{(0)}_t$, as being the probability of finding the tagged particle at a certain position at time $t$ such that it has experienced exactly $j$ collisions up to time $t$. The $P_t^{(j)}$ will be required to show the connection between $P_t$ and the solution of the linear Boltzmann equation. \[def-pj\] Let $t\in [0,T]$ and $\Omega \subset U \times \mathbb{R}^3$ be measurable. Recall in theorem \[thm-ideq\] we define the set, $S_t(\Omega)=\{ \Phi \in \mathcal{MT} : (x(t),v(t))\in \Omega \}$ - the set of all trees such that the tagged particle at time $t$ is in $\Omega$. Define for all $j \in \mathbb{N}\cup \{0\}$, $$S^j_t(\Omega):=\mathcal{T}_j \cap S_t(\Omega) .$$ Then for $j \geq 1$, define $P^{(j)}(\Omega)$ by, $$P^{(j)}_t(\Omega):= \int_{ S^j_t(\Omega) } P_t(\Phi) \, \mathrm{d}\Phi.$$ \[lem-Ptjmble\] Let $t\in [0,T]$, $j \geq 1$. Then $P_t^{(j)}$ is absolutely continuous with respect to the Lebesgue measure on $U \times \mathbb{R}^3$. Let $j=1$. Then we have by , $$\begin{aligned} \label{eq-ptjomegaform} P^{(1)}_t(\Omega)& = \int_{ S^1_t(\Omega) } P_t(\Phi) \nonumber \\ & = \int_0^t \int_{\mathbb{R}^3} \int_{\mathbb{R}^3} \int_{\mathbb{S}^2} \int_U \exp \left( -(t-\tau) \int_{\mathbb{S}^2} \int_{\mathbb{R}^3} g_0(\bar{v}) [(v(\tau)-\bar{v})\cdot \nu]_+ \, \mathrm{d}\bar{v} \, \mathrm{d}\nu \right) \nonumber \\ & \qquad \qquad P_\tau^{(0)}(x_0+ \tau v_0,v_0)g_0(v') [(v_0-v')\cdot \nu]_+ \mathbbm{1}_{(x(t),v(t)) \in \Omega)} \, \mathrm{d}x_0 \, \mathrm{d}\nu \, \mathrm{d}v' \, \mathrm{d}v_0 \, \mathrm{d}\tau\end{aligned}$$ We define a coordinate transform $(\nu, x_0,v_0,v') \mapsto (\nu, x,v,\bar{w})$ given by, $$\begin{aligned} v & :=v_0 + \nu (v'-v_0)\cdot \nu \\ x & := x_0 + \tau v_0 + (t-\tau) v \\ \bar{w} & := v' - \nu (v'-v_0)\cdot \nu.\end{aligned}$$ This transformation has Jacobi matrix, $$\begin{pmatrix} \textrm{Id} & 0 & 0 & 0 \\ & \textrm{Id} & & \\ & 0 & \textrm{Id}-\nu \otimes \nu & \nu \otimes \nu \\ & 0 & \nu \otimes \nu & \textrm{Id}-\nu \otimes \nu \\ \end{pmatrix}$$ where the blank entries are not required for the computation of the matrix’s determinant. The 2x2 matrix in the bottom right has determinant $- 1$ and hence the absolute value of the determinant of the entire matrix is $1$. With this becomes, $$\begin{aligned} P^{(1)}_t(\Omega)& = \int_{\Omega} \int_0^t \int_{\mathbb{R}^3} \int_{\mathbb{S}^2} \exp \left( -(t-\tau) \int_{\mathbb{S}^2} \int_{\mathbb{R}^3} g_0(\bar{v}) [(v-\bar{v})\cdot \nu]_+ \, \mathrm{d}\bar{v} \, \mathrm{d}\nu \right) \nonumber \\ & \qquad \qquad P_\tau^{(0)}(x-(t-\tau)v,w')g_0(\bar{w}') [(v-\bar{w})\cdot \nu]_+ \, \mathrm{d}\nu \, \mathrm{d}\bar{w} \, \mathrm{d}\tau \, \mathrm{d}x \, \mathrm{d}v \nonumber,\end{aligned}$$ where $w'=v + \nu (\bar{w} - v) \cdot \nu$ and $\bar{w}'=\bar{w} - \nu (\bar{w} - v) \cdot \nu$. Hence we see that if the Lebesgue measure of $\Omega$ equals zero then so does $P_t^{(1)}(\Omega)$. For $j \geq 2$ we use a similar approach, using the iterative formula for $P_t(\Phi)$ . Since $P_t^{(j)}$ is an absolutely continuous measure on $U\times \mathbb{R}^3$, the Radon-Nikodym theorem (see Theorem 4.2.2 [@cohn13]) implies that $P_t^{(j)}$ has a density, which we denote by $P_t^{(j)}$ also. This gives, $$\int_{\Omega} P_t^{(j)}(x,v) \, \mathrm{d}x \, \mathrm{d}v = \int_{ S^j_t(\Omega) } P_t(\Phi) \, \mathrm{d}\Phi.$$ Hence for almost all $(x,v) \in U \times \mathbb{R}^3$, $$P_t^{(j)}(x,v) = \int_{S^j_t(x,v) } P_t(\Phi) \, \mathrm{d}\Phi.$$ \[rmk-p0pt\] A similar formula holds for $P^{(0)}_t$ since the set $S^0_t(x,v)$ contains exactly one tree: the tree $\Phi$ with initial root data $(x-tv,v)$ and such that the root has no collisions, $$\begin{aligned} \int_{S^0_t(x,v) } P_t(\Phi) \, \mathrm{d}\Phi &= P_t((x-tv,v)) = P^{(0)}_t(x,v).\end{aligned}$$ \[prop-pj\] For $j\geq 1$, $P^{(j)}_t$ as defined above is almost everywhere the unique mild solution to the following differential equation, $$\begin{cases} \partial_t P^{(j)}_t(x,v) & = AP_t^{(j)}(x,v) + BP_t^{(j-1)}(x,v), \\ P^{(j)}_0(x,v)& =0, \end{cases}$$ where $A$ is given in and $B$ in . The following lemma helps prove the proposition for the case $j=1$ which allows the use of an inductive argument to prove the proposition in full. \[lem-p1alt\] For any $t \in [0,T]$ and almost all $(x,v)\in U \times \mathbb{R}^3$, $$P^{(1)}_t(x,v)= \int_0^t T(t-\theta)BP_\theta^{(0)}(x,v) \, \mathrm{d}\theta,$$ where the semigroup $T(t)$ is as in . The right hand side is well defined since , and imply $P_t^{(0)} \in D(B)$. We show that for any $\Omega \subset U \times \mathbb{R}^3$ measurable we have, $$\label{eq-p1alteq} \int_{ \Omega } P_t^{(1)}(x,v) \, \mathrm{d}x \, \mathrm{d}v = \int_{\Omega} \int_0^t T(t-\theta)BP_\theta^{(0)}(x,v) \, \mathrm{d}\theta \, \mathrm{d}x\, \mathrm{d}v. \\$$ By the definition of $T(t)$ in equation , the definition of $B$ in , and the proof of lemma \[lem-Ptjmble\], for $w'=v + \nu (\bar{w} - v) \cdot \nu$ and $\bar{w}'=\bar{w} - \nu (\bar{w} - v) \cdot \nu$, we have, $$\begin{aligned} \label{eq-p1altrhs} \int_{\Omega} \int_0^t & T(t-\theta)BP_\theta^{(0)}(x,v) \, \mathrm{d}\theta \, \mathrm{d}x\, \mathrm{d}v \\ & = \int_{\Omega} \int_0^t \exp \left( -(t-\theta) \int_{\mathbb{S}^2} \int_{\mathbb{R}^3} g_0(\bar{v}) [(v-\bar{v})\cdot \nu]_+ \, \mathrm{d}\bar{v} \, \mathrm{d}\nu \right) \nonumber \\ & \qquad \qquad \times BP_\theta^{(0)}(x-(t-\theta)v,v) \, \mathrm{d}\theta \, \mathrm{d}x\, \mathrm{d}v \\ & = \int_{\Omega} \int_0^t \exp \left( -(t-\theta) \int_{\mathbb{S}^2} \int_{\mathbb{R}^3} g_0(\bar{v}) [(v-\bar{v})\cdot \nu]_+ \, \mathrm{d}\bar{v} \, \mathrm{d}\nu \right) \nonumber \\ & \qquad \int_{\mathbb{R}^3} \int_{\mathbb{S}^2} P_\theta^{(0)}(x-(t-\theta)v,w')g_0(\bar{w}') [(v-\bar{w})\cdot \nu]_+ \, \mathrm{d}\nu \, \mathrm{d}\bar{w} \, \mathrm{d}\theta \, \mathrm{d}x \, \mathrm{d}v \nonumber \\ & = \int_{\Omega} \int_0^t \int_{\mathbb{R}^3} \int_{\mathbb{S}^2} \exp \left( -(t-\theta) \int_{\mathbb{S}^2} \int_{\mathbb{R}^3} g_0(\bar{v}) [(v-\bar{v})\cdot \nu]_+ \, \mathrm{d}\bar{v} \, \mathrm{d}\nu \right) \nonumber \\ & \qquad \qquad P_\theta^{(0)}(x-(t-\theta)v,w')g_0(\bar{w}') [(v-\bar{w})\cdot \nu]_+ \, \mathrm{d}\nu \, \mathrm{d}\bar{w} \, \mathrm{d}\theta \, \mathrm{d}x \, \mathrm{d}v \nonumber \\ & = \int_{ \Omega } P_t^{(1)}(x,v) \, \mathrm{d}x \, \mathrm{d}v.\end{aligned}$$ Consider induction on $j$. First let $j=1$. We seek to apply [@arendt11 Prop 3.1.16]. If $\int_0^t BP_\theta^{(0)} \, \mathrm{d}\theta \in L^1(U\times \mathbb{R}^3)$ then the proposition holds so by the above lemma $P_t^{(1)}$ is the unique mild solution. To this aim note that since $P_t^{(0)}$ is the unique mild solution of , $$\int_0^t P_\theta^{(0)} \, \mathrm{d}\theta \in D(A).$$ By [@arlotti06 Section 10.4.3] $D(A) \subset D(B)$ and hence $$\int_0^t P_\theta^{(0)} \, \mathrm{d}\theta \in D(B).$$ This implies, $$B\int_0^t P_\theta^{(0)} \, \mathrm{d}\theta \in L^1(U\times \mathbb{R}^3).$$ as required. Now consider $j\geq 2$ and assume the proposition is true for $j-1$. By setting $F_t^{(j-1)}:=f_t-P_t^{(j-1)}$ a similar argument to lemma \[lem-p0leqft\] shows that $P_t^{(j-1)} \leq f_t$. By and , $P_t^{(j-1)} \in D(B)$ so the right hand side is well defined. A similar approach to lemma \[lem-p1alt\] shows that for any $t \in [0,T]$ and almost all $(x,v)\in U \times \mathbb{R}^3$, $$P^{(j)}_t(x,v)= \int_0^t (T(t-\theta)BP_\theta^{(j-1)})(x,v) \, \mathrm{d}\theta,$$ where $T(t)$ is the semigroup given in . The rest follows by the same argument as in the $j=1$ case. \[prop-pjsum\] For all $t \in [0,T]$ and almost all $(x,v) \in U \times \mathbb{R}^3$, $$\label{eq-pjsum} \sum_{j=0}^\infty P^{(j)}_t(x,v) = f_t(x,v),$$ where $f_t$ is the unique mild solution of the linear Boltzmann equation given in proposition \[prop-linboltz\]. Since $P_t^{(0)}$ is a mild solution of , $$\label{eq-p0mildsolda} \int_0^t P_\theta^{(0)}(x,v) \, \mathrm{d}\theta \in D(A),$$ and, $$\label{eq-p0mildsol} P_t^{(0)}(x,v) = f_0(x,v) + A\int_0^t P_\theta^{(0)}(x,v) \, \mathrm{d}\theta.$$ Further for $j\geq 1$ by proposition \[prop-pj\] and [@arlotti06 Prop 3.31], $$\label{eq-pjmildsolda} \int_0^t P_\theta^{(j)}(x,v) \, \mathrm{d}\theta \in D(A),$$ and, $$\label{eq-pjmildsol} P_t^{(j)}(x,v) = A\int_0^t P_\theta^{(j)}(x,v) \, \mathrm{d}\theta + \int_0^t BP_\theta^{(j-1)}(x,v) \, \mathrm{d}\theta.$$ Combining and , $$\int_0^t \sum_{j=0}^\infty P_\theta^{(j)}(x,v) \, \mathrm{d}\theta \in D(A).$$ Recalling from the proof of proposition \[prop-pj\] that $D(A) \subset D(B)$, $$\int_0^t \sum_{j=0}^\infty P_\theta^{(j)}(x,v) \, \mathrm{d}\theta \in D(A) \cap D(B)=D(A+B).$$ Further summing and for $j \geq 1$, $$\begin{aligned} \sum_{j=0}^\infty P_t^{(j)}(x,v) & = f_0(x,v) + \sum_{j=0}^\infty A\int_0^t P_\theta^{(j)}(x,v) \, \mathrm{d}\theta + \sum_{j=1}^\infty \int_0^t BP_\theta^{(j-1)}(x,v) \, \mathrm{d}\theta \\ & = f_0(x,v) + A \int_0^t \sum_{j=0}^\infty P_\theta^{(j)}(x,v) \, \mathrm{d}\theta + \int_0^t B \sum_{j=0}^\infty P_\theta^{(j)}(x,v) \, \mathrm{d}\theta \\ & = f_0(x,v) + (A+B)\int_0^t \sum_{j=0}^\infty P_\theta^{(j)}(x,v).\end{aligned}$$ Hence by definition \[deff-mildsol\], $\sum_{j=0}^\infty P_t^{(j)}(x,v)$ is a mild solution of and therefore since $f_t$ is the unique mild solution the proof is complete. We now have all the results needed to prove that $P_t$ satisfies all the requirements of theorem \[thm-ideq\]. Using definition \[def-pj\], proposition \[prop-pjsum\], and, since each $P_t^{(j)}$ is positive, the monotone convergence theorem we have for any measurable $\Omega \subset U \times \mathbb{R}^3$, $$\begin{aligned} \label{eq-ptftconn} \int_{S_t(\Omega)} P_t(\Phi) \, \mathrm{d}\Phi & = \sum_{j=0}^\infty \int_{S_t^{j}(\Omega)} P_t(\Phi) \, \mathrm{d}\Phi = \sum_{j=0}^\infty \int_{\Omega} P^{(j)}_t(x,v) \, \mathrm{d}x \, \mathrm{d}v \nonumber \\ & = \int_{\Omega} f_t(x,v) \, \mathrm{d}x \, \mathrm{d}v.\end{aligned}$$ In particular, $$\int_{\mathcal{MT}} P_t(\Phi) \, \mathrm{d}\Phi = \int_{U \times \mathbb{R}^3} f_t(x,v) \, \mathrm{d}x \, \mathrm{d}v < \infty.$$ Hence $P_t \in L^1(\mathcal{MT})$. To show that $P_t$ is a solution of first consider $\Phi \in \mathcal{T}_0$. Since $n(\Phi)=0$, $$P_0(\Phi) = P^{(0)}_0 (x(0),v(0)) = f_0(x_0,v_0) = f_0(x_0,v_0)\mathbbm{1}_{n(\Phi)=0}.$$ Hence it solves the initial condition. Now for $t>0$, since $\Phi \in \mathcal{T}_0$, $v(t)=v_0$ and $x(t)=x_0+tv_0$. Hence by and , $$\begin{aligned} P_t(\Phi) & = P_t^{(0)}(x(t),v(t)) = P_t^{(0)}(x_0+tv_0,v_0) \\ & = \exp \left( -t \int_{\mathbb{S}^2} \int_{\mathbb{R}^3} g_0(\bar{v}) [(v_0-\bar{v})\cdot \nu]_+ \, \mathrm{d}\bar{v} \, \mathrm{d}\nu \right) f_0(x_0,v_0).\end{aligned}$$ The only dependence on $t$ here is in the exponential term so we differentiate $P_t(\Phi)$ with respect to $t$, $$\begin{aligned} \partial_t P_t(\Phi) & = \partial_t \left( \exp \left( -t \int_{\mathbb{S}^2} \int_{\mathbb{R}^3} g_0(\bar{v}) [(v_0-\bar{v})\cdot \nu]_+ \, \mathrm{d}\bar{v} \, \mathrm{d}\nu \right) f_0(x_0,v_0) \right) \\ & = -\int_{\mathbb{S}^2} \int_{\mathbb{R}^3} g_0(\bar{v}) [(v_0-\bar{v})\cdot \nu]_+ \, \mathrm{d}\bar{v} \, \mathrm{d}\nu \times P_t(\Phi) = -\mathcal{Q}_t^-[P_t](\Phi).\end{aligned}$$ Hence $P_t$ solves on $\mathcal{T}_0$. We now consider $ \Phi \in \mathcal{T}_j$ for $j \geq 1$. Since $\Phi \in \mathcal{T}_j$ we have $n(\Phi)=j$ and $\tau >0$. Hence $$\label{eq-pt01} P_0(\Phi) = 0 = f_0(x_0,v_0)\mathbbm{1}_{n(\Phi)=0}.$$ For $t=\tau$, $$\begin{aligned} \label{eq-pt02} P_\tau(\Phi) & = P_\tau(\bar{\Phi})g_0(v')[v(\tau^-)_-v')\cdot \nu]_+.\end{aligned}$$ Further for $t>\tau$ the only dependence on $t$ is inside the exponential term and hence differentiating gives, $$\begin{aligned} \label{eq-pt03} \partial_t P_t(\Phi) & = \partial_t \Big( \exp \big(-(t-\tau) \int_{\mathbb{S}^2} \int_{\mathbb{R}^3} g_0(\bar{v}) [(v(\tau)-\bar{v})\cdot \nu']_+ \, \mathrm{d}\bar{v} \, \mathrm{d}\nu' \big) \nonumber \\ & \qquad \qquad P_\tau (\bar{\Phi})g_0(v')[(v_0-v')\cdot \nu]_+ \Big) \nonumber \\ & = \exp \left( -(t-\tau) \int_{\mathbb{S}^2} \int_{\mathbb{R}^3} g_0(\bar{v}) [(v(\tau)-\bar{v})\cdot \nu']_+ \, \mathrm{d}\bar{v} \, \mathrm{d}\nu' \right)\nonumber \\ & \qquad \qquad P_\tau (\bar{\Phi})g_0(v')[(v_0-v')\cdot \nu]_+ \nonumber \\ & \qquad \qquad \times \left( - \int_{\mathbb{S}^2} \int_{\mathbb{R}^3} g_0(\bar{v}) [(v(\tau)-\bar{v})\cdot \nu']_+ \, \mathrm{d}\bar{v} \, \mathrm{d}\nu' \right) \nonumber \\ & = - P_t(\Phi) \int_{\mathbb{S}^2} \int_{\mathbb{R}^3} g_0(\bar{v}) [(v(\tau)-\bar{v})\cdot \nu']_+ \, \mathrm{d}\bar{v} \, \mathrm{d}\nu'= -\mathcal{Q}^-_t[P_t](\Phi).\end{aligned}$$ Equations , and prove that $P_t$ solves on $\mathcal{T}_j$. Since $\mathcal{MT}$ is the disjoint union of $\mathcal{T}_j$ for $j \geq 0$, $P_t$ is a solution of on $\mathcal{MT}$. Finally, the required connection between $P_t$ and the solution of the linear Boltzmann equation has been shown in . The Empirical Distribution {#sec:emp} ========================== We now consider the empirical distribution of trees $\hat{P}_t^\varepsilon$ defined by the dynamics of the particle system for particles with diameter $\varepsilon$. To ease notation we drop the dependence on $\varepsilon$ and write $\hat{P}_t$. The key result of this section is that $\hat{P}_t$ solves the differential equation which is similar to the idealized equation . The similarity between the two equations is exploited in the next section to prove the required convergence as $\varepsilon$ tends to zero. We do this by restricting our attention to trees that are well controlled in various ways, calling these trees good trees. For a tree $\Phi \in \mathcal{MT}$ define $ \mathcal{V}(\Phi) \in [0,\infty)$ to be the maximum velocity involved in the tree. That is, $$\mathcal{V}(\Phi):= \max \left\{ \max_{j=1,\dots,n(\Phi)}\{|v_j|\}, \max_{s \in [0,T]}\{|v(s)| \} \right\}.$$ \[def-recollisionfree\] A tree $\Phi \in \mathcal{MT}$ is called re-collision free at diameter $\varepsilon$ if for all $0 \leq \varepsilon'\leq \varepsilon$, for all $1 \leq j \leq n(\Phi)$ and for all $t < t_j$, $$|x(t)-(x_j+tv_j)|>\varepsilon'.$$ That is to say, if the tree involves a collision between the root particle and background particle $j$ at time $t_j$ then the root particle has not previously collided with background particle $j$. So if a tree is re-collision free then it involves at most one collision per background particle. Further define $$R(\varepsilon) : = \{ \Phi \in \mathcal{MT} : \Phi \textrm{ is re-collision free at diameter } \varepsilon \}.$$ A tree $\Phi \in \mathcal{MT}$ is called non-grazing if all collisions in $\Phi$ are non-grazing, that is if, $$\min_{1\leq j\leq n(\Phi)} \nu_j \cdot (v(t_j^-)-v_j) >0.$$ A tree is $\Phi \in \mathcal{MT}$ is called free from initial overlap at diameter $\varepsilon>0$ if initially the root is at least $\varepsilon$ away from the centre of each background particle. Explicitly if, for $j=1,\dots,N$, $$|x_0-x_j|>\varepsilon.$$ Define $S(\varepsilon) \subset \mathcal{MT}$ to be the set of all trees that are free from initial overlap at radius $\varepsilon$. \[def-goodtrees\] For any pair of decreasing functions $V,M:(0,\infty) \to [0,\infty)$ such that $\lim_{\varepsilon \to 0} V(\varepsilon) = \lim_{\varepsilon \to 0}M(\varepsilon) =\infty $, the set of good trees of diameter $\varepsilon$, $\mathcal{G}(\varepsilon)$, is defined as, $$\begin{aligned} \mathcal{G}(\varepsilon) &: = \Big\{ \Phi \in \mathcal{MT} : n(\Phi) \leq M(\varepsilon), \, \mathcal{V}(\Phi) \leq V(\varepsilon), \\ & \qquad \qquad \Phi \in R(\varepsilon) \cap S(\varepsilon) \, \textrm{ and } \Phi \textrm{ is non-grazing} \Big\}\end{aligned}$$ Since $M,V$ are decreasing for $\varepsilon' < \varepsilon$ we have $\mathcal{G}(\varepsilon) \subset \mathcal{G}(\varepsilon')$. Later some conditions on $M$ and $V$ are required to prove that $\hat{P}_t$ solves the relevant equation and to prove convergence. Now define the operator $\hat{\mathcal{Q}}_t$ which mirrors the idealized operator $\mathcal{Q}_t$ in the empirical case. Fix $\hat{C}_1>0$, a constant depending only on $\Phi$ described later. Define the gain operator, $$\hat{\mathcal{Q}}_t^+ [\hat{P}_t](\Phi): =\left\{ \begin{array}{ll} \delta (t - \tau) \hat{P}_t(\bar{\Phi}) \frac{ g_0(v')[(v(\tau^-)-v')\cdot \nu ]_+}{1- \pi \varepsilon^2 \int_0^\tau |v(s)-v'| \, \mathrm{d}s + \hat{C}_1\varepsilon^3}& \mbox{if } n\geq 1\\0& \mbox{if } n = 0.\end{array}\right.$$ Next for a given tree $\Phi$, a time $0<t<T$ and $\varepsilon>0$, define the function $\mathbbm{1}_t^\varepsilon[\Phi]: U \times \mathbb{R}^3 \rightarrow \{0,1\}$ by $$\label{eq-deff1phi} \mathbbm{1}_t^\varepsilon[\Phi] (\bar{x},\bar{v}):= \begin{cases} 1 \textrm{ if for all } s \in (0,t), \, |x(s)-(\bar{x}+s\bar{v})| > \varepsilon, \\ 0 \textrm{ else}. \end{cases}$$ That is $\mathbbm{1}_t^\varepsilon[\Phi] (\bar{x},\bar{v})$ is $1$ if a background particle starting at the position $(\bar{x},\bar{v})$ avoids colliding with the root particle of the tree $\Phi$ up to the time $t$. This allows us to define the loss operator, $$\hat{\mathcal{Q}}_t^- [\hat{P}_t](\Phi): = \hat{P}_t(\Phi)\frac{\int_{\mathbb{S}^{2}} \int_{\mathbb{R}^3} g_0(\bar{v})[(v(\tau)-\bar{v})\cdot \nu]_+ \, \mathrm{d}\bar{v} \, \mathrm{d}\nu - \hat{C}_2(\varepsilon)}{\int_{ U \times {R}^3} g_0(\bar{v}) \mathbbm{1}_t^\varepsilon[\Phi] (\bar{x},\bar{v}) \, \mathrm{d}\bar{x} \, \mathrm{d}\bar{v} }.$$ For some $\hat{C}_2(\varepsilon)>0$ depending on $t$ and $\Phi$ of $o(1)$ as $\varepsilon$ tends to zero. Finally define the operator $\hat{\mathcal{Q}}_t$ as follows, $$\hat{\mathcal{Q}}_t = \hat{\mathcal{Q}}_t^+ - \hat{\mathcal{Q}}_t^- .$$ \[thm-emp\] For $\varepsilon$ sufficiently small and for $\Phi \in \mathcal{G}(\varepsilon)$, $\hat{P}_t$ solves the following $$\label{eq-emp} \begin{cases} \partial_t \hat{P}_t(\Phi) & = (1-\gamma(t)) \hat{\mathcal{Q}}_t [\hat{P_t}](\Phi) \\ \hat{P}_0 (\Phi)& = \zeta(\varepsilon) f_0 (x_0,v_0) \mathbbm{1}_{n(\Phi) = 0}. \end{cases}$$ The functions $\gamma$ and $\zeta$ are given by $$\label{eq-zetadeff} \zeta(\varepsilon) := (1- \frac{4}{3}\pi \varepsilon^3)^N,$$ and, $$\gamma(t) := \left\{ \begin{array}{ll} n(\bar{\Phi})\varepsilon^{2}&\mbox{if } t=\tau,\\ n(\Phi)\varepsilon^2& \mbox{if } t>\tau. \end{array}\right.$$ When choosing the background particles according to some Poisson point process some of these terms simplify as in [@matthies10]. The proof is developed by a series of lemmas in which we prove the gain term, loss term and initial condition separately. \[deff-omega\] Define $\omega_0:=(u_0,w_0) \in U \times \mathbb{R}^3$ to be the random initial position of the test particle. By our model $\omega_0$ has distribution $f_0$. Further for $j=1,\dots,N$ define $\omega_j:=(u_j,w_j)$ to be the random initial position and velocity of background particle $j$. Note that $\omega_j$ has distribution $\textrm{Unif}(U) \times g_0$. Finally define $\omega := (\omega_1,\dots,\omega_N)$. \[lem-phatabscts\] Let $\varepsilon>0$ and $\Psi \in \mathcal{G}(\varepsilon)$ then $\hat{P}_t$ is absolutely continuous with respect to the Lebesgue measure $\lambda$ on a neighbourhood of $\Psi$. Recall the definition of $B_h(\Psi)$ . Since $\mathcal{G}(\varepsilon)$ is open there exists a $h>0$ such that $B_h(\Psi) \subset \mathcal{G}(\varepsilon)$. In the case $n(\Psi) =0$ for all $t \geq 0$, $\hat{P}_t(\Psi) \leq f_0(x_0,v_0)$ and hence absolute continuity follows. Suppose $n(\Phi) \geq 1$. Define a map $\varphi:B_h(\Psi) \to \mathcal{MT} \times U\times \mathbb{R}^3$, $$\varphi(\Phi) := (\bar{\Phi},(x(\tau)+\varepsilon \nu - \tau v',v')).$$ We view $\varphi$ as having $n(\Phi)+1$ components, the first being the initial root position $(x_0,v_0)$, components $j=2\dots,n$ being the marker $(t_j,\nu_j,\nu_j)$ and the final component being $(x(\tau)+\varepsilon \nu - \tau v',v')$ - the initial position of the background particle that leads to the final collision with the root in $\Phi$. We claim that, $$\label{eq-detgradphi} \det(\nabla \varphi)(\Phi) = \varepsilon^2 (v(\tau^-) - v')\cdot \nu.$$ To prove this we first rotate our coordinate axis so that $\nu = \textbf{e}_1$. Then for $k=0,\dots,n$ define $F_{0,k}: = \nabla_{x_0} \varphi_k(\Phi) $ and for $j=1,\dots, n$ define $F_{j,k}: = \nabla_{t_k,\nu_k}\varphi_j(\Phi)$. We calculate, $$F_{n+1,n+1} = \nabla_{\tau,\nu}\varphi_{n+1}(\Phi) = \begin{pmatrix} (v(\tau^-)-v')\cdot \nu & 0 & 0 \\ & \varepsilon & 0 \\ & 0 & \varepsilon \\ \end{pmatrix},$$ where the blank components are not needed. Also, $ F_{0,n} = \nabla_{x_0} \varphi_{n}(\Phi) = \textrm{Id}(2). $ Further for $j=2,\dots,n+1$, $F_{j,j} = \nabla_{t_j,\nu_j}\varphi_{j}(\Phi) = \textrm{Id}(2). $ For any other $j,k$ not already calculated, $ F_{j,k} =0.$ Hence $\det(\nabla \varphi)(\Phi)$ is the product of the determinants of all $F_{k,k}$ for $k=1,\dots,n+1$, proving the claim. Now define a second map, $\tilde{\varphi}: B_h(\Psi) \to (U \times \mathbb{R})^{n+1}$, $$\tilde{\varphi}(\Phi) : = ((x_0,v_0),(x_1,v_1),\dots,(x_n,v_n)).$$ This maps $\Phi$ to the initial position of each particle in $\Phi$. By repeatedly applying , $$\label{eq-dettildphi} \det (\nabla \tilde{\varphi}) (\Phi) = \prod_{j=1}^{n(\Phi)} \big(\varepsilon^2 (v(t_j^-) - v_j)\cdot \nu_j \big).$$ For $h>0$ and $j=0,\dots,n$ define $C_{h,j}(\Phi) = C_{h,j} \subset U\times \mathbb{R}^3$ to be the cube with side length $h$ centred at $\tilde{\varphi}_j(\Phi)$. Further for $h>0$ define, $$C_h(\Phi) := \prod_{j=0}^{n} C_{h,j}(\Phi).$$ By the fact that the probability of finding a tree is less than the probability that initially there is a particles at the required initial position, $$\begin{aligned} \label{eq-phatchbound} \hat{P}_t(\tilde{\varphi}^{-1}(C_h)) & \leq \int_{C_{h,0}}f_0(x,v) \, \mathrm{d}x \, \mathrm{d}v \nonumber \\ & \qquad \qquad \times \prod_{j=1}^n N \int_{C_{h,j}}g_0(v) \, \mathrm{d}x \, \mathrm{d}v.\end{aligned}$$ Recalling that $\lambda$ denotes the Lebesgue measure on $\mathcal{MT}$, by it follows, $$\label{eq-lambdabound} \lambda (\tilde{\varphi}^{-1}(C_h)) = \frac{h^{6(n+1)}}{\prod_{j=1}^{n} \big(\varepsilon^2 (v(t_j^-) - v_j)\cdot \nu_j \big)}(1+o(1)).$$ Hence combining and and recalling , $$\begin{aligned} \frac{\hat{P}_t(\tilde{\varphi}^{-1}(C_h))}{\lambda (\tilde{\varphi}^{-1}(C_h))} & \leq \frac{1}{h^6} \int_{C_{h,0}}f_0(x,v) \, \mathrm{d}x \, \mathrm{d}v \nonumber \\ & \qquad \times \prod_{j=1}^n \left( \frac{(v(t_j^-) - v_j)\cdot \nu_j}{h^6(1+o(1))} \int_{C_{h,j}}g_0(v) \, \mathrm{d}x \, \mathrm{d}v \right).\end{aligned}$$ Since $f_0\in L^1(U\times \mathbb{R}^3)$ and $g_0\in L^1(\mathbb{R}^3)$ let $h$ tend to zero and the left hand side, which becomes $\hat{P}_t(\Phi) / \lambda (\Phi)$ in the limit, remains bounded. This completes the proof. We now prove the initial condition requirement on $\hat{P}_t$. \[lem-emp1\] Under the assumptions and set up of theorem \[thm-emp\] we have $$\hat{P}_0(\Phi) = \zeta(\varepsilon) f_0 (x_0,v_0) \mathbbm{1}_{n(\Phi) = 0}.$$ In the case $n(\Phi) > 0$, we have $\hat{P}_0(\Phi)=0$. This is because the tree is free from initial overlap and the tree involves collisions happening at some positive time therefore the collisions cannot have occurred at time 0. Now consider $n(\Phi)=0$. In this situation the tree $\Phi$ contains only the root particle and the probability of finding the root at the given initial data $(x_0,v_0)$ is given by $f_0(x_0,v_0)$. However this must be multiplied by a factor less than one because we must rule out situations that would give initial overlap of the root particle with a background particle. So we calculate the probability that there is no overlap. Firstly, $$\begin{aligned} \mathbb{P}(|x_0 - x_1| > \varepsilon ) & = 1 - \mathbb{P}(|x_0 - x_1| < \varepsilon ) = 1 - \int_{\mathbb{R}^3} \int_{|x_0 - x_1| <\varepsilon} g_0(\bar{v}) \, \mathrm{d}{x_1} \, \mathrm{d}\bar{v} \\ &= 1- \frac{4}{3}\pi \varepsilon^3 \int_{\mathbb{R}^3} g_0(\bar{v}) \, \mathrm{d}\bar{v} = 1 - \frac{4}{3}\pi \varepsilon^3.\end{aligned}$$ Hence, $$\begin{aligned} \mathbb{P}(|x_0 - x_j| > \varepsilon, \forall j = 1 ,\dots ,N ) & = \mathbb{P}(|x_0 - x_1| > \varepsilon ) ^N =( 1 - \frac{4}{3}\pi \varepsilon^3) ^N = \zeta (\varepsilon),\end{aligned}$$ as required. Before we prove the loss term lemma we require a few technical estimates to calculate the rate at which the root particle experiences a collision. \[def-wht\] Let $\Phi \in \mathcal{G}(\varepsilon)$. Define for $h>0$, $$\begin{aligned} W_h(t):= \Big\{ (\bar{x},\bar{v}) & \in U \times \mathbb{R}^3 : \exists (\nu',t') \in \mathbb{S}^2 \times (t,t+h) \\ & \textrm{ such that } x(t')+ \varepsilon \nu' = \bar{x }+t' \bar{v} \textrm{ and } (v(t'^-) - \bar{v})\cdot \nu' >0 \Big\}.\end{aligned}$$ That is $W_h(t)$ is the set of initial points in $U \times \mathbb{R}^3 $ for the background particles that lead to a collision with the root particle of $\Phi$ between the time $t$ and $t+h$. Further define, $$\label{eq-iht} I_h(t):= \frac{\int_{U \times \mathbb{R}^{3}} g_0(\bar{v}) \mathbbm{1}_{W_h(t)} (\bar{x},\bar{v}) \mathbbm{1}_t^\varepsilon[\Phi] (\bar{x},\bar{v}) \, \mathrm{d}\bar{x} \, \mathrm{d}\bar{v} }{\int_{ U \times \mathbb{R}^3} g_0(\bar{v}) \mathbbm{1}_t^\varepsilon[\Phi] (\bar{x},\bar{v}) \, \mathrm{d}\bar{x} \, \mathrm{d}\bar{v} }.$$ From now on assume that the functions $V$ and $M$ in definition \[def-goodtrees\] satisfy, for any $0<\varepsilon<1$, $$\label{eq-vassump2} \varepsilon V(\varepsilon)^3 \leq \frac{1}{8},$$ and, $$\label{eq-massump} M(\varepsilon) \leq \frac{1}{\sqrt[]{\varepsilon}}.$$ \[lem-phwht2coll\] Recall definition \[deff-omega\]. For $\varepsilon$ sufficiently small, $\Phi \in \mathcal{G}(\varepsilon)$ and $t>\tau$, $$\lim_{h \to 0}\frac{1}{h} \hat{P}_{t}(\#(\omega \cap W_h(t))\geq 2 \, | \, \Phi ) =0 .$$ Note that by the inclusion exclusion principle the fact that the background particles are independent, $$\begin{aligned} \label{eq-phatwht2est} \hat{P}_{t}(&\#(\omega \cap W_h(t))\geq 2 \, | \, \Phi ) \leq \sum_{1\leq i < j \leq N-n(\Phi)} \hat{P}_t( (x_i,v_i)\in W_h(t) \textrm{ and } (x_j,v_j) \in W_h(t) \, | \, \Phi) \nonumber \\ & \leq N(N-1) \hat{P}_t( (x_1,v_1)\in W_h(t) \textrm{ and } (x_2,v_2) \in W_h(t) \, | \, \Phi) \nonumber \\ &= N(N-1) \hat{P}_t( (x_1,v_1)\in W_h(t) \, | \, \Phi ) ^2.\end{aligned}$$ Recalling , $$\label{eq-phatwhest} \hat{P}_t( (x_1,v_1)\in W_h(t) \, | \, \Phi )= I_h(t).$$ Now we estimate the right hand side of by estimating the numerator and denominator. Firstly by calculating the volume of the appropriate cylinder, for any $\bar{v} \in \mathbb{R}^3$, $$\label{eq-whtest} \int_{U} \mathbbm{1}_{W_h(t)} (\bar{x},\bar{v}) \, \mathrm{d}\bar{x} \leq \pi \varepsilon^2 \int_t^{t+h} |v(s)-\bar{v}| \, \mathrm{d}s.$$ Define $$\label{eq-beta} \beta := \int_{\mathbb{R}^3} g_0(v)(1+|v|)\,\mathrm{d}v.$$ Note that by assumption , $\beta<\infty$. Since $\Phi \in \mathcal{G}(\varepsilon)$ it follows that $|v(t)| \leq \mathcal{V}(\Phi) \leq V(\varepsilon)$. Using these and we estimate the numerator in , $$\begin{aligned} \label{eq-whtnumest} \int_{U \times \mathbb{R}^{3}} g_0(\bar{v}) \mathbbm{1}_{W_h(t)}& (\bar{x},\bar{v}) \mathbbm{1}_t^\varepsilon[\Phi] (\bar{x},\bar{v}) \, \mathrm{d}\bar{x} \, \mathrm{d}\bar{v} \leq \int_{U \times \mathbb{R}^{3}} g_0(\bar{v}) \mathbbm{1}_{W_h(t)} (\bar{x},\bar{v}) \, \mathrm{d}\bar{x} \, \mathrm{d}\bar{v} \nonumber \\ & \leq \int_{\mathbb{R}^3} g_0(\bar{v}) \pi \varepsilon^2 \int_t^{t+h} |v(s)-\bar{v}| \, \mathrm{d}s \, \mathrm{d}\bar{v} \leq \pi \varepsilon^2 \int_{\mathbb{R}^3} g_0(\bar{v}) \int_t^{t+h} |v(s)| + | \bar{v}| \, \mathrm{d}s \, \mathrm{d}\bar{v} \nonumber \\ & \leq \pi \varepsilon^2 \int_{\mathbb{R}^3} g_0(\bar{v}) \int_t^{t+h} V(\varepsilon) + | \bar{v}| \, \mathrm{d}s \, \mathrm{d}\bar{v} \leq \pi \varepsilon^2 \int_{\mathbb{R}^3} g_0(\bar{v}) h\left(V(\varepsilon) + | \bar{v}| \right) \, \mathrm{d}\bar{v} \nonumber \\ & \leq h \pi \varepsilon^2 \int_{\mathbb{R}^3} g_0(\bar{v}) \left( V(\varepsilon) + | \bar{v}| \right) \, \mathrm{d}\bar{v} \leq h \pi \varepsilon^2 (V(\varepsilon)+\beta).\end{aligned}$$ Turning to the denominator of . Firstly note that, $$\begin{aligned} \label{eq-whtdenom1} \int_{ U \times \mathbb{R}^3} g_0(\bar{v}) \mathbbm{1}_t^\varepsilon[\Phi] (\bar{x},\bar{v}) \, \mathrm{d}\bar{x} \, \mathrm{d}\bar{v} & = \int_{ U \times \mathbb{R}^3} g_0(\bar{v}) \left ( 1 - \mathbbm{1}_{W_t(0)} (\bar{x},\bar{v}) \right) \, \mathrm{d}\bar{x} \, \mathrm{d}\bar{v} \nonumber \\ & = 1- \int_{ U \times \mathbb{R}^3} g_0(\bar{v}) \mathbbm{1}_{W_t(0)} (\bar{x},\bar{v}) \, \mathrm{d}\bar{x} \, \mathrm{d}\bar{v}.\end{aligned}$$ By using , $t\leq T$ and the same estimates from the numerator estimate, $$\begin{aligned} \label{eq-whtdenom2} \int_{ U \times {R}^3} g_0(\bar{v}) \mathbbm{1}_{W_t(0)} & (\bar{x},\bar{v}) \, \mathrm{d}\bar{x} \, \mathrm{d}\bar{v} \nonumber \\ & \leq \pi \varepsilon^2 \int_{\mathbb{R}^3} g_0(\bar{v}) \int_0^{t} |v(s)-\bar{v}| \, \mathrm{d}s \, \mathrm{d}\bar{v} \nonumber \leq \pi \varepsilon^2 \int_{\mathbb{R}^3} g_0(\bar{v}) \int_0^{t} \mathcal{V}(\Phi)+|\bar{v}| \, \mathrm{d}s \, \mathrm{d}\bar{v} \nonumber \\ &\leq \pi \varepsilon^2 \int_{\mathbb{R}^3} g_0(\bar{v}) t \left( V(\varepsilon)+|\bar{v}| \right) \, \mathrm{d}\bar{v} \leq \pi \varepsilon^2 T (V(\varepsilon)+\beta).\end{aligned}$$ Hence for $\varepsilon$ sufficiently small by , $$\pi \varepsilon^2 T (V(\varepsilon)+\beta) \leq 1/2,$$ so by and , $$\label{eq-whtdenomest} \int_{ U \times \mathbb{R}^3} g_0(\bar{v}) \mathbbm{1}_t^\varepsilon[\Phi] (\bar{x},\bar{v}) \, \mathrm{d}\bar{x} \, \mathrm{d}\bar{v} \geq 1/2.$$ Bounds for both the numerator and the denominator of have been found in equations and respectively. Hence, $$\label{eq-ihtbound} I_h(t) \leq 2 h \pi \varepsilon^2 (V(\varepsilon)+\beta).$$ Substituting this into and recalling , $$\begin{aligned} \hat{P}_{t}(\#(\omega \cap W_h(t))\geq 2 \, | \, \Phi ) & \leq N(N-1 ) \times 4 h^2 \pi^2 \varepsilon^4 (V(\varepsilon)+\beta)^2 \\ & \leq 4 h^2 \pi^2 N^2 \varepsilon^4 (V(\varepsilon)+\beta)^2 \leq 4 h^2 \pi^2 (V(\varepsilon)+\beta)^2.\end{aligned}$$ This gives finally that, $$\begin{aligned} \lim_{h \to 0} \frac{1}{h} \left( \hat{P}_{t}(\#(\omega \cap W_h(t))\geq 2 \, | \, \Phi ) \right) & \leq \lim_{h \to 0} \frac{1}{h} \times 4 h^2 \pi^2 (V(\varepsilon)+\beta)^2 \\ & \leq 4 \pi^2 (V(\varepsilon)+\beta)^2 \lim_{h \to 0} h= 0,\end{aligned}$$ completing the proof of the lemma. The previous lemma shows that the rate of seeing two collisions in a short time converges to zero. We now show that the rate of seeing one collision converges to the required loss term. Before we do this we first estimate the error caused by re-collisions. \[def-bht\] For $\Phi \in \mathcal{G}(\varepsilon)$, $t>\tau$ and $h>0$ recall the definition of $W_h(t)$ in definition \[def-wht\] and $\mathbbm{1}_t^\varepsilon[\Phi] (\bar{x},\bar{v})$ . Define, $B_{h,t}(\Phi) \subset U \times \mathbb{R}^3$, $$\begin{aligned} B_{h,t}(\Phi) := \big\{ (\bar{x},\bar{v})\in U \times \mathbb{R}^3 : \mathbbm{1}_t^\varepsilon[\Phi] (\bar{x},\bar{v}) = 0 \textrm{ and } \mathbbm{1}_{W_h(t)} (\bar{x},\bar{v}) =1 \big\}.\end{aligned}$$ Notice that $B_{h,t}(\Phi)$ is the set of all initial positions that a background particle can take such that it collides with the root once during $(0,t)$ and once during $(t,t+h)$. \[lem-intg0bhtbound\] For $\varepsilon$ sufficiently small, $\Phi \in \mathcal{G}(\varepsilon)$, $t>\tau$ and $h>0$ sufficiently small there exists a $\hat{C}_2(\varepsilon)>0$ depending on $t$ and $\Phi$ with $\hat{C}_2(\varepsilon)= o(1)$ as $\varepsilon$ tends to zero such that, $$\int_{B_{h,t}(\Phi)} g_0(\bar{v}) \, \mathrm{d}\bar{x} \, \mathrm{d}\bar{v} = h \varepsilon^2\hat{C}_2(\varepsilon).$$ Recall that $(x(t),v(t))$ is the trajectory of the tagged particle defined by the tree $\Phi$. $B_{h,t}(\Phi)$ is given by, $$\begin{aligned} B_{h,t}(\Phi) = \big\{ (\bar{x}, \bar{v}) : & \, \exists s\in (0,t), \sigma \in (t,t+h), \nu_1,\nu_2 \in \mathbb{S}^2 \textrm{ such that } \\ & \bar{x} + s \bar{v} - x(s) = \varepsilon \nu_1, \, \, \bar{x} + \sigma \bar{v} - x(\sigma) =\varepsilon \nu_2 \textrm{ and } \\ & (v(s)-\bar{v})\cdot \nu_1 >0, \, \, (v(\sigma)-\bar{v})\cdot \nu_2 >0 \big\}.\end{aligned}$$ Define $\delta := \varepsilon^{1/3}$. We split the set $B_{h,t}(\Phi)$ into two parts, the first, denoted $B_{h,t}^\delta(\Phi)$, which considers $s\in(0,t-\delta]$ and the second, denoted $B_{h,t}^2(\Phi)$, which considers $s \in (t-\delta,t)$. We evaluate the bounds on these two sets separately. Consider $s\in (0,t-\delta]$ and $ \sigma\in(t,t+h)$ fixed. The conditions defined by $B_{h,t}$ require that, $$\bar{v} = \frac{x(\sigma)- x(s)}{\sigma - s} + \frac{\varepsilon\nu_2 - \varepsilon \nu_1}{\sigma - s}.$$ For $\sigma$ fixed this implies that $\bar{v}$ is contained in a cylinder of radius $2\varepsilon/\delta$ around the curve defined by $\frac{x(\sigma)- x(s)}{\sigma - s}$ for $s \in (0,t-\delta]$. Recalling the definition of $V(\varepsilon)$ from definition \[def-goodtrees\], taking $h \ll \varepsilon/ V(\varepsilon)$ implies $h|v(\sigma)| \ll \varepsilon$ so the dependence on $\sigma \in (t,t+h)$ gives only a small perturbation around the curve defined by $\frac{x(t)- x(s)}{t - s}$ for $s \in (0,t-\delta]$. Hence $\bar{v}$ is contained in the cylinder with radius $4\varepsilon/\delta$ around the piecewise differentiable curve $r(s):=\frac{x(t)- x(s)}{t - s}$ for $s \in (0,t-\delta]$. Denote this cylinder in $\mathbb{R}^3$ by $E=E(t,\Phi,\delta)$. We seek a bound on the volume of $E$, $|E|$. First consider the length of the curve $r$. For almost all $s \in (0,t-\delta)$, $$\frac{\mathrm{d}}{\mathrm{d}s}r(s )= \frac{x(t)-x(s)}{(t-s)^2} + \frac{v(s)}{t-s} .$$ Hence, $$\begin{aligned} |\frac{\mathrm{d}}{\mathrm{d}s}r(s )| & \leq \frac{|x(t)-x(s)|}{(t-s)^2}+ \frac{|v(s)|}{t-s} \leq \frac{3}{(t-s)^2} + \frac{V(\varepsilon)}{t-s}.\end{aligned}$$ Thus the length of the curve is bounded by, $$\begin{aligned} \int_0^{t-\delta} |\frac{\mathrm{d}}{\mathrm{d}s}r(s )| \, \mathrm{d}s & \leq \int_0^{t-\delta} \frac{3}{(t-s)^2} + \frac{V(\varepsilon)}{t-s} \, \mathrm{d}s = \frac{3}{\delta} - \frac{3}{t} - V(\varepsilon)(\log(\delta) - \log(t)).\end{aligned}$$ Therefore for some $C>0$, $$\label{eq-volumeE} |E| \leq C \left(\frac{\varepsilon}{\delta}\right)^2 \left(\frac{3}{\delta} - \frac{3}{t} - V(\varepsilon)(\log(\delta) - \log(t)) \right).$$ Noting that $x(\sigma) = x(t)+(\sigma-t) v(t)$, for $\bar{v}$ given, $(\bar{x},\bar{v}) \in B_{h,t} (\Phi)$ requires that, $$\begin{aligned} \bar{x} & = x(\sigma) -\sigma\bar{v} + \varepsilon\nu_2 = x(t) + (\sigma -t) - \sigma \bar{v} + \varepsilon\nu_2 \\ & = x(t)-t\bar{v} +(\sigma- t)(v(t)-\bar{v})+\varepsilon\nu_2.\end{aligned}$$ Hence for $\bar{v}$ given $\bar{x}$ is contained in cylinder of radius $\varepsilon$ and length $h|v(t)-\bar{v}|$. Denote this cylinder by $C(\bar{v})$. By or , for constants $C$ that change on each line, $$\begin{aligned} \int_{B_{h,t}^\delta(\Phi) }g(\bar{v}) \, \mathrm{d}\bar{x} \, \mathrm{d}\bar{v} & \leq \int_E g(\bar{v}) \int_{C(\bar{v})} \, \mathrm{d}\bar{x} \, \mathrm{d}\bar{v} \leq C \varepsilon^2 h \int_E g(\bar{v}) |v(t)-\bar{v}| \, \mathrm{d}\bar{v} \\ & \leq C \varepsilon^2 h \int_E g(\bar{v}) (V(\varepsilon)+ |\bar{v}|) \, \mathrm{d}\bar{v} \leq C \varepsilon^2 h (V(\varepsilon)+1 ) |E|. \end{aligned}$$ It remains to show that $(V(\varepsilon)+1)|E|$ is $o(1)$ as $\varepsilon$ tends to zero. Recall , that $\delta = \varepsilon^{1/3}$ and , $$\begin{aligned} (V(\varepsilon)+1)|E| & = (V(\varepsilon)+1) \times C \left(\frac{\varepsilon}{\delta}\right)^2 \left(\frac{3}{\delta} - \frac{3}{t} - V(\varepsilon)(\log(\delta) - \log(t)) \right) \\ & \leq C\left(\frac{1}{2\varepsilon^{1/3}}+1\right)\varepsilon^{4/3} \left(\frac{3}{\varepsilon^{1/3}} +\frac{1}{2\varepsilon^{1/3}}(|\log\varepsilon^{1/3}|+|\log t|)\right) \\ & \leq C \left(\frac{1}{2\varepsilon^{1/3}}+1\right) \left(3\varepsilon +\frac{1}{2}\varepsilon\left(\frac{1}{3}|\log\varepsilon|+|\log t|\right)\right)\end{aligned}$$ as required. Now consider the second part of $B_{h,t}(\Phi)$ for $s \in (t-\delta,t)$ denoted $B_{h,t}^2(\Phi)$. Since $\Phi$ is fixed, $t > \tau$ and $\delta = \varepsilon^{1/3}$ let $\varepsilon$ sufficiently small such that $t-\delta >\tau$. Hence for $s \in (t-\delta,t)$, $v(s)=v(t)$. We change the velocity space coordinates so that $v(t)=0$. If we require that a particle starting at $(\bar{x},\bar{v})$ collides with the tagged particle in $(t-\delta,t)$ and again in $(t,t+h)$ we require in the new coordinates that either $\bar{v}=0$ or that $|\bar{v}|$ is sufficiently large so that the background particle wraps round the torus having travelled at least distance $3/4$ (for $\varepsilon$ sufficiently small) within time $(\delta +h)$ to re-collide with the tagged particle. That is, $$|\bar{v}| \geq \frac{3}{4(\delta + h)} .$$ For $h \leq 1/4 \delta $, this implies $|\bar{v}| \geq \frac{3}{5\delta}$. Changing back to the original coordinates, this means it is required that $\bar{v}=v(t)$ or $|\bar{v}-v(t)| \geq 3/5\delta$. For a given $\bar{v}$, the same conditions as before on the $\bar{x}$ coordinate must hold and so $\bar{x}$ is in the cylinder $C(\bar{v})$. Recalling , , $\delta=\varepsilon^{1/3}$ and that $|v(t)| \leq V(\varepsilon) \leq 1/2\varepsilon^{-1/3} = 1/2\delta$ it follows for constants $C$ that change on each line, $$\begin{aligned} \int_{B_{h,t}^2(\Phi)} g_0(\bar{v}) \, &\mathrm{d}\bar{x} \, \mathrm{d}\bar{v} \leq \int_{\mathbb{R}^3 \setminus B_{3/5\delta}(v(t))} g_0(\bar{v}) \int_{C(\bar{v})} \, \mathrm{d}\bar{x} \, \mathrm{d}\bar{v} \leq C\varepsilon^2h \int_{\mathbb{R}^3 \setminus B_{3/5\delta}(v(t))} g_0(\bar{v})|v(t)-\bar{v}| \, \mathrm{d}\bar{v} \\ & \leq C\varepsilon^2h \int_{\mathbb{R}^3 \setminus B_{3/5\delta}(v(t))} g_0(\bar{v})(V(\varepsilon)+|\bar{v}|) \, \mathrm{d}\bar{v} \leq C\varepsilon^2h \int_{\mathbb{R}^3 \setminus B_{1/10\delta}(0)} g_0(\bar{v})(V(\varepsilon)+|\bar{v}|) \, \mathrm{d}\bar{v} \\ & \leq C\varepsilon^2h \int_{\mathbb{R}^3 \setminus B_{1/10\delta}(0)} g_0(\bar{v})(100\delta^2 |\bar{v}|^2 V(\varepsilon)+10\delta|\bar{v}|^2) \, \mathrm{d}\bar{v} \leq C\varepsilon^2h (10\delta^2 V(\varepsilon)+\delta )\\ & \leq C\varepsilon^2 h \left(5 \frac{\varepsilon^{2/3}}{\varepsilon^{1/3}}+\varepsilon^{1/3} \right) = C\varepsilon^2 h \times \varepsilon^{1/3},\end{aligned}$$ as required. Since $B_{h,t}(\Phi) = B_{h,t}^\delta(\Phi) \cup B_{h,t}^2(\Phi)$ the proof of the lemma is complete. \[lem-phwht1coll\] For $\varepsilon$ sufficiently small, $\Phi \in \mathcal{G}(\varepsilon)$, $t>\tau$ and $\hat{C}_2(\varepsilon)$ as in the above lemma, $$\begin{aligned} \lim_{h \to 0}\frac{1}{h} \hat{P}_{t} & (\#(\omega \cap W_h(t)) = 1 \, | \, \Phi ) \\ & =(1- \gamma(t)) \frac{\int_{\mathbb{S}^{2}} \int_{\mathbb{R}^3} g_0(\bar{v})[(v(t)-\bar{v})\cdot \nu]_+ \, \mathrm{d}\bar{v} \, \mathrm{d}\nu - \hat{C}_2(\varepsilon) }{\int_{ U \times {R}^3} g_0(\bar{v}) \mathbbm{1}_t^\varepsilon[\Phi] (\bar{x},\bar{v}) \, \mathrm{d}\bar{x} \, \mathrm{d}\bar{v} }.\end{aligned}$$ Since the initial data for each background particle is independent of the other background particles, $$\begin{aligned} \label{eq-omegawht1} \hat{P}_{t}(\#(\omega \cap & W_h(t)) = 1 \, | \, \Phi ) =\sum_{i=1}^{N-n(\Phi)} \hat{P}_t \Big( (x_i,v_i) \in W_h(t) \textrm{ and } (x_1,v_1), \dots, \nonumber \\ & \qquad \qquad (x_{i-1},v_{i-1}),(x_{i+1},v_{i+1}), \dots, (x_{N-n(\Phi)},v_{N-n(\Phi)}) \notin W_h(t) \, | \, \Phi \Big) \nonumber \\ & = (N-n(\Phi))\hat{P}_t((x_1,v_1) \in W_h(t) \, | \, \Phi)\hat{P}_t((x_2,v_2) \notin W_h(t) \, | \, \Phi)^{N-n(\Phi) -1}\nonumber \\ & = (N-n(\Phi))I_h(t) \left( 1-I_h(t) \right)^{N-n(\Phi) -1} \nonumber \\& = (N-n(\Phi))I_h(t) \sum_{j=0}^{N-n(\Phi)-1} (-1)^{j} \binom{N-n(\Phi)-1}{j} I_h(t)^j \nonumber \\ & = (N-n(\Phi))\sum_{j=0}^{N-n(\Phi)-1} (-1)^{j} \binom{N-n(\Phi)-1}{j} I_h(t)^{j+1}.\end{aligned}$$ By , $$\lim_{h \to 0} \frac{1}{h}I_h(t)^2 = 0.$$ Hence dividing by $h$ and taking $h$ to zero we see that all terms in the sum for $j\geq 1$ tend to zero, leaving only the contribution from the term $j=0$. Hence, $$\begin{aligned} \label{eq-wht1calc} \lim_{h \to 0} \frac{1}{h} \hat{P}_{t} & (\#(\omega \cap W_h(t)) = 1 \, | \, \Phi ) \nonumber \\ & = \lim_{h \to 0} \frac{1}{h} (N-n(\Phi))\sum_{j=0}^{N-n(\Phi)-1} (-1)^{j} \binom{N-n(\Phi)-1}{j} I_h(t)^{j+1} = \lim_{h \to 0} \frac{1}{h} (N-n(\Phi)) I_h(t) .\end{aligned}$$ It remains to investigate, $$\lim_{h \to 0} \frac{1}{h} I_h(t) = \lim_{h \to 0} \frac{1}{h} \frac{\int_{U \times \mathbb{R}^{3}} g_0(\bar{v}) \mathbbm{1}_{W_h(t)} (\bar{x},\bar{v}) \mathbbm{1}_t^\varepsilon[\Phi] (\bar{x},\bar{v}) \, \mathrm{d}\bar{x} \, \mathrm{d}\bar{v} }{\int_{ U \times \mathbb{R}^3} g_0(\bar{v}) \mathbbm{1}_t^\varepsilon[\Phi] (\bar{x},\bar{v}) \, \mathrm{d}\bar{x} \, \mathrm{d}\bar{v} }.$$ For $B_{h,t}(\Phi)$ as defined in definition \[def-bht\] we have, $$\begin{aligned} \int_{U \times \mathbb{R}^{3}} g_0(\bar{v}) & \mathbbm{1}_{W_h(t)} (\bar{x},\bar{v}) \mathbbm{1}_t^\varepsilon[\Phi] (\bar{x},\bar{v}) \, \mathrm{d}\bar{x} \, \mathrm{d}\bar{v} \\ & = \int_{U \times \mathbb{R}^{3}} g_0(\bar{v}) \mathbbm{1}_{W_h(t)} (\bar{x},\bar{v}) \, \mathrm{d}\bar{v} - \int_{B_{h,t}(\Phi)} g_0(\bar{v}) \, \mathrm{d}\bar{x} \, \mathrm{d}\bar{v}\end{aligned}$$ It then follows from lemma \[lem-intg0bhtbound\], $$\begin{aligned} \lim_{h \to 0} \frac{1}{h} I_h(t) & = \lim_{h \to 0} \frac{1}{h} \frac{\int_{U \times \mathbb{R}^{3}} g_0(\bar{v}) \mathbbm{1}_{W_h(t)} (\bar{x},\bar{v}) \mathbbm{1}_t^\varepsilon[\Phi] (\bar{x},\bar{v}) \, \mathrm{d}\bar{x} \, \mathrm{d}\bar{v} }{\int_{ U \times \mathbb{R}^3} g_0(\bar{v}) \mathbbm{1}_t^\varepsilon[\Phi] (\bar{x},\bar{v}) \, \mathrm{d}\bar{x} \, \mathrm{d}\bar{v} } \\ & = \lim_{h \to 0} \frac{1}{h} \left( \frac{h \varepsilon^2 \int_{\mathbb{S}^2} \int_{\mathbb{R}^{3}} g_0(\bar{v}) [(v(t)-\bar{v})\cdot \nu ]_+ \, \mathrm{d}\bar{v} \, \mathrm{d}\nu -h\varepsilon^2\hat{C}_2(\varepsilon)}{\int_{ U \times \mathbb{R}^3} g_0(\bar{v}) \mathbbm{1}_t^\varepsilon[\Phi] (\bar{x},\bar{v}) \, \mathrm{d}\bar{x} \, \mathrm{d}\bar{v}} \right) \\ & = \varepsilon^2 \frac{ \int_{\mathbb{S}^2} \int_{\mathbb{R}^{3}} g_0(\bar{v}) [(v(t)-\bar{v})\cdot \nu ]_+ \, \mathrm{d}\bar{v} \, \mathrm{d}\nu - \hat{C}_2(\varepsilon) }{\int_{ U \times \mathbb{R}^3} g_0(\bar{v}) \mathbbm{1}_t^\varepsilon[\Phi] (\bar{x},\bar{v}) \, \mathrm{d}\bar{x} \, \mathrm{d}\bar{v}}.\end{aligned}$$ Substituting this into , $$\begin{aligned} \lim_{h \to 0} \frac{1}{h} \hat{P}_{t} & (\#(\omega \cap W_h(t)) = 1 \, | \, \Phi ) = \lim_{h \to 0} \frac{1}{h} (N-n(\Phi)) I_h(t) \\ & = (N-n(\Phi))\varepsilon^2 \frac{ \int_{\mathbb{S}^2} \int_{\mathbb{R}^{3}} g_0(\bar{v}) [(v(t)-\bar{v})\cdot \nu ]_+ \, \mathrm{d}\bar{v} \, \mathrm{d}\nu - \hat{C}_2(\varepsilon) }{\int_{ U \times \mathbb{R}^3} g_0(\bar{v}) \mathbbm{1}_t^\varepsilon[\Phi] (\bar{x},\bar{v}) \, \mathrm{d}\bar{x} \, \mathrm{d}\bar{v}} \\ & = (1-\gamma(t))\frac{ \int_{\mathbb{S}^2} \int_{\mathbb{R}^{3}} g_0(\bar{v}) [(v(t)-\bar{v})\cdot \nu ]_+ \, \mathrm{d}\bar{v} \, \mathrm{d}\nu- \hat{C}_2(\varepsilon) }{\int_{ U \times \mathbb{R}^3} g_0(\bar{v}) \mathbbm{1}_t^\varepsilon[\Phi] (\bar{x},\bar{v}) \, \mathrm{d}\bar{x} \, \mathrm{d}\bar{v}},\end{aligned}$$ which proves the lemma. With these estimates we can now prove that the loss term of holds. \[lem-emp2\] Under the assumptions and set up of theorem \[thm-emp\], for $t > \tau$, $$\partial_t \hat{P}_t(\Phi) = (1-\gamma(t))\hat{\mathcal{Q}}_t^- [\hat{P}_t] (\Phi).$$ We calculate, for $t>\tau$, $$\partial_t \hat{P}_t (\Phi) = \lim_{h \to 0} \frac{1}{h} \left(\hat{P}_{t+h}(\Phi) - \hat{P}_t(\Phi) \right).$$ Noting that, $$\hat{P}_{t+h}(\Phi) = \left(1-\hat{P}_{t}\left(\#(\omega \cap W_h(t))>0 \, | \, \Phi \right) \right) \hat{P}_t(\Phi).$$ It follows, $$\frac{1}{h}\left( \hat{P}_{t+h}(\Phi) - \hat{P}_t(\Phi)\right) = -\frac{1}{h}\hat{P}_{t}(\#(\omega \cap W_h(t))>0 \, | \, \Phi ) \hat{P}_t(\Phi).$$ Using lemma \[lem-phwht2coll\] and lemma \[lem-phwht1coll\], $$\begin{aligned} \partial_t \hat{P}_t (\Phi) & = \lim_{h \to 0} \frac{1}{h} \left(\hat{P}_{t+h}(\Phi) - \hat{P}_t(\Phi)\right) \\ & = - \lim_{h \to 0} \frac{1}{h}\hat{P}_{t}(\#(\omega \cap W_h(t))>0 \, | \, \Phi ) \hat{P}_t(\Phi) = -\hat{P}_t(\Phi) \lim_{h \to 0} \frac{1}{h}\hat{P}_{t}(\#(\omega \cap W_h(t))=1 \, | \, \Phi ) \\ &= -\hat{P}_t(\Phi) (1- \gamma(t)) \frac{\int_{\mathbb{S}^{2}} \int_{\mathbb{R}^3} g_0(\bar{v})[(v(t)-\bar{v})\cdot \nu]_+ \, \mathrm{d}\bar{v} \, \mathrm{d}\nu - \hat{C}_2(\varepsilon) }{\int_{ U \times {R}^3} g_0(\bar{v}) \mathbbm{1}_t^\varepsilon[\Phi] (\bar{x},\bar{v}) \, \mathrm{d}\bar{x} \, \mathrm{d}\bar{v} } = (1- \gamma(t))\hat{\mathcal{Q}}_t^- [\hat{P}_t] (\Phi),\end{aligned}$$ which proves the lemma. We next move to proving the gain term in . \[lem-emp3\] Under the assumptions and set up of theorem \[thm-emp\], for $n(\Phi)\geq 1$ $$\hat{P}_\tau (\Phi) =(1-\gamma(\tau))\hat{P}_\tau(\bar{\Phi}) \frac{g_0(v')[(v(\tau^-)-v')\cdot \nu ]_+}{1- \pi \varepsilon^2 \int_0^\tau |v(s)-v'| \, \mathrm{d}s + \hat{C}_1\varepsilon^3}.$$ For some $\hat{C}_1>0$ depending only on $\Phi$. Firstly, $$\begin{aligned} \hat{P}_\tau(\Phi) & = \hat{P}_\tau(\Phi \cap \bar{\Phi}) = \hat{P}_\tau(\Phi \, |\, \bar{\Phi}) \hat{P}_\tau( \bar{\Phi}) .\end{aligned}$$ It remains to show that $$\label{eq-phatgain} \hat{P}_\tau(\Phi \, |\, \bar{\Phi})=(1-\gamma(\tau)) \frac{g_0(v')[(v(\tau^-)-v')\cdot \nu ]_+}{1- \pi \varepsilon^2 \int_0^\tau |v(s)-v'| \, \mathrm{d}s + \hat{C}_1\varepsilon^3}.$$ We do this by proving upper and lower bounds. For $h \geq 0$ define, $$U_h:= \{ \Psi \in \mathcal{MT} : \bar{\Psi}_{\tau(\Phi)} = \bar{\Phi}_{\tau(\Phi)} \textrm{ and } \Psi \in B_h(\Phi) \}.$$ Note that $U_0:= \{ \Phi \}$. Then by lemma \[lem-phatabscts\], $$\label{eq-phatlim} \hat{P}_\tau (\Phi \, | \, \bar{\Phi}) = \lim_{h \to 0} h^{-6}\hat{P}_\tau (U_h \, | \, \bar{\Phi}).$$ For $\Psi \in U_h$ define $V_h(\Psi) \in U \times \mathbb{R}^3$ to be the initial position of the background particle that leads to the final collision of $\Psi$ and define $V_h \subset U \times \mathbb{R}^3$ by, $$V_h = \cup_{\Psi \in U_h} V_h(\Psi).$$ Note that $V_0 = \{ (x(\tau)+\varepsilon \nu - \tau v',v') \}$ that is $V_0$ contains only the initial point of the background particle that gives the final collision in $\Phi$. Then by a change of coordinates, recalling , $$\begin{aligned} \label{eq-phatuhlower} \hat{P}_\tau(U_h \, | \, \bar{\Phi}) & \leq \sum_{i=1}^{N-(n(\Phi)-1)} \hat{P}_\tau ((x_i,v_i) \in V_h \, | \, \bar{\Phi} _\tau) \varepsilon^2 [(v(\tau^-) - v')\cdot \nu]_+ \nonumber \\ & = (N-(n(\Phi)-1))\varepsilon ^2\hat{P}_\tau ((x_1,v_1) \in V_h \, | \, \bar{\Phi} ) [(v(\tau^-) - v')\cdot \nu]_+ \nonumber \\ & =(1- \gamma(\tau) )\hat{P}_\tau ((x_1,v_1) \in V_h \, | \, \bar{\Phi} ) [(v(\tau^-) - v')\cdot \nu]_+.\end{aligned}$$ By absolute continuity of $\hat{P}_\tau$, $$\lim_{h \to 0} h^{-6}\hat{P}_\tau ((x_1,v_1) \in V_h \, | \, \bar{\Phi}) = \hat{P}_\tau ((x_1,v_1) \in V_0 \, | \, \bar{\Phi}).$$ Combining these into , $$\begin{aligned} \label{eq-phatgivenlower} \hat{P}_\tau (\Phi \, | \, \bar{\Phi}) &= \lim_{h \to 0} h^{-6}\hat{P}_\tau (U_h \, | \, \bar{\Phi}) \nonumber \\ & \leq \lim_{h \to 0} h^{-6} (1- \gamma(\tau)) \hat{P}_\tau ((x_1,v_1) \in V_h \, | \, \bar{\Phi} ) [(v(\tau^-) - v')\cdot \nu]_+\nonumber \\ & = (1- \gamma(\tau)) \hat{P}_\tau ((x_1,v_1) \in V_0 \, | \, \bar{\Phi} _\tau) [(v(\tau^-) - v')\cdot \nu]_+.\end{aligned}$$ Next consider the lower bound. By the inclusion-exclusion principle, $$\begin{aligned} \label{eq-phatuhupper} \hat{P} _\tau (U_h \, | \, \bar{\Phi}) & \geq \sum_{i=1}^{N-(n(\Phi)-1)} \hat{P}_\tau ((x_i,v_i) \in V_h \, | \, \bar{\Phi}) \varepsilon^2 [(v(\tau^-) - v')\cdot \nu]_+ \nonumber \\ & \quad - \sum_{1 \leq i < j \leq N-(n(\Phi)-1)} \hat{P}_\tau ((x_i,v_i), (x_j,v_j) \in V_h \, | \, \bar{\Phi}) \varepsilon^2 [(v(\tau^-) - v')\cdot \nu]_+ .\end{aligned}$$ As in it follows, $$\begin{aligned} \sum_{i=1}^{N-(n(\Phi)-1)} \hat{P}_\tau & ((x_i,v_i) \in V_h \, | \, \bar{\Phi}) \varepsilon^2 [(v(\tau^-) - v')\cdot \nu]_+ \\ & = (1-\gamma(\tau))\hat{P}_\tau ((x_1,v_1) \in V_h \, | \, \bar{\Phi}) [(v(\tau^-) - v')\cdot \nu]_+.\end{aligned}$$ Further, $$\begin{aligned} \sum_{1 \leq i < j \leq N-(n(\Phi)-1)} \hat{P}_\tau & ((x_i,v_i), (x_j,v_j) \in V_h \, | \, \bar{\Phi}) \varepsilon^2 [(v(\tau^-) - v')\cdot \nu]_+ \nonumber \\ & \leq N(N-1) \hat{P}_\tau ((x_1,v_1), (x_2,v_2) \in V_h \, | \, \bar{\Phi}) \varepsilon^2 [(v(\tau^-) - v')\cdot \nu]_+ \nonumber \\ & = (N-1)\hat{P}_\tau ((x_1,v_1) \in V_h \, | \, \bar{\Phi})^2 [(v(\tau^-) - v')\cdot \nu]_+.\end{aligned}$$ By the absolute continuity of $\hat{P}_t$ this implies, $$\begin{aligned} \lim_{h \to 0} \frac{1}{h^6} \sum_{1 \leq i < j \leq N-(n(\Phi)-1)} \hat{P}_\tau & ((x_i,v_i), (x_j,v_j) \in V_h \, | \, \bar{\Phi}) \varepsilon^2 [(v(\tau^-) - v')\cdot \nu]_+ =0.\end{aligned}$$ Hence by , $$\begin{aligned} \label{eq-phatgivenupper} \hat{P}_\tau (\Phi \, | \, \bar{\Phi}) &= \lim_{h \to 0} h^{-6}\hat{P}_\tau (U_h \, | \, \bar{\Phi}) \nonumber \\ & \geq \lim_{h \to 0} h^{-6} (1-\gamma(\tau)) \hat{P}_\tau ((x_1,v_1) \in V_h \, | \, \bar{\Phi}) [(v(\tau^-) - v')\cdot \nu]_+ \nonumber \\ & = (1-\gamma(\tau)) \hat{P}_\tau ((x_1,v_1) \in V_0 \, | \, \bar{\Phi}) [(v(\tau^-) - v')\cdot \nu]_+.\end{aligned}$$ Recalling that we need to prove to prove the lemma, we see that with and we now need only to show that, $$\hat{P}_\tau ((x_1,v_1) \in V_0 \, | \, \bar{\Phi}) = \frac{g_0(v')}{1- \pi \varepsilon^2 \int_0^\tau |v(s)-v'| \, \mathrm{d}s + \hat{C}_1\varepsilon^3}.$$ That is, $$\label{eq-phatx1v1} \hat{P}_\tau \left( (x_1,v_1)=(x(\tau)+\varepsilon\nu - \tau v',v')\, | \, \bar{\Phi} \right) = \frac{g_0(v')}{1- \pi \varepsilon^2 \int_0^\tau |v(s)-v'| \, \mathrm{d}s + \hat{C}_1\varepsilon^3}.$$ Since we are conditioning on $\bar{\Phi}$ occurring there is a region of $U$ that we must rule out for the initial position of the background particle if we know that is has velocity $v'$. That is to say there is a region of $U$ where we know the background particle cannot have started with initial velocity $v'$ because if it did it would have interfered with $\bar{\Phi}$. Denote this region of $U$ by $\Delta$. To calculate the volume of $\Delta$ we imagine it as cylinders to get, for fixed incoming velocity $v'$, $$\label{eq-voldelta} |\Delta| = \pi \varepsilon^2 \int_0^\tau |v(s)-v'| \, \mathrm{d}s - \hat{C}_1\varepsilon^3,$$ where the first term calculates the volume of each cylinder as the root particle changes direction and the second term subtracts the over-estimate error caused counting certain parts twice as the particle changes direction. ![In the case $v'=0$ we are calculating the volume of $\Delta$, since we know the background particle cannot start in $\Delta$. For $v'\neq 0$ the cylinders get shifted but the principle is the same. (Diagram not to scale)[]{data-label="fig-delta"}](delta) Therefore, recalling that $|U|=1$, $$|U \setminus \Delta| = 1- \left( \pi \varepsilon^2 \int_0^\tau |v(s)-v'| \, \mathrm{d}s -\hat{C}_1\varepsilon^3\right),$$ and this together with the fact that the velocity of the background particle has initial distribution $g_0$ gives the required . Combining lemmas \[lem-emp1\], \[lem-emp2\] and \[lem-emp3\] gives the required proof of the theorem. Convergence {#sec:conv} =========== Having proven the existence of the idealized distribution $P_t$ and shown that the empirical distribution $\hat{P}_t$ solves the appropriate equation, we seek to show the convergence results that will help prove our main result. Following [@matt12], the idea is to establish a differential inequality in . In combination with the fact that $P_t$ is a probability measure and that $\lim_{\varepsilon \to 0}P_t(\mathcal{G}(\varepsilon))=1$ in proposition \[prop-goodtreefull\] the inequality delivers the convergence result theorem \[thm-ptphatcomp\]. The main theorem \[thm-main\] is a direct consequence. We first introduce some notation. Recall and . For $\varepsilon>0$, $\Phi \in \mathcal{G}(\varepsilon)$, $t \in [0,T]$, define the following, $$\begin{aligned} \eta_t^\varepsilon(\Phi) & := \int_{U \times \mathbb{R}^3} g_0(\bar{v})(1- \mathbbm{1}_t^\varepsilon[\Phi](\bar{x},\bar{v})) \, \mathrm{d}\bar{x} \, \mathrm{d}\bar{v}, \nonumber \\ R_t^\varepsilon(\Phi) &: = \zeta(\varepsilon)P_t(\Phi), \nonumber \\ L(\Phi) &: = -\int_{\mathbb{S}^{2}} \int_{\mathbb{R}^3} g_0(\bar{v})[(v(\tau)-\bar{v})\cdot \nu ]_+ \, \mathrm{d}\bar{v} \, \mathrm{d}\nu, \nonumber \\ C(\Phi)& := 2 \sup_{t \in [0,T]} \left\{ \int_{\mathbb{S}^{2}} \int_{\mathbb{R}^3} g_0(\bar{v})[(v(t)-\bar{v})\cdot \nu ]_+ \right\} \label{eq-cdeff} \\ \rho^{\varepsilon,0}_t (\Phi) &: = \eta_t^\varepsilon(\Phi)C(\Phi)t . \nonumber\end{aligned}$$ Further for $k \geq 1$ define, $$\rho^{\varepsilon,k}_t (\Phi) := (1-\varepsilon)\rho^{\varepsilon,k-1}_t (\Phi) + \rho^{\varepsilon,0}_t (\Phi) + \varepsilon.$$ Note that this implies that for $k \geq 1$, $$\label{eq-rhoalt} \rho^{\varepsilon,k}_t(\Phi) =(1-\varepsilon)^k \rho^{\varepsilon,0}_t (\Phi) + (\rho^{\varepsilon,0}_t (\Phi) + \varepsilon) \sum_{j=1}^k(1-\varepsilon)^{k-j}.$$ Finally define, $$\hat{\rho}^\varepsilon_t(\Phi) := \rho^{\varepsilon,n(\Phi)}_t(\Phi).$$ \[prop-ptphatcomp\] For $\varepsilon$ sufficiently small, $\Phi \in \mathcal{G}(\varepsilon)$ and $t\in [0,T]$, $$\label{eq-phatrcompeq} \hat{P}^\varepsilon_t(\Phi)-R^\varepsilon_t(\Phi) \geq - \hat{\rho}^\varepsilon_t(\Phi)R_t^\varepsilon(\Phi).$$ To prove this proposition we use a number of lemmas. \[lem-pphatcomphelp\] For $\Phi \in \mathcal{G}(\varepsilon)$ and $t \geq \tau$, $$\begin{aligned} \hat{P}_t^\varepsilon(\Phi) - R_t^\varepsilon(\Phi) & \geq \exp\left(L(\Phi) \int_\tau^t (1+2\eta_s^\varepsilon(\Phi)) \, \mathrm{d}s \right)(\hat{P}_\tau^\varepsilon(\Phi) - R_\tau^\varepsilon(\Phi)) \\ & \qquad + 2\eta_t^\varepsilon(\Phi) L(\Phi) R_t^\varepsilon(\Phi) \int_\tau^t \exp \left(2\eta_s^\varepsilon(\Phi) (t-s) L(\Phi) \right) \, \mathrm{d}s.\end{aligned}$$ For $t=\tau$ the result holds trivially. For $t >\tau $, by theorem \[thm-ideq\] and theorem \[thm-emp\], $$\label{eq-rtptcomp} \partial_t \left( \hat{P}_t^\varepsilon(\Phi) -R_t^\varepsilon(\Phi) \right) = (1-\gamma(t))\hat{L}_t(\Phi)\hat{P}_t(\Phi) -L(\Phi)R_t^\varepsilon(\Phi),$$ where $$\hat{L}_t(\Phi) := - \left( \frac{\int_{\mathbb{S}^{2}} \int_{\mathbb{R}^3} g_0(\bar{v})[(v(\tau)-\bar{v})\cdot \nu]_+ \, \mathrm{d}\bar{v} \, \mathrm{d}\nu - \hat{C}_2(\varepsilon)}{\int_{ U \times \mathbb{R}^3} g_0(\bar{v}) \mathbbm{1}_t^\varepsilon[\Phi] (\bar{x},\bar{v}) \, \mathrm{d}\bar{x} \, \mathrm{d}\bar{v} } \right).$$ Recalling and , for $\varepsilon$ sufficiently small, $$\begin{aligned} \label{eq-etabound} \int_{U \times \mathbb{R}^3} g_0(\bar{v})(1 & - \mathbbm{1}_t^\varepsilon[\Phi](\bar{x},\bar{v})) \, \mathrm{d}\bar{x} \, \mathrm{d}\bar{v} \leq \int_{\mathbb{R}^3} g_0(\bar{v})\left(\pi \varepsilon^2 \int_0^t |v(s)-\bar{v}| \, \mathrm{d}s - \hat{C}_1\varepsilon^3 \right) \, \mathrm{d}\bar{v} \nonumber \\ &\leq \int_{\mathbb{R}^3} g_0(\bar{v})\left(\pi \varepsilon^2 T(\mathcal{V}(\Phi)+ | \bar{v}|) - \hat{C}_1\varepsilon^3 \right) \, \mathrm{d}\bar{v} \leq \int_{\mathbb{R}^3} g_0(\bar{v})\left(\pi \varepsilon^2 T(V(\varepsilon)+ | \bar{v}|) \right) \, \mathrm{d}\bar{v} \nonumber \\ & \leq \pi \varepsilon^2 T \int_{\mathbb{R}^3} g_0(\bar{v})\left(V(\varepsilon)+ | \bar{v}| \right) \, \mathrm{d}\bar{v} \leq \pi \varepsilon^2 T (V(\varepsilon)+\beta) < 1/2.\end{aligned}$$ Noting that for $0\leq z \leq 1/2$, $\sum_{i=0}^\infty z^i \leq 2$ which gives, $$\begin{aligned} \frac{1}{1-z} & = \sum_{i=0}^\infty z^i = 1 + z\left( \sum_{i=0}^\infty z^i \right) \leq 1+ 2z.\end{aligned}$$ It follows, $$\begin{aligned} \frac{1}{\int_{ U \times \mathbb{R}^3} g_0(\bar{v}) \mathbbm{1}_t^\varepsilon[\Phi] (\bar{x},\bar{v}) \, \mathrm{d}\bar{x} \, \mathrm{d}\bar{v}} & = \frac{1}{1- \int_{ U \times \mathbb{R}^3} g_0(\bar{v}) (1-\mathbbm{1}_t^\varepsilon[\Phi] (\bar{x},\bar{v})) \, \mathrm{d}\bar{x} \, \mathrm{d}\bar{v}} \nonumber \\ & \leq 1+2\left( \int_{ U \times \mathbb{R}^3} g_0(\bar{v}) (1-\mathbbm{1}_t^\varepsilon[\Phi] (\bar{x},\bar{v})) \, \mathrm{d}\bar{x} \, \mathrm{d}\bar{v} \right) = 1+2\eta_t^\varepsilon(\Phi).\end{aligned}$$ This gives that, $$\begin{aligned} (1-\gamma(t)) & \left( \frac{\int_{\mathbb{S}^{2}} \int_{\mathbb{R}^3} g_0(\bar{v})[(v(\tau)-\bar{v})\cdot \nu]_+ \, \mathrm{d}\bar{v} \, \mathrm{d}\nu - \hat{C}_2(\varepsilon)}{\int_{ U \times \mathbb{R}^3} g_0(\bar{v}) \mathbbm{1}_t^\varepsilon[\Phi] (\bar{x},\bar{v}) \, \mathrm{d}\bar{x} \, \mathrm{d}\bar{v}} \right) \\ & \leq \frac{\int_{\mathbb{S}^{2}} \int_{\mathbb{R}^3} g_0(\bar{v})[(v(\tau)-\bar{v})\cdot \nu]_+ \, \mathrm{d}\bar{v} \, \mathrm{d}\nu }{\int_{ U \times \mathbb{R}^3} g_0(\bar{v}) \mathbbm{1}_t^\varepsilon[\Phi] (\bar{x},\bar{v}) \, \mathrm{d}\bar{x} \, \mathrm{d}\bar{v}} \\ & \leq (1+ 2\eta_t^\varepsilon(\Phi)) \int_{\mathbb{S}^{2}} \int_{\mathbb{R}^3} g_0(\bar{v})[(v(\tau)-\bar{v})\cdot \nu]_+ \, \mathrm{d}\bar{v} \, \mathrm{d}\nu.\end{aligned}$$ Finally giving that, $$(1-\gamma(t))\hat{L}_t(\Phi) \geq (1+2\eta_t^\varepsilon(\Phi))L(\Phi).$$ Substituting this into , $$\begin{aligned} \label{eq-phatrode} \partial_t \left( \hat{P}_t^\varepsilon(\Phi) -R_t^\varepsilon(\Phi) \right) & = (1-\gamma(t))\hat{L}_t(\Phi)\hat{P}_t(\Phi) -L(\Phi)R_t^\varepsilon(\Phi)\nonumber \\ & \geq (1+2\eta_t^\varepsilon(\Phi))L(\Phi)\hat{P}_t(\Phi) -L(\Phi)R_t^\varepsilon(\Phi)\nonumber \\ & = (1+2\eta_t^\varepsilon(\Phi))L(\Phi)\left(\hat{P}_t(\Phi) - R_t^\varepsilon(\Phi) \right) +2\eta_t^\varepsilon(\Phi)L(\Phi) R_t^\varepsilon(\Phi).\end{aligned}$$ For fixed $\Phi$ this is simply a $1d$ ODE in $t$. If $y:[\tau,\infty) \to \mathbb{R}$ satisfies, $$\begin{cases} \frac{d}{dt}y(t) \geq a(t)y(t)+b(t), \\ y(\tau) = y_0. \end{cases}$$ Then it follows by the variation of constants, $$\begin{aligned} y(t) \geq \exp\left( \int_\tau^t a(s) \, \mathrm{d}s \right)y_0 + \int_\tau^t \exp \left( \int_s^t a(\sigma) \, \mathrm{d}\sigma \right) b(s) \, \mathrm{d}s.\end{aligned}$$ Applying this to , $$\begin{aligned} \label{eq-phatrcomp} \hat{P}_t^\varepsilon(\Phi) -R_t^\varepsilon(\Phi) & \geq \exp\left( \int_\tau^t (1+2\eta^\varepsilon_s(\Phi))L(\Phi) \, \mathrm{d}s \right)(\hat{P}_\tau^\varepsilon(\Phi) -R_\tau^\varepsilon(\Phi))\nonumber \\ & \qquad + \int_\tau^t \exp \left( \int_s^t (1+2\eta^\varepsilon_\sigma(\Phi))L(\Phi) \, \mathrm{d}\sigma \right) 2\eta^\varepsilon_s(\Phi)L(\Phi) R_s^\varepsilon(\Phi) \, \mathrm{d}s.\end{aligned}$$ Recall the definition of $\mathbbm{1}_t^\varepsilon[\Phi]$ and note that it is non-increasing. Hence $\eta_t^\varepsilon(\Phi)$ is non-decreasing. So for $\tau\leq \sigma \leq t$, $\eta^\varepsilon_\sigma(\Phi) \leq \eta_t^\varepsilon(\Phi)$. Recalling that $L(\Phi)$ is non-positive, becomes, $$\begin{aligned} \hat{P}_t^\varepsilon(\Phi) -R_t^\varepsilon(\Phi) & \geq \exp\left( \int_\tau^t (1+2\eta_s^\varepsilon(\Phi))L(\Phi) \, \mathrm{d}s \right)(\hat{P}_\tau^\varepsilon(\Phi) -R_\tau^\varepsilon(\Phi))\nonumber \\ & \qquad +2\eta_t^\varepsilon(\Phi) \int_\tau^t \exp \left( \int_s^t (1+2\eta^\varepsilon_\sigma(\Phi))L(\Phi) \, \mathrm{d}\sigma \right) L(\Phi) R_s^\varepsilon(\Phi) \, \mathrm{d}s \nonumber \\ & \geq \exp\left( \int_\tau^t (1+2\eta_s^\varepsilon(\Phi))L(\Phi) \, \mathrm{d}s \right)(\hat{P}_\tau^\varepsilon(\Phi) -R_\tau^\varepsilon(\Phi))\nonumber \\ & \qquad +2\eta_t^\varepsilon(\Phi)L(\Phi) \int_\tau^t \exp \Big( (1+2\eta_s^\varepsilon(\Phi)) (t-s) \, L(\Phi) \Big) R_s^\varepsilon(\Phi) \, \mathrm{d}s \label{eq-phatrcomp3} .\end{aligned}$$ Finally for $t>\tau$, $$\begin{aligned} \partial_t R_t^\varepsilon(\Phi) & =\zeta(\varepsilon) \partial_t P_t(\Phi) = \zeta(\varepsilon)P_t(\Phi)L(\Phi) = R_t^\varepsilon(\Phi)L(\Phi).\end{aligned}$$ Hence for $\tau \leq s \leq t$, $$R_t^\varepsilon(\Phi) = \exp \left( (t-s) L (\Phi) \right) R_s^\varepsilon(\Phi).$$ Which implies, $$\label{eq-rexpdecay} R_s^\varepsilon(\Phi) = \exp \left( - (t-s)L(\Phi) \right) R_t^\varepsilon(\Phi).$$ Substituting this into , $$\begin{aligned} \hat{P}_t^\varepsilon(\Phi) -R_t^\varepsilon(\Phi) & \geq \exp\left( \int_\tau^t (1+2\eta_s^\varepsilon(\Phi))L(\Phi) \, \mathrm{d}s \right)(\hat{P}_\tau^\varepsilon(\Phi) -R_\tau^\varepsilon(\Phi))\nonumber \\ & \qquad + 2\eta_t^\varepsilon(\Phi)L(\Phi) \int_\tau^t \exp \Big( (1+2\eta_s^\varepsilon(\Phi)) (t-s) L(\Phi) \Big) \\ & \qquad \times \exp \left( - (t-s)L(\Phi) \right) R_t^\varepsilon(\Phi) \, \mathrm{d}s \\ & \geq \exp\left( \int_\tau^t (1+2\eta_s^\varepsilon(\Phi))L(\Phi) \, \mathrm{d}s \right)(\hat{P}_\tau^\varepsilon(\Phi) -R_\tau^\varepsilon(\Phi))\nonumber \\ & \qquad +2\eta_t^\varepsilon(\Phi) L(\Phi) R_t^\varepsilon(\Phi) \int_\tau^t \exp \left(2\eta_s^\varepsilon(\Phi) (t-s) L(\Phi) \right) \, \mathrm{d}s.\end{aligned}$$ This completes the proof of the lemma. \[lem-rho0bound\] For $\Phi \in \mathcal{G}(\varepsilon)$ and $t \geq \tau$, $$2\eta_t^\varepsilon(\Phi) L(\Phi) \int_\tau^t \exp \left(2\eta_s^\varepsilon(\Phi) (t-s) L(\Phi) \right) \, \mathrm{d}s \geq -\rho^{\varepsilon,0}_t(\Phi).$$ Since $L(\Phi) \leq 0$, $$\int_\tau^t \exp \left(2\eta_s^\varepsilon(\Phi) (t-s) L(\Phi) \right) \, \mathrm{d}s \leq t-\tau \leq t.$$ Recalling , $$-L(\Phi) \leq \frac{C(\Phi)}{2}.$$ Hence combining these, $$2 L(\Phi) \int_\tau^t \exp \left(2\eta_s^\varepsilon(\Phi) (t-s) L(\Phi) \right) \, \mathrm{d}s \geq -C(\Phi)t.$$ Multiplying both sides by $\eta_t^\varepsilon(\Phi)$ gives the required identity. \[lem-gammavolbound\] For $\varepsilon$ sufficiently small, for any $\Phi \in \mathcal{G}(\varepsilon)$ and any $t \in [0,T]$, $$1- \frac{1-\gamma(t)}{1-|\Delta|} \leq \varepsilon .$$ We show that for $\varepsilon$ sufficiently small, $$\frac{1}{\varepsilon}\left(1- \frac{1-\gamma(t)}{1-|\Delta|}\right) \leq 1.$$ Note, $$\begin{aligned} |\Delta| & = \pi \varepsilon^2 \int_0^\tau |v(s)-v'| \, \mathrm{d}s - \hat{C}\varepsilon^3 \leq \pi \varepsilon^2 \int_0^\tau |v(s)-v'| \, \mathrm{d}s \leq \pi \varepsilon^2 2T V(\varepsilon).\end{aligned}$$ Recalling , $$\begin{aligned} 1- \frac{1-\gamma(t)}{1-|\Delta|} & = \frac{1-|\Delta|}{1-|\Delta|}- \frac{1-\gamma(t)}{1-|\Delta|} = \frac{\gamma(t)-|\Delta|}{1-|\Delta|} \\ & \leq \frac{\gamma(t)}{1-|\Delta|} \leq \frac{\varepsilon^2n(\Phi)}{1-2\varepsilon^2 \pi TV(\varepsilon) } \leq \frac{\varepsilon^2M(\varepsilon)}{1-2\varepsilon^2 \pi TV(\varepsilon) } \leq \frac{\varepsilon^{3/2}}{1-2\varepsilon^2 \pi TV(\varepsilon) }.\end{aligned}$$ Therefore $$\frac{1}{\varepsilon}\left(1- \frac{1-\gamma(t)}{1-|\Delta|}\right) \leq \frac{\varepsilon^{1/2}}{1-2\varepsilon^2 \pi TV(\varepsilon) },$$ which, by recalling , converges to zero as $\varepsilon$ converges to zero. Hence for $\varepsilon$ sufficiently small the right hand side is less than $1$. We prove by induction on the degree of $\Phi$. Firstly we show that the proposition holds for $\Phi \in \mathcal{T}_0 \cap \mathcal{G}(\varepsilon)$. Now if $\Phi \in \mathcal{T}_0 \cap \mathcal{G}(\varepsilon)$ it follows that $\tau =0$ and hence, $$\begin{aligned} \hat{P}_\tau^\varepsilon(\Phi) & = \hat{P}_0^\varepsilon(\Phi) = \zeta(\varepsilon)f_0(x_0,v_0) = \zeta(\varepsilon)P_0(\Phi) = R_\tau^\varepsilon(\Phi).\end{aligned}$$ By lemma \[lem-pphatcomphelp\] and \[lem-rho0bound\] for $t \geq 0$, $$\begin{aligned} \hat{P}_t^\varepsilon(\Phi) - R_t^\varepsilon(\Phi) & \geq 2\eta_t^\varepsilon(\Phi) L(\Phi) R_t^\varepsilon(\Phi) \int_\tau^t \exp \left(2\eta_s^\varepsilon(\Phi) (t-s) L(\Phi) \right) \, \mathrm{d}s \geq - \rho^{\varepsilon,0}_t(\Phi)R_t^\varepsilon(\Phi) \\ & = - \hat{\rho}^{\varepsilon}_t(\Phi)R_t^\varepsilon(\Phi).\end{aligned}$$ Proving the proposition in the base case. Now suppose that the proposition holds true for all trees $\Phi \in \mathcal{T}_{k-1}\cap \mathcal{G}(\varepsilon)$ for some $k \geq 1$ and let $\Psi \in \mathcal{T}_{k}\cap \mathcal{G}(\varepsilon)$. For $t<\tau$ the proposition holds trivially, so we consider $t\geq \tau$. Now recall that, $$\hat{P}_\tau^\varepsilon(\Psi) = \frac{1-\gamma(t)}{1-|\Delta|}\hat{P}_\tau^\varepsilon(\bar{\Psi})g_0(v')[(v(\tau^-)-v')\cdot \nu]_+ ,$$ and, $$R_\tau^\varepsilon(\Psi) = R_\tau^\varepsilon(\bar{\Psi})g_0(v')[(v(\tau^-)-v')\cdot \nu]_+ .$$ Further since $\bar{\Psi} \in \mathcal{T}_{k-1}$ we know by our inductive assumption that the proposition holds for $\bar{\Psi}$, which implies. $$\hat{P}^\varepsilon_t(\bar{\Psi}) \geq R^\varepsilon_t(\bar{\Psi}) - \hat{\rho}^\varepsilon_t(\bar{\Psi})R_t^\varepsilon(\bar{\Psi}).$$ Hence by the estimate in lemma \[lem-gammavolbound\] for $\varepsilon$ sufficiently small, $$\begin{aligned} \label{eq-ptauindstep} \hat{P}_\tau^\varepsilon(\Psi) - R_\tau^\varepsilon(\Psi) & = g_0(v')[(v(\tau^-)-v')\cdot \nu]_+ \left(\frac{1-\gamma(t)}{1-|\Delta|}\hat{P}_\tau^\varepsilon(\bar{\Psi}) - R_\tau^\varepsilon(\bar{\Psi}) \right) \nonumber \\ & \geq g_0(v')[(v(\tau^-)-v')\cdot \nu]_+ \left((1-\varepsilon)\hat{P}_\tau^\varepsilon(\bar{\Psi}) - R_\tau^\varepsilon(\bar{\Psi}) \right) \nonumber \\ & \geq g_0(v')[(v(\tau^-)-v')\cdot \nu]_+ \left((1-\varepsilon)(R^\varepsilon_\tau(\bar{\Psi}) - \hat{\rho}_t(\bar{\Psi})R_\tau^\varepsilon(\bar{\Psi})) - R_\tau^\varepsilon(\bar{\Psi}) \right) \nonumber \\ & = g_0(v')[(v(\tau^-)-v')\cdot \nu]_+ R_\tau^\varepsilon(\bar{\Psi}) \left(1-\varepsilon - (1-\varepsilon)\hat{\rho}^\varepsilon_\tau(\bar{\Psi}) -1 \right)\nonumber \\ & = g_0(v')[(v(\tau^-)-v')\cdot \nu]_+ R_\tau^\varepsilon(\bar{\Psi}) \left(-\varepsilon - (1-\varepsilon)\hat{\rho}^\varepsilon_\tau(\bar{\Psi}) \right) = R_\tau^\varepsilon(\Psi) \left(-\varepsilon -(1-\varepsilon) \hat{\rho}^\varepsilon_\tau(\bar{\Psi}) \right) .\end{aligned}$$ Since the trajectory of the root particle of $\Psi$ up to time $\tau$ is equal to the trajectory of the root particle of $\bar{\Psi}$ up to time $\tau$ and recalling that for any $\Phi$, $\eta_t^\varepsilon(\Phi)$ is non-decreasing in $t$, it follows, $$\begin{aligned} \eta_\tau^\varepsilon(\bar{\Psi}) & =\int_{U \times \mathbb{R}^3} g_0(\bar{v})(1- \mathbbm{1}_\tau^\varepsilon[\bar{\Psi}](\bar{x},\bar{v})) \, \mathrm{d}\bar{x} \, \mathrm{d}\bar{v} \nonumber \\ & = \int_{U \times \mathbb{R}^3} g_0(\bar{v})(1- \mathbbm{1}_\tau^\varepsilon[\Psi](\bar{x},\bar{v})) \, \mathrm{d}\bar{x} \, \mathrm{d}\bar{v}, = \eta_\tau^\varepsilon(\Psi) \leq \eta_t^\varepsilon(\Psi),\end{aligned}$$ and recalling , $$\begin{aligned} C(\bar{\Psi}) \leq C(\Psi).\end{aligned}$$ Hence, $$\begin{aligned} \rho^{\varepsilon,0}_\tau(\bar{\Psi}) & = \eta^\varepsilon_\tau(\bar{\Psi}) C(\bar{\Psi})\tau \leq \eta_t^\varepsilon(\Psi)C(\Psi) t= \rho^{\varepsilon,0}_t(\Psi).\end{aligned}$$ Recalling , this implies, $$\begin{aligned} \hat{\rho}^{\varepsilon}_\tau(\bar{\Psi}) & = \rho^{\varepsilon,k-1}_\tau(\bar{\Psi}) \leq \rho^{\varepsilon,k-1}_t(\Psi).\end{aligned}$$ Substituting this into , $$\begin{aligned} \label{eq-phatrindstep2} \hat{P}_\tau^\varepsilon(\Psi) - R_\tau^\varepsilon(\Psi) & \geq R_\tau^\varepsilon(\Psi) \left(-\varepsilon -(1-\varepsilon) \hat{\rho}^\varepsilon_\tau(\bar{\Psi}) \right) \nonumber \\ & \geq R_\tau^\varepsilon(\Psi) \left(-\varepsilon -(1-\varepsilon) \rho^{\varepsilon,k-1}_t(\Psi) \right) \nonumber \\ & = -R_\tau^\varepsilon(\Psi) \left(\varepsilon +(1-\varepsilon) \rho^{\varepsilon,k-1}_t(\Psi) \right) .\end{aligned}$$ Recalling , $$\begin{aligned} \exp & \left( L(\Psi) \int_\tau^t(1+2\eta_s^\varepsilon(\Psi)) \, \mathrm{d}s \right) (\hat{P}_\tau^\varepsilon(\Psi) - R_\tau^\varepsilon(\Psi)) \\ & \geq - \exp\left( L(\Psi) \int_\tau^t (1+2\eta_s^\varepsilon(\Psi)) \, \mathrm{d}s \right)R_\tau^\varepsilon(\Psi) \left(\varepsilon +(1-\varepsilon) \rho^{\varepsilon,k-1}_t(\Psi) \right) \\ & = - \exp\left(L(\Psi) \int_\tau^t 2\eta_s^\varepsilon(\Psi) \, \mathrm{d}s \right)R_t^\varepsilon(\Psi) \left(\varepsilon +(1-\varepsilon) \rho^{\varepsilon,k-1}_t(\Psi) \right) \\ & \geq - R_t^\varepsilon(\Psi) \left(\varepsilon +(1-\varepsilon) \rho^{\varepsilon,k-1}_t(\Psi) \right).\end{aligned}$$ Recalling lemma \[lem-pphatcomphelp\] and \[lem-rho0bound\] this gives, $$\begin{aligned} \hat{P}_t^\varepsilon(\Psi) - R_t^\varepsilon(\Psi) & \geq \exp\left( \int_\tau^t L(\Psi)(1+2\eta_s^\varepsilon(\Psi)) \, \mathrm{d}s \right)(\hat{P}_\tau^\varepsilon(\Psi) - R_\tau^\varepsilon(\Psi)) \\ & \qquad +2\eta_t^\varepsilon(\Psi) L(\Psi) R_t^\varepsilon(\Psi) \int_\tau^t \exp \left(2\eta_s^\varepsilon(\Psi) (t-s) L(\Psi) \right) \, \mathrm{d}s \\ & \geq - R_t^\varepsilon(\Psi) \left(\varepsilon +(1-\varepsilon) \rho^{\varepsilon,k-1}_t(\Psi) \right) - \rho_t^{\varepsilon,0}(\Psi)R_t^\varepsilon(\Psi) \\ & \geq -R_t^\varepsilon(\Psi) \left( \varepsilon +(1-\varepsilon) \rho^{\varepsilon,k-1}_t(\Psi) +\rho_t^{\varepsilon,0}(\Psi) \right) \\ & = -R_t^\varepsilon(\Psi) \rho_t^{\varepsilon,k}(\Psi) = -R_t^\varepsilon(\Psi) \hat{\rho}_t^{\varepsilon}(\Psi).\end{aligned}$$ This completes the proof of the inductive step which concludes the proof of the proposition. \[prop-goodtreefull\] Good trees have full measure in the sense that $$\lim_{\varepsilon \to 0}P_t(\mathcal{MT} \setminus \mathcal{G}(\varepsilon)) =0.$$ Firstly we prove that $\mathcal{G}(0)$ is of measure 1. To this aim recall the definition of $R(\varepsilon)$ from definition \[def-recollisionfree\] and the definition of $\mathcal{T}_j$ . Define for $j \in \mathbb{N} \cup \{0 \}$, $$R_j(\varepsilon) := \mathcal{T}_j \cap R(\varepsilon).$$ Now note that $\mathcal{T}_0 \setminus R_0(\varepsilon)$ and $\mathcal{T}_1 \setminus R_1(\varepsilon)$ are empty, since it is not possible for a tree in $\mathcal{T}_0$ or $\mathcal{T}_1$ to have a re-collision. For a general $\Phi \in \mathcal{T}_2 $, $\Phi$ is of the form, $$\Phi = \big( (x_0,v_0),(s_1,\nu_1,v_1),(s_2,\nu_2,v_2) \big).$$ If we require that $\Phi \in \mathcal{T}_2 \setminus R_2(0)$ then there must be a re-collision and since the background particles do not change velocity this requires that $v_1=v_2$. Hence $\mathcal{T}_2 \setminus R_2(0)$ is a submanifold of $\mathcal{T}_2$ with, $$\dim(\mathcal{T}_2 \setminus R_2(0)) < \dim(\mathcal{T}_2).$$ This argument holds true for all $j\geq 2$ and hence, because $\mathcal{MT} = \cup _{j=0}^{\infty} \mathcal{T}_j$ we see that, $$P_t(\mathcal{MT}\setminus R(0))=1.$$ Therefore it follows, since the other requirements on $\mathcal{G}(0)$ are clear, that $P_t(\mathcal{MT}\setminus \mathcal{G}(0))=1$, and hence also that $P_t( \mathcal{G}(0))=1$. Since $\mathcal{G}(\varepsilon)$ is increasing as $\varepsilon$ decreases and $\lim_{\varepsilon \to 0}\mathcal{G}(\varepsilon)=\mathcal{G}(0)$ it follows by the dominated convergence theorem that, $$\lim_{\varepsilon \to 0}P_t( \mathcal{G}(\varepsilon)) =P_t(\mathcal{G}(0)) =1.$$ Hence $$\lim_{\varepsilon \to 0}P_t( \mathcal{MT} \setminus \mathcal{G}(\varepsilon)) =0,$$ as required. \[lem-convbounds\] Recall $\beta$ . For $\varepsilon>0$, $\Phi \in \mathcal{G}(\varepsilon)$ there exists constants $C_1,C_2> 0$ such that, $$\begin{aligned} \eta_t^\varepsilon(\Phi)& \leq C_1\varepsilon^2(\beta+V(\varepsilon))\label{eq-convbounds1}, \\ C(\Phi) & \leq C_2 (\beta+V(\varepsilon)). \label{eq-convbounds2} \end{aligned}$$ Firstly by , $$\begin{aligned} \eta_t^\varepsilon(\Phi) & = \int_{U \times \mathbb{R}^3} g_0(\bar{v})(1- \mathbbm{1}_t^\varepsilon[\Phi](\bar{x},\bar{v})) \, \mathrm{d}\bar{x} \, \mathrm{d}\bar{v} \leq \pi \varepsilon^2 T (\beta+V(\varepsilon)).\end{aligned}$$ So take $C_1:= \pi T$ proving . Next note, $$\begin{aligned} \int_{\mathbb{S}^{2}} \int_{\mathbb{R}^3} g_0(\bar{v})[(v(t)-\bar{v})\cdot \nu ]_+ \, \mathrm{d}\bar{v} \, \mathrm{d}\nu & \leq \pi \int_{\mathbb{R}^3} g_0(\bar{v})|v(t)-\bar{v}| \, \mathrm{d}\bar{v} \\ & \leq \pi \int_{\mathbb{R}^3} g_0(\bar{v})(\mathcal{V}(\Phi) + |\bar{v}|) \, \mathrm{d}\bar{v} \leq \pi (V(\varepsilon)+\beta).\end{aligned}$$ Hence by , $$\begin{aligned} C(\Phi) & \leq 2 \pi (V(\varepsilon)+\beta).\end{aligned}$$ so take $C_2 :=2 \pi$ which proves . \[lem-rhokbound\] For any $\delta > 0$, there exists a $\varepsilon'>0$ such that for $0<\varepsilon < \varepsilon'$ and for any $\Phi \in \mathcal{G}(\varepsilon)$, $$\hat{\rho}_t^{\varepsilon}(\Phi) < \delta.$$ Fix $\delta > 0$. Firstly by the above lemma, $$\begin{aligned} \rho_t^{\varepsilon,0}(\Phi) & = \eta_t(\Phi)C(\Phi)t \leq C_1C_2T \varepsilon^2(\beta+V(\varepsilon))^2.\end{aligned}$$ Recalling there exists an $\varepsilon_1>0$ such that for $0<\varepsilon<\varepsilon_1$, $$\rho_t^{\varepsilon,0}(\Phi)<\frac{\delta}{3}.$$ Further there exists an $\varepsilon_2>0$ such that for $0<\varepsilon<\varepsilon_2$, $$\frac{1}{\sqrt[]{\varepsilon}}\rho_t^{\varepsilon,0}(\Phi) \leq C_1C_2T \varepsilon^{3/2}(\beta+V(\varepsilon))^2 < \frac{\delta}{3}.$$ Finally there exists an $\varepsilon_3>0$ such that for $0<\varepsilon<\varepsilon_3$, $\sqrt[]{\varepsilon}<\frac{\delta}{3}$. Hence take $\varepsilon'=\min\{\varepsilon_1,\varepsilon_2,\varepsilon_3,1\}$ then for any $\Phi \in \mathcal{G}(\varepsilon)$, $$\begin{aligned} \hat{\rho}^{\varepsilon}_t(\Phi) & = \rho^{\varepsilon,n(\Phi)}_t(\Phi) = (1-\varepsilon)^{n(\Phi)} \rho^{\varepsilon,0}_t (\Phi) + (\rho^{\varepsilon,0}_t (\Phi) + \varepsilon) \sum_{j=1}^{n(\Phi)}(1-\varepsilon)^{n(\Phi)-j} \\ & \leq \rho^{\varepsilon,0}_t (\Phi) + (\rho^{\varepsilon,0}_t (\Phi) + \varepsilon)\times n(\Phi) \leq \rho^{\varepsilon,0}_t (\Phi) + M(\varepsilon) (\rho^{\varepsilon,0}_t (\Phi) + \varepsilon) \\ & = \rho^{\varepsilon,0}_t (\Phi) + M(\varepsilon) \rho^{\varepsilon,0}_t (\Phi) + M(\varepsilon)\varepsilon \leq \rho^{\varepsilon,0}_t (\Phi) + \frac{1}{\sqrt[]{\varepsilon}} \rho^{\varepsilon,0}_t (\Phi) + \sqrt[]{\varepsilon} < \delta.\end{aligned}$$ \[thm-ptphatcomp\] Uniformly for $t \in [0,T]$, $$\lim_{\varepsilon \to 0} \|P_t - \hat{P}^\varepsilon _t\|_{TV}=0.$$ Let $\delta > 0$ and $S \subset \mathcal{MT}$ then, $$\begin{aligned} P_t(S) - \hat{P}^\varepsilon _t(S) &= P_t(S \cap \mathcal{G}(\varepsilon))+ P_t(S \setminus \mathcal{G}(\varepsilon)) - \hat{P}^\varepsilon _t(S \cap \mathcal{G}(\varepsilon)) - \hat{P}^\varepsilon _t(S \setminus \mathcal{G}(\varepsilon)) \nonumber \\ & \leq P_t(S \cap \mathcal{G}(\varepsilon))+ P_t(S\setminus \mathcal{G}(\varepsilon)) - \hat{P}^\varepsilon _t(S \cap \mathcal{G}(\varepsilon)).\end{aligned}$$ By proposition \[prop-goodtreefull\] for $\varepsilon$ sufficiently small, $$\begin{aligned} P_t(S \setminus \mathcal{G}(\varepsilon)) \leq P_t(\mathcal{MT} \setminus \mathcal{G}(\varepsilon))< \frac{\delta}{3}.\end{aligned}$$ Hence, $$\label{eq-ptphatcomp1} P_t(S) - \hat{P}^\varepsilon_t(S) < P_t(S \cap \mathcal{G}(\varepsilon)) - \hat{P}^\varepsilon_t(S \cap \mathcal{G}(\varepsilon)) +\frac{\delta}{3}.$$ Recall the definition of $\zeta(\varepsilon)$ in . It is clear that this implies $$\label{eq-zetaupbound} \zeta(\varepsilon) \leq 1.$$ Hence by the above lemma, for $\varepsilon$ sufficiently small and $\Phi \in \mathcal{G}(\varepsilon)$, $$\zeta(\varepsilon) \hat{\rho}^{\varepsilon}_t(\Phi)\leq \hat{\rho}^{\varepsilon}_t(\Phi) < \frac{\delta}{3}.$$ Further by proposition \[prop-ptphatcomp\], for $\Phi \in \mathcal{G}(\varepsilon)$, $$\label{eq-minushatpbound} -\hat{P}^\varepsilon_t(\Phi) \leq - R^\varepsilon_t(\Phi) +\hat{\rho}^\varepsilon_t(\Phi)R_t^\varepsilon(\Phi) .$$ The Binomial inequality states that for $x \geq -1$ and $ N \in \mathbb{N}$, $$(1+x)^N \geq 1+Nx.$$ Hence for $\varepsilon>0$ such that $\frac{4}{3}\pi \varepsilon^3 \leq 1$ we apply this to $\zeta(\varepsilon)$ recalling , $$\begin{aligned} \label{eq-zetalowbound} \zeta(\varepsilon) & = (1-\frac{4}{3}\pi \varepsilon^3)^N \geq 1-N\frac{4}{3}\pi \varepsilon^3 = 1-\frac{4}{3}\pi\varepsilon.\end{aligned}$$ Hence for $\varepsilon$ sufficiently small gives, $$\begin{aligned} P_t(\Phi)-\hat{P}^\varepsilon_t(\Phi) & \leq P_t(\Phi) - R^\varepsilon_t(\Phi) + \hat{\rho}^\varepsilon_t(\Phi)R_t^\varepsilon(\Phi) \nonumber \\ & = P_t(\Phi) - \zeta(\varepsilon)P_t(\Phi) + \hat{\rho}^\varepsilon_t(\Phi)\zeta(\varepsilon)P_t(\Phi) \leq \frac{4}{3}\pi\varepsilon P_t(\Phi) + \frac{\delta}{3} P_t(\Phi) \leq \frac{2\delta}{3} P_t(\Phi).\end{aligned}$$ This holds for all $\Phi \in \mathcal{G}(\varepsilon)$ with $\varepsilon$ sufficiently small hence $$\begin{aligned} \label{eq-ptphatcomp2} P_t(S \cap \mathcal{G}(\varepsilon))-\hat{P}^\varepsilon_t(S \cap \mathcal{G}(\varepsilon)) \leq \frac{2\delta}{3} P_t(S \cap \mathcal{G}(\varepsilon)) \leq \frac{2\delta}{3}.\end{aligned}$$ By substituting into , for $\varepsilon$ sufficiently small, $$\label{eq-pphatcompdeltalower} P_t(S) - \hat{P}^\varepsilon_t(S) < \delta.$$ Since $\varepsilon$ did not depend on $S$ this holds true for every $S \subset \mathcal{MT}$. Hence for any $S \subset \mathcal{MT}$, $$\begin{aligned} \hat{P}^\varepsilon_t(S) - P_t(S) & = (1-\hat{P}^\varepsilon_t(\mathcal{MT}\setminus S) ) - (1-P_t(\mathcal{MT}\setminus S)) & = P_t(\mathcal{MT}\setminus S) - \hat{P}^\varepsilon_t(\mathcal{MT}\setminus S) < \delta.\end{aligned}$$ This together with gives that, for $\varepsilon$ sufficiently small, for any $S \subset \mathcal{MT}$, $$|P_t(S) - \hat{P}^\varepsilon_t(S) |< \delta,$$ which completes the proof of the theorem. We can now prove the main theorem of the paper, theorem \[thm-main\], which follows from the above theorem. Recall for $\Omega \subset U\times \mathbb{R}^3$, $$S_t(\Omega):=\{ \Phi \in \mathcal{MT} : (x(t),v(t)) \in \Omega \}.$$ By theorem \[thm-ideq\], $$\int_\Omega f_t(x,v) \, \mathrm{d}x \, \mathrm{d}v = \int_{S_t(\Omega)} P_t(\Phi) \, \mathrm{d}\Phi = P_t(S_t(\Omega)) .$$ Also by definition of $\hat{P}_t^\varepsilon$, $$\int_\Omega \hat{f}^N_t(x,v) \, \mathrm{d}x \, \mathrm{d}v = \int_{S_t(\Omega)} \hat{P}_t^\varepsilon(\Phi) \, \mathrm{d}\Phi = \hat{P}_t^\varepsilon(S_t(\Omega)) .$$ Hence by theorem \[thm-ptphatcomp\], $$\begin{aligned} \lim_{N \to \infty} \sup_{\Omega \subset U \times \mathbb{R}^3} & | \int_\Omega \hat{f}^N_t(x,v) - f_t(x,v) \, \mathrm{d}x \, \mathrm{d}v | = \lim_{\varepsilon \to 0} \sup_{\Omega \subset U \times \mathbb{R}^3} |P_t(S_t(\Omega)) - \hat{P}_t^\varepsilon(S_t(\Omega))| = 0,\end{aligned}$$ which completes the proof. Proof of auxiliary results {#sec:aux} ========================== Particle dynamics ----------------- The dynamics become undefined if there is instantaneously more than one background particle colliding with the tagged particle or if the tagged particle experiences an infinite number of collisions in finite time. We adapt a similar proof for the full hard-spheres dynamics from [@saintraymond13 Prop 4.1.1]. Let $R>0$ and $\delta<\varepsilon/2$ such that there exists a $K\in \mathbb{N}$ with $T=K\delta$. Denote the ball of radius $R$ about $x$ in $\mathbb{R}^3$ by $B_R(x)$. For the initial position of the tagged particle $(x_0,v_0)\in U \times B_R(0)$ fixed define $I(x_0,v_0) \subset (U \times \mathbb{R}^3)^N $ by, $$\begin{aligned} I(x_0,v_0) & := \{(x_1,v_1),\dots,(x_N,v_N) \in (U \times B_R(0))^N : \textrm{ the tagged particle collides} \\ & \qquad \textrm{ with at least two background particles in the time interval } [0,\delta] \}.\end{aligned}$$ We bound the volume of this set. Firstly define $$I^1(x_0,v_0): = \{ (x_1,v_1) \in U \times B_R(0): \varepsilon \leq |x_0-x_1| \leq \varepsilon+2R\delta \}.$$ It can be seen that for some $C$, $$|I^1(x_0,v_0)| \leq CR^3 \times (2R\delta)^3.$$ Since $I(x_0,v_0)$ is a subset of, $$\begin{aligned} \{ (x_1,v_1) & ,\dots,(x_N,v_N) \in (U \times B_R(0))^N : \exists \, 1 \leq i < j \leq N \textrm{ such that }\\ & \qquad \varepsilon \leq |x_0-x_i| \leq \varepsilon+2R\delta \textrm{ and } \varepsilon \leq |x_0-x_j| \leq \varepsilon+2R\delta \}\end{aligned}$$ The above estimate gives, for some constant $C=C(N,\varepsilon)$, $$\begin{aligned} |I(x_0,v_0)| & \leq CR^{3(N-2)} \times (R^6\delta^3)^2 \\ & \leq CR^{3(N+2)} \delta^6.\end{aligned}$$ Hence if we define, $$I:= \cup \{I(x_0,v_0) : (x_0,v_0) \in U \times B_R(0)\},$$ it follows, $$|I| \leq CR^{3(N+3)} \delta^6.$$ Hence there exists a subset $I_0(\delta,R)$ of measure at most $CR^{3(N+3)} \delta^6$ such that for any initial configuration in $(U\times B_R(0))^{N+1} \setminus I_0(\delta,R)$ the tagged particle experiences at most one collision in $[0,\delta]$. Now consider the system at time $\delta$. Since all particles had initial velocity in $B_R(0)$ and the tagged particle had at most one collision in time $[0,\delta]$ the velocity of the tagged particle at time $\delta$ is in $B_{2R}(0)$. By the same arguments above there exists a set $I_1(\delta,R)$ of measure at most $CR^{3(N+3)} \delta^6$ for some new constant $C$ such that for any initial configuration in $(U\times B_R(0))^{N+1} \setminus I_0(\delta,R)\cup I_1(\delta,R)$ the tagged particle experiences at most one collision in $[0,\delta]$ and at most one collision in $[\delta,2\delta]$ and thus the dynamics are well defined up to $2\delta$. Continue this process $K$ times defining the set, $$I(\delta,R) := \cup_{j=0}^{K-1} I_j(\delta,R),$$ which has measure at most $CR^{3(N+3)} \delta^6$ for some new constant $C$ and such that for any initial configuration in $(U\times B_R(0))^{N+1} \setminus I(\delta,R)$ the tagged particle has at most one collision per time interval $[j\delta, (j+1)\delta ]$ and hence the dynamics are well defined up to time $T$. Defining, $$I(T,R):= \cap_{\delta > 0 } I(\delta,R),$$ if follows $I(T,R)$ is of measure zero and for any any initial configuration in $(U\times B_R(0))^{N+1} \setminus I(T,R)$ the dynamics are well defined up to time $T$. Finally take, $$I := \cup_{R \in \mathbb{N} } I(T,R)$$ and note that $I$ is a countable union of measure zero sets and for any initial configuration in $(U\times \mathbb{R}^3) \setminus I$ the dynamics are well defined up to time $T$. Semigroup theory ---------------- \[lem-tsemi\] $A$ is the generator of the substochastic semigroup (c.f. [@arlotti06 Section 10.2]) $T(t):L^1(U \times \mathbb{R}^3) \to L^1(U \times \mathbb{R}^3)$ given by, $$\label{eq-tsemi} (T(t)f)(x,v) := \exp \left( -t \int_{\mathbb{S}^2} \int_{\mathbb{R}^3} g_0(\bar{v}) [(v-\bar{v})\cdot \nu]_+ \, \mathrm{d}\bar{v} \, \mathrm{d}\nu \right) f(x-tv,v).$$ We seek to apply theorem 10.4 of [@arlotti06]. Conditions (A1), (A2) trivially hold since $F=0$ in our situation. As for (A3), $$\label{eq-arlottinu} \nu(x,v)= \int_{\mathbb{S}^2} \int_{\mathbb{R}^3} g_0(\bar{v}) [(v-\bar{v})\cdot \nu]_+ \, \mathrm{d}\bar{v} \, \mathrm{d}\nu.$$ This is locally integrable so (A3) holds. Hence we can apply the theorem. In our case, $$\varphi(x,v,t,s)=x-(t+s)v.$$ So the semigroup is given by $$(T(t)f)(x,v) := \exp \left( -t \int_{\mathbb{S}^2} \int_{\mathbb{R}^3} g_0(\bar{v}) [(v-\bar{v})\cdot \nu]_+ \, \mathrm{d}\bar{v} \, \mathrm{d}\nu \right) f(x-tv,v),$$ as required. For $v,v_* \in \mathbb{R}^3, v \neq v_*$ define the Boltzmann kernel $k$ by, $$\label{eq-k} k(v,v_*) := \frac{1}{|v-v_*|} \int_{E_{v,v_*}} g_0(w) \, \mathrm{d}w,$$ where $E_{v,v_*} = \{ w \in \mathbb{R}^3 : w\cdot (v-v_*) = v\cdot (v-v_*) \}$. By the use of Carleman’s representation, first described in [@carleman57] (see also [@bisi14 Section 3]), $$\begin{aligned} \label{eq-bdeffalt} (Bf)(x,v)& = Q^+[f](x,v) \nonumber \\ & = \int_{\mathbb{S}^2} \int_{\mathbb{R}^3} f(x,v')g_0(\bar{v}')[(v-\bar{v})\cdot \nu]_+ \, \mathrm{d}\bar{v} \, \mathrm{d}\nu \nonumber \\ & = \int_{\mathbb{R}^3} k(v,v_*)f(x,v_*) \, \mathrm{d}v_*.\end{aligned}$$ \[lem-mya7\] There exists a $C>0$ such that for any $V>0$, $$\label{eq-mya7} \int_{|v|>V} k(v,v_*) \, \mathrm{d}v \leq C,$$ for almost all $|v_*| \leq V$. A similar calculation to the proof of theorem 2.1 in [@arlott07] (see also [@molinet77]) allows us to show that in our case, for $v \neq v_*$, $$k(v,v_*) \leq \frac{2\pi}{|v-v_*|} h \left( \frac{1}{4} \left[ |v-v_*|+ \frac{|v|^2 - |v_*|^2}{|v-v_*|} \right]^2 \right).$$ Where $h:[0,\infty) \rightarrow \mathbb{R}$ is given by, $$h(r):= \int_r^\infty z \sup_{ \{ v \in \mathbb{R}^3: |v|=z\} } g_0(v) \, \mathrm{d}z.$$ Thus we can follow the calculations in [@arlotti06 Ex 10.29] to see that, for $d=3$, for almost all $|v^*| \leq V$ and using assumption , $$\int_{|v|>V} k(v,v_*) \, \mathrm{d}v \leq C\Gamma <\infty,$$ where $C>0$ is some geometric constant and $$\Gamma: = \int_0^\infty rh(r^2) \, \mathrm{d}r.$$ By the definition of $A$ and $B$, is now, $$\label{eq-linboltzoperatorform} \begin{cases} \partial_t f_t(x,v)&=(Af+Bf)(x,v) \\ f_{t=0}(x,v) &= f_0(x,v). \end{cases}$$ By lemma \[lem-tsemi\] and equations (10.86), (10.88) of [@arlotti06], [@arlotti06 corollary 5.17] holds for our equation, so we apply [@arlotti06 theorem 5.2]. Lemma \[lem-mya7\] allow us to further apply [@arlotti06 theorem 10.28]. Therefore we have an honest $C_0$ semigroup of contractions generated by $\overline{A+B}$, which we denote by $G(t)$. Finally by [@arendt11 theorem 3.1.12], has a unique mild solution for each $f_0 \in L^1(U \times \mathbb{R}^3)$. It remains to show and . Firstly we find a bound for the operator $B$. Then we prove for $f_0 \in D(A)$ first before generalising to all $f_0$. Recall . As $g_0$ is normalized, $$\begin{aligned} \label{eq-g0betabound} \int_{\mathbb{S}^2} \int_{\mathbb{R}^3} g_0(\bar{v}) [(v-\bar{v})\cdot \nu]_+ \, \mathrm{d}\bar{v} \, \mathrm{d}\nu & \leq \int_{\mathbb{S}^2} \int_{\mathbb{R}^3} g_0(\bar{v}) |v-\bar{v}| \, \mathrm{d}\bar{v} \, \mathrm{d}\nu \nonumber \\ & \leq \pi \int_{\mathbb{R}^3} g_0(\bar{v}) (|v| + |\bar{v}|) \, \mathrm{d}\bar{v} \, \mathrm{d}\nu \leq \pi (|v| + \beta).\end{aligned}$$ By (10.6) in [@arlotti06] for $f \in L^1(U \times \mathbb{R}^3)$ and by recalling , $$\begin{aligned} \label{eq-bfl1bound} \int_{U\times \mathbb{R}^3} Bf (x,v) \, \mathrm{d}x \, \mathrm{d}v &= \int_{U\times \mathbb{R}^3} \int_{\mathbb{R}^3} k(v,v_*)f (x,v_*) \, \mathrm{d}v_* \, \mathrm{d}x \, \mathrm{d}v \nonumber \\ & = \int_{U\times \mathbb{R}^3} \nu(x,v)f(x,v) \, \mathrm{d}x \, \mathrm{d}v \nonumber \\ & = \int_{U\times \mathbb{R}^3} f(x,v) \int_{\mathbb{S}^2} \int_{\mathbb{R}^3} g_0(\bar{v}) [(v-\bar{v})\cdot \nu]_+ \, \mathrm{d}\bar{v} \, \mathrm{d}\nu \, \mathrm{d}x \, \mathrm{d}v \nonumber \\ & \leq \int_{U\times \mathbb{R}^3} f(x,v) \pi (|v| + \beta)\, \mathrm{d}x \, \mathrm{d}v.\end{aligned}$$ Now suppose that $f_0 \in D(A)$. Then [@arlotti06 corollary 5.17] holds and so we apply [@arlotti06 corollary 5.8] which gives, $$\label{eq-linboltzduhamelform} f_t= T(t)f_0 + \int_0^t G(t-\theta)BT(\theta)f_0 \, \mathrm{d}\theta.$$ Where $G(t)$ is the contraction semigroup generated by $\overline{A+B}$ and $T(t)$ the contraction semigroup generated by $A$. Hence by , $$\begin{aligned} \int_{U\times \mathbb{R}^3} f_t(x,v)& (1+|v|) \, \mathrm{d}x \, \mathrm{d}v \\ & \leq \int_{U\times \mathbb{R}^3} T(t)f_0(x,v)(1+|v|) + \int_0^t G(t-\theta)BT(\theta)f_0 \, \mathrm{d}\theta (x,v)(1+|v|) \, \mathrm{d}x \, \mathrm{d}v \\ & \leq \int_{U\times \mathbb{R}^3} f_0(x,v)(1+|v|) + t f_0 (x,v)(1+|v|) \pi (|v| + \beta) \, \mathrm{d}x \, \mathrm{d}v. \end{aligned}$$ Noting that $t\in [0,T]$ and recalling our assumption on $f_0$ in , this is bounded. Consider a general $f_0$ in , not necessarily in $D(A)$. Suppose for contradiction that is not true. Hence there exists a $t \in [0,T]$ such that for any $C>0$ there exists an $R>0$ such that, $$\int_U \int_{B_R(0)} f_t(x,v)(1+|v|) \, \mathrm{d}v \, \mathrm{d}x \geq C.$$ $D(A)$ is dense in $L^1(U\times \mathbb{R}^3)$ because it contains, for example, smooth compactly supported functions. Hence for any $n \in \mathbb{N}$ there exists an $f_0^n \in L^1(U\times \mathbb{R}^3)$ such that $f_0^n \in D(A)$, $f_0^n \geq 0$, there exists a $C_1>0$ such that $$\label{eq-f0n2ndmombound} \int_{U \times \mathbb{R}^3} f_0^n (x,v)(1+|v|^2) \, \mathrm{d}x \, \mathrm{d}v \leq C_1,$$ and that, $$\label{eq-f0f0nbound} \int_{U \times \mathbb{R}^3} |f_0^n (x,v)-f_0(x,v)|(1+|v|) \, \mathrm{d}x \, \mathrm{d}v \leq \frac{1}{n}.$$ Now define $f_t^n$ to be the solution of with initial data given by $f_0^n$. In this case gives, $$f_t^n= T(t)f_0^n + \int_0^t G(t-\theta)BT(\theta)f_0^n \, \mathrm{d}\theta.$$ The argument above for $f_0^n \in D(A)$ together with gives that there exists a $C_2>0$ independent of $n$ such that, $$\label{eq-ftnlower} \int_{U \times \mathbb{R}^3} f_t^n (x,v)(1+|v|) \, \mathrm{d}x \, \mathrm{d}v \leq C_2.$$ By our contradiction assumption there exists an $R>0$ such that, $$\int_U \int_{B_R(0)} f_t(x,v)(1+|v|) \, \mathrm{d}v \, \mathrm{d}x \geq 2C_2.$$ These two bounds together give that, $$\begin{aligned} \label{eq-ftcontupper} \int_{U\times \mathbb{R}^3} |f_t(x,v)& -f_t^n(x,v)|(1+|v|) \, \mathrm{d}x \, \mathrm{d}v \nonumber \\ & \geq \int_U \int_{B_R(0)} |f_t(x,v) -f_t^n(x,v)|(1+|v|) \, \mathrm{d}v \, \mathrm{d}x \nonumber \\ & \geq \bigg| \int_U \int_{B_R(0)} f_t(x,v)(1+|v|)\, \mathrm{d}v \, \mathrm{d}x -\int_U \int_{B_R(0)} f_t^n(x,v)(1+|v|) \, \mathrm{d}v \, \mathrm{d}x \bigg| \geq C_2.\end{aligned}$$ However since $G(t)$ is a contraction semigroup it follows for $n > 1/C_2$ by , $$\begin{aligned} \int_{U\times \mathbb{R}^3} (f_t(x,v)& -f_t^n(x,v))(1+|v|) \, \mathrm{d}x \, \mathrm{d}v = \int_{U\times \mathbb{R}^3} G(t)(f_0 -f_0^n)(x,v)(1+|v|) \, \mathrm{d}x \, \mathrm{d}v \\ & \leq \int_{U\times \mathbb{R}^3} (f_0(x,v) -f_0^n(x,v))(1+|v|) \, \mathrm{d}x \, \mathrm{d}v \leq \frac{1}{n} < C_2.\end{aligned}$$ By the same argument, $$\int_{U\times \mathbb{R}^3} (f_t^n(x,v) -f_t(x,v))(1+|v|) \, \mathrm{d}x \, \mathrm{d}v < C_2.$$ Hence we have a contradiction with which completes the proof of . To show fix $t \in [0,T]$. By , $$\int_{U \times \mathbb{R}^3} Bf_t(x,v) \, \mathrm{d}x \, \mathrm{d}v \leq \int_{U \times \mathbb{R}^3} f_t(x,v) \pi (|v|+\beta) \, \mathrm{d}x \, \mathrm{d}v.$$ By the above calculations $f_t$ has finite first moment so this is finite as required.

--- abstract: 'Deformations of the structure constants for a class of associative noncommutative algebras generated by Deformation Driving Algebras (DDA’s) are defined and studied. These deformations are governed by the Central System (CS). Such a CS is studied for the case of DDA being the algebra of shifts. Concrete examples of deformations  for the three-dimensional algebra governed by discrete and mixed continuous-discrete Boussinesq (BSQ) and WDVV equations are presented. It is shown that the theory of the Darboux transformations, at least for the BSQ case, is completely incorporated into the proposed scheme of deformations.' author: - | B.G.Konopelchenko\ \ Dipartimento di Fisica, Universita del Salento\ and INFN, Sezione di Lecce, 73100, Lecce, Italy title: 'Continuous-discrete integrable equations and Darboux transformations as deformations of associative algebras' --- Introduction ============ Deep interrelation between integrable systems and deformations of the structure constants for associative algebras is now well established due to the discovery of the WDVV equation by Witten \[1\] and Dijkgraaf-Verlinde-Verlinde \[2\], its beautiful formalization by Dubrovin \[3,4\] within the theory of Frobenius manifolds and subsequent extensions (see e.g. \[5,6\]). Novel approach to the deformation theory of the structure constants for associative algebras  has been proposed recently in \[7-11\].  In contrast with the theory of Frobenius manifolds and F-manifolds \[3-6\] where the algebra action is defined in the tangent bundle the approach \[7-11\] is formulated basically in terms of classical and quantum mechanics and provides us with the coisotropic and quantum deformations of structure constants. One of its advantages is that it naturally admits different extensions. To formulate them we recall briefly that of quantum deformations \[10\]. It consists basically in three steps: 1) to take the table of multiplication for an associative algebra in the basis $\mathbf{P}_{0},\mathbf{P}_{1},...,\mathbf{P}_{N-1},$ with the structure constants $C_{jk}^{l},$ i.e. $$\mathbf{P}_{j}\mathbf{P}_{k}=C_{jk}^{l}\mathbf{P}_{l},\quad j,k=0,1,...N-1$$ and associate the set of operators $$f_{jk}=-p_{j}p_{k}+C_{jk}^{l}(x^{0},x^{1},...,x^{N-1})p_{l},\quad j,k=0,1,...,N-1$$ with it where $x^{0},x^{1},...,x^{N-1}$ stand for the deformation parameters, $\mathbf{P}_{0}$ denotes the unite element of the basis and summation over repeated index ( from 0 to N-1 ) is assumed, 2) to require that the operators $p_{0},p_{1},...,p_{N-1}$and $x^{0},x^{1},...,x^{N-1}$ are elements of the Heisenberg algebra, i.e. $$\lbrack p_{j},p_{k}]=0, \quad [x^{j},x^{k}]=0, \quad [p_{j},x^{k}]=\hbar \delta _{j}^{k}, \quad j,k=0,1,...,N-1$$ where $\hbar $ is a constant and $\delta _{j}^{k}$ is the Kronecker symbol and 3) the requirement that the functions $C_{jk}^{l}(x)$ are such that the set of equations $$f_{jk}\mid \Psi \rangle =0, \quad j,k=0,1,...,N-1$$ has a nontrivial common solution ( Dirac’s prescription) where $\mid \Psi \rangle $ are elements of a certain linear space. The requirement (4) gives rise to the set of equations ( quantum central system (QCS)) \[10\] $$\hbar \frac{\partial C_{jk}^{n}}{\partial x^{l}}-\hbar \frac{\partial C_{kl}^{n}}{\partial x^{j}}+C_{jk}^{m}C_{lm}^{n}-C_{kl}^{m}C_{jm}^{n}=0,\quad j,k,l,n=0,1,...,N-1$$ which governs quantum deformations of the structure constants $C_{jk}^{l}(x^{0},...,x^{N-1})$ . The QCS (5)  contains the oriented associativity equation, WDVV equation, Boussinesq equation, Gelfand-Dikii and Kadomtsev-Petviashvili (KP) hierarchies as the particular cases \[10\].  In this paper we will present an extension of the above aprroach. The basic idea is to consider other algebras formed by the elements $p_{0},...,p_{N-1}$ of the basis and deformation parameters $x^{0},...,x^{N-1} $ instead of the Heisenberg algebra. We will refer to such algebras as the Deformation Driving Algebras (DDA’s) . We will show that choosing DDA as the algebra of differences, one gets discrete CS and subsequently discrete integrable equations and hierarchies. DDA which contains both Heisenberg and differences subalgebras gives rise to the mixed continuous-discrete equations and Darboux transformations for continuous systems. We consider the deformations of the three and four-dimensional  associative algebras as the illustration of the general approach. The corresponding CS contains, in particular, the continuous Boussinesq (BSQ) equation, discrete and mixed discrete-continuous BSQ equations and corresponding Darboux transformation. We demonstrate that the Darboux transformations for the BSQ equation are completely incorporated into the scheme for deformations of associative algebras presented in the paper. Another illustrative example is given by the WDVV equation, its discrete and mixed discrete-continuous versions. The paper is organized as follows. In section 2 definition of deformations generated by DDA is given and general CS is derived. Continuous, discrete and general mixed BSQ deformations are considered in section 3. Section 4 is devoted to the study of WDVV, discrete and mixed WDVV deformations and equations. Mixed DDA and corresponding Darboux transformations are studied in section 5. Darboux transformations for the BSQ equation and deformations for the four-dimensional algebra are discussed in section 6. Appendix contains some explicit formulas. Deformations of the structure constants generated by DDA ========================================================  So, let us consider a finite-dimensional associative algebra *A* with (or without) unite element $\mathbf{P}_{0}$ . We choose a basis $\mathbf{P}_{0},\mathbf{P}_{1},...,\mathbf{P}_{N-1}$ and introduce structure constants via the table of multiplication $$\mathbf{P}_{j}\mathbf{P}_{k}=C_{jk}^{l}\mathbf{P}_{l},\quad j,k=0,1,...,N-1.$$ In order to define deformations $C_{jk}^{l}(x^{0},x^{1},...,x^{N-1})$ of the structure constants we first identify the elements of the basis $P_{j}$ and deformation parameters $x^{j}$ with the elements of a certain  algebra which we will call the Deformation Driving Algebra (DDA) . Then we introduce the operators $$f_{jk}\doteqdot -p_{j}p_{k}+C_{jk}^{l}(x)p_{l},\quad j,k=0,1,...,N-1$$ and finally we require that the functions $C_{jk}^{l}(x)$ are such that these operators have common nontrivial kernel or, equivalently, that the equations $$f_{jk}\mid \Psi \rangle =0,\quad j,k=0,1,...,N-1$$ have nontrivial common solutions ( Dirac’s prescription). If the structure constants $C_{jk}^{l}(x)$ are such that this requirement is satisfied we say that they define deformations generated by DDA. The requirement (8) can be converted into the set of equations for $C_{jk}^{l}(x)$ which we will call the central system (CS) for deformations. The concrete form of CS depends on DDA. For instance, for the Heisenberg DDA (3) the CS is given by QCS (5). In the present paper we restrict ourselfs to associative algebras which have commutative basis, i.e. for which $C_{jk}^{l}(x)=C_{kj}^{l}(x).$  For such algebras DDA should have a commutative sublagebra generated by $p_{0},...,p_{N-1},$ i.e. $[p_{j},p_{k}]=0$. Examples of such algebras close to the Heisenberg DDA (3) are given by the well-known algebra of shifts$$\lbrack p_{j},p_{k}]=0,\quad \lbrack x^{j},x^{k}]=0,\quad \lbrack p_{j},x^{k}]=\delta _{j}^{k}p_{j},\quad j,k=0,1,...,N-1$$algebra of differences $$\lbrack p_{j},p_{k}]=0,\quad \lbrack x^{j},x^{k}]=0,\quad \lbrack p_{j},x^{k}]=\delta _{j}^{k}(\hat{I}+p_{j}),\quad j,k=0,1,...,N-1$$ algebra of q-differences$$\lbrack p_{j},p_{k}]=0,\quad \lbrack x^{j},x^{k}]=0,\quad \lbrack p_{j},x^{k}]=\delta _{j}^{k}(\hat{I}+qx^{j}p_{j}),\quad j,k=0,1,...,N-1$$ and their mixtures. Here $\hat{I}$ denotes the identity operator. Here we will concentrate on DDA of the types (3),(9), (10) and DDA which contains them as subalgebras. All these algebras can be presented in the unified form as the algebra with the following determining commutation relations $$\lbrack p_{j},p_{k}]=0,\quad \lbrack x^{j},x^{k}]=0,\quad \lbrack p_{j},x^{k}]=\delta _{j}^{k}(\hat{I}+\varepsilon _{j}p_{j}),\quad j,k=0,1,...,N-1$$ where $\varepsilon _{j}$ are arbitrary parameters. For $\varepsilon _{j}\rightarrow 0$ one has the Heisenberg subalgebra while at $\varepsilon _{j}=1$ one has algebra (10). A realization of such DDA is given by the algebra of differences $p_{j}=\Delta _{j}.$ where $\Delta _{j}=\frac{1}{\varepsilon _{j}}(T_{j}-1)$ and $T_{j}(x^{k})=x^{k}+\delta _{j}^{k}\varepsilon _{j},j,k=0,1,...,N-1.$ At $\varepsilon _{j}\rightarrow 0$ obviously $\Delta _{j}\rightarrow \frac{\partial }{\partial x^{j}}.$ For the DDA (12) the derivation of the CS is based on the following identities. The first is $$\lbrack p_{j},\varphi (x)]=\Delta _{j}\varphi (x)\cdot (\hat{I}+\varepsilon _{j}p_{j}),\quad j=1,2,...,N-1$$ where $\varphi (x)$ is an arbitrary function, $\Delta _{j}\varphi (x^{0},x^{1},...,x^{N-1})=(T_{j}-1)\varphi (x^{0},x^{1},...,x^{N-1})$ and $T_{j}\varphi (x^{0},...,x^{j},...,x^{N-1})=\varphi (x^{0},...,x^{j}+\varepsilon _{j},....,x^{N-1}).$ The second is $$\begin{aligned} \left( p_{j}p_{k}\right) p_{l}-p_{j}\left( p_{k}p_{l}\right) =-p_{l}f_{jk}+p_{j}f_{kl}-T_{l}C_{jk}^{m}\cdot f_{lm}+T_{j}C_{kl}^{m}\cdot f_{jm}+\quad && \notag \\ +(\Delta _{l}C_{jk}^{n}-\Delta _{j}C_{kl}^{n}+T_{l}C_{jk}^{m}\cdot C_{lm}^{n}-T_{j}C_{kl}^{m}\cdot C_{jm}^{n})p_{n},\quad j,k,l=1,2,...,N-1.\end{aligned}$$ The identity (14) implies that $$\begin{aligned} \left\{ \left( p_{j}p_{k}\right) p_{l}-p_{j}\left( p_{k}p_{l}\right) \right\} \mid \Psi \rangle =A_{klj}^{n}(x)p_{n}\mid \Psi \rangle =\quad && \notag \\ =\left( \Delta _{l}C_{jk}^{n}-\Delta _{j}C_{kl}^{n}+T_{l}C_{jk}^{m}\cdot C_{lm}^{n}-T_{j}C_{kl}^{m}\cdot C_{jm}^{n}\right) p_{n}\mid \Psi\rangle\end{aligned}$$ where $\mid \Psi \rangle \subset $ linear subspace $\mathit{H}_{\Gamma }$ defined by equations (8) and $A_{klj}^{n}$ is the associator for the algebra* A.* * *For an associative algebra l.h.s. of (15) vanishes  and under the condition that subspace $\mathit{H}_{\Gamma }$ does not contain elements linear in $p_{j}\mid \Psi \rangle $ equation (15) is satisfied iff $$\Delta _{l}C_{jk}^{n}-\Delta _{j}C_{kl}^{n}+T_{l}C_{jk}^{m}\cdot C_{lm}^{n}-T_{j}C_{kl}^{m}\cdot C_{jm}^{n}=0,\quad j,k,l,n=0,1,...,N-1.$$ Thus, we have **Proposition.**  The structure constants $C_{jk}^{l}(x)$ define deformations generated by the DDA (12) if they obey the CS (16). We would like to emphasize that the CS (16) is, in fact, the consequence of the weak associativity condition $$\left\{ \left( p_{j}p_{k}\right) p_{l}-p_{j}\left( p_{k}p_{l}\right) \right\} \mid \Psi \rangle =0.$$ In standard matrix notations with the matrices $C_{j}$,$A_{lj}$ defined by $\left( C_{j}\right) _{k}^{l}=C_{jk}^{l},$ $\left( A_{lj}\right) _{k}^{n}=A_{klj}^{n}$ the CS (16) takes the form $$A_{lj}^{d}\doteqdot \Delta _{l}C_{j}-\Delta _{j}C_{l}+C_{l}T_{l}C_{j}-C_{j}T_{j}C_{l}=0$$ or $$A_{lj}^{d}\doteqdot (1+\varepsilon _{l}C_{l})T_{l}(1+\varepsilon _{j}C_{j})-(1+\varepsilon _{j}C_{j})T_{j}(1+\varepsilon _{l}C_{l})=0.$$ There is no summation over repeated indices in these formulae. In the case of all $\varepsilon _{j}=0,T_{j}=1,\Delta _{j}=\frac{\partial }{\partial x^{j}}$ and CS (16) coincides with the QCS ( 5). For all $\varepsilon _{j}=1$ one has a pure discrete CS which has been discussed in \[11\]. Here we have general mixed CS which will be quite useful in discussion of relation between continuous and discrete equations and Darboux transformations as well. We note also that the results of the paper \[11\] concerning the relation between the associator and discrete curvature tensor are valid in the general case too. Continuous and discrete Boussinesq equations ============================================ A simple nontrivial example of the proposed scheme corresponds to the three-dimensional algebra with  the unite element and the basis $\mathbf{P}_{0},\mathbf{P}_{1},\mathbf{P}_{2}.$ The table of multiplication is given by the trivial part $\mathbf{P}_{0}\mathbf{P}_{j}=\mathbf{P}_{j},j=0,1,2$ and by $$\begin{aligned} \mathbf{P}_{1}^{2} =A\mathbf{P}_{0}+B\mathbf{P}_{1}+C\mathbf{P}_{2}, \notag \\ \mathbf{P}_{1}\mathbf{P}_{2} =D\mathbf{P}_{0}+E\mathbf{P}_{1}+G\mathbf{P}_{2}, \\ \mathbf{P}_{2}^{2} =L\mathbf{P}_{0}+M\mathbf{P}_{1}+N\mathbf{P}_{2} \notag\end{aligned}$$ where the structure constants A,B,...,N depend only on the deformation parameters $x^{1},x^{2}$ due to the cyclicity of the variable $x^{0}$, associated with the unite element  $\mathbf{P}_{0}$.  It is the easy consequence of the CS at l=0 and $\ C_{j0}^{l}=\delta _{j}^{l}.$ The CS (16) generated by the DDA (12) in this case is of the form $$\begin{aligned} \Delta _{1}D-\Delta _{2}A+AE_{1}+DG_{1}-DB_{2}-LC_{2} =0, \notag \\ \Delta _{1}E-\Delta _{2}B+D_{1}+BE_{1}+EG_{1}-EB_{2}-MC_{2} =0, \notag \\ \Delta _{1}G-\Delta _{2}C+CE_{1}+GG_{1}-A_{2}-GB_{2}-NC_{2} =0, \notag \\ \Delta _{1}L-\Delta _{2}D+AM_{1}+DN_{1}-DE_{2}-LG_{2} =0, \notag \\ \Delta _{1}M-\Delta _{2}E+L_{1}+BM_{1}+EN_{1}-EE_{2}-MG_{2} =0, \notag \\ \Delta _{1}N-\Delta _{2}G+CM_{1}+GN_{1}-D_{2}-GE_{2}-NG_{2} =0\end{aligned}$$ where $A_{j}\doteqdot T_{j}A$ etc. The CS (20) which governs the deformations of the structure constants A,B,... is the underdetermined one and , hence, admits the gauge  freedom. One of the gauges, namely, $B=0,C=1,G=0$ has been discussed in \[10\] for the quantum deformations and it was shown that in this case the QCS is reduced to the BSQ equation. Let us consider the same gauge here.  The first three equations (20) give $$\begin{aligned} L &=&\Delta _{1}D-\Delta _{2}A+AE_{1}, \notag \\ M &=&\Delta _{1}E+D_{1}, \notag \\ N &=&E_{1}-A_{2}\end{aligned}$$ and the rest of the system (20) takes the form $$\begin{aligned} \left( \Delta _{1}^{2}-\Delta _{2}+E_{11}-E_{2}-A_{12}\right) E+2\Delta _{1}D_{1}-\Delta _{2}A_{1}+A_{1}E_{11} =0, \notag \\ \left( \Delta _{1}^{2}-\Delta _{2}+E_{11}-E_{2}-A_{12}\right) D+\left( -\Delta _{1}\Delta _{2}+\Delta _{1}E_{1}+D_{11}\right) A+\Delta _{1}(AE_{1}) =0, \notag \\ \Delta _{1}(2E_{1}-A_{2})+D_{11}-D_{2} =0.\end{aligned}$$ The CS (22) defines the Boussinesq  deformations of the structure constants in (19) generated by the DDA (12). Indeed, in the case $\varepsilon _{1}=\varepsilon _{2}=0$, i.e. for the Heisenberg DDA the CS (22) becomes ( $A_{x_{j}}=\frac{\partial A}{\partial x^{j}},$ etc) $$\begin{aligned} E_{x_{1}x_{1}}-E_{x_{2}}+2D_{x_{1}}-A_{x_{2}} =0, \notag \\ D_{x_{1}x_{1}}-D_{x_{2}}-A_{x_{1}x_{2}}+AE_{x_{1}}+(AE)_{x_{1}} =0, \notag \\ 2E-A =0\end{aligned}$$ where all integration constants have been choosen to be equal to zero. Hence , one has the system $$\begin{aligned} \frac{1}{2}A_{x_{1}x_{1}}-\frac{3}{2}A_{x_{2}}+2D_{x_{1}} =0, \notag \\ D_{x_{1}x_{1}}-D_{x_{2}}-A_{x_{1}x_{2}}+\frac{3}{4}(A^{2})_{x_{1}} =0.\end{aligned}$$ Eliminating D, one gets the Boussinesq (BSQ) equation $$A_{x_{2}x_{2}}+\frac{1}{3}A_{x_{1}x_{1}x_{1}x_{1}}-(A^{2})_{x_{1}x_{1}}=0.$$ The BSQ equation (25) defines quantum deformations of the structure constants in (19) with $B=0,C=1,G=0,L=D_{x_{1}}-A_{x_{2}}+\frac{1}{2}A^{2},M=\frac{1}{2}A_{x_{1}}+D,N=-\frac{1}{2}A$ and A, D defined by (24). In the pure discrete case $\varepsilon _{1}=\varepsilon _{2}=1$ the CS (22) represents the discrete version of the BSQ system (24). It defines the discrete deformations of the same structure constants. There are also the mixed cases. The first is $\varepsilon _{1}=0,\varepsilon _{2}=1(T_{1}=1,\Delta _{1}=\frac{\partial }{\partial x^{1}},\Delta _{2}=T_{2}-1)$ . The CS (22) is $$\begin{aligned} E_{x_{1}x_{1}}+2D_{x_{1}}-(1+E)\Delta _{2}(E+A) =0, \notag \\ D_{x_{1}x_{1}}-D_{x_{2}}+AE_{x_{1}}+(AE)_{x_{1}}-\Delta _{2}A_{x_{1}}-D\Delta _{2}(E+A) =0, \\ (2E-A_{2})_{x_{1}}-\Delta _{2}D_{2} =0. \notag\end{aligned}$$ The second case corresponds to $\varepsilon _{1}=1,\varepsilon _{2}=0(\Delta _{1}=T_{1}-1,T_{2}=1,\Delta _{2}=\frac{\partial }{\partial x^{2}})$ and CS takes the form $$\begin{aligned} \Delta _{1}^{2}E+(E_{11}-E-A_{1})E+A_{1}E_{11}+2\Delta _{1}D_{1}-(E+A_{1})_{x_{2}} =0, \notag \\ \Delta _{1}^{2}D+(E_{11}-E-A_{1})D+A\Delta _{1}E_{1}+\Delta _{1}(AE_{1})+D_{11}A-\Delta _{1}A_{x_{2}}-D_{x_{2}} =0, \notag \\ \Delta _{1}(2E_{1}-A)+D_{11}-D =0.\end{aligned}$$ Equations (8) provide us with the linear problems for the CS (22). They are $$\begin{aligned} \left( \Delta _{2}-\Delta _{1}^{2}+A\right)\mid\Psi \rangle =0, \notag \\ \left( \Delta _{1}\Delta _{2}-E\Delta _{1}-D\right)\mid\Psi \rangle =0\end{aligned}$$ or equivalently $$\begin{aligned} \left( \Delta _{2}-\Delta _{1}^{2}+A\right)\mid \Psi \rangle =0, \notag \\ \left( \Delta _{1}^{3}-(E+A)\Delta _{1}-(D+\Delta _{1}A)\right)\mid \Psi \rangle =0.\end{aligned}$$ One can check that the equation $f_{22}\mid \Psi \rangle =0$ is a consequence of (28). In the quantum case ($\Delta _{j}=\frac{\partial }{\partial x^{j}})$ the system (29) is the well-known linear system for the continuous BSQ equation \[12\]. We note that in this case the third equation $f_{22}\mid \Psi \rangle =0$ , i.e. $\left( \Delta _{2}^{2}-L-D\Delta _{1}+\frac{1}{2}A\right) \mid \Psi \rangle =0$ is the consequence of two equations (29). In the pure discrete case equations (29) represent the linear problems for the discrete BSQ equation. The first mixed case considered above can be treated as the BSQ equation with the discrete time. The second case instead represents a continuous isospectral flow for the third order difference problem, i.e. a sort of the difference BSQ equation with continuous time. WDVV, discrete and continuous-discrete WDVV equations ===================================================== Another interesting reduction of the CS (20) corresponds to the constraint $C=1,G=0,N=0.$ In this case the CS is $$\begin{aligned} \Delta _{1}D-\Delta _{2}A+AE_{1}-DB_{2}-L =0, \notag \\ \Delta _{1}E-\Delta _{2}B+D_{1}+BE_{1}+EG_{1}-EB_{2}-M =0, \notag \\ E_{1}-A_{2} =0, \notag \\ \Delta _{1}L-\Delta _{2}D+AM_{1}-DE_{2} =0, \notag \\ \Delta _{1}M-\Delta _{2}E+L_{1}+BM_{1}-EE_{2} =0, \notag \\ M_{1}-D_{2} =0.\end{aligned}$$ Third and sixth equations (30) imply the existence of the functions U and V such that $$A=U_{1},\quad E=U_{2},\quad D=V_{1},\quad M=V_{2}.$$ Substituting the expressions for L and M given by the first two equations (30), i.e. $$\begin{aligned} L =\Delta _{1}V_{1}-\Delta _{2}U_{1}+U_{1}U_{12}-V_{1}B_{2}, \notag \\ M =\Delta _{1}U_{2}-\Delta _{2}B+V_{11}+BU_{12}-U_{2}B_{2}\end{aligned}$$ into the rest of the system , one gets the following three equations $$\begin{aligned} \Delta _{1}U_{1}-\Delta _{2}B+V_{11}+BU_{12}-U_{2}B_{2}-V_{2} =0, \\ (\Delta _{1}^{2}-\Delta _{2})V_{1}-\Delta _{1}\Delta _{2}U_{1}+\Delta _{1}(U_{1}U_{12})-\Delta _{1}(V_{1}B_{2})-V_{1}U_{22}+U_{1}V_{12} =0, \notag \\ \Delta _{1}(V_{2}+V_{11})-\Delta _{2}(U_{2}+U_{11})+U_{11}U_{112}-U_{2}U_{22}-V_{11}B_{12}+BV_{12} =0. \notag\end{aligned}$$ Solution of this system together with the formulas (31) define deformations of the structure constants A,B,... generated by the DDA (12). In the pure continuous case ( $\varepsilon _{1}=\varepsilon _{2}=0$ ) the system (32) takes the form $$\begin{aligned} U_{x_{1}}-B_{x_{2}} =0, \notag \\ (V_{x_{1}}+U_{x_{2}}+U^{2}-VB)_{x_{1}}-V_{x_{2}} =0, \\ V_{x_{1}}-U_{x_{2}} =0. \notag\end{aligned}$$ This system of three conservation laws implies the existence of the function F such that $$U=F_{x_{1}x_{1}x_{2}},\quad V=F_{x_{1}x_{2}x_{2}}, \quad B=F_{x_{1}x_{1}x_{1}}.$$ In terms of F the system (33) is $$F_{x_{2}x_{2}x_{2}}-(F_{x_{1}x_{1}x_{2}})^{2}+F_{x_{1}x_{1}x_{1}}F_{x_{1}x_{2}x_{2}}=0.$$ It is the famous WDVV equation \[1,2\]. It is a well-known fact the WDVV equation describes deformations of the three-dimensional algebra (19) under the reduction $C=1,G=0,N=0$ \[3-10\]. So, the system (32) represents the generalization of the WDVV equation to the case of deformations of the same algebra generated by DDA (12). In the pure discrete case $\varepsilon _{1}=\varepsilon _{2}=1$ the system (32) gives us a pure discrete version of the WDVV equation. In the first mixed case $\varepsilon _{1}=0,\varepsilon _{2}=1$ we have the system $$\begin{aligned} U_{x_{1}}-\Delta _{2}(B+V)-U_{2}\Delta _{2}B =0, \\ (V_{x_{1}}+UU_{2}-VB_{2})_{x_{1}}-\Delta _{2}(V+U_{x_{1}})-VU_{22}+UV_{2} =0, \notag \\ (V+V_{2})_{x_{1}}-\Delta _{2}(U+U_{2}+UU_{2})-VB_{2}+BV_{2} =0 \notag\end{aligned}$$ while in the case $\varepsilon _{1}=1,\varepsilon _{2}=0$ one gets $$\begin{aligned} \Delta _{1}U_{1}+B\Delta _{1}U+V_{11}-V-B_{x_{2}} =0, \\ \Delta _{1}(\Delta _{1}V_{1}-U_{1x_{2}}+U_{1}^{2}-V_{1}B)-V_{1}U+U_{1}V_{1}-V_{1x_{2}} =0, \notag \\ \Delta _{1}(V+V_{11})+U_{11}^{2}-U^{2}-V_{11}B_{1}+BV_{1}-(U+U_{11})_{x_{2}} =0. \notag\end{aligned}$$ Equations $f_{jk}\mid \Psi \rangle =0$ for the system (30) in the coordinate representation $p_{j}=\Delta _{j}$ have the form $$\begin{aligned} \left( \Delta _{2}-\Delta _{1}^{2}+B\Delta _{1}+A\right)\mid\Psi \rangle =0, \notag \\ \left( \Delta _{1}\Delta _{2}-E\Delta _{1}-D\right)\mid\Psi \rangle =0, \notag \\ \left( \Delta _{2}^{2}-M\Delta _{1}-L\right)\mid\Psi \rangle =0.\end{aligned}$$ The system (38) in its turn is equivalent to the following $$\begin{aligned} \left( \Delta _{2}-\Delta _{1}^{2}+B\Delta _{1}+A\right)\mid\Psi \rangle =0, \notag \\ \left( \Delta _{1}^{3}-B\Delta _{1}^{2}-(\Delta _{1}B+A_{1}+E)\Delta _{1}-(\Delta _{1}A+D)\right)\mid\Psi \rangle =0, \notag \\ \left( \Delta _{2}^{2}-M\Delta _{1}-L\right)\mid\Psi \rangle =0.\end{aligned}$$ The compatibility condition for the above linear problems are equivalent to the ”discretized” WDVV equations (30) or (32). In particular, in the pure continuous case the problems (38) in the coordinate representation are of the well-known form \[3,4\] $$\begin{aligned} \Psi _{x_{1}x_{1}}&=&F_{x_{1}x_{1}x_{2}}\Psi +F_{x_{1}x_{1}x_{1}}\Psi _{x_{1}}+\Psi _{x_{2}}, \notag \\ \Psi _{x_{1}x_{2}}& =&F_{x_{1}x_{2}x_{2}}\Psi +F_{x_{1}x_{1}x_{2}}\Psi _{x_{1}}, \notag \\ \Psi _{x_{2}x_{2}}& =&F_{x_{2}x_{2}x_{2}}\Psi +F_{x_{1}x_{2}x_{2}}\Psi _{x_{1}}\end{aligned}$$ or equivalently $$\begin{aligned} \Psi _{x_{2}}& =&\Psi _{x_{1}x_{1}}-F_{x_{1}x_{1}x_{1}}\Psi _{x_{1}}-F_{x_{1}x_{1}x_{2}}\Psi , \\ \Psi _{x_{1}x_{1}x_{1}}& =&F_{x_{1}x_{1}x_{1}}\Psi _{x_{1}x_{1}}+(2F_{x_{1}x_{1}x_{2}}+F_{x_{1}x_{1}x_{1}x_{1}})\Psi _{x_{1}}+(F_{x_{1}x_{2}x_{2}}+F_{x_{1}x_{1}x_{1}x_{2}})\Psi , \\ \Psi _{x_{1}x_{1}x_{1}x_{1}}& =&F_{x_{1}x_{1}x_{1}}\Psi _{x_{1}x_{1}x_{1}}+(2F_{x_{1}x_{1}x_{2}}+F_{x_{1}x_{1}x_{1}x_{1}})\Psi _{x_{1}x_{1}}+ \\ &&+(3F_{x_{1}x_{1}x_{1}x_{2}}+F_{x_{1}x_{1}x_{1}x_{1}x_{1}}+F_{x_{1}x_{2}x_{2}})\Psi _{x_{1}}+ \\ &&+(F_{x_{1}x_{1}x_{1}x_{1}x_{2}}+F_{x_{1}x_{1}x_{2}x_{2}}+F_{x_{2}x_{2}x_{2}}-(F_{x_{1}x_{1}x_{2}})^{2}+F_{x_{1}x_{1}x_{1}}F_{x_{1}x_{2}x_{2}})\Psi .\end{aligned}$$ The first of these equations represents itself the reduction of the linear problem $$\Psi _{x_{2}}=\Psi _{x_{1}x_{1}}+\mathit{V}\Psi _{x_{1}}+\mathit{U}\Psi$$ associated with the generalized KP-modified KP hierarchy. So, the WDVVequation is a very special reduction of the generalized KP-mKP hierarchy stationary with the respect to the times $x_{3},x_{4}$.  Note that the second and the third of above equations are equivalent if F obeys WDVVequation. For the similar results see \[13\]. Mixed DDA and Darboux transformations ===================================== For $N\geq 4$ the CS (16) ( or (18)) generated by  the DDA (12) contains much more mixed cases between the pure continuous and pure discrete equations. The situation with N-1 continuous variables and one discrete , i.e. when $\varepsilon _{0}=\varepsilon _{1}=...=\varepsilon _{N-2}=0$ and $\varepsilon _{N-1}=1$ is of particular interest. The commutation relations (12) defining the DDA of this type can be presented as $$\begin{aligned} \lbrack p_{j},p_{k}] =0,\quad \lbrack x^{j},x^{k}]=0,\quad \lbrack p_{j},x^{k}]=\hbar \delta _{j}^{k},\quad j,k=0,1,...,N-2, \\ \lbrack p_{j},T] =0,\quad \lbrack x^{j},n]=0,\quad \lbrack p_{j},n]=0,\quad \lbrack T,n]=T\end{aligned}$$ where $T=p_{N-1}+1$ and n is a discrete variable associated with T. The table of multiplication in the general case is $$\begin{aligned} \mathbf{P}_{j}\mathbf{P}_{k}& =&\sum_{l=0}^{N-2}C_{jk}^{l}\mathbf{P}_{l}+C_{jk}^{N-1}T,\quad j,k=0,1,...N-2 \notag \\ \mathbf{TP}_{k}& =&\sum_{l=0}^{N-2}C_{N-1k}^{l}\mathbf{P}_{l}+C_{N-1k}^{N-1}T,\quad k=0,1,...N-2 \notag \\ \mathbf{T}^{2}& =&\sum_{l=0}^{N-2}C_{N-1N-1}^{l}\mathbf{P}_{l}+C_{N-1N-1}^{N-1}T.\quad\end{aligned}$$ The change of the basis $p_{N-1}\rightarrow T$ in the table (31) is performed for the further convenience. Under the constraint $C_{jk}^{N-1}=0,j,k=0,1,...,N-2$ such an algebra has  a special meaning. Namely, it contains the subalgebra spanned by the elements $p_{0},p_{1},...,p_{N-2}$.  As far as the CS is concerned, one can show that it separates into two parts, namely,$$\frac{\partial C_{j}}{\partial x^{k}}-\frac{\partial C_{k}}{\partial x^{j}}+[C_{k},C_{j}]=0,\quad j,k=0,1...,N-2$$ and $$\frac{\partial C_{N-1}}{\partial x^{l}}+C_{l}C_{N-1}-C_{N-1}TC_{l}=0, \quad l=0,1,...,N-2$$ where $TC_{l}(x^{1},x^{2},...,x^{N-2},n)=C_{l}$ $(x^{1},x^{2},...,x^{N-2},n+1).$ Note the difference with the CS (16) in virtue of the change $p_{N-1}\rightarrow T$. Compatibility conditions for equations (46) are satisfied due to the fact that the matrices $TC_{j}(j=0,1,...,N-2)$ obey equations (45) together with $C_{j}$. It is not difficult to show that the system (45) contains a closed subsystem for the first $(N-2)\times (N-2)$ principal minors $\widetilde{C}_{j}$ of the matrices $C_{j}$ $((\widetilde{C_{j}})_{k}^{l}=C_{jk}^{l},l,k=0,1,...,N-2)$, i.e. $$\frac{\partial \widetilde{C}_{j}}{\partial x^{k}}-\frac{\partial \widetilde{C}_{k}}{\partial x^{j}}+[\widetilde{C}_{k},\widetilde{C}_{j}]=0,\quad j,k=0,1...,N-2$$ This subsystem defines an integrable system of differential equations which coincides with the QCS (5) with $j,k,l,n=0,1,...,N-2$ and the independent variables $x^{1},x^{2},...,x^{N-2}$. The rest of the system (45) and equations (46) determine a tranformation $\widetilde{C}_{l}\rightarrow T\widetilde{C}_{l}$ which converts solutions of the system (47) into the solutions of the same system.Thus, equations (45)-(46) define a Darboux tranformation for the continuous system (47). We will consider the simplest case N=4 to illustrate this general scheme. Thus, we consider the four-dimensional associative algebra with the commutative basis $\mathbf{P}_{0},\mathbf{P}_{1},\mathbf{P}_{2},\mathbf{T}$ . The table of multiplication is given by (nontrivial part)$$\begin{aligned} \mathbf{P}_{1}^{2} &=&A\mathbf{P}_{0}+B\mathbf{P}_{1}+C\mathbf{P}_{2}, \notag \\ \mathbf{P}_{1}\mathbf{P}_{2}& =&D\mathbf{P}_{0}+E\mathbf{P}_{1}+G\mathbf{P}_{2}, \notag \\ \mathbf{P}_{2}^{2} &=&L\mathbf{P}_{0}+M\mathbf{P}_{1}+N\mathbf{P}_{2}, \notag \\ \mathbf{P}_{1}\mathbf{T}& =&a_{1}\mathbf{P}_{0}+a_{2}\mathbf{P}_{1}+a_{3}\mathbf{P}_{2}+a_{4}\mathbf{T,} \notag \\ \mathbf{P}_{2}\mathbf{T}& =&b_{1}\mathbf{P}_{0}+b_{2}\mathbf{P}_{1}+b_{3}\mathbf{P}_{2}+b_{4}\mathbf{T,} \notag \\ \mathbf{T}^{2}& =&c_{1}\mathbf{P}_{0}+c_{2}\mathbf{P}_{1}+c_{3}\mathbf{P}_{2}+c_{4}\mathbf{T.}\end{aligned}$$The matrices $C_{1},C_{2},C_{3}(C_{0}=1)$ are $$C_{1}=\left( \begin{array}{cccc} 0 & A & D & a_{1} \\ 1 & B & E & a_{2} \\ 0 & C & G & a_{3} \\ 0 & 0 & 0 & a_{4}\end{array}\right) ,C_{2}=\left( \begin{array}{cccc} 0 & D & L & b_{1} \\ 0 & E & M & b_{2} \\ 1 & G & N & b_{3} \\ 0 & 0 & 0 & b_{4}\end{array}\right) ,C_{3}=\left( \begin{array}{cccc} 0 & a_{1} & b_{1} & c_{1} \\ 0 & a_{2} & b_{2} & c_{2} \\ 0 & a_{3} & b_{3} & c_{3} \\ 1 & a_{4} & b_{4} & c_{4}\end{array}\right) .$$ Equation (47) with $j=1,k=2$ is the pure continuous system (20). The rest of equations (45) have the following form $$\begin{aligned} a_{1x_{2}}-b_{1x_{1}}+Da_{2}+La_{3}-Ab_{2}-Db_{3}+b_{1}a_{4}-a_{1}b_{4} =0, \\ a_{2x_{2}}-b_{2x_{1}}+Ea_{2}+Ma_{3}-Bb_{2}-Eb_{3}-b_{1}+b_{2}a_{4}-a_{2}b_{4} =0, \\ a_{3x_{2}}-b_{3x_{1}}+Ga_{2}+Na_{3}-Cb_{2}-Gb_{3}+a_{1}+b_{3}a_{4}-a_{3}b_{4} =0, \\ a_{4x_{2}}-b_{4x_{1}} =0.\end{aligned}$$ Matrix equations (46) are equivalent to 24 scalar differential equations. They are presented in the Appendix. Analyzing equations (49)-(52) and those in the Appendix, one concludes that they are not all independent. For instance, equation (49) coincides with the difference of equations (87) and (78). Equation (50) coincides with the difference of equations (88) and (79) while equations (51) and (52) are similar consequences of equations (89), (80) and equations (90), (81), respectively. Further , it follows that all these equations, in principle, allow us to reconstruct $a_{j},b_{j},c_{j}(j=1,2,3,4)$ and finally $TA,TB$ etc in terms of A,B etc. Darboux transformation for the Boussinesq equation as the discrete deformation ============================================================================== Here we will present solution of the above equations for the Boussinesq reduction $B=0,C=1,G=0$. In this case $L=D_{x_{1}}-A_{x_{2}}+\frac{1}{2}A^{2},M=\frac{1}{2}A_{x_{1}}+D,N=-\frac{1}{2}A$ (see section 3 ). Let us begin with equations (77) and (81) which in the BSQ gauge are of the form $$\begin{aligned} a_{4x_{1}}+a_{4}^{2}-b_{4}-TA =0, \\ b_{4x_{1}}+a_{4}b_{4}-\frac{1}{2}a_{4}TA-TD =0.\end{aligned}$$ Equation (52) implies that there exists a function $\varphi $ such that $$a_{4}=(\ln \varphi )_{x_{1}},\quad b_{4}=(\ln \varphi )_{x_{2}}.$$ Substituting these expressions into (53) and (54), one gets the equations $$\begin{aligned} \varphi _{x_{2}} =\varphi _{x_{1}x_{1}}-TA\varphi , \\ \varphi _{x_{1}x_{2}} =\frac{1}{2}TA\varphi _{x_{1}}+TD\varphi .\end{aligned}$$ These equations are just the linear problems (29) for the continuous BSQ equation for the shifted solutions TA and TD.  This observation serves a lot for the further calculations. To simplify them we consider a special case for which $a_{3}=1,c_{3}=1.$  With such a choice equations (51), (76), (80), (84), (89) take the form $$\begin{aligned} -b_{3x_{1}}+a_{1}+\frac{1}{2}A+b_{3}a_{4}-b_{2}-b_{4} =0, \\ a_{2}+a_{4}-b_{3} =0, \\ b_{3x_{1}}+b_{2}+b_{4}-\frac{1}{2}TA =0, \\ c_{2}+c_{4}-b_{3}-T(a_{2}+a_{4}) =0, \\ a_{1}-\frac{1}{2}A+b_{3}a_{4}-\frac{1}{2}TA =0.\end{aligned}$$  Rather involved analysis, which we omit, shows that equations (58)-(62) and others have a solution $$\begin{aligned} a_{1}& =&A-(\ln \widetilde{\Psi })_{x_{1}x_{1}}+(\ln \widetilde{\Psi })_{x_{1}}(\ln \varphi )_{x_{1}},\quad a_{2}=-(\ln \widetilde{(\Psi }\varphi ))_{x_{1}}, \notag \\ a_{3}& =&1,\quad a_{4}=(\ln \varphi )_{x_{1}}, \notag \\ b_{1}& =&D-(\ln \widetilde{\Psi })_{x_{1}x_{2}}+(\ln \widetilde{\Psi })_{x_{1}}(\ln \varphi )_{x_{2}},\quad b_{2}=\frac{1}{2}A-(\ln \varphi )_{x_{2}}, \notag \\ b_{3}& =&-(\ln \widetilde{\Psi })_{x_{1}},\quad b_{4}=(\ln \varphi )_{x_{2}}, \\ c_{1}& =&A-(\ln \widetilde{\Psi })_{x_{1}x_{1}}-(\ln \widetilde{\Psi })_{x_{1}}(\ln \varphi )_{x_{1}}+(\ln \widetilde{\Psi })_{x_{1}}\frac{T\varphi }{\varphi }, \notag \\ c_{2}& =&-(\ln (\widetilde{\Psi }\varphi ))_{x_{1}}-\frac{T\varphi }{\varphi },\quad c_{3}=1,\quad c_{4}=\frac{T\varphi }{\varphi } \notag\end{aligned}$$ and $$\begin{aligned} TA =A-2(\ln \widetilde{\Psi })_{x_{1}x_{1}}, \\ TD =D+\frac{1}{2}(\ln \widetilde{\Psi })_{x_{1}x_{1}x_{1}}-\frac{3}{2}(\ln \widetilde{\Psi })_{x_{1}x_{2}}\end{aligned}$$ where $\widetilde{\Psi }$ is a solution of the linear problems for the BSQ equation $$\begin{aligned} \Psi _{x_{2}} =\Psi _{x_{1}x_{1}}-A\Psi , \notag \\ \Psi _{x_{1}x_{2}} =\frac{1}{2}A\Psi _{x_{1}}+D\Psi\end{aligned}$$ and $\varphi $ solves equations (56-57).  The formulae (64) and (65) represent the well-known Darboux transformation for BSQ equation (23) ( see e.g.\[14\] ). Recall that in the terms of the $\tau -$ function defined via $$A=-2F_{x_{1}x_{1}},\quad D=-\frac{3}{2}F_{x_{1}x_{2}}+\frac{1}{2}F_{x_{1}x_{1}x_{1}}$$ the Darboux transformation  is a very simple one $$T\tau =\widetilde{\Psi }\tau .$$ Thus , we have constructed deformations of the structure constants from the table of multiplication (48) parametrized by two continuous variables $x^{1},x^{2}$ and one discrete variable n. They are governed by the BSQ equation (23), formulae (21), (63) and transformations (64),(65). From the viewpoint of integrable systems the corresponding CS  represents a composition of the BSQ equation and its Darboux transformation. These deformations have an important algebraic property. Using expressions (63), one can show that the linear problems $f_{jk}\mid \Psi \rangle =0$ are equivalent to the following one $$\begin{aligned} \left( p_{2}-p_{1}^{2}+Ap_{0}\right)\mid\Psi \rangle =0, \notag \\ \left( p_{1}p_{2}-\frac{1}{2}Ap_{1}-Dp_{0}\right)\mid\Psi \rangle =0,\end{aligned}$$ and $$\begin{aligned} (p_{1}-a_{4}p_{0})(T-p_{1}+(\ln \widetilde{\Psi })_{x_{1}}p_{0})\mid\Psi \rangle =0, \notag \\ (p_{2}-b_{4}p_{0})(T-p_{1}+(\ln \widetilde{\Psi })_{x_{1}}p_{0})\mid\Psi \rangle =0, \\ (T-c_{4}p_{0})(T-p_{1}+(\ln \widetilde{\Psi })_{x_{1}}p_{0})\mid\Psi \rangle =0. \notag\end{aligned}$$ Factorization of the linear operators in the problems (70) which correspond to the discrete part of the basic equations $f_{jk}\mid \Psi \rangle =0$ is the characteristic property of the deformations considered above. In the coordinate representation for which $p_{1}=\partial _{x_{1}},p_{2}=\partial _{x_{2}},Tf(x,n)=f(x,n+1)$ the linear problems (70) have the form $$\begin{aligned} (\partial _{x_{1}}-(\ln \varphi )_{x_{1}})\left( T\Psi -(\Psi _{x_{1}}-(\ln \widetilde{\Psi })_{x_{1}}\Psi )\right) =0, \notag \\ (\partial _{x_{2}}-(\ln \varphi )_{x_{2}})\left( T\Psi -(\Psi _{x_{1}}-(\ln \widetilde{\Psi })_{x_{1}}\Psi )\right) =0, \\ (T-\frac{T\varphi }{\varphi })\left( T\Psi -(\Psi _{x_{1}}-(\ln \widetilde{\Psi })_{x_{1}}\Psi )\right) =0. \notag\end{aligned}$$ These equations imply that $$T\Psi =\Psi _{x_{1}}-(\ln \widetilde{\Psi })_{x_{1}}\Psi +\alpha \varphi$$ where $\alpha $ is an arbitrary constant. At $\ \alpha =0$ the formula (72) is nothing but  the standard transformation of the BSQ wavefunction $\Psi $ under the Darboux transformation  (see e.g. \[14\] ). For $\alpha \neq 0$ the function $T\Psi -\alpha \varphi $ also solves linear problems (66) with the potentials TA, TD. Thus, the theory of the Darboux transformations ( at least, for the BSQ equation) is completely incorporated into the deformation theory presented in this paper. The situation with the WDVV equation is quite different. At the reduction $C=1,G=N=0$ and consequently $E=A,M=D$ one gets equations (49)-(52) and others which are not too different from those for the BSQ equation.  But in contrast with the former one these equations are equivalent to the overdetermined  system of nonlinear PDEs which are doubtly compatible. The conjecture is that in order to construct Darboux type transformation for the WDVV equation one should consider higher-dimensional algebras ( $N\geq 5).$ Appendix ======== In the case N=2 the matrix equation $$\frac{\partial C_{3}}{\partial x^{1}}+C_{1}C_{3}-C_{3}TC_{1}=0$$ is equivalent to the following twelve equations $$\frac{\partial a_{1}}{\partial x_{1}}+Aa_{2}+Da_{3}+a_{1}a_{4}-a_{1}TB-b_{1}TC=0,$$ $$\frac{\partial a_{2}}{\partial x_{1}}+a_{1}+Ba_{2}+Ea_{3}+a_{2}a_{4}-a_{2}TB-b_{2}TC=0,$$ $$\frac{\partial a_{3}}{\partial x_{1}}+Ca_{2}+Ga_{3}+a_{3}a_{4}-a_{3}TB-b_{3}TC=0,$$ $$\frac{\partial a_{4}}{\partial x_{1}}+a_{4}^{2}-TA-a_{4}TB-b_{4}TC=0,$$ $$\frac{\partial b_{1}}{\partial x_{1}}+Ab_{2}+Db_{3}+a_{1}b_{4}-a_{1}TE-b_{1}TG=0,$$ $$\frac{\partial b_{2}}{\partial x_{1}}+b_{1}+Bb_{2}+Eb_{3}+a_{2}b_{4}-a_{2}TE-b_{2}TG=0,$$ $$\frac{\partial b_{3}}{\partial x_{1}}+Cb_{2}+Gb_{3}+a_{3}b_{4}-a_{3}TE-b_{3}TG=0,$$ $$\frac{\partial b_{4}}{\partial x_{1}}+a_{4}b_{4}-TD-a_{4}TE-b_{4}TG=0,$$ $$\frac{\partial c_{1}}{\partial x_{1}}+Ac_{2}+Dc_{3}+a_{1}c_{4}-a_{1}Ta_{2}-b_{1}Ta_{3}-c_{1}Ta_{4}=0,$$ $$\frac{\partial c_{2}}{\partial x_{1}}+c_{1}+Bc_{2}+Ec_{3}+a_{2}c_{4}-a_{2}Ta_{2}-b_{2}Ta_{3}-c_{2}Ta_{4}=0,$$ $$\frac{\partial c_{3}}{\partial x_{1}}+Cc_{2}+Gc_{3}+a_{3}c_{4}-a_{3}Ta_{2}-b_{3}Ta_{3}-c_{3}Ta_{4}=0,$$ $$\frac{\partial c_{4}}{\partial x_{1}}+a_{4}c_{4}-Ta_{1}-a_{4}Ta_{2}-b_{4}Ta_{3}-c_{4}Ta_{4}=0.$$ Equation $$\frac{\partial C_{3}}{\partial x^{2}}+C_{2}C_{3}-C_{3}TC_{2}=0$$ takes the form $$\frac{\partial a_{1}}{\partial x_{2}}+Da_{2}+La_{3}+b_{1}a_{4}-a_{1}TE-b_{1}TG=0,$$ $$\frac{\partial a_{2}}{\partial x_{2}}+Ea_{2}+Ma_{3}+b_{2}a_{4}-a_{2}TE-b_{2}TG=0,$$ $$\frac{\partial a_{3}}{\partial x_{2}}+a_{1}+Ga_{2}+Na_{3}+b_{3}a_{4}-a_{3}TE-b_{3}TG=0,$$ $$\frac{\partial a_{4}}{\partial x_{2}}+b_{4}a_{4}-TD-a_{4}TE-b_{4}TG=0,$$ $$\frac{\partial b_{1}}{\partial x_{2}}+Db_{2}+Lb_{3}+b_{1}b_{4}-a_{1}TM-b_{1}TN=0,$$ $$\frac{\partial b_{2}}{\partial x_{2}}+Eb_{2}+Mb_{3}+b_{2}b_{4}-a_{2}TM-b_{2}TN=0,$$ $$\frac{\partial b_{3}}{\partial x_{2}}+b_{1}+Gb_{2}+Nb_{3}+b_{3}b_{4}-a_{3}TM-b_{3}TN=0,$$ $$\frac{\partial b_{4}}{\partial x_{2}}+b_{4}^{2}-TL-a_{4}TM-b_{4}TN=0,$$ $$\frac{\partial c_{1}}{\partial x_{2}}+Dc_{2}+Lc_{3}+b_{1}c_{4}-a_{1}Tb_{2}-b_{1}Tb_{3}-c_{1}Tb_{4}=0,$$ $$\frac{\partial c_{2}}{\partial x_{2}}+Ec_{2}+Mc_{3}+b_{2}c_{4}-a_{2}Tb_{2}-b_{2}Tb_{3}-c_{2}Tb_{4}=0,$$ $$\frac{\partial c_{3}}{\partial x_{2}}+c_{1}+Gc_{2}+Nc_{3}+b_{3}c_{4}-a_{3}Tb_{2}-b_{3}Tb_{3}-c_{3}Tb_{4}=0,$$ $$\frac{\partial c_{4}}{\partial x_{2}}+b_{4}c_{4}-Tb_{1}-a_{4}Tb_{2}-b_{4}Tb_{3}-c_{4}Tb_{4}=0.$$ References ========== 1\. Witten E., On the structure of topological phase of two-dimensional gravity, Nucl. Phys., 340, 281-332 (1990). 2\. Dijkgraaf R., Verlinde H. and Verlinde E., Topological strings in d$\prec $1, Nucl. Phys., B 352, 59-86 (1991). 3\. Dubrovin B., Integrable systems in topological field theory, Nucl. Phys., B 379, 627-689 (1992). 4\. Dubrovin B., Geometry of 2D topological field theories, Lecture Notes in Math., 1620, 120-348 (1996), Springer, Berlin. 5\. Manin Y.I., Frobenius manifolds, quantum cohomology and moduli spaces, AMS, Providence, 1999. 6\. Hertling C., Frobenius manifolds and moduli spaces for singularities, Cambridge Univ. Press, 2002. 7\. Konopelchenko B.G. and Magri F., Coisotropic deformations of associative algebras and dispersionless integrable hierarchies, Commun. Math. Phys., 274, 627-658 (2007). 8\. Konopelchenko B.G. and Magri F., Dispersionless integrable equations as coisotropic deformations: extensions and reductions, Theor. Math. Phys., 151, 803-819 (2007). 9\. Konopelchenko B.G. and Magri F., Coisotropic deformations of associative algebras,Yano manifolds and integrable systems, to appear. 10\. Konopelchenko B.G., Quantum deformations of associative algebras and integrable systems, arXiv:0802.3022,2008. 11\. Konopelchenko B.G., Discrete, q-difference deformations of associative algebras and integrable systems, arXiv:0809.1938, 2008. 12\. Zakharov V.E., On stochastization of one-dimensional chain of nonlinear oscillators, Sov. Phys. JETP, 35, 908-914 (1974). 13\. Givental A., $A_{n-1}$ singularities and nKdV hierarchies, Mosc. Math. J., 3, 475-505 (2003). 14\. Matveev V.B. and Salle M.A., Darboux transformations and solitons, Springer-Verlag, Berlin, 1991.

--- abstract: 'In a real Hilbert space $\mathcal H$, we study the fast convergence properties as $t \to + \infty$ of the trajectories of the second-order evolution equation $$\ddot{x}(t) + \frac{\alpha}{t} \dot{x}(t) + \nabla \Phi (x(t)) = 0,$$ where $\nabla \Phi$ is the gradient of a convex continuously differentiable function $\Phi : \mathcal H \rightarrow \mathbb R$, and $\alpha$ is a positive parameter. In this inertial system, the viscous damping coefficient $\frac{\alpha}{t}$ vanishes asymptotically in a moderate way. For $\alpha > 3$, we show that any trajectory converges weakly to a minimizer of $\Phi$, just assuming that $\operatorname{argmin}\Phi \neq \emptyset$. The strong convergence is established in various practical situations. These results complement the $\mathcal O(t^{-2})$ rate of convergence for the values obtained by Su, Boyd and Candès. Time discretization of this system, and some of its variants, provides new fast converging algorithms, expanding the field of rapid methods for structured convex minimization introduced by Nesterov, and further developed by Beck and Teboulle. This study also complements recent advances due to Chambolle and Dossal.' address: - 'Institut de Mathématiques et Modélisation de Montpellier, UMR 5149 CNRS, Université Montpellier 2, place Eugène Bataillon, 34095 Montpellier cedex 5, France' - 'Universidad Técnica Federico Santa María, Av España 1680, Valparaíso, Chile' - 'Institut de Mathématiques et Modélisation de Montpellier, UMR 5149 CNRS, Université Montpellier 2, place Eugène Bataillon, 34095 Montpellier cedex 5, France' author: - Hedy Attouch - Juan Peypouquet - Patrick Redont date: 'June 11, 2015' title: 'ON THE FAST CONVERGENCE OF AN INERTIAL GRADIENT-LIKE DYNAMICS WITH VANISHING VISCOSITY' --- [^1] Introduction ============ Let $\mathcal H$ be a real Hilbert space, which is endowed with the scalar product $\langle \cdot,\cdot\rangle$ and norm $\|\cdot\|$, and let $\Phi : \mathcal H \rightarrow \mathbb R$ be a (smooth) convex function. In this paper, we study the solution trajectories of the second-order differential equation $$\label{edo01} \ddot{x}(t) + \frac{\alpha}{t} \dot{x}(t) + \nabla \Phi (x(t)) = 0,$$ with $\alpha>0$, in terms of their asymptotic behavior as $t \to + \infty$. Although this is not our main concern, we point out that, given $t_0>0$, for any $x_0 \in \mathcal H, \ v_0 \in \mathcal H $, the existence of a unique global solution on $[t_0,+\infty[$ for the Cauchy problem with initial condition $x(t_0)=x_0$ and $\dot x(t_0)=v_0$ can be guaranteed, for instance, if $\nabla\Phi$ is Lipschitz-continuous on bounded sets. The importance of this evolution system is threefold: [*1. Mechanical interpretation:*]{} It describes the position of a particle subject to a potential energy function $\Phi$, and an isotropic linear damping with a viscosity parameter that vanishes asymptotically. This provides a simple model for a progressive reduction of the friction, possibly due to material fatigue. [*2. Fast minimization of function $\Phi$:*]{} Equation is a particular case of the inertial gradient-like system $$\label{edo2} \ddot{x}(t) + a(t) \dot{x}(t) + \nabla \Phi (x(t)) = 0,$$ with [*asymptotic vanishing damping*]{}, studied by Cabot, Engler and Gadat in [@CEG1; @CEG2]. As shown in [@CEG1 Corollary 3.1] (under some additional conditions on $\Phi$), every solution $x(\cdot)$ of satisfies $\lim_{t\to+\infty}\Phi(x(t))=\min\Phi$, provided $\int_0^{\infty}a(t)dt = + \infty$. The specific case was studied by Su, Boyd and Candès in [@SBC] in terms of the rate of convergence for the values. More precisely, [@SBC Theorem 4.1] establishes that $\Phi(x(t))-\min\Phi =\mathcal O(t^{-2})$, whenever $\alpha\ge 3$. Unfortunately, their analysis does not entail the convergence of the trajectory itself. [*3. Relationship with fast numerical optimization methods:*]{} As pointed out in [@SBC Section 2], for $\alpha=3$, can be seen as a continuous version of the fast convergent method of Nesterov (see [@Nest1; @Nest2; @Nest3; @Nest4]), and its widely used successors, such as the [*Fast Iterative Shrinkage-Thresholding Algorithm*]{} (FISTA), studied in [@BT]. These methods have a convergence rate of $\Phi(x^k)-\min\Phi=\mathcal O(k^{-2})$, where $k$ is the number of iterations. As for the continuous-time system , convergence of the sequences generated by FISTA and related methods has not been established so far. This is a central and long-standing question in the study of numerical optimization methods. The purpose of this research is to establish the convergence of the trajectories satisfying , as well as the sequences generated by the corresponding numerical methods with Nesterov-type acceleration. We also complete the study with several stability properties concerning both the continuous-time system and the algorithms. More precisely, the main contributions of this work are the following: In Section \[S:minimizing\], we first establish the minimizing property in the general case where $\alpha>0$ and $\inf\Phi$ is not necessarily attained. As a consequence, every weak limit point of the trajectory must be a minimizer of $\Phi$, and so, the existence of a bounded trajectory characterizes the existence of minimizers. Next, assuming $\operatorname{argmin}\Phi\neq\emptyset$ and $\alpha\ge 3$, we recover the $\mathcal O(t^{-2})$ convergence rates and give several examples and counterexamples concerning the optimality of these results. Next, we show that every solution of converges weakly to a minimizer of $\Phi$ provided $\alpha >3$ and $\operatorname{argmin}\Phi \neq \emptyset$. We rely on a Lyapunov analysis, which was first used by Alvarez [@Al] in the context of the heavy ball with friction. For the limiting case $\alpha=3$, which corresponds exactly to Nesterov’s method, the convergence of the tra! jectories is still a puzzling open question. We finish this section by providing an ergodic convergence result for the acceleration of the system in case $\nabla\Phi$ is Lipschitz-continuous on sublevel sets of $\Phi$. Strong convergence is established in various practical situations enjoying further geometric features, such as strong convexity, symmetry, or nonemptyness of the interior of the solution set (see Section \[S:strong\]). In the strongly convex case, we obtained a surprising result: convergence of the values occurs at a rate of $\mathcal O(t^{-\frac{2\alpha}{3}})$. Section \[S:algorithm\] contains the analogous results for the associated Nesterov-type algorithms (which also correspond to the case $\alpha>3$). As we were preparing the final version of this manuscript, we discovered the preprint [@CD] by Chambolle and Dossal, where the weak convergence result is obtained by a similar, but different, argument (see [@CD Theorem 3]). Minimizing property, convergence rates and weak convergence of the trajectories {#S:minimizing} =============================================================================== We begin this section by providing some preliminary estimations concerning the global energy of the system and the distance to the minimizers of $\Phi$. These allow us to show the minimizing property of the trajectories under minimal assumptions. Next, we recover the convergence rates for the values originally given in [@SBC] and obtain further decay estimates that ultimately imply the convergence of the solutions of . We finish the study by proving an ergodic convergence result for the acceleration. Several examples and counterexamples are given throughout the section. Preliminary remarks and estimations ----------------------------------- The existence of global solutions to has been examined, for instance, in [@CEG1 Proposition 2.2.] in the case of a general asymptotic vanishing damping coefficient. In our setting, for any $t_0 >0$, $\alpha >0$, and $(x_0,v_0)\in \mathcal H\times\mathcal H$, there exists a unique global solution $x : [t_0, +\infty[ \rightarrow \mathcal H$ of , satisfying the initial condition $x(t_0)= x_0$, $\dot{x}(t_0)= v_0$, under the sole assumption that $\inf\Phi > - \infty$. Taking $t_0>0$ comes from the singularity of the damping coefficient $a(t) = \frac{\alpha}{t}$ at zero. Indeed, since we are only concerned about the asymptotic behavior of the trajectories, we do not really care about the origin of time. If one insists in starting from $t_0 =0$, then all the results remain valid with $a(t)=\frac{\alpha}{t+1}$. At different points, we shall use the [*global energy*]{} of the system, given by $W:[t_0,+\infty [\rightarrow{{\mathbb R}}$ $$\label{E:W} W(t) = \frac{1}{2}\|\dot{x}(t)\|^2 + \Phi (x(t)).$$ Using , we immediately obtain \[L:W\] Let $W$ be defined by . For each $t>t_0$, we have $$\dot{W}(t) = -\frac{\alpha}{t}\|\dot{x}(t)\|^2.$$ Hence, $W$ is nonincreasing[^2], and $W_\infty=\lim_{t\rightarrow+\infty}W(t)$ exists in ${{\mathbb R}}\cup\{-\infty\}$. If $\Phi$ is bounded from below, $W_\infty$ is finite. Now, given $z\in\mathcal H$, we define $h_z:[t_0,+\infty [ \to{{\mathbb R}}$ by $$\label{E:h_z} h_z(t)=\frac{1}{2}\|x(t)-z\|^2.$$ By the Chain Rule, we have $$\dot{h}_z(t) = \langle x(t) - z , \dot{x}(t) \rangle\qquad\hbox{and}\qquad \ddot{h}_z(t) = \langle x(t) - z , \ddot{x}(t) \rangle + \| \dot{x}(t) \|^2.$$ Using , we obtain $$\label{wconv3} \ddot{h}_z(t) + \frac{\alpha}{t} \dot{h}_z(t) = \| \dot{x}(t) \|^2 + \langle x(t) - z , \ddot{x}(t) + \frac{\alpha}{t} \dot{x}(t) \rangle = \| \dot{x}(t) \|^2 + \langle x(t) - z , -\nabla \Phi (x(t)) \rangle .$$ The convexity of $ \Phi$ implies $$\langle x(t) - z , \nabla \Phi (x(t)) \rangle \geq \Phi (x(t)) - \Phi (z),$$ and we deduce that $$\label{E:ineq_h_z} \ddot{h}_z(t) + \frac{\alpha}{t} \dot{h}_z(t) +\Phi (x(t)) - \Phi (z) \leq \|\dot{x}(t)\|^2.$$ We have the following relationship between $h_z$ and $W$: \[L:Wh\_z\] Take $z\in \mathcal H$, and let $W$ and $h_z$ be defined by and , respectively. There is a constant $C$ such that $$\int_{t_0}^t \frac{1}{s}\left(W(s) - \Phi(z)\right)ds \leq C-\frac{1}{t} \dot{h}_{z}(t)-\frac{3}{2\alpha}W(t).$$ Divide by $t$, and use the definition of $W$ given in , to obtain $$\frac{1}{t} \ddot{h}_{z}(t) + \frac{\alpha}{t^2} \dot{h}_{z}(t) + \frac{1}{t}\left( W(t) - \Phi (z) \right) \leq \frac{3}{2t}\|\dot{x}(t)\|^2.$$ Integrate this expression from $t_0$ to $t>t_0$ (use integration by parts for the first term), to obtain $$\label{conv3-8} \int_{t_0}^t \frac{1}{s}\left( W(s) - \Phi (z) \right)ds \leq \frac{1}{t_0} \dot{h}_{z}(t_0) - \frac{1}{t} \dot{h}_{z}(t) - (\alpha + 1)\int_{t_0}^t \frac{1}{s^2}\dot{h}_{z}(s)ds + \int_{t_0}^t \frac{3}{2s}\|\dot{x}(s)\|^2 ds.$$ On the one hand, Lemma \[L:W\] gives $$\int_{t_0}^t \frac{3}{2s}\|\dot{x}(s)\|^2 ds = \frac{3}{2\alpha}(W(t_0)-W(t)).$$ On the other hand, another integration by parts yields $$\int_{t_0}^t \frac{1}{s^2}\dot{h}_{z}(s)ds = \frac{1}{t^2}h_z(t) - \frac{1}{t_0^2}h_z(t_0) + \int_{t_0}^t\frac{2}{s^3}h_z(s)ds \geq - \frac{1}{t_0^2}h_z(t_0).$$ Combining these inequalities with , we get $$\int_{t_0}^t \frac{1}{s}\left( W(s) - \Phi (z) \right)ds \leq \frac{1}{t_0} \dot{h}_{z}(t_0) - \frac{1}{t} \dot{h}_{z}(t) + (\alpha + 1)\frac{1}{t_0^2}h_z(t_0) + \frac{3}{2\alpha}(W(t_0)-W(t)) = C-\frac{1}{t} \dot{h}_{z}(t)-\frac{3}{2\alpha}W(t),$$ where $C$ collects the constant terms. Minimizing property ------------------- It turns out that the trajectories of minimize $\Phi$ in the completely general setting, where $\alpha>0$, $\operatorname{argmin}\Phi$ is possibly empty and $\Phi$ is not necessarily bounded from below. This property was obtained by Alvarez in [@Al Theorem 2.1] for the heavy ball with friction (where the damping is constant). Similar results can be found in [@CEG1]. We have the following: \[Thm-weak-conv2\] Let $\alpha>0$ and suppose $x:[t_0,+\infty[\rightarrow{\mathcal H}$ is a solution of . Then 1. $W_\infty=\lim_{t \rightarrow+\infty}W(t)=\lim_{t \rightarrow+\infty}\Phi(x(t))=\inf\Phi\in{{\mathbb R}}\cup\{-\infty\}$. 2. As $t\to+\infty$, every weak limit point of $x(t)$ lies in $\operatorname{argmin}\Phi$. 3. If $\operatorname{argmin}\Phi=\emptyset$, then $\lim_{t\to+\infty}\|x(t)\|=+\infty$. 4. If $x$ is bounded, then $\operatorname{argmin}\Phi\neq\emptyset$. 5. If $\Phi$ is bounded from below, then $\lim_{t\to+\infty}\|\dot x(t)\|=0$. 6. If $\Phi$ is bounded from below and $x$ is bounded, then $\lim_{t\to+\infty}\dot h_z(t)=0$ for each $z\in\mathcal H$. Moreover, $$\int_{t_0}^\infty\frac{1}{t}(\Phi(x(t))-\min\Phi)dt<+\infty.$$ To prove i), first set $z\in{\mathcal H}$ and $\tau\geq t>t_0$. By Lemma \[L:W\], $W$ in nonincreasing. Hence, Lemma \[L:Wh\_z\] gives $$(W(\tau)-\Phi(z))\int_{t_0}^t\frac{ds}{s}+\frac{3}{2\alpha}W(\tau) \leq C-\frac{1}{t}\dot h_z(t),$$ which we rewrite as $$(W(\tau)-\Phi(z))\left(\int_{t_0}^t\frac{ds}{s}+\frac{3}{2\alpha}\right) \leq C-\frac{3}{2\alpha}\Phi(z)-\frac{1}{t}\dot h_z(t),$$ and then $$(W(\tau)-\Phi(z))\left(\ln(t)+\frac{3}{2\alpha}-\ln(t_0)\right) \leq C-\frac{3}{2\alpha}\Phi(z)-\frac{1}{t}\dot h_z(t).$$ Integrate from $t=t_0$ to $t=\tau$ to obtain $$(W(\tau)-\Phi(z)) \left( \tau\ln(\tau)-t_0\ln(t_0)+t_0 -\tau+ \left(\frac{2}{3\alpha}-\ln(t_0)\right)(\tau-t_0)\right) \leq \left(C-\frac{3}{2\alpha}\Phi(z)\right)(\tau-t_0)- \int_{t_0}^\tau\frac{1}{t}\dot h_z(t)dt.$$ But $$\int_{t_0}^\tau\frac{\dot h_z(t)}{t}dt =\frac{h_z(\tau)}{\tau}-\frac{h_z(t_0)}{t_0}+\int_{t_0}^\tau\frac{h_z(t)}{t^2}dt \geq-\frac{h_z(t_0)}{t_0}.$$ Hence, $$(W(\tau)-\Phi(z))(\tau\ln(\tau)+A\tau+B) \leq \widetilde C\tau+D,$$ for suitable constants $A$, $B$, $\widetilde C$ and $D$. This immediately yields $W_\infty\leq\Phi(z)$, and hence $W_\infty\leq \inf\Phi$. It suffices to observe that $$\inf\Phi\leq \liminf_{t\to+\infty}\Phi(x(t))\leq \limsup_{t\to+\infty}\Phi(x(t))\leq \lim_{t\to+\infty}W(t)=W_\infty$$ to obtain i). Next, ii) follows from i) by the weak lower-semicontinuity of $\Phi$. Clearly, iii) and iv) are immediate consequences of ii). We obtain v) by using i) and the definition of $W$ given in . For vi), since $\dot h_z(t)=\langle x(t) - z , \dot{x}(t) \rangle$ and $x$ is bounded, v) implies $\lim\limits_{t\to+\infty}\dot h_z(t)=0$. Finally, using the definition of $W$ together with Lemma \[L:Wh\_z\] with $z\in\operatorname{argmin}\Phi$, we get $$\int_{t_0}^\infty\frac{1}{t}(\Phi(x(t))-\min\Phi)dt\le C-\frac{3}{2\alpha}\min\Phi<+\infty,$$ which completes the proof. We shall see in Theorem \[T:alpha3\] that, for $\alpha\ge 3$, the existence of minimizers implies that every solution of is bounded. This gives a converse to part iv) of Theorem \[Thm-weak-conv2\]. If $\Phi$ is not bounded from below, it may be the case that $\|\dot x(t)\|$ does not tend to zero, as shown in the following example: Let $\mathcal H={{\mathbb R}}$ and $\alpha>0$. The function $x(t)=t^2$ satisfies with $\Phi(x)=-2(\alpha+1)x$. Then $\lim_{t\to+\infty}\Phi(x(t))=-\infty=\inf\Phi$, and $\lim_{t\to+\infty}\|\dot x(t)\|=+\infty$. Two “anchored" energy functions {#DefEnerg} ------------------------------- We begin by introducing two important auxiliary functions and showing their basic properties. From now on, we assume $\operatorname{argmin}\Phi\neq\emptyset$. Fix $\lambda\geq0$, $\xi\geq0$, $p\geq0$ and $x^*\in\operatorname{argmin}\Phi$. Let $x:[t_0,+\infty [\to\mathcal H$ be a solution of . For $t\geq t_0$ define $$\begin{aligned} {\mathcal E_{\lambda,\xi}}(t) & = & t^2(\Phi(x(t))-\min\Phi)+\frac{1}{2}\|\lambda(x(t)-x^*)+t\dot x(t)\|^2+ \frac{\xi}{2}\|x(t)-x^*\|^2, \\ {\mathcal E_\lambda^p}(t) & = & t^p\mathcal E_{\lambda,0}(t) \; = \; t^p\left(t^2(\Phi(x(t))-\min\Phi)+ \frac{1}{2}\|\lambda(x(t)-x^*)+t\dot x(t)\|^2\right),\end{aligned}$$ and notice that ${\mathcal E_{\lambda,\xi}}$ and ${\mathcal E_\lambda^p}$ are sums of nonnegative terms. These generalize the energy functions $\mathcal E$ and $\tilde{\mathcal E}$ introduced in [@SBC]. More precisely, $\mathcal E=\mathcal E_{\alpha-1,0}$ and $\tilde{\mathcal E}=\mathcal E_{(2\alpha-3)/3}^1$. We need some preparatory calculations prior to differentiating ${\mathcal E_{\lambda,\xi}}$ and ${\mathcal E_\lambda^p}$. For simplicity of notation, we do not make the dependence of $x$ or $\dot x$ on $t$ explicit. Notice that we use in the second line to dispose of $\ddot x$. $$\begin{aligned} \frac{d}{dt}t^2(\Phi(x)-\min\Phi) & = & 2t(\Phi(x)-\min\phi)+t^2\langle\dot x,\nabla\Phi(x)\rangle \\ \frac{d}{dt}\frac{1}{2}\|\lambda(x-x^*)+t\dot x\|^2 & = & - \lambda t\langle x-x^*,\nabla\Phi (x)\rangle - \lambda(\alpha-\lambda-1)\langle x-x^*,\dot x\rangle - (\alpha-\lambda-1)t\|\dot x\|^2 - t^2\langle\dot x,\nabla\Phi(x)\rangle \\ \frac{d}{dt}\frac{1}{2}\|x-x^*\|^2 & = & \langle x-x^*,\dot x\rangle.\end{aligned}$$ Whence, we deduce $$\begin{aligned} \label{dElambdaxi} \frac{d}{dt}{\mathcal E_{\lambda,\xi}}(t) & = & 2t(\Phi(x)-\min\Phi) - \lambda t\langle x-x^*,\nabla\Phi(x)\rangle + (\xi-\lambda(\alpha-\lambda-1))\langle x-x^*,\dot x\rangle - (\alpha-\lambda-1)t\|\dot x\|^2 \\ \label{dElambdap} \frac{d}{dt}{\mathcal E_\lambda^p}(t) & = & (p+2)t^{p+1}(\Phi(x)-\min\Phi) - \lambda t^{p+1}\langle x-x^*,\nabla\Phi(x)\rangle - \lambda(\alpha-\lambda-1-p)t^p\langle x-x^*,\dot x\rangle \\ \nonumber & & + \frac{\lambda^2 p}{2}t^{p-1}\|x-x^*\|^2 - \left(\alpha-\lambda-1-\frac{p}{2}\right)t^{p+1}\|\dot x\|^2.\end{aligned}$$ \[R:E\_decreasing\] If $y\in\mathcal H$ and $x^*\in\operatorname{argmin}\Phi$, the convexity of $\Phi$ gives $\min\Phi=\Phi(x^*)\ge\Phi(y)+\langle\nabla\Phi(y),x^*-y\rangle$. Using this in with $y=x(t)$, we obtain $$\frac{d}{dt}{\mathcal E_{\lambda,\xi}}(t) \leq (2-\lambda)\,t\,(\Phi(x)-\min\Phi) + (\xi-\lambda(\alpha-\lambda-1))\langle x-x^*,\dot x\rangle - (\alpha-\lambda-1)\,t\,\|\dot x\|^2.$$ If one chooses $\xi^*=\lambda(\alpha-\lambda-1)$, then $$\frac{d}{dt}\mathcal E_{\lambda,\xi^*}(t) \leq(2-\lambda)\,t\,(\Phi(x)-\min\Phi) - (\alpha-\lambda-1)\,t\,\|\dot x\|^2.$$ Therefore, if $\alpha\ge 3$ and $2\le\lambda\le\alpha-1$, then $\mathcal E_{\lambda,\xi^*}$ is nonincreasing. The extreme cases $\lambda=2$ and $\lambda=\alpha-1$ are of special importance, as we shall see shortly. Rate of convergence for the values ---------------------------------- We now recover convergence rate results for the value of $\Phi$ along a trajectory, already established in [@SBC Theorem 4.1]: \[T:SBC\] Let $x:[t_0,+\infty[ \to\mathcal H$ be a solution of and assume $\operatorname{argmin}\Phi\neq\emptyset$. If $\alpha\ge 3$, then $$\Phi(x(t))-\min\Phi \leq \frac{\mathcal E_{\alpha-1,0}(t_0)}{t^2}.$$ If $\alpha>3$, then $$\int_{t_0}^{+\infty}t\big(\Phi(x(t))-\min\Phi\big)\,dt\le \frac{\mathcal E_{\alpha-1,0}(t_0)}{\alpha-3}<+\infty.$$ Suppose $\alpha\ge 3$. Choose $\lambda=\alpha-1$ and $\xi=0$, so that $\xi-\lambda(\alpha-\lambda-1)=\alpha-\lambda-1=0$ and $\lambda-2=\alpha-3$. Remark \[R:E\_decreasing\] gives $$\label{DE1} \frac{d}{dt}\mathcal E_{\alpha-1,0}(t) \leq - (\alpha-3)\,t\,(\Phi(x)-\min\Phi),$$ and $\mathcal E_{\alpha-1,0}$ is nonincreasing. Since $t^2(\Phi(x)-\min\Phi)\leq \mathcal E_{\alpha-1,0}(t)$, we obtain $$\Phi(x(t))-\min\Phi \leq \frac{\mathcal E_{\alpha-1,0}(t_0)}{t^2}.$$ If $\alpha>3$, integrating from $t_0$ to $t$ we obtain $$\int_{t_0}^t s(\Phi(x(s))-\min\Phi)ds \leq \frac{1}{\alpha-3}(\mathcal E_{\alpha-1,0}(t_0)-\mathcal E_{\alpha-1,0}(t))\le \frac{1}{\alpha-3}\mathcal E_{\alpha-1,0}(t_0),$$ which allows us to conclude. It would be interesting to know whether $\alpha=3$ is critical for the convergence rate given above. For the (first-order) steepest descent dynamical system, the typical rate of convergence is $\mathcal O(1/t)$ (see, for instance, [@Pey_Sor Section 3.1]). For the second-order system , we have obtained a rate of $\mathcal O(1/t^2)$. It would be interesting to know whether higher-order systems give the corresponding rates of convergence. Another challenging question is the convergence rate of the trajectories defined by differential equations involving fractional time derivatives, as well as integro-differential equations. Some examples and counterexamples --------------------------------- A convergence rate of $\mathcal O(1/t^2)$ may be attained, even if $\operatorname{argmin}\Phi = \emptyset$ and $\alpha<3$. This is illustrated in the following example: Let $\mathcal H = \mathbb R$ and take $\Phi (x) = \frac{\alpha -1}{2}e^{-2x}$ with $\alpha\geq1$. Let us verify that $x(t)= \ln t$ is a solution of (\[edo01\]). On the one hand, $$\ddot{x}(t) + \frac{\alpha}{t} \dot{x}(t) = \frac{\alpha -1}{t^2} .$$ On the other hand, $\nabla \Phi (x)= -(\alpha -1)e^{-2x}$ which gives $$\nabla \Phi (x(t))= -(\alpha -1)e^{-2 \ln t}= -\frac{\alpha -1}{t^2} .$$ Thus, $x(t)= \ln t$ is a solution of (\[edo01\]). Let us examine the minimizing property. We have $\inf \Phi = 0$, and $$\Phi (x(t)) = \frac{\alpha -1}{2}e^{-2 \ln t}= \frac{\alpha -1}{t^2} .$$ Therefore, one may wonder whether the rapid convergence of the values is true in general. The following example shows that this is not the case: \[X:xtheta\] Let $\mathcal H = \mathbb R$ and take $\Phi (x) = \frac{c}{x^{\theta}}$ , with $\theta>0$ , $\alpha\geq\frac{\theta}{(2+\theta)}$ and $c= \frac{2( 2\alpha + \theta (\alpha -1))}{\theta ( 2+ \theta )^2}$. Let us verify that $x(t)= t^{\frac{2}{2 + \theta}}$ is a solution of . On the one hand, $$\ddot{x}(t) + \frac{\alpha}{t} \dot{x}(t) = \frac{2}{(2+ \theta)^2}(2\alpha + \theta (\alpha -1)) t^{- \frac{2(1+ \theta)}{2 + \theta}}.$$ On the other hand, $\nabla \Phi (x)= -c \theta x^{-\theta -1} $ which gives $$\nabla \Phi (x(t))= -c \theta t^{- \frac{2(1+ \theta)}{2 + \theta}}= -\frac{2}{(2+ \theta)^2}(2\alpha + \theta (\alpha -1)) t^{- \frac{2(1+ \theta)}{2 + \theta}}.$$ Thus, $x(t)= t^{\frac{2}{2 + \theta}}$ is solution of . Let us examine the minimizing property. We have $\inf \Phi = 0$, and $$\Phi (x(t)) = c \frac{1}{t^{\frac{2 \theta}{2 + \theta}}}\ , \mbox{ with }\ \frac{2 \theta}{2 + \theta}<2.$$ We conclude that the order of convergence may be strictly slower than $\mathcal O(1/t^2)$ when $\operatorname{argmin}\Phi=\emptyset$. In the Example \[X:xtheta\], this occurs no matter how large $\alpha$ is. The speed of convergence of $\Phi (x(t))$ to $\inf \Phi$ depends on the behavior of $\Phi (x)$ as $\| x\| \to +\infty$. The above examples suggest that, when $\Phi (x)$ decreases rapidly and attains its infimal value as $\| x\| \to \infty$, we can expect fast convergence of $\Phi (x(t))$. Even when $\operatorname{argmin}\Phi\neq\emptyset$, $\mathcal O (1/t^2)$ is the worst possible case for the rate of convergence, attained as a limit in the following example: Take $\mathcal H = \mathbb R$ and $\Phi (x) = c|x|^{\gamma}$, where $c$ and $\gamma$ are positive parameters. Let us look for nonnegative solutions of of the form $x(t)= \frac{1}{t^{\theta}}$, with $\theta >0$. This means that the trajectory is not oscillating, it is a completely damped trajectory. We begin by determining the values of $c$, $\gamma$ and $\theta$ that provide such solutions. On the one hand, $$\ddot{x}(t) + \frac{\alpha}{t} \dot{x}(t) = \theta (\theta +1 -\alpha) \frac{1}{t^{\theta+ 2}}.$$ On the other hand, $\nabla \Phi (x)= c \gamma |x|^{\gamma -2}x $, which gives $$\nabla \Phi (x(t))= c \gamma \frac{1}{t^{\theta (\gamma -1)}}.$$ Thus, $x(t)= \frac{1}{t^{\theta}}$ is solution of if, and only if, - $\theta+ 2 = \theta (\gamma -1)$, which is equivalent to $\gamma >2$ and $\theta= \frac{2}{\gamma -2}$; and - $c \gamma = \theta (\alpha -\theta -1)$, which is equivalent to $\alpha > \frac{\gamma}{\gamma -2}$ and $c= \frac{2}{\gamma(\gamma -2)}( \alpha - \frac{\gamma}{\gamma -2})$. We have $\min\Phi = 0$ and $$\Phi (x(t)) =\frac{2}{\gamma(\gamma -2)}( \alpha - \frac{\gamma}{\gamma -2} ) \frac{1}{t^{\frac{2 \gamma}{\gamma -2 }}}.$$ The speed of convergence of $\Phi (x(t))$ to $0$ depends on the parameter $\gamma$. As $\gamma$ tends to infinity, the exponent $\frac{2 \gamma}{\gamma -2 }$ tends to $2$. This limiting situation is obtained by taking a function $\Phi$ that becomes very flat around the set of its minimizers. Therefore, without other geometric assumptions on $\Phi$, we cannot expect a convergence rate better than $\mathcal O (1/t^2)$. By contrast, in Section \[S:strong\], we will show better rates of convergence under some geometrical assumptions, like strong convexity of $\Phi$. Weak convergence of the trajectories ------------------------------------ In this subsection, we show the convergence of the solutions of , provided $\alpha>3$. We begin by establishing some preliminary estimations that cannot be derived from the analysis carried out in [@SBC]. The first statement improves part v) of Theorem \[Thm-weak-conv2\], while the second one is the key to proving the convergence of the trajectories of : \[T:alpha3\] Let $x:[t_0,+\infty[ \to\mathcal H$ be a solution of with $\operatorname{argmin}\Phi\neq\emptyset$. - If $\alpha\ge 3$ and $x$ is bounded, then $\|\dot x(t)\|=\mathcal O(1/t)$. More precisely, $$\label{bornedx} \|\dot x(t)\| \leq \frac{1}{t} \left(\sqrt{2\mathcal E_{\alpha-1,0}(t_0)} + (\alpha-1)\sup_{t\geq t_0}\|x(t)-x^*\|\right).$$ - If $\alpha>3$, then $x$ is bounded and $$\label{energy1} \int_{t_0}^{+\infty}t\|\dot x(t)\|^2\,dt\le \frac{\mathcal E_{2,2(\alpha-3)}(t_0)}{\alpha-3}<+\infty.$$ To prove i), assume $\alpha\ge 3$ and $x$ is bounded. From the definition of ${\mathcal E_{\lambda,\xi}}$, we have $\frac{1}{2}\|\lambda(x-x^*)+t\dot x\|^2\leq{\mathcal E_{\lambda,\xi}}(t)$, and so $\|t\dot x\| \leq \sqrt{2{\mathcal E_{\lambda,\xi}}(t)}+\lambda\|x-x^*\|$. By Remark \[R:E\_decreasing\], $\mathcal E_{\alpha-1,0}$ is nonincreasing, and we immediately obtain . In order to show ii), suppose now that $\alpha>3$. Choose $\lambda=2$ and $\xi^*=2(\alpha-3)$. By Remark \[R:E\_decreasing\], we have $$\label{DE2} \frac{d}{dt}\mathcal E_{\lambda,\xi^*}(t) \leq - (\alpha-3)\,t\,\|\dot x\|^2,$$ and $\mathcal E_{\lambda,\xi^*}$ is nonincreasing. From the definition of ${\mathcal E_{\lambda,\xi}}$, we deduce that $\|x(t)-x^*\|^2 \leq \frac{2}{\xi} {\mathcal E_{\lambda,\xi}}(t)$, which gives $$\label{E:x_bounded} \|x(t)-x^*\|^2\leq \frac{\mathcal E_{2,2(\alpha-3)}(t)}{\alpha-3}\le\frac{\mathcal E_{2,2(\alpha-3)}(t_0)}{\alpha-3},$$ and establishes de boundedness of $x$. Integrating from $t_0$ to $t$, and recalling that $\mathcal E_{\lambda,\xi^*}$ is nonnegative, we obtain $$\int_{t_0}^t s\|\dot x(s)\|^2 ds \leq \frac{\mathcal E_{2,2(\alpha-3)}(t_0)}{\alpha-3},$$ as required. In view of and , when $\alpha >3$, we obtain the following explicit bound for $\|\dot x\|$, namely $$\|\dot x(t)\| \leq \frac{1}{t}\left(\sqrt{2\mathcal E_{\alpha-1,0}(t_0)} + (\alpha-1)\sqrt{\frac{\mathcal E_{2,2(\alpha-3)}(t_0)}{\alpha-3}}\right).$$ Since $\lim_{t\to+\infty}\|\dot x(t)\|=0$ by Theorem \[Thm-weak-conv2\], we also have $\lim_{t\to+\infty}t\,\|\dot x(t)\|^2=0$. We are now in a position to prove the weak convergence of the trajectories of , which is the main result of this section: \[T:weak\_convergence\] Let $\Phi : \mathcal H \rightarrow \mathbb R$ be a continuously differentiable convex function. Let $\operatorname{argmin}\Phi\neq\emptyset$ and let $x:[t_0,+\infty[\to\mathcal H$ be a solution of with $\alpha>3$. Then $x(t)$ converges weakly, as $t\to+\infty$, to a point in $\operatorname{argmin}\Phi$. We shall use Opial’s Lemma \[Opial\]. To this end, let $x^*\in\operatorname{argmin}\Phi$ and recall from that $$\ddot{h}_{x^*}(t) + \frac{\alpha}{t} \dot{h}_{x^*}(t) +\Phi (x(t)) - \min\Phi \leq \|\dot{x}(t)\|^2,$$ where $h_z$ is given by . This yields $$t\ddot{h}_{x^*}(t) + \alpha\dot{h}_{x^*}(t)\leq t\|\dot{x}(t)\|^2.$$ In view of Theorem \[T:alpha3\], part ii), the right-hand side is integrable on $[t_0,+\infty[$. Lemma \[basic-edo\] then implies $\lim_{t\to+\infty}h_{x^*}(t)$ exists. This gives the first hypothesis in Opial’s Lemma. The second one was established in part ii) of Theorem \[Thm-weak-conv2\]. A puzzling question concerns the convergence of the trajectories for $\alpha=3$, a question which is directly related to the convergence of the sequences generated by Nesterov’s method. Further stabilization results ----------------------------- Let us complement the study of equation by examining the asymptotic behavior of the acceleration $\ddot{x}$. To this end, we shall use an additional regularity assumption on the gradient of $\Phi$. \[P:acceleration\] Let $\alpha>3$ and let $x:[t_0,+\infty[ \to\mathcal H$ be a solution of with $\operatorname{argmin}\Phi\neq\emptyset$. Assume $\nabla\Phi$ Lipschitz-continuous on bounded sets. Then $\ddot x$ is bounded, globally Lipschitz continuous on $[t_0,+\infty[$, and satisfies $$\lim_{t\to+\infty}\frac{1}{t^\alpha} \int_{t_0}^t s^\alpha \|\ddot x(s)\|^2 ds =0.$$ First recall that $x$ and $\dot x$ are bounded, by virtue of Theorems \[T:alpha3\] and \[Thm-weak-conv2\], respectively. By , we have $$\label{scale-energy7} \ddot{x}(t) = -\frac{\alpha}{t} \dot{x}(t) - \nabla \Phi (x(t)).$$ Since $\nabla \Phi$ is Lipschitz-continuous on bounded sets, it follows from , and the boundedness of $x$ and $\dot x$, that $\ddot {x}$ is bounded on $[t_0, +\infty[$. As a consequence, $\dot {x}$ is Lipschitz-continuous on $[t_0, +\infty[$. Returning to , we deduce that $\ddot{x}$ is Lipschitz-continuous on $[t_0, +\infty[$. Pick $x^*\in\operatorname{argmin}\Phi$, set $h=h_{x^*}$ (to simplify the notation) and use to obtain $$\label{E:h_x*} \ddot{h}(t) + \frac{\alpha}{t} \dot{h}(t) + \langle x(t) - x^{*} , \nabla \Phi (x(t)) \rangle =\| \dot{x}(t) \|^2.$$ Let $L$ be a Lipschitz constant for $\nabla\Phi$ on some ball containing the minimizer $x^*$ and the trajectory $x$. By virtue of the Baillon-Haddad Theorem (see, for instance, [@BaHa], [@Pey Theorem 3.13] or [@Nest2 Theorem 2.1.5]), $\nabla \Phi$ is $\frac{1}{L}$-cocoercive on that ball, which means that $$\langle x(t) - x^{*} , \nabla \Phi (x(t))- \nabla \Phi (x^{*})\rangle \geq \frac{1}{L} \| \nabla \Phi (x(t))- \nabla \Phi (x^{*}) \|^2.$$ Substituting this inequality in , and using the fact that $\nabla \Phi (x^{*}) =0$, we obtain $$\ddot{h}(t) + \frac{\alpha}{t} \dot{h}(t) + \frac{1}{L} \| \nabla \Phi (x(t)) \|^2 \leq \| \dot{x}(t) \|^2.$$ In view of , this gives $$\ddot{h}(t) + \frac{\alpha}{t} \dot{h}(t) + \frac{1}{L} \| \ddot{x}(t) +\frac{\alpha}{t} \dot{x}(t) \|^2 \leq \| \dot{x}(t) \|^2.$$ Developing the square on the left-hand side, and neglecting the nonnegative term $(\alpha\|\dot x(t)\|/t)^2/L$, we obtain $$\ddot{h}(t) + \frac{\alpha}{t} \dot{h}(t) + \frac{1}{L}\|\ddot{x}(t)\|^2 + \frac{\alpha}{Lt}\frac{d}{dt}\|\dot{x}(t)\|^2 \leq \|\dot{x}(t)\|^2.$$ We multiply this inequality by $t^\alpha$ to obtain $$\frac{d}{dt}\left( t^{\alpha}\dot{h}(t)\right) + \frac{1}{L} t^{\alpha}\|\ddot{x}(t)\|^2 + \frac{\alpha}{L}t^{\alpha-1}\frac{d}{dt}\|\dot{x}(t)\|^2 \leq t^{\alpha} \|\dot{x}(t)\|^2.$$ Integration from $t_0$ to $t$ yields $$t^{\alpha} \dot{h}(t) - t_0^{\alpha} \dot{h}(t_0) + \frac{1}{L} \int_{t_0}^t s^{\alpha}\| \ddot{x}(s) \|^2 ds + \frac{\alpha}{L}\Big ( t^{\alpha -1} \|\dot{x}(t) \|^2 - {t_0}^{\alpha -1} \|\dot{x}(t_0) \|^2 - (\alpha -1)\int_{t_0}^t \|\dot{x}(s) \|^2 s^{\alpha -2} ds \Big ) \leq \int_{t_0}^t s^{\alpha} \| \dot{x}(s) \|^2 ds.$$ Neglecting the nonnegative term $\alpha t^{\alpha-1}\|\dot x(t)\|^2/L$, we obtain $$\label{E:stabilization} t^\alpha\dot h(t) + \frac{1}{L} \int_{t_0}^t s^{\alpha}\|\ddot{x}(s)\|^2 ds \leq C + (\alpha -1)\int_{t_0}^t \|\dot{x}(s) \|^2 s^{\alpha -2} ds + \int_{t_0}^t s^{\alpha} \| \dot{x}(s) \|^2 ds,$$ where $C=t_0^\alpha\dot h(t_0)+\alpha t_0^{\alpha-1}\|\dot x(t_0)\|^2/L$. If $t_0<1$, we have $$\frac{1}{t^\alpha} \int_{t_0}^t s^\alpha \|\ddot x(s)\|^2 ds = \frac{1}{t^\alpha} \int_{t_0}^1 s^\alpha \|\ddot x(s)\|^2 ds + \frac{1}{t^\alpha} \int_1^t s^\alpha \|\ddot x(s)\|^2 ds$$ for all $t\geq 1$. Since the first term on the right-hand side tends to $0$ as $t\to+\infty$, we may assume, without loss of generality, that $t_0\ge 1$. Observe now that $s^{\alpha -2} \leq s^{\alpha}$, whenever $s\geq1$. Whence, inequality simplifies to $$t^\alpha\dot h(t) + \frac{1}{L} \int_{t_0}^t s^{\alpha}\|\ddot{x}(s)\|^2 ds \leq C + \alpha \int_{t_0}^t s^{\alpha} \| \dot{x}(s) \|^2 ds.$$ Dividing by $t^\alpha$ and integrating again, we obtain $$h(t)-h(t_0) + \frac{1}{L} \int_{t_0}^t \tau^{-\alpha } \left( \int_{t_0}^\tau {s}^{\alpha}\| \ddot{x}(s) \|^2 ds \right) d\tau \leq \frac{C}{\alpha-1}(t_0^{-\alpha+1}-t^{-\alpha+1}) +\alpha \int_{t_0}^t \tau^{-\alpha } \left( \int_{t_0}^\tau {s}^{\alpha}\| \dot{x}(s) \|^2 ds \right) d\tau.$$ Setting $C'=h(t_0)+C t_0^{-\alpha+1}/(\alpha-1)$, and neglecting the nonnegative term $h(t)$ of the left-hand side and the nonpositive term $-C t^{-\alpha+1}/(\alpha-1)$ of the right-hand side, we get $$\frac{1}{L} \int_{t_0}^t \tau^{-\alpha } \left( \int_{t_0}^\tau {s}^{\alpha}\| \ddot{x}(s) \|^2 ds \right) d\tau \leq C' +\alpha \int_{t_0}^t \tau^{-\alpha } \left( \int_{t_0}^\tau {s}^{\alpha}\| \dot{x}(s) \|^2 ds \right) d\tau.$$ Set $\displaystyle g(\tau)= \tau^{-\alpha}\left(\int_{t_0}^\tau s^\alpha\|\ddot x(s)\|^2 ds\right)$ and use Fubini’s Theorem on the second integral to get $$\frac{1}{L}\int_{t_0}^t g(\tau)d\tau \leq C' + \frac{\alpha}{\alpha-1}\int_{t_0}^t s^\alpha\|\dot x(s)\|^2 (s^{-\alpha+1}-t^{-\alpha+1})\ ds \leq C' + \frac{\alpha}{\alpha-1}\int_{t_0}^t s\|\dot x(s)\|^2 ds.$$ By part ii) of Theorem \[T:alpha3\], the integral on the right-hand side is finite. We have $$\label{gl1} \int_{t_0}^{+\infty} g(\tau)d\tau < +\infty.$$ The derivative of $g$ is $$\dot g(\tau) = - \alpha\tau^{-\alpha-1}\int_{t_0}^\tau s^\alpha\|\ddot x(s)\|^2 ds +\|\ddot x(\tau)\|^2.$$ Let $C''$ be an upper bound for $\|\ddot x\|^2$. We have $$\label{glip} |\dot g(\tau)| \leq C''\left(1+\alpha\tau^{-\alpha-1}\int_{t_0}^\tau s^\alpha ds\right) = C''\left( 1+\frac{\alpha}{\alpha+1}\tau^{-\alpha-1} \left(\tau^{\alpha+1}-t^{\alpha+1}\right) \right) \leq C''\left(1+\frac{\alpha}{\alpha+1}\right).$$ From and we deduce that $\lim_{\tau\to+\infty}g(\tau)=0$ by virtue of Lemma \[lm:aux\]. Since $\int_{t_0}^ts^\alpha ds= \frac{1}{\alpha+1}\left(t^{\alpha+1}-t_0^{\alpha+1}\right)$, Proposition \[P:acceleration\] expresses a fast ergodic convergence of $\|\ddot x(s)\|^2$ to 0 with respect to the weight $s^\alpha$ as $t\to+\infty$, namely $$\frac{\int_{t_0}^t s^\alpha\|\ddot x(s)\|^2 ds}{\int_{t_0}^t s^\alpha ds} = o\left(\frac{1}{t}\right).$$ Strong convergence results {#S:strong} ========================== A counterexample due to Baillon [@Ba] shows that the trajectories of the steepest descent dynamical system may converge weakly but not strongly. Nevertheless, under some additional geometrical or topological assumptions on $\Phi$, the steepest descent trajectories do converge strongly. This has been proved in the case where the function $\Phi$ is either even or strongly convex (see [@Bruck]), or when ${{\rm int}\kern 0.06em}(\operatorname{argmin}\Phi) \neq \emptyset$ (see [@Bre1 theorem 3.13]). Some of these results have been extended to inertial dynamics, see [@Al] for the heavy ball with friction, and [@aabr] for an inertial version of Newton’s method. This suggests that convexity alone may not be sufficient for the trajectories of to converge strongly, but one can reasonably expect it to be the case under some additional conditions. The purpose of this section is to establish this fact. The different types of hypotheses will be studied in independent subsections s! ince different techniques are required. Set of minimizers with nonempty interior ---------------------------------------- Let us begin by studying the case where ${{\rm int}\kern 0.06em}(\operatorname{argmin}\Phi)\neq\emptyset$. \[Thm-strong-int\] Let $\Phi : \mathcal H \rightarrow \mathbb R$ be a continuously differentiable convex function. Let ${{\rm int}\kern 0.06em}(\operatorname{argmin}\Phi)\neq\emptyset$ and let $x:[t_0,+\infty[\to\mathcal H$ be a solution of with $\alpha>3$. Then $x(t)$ converges strongly, as $t\to+\infty$, to a point in $\operatorname{argmin}\Phi$. Moreover, $$\int_{t_0}^{\infty} t\| \nabla \Phi (x(t)) \|dt < +\infty.$$ Since ${{\rm int}\kern 0.06em}(\operatorname{argmin}\Phi)\neq\emptyset$, there exist $x^*\in\operatorname{argmin}\Phi$ and some $\rho>0$ such that $\nabla\Phi(z)=0$ for all $z\in\mathcal H$ such that $\|z-x^*\|<\rho$. By the monotonicity of $\nabla\Phi$, for all $y\in\mathcal H$, we have $$\langle \nabla \Phi (y), y- z \rangle \geq 0.$$ Hence, $$\langle \nabla \Phi (y), y-x^* \rangle \geq \langle \nabla \Phi (y), z-x^* \rangle .$$ Taking the supremum with respect to $z \in \mathcal H$ such that $\|z-x^*\| < \rho$, we infer that $$\langle \nabla \Phi (y), y-x^* \rangle \geq \rho \|\nabla \Phi (y)\|$$ for all $ y \in \mathcal H $. In particular, $$\langle \nabla \Phi ( x(t)), x(t)-x^* \rangle \geq \rho \|\nabla \Phi ( x(t))\|.$$ By using this inequality in with $\lambda=\alpha-1$ and $\xi=0$, we obtain $$\frac{d}{dt}\mathcal E_{\alpha-1,0}(t)+(\alpha-1)\rho t\|\nabla\Phi(x(t))\| \le 2\,t\big(\Phi(x(t))-\min\Phi\big),$$ whence we derive, by integrating from $t_0$ to $t$ $$\mathcal E_{\alpha-1,0}(t)-\mathcal E_{\alpha-1,0}(t_0) +(\alpha-1)\rho\int_{t_0}^t s\|\nabla\Phi(x(s))\|ds \le 2\int_{t_0}^t s\big(\Phi(x(s))-\min\Phi\big)\,ds.$$ Since $\mathcal E_{\alpha-1,0}(t)$ is nonnegative, part ii) of Theorem \[T:SBC\] gives $$\label{strong-conv-int8} \int_{t_0}^{\infty} t\| \nabla \Phi (x(t)) \|dt < +\infty.$$ Finally, rewrite as $$t\ddot{x}(t) + \alpha \dot{x}(t) = - t\nabla \Phi (x(t)).$$ Since the right-hand side is integrable, we conclude by applying Lemma \[lemma-edo1\] and Theorem \[T:weak\_convergence\]. Even functions -------------- Let us recall that $\Phi : \mathcal H \to \mathbb R$ is [*even*]{} if $\Phi (-x)= \Phi (x) $ for every $x\in \mathcal H$. In this case the set $\operatorname{argmin}\Phi$ is nonempty and contains the origin. \[Thm-strong-conv\] Let $\Phi:\mathcal H\rightarrow\mathbb R$ be a continuously differentiable convex even function and let $x:[t_0,+\infty[\to\mathcal H$ be a solution of with $\alpha>3$. Then $x(t)$ converges strongly, as $t\to+\infty$, to a point in $\operatorname{argmin}\Phi$. For $t_0 \leq \tau \leq s$, set $$q(\tau)= \| x(\tau) \|^2 -\| x(s) \|^2 - \frac{1}{2} \|x(\tau)- x(s) \|^2 .$$ We have $$\dot{q}(\tau)= \langle \dot{x}(\tau),x(\tau) + x(s) \rangle \qquad\hbox{and}\qquad \ddot{q}(\tau) = \|\dot{x}(\tau)\|^2 + \langle \ddot{x}(\tau),x(\tau) + x(s) \rangle.$$ Combining these two equalities and using , we obtain $$\label{E:q_ddot} \ddot{q}(\tau) + \frac{\alpha}{\tau} \dot{q}(\tau) = \|\dot{x}(\tau)\|^2 + \langle \ddot{x}(\tau) + \frac{\alpha}{\tau} \dot{x}(\tau) ,x(\tau) + x(s) \rangle = \|\dot{x}(\tau)\|^2 - \langle \nabla \Phi (x(\tau)) ,x(\tau) + x(s) \rangle .$$ Recall that the energy $W(\tau)= \frac{1}{2} \| \dot{x}(\tau) \|^2 + \Phi (x(\tau))$ is nonincreasing. Therefore, $$\begin{aligned} \frac{1}{2} \| \dot{x}(\tau) \|^2 + \Phi (x(\tau)) & \geq & \frac{1}{2} \| \dot{x}(s) \|^2 + \Phi (x(s)) \\ & = & \frac{1}{2} \| \dot{x}(s) \|^2 + \Phi (-x(s))\\ & \ge & \frac{1}{2} \| \dot{x}(s) \|^2 + \Phi (x(\tau)) - \langle \nabla \Phi (x(\tau)) ,x(\tau) + x(s) \rangle,\end{aligned}$$ by convexity. After simplification, we obtain $$\label{even6} \frac{1}{2} \| \dot{x}(\tau) \|^2 \geq - \langle \nabla \Phi (x(\tau)),x(\tau) + x(s) \rangle .$$ Combining and , we obtain $$\tau\ddot{q}(\tau) + \alpha\dot{q}(\tau) \leq \frac{3}{2}\tau \|\dot{x}(\tau)\|^2.$$ As in the proof of Lemma \[basic-edo\], we have $$\dot{q}(\tau) \leq k(\tau):= \frac{C}{\tau^{\alpha} } + \frac{3}{2\tau^{\alpha} } \int_{t_0}^\tau u^{\alpha} \|\dot{x}(u)\|^2du,$$ where $C=2\|\dot x(t_0)\|\,\|x\|_{\infty}$. The function $k$ does not depend on $s$. Moreover, using Fubini’s Theorem, we deduce that $$\int_{t_0}^{+\infty}k(\tau)\,d\tau\le \frac{C}{t_0^{\alpha-1}(\alpha-1)}+\frac{3}{2(\alpha-1)}\int_{t_0}^{+\infty}u\|\dot x(u)\|^2\,du<+\infty,$$ by part ii) of Theorem \[T:alpha3\]. Integrating $\dot{q}(\tau) \leq k(\tau)$ from $t$ to $s$, we obtain $$\frac{1}{2} \|x(t)- x(s) \|^2 \leq \| x(t) \|^2 -\| x(s) \|^2 + \int_t^s k(\tau) d\tau.$$ Since $\Phi$ is even, we have $0 \in\operatorname{argmin}\Phi$. Hence $\lim_{t\to +\infty }\| x(t) \|^2$ exists (see the proof of Theorem \[T:weak\_convergence\]). As a consequence, $x(t)$ has the Cauchy property as $t \to + \infty$, and hence converges. Uniformly convex functions -------------------------- Following [@BC], a function $\Phi : \mathcal H \rightarrow \mathbb R$ is [*uniformly convex on bounded sets*]{} if, for each $r>0$, there is an increasing function $\omega_r:[0,+\infty[\to[0,+\infty[$ vanishing only at $0$, and such that $$\label{E:uniform} \Phi(y) \geq \Phi (x) + \langle \nabla \Phi (x) , y-x \rangle + \omega_r(\|x-y\|)$$ for all $x, y \in \mathcal H$ such that $\| x \|\leq r$ and $\| y \|\leq r$. Uniformly convex functions are strictly convex and coercive. \[T:unif\_convex\] Let $\Phi$ be uniformly convex on bounded sets, and let $x:[t_0,+\infty[\to\mathcal H$ be a solution of with $\alpha>3$. Then $x(t)$ converges strongly, as $t\to+\infty$, to the unique $x^*\in\operatorname{argmin}\Phi$. Recall that the trajectory $x(\cdot)$ is bounded, by part ii) in Theorem \[T:alpha3\]. Let $r >0$ be such that $x$ is contained in the ball of radius $r$ centered at the origin. This ball also contains $x^*$, which is the weak limit of the trajectory in view of the weak lower-semicontinuity of the norm and Theorem \[T:weak\_convergence\]. Writing $y=x(t)$ and $x=x^*$ in , we obtain $$\omega_r (\|x(t)-x^{*} \|)\le \Phi(x(t))-\min\Phi.$$ The right-hand side tends to $0$ as $t\to+\infty$ by virtue of Theorem \[Thm-weak-conv2\]. It follows that $x(t)$ converges strongly to $x^*$ as $t\to+\infty$. Let us recall that a function $\Phi : \mathcal H \rightarrow \mathbb R$ is [*strongly convex*]{} if there exists a positive constant $\mu$ such that $$\Phi(y) \geq \Phi (x) + \langle \nabla \Phi (x) , y-x \rangle + \frac{\mu}{2} \|x-y \|^2$$ for all $x, y \in \mathcal H$. Clearly, strongly convex functions are uniformly convex on bounded sets. However, a striking fact is that convergence rates increase indefinitely with larger values of $\alpha$ for these functions. Let $\Phi:\mathcal H\rightarrow{{\mathbb R}}$ be strongly convex, and let $x:[t_0,+\infty[\to\mathcal H$ be a solution of with $\alpha>3$. Then $x(t)$ converges strongly, as $t\to+\infty$, to the unique element $x^*\in\operatorname{argmin}\Phi$. Moreover $$\label{elliptique} \Phi(x(t))-\min\Phi=\mathcal O\left(t^{-{\frac{2}{3}\alpha}}\right),\qquad \|x(t)-x^*\|^2=\mathcal O\left(t^{-{\frac{2}{3}\alpha}}\right),\qquad\hbox{and}\qquad \|\dot x(t)\|^2=\mathcal O\left(t^{-{\frac{2}{3}\alpha}}\right).$$ Strong convergence follows from Theorem \[T:unif\_convex\] because strongly convex functions are uniformly convex on bounded sets. From and the strong convexity of $\Phi$, we deduce that $$\begin{aligned} \frac{d}{dt}{\mathcal E_\lambda^p}(t) & \leq & (p+2-\lambda)t^{p+1}(\Phi(x)-\min\Phi) - \lambda(\alpha-\lambda-1-p)t^p\langle x-x^*,\dot x\rangle \\ \nonumber & & -\frac{\lambda}{2}(\mu t^2-p\lambda)t^{p-1}\|x-x^*\|^2 - \left(\alpha-\lambda-1-\frac{p}{2}\right)t^{p+1}\|\dot x\|^2\end{aligned}$$ for any $\lambda\geq 0$ and any $p\geq 0$. Now fix $p={\frac{2}{3}(\alpha-3)}$ and $\lambda={\frac{2}{3}\alpha}$, so that $p+2-\lambda=\alpha-\lambda-1-p/2=0$ and $\alpha-\lambda-1-p=-p/2$. The above inequality becomes $$\frac{d}{dt}{\mathcal E_\lambda^p}(t) \leq \frac{\lambda p}{2}t^p\langle x-x^*,\dot x\rangle -\frac{\lambda}{2}(\mu t^2-p\lambda)t^{p-1}\|x-x^*\|^2.$$ Define $t_1=\max\left\{t_0,\sqrt{\frac{p\lambda}{\mu}}\right\}$, so that $$\frac{d}{dt}{\mathcal E_\lambda^p}(t) \leq \frac{\lambda p}{2}t^p\langle x-x^*,\dot x\rangle$$ for all $t\geq t_1$. Integrate this inequality from $t_1$ to $t$ (use integration by parts on the right-hand side) to get $${\mathcal E_\lambda^p}(t) \leq {\mathcal E_\lambda^p}(t_1) + \frac{\lambda p}{4}\left( t^p\|x(t)-x^*\|^2 - t_1^p\|x(t_1)-x^*\|^2 - p\int_{t_1}^t s^{p-1}\|x(s)-x^*\|^2 ds \right).$$ Hence, $$\label{Ett} {\mathcal E_\lambda^p}(t) \leq {\mathcal E_\lambda^p}(t_1) + \frac{\lambda p}{4} t^p\|x(t)-x^*\|^2 \leq {\mathcal E_\lambda^p}(t_1) + \frac{\lambda p}{2\mu} t^p(\Phi(x(t))-\min\Phi),$$ in view of the strong convexity of $\Phi$. By the definition of ${\mathcal E_\lambda^p}$, we have $$t^{p+2}((\Phi(x(t))-\min\Phi) \leq {\mathcal E_\lambda^p}(t) \leq {\mathcal E_\lambda^p}(t_1) + \frac{\lambda p}{2\mu} t^p(\Phi(x(t))-\min\Phi).$$ Dividing by $t^{p+2}$ and using the definition of $t_1$, along with the fact that $t\geq t_1$, we obtain $$\begin{aligned} \Phi(x(t))-\min\Phi & \leq & {\mathcal E_\lambda^p}(t_1)t^{-p-2} + \frac{\lambda p}{2\mu} t^{-2}(\Phi(x(t))-\min\Phi)\\ & \leq & {\mathcal E_\lambda^p}(t_1)t^{-p-2}+\frac{\lambda p}{2\mu}t_1^{-2}(\Phi(x(t))-\min\Phi)\\ & \leq & {\mathcal E_\lambda^p}(t_1)t^{-p-2}+\frac{1}{2}(\Phi(x(t))-\min\Phi).\end{aligned}$$ Recalling that $p={\frac{2}{3}(\alpha-3)}$ and $\lambda={\frac{2}{3}\alpha}$, we deduce that $$\label{Phit} \Phi(x(t))-\min\Phi \leq 2{\mathcal E_\lambda^p}(t_1)t^{-p-2} = \left[2\mathcal E_{\frac{2}{3}\alpha}^{\frac{2}{3}(\alpha-3)}(t_1)\right]t^{-\frac{2}{3}\alpha}.$$ The strong convexity of $\Phi$ then gives $$\label{xxt} \|x(t)-x^*\|^2 \leq \frac{2}{\mu}(\Phi(x(t))-\min\Phi) \leq \left[\frac{4}{\mu}{\mathcal E_\lambda^p}(t_1)\right]t^{-p-2} = \left[\frac{4}{\mu} \mathcal E_{\frac{2}{3}\alpha}^{\frac{2}{3}(\alpha-3)}(t_1)\right]t^{-\frac{2}{3}\alpha}.$$ Inequalities and settle the first two points in . Now, using and , we derive $${\mathcal E_\lambda^p}(t) \leq {\mathcal E_\lambda^p}(t_1) + \frac{\lambda p}{2\mu} t^p(\Phi(x(t))-\min\Phi)\le {\mathcal E_\lambda^p}(t_1)+\frac{\lambda p}{\mu}{\mathcal E_\lambda^p}(t_1)t^{-2}\le {\mathcal E_\lambda^p}(t_1)+\frac{\lambda p}{\mu}{\mathcal E_\lambda^p}(t_1)t_1^{-2}\le 2{\mathcal E_\lambda^p}(t_1).$$ The definition of ${\mathcal E_\lambda^p}$ then gives $$\frac{t^p}{2}\|\lambda(x(t)-x^*)+t\dot x(t)\|^2 \le {\mathcal E_\lambda^p}(t) \le 2{\mathcal E_\lambda^p}(\tau).$$ Hence $$\|\lambda(x(t)-x^*)+t\dot x(t)\|^2 \leq 4 t^{-p} {\mathcal E_\lambda^p}(t_1),$$ and $$t\|\dot x(t)\| \leq 2 t^{-p/2} \sqrt{{\mathcal E_\lambda^p}(t_1)} + \lambda\|x(t)-x^*\|.$$ But using , we deduce that $$\lambda\|x(t)-x^*\|\le \frac{2\lambda}{\sqrt{\mu}}t^{-p/2-1}\sqrt{{\mathcal E_\lambda^p}(t_1)}.$$ The last two inequalities together give $$t\|\dot x(t)\| \le 2 t^{-p/2} \sqrt{{\mathcal E_\lambda^p}(t_1)} \left(1+\frac{\lambda t^{-1}}{\sqrt{\mu}}\right) \le 2 t^{-p/2}\sqrt{{\mathcal E_\lambda^p}(t_1)}\left(1+\sqrt{\frac{\lambda}{p}}\right).$$ Taking squares, and rearranging the terms, we obtain $$\|\dot x(t)\|^2 \leq \left[4 \left(1+\sqrt{\frac{\alpha}{\alpha-3}}\right)^2\mathcal E_{\frac{2}{3}\alpha}^{\frac{2}{3}(\alpha-3)}(t_1) \right] t^{-\frac{2}{3}\alpha},$$ which shows the last point in and completes the proof. The preceding theorem extends [@SBC Theorem 4.2], which states that if $\alpha>9/2$, then $\Phi(x(t))-\min\Phi=\mathcal O(1/t^{3})$. Convergence of the associated algorithms {#S:algorithm} ======================================== In many situations, one is faced with a non-smooth convex minimization problems with an additive structure of the form $$\label{algo1} \min \left\lbrace \Phi (x) + \Psi (x): \ x \in \mathcal H \right\rbrace,$$ where $\Phi: \mathcal H \to \mathbb R \cup \lbrace + \infty \rbrace $ is proper, lower-semicontinuous and convex, and $\Psi: \mathcal H \to \mathbb R$ is convex and continuously differentiable. Following the analysis carried out in the previous sections, it seems reasonable to consider the differential inclusion $$\label{algo2} \ddot{x}(t) + \frac{\alpha}{t} \dot{x}(t) + \partial \Phi (x(t)) + \nabla \Psi (x(t)) \ni 0,$$ in order to approximate optimal solutions for . This differential inclusion is a special instance of $$\ddot{x}(t) + a(t) \dot{x}(t) + \partial \Theta (x(t)) \ni 0,$$ where $\Theta: \mathcal H \to \mathbb R \cup \lbrace + \infty \rbrace $ is a convex lower-semicontinuous proper function, and $a(\cdot)$ is a positive damping parameter. This differential inclusion has been studied in [@ACR] in the case of a fixed positive damping parameter $a(t)\equiv\gamma >0$. In that setting, and at least localy, each trajectory is Lipschitz continuous, its velocity has bounded variation, and its acceleration is a bounded vectorial measure. Thus, setting $\Theta (x)= \Phi (x) + \Psi (x)$, we can reasonably expect that the rapid convergence properties, studied in Section \[S:minimizing\] for the solutions of , should hold for the solutions of as well. Especially, that $\Theta(x(t))-\min\Theta=\mathcal O(1/t^2)$, and that each trajectory converges to an optimal solution. A detailed analysis is an interesting topic for further research, but goes beyond the scope of this paper. Using these ideas as a guideline, we shall introduce corresponding fast converging algorithms, making the link with Nesterov [@Nest1]-[@Nest4], Beck-Teboulle [@BT], and the recent works of Chambolle-Dossal [@CD], and Su-Boyd-Candes [@SBC]. More precisely, it is possible to discretize implicitely with respect to the nonsmooth function $\Phi$, and explicitly with respect to the smooth function $\Psi$. Indeed, taking a time step size $h>0$, and $t_k= kh$, $x_k = x(t_k)$ the classical finite difference scheme for (\[algo2\]) gives $$\label{algo3} \frac{1}{h^2}(x_{k+1} -2 x_{k} + x_{k-1} ) +\frac{\alpha}{kh^2}( x_{k} - x_{k-1}) + \partial \Phi (x_{k+1} ) + \nabla \Psi (y_k) \ni 0,$$ where $y_k$ is a linear combination of $x_k$ and $x_{k-1}$, that will be made precise later. After developing , we obtain $$\label{algo4} x_{k+1} + h^2 \partial \Phi (x_{k+1}) \ni x_{k} + \left( 1- \frac{\alpha}{k}\right) ( x_{k} - x_{k-1}) - h^2 \nabla \Psi (y_k).$$ A natural choice for $y_k$ leading to a simple formulation of the algorithm is $$\label{algo5} y_k= x_{k} + \left( 1- \frac{\alpha}{k}\right) ( x_{k} - x_{k-1}).$$ Of course, other choices are possible, an interesting topic for further research. Using the classical [*proximity operator*]{} $$\mbox{prox}_{ \gamma \Phi } (x)= {\operatorname{argmin}}_{\xi} \left\lbrace \Phi (\xi) + \frac{1}{2 \gamma} \| \xi -x \|^2 \right\rbrace = \left(I + \gamma \partial \Phi \right)^{-1} (x),$$ and setting $\gamma=h^2$, the algorithm can be written as $$\label{algo7} \left\{ \begin{array}{l} y_k= x_{k} + \left( 1- \frac{\alpha}{k}\right) ( x_{k} - x_{k-1}); \\ \rule{0pt}{20pt} x_{k+1} = \mbox{prox}_{\gamma\Phi} \left( y_k- \gamma \nabla \Psi (y_k) \right). \end{array}\right.$$ For practical purposes, and in order to fit with the existing literature on the subject, it is convenient to work with the following equivalent formulation $$\label{algo7b} \left\{ \begin{array}{l} y_k= x_{k} + \frac{k -1}{k + \alpha -1} ( x_{k} - x_{k-1}); \\ \rule{0pt}{20pt} x_{k+1} = \mbox{prox}_{\gamma\Phi} \left( y_k- \gamma\nabla \Psi (y_k) \right). \end{array}\right.$$ This algorithm is within the scope of the proximal-based inertial algorithms [@AA1], [@MO], [@LP] and forward-backward methods. It has been recently introduced by Chambolle-Dossal [@CD], and Su-Boyd-Candès [@SBC]. For $\alpha = 3$, we recover the classical algorithm based on Nesterov and Güler ideas, and developed by Beck-Teboulle (FISTA) $$\label{algo7c} \ \left\{ \begin{array}{l} y_k= x_{k} + \frac{k -1}{k + 2} ( x_{k} - x_{k-1}); \\ \rule{0pt}{20pt} x_{k+1} = \mbox{prox}_{h^2 \Phi} \left( y_k- h^2 \nabla \Psi (y_k) \right). \end{array}\right.$$ The fast convergence properties of the algorithm were recently highlighted by Su-Boyd-Candès [@SBC] and Chambolle-Dossal [@CD]. An important $-$ and still open $-$ question regarding the FISTA method, as described in , is the convergence of sequences $(x_k)$ and $(y_k)$. The main interest of considering the broader context given in is that, for $\alpha >3$, these sequences converge. This has recently been obtained by Chambolle-Dossal [@CD]. Following [@SBC], we will see that the proof of the convergence properties of can be obtained in a parallel way with the convergence analysis in the continuous case. More precisely, following the arguments in the preceding sections, one is able to prove the following: If $\alpha>0$, then $\lim\Phi(x_k)=\inf\Phi$ and every weak limit point of $(x_k)$, as $k\to+\infty$, belongs to $\operatorname{argmin}\Phi$. For $\operatorname{argmin}\Phi\neq\emptyset$, we have the following: - If $\alpha\ge 3$, then $\Phi(x_k)-\min\Phi=\mathcal O(1/k^2)$ and $\|x_{k+1}-x_k\|=\mathcal O(1/k)$. - If $\alpha>3$, then $\sum k(\Phi(x_k)-\min\Phi)<+\infty$, $\sum k\|x_{k+1}-x_k\|^2<+\infty$, and $x_k$ converges weakly, as $k\to+\infty$, to some $x^*\in\operatorname{argmin}\Phi$. Strong convergence holds if $\Phi$ is even, uniformly convex, or if $\operatorname{argmin}\Phi\neq\emptyset$. Further remarks {#S:conclusion} =============== Nonsmooth objective function ---------------------------- As mentioned at the beginning of Section \[S:algorithm\], it is interesting to establish the asymptotic properties, as $t\to+\infty$, of the solutions of the differential inclusion . Beyond global existence issues, one must check that the Lyapunov analysis is still valid. In view of the validity of the subdifferential inequality for convex functions, the (generalized) chain rule for derivatives over curves (see [@Bre1]), our conjecture is that most results presented here can be transposed to this more general context, except for the stabilization of the acceleration, which relies on the Lipschitz character of the gradient. Time reparameterization ----------------------- Let us examine briefly the effect of some simple rescaling procedures: ### Invariance {#invariance .unnumbered} The condition $\alpha >3$ is not affected by an affine time rescaling. Indeed, if $a>0$ and we take $t=as$ in , we obtain $$\ddot{y}(s) + \frac{\alpha}{s} \dot{y}(s) + a^2\nabla \Phi (y(s)) = 0,$$ where $y(s)= x(as)$. This produces an analogue system with $\Phi$ replaced by $\Phi_a:=a^2\Phi$. ### Variable damping versus variable mass {#variable-damping-versus-variable-mass .unnumbered} If we take $t=\sqrt{s}$ in we obtain $$2s\ddot{z} (s) + (2\alpha + 1) \dot{z}(s) + \nabla \Phi (z(s)) = 0,$$ where $z(s)= x(\sqrt{s})$. In this alternative formulation, the viscous damping parameter is fixed, but the mass coefficient becomes infinitely large as $t\to + \infty$. This suggests that a parallel analysis can be performed by controlling the mass coefficient, instead of the viscosity coefficient. Selection of the initial conditions ----------------------------------- The constant in the order of convergence given by Theorem \[T:SBC\] is $$K(x_0,v_0)=\mathcal E_{\alpha-1,0}(t_0)=t_0^2(\Phi(x_0)-\min\Phi)+\frac{1}{2}\|(\alpha-1)(x_0-x^*)+t_0v_0\|^2,$$ where $x_0=x(t_0)$ and $v_0=\dot x(t_0)$. This quantity is minimized when $x_0^*\in\operatorname{argmin}\Phi$ and $v_0^*=\frac{(\alpha-1)}{t_0}(x^*-x_0^*)$, with $\min K=0$. If $x_0^*\neq x^*$, the trajectory will not be stationary, but the value $\Phi(x(t))$ will be constantly equal to $0$. Of course, selecting $x_0^*\in\operatorname{argmin}\Phi$ is not realistic, and the point $x^*$ is unknown. Keeping $\hat x_0$ fixed, the function $v_0\mapsto K(\hat x_0,v_0)$ is minimized at $\hat v_0=\frac{(\alpha-1)}{t_0}(x^*-\hat x_0)$. This suggests taking the initial velocity as a multiple of an approximation of $x^*-\hat x_0$, such as the gradient direction $\hat v_0=\nabla\Phi(\hat x_0)$, Newton or Levenberg-Marquardt direction $\hat v_0=[\varepsilon I+\nabla^2\Phi(\hat x_0)^{-1}]\nabla\Phi(\hat x_0)$ ($\varepsilon\ge 0$), or the proximal point direction $\hat v_0=\left[(I+\gamma\nabla\Phi)^{-1}(\hat x_0)-\hat x_0\right]$ ($\gamma>>0$). Continuous versus discrete -------------------------- The analysis carried out in Section \[S:algorithm\] for inertial forward-backward algorithm is a reinterpretation of the proof of the corresponding results in the continuous case. In other words, we built a complete proof having the continuous setting as a guideline. It would be interesting to know whether the results in [@Alv_Pey2; @Alv_Pey3] can be applied in order to deduce the asymptotic properties without repeating the proofs. Hessian-driven damping ---------------------- In the dynamical system studied here, second-order information with respect to time ultimately induces fast convergence properties. On the other hand, in Newton-type methods, second-order information in space, has a similar consequence. In a forthcoming paper, we analyze the solutions of the second-order evolution equation $$\ddot{x}(t) + \frac{\alpha}{t} \dot{x}(t) + \beta\,\nabla^2 \Phi (x(t))\,\dot{x} (t) + \nabla \Phi (x(t)) = 0,$$ where $\Phi$ is a smooth convex function, and $\alpha$, $\beta$ are positive parameters. This inertial system combines an isotropic viscous damping which vanishes asymptotically, and a geometrical Hessian-driven damping, which makes it naturally related to Newton and Levenberg-Marquardt methods. Appendix: Some auxiliary results ================================ In this section, we present some auxiliary lemmas to be used later on. The following result can be found in [@AAS]: \[lm:aux\] Let $\delta >0$, $1\leq p<\infty$ and $1\leq r\leq\infty$. Suppose $F\in L^{p}([\delta,\infty[)$ is a locally absolutely continuous nonnegative function, $G\in L^{r}([\delta,\infty[)$ and $$\frac{d}{dt}F(t)\leq G(t)$$ for almost every $t>\delta$. Then $\lim_{t\to\infty}F(t)=0$. To establish the weak convergence of the solutions of , we will use Opial’s Lemma [@Op], that we recall in its continuous form. This argument was first used in [@Bruck] to establish the convergence of nonlinear contraction semigroups. \[Opial\] Let $S$ be a nonempty subset of $\mathcal H$ and let $x:[0,+\infty)\to \mathcal H$. Assume that - for every $z\in S$, $\lim_{t\to\infty}\|x(t)-z\|$ exists; - every weak sequential limit point of $x(t)$, as $t\to\infty$, belongs to $S$. Then $x(t)$ converges weakly as $t\to\infty$ to a point in $S$. The following allows us to establish the existence of a limit for a real-valued function, as $t\to+\infty$: \[basic-edo\] Let $\delta >0$, and let $w: [\delta, +\infty[ \rightarrow \mathbb R$ be a continuously differentiable function which is bounded from below. Assume $$\label{basic-edo1} t\ddot{w}(t) + \alpha \dot w(t) \leq g(t),$$ for some $\alpha > 1$, almost every $t>\delta$, and some nonnegative function $g\in L^1 (\delta, +\infty)$. Then, the positive part $[\dot w]_+$ of $\dot w$ belongs to $L^1(t_0,+\infty)$ and $\lim_{t\to+\infty}w(t)$ exists. Multiply by $t^{\alpha-1}$ to obtain $$\frac{d}{dt} \big(t^{\alpha} \dot w(t)\big) \leq t^{\alpha-1} g(t).$$ By integration, we obtain $$\dot w(t) \leq \frac{{\delta}^{\alpha}|\dot w(\delta)|}{t^{\alpha} } + \frac{1}{t^{\alpha} } \int_{\delta}^t s^{\alpha-1} g(s)ds.$$ Hence, $$[\dot w]_{+}(t) \leq \frac{{\delta}^{\alpha}|\dot w(\delta)|}{t^{\alpha} } + \frac{1}{t^{\alpha} } \int_{\delta}^t s^{\alpha-1} g(s)ds,$$ and so, $$\int_{\delta}^{\infty} [\dot w]_{+}(t) dt \leq \frac{{\delta}^{\alpha}|\dot w(\delta)|}{(\alpha -1) \delta^{\alpha -1}} + \int_{\delta}^{\infty}\frac{1}{t^{\alpha}} \left( \int_{\delta}^t s^{\alpha-1} g(s) ds\right) dt.$$ Applying Fubini’s Theorem, we deduce that $$\int_{\delta}^{\infty}\frac{1}{t^{\alpha}} \left( \int_{\delta}^t s^{\alpha-1} g(s) ds\right) dt = \int_{\delta}^{\infty} \left( \int_{s}^{\infty} \frac{1}{t^{\alpha}} dt\right) s^{\alpha-1} g(s) ds = \frac{1}{\alpha -1} \int_{\delta}^{\infty}g(s) ds.$$ As a consequence, $$\int_{\delta}^{\infty} [\dot w]_{+}(t) dt \leq \frac{{\delta}^{\alpha}|\dot w(\delta) |}{(\alpha -1) \delta^{\alpha -1}} + \frac{1}{\alpha -1} \int_{\delta}^{\infty}g(s) ds < + \infty.$$ Finally, the function $\theta:[\delta,+\infty)\to{{\mathbb R}}$, defined by $$\theta(t)=w(t)-\int_{\delta}^{t}[\dot w]_+(\tau)\,d\tau,$$ is nonincreasing and bounded from below. It follows that $$\lim_{t\to+\infty}w(t)=\lim_{t\to+\infty}\theta(t)+\int_{\delta}^{+\infty}[\dot w]_+(\tau)\,d\tau$$ exists. The following is a vector-valued version of Lemma \[basic-edo\]: \[lemma-edo1\] Take $\delta >0$, and let $F \in L^1 (\delta , +\infty; \mathcal H)$ be continuous. Let $x:[\delta, +\infty[ \rightarrow \mathcal H$ be a solution of $$\label{strong-conv-int10} t\ddot{x}(t) + \alpha \dot{x}(t) = F(t)$$ with $\alpha >1$. Then, $x(t)$ converges strongly in $\mathcal H$ as $t \to + \infty$. As in the proof of Lemma \[basic-edo\], multiply by $t^{\alpha -1}$ and integrate to obtain $$\dot{x}(t) = \frac{{\delta}^{\alpha}\dot{x}(\delta)}{t^{\alpha}} + \frac{1}{t^{\alpha}} \int_{\delta}^t s^{\alpha -1}F(s)ds.$$ Integrate again to deduce that $$x(t) = x(\delta) + {\delta}^{\alpha}\dot{x}(\delta) \int_{\delta}^t \frac{1}{s^{\alpha}} ds + \int_{\delta}^t \frac{1}{s^{\alpha}} \left( \int_{\delta}^s {\tau}^{\alpha -1}F(\tau)d\tau\right) ds.$$ Fubini’s Theorem applied to the last integral gives $$\label{strong-conv-int14} x(t) = x(\delta) + \frac{{\delta}^{\alpha}\dot{x}(\delta)}{\alpha-1}\left(\frac{1}{\delta^{\alpha-1}}-\frac{1}{t^{\alpha-1}}\right)+ \frac{1}{\alpha -1} \left( \int_{\delta}^t F(\tau)d\tau - \frac{1}{t^{\alpha-1}} \int_{\delta}^t {\tau}^{\alpha-1} F(\tau)d\tau \right).$$ Finally, apply Lemma \[basic-int\] to the last integral with $\psi(s)=s^{\alpha-1}$ and $f(s)=\|F(s)\|$ to conclude that all the terms in the right-hand side of have a limit as $t \to + \infty$. The following is a continuous version of Kronecker’s Theorem for series (see, for example, [@Kno page 129]): \[basic-int\] Take $\delta >0$, and let $f \in L^1 (\delta , +\infty)$ be nonnegative and continuous. Consider a nondecreasing function $\psi:(\delta,+\infty)\to(0,+\infty)$ such that $\lim\limits_{t\to+\infty}\psi(t)=+\infty$. 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--- author: - Antoine Amarilli - '[İ]{}smail [İ]{}lkan Ceylan' bibliography: - 'main.bib' title: 'A Dichotomy for Homomorphism-Closed Queries on Probabilistic Graphs' --- Introduction {#sec:intro} ============ Preliminaries {#sec:prelim} ============= Result Statement {#sec:result} ================ Hardness with Non-Iterable Edges {#sec:pp2dnf} ================================ \[apx:pp2dnf\] Finding a Minimal Tight Pattern {#sec:findhard} =============================== Hardness with Tight Iterable Edges {#sec:ustcon} ================================== \[apx:ustcon\] Conclusion {#sec:conc} ==========

--- abstract: 'Finite-${\mathbf Q}$ magnetic instabilities are rather common in frustrated magnets. When the magnetic susceptibility is maximized at multiple-${\mathbf Q}$ vectors related through lattice symmetry operations, exotic magnetic orderings such as vortex and skyrmion crystals may follow. Here we show that a periodic array of nonmagnetic impurities, which can be realized through charge density wave ordering, leads to a rich phase diagram featuring a plethora of chiral magnetic phases, especially when there is a simple relation between the reciprocal vectors of the impurity superlattice and the magnetic ${\mathbf Q}$-vectors. We also investigate the effect of changing the impurity concentration or disturbing the impurity array with small quenched randomness. Alternative realizations of impurity superlattices are briefly discussed.' author: - Satoru Hayami - 'Shi-Zeng Lin' - Yoshitomo Kamiya - 'Cristian D. Batista' bibliography: - 'my-refcontrol.bib' - 'ref.bib' nocite: '[@apsrev41Control]' title: | Vortices, skyrmions, and chirality waves in frustrated Mott insulators\ with a quenched periodic array of impurities --- Introduction {#sec:Introduction} ============ The emergence of nonzero bulk spin-scalar chirality, known as chiral order, has drawn considerable interest in condensed matter physics. Various consequences of a chiral order have been discussed in different fields ranging from superconductivity to Mott insulators [@Wen_PhysRevB.39.11413; @Kawamura_PhysRevLett.68.3785; @Momoi_PhysRevLett.79.2081; @Bulaevskii_PhysRevB.78.024402]. An attractive area of research is generated by potential realizations of chiral liquid states, i.e., states that exhibit chiral order in absence of magnetic order [@Kawamura_0953-8984-10-22-004; @Kawamura_PhysRevLett.68.3785; @Domenge_PhysRevB.72.024433; @Hassanieh_PhysRevLett.103.216402]. Another attractive aspect of chiral states is their potential for inducing nontrivial topological phenomena, such as topological anomalous Hall effect for electrons coupled to a chiral spin state through the Berry phase mechanism [@berry1984quantal; @Aharonov_PhysRev.115.485; @Loss_PhysRevB.45.13544; @Ye_PhysRevLett.83.3737; @Nagaosa_RevModPhys.82.1539; @Martin_PhysRevLett.101.156402]. The very large fictitious magnetic field produced by this mechanism ($10^3$–$10^4$ T) may bring major advances for spintronics applications [@Zutic_RevModPhys.76.323]. It is then important to understand the physical mechanisms to stabilize the chiral order. Noncoplanar magnetic orderings are accompanied by a nonzero local scalar chirality $\langle \chi_{jkl} \rangle = \langle \mathbf{S}_j \cdot (\mathbf{S}_k \times \mathbf{S}_l) \rangle \neq 0$, where $j$, $k$, and $l$ are three neighboring lattice sites. Recent theoretical studies on frustrated Kondo lattice models have unveiled a general mechanism for stabilizing noncoplanar magnetic orderings in itinerant magnets comprising conduction electrons coupled to localized magnetic moments [@Martin_PhysRevLett.101.156402; @Chern:PhysRevLett.105.226403; @Kato_PhysRevLett.105.266405; @Akagi_PhysRevLett.108.096401; @Hayami_PhysRevB.90.060402; @Ozawa15; @Hayami_PhysRevB.94.024424]. The mechanism relies on the generation of four and higher spin interaction terms, which appear upon expanding in the small Kondo interaction beyond the Ruderman-Kittel-Kasuya-Yosida (RKKY) level [@Akagi_PhysRevLett.108.096401; @Hayami_PhysRevB.90.060402; @Ozawa15]. These multi-spin interactions are relatively weak in strongly coupled Mott insulators. In terms of a Hubbard model description with hopping amplitude $t$ and on-site Coulomb potential $U$, four spin interactions are of order ${\cal O}(t^4/U^3)$, while two-spin interactions are ${\cal O}(t^2/U)$. Consequently, chiral spin textures are less common in Mott insulators with isotropic exchange interactions. Indeed, these systems usually exhibit a conical spiral order with zero net scalar chirality even in an external magnetic field; otherwise either collinear or coplanar orderings. However, recent theoretical studies in both classical [@Okubo_PhysRevLett.108.017206; @Rousochatzakis2016; @leonov2015multiply; @Shizeng_PhysRevB.93.064430; @Hayami_PhysRevB.93.184413] and quantum [@Kamiya_PhysRevX.4.011023; @Wang_PhysRevLett.115.107201; @Marmorini2014] frustrated spin systems, have shown that the interplay between geometric frustration, thermal [@Okubo_PhysRevLett.108.017206] or quantum fluctuations, [@Kamiya_PhysRevX.4.011023; @Wang_PhysRevLett.115.107201; @Marmorini2014] magnetic anisotropy [@Rousochatzakis2016; @leonov2015multiply; @Shizeng_PhysRevB.93.064430; @Hayami_PhysRevB.93.184413; @Binz_PhysRevLett.96.207202; @Binz_PhysRevB.74.214408; @Park_PhysRevB.83.184406], and long-range (dipolar) interactions [@lin1973bubble; @malozemoff1979magnetic; @kiselev2011chiral], can stabilize a plethora of multiple-$Q$ spin textures in Mott insulators, some of which have net scalar spin chirality. In this context, it is worth mentioning a recent experimental confirmation of a triple-$Q$ vortex crystal in the scandium thiospinel MnSc$_2$S$_4$ induced by a magnetic field, where geometric frustration and anisotropy seem to play the key role. [@Gao2016Spiral] In this article we demonstrate that frustrated Mott insulators coupled to a superlattice of nonmagnetic impurities can generate chiral states in the presence of an external magnetic field. In contrast to Ref. , where we have shown that a single nonmagnetic impurity nucleates a magnetic vortex over a finite range of magnetic field values above the bulk saturation field ${H_\text{sat}}$, here we focus on the effects of a [*periodic array*]{} of non-magnetic impurities [*below*]{} the saturation field. A crucial observation is that the local saturation field, ${H_\text{sat}}^I$, around an impurity can be higher than ${H_\text{sat}}$ in frustrated magnets with competing ferro- and antiferromagnetic interactions. Given that a single impurity nucleates a magnetic vortex around it for ${H_\text{sat}}< H < {H_\text{sat}}^I$, it is natural to ask about the effect of an array of impurities when $H < {H_\text{sat}}$. It is known that nonmagnetic impurities tend to reorient the surrounding spins into a less collinear fashion by inducing an effective biquadratic interaction $(\mathbf{S}_j \cdot \mathbf{S}_k)^2$ with a positive (hence antiferroquadrupolar) coupling constant [@Wollny_PhysRevLett.107.137204; @Sen_PhysRevB.86.205134; @Maryasin_PhysRevLett.111.247201; @maryasin2015collective]. This rather general observation provides an alternative motivation for studying the magnetic effects of periodic, and nearly periodic, arrays of impurities. Motivated by these observations we focus on the low temperature ($T$) physics of a classical spin model. An important prerequisite is to include competing ferro- and antiferromagnetic exchange interactions so that the Fourier transform, $J(\mathbf{q})$, has multiple degenerate minima, $\mathbf{Q}_1, \mathbf{Q}_2,\cdots$ connected by lattice symmetry transformations. The ordered array of nonmagnetic impurities may be realized in a hole- or electron-doped system with sufficiently strong off-site Coulomb interactions so that a charge density wave (CDW) order leads to an array of holes or doubly-occupied sites. For instance, certain high-$T_{c}$ superconductors and related materials are known to have a CDW of holes at a hole concentration $x=1/8$ (Refs. ). Other possible realizations will be discussed later. The rest of the paper is organized as follows. In Sec. \[sec: model\], we present our model, outline our Monte Carlo (MC) method, and list the observables that we evaluate. In Sec. \[sec: perfect array\], we show the temperature ($T$)-magnetic field ($H$) phase diagram for a perfectly periodic array of nonmagnetic impurities and a simple relation between the superlattice reciprocal unit vectors and the $\mathbf{Q}$-vectors (a preliminary account of the discussion in Sec. \[sec:a8\] was presented in Ref. ). A plethora of multi-$\mathbf{Q}$ phases are obtained from our MC simulations. In Sec. \[sec:variational\], we provide an analytical analysis that explains all of the numerically obtained phases close to $T = 0$. In Sec. \[sec: more realistic\], we discuss the effects of changing the impurity concentration or introducing small quenched randomness to the impurity lattice. Sec. \[sec: summary\] includes a discussion of potential realizations of periodic arrays of nonmagnetic impurities. \[sec: model\] Model ===================== Model ----- We consider a two-dimensional classical $J_1$-$J_3$ triangular-lattice Heisenberg magnet in a magnetic field with nonmagnetic impurities. In the absence of impurities, the Hamiltonian is $$\begin{aligned} {\mathcal{H}}_\text{pure} = J_1\sum_{\langle j,l \rangle} \mathbf{S}_j \cdot \mathbf{S}_l + J_3 \sum_{\langle \langle j,l \rangle \rangle} \mathbf{S}_j \cdot \mathbf{S}_l-H\sum_j S^z_j, \label{Eq:J1J3} \end{aligned}$$ with ferromagnetic nearest-neighbor (NN) exchange, $J_1 < 0$, and antiferromagnetic third NN exchange $J_3 > 0$. $\mathbf{S}_j$ represents a classical spin located at the site $j$ with $|\mathbf{S}_j|=1$. The thermodynamic phase diagram of this model Hamiltonian includes a magnetic field induced finite temperature skyrmion crystal phase for [@Okubo_PhysRevLett.108.017206; @leonov2015multiply; @Shizeng_PhysRevB.93.064430; @Hayami_PhysRevB.93.184413] $$\begin{aligned} J_3 / \lvert{J_1}\rvert > J^c_3 / \lvert{J_1}\rvert = 1.0256(53). \label{cond}\end{aligned}$$ The skyrmion crystal is a state with spontaneously broken chiral symmetry, corresponding to the superposition of harmonic waves with sixfold-degenerate incommensurate wave vectors $\pm\mathbf{Q}_{\nu}$ ($\nu=1$–$3$). These are the wave-vectors that minimize the Fourier transform of the exchange interaction $$\begin{aligned} \label{Eq:Jq} J(\mathbf{q})=\sum_{1 \le j \le 3} (J_1 \cos \mathbf{q}\cdot \mathbf{e}_j + J_3 \cos 2 \mathbf{q} \cdot \mathbf{e}_j).\end{aligned}$$ Here $\mathbf{e}_1 = \mathbf{\hat{x}}$, $\mathbf{e}_2=-\mathbf{\hat{x}}/2+\sqrt{3} \mathbf{\hat{y}}/2$, and $\mathbf{e}_3 = -\mathbf{e}_1 - \mathbf{e}_2$. The incommensurate minima emanate from the $\Gamma$ point (Lifshitz transition) for $J_3 / \lvert{J_1}\rvert > 1/4$ (we will adopt a convention where the lattice spacing is $a = a^{-1} = 1$): $$\begin{aligned} \pm \mathbf{Q}_1 &= \pm Q \mathbf{\hat{x}}, \notag\\ \pm \mathbf{Q}_2 &= \pm Q (-\mathbf{\hat{x}}/2+\sqrt{3} \mathbf{\hat{y}}/2), \notag\\ \pm \mathbf{Q}_3 &= \pm Q (-\mathbf{\hat{x}}/2-\sqrt{3} \mathbf{\hat{y}}/2),\end{aligned}$$ with $$\begin{aligned} Q = 2 \arccos \left[ \frac{1}{4} \left( 1 + \sqrt{1-\frac{2J_1}{J_3}} \right) \right]. \label{Eq:Qvalue}\end{aligned}$$ Thus, according to Eqs.  and , the skyrmion crystal phase is only stable for $Q / (2\pi) > Q_c / (2\pi) = 0.2623(3)$. For $\lvert{J_1}\rvert/4<$ $J_3 \lesssim J_3^c$, $J(\mathbf{q})$ resembles the bottom of a wine bottle near the $\Gamma$ point, with a weaker $C_6$ anisotropy as the system approaches the Lifshitz transition point $J_3 = |J_1|/4$. In this regime, the phase diagram comprises only the high-temperature paramagnetic phase and the single-${\bf Q}$ conical spiral phase, both of which have no net scalar spin chirality. [@Hayami_PhysRevB.93.184413] We will first consider the effect of a periodic array of nonmagnetic impurities forming a perfect triangular superlattice. The primitive reciprocal superlattice vectors are $$\begin{aligned} \mathbf{K}_{\pm} = \frac{2\pi}{a_\text{imp}} (1,\pm 1/\sqrt{3}),\end{aligned}$$ where $a_\text{imp}$ is the impurity superlattice constant. The impurities are introduced in the Hamiltonian by replacing $$\begin{aligned} \mathbf{S}_{j} &\to p_j \mathbf{S}_{j}, \label{Eq:imp}\end{aligned}$$ where $p_j=0$ ($1$) for a nonmagnetic impurity (magnetic) site. This amounts to introducing the impurity contribution to the Hamiltonian ${\mathcal{H}}_\text{imp} = {\mathcal{H}}^{J}_\text{imp} + {\mathcal{H}}^{H}_\text{imp}$, so that $$\begin{aligned} {\mathcal{H}}= {\mathcal{H}}_\text{pure} + {\mathcal{H}}^{J}_\text{imp} + {\mathcal{H}}^{H}_\text{imp}, \label{hamil} \end{aligned}$$ with $$\begin{aligned} {\mathcal{H}}^{J}_\text{imp} &= - \sum_j (1 - p_j) \sum_\eta J_{j,\eta} \mathbf{S}_j \cdot \mathbf{S}_{j+\eta}, \notag\\ {\mathcal{H}}^{H}_\text{imp} &= H \sum_{j}(1-p_j) S_j^z. \label{eq:Himp}\end{aligned}$$ Here $\eta$ is the index of the coordination vectors and the different notation $J_{j,\eta} = J_1,J_3$ for the exchange coupling is associated with the $\eta$th coordination vector at each site $j$. After presenting phase diagrams for periodic arrays of impurities (Sec. \[sec: perfect array\]), we will introduce a small randomness in the impurity locations around the superlattice sites (Sec. \[subsec:Skyrmion state by slight charge fluctuations\]). Monte Carlo method ------------------ We perform classical MC simulations of our spin Hamiltonian ${\mathcal{H}}$ given in Eq.  for several impurity configurations to be specified below. Our simulations are carried out with the standard Metropolis local updates supplemented with the over-relaxation method. [@Brown1987] The lattice has $N=L^2$ sites (including impurity sites) with $L = 48$, 64, 72, 80, and 96 and we impose the periodic boundary conditions in both directions. We first perform simulated annealing for $10^5$–$10^6$ MC sweeps (MCS) to find low energy configuration, which is then followed by equilibration steps and a subsequent sampling process of $10^5$–$10^7$ MCS at the target temperature. The statistical errors are estimated from $5$–$64$ independent runs. We calculate the specific heat, $C$, the uniform magnetic susceptibility $\mathrm{d}M/\mathrm{d}H$, and the spin and the chirality structure factors. The spin structure factor $S_{s}^{\alpha\alpha}(\mathbf{q})$ ($\alpha=x,y,z$) is given by $$\begin{aligned} \label{Eq:spinstructure} S_{s}^{\alpha\alpha}(\mathbf{q}) &=\frac{1}{N} \sum_{j,l} \langle S_j^{\alpha} S_{l}^{\alpha} \rangle e^{i \mathbf{q}\cdot(\mathbf{r}_j-\mathbf{r}_l)}, \notag\\ S_{s}^{\perp}(\mathbf{q}) &= S_{s}^{xx}(\mathbf{q}) + S_{s}^{yy}(\mathbf{q}).\end{aligned}$$ We define the chirality structure factor $S_{\chi}^{\mu}(\mathbf{q})$ for the upward and downward triangles ($\mu = u,d$, respectively) as $$\begin{aligned} \label{Eq:chiralstructure} S_{\chi}^{\mu}(\mathbf{q})=\frac{1}{N} \sum_{\mathbf{R},\mathbf{R}' \in \mu} \langle \chi^{}_{\mathbf{R}} \chi^{}_{\mathbf{R}'} \rangle e^{i \mathbf{q}\cdot(\mathbf{R} - \mathbf{R}')}, \end{aligned}$$ where $\mathbf{R},\mathbf{R}'$ run over sites of the specified sublattice, $\mu = u, d$, of the dual honeycomb lattice. $\chi^{}_{\mathbf{R}} =\mathbf{S}_j \cdot \left(\mathbf{S}_k \times \mathbf{S}_l\right)$ is the spin scalar chirality associated with a triangle centered at $\mathbf{R}$, where $j,k,l$ are the sites aligned counterclockwise on the triangle. We also introduce the following notations for the total scalar chirality associated with the up ($\chi^u$) and the down ($\chi^d$) triangles, $$\begin{aligned} \chi^{\mu} &= \frac{1}{N}\sum_{\mathbf{R}\in \mu} \chi^{}_{\mathbf{R}}~~\text{for $\mu=u,d$},\end{aligned}$$ and their sum, $$\begin{aligned} \chi^\text{tot} &= \chi^{u} + \chi^{d},\end{aligned}$$ which is the total scalar chirality. Periodic array of impurities {#sec: perfect array} ============================ \[tab\_a8\] We start our discussion with the case where magnetic impurities form a perfect triangular superlattice. We first require commensurability between the superlattice reciprocal vectors, $\mathbf{K}_\pm$, and $\pm\mathbf{Q}_{1 \le \nu \le 3}$, which is expected to enhance the constructive interference between the impurity superlattice and the spin texture (we will relax this condition later). For the sake of concreteness, we fix $J_3 / \lvert{J_1}\rvert = 1/(4 - 2\sqrt{2}) \approx 0.854$ unless otherwise specified, which corresponds to $Q / (2\pi) = 1/4$ according to Eq. . Given that $J_3<J_3^c$, the phase diagram in absence of impurities only includes the single-${\boldmath Q}$ conical spiral state shown in Fig. \[Fig:phase-diagram\_a8\](b). Such a state is not chiral because $\chi^u$ and $\chi^d$ cancel each other. For nonzero $H$, this quasi-long-range ordered state completely breaks the $C_6$ symmetry of ${\mathcal{H}}$ because of the associated bond ordering, whereas the $C_6$ symmetry is broken down to $C_3$ for $H = 0$ as there is a continuous symmetry operation that can change the sign of the vector chirality. In both cases, our results are consistent with the single-step first-order transition found for the zero-field $J_1$-$J_3$ model with classical XY and Heisenberg spins in Refs. , where the symmetry is O(2) and O(3), respectively, corresponding to the cases with and without the magnetic field in the present consideration. As described in the introduction, the nonmagnetic impurities in a frustrated magnet with competing ferro- and antiferromagnetic exchanges make the local saturation field for the surrounding spins larger than the bulk value (${H_\text{sat}}^I > {H_\text{sat}}$). This is so because the nearest neighbor spins of the non-magnetic impurity feel a molecular field, parallel to the applied field, which is lower than the molecular field acting on other spins. This effect is of course present for any value of the external field: the spins that surround a non-magnetic impurity have a lower magnetic susceptibility because the impurity removes the ferromagnetic ($J_1$) bonds connecting them to the missing spin (Fig. \[Fig:vortex\]). Because the field inducing the $z$ spin component is smaller than the average for these spins, their $xy$ component becomes larger at low energies. Moreover, we can anticipate that the resulting local spin configuration near each impurity is likely to be a vortex as in the case with ${H_\text{sat}}< H < {H_\text{sat}}^I$, [@Lin_PhysRevLett.116.187202] because of the competition between $J_1$ and $J_3$ for the six spins surrounding the impurity (Fig. \[Fig:vortex\]). The rest of the spins have to accommodate their configuration to the local “boundary condition" imposed by each impurity. Below, we consider two representative cases where the superlattice constant for the periodic impurities is $a_\text{imp} = 8$ and $a_\text{imp} = 4$. They correspond to simple relations between the superlattice reciprocal vectors and the ${\boldmath Q}$ vectors, namely, $\mathbf{K}_+ + \mathbf{K}_- = \mathbf{Q}_1$ and $\mathbf{K}_+ + \mathbf{K}_- = 2\mathbf{Q}_1$, respectively. We will demonstrate that such a commensurate impurity superlattice produces a drastic change of the magnetic phase diagram. Our finite-size scaling analysis to characterize the phases is summarized in Appendix \[sec:Finite-size scaling of each phase\]. {width="0.85\hsize"} {width="0.85\hsize"} Case with $a_\text{imp} = 8$ {#sec:a8} ---------------------------- The configuration of impurities is shown in the inset of Fig. \[Fig:phase-diagram\_a8\](a). The impurities are separated by $a_\text{imp} = 8$ sites along the lattice principal directions. This superlattice spacing is exactly twice as large as $2\pi/ Q =4$. Figure \[Fig:phase-diagram\_a8\](a) shows the $T$-$H$ phase diagram obtained with our MC simulations, which features eight different phases other than the paramagnetic state (Table \[tab\_a8\]). The phase boundaries are determined by analyzing the structure factors in Eqs.  and (Figs. \[Fig:phase1-4\_a8\] and \[Fig:phase5-8\_a8\]) and the peak in the uniform magnetic susceptibility \[Fig. \[Fig:phase-diagram\_a8\](d)\] as a function of $H$. Notably, many phases are multiple-$Q$ states that support long-wavelength modulation of local scalar chirality (chirality wave), which is not necessarily single-$Q^\chi$ but some of them are actually of the multiple-$Q^\chi$ type (our convention is to use $Q^{M}$ and $Q^{\chi}$, respectively, when it is necessary to make an unambiguous distinction between spin and chirality textures). The commensuration between the chirality waves and the impurity superlattice leads to a majority of magnetically ordered phases with net scalar spin chirality \[see Figs. \[Fig:phase-diagram\_a8\](c) and \[Fig:phase-diagram\_a8\](e)\]. The uniform component arises from uncompensated positive and negative components of the chirality wave texture: the impurities remove spins contributing to only one sign of the modulated chiral structure. Meanwhile, the $xy$ spin components only exhibit quasi-long-range correlations in $d = 2$, as expected from Mermin-Wagner’s theorem. [@Mermin_PhysRevLett.17.1133] In agreement with our discussion above, the spins around the impurities form vortices with enhanced $xy$ components. In what follows, we describe details of the obtained phases. #### Ferrochiral 3$Q^M$-6$Q^\chi$ vortex crystal [^1] This phase appears right below the saturation field near $T = 0$. Upon entering this phase via the thermal transition, the specific heat shows a single weak anomaly as shown in Fig. \[Fig:phase-diagram\_a8\](f). The spin configuration is a triple-$Q^M$ vortex crystal similar to the one reported recently in 3D frustrated quantum magnets [@Kamiya_PhysRevX.4.011023; @Wang_PhysRevLett.115.107201]. This phase may be understood as the natural extension of the single-impurity vortex for ${H_\text{sat}}< H < {H_\text{sat}}^I$ [@Lin_PhysRevLett.116.187202]. In fact, as shown in Fig. \[Fig:phase1-4\_a8\](a), each nonmagnetic impurity induces a vortex very similar to the schematic picture in Fig. \[Fig:vortex\], forming a triangular vortex lattice as a whole. Upon closer inspection, it is possible to see that antivortices appear right in the middle of nearest neighbor vortices, so that the “boundary condition" by the impurity array is satisfied. The vortices and antivortices give opposite contributions to the scalar spin chirality. While this would lead to a total cancellation of $\chi^\text{tot}$ in a system without impurities [@Kamiya_PhysRevX.4.011023], the present vortex crystal has net scalar spin chirality because the local scalar chirality around each impurity has the same sign. The net chirality for up and down triangles is equal, $\chi^u = \chi^d$, which we refer to as “ferrochiral.” Interestingly, the chirality wave is primarily characterized by higher harmonics relative to the wave vectors that minimize $J(\mathbf{q})$, namely, $\mathbf{Q}_1^\chi = \mathbf{Q}_1 + 2\mathbf{Q}_2$, etc. #### Ferrochiral 3$Q^M$ vortex crystal This phase can be seen as another type of crystallization of vortices found in Ref.  for ${H_\text{sat}}< H < {H_\text{sat}}^I$. It occupies a small corner of the entire region of the ordered phases, which appears right below the saturation field in the range of intermediate temperature, $0.37 \lesssim T / J_3 \lesssim 0.43$. The spin configuration is characterized by the triple-$Q^M$ modulation and net spin scalar chirality ($\chi^u = \chi^d \ne 0$) \[Fig. \[Fig:phase1-4\_a8\](b)\]. However, unlike the ferrochiral 3$Q^M$-6$Q^\chi$ vortex crystal phase discussed above, the chirality texture does not support a static finite-$Q^\chi$ component in the thermodynamic limit (namely, the chirality texture is homogeneous) as shown in Fig. \[Fig:sizedep\_opt1-4\](b) in Appendix \[sec:Finite-size scaling of each phase\]; this aspect distinguishes this phase from the skyrmion crystal phase [@Okubo_PhysRevLett.108.017206], although both phases are triple-$Q^M$ and chiral. Moreover, the $z$ component near the impurities is $S^z \sim 0$ in the ferrochiral 3$Q^M$ vortex crystal phase, while it is $S^z \sim -1$ in the skyrmion crystal phase, which implies a different topological nature. The scalar chirality shows a peak near the phase boundary between this and the ferrichiral 3$Q^M$-2$Q^\chi$ spiral II phase discussed below \[Fig. \[Fig:phase-diagram\_a8\](c)\]. #### Ferrochiral 3$Q^M$-1$Q^\chi$ spiral This phase occupies a smaller $H$ region next to the ferrochiral 3$Q^M$-6$Q^\chi$ vortex crystal phase at low $T$. As shown in Fig. \[Fig:phase1-4\_a8\](c), it has a triple-$Q^M$ spin texture. In the structure factor for the $xy$ component, there are two dominant peaks and a single subdominant peak, while a single-$Q$ peak is in the structure factor for the $z$ component. Upon closer looking, it becomes clear that the spin texture is pinned by the impurities via the parts with local chirality having the same sign; for this reason this state is chiral, $\chi^u = \chi^d \ne 0$. In addition to the uniform component, the chirality texture has a single-$Q^\chi$ component corresponding to the stripe modulation. #### Ferrichiral 3$Q^M$-2$Q^\chi$ spiral I This phase occupies the largest portion of the phase diagram among the ordered phases and it is found next to the ferrochiral 3$Q^M$-1$Q^\chi$ spiral upon decreasing $H$. The spin configuration is characterized by the triple-$Q^M$ noncoplanar modulation. $S^{zz}_s(\mathbf{q})$ has two dominant peaks with an additional smaller peak, which in $S^{\perp}_s(\mathbf{q})$ in turn correspond to two subdominant peaks and the major peak, respectively \[Fig. \[Fig:phase1-4\_a8\](d)\]. We find that the impurities are on the contour $S^z \approx 0$ as shown in the left panel of Fig. \[Fig:phase1-4\_a8\](d) in accordance with the general argument that the effect of the magnetic field is effectively weakened for spins surrounding impurities. The chirality wave is mainly characterized by the double-$Q^\chi$ modulation, which can be seen as the “checkerboard" pattern. Note that the ferrichiral 3$Q^M$-2$Q^\chi$ spiral I phase also possesses a small subdominant peak at $\mathbf{Q}_1+\mathbf{Q}_2$ as shown in the chiral structure factor in Fig. \[Fig:phase1-4\_a8\](d). A subtle difference is that $S^{u}_{\chi}(\mathbf{q})$ and $S^{d}_{\chi}(\mathbf{q})$ have different profiles in this state. Based on this observation, we call this state “ferrichiral.” #### Ferrichiral 3$Q^M$-2$Q^\chi$ spiral II This state is found in the intermediate-$H$ regime, next to the ferrichiral 3$Q^M$-2$Q^\chi$ spiral I phase upon increasing $T$. While this is very similar to the ferrichiral 3$Q^M$-2$Q^\chi$ spiral I, the intensities of the two dominant components of the chiral structure factor are different in this phase \[Fig. \[Fig:phase5-8\_a8\](e)\] , while the ones for the ferrichiral 3$Q^M$-2$Q^\chi$ spiral I phase are the same. #### Antiferrochiral single-$Q^M$ spiral This state appears next to the ferrichiral 3$Q^M$-2$Q^\chi$ spiral II state upon decreasing $H$. The spin configuration shown in Fig. \[Fig:phase5-8\_a8\](f) resembles the conical spiral state that is obtained without impurities \[Fig. \[Fig:phase-diagram\_a8\](b)\]. The difference, however, is that the impurities introduce the additional weak longitudinal modulation. The $C_6$ symmetry is broken in this phase as in the single-$Q$ spiral phase in the pure $J_1$-$J_3$ model with easy-plane anisotropy, where the symmetry of the global spin rotation, U(1), is the same as in the present case. [@tamura2011first] The obtained specific heat curve \[Fig. \[Fig:phase-diagram\_a8\](g)\] is consistent with the single first order phase transition reported by Tamura [*et al.* ]{} [@tamura2011first] in the pure easy-plane model, although a more careful finite size scaling is required to settle this point. This state is not chiral because $\chi^u$ and $\chi^d$ cancel each other out: $\chi^u = -\chi^d$. For this reason we refer to this phase as “antiferrochiral.” #### Ferrochiral 3$Q^M$-2$Q^\chi$ spiral This phase occupies the low-field and low-$T$ region of the phase diagram \[Fig. \[Fig:phase-diagram\_a8\](a)\] and appears next to the ferrichiral 3$Q^M$-2$Q^\chi$ spiral I phase with decreasing $H$. The spin configuration \[see Fig. \[Fig:phase5-8\_a8\](g)\] closely resembles a single-$Q^M$ vertical spiral state (see below) though the small additional $\mathbf{Q}^M$ components render the spin configuration noncoplanar with the triple-$Q^M$ modulation. Meanwhile, the chirality wave texture shows the double-$Q^\chi$ modulation with a single-$Q^\chi$ subdominant component, which is very similar to that of the ferrichiral 3$Q^M$-2$Q^\chi$ spiral I state. The difference is that the net chirality for up and down triangles in the present state is equal, $\chi^u = \chi^d$, while it is different in the ferrichiral 3$Q^M$-2$Q^\chi$ spiral state. Once again, it is evident that the spin texture is pinned by the impurities where the local chirality has the same sign, which leads to nonzero uniform scalar chirality. The ferrochiral 3$Q^M$-2$Q^\chi$ spiral state extends its stability down to $H = 0$, as shown in Fig. \[Fig:phase-diagram\_a8\](e). #### Vertical single-$Q^M$ spiral This phase occupies a region at higher $T$ next to the ferrochiral 3$Q^M$-2$Q^\chi$ spiral phase. This is a single-$Q^M$ coplanar state in an arbitrary plane containing the vertical $c$-axis when the magnetic field is applied in the $c$ direction. Thus, this state has no net chirality. The difference relative to the ferrochiral 3$Q^M$-2$Q^\chi$ spiral state is the disappearance of the subdominant components in the spin structure factor induced by thermal fluctuations \[Fig. \[Fig:phase5-8\_a8\](h)\]. The transition from the paramagnetic state is suggested to be a single-step first-order phase transition (not shown). This is similar to the case in the pure $J_1$-$J_3$ model, [@Tamura2008; @tamura2011first; @tamura2014phase] albeit with a subtle difference that the state in the latter is the conical spiral. Case with $a_\text{imp} = 4$ \[sec:a4\] ---------------------------------------- \[tab\_a4\] ![ \[Fig:phase-diagram\_a4\] (a) Phase diagram of the model with a periodic array of impurities with $a_\text{imp} = 4$ and $J_3 / |J_1| \approx 0.854$ ($Q = 2\pi/4$). The inset shows the impurity configuration. (b), (c) $T$ dependence of the specific heat at (b) $H/J_3=3.3$ and (c) $H/J_3=0.5$. ](fig5){width="1.0\hsize"} {width="0.85\hsize"} Next, we briefly discuss the case where the superlattice constant $a_\text{imp} = 4$ is half of the previous case, as illustrated in the inset of Fig. \[Fig:phase-diagram\_a4\](a). This corresponds to the relation $\mathbf{K}_+ + \mathbf{K}_- = 2\mathbf{Q}_1$. Even though the commensurability still holds, the obtained phases summarized in Table \[tab\_a4\] are very different from the previous case, except for the fact that a vortex crystal still appears right below the saturation field. #### Nonchiral 2$Q^M$-3$Q^\chi$ vortex crystal As it is shown in Fig. \[Fig:phases\_a4\](a), the high field state is another vortex crystal, in which both the vortices and anti-vortices are nucleated by the impurities. The vortex and antivortex-chains alternate creating the stripe pattern shown in the middle of Fig. \[Fig:phases\_a4\](a). This stripe pattern breaks the translational symmetry of the system. Vortices and antivortices have opposite scalar spin chirality, producing a chirality wave with a wave vector equal to half of the impurity lattice reciprocal vector: $\mathbf{Q}^\chi = \mathbf{K}^+ / 2$ \[Fig. \[Fig:phases\_a4\](a)\]. Given that both vortex structures, the vortices and the antivortices, are nucleated around the impurities, the net chirality is perfectly cancelled, rendering this state non-chiral. Figure \[Fig:phase-diagram\_a4\](b) shows the specific curve heat near the thermal phase transition. Though it is suggestive of a single-step continuous phase transition, a more careful analysis will be required to draw a final conclusion about the critical behavior. #### Nonchiral 3$Q^M$-2$Q^\chi$ spiral The spin configuration in the low field phase has a single-$Q^M$ longitudinal modulation and a double-$Q^M$ transverse modulation \[Fig. \[Fig:phases\_a4\](b)\]. A closer inspection reveals that both $S^{\perp}_s(\mathbf{q})$ and $S^{zz}_s(\mathbf{q})$ have additional peaks induced by the impurities, such as the peaks at $q=\pi/(\sqrt{3}) \approx 1.814$ and $q=\pi/(2\sqrt{3}) \approx 0.907$. This fact is more evident for the double-$Q^\chi$ chirality wave texture, in which the subdominant component is induced by the impurities. The spin configuration accommodates itself to the impurity superlattice in such a way that the impurities are on the nodal line of the chirality wave texture \[Fig. \[Fig:phases\_a4\](b)\]. As depicted in Fig. \[Fig:phases\_a4\](b), the net chirality vanishes. The specific heat near the thermal phase transition is suggestive of a single-step continuous phase transition also in this case \[Fig. \[Fig:phase-diagram\_a4\](c)\]. Variational analysis \[sec:variational\] ========================================= Our numerical calculation indicate that multiple-$\mathbf{Q}$ ground states are realized instead of the single-${\boldmath Q}$ conical ground state that is obtained for the pure system. Below we provide a variational analysis confirming that the commensurability relation between the ${\boldmath Q}$-vectors and the superlattice reciprocal vectors $\mathbf{K}_\pm$ renders the single-$Q$ conical state unstable towards more complex multiple-$\mathbf{Q}$ structures. For the sake of concreteness, we consider below the case $a^{}_\text{imp} = 8$ where $\mathbf{K}^{}_{+} + \mathbf{K}^{}_{-} = \mathbf{Q}_1$. First we note that ${\mathcal{H}}^{J}_\text{imp}$ in Eq.  can be written as $$\begin{aligned} {\mathcal{H}}^{J}_\text{imp} = -\rho^{}_\text{imp} \sum_{\mathbf{q},\mathbf{q}' \in \text{FBZ}} 2J(\mathbf{q}) \left(\sum_{\mathbf{G}_\text{imp}} \delta_{\mathbf{q}', \mathbf{q} + \mathbf{G}_\text{imp}}\right) \mathbf{S}_{\mathbf{q}} \cdot \mathbf{S}_{-\mathbf{q}'}, \label{eq:Himp:2}\end{aligned}$$ after taking the Fourier transform $\mathbf{S}_\mathbf{q} = \sqrt{{1}/{N}} \sum_j e^{-i\mathbf{q}\cdot\mathbf{r}_j} \mathbf{S}_j$. $\mathbf{G}_\text{imp}$ runs over the set of impurity superlattice reciprocal vectors and $\rho^{}_\text{imp} = a^{-2}_\text{imp}$ is impurity concentration. Thus, because of the commensurability relation, ${\mathcal{H}}^{J}_\text{imp}$ couples the different ${\mathbf Q}$ vectors and the $\mathbf{q} = 0$ component induced by the magnetic field. Likewise, ${\mathcal{H}}^{H}_\text{imp}$ is written as $$\begin{aligned} {\mathcal{H}}^{H}_\text{imp} = \sqrt{N} \rho^{}_\text{imp} H \sum_{\mathbf{G}_\text{imp} \in \text{FBZ}} S_{\mathbf{G}_\text{imp}}^z,\end{aligned}$$ where the $Q$ vectors and $\mathbf{q} = 0$ are both included in the summation. Stability analysis of the conical spiral ---------------------------------------- We start from showing that the impurity-induced coupling makes the single-${\boldmath Q}$ state unstable at $T = 0$. To this end, we consider the following deformation of the single-${\boldmath Q}$ conical state, which satisfies the fixed-length constraint required for classical spins: $$\begin{aligned} S_j^x &= \sqrt{\sin^2\tilde{\theta} - \delta^2} \cos(\mathbf{Q}_1\cdot\mathbf{r}_j) + \delta \cos(\mathbf{Q}_2\cdot\mathbf{r}_j), \notag\\ S_j^y &= \sqrt{\sin^2\tilde{\theta} - \delta^2} \sin(\mathbf{Q}_1\cdot\mathbf{r}_j) - \delta \sin(\mathbf{Q}_2\cdot\mathbf{r}_j), \notag\\ S_j^z &= \sqrt{\cos^2\tilde{\theta} - 2\delta \sqrt{\sin^2\tilde{\theta} - \delta^2} \cos(\mathbf{Q}_3\cdot\mathbf{r}_j)}, \label{eq:ansatz}\end{aligned}$$ where $\delta$ parametrizes the magnitude of the deformation and $\tilde{\theta}$ \[which is equal to $\cos^{-1}(S_j^z),~\forall j$ for $\delta \to 0$\] is a variational parameter. Following Ref. , we introduce $$\begin{aligned} x = \delta \cos^{-2}\tilde{\theta} \sqrt{\sin^2\tilde{\theta} - \delta^2},\end{aligned}$$ and $S_j^z$ can be expanded as $$\begin{aligned} S_j^z = \cos\tilde{\theta} \sum_{n\ge 0} f_n(x) \cos\left(n\mathbf{Q}_3 \cdot \mathbf{r}_j\right),\end{aligned}$$ where, to ${\mathcal{O}\bigl({\delta^5}\bigr)}$, $f_0(x) = 1 - {x^2/4} - {15x^4/64}$, $f_1(x) = -x - {3x^3/8}$, $f_2(x) = -{x^2/4} - {5x^3/16}$, $f_3(x) = - {x^3/8}$, $f_4(x) = - {5x^4/64}$, and $f_{n \ge 5}(x)$ can be neglected at this order. We find $$\begin{aligned} \left\langle{{\mathcal{H}}^{H}_\text{imp}}\right\rangle &= N \rho^{}_\text{imp} H \cos \tilde{\theta} \left(1 - x - \frac{x^2}{2} - \frac{x^3}{2} - \frac{35 x^4}{64} \right) \notag\\ &+ {\mathcal{O}\bigl({\delta^5}\bigr)}.\end{aligned}$$ Also, by splitting Eq.  into different spin components as ${\mathcal{H}}^{J}_{\text{imp}} = {\mathcal{H}}_{\text{imp}}^{xx} + {\mathcal{H}}_{\text{imp}}^{yy} + {\mathcal{H}}_{\,\text{imp}}^{zz}$ and denoting $J_{n Q} = J(n\mathbf{Q}_1) = J(n\mathbf{Q}_2) = J(n\mathbf{Q}_3)$ and $J_0 = J(0)$, we find $$\begin{aligned} \left\langle{{\mathcal{H}}_{\text{imp}}^{xx}}\right\rangle &= -2 N \rho^{}_\text{imp} J_Q \left( \sin^2\tilde{\theta} + 2 x \cos^2\tilde{\theta} \right), \notag\\ \left\langle{{\mathcal{H}}_{\text{imp}}^{yy}}\right\rangle &= 0,\end{aligned}$$ which are independent of the deformation, and $$\begin{aligned} \left\langle{{\mathcal{H}}_{\text{imp}}^{zz}}\right\rangle &= -2 N \rho^{}_\text{imp} \cos^2\tilde{\theta} \left[ \frac{1}{4} J_0 f^2_0(x) + \frac{1}{2} f_0(x) \sum_{n \ge 1} \left( J_0 + J_{n Q} \right) f_n(x) + \sum_{n \ge 1} J_{nQ} f_n(x) \sum_{m \ge 1} f_m(x) \right] \notag\\ &= -2 N \rho^{}_\text{imp} \cos^2\tilde{\theta}~ \Biggl[ \frac{J_0}{4} - \frac{J_0 + J_{Q}}{2} x - \frac{2 J_0 - 8 J_{Q} + J_{2Q}}{8} x^2 + \frac{-2 J_0 + 3 J_{Q} + 4 J_{2Q} - J_{3Q}}{16} x^3 \notag\\[-2pt] &\hspace{185pt} + \frac{-34 J_0 + 112 J_Q - 8 J_{2Q} + 16 J_{3Q} - 5 J_{4Q}}{128} x^4 \,\Biggr] + {\mathcal{O}\bigl({\delta^5}\bigr)}. \end{aligned}$$ These results show that $\left\langle{{\mathcal{H}}_{\,\text{imp}}}\right\rangle_\delta - \left\langle{{\mathcal{H}}_{\,\text{imp}}}\right\rangle_{\delta=0}$ includes a *linear* contribution in the deformation parameter $\delta$. In the meantime, the change in ${\mathcal{H}}_\text{pure}$ was evaluated in Ref.  as $\left\langle{{\mathcal{H}}_{\text{pure}}}\right\rangle_\delta - \left\langle{{\mathcal{H}}_{\text{pure}}}\right\rangle_{\delta=0} = N \cos^2\tilde{\theta}\,(J_{2Q} - J_0) x^4 /32 + {\mathcal{O}\bigl({\delta^5}\bigr)}$. Thus, $\left\langle{{\mathcal{H}}}\right\rangle_\delta - \left\langle{{\mathcal{H}}}\right\rangle_{\delta=0}$ decreases linearly in $\delta$, implying that the conical spiral is indeed unstable in the presence of periodic array of impurities, which is commensurate with the ordering wave vectors. Luttinger-Tisza analysis ------------------------ The next question is whether the numerically found field-induced phases can be analytically explained in a simple manner. Below we first perform a soft-spin variational analysis at $T = 0$ by adopting the following ansatz, $$\begin{aligned} {\widetilde{\mathbf{S}}}_j &= \mathbf{M}_{0} + \sum_{1 \le \mu \le 3} \left( \mathbf{M}_{{\mu}} e^{i\mathbf{Q}_\mu \cdot \mathbf{r}_j} + \text{c.c.}\right),\end{aligned}$$ where $\mathbf{M}_{0} = N^{-1} \sum_{j} {\widetilde{\mathbf{S}}}_j$ is a three-component real vector for the uniform component and $\mathbf{M}_{{\mu}} = N^{-1}\sum_{j} e^{-i\mathbf{Q}_{\mu}\cdot\mathbf{r}_j}{\widetilde{\mathbf{S}}}_j$ ($1\le\mu\le3$) are three-component complex vectors. The tilde attached to the spin variable indicates that the fixed-length constraint is replaced by the average normalization condition given by a quadratic function $$\begin{aligned} N^{-1} \sum_j \left\lvert{{\widetilde{\mathbf{S}}}_j}\right\rvert^2 &= {\left\lvert{\mathbf{M}_{0}}\right\rvert_{}^2} + 2 \sum_{1 \le \mu \le 3} {\left\lvert{\mathbf{M}_{{\mu}}}\right\rvert_{}^2} = 1.\end{aligned}$$ This average constraint can be easily taken into account with the Lagrange multiplier method. The variational energy density extended by the Lagrange multiplier $\lambda$ is $E_\text{var} = E_J^\text{pure} + E_J^\text{imp} + E_{H} + E_{\lambda}$ with $$\begin{aligned} E_J^\text{pure} &= J_0 {\left\lvert{\mathbf{M}_{0}}\right\rvert_{}^2} + 2 J_Q \sum_{1 \le \mu \le 3} {\left\lvert{\mathbf{M}_{{\mu}}}\right\rvert_{}^2}, \notag\\ E_J^\text{imp} &= -2 \rho^{}_\text{imp} J_0 {\left\lvert{\mathbf{M}_{0}}\right\rvert_{}^2} \notag\\ &\hspace{0pt} -2 \rho^{}_\text{imp} J_Q {\left\lvert{\sum_{1 \le \mu \le 3} \left( \mathbf{M}_{{\mu}} + \mathbf{M}^{\ast}_{{\mu}} \right)}\right\rvert_{}^2} \notag\\ &\hspace{0pt} -2 \rho^{}_\text{imp} \left(J_0 + J_Q\right) \sum_{1 \le \mu \le 3} \mathbf{M}_{0} \cdot \left( \mathbf{M}_{{\mu}} + \mathbf{M}^{\ast}_{{\mu}} \right), \allowdisplaybreaks[2] \notag\\ E_H &=- (1 - \rho^{}_\text{imp}) H M^{z}_{0} \notag\\ &\hspace{10pt}+ \rho^{}_\text{imp} H \sum_{1 \le \mu \le 3} \left[ M^z_{{\mu}} + \bigl(M^z_{{\mu}}\bigr)^\ast \right], \allowdisplaybreaks[2] \notag\\ E_{\lambda} &= - \lambda \left({\left\lvert{\mathbf{M}_{0}}\right\rvert_{}^2} + 2 \sum_{1 \le \mu \le 3} {\left\lvert{\mathbf{M}_{{\mu}}}\right\rvert_{}^2} - 1 \right).\end{aligned}$$ By rewriting $\mathbf{M}_0 = \mathbf{A}_0$ and $({\mathrm{Re}\,{\mathbf{M}_{{\mu}}}},{\mathrm{Im}\,{\mathbf{M}_{{\mu}}}}) = (\mathbf{A}_{\mu}, \mathbf{B}_{\mu})$ for $1 \le \mu \le 3$, we first look into the quadratic part $E_\text{var}^\text{quad} = E_\text{var}(\lambda) - E_H$, $$\begin{aligned} E_\text{var}^\text{quad} &= \begin{pmatrix} \mathbf{A}_0 & \mathbf{A}_1 & \mathbf{A}_2 & \mathbf{A}_3 \end{pmatrix} \begin{pmatrix} \omega^{}_{0} & \Delta_{0Q} & \Delta_{0Q} & \Delta_{0Q} \\ \Delta_{0Q} & \omega^{}_Q & \Delta_Q & \Delta_Q \\ \Delta_{0Q} & \Delta_Q & \omega^{}_Q & \Delta_Q \\ \Delta_{0Q} & \Delta_Q & \Delta_Q & \omega^{}_Q \end{pmatrix} \begin{pmatrix} \mathbf{A}_0 \\ \mathbf{A}_1 \\ \mathbf{A}_2 \\ \mathbf{A}_3 \end{pmatrix} \notag\\ &\hspace{10pt}+ 2 J_Q \sum_{1\le\mu\le3} {\left\lvert{\mathbf{B}_\mu}\right\rvert_{}^2} \notag\\ &= \sum_{0 \le \kappa \le 3}{\varepsilon}_\kappa {\left\lvert{ \bm{\Phi}_\kappa }\right\rvert_{}^2} + 2 J_Q \sum_{1\le\mu\le3} {\left\lvert{\mathbf{B}_\mu}\right\rvert_{}^2},\end{aligned}$$ where $\omega^{}_{0} = (1 - 2\rho^{}_\text{imp})J_0 - \lambda$, $\omega^{}_Q = (2 - 8\rho^{}_\text{imp})J_Q - 2 \lambda$, $\Delta_{0Q} = -2 \rho^{}_\text{imp} (J_0 + J_Q)$, and $\Delta_Q = -8\rho^{}_\text{imp} J_Q$. Here we have diagonalized the $4\times4$ real symmetric coefficient matrix, obtaining the eigenvalues, ${\varepsilon}_0 = {\varepsilon}_1 = 2 (J_Q - \lambda)$, ${\varepsilon}_2 = {\varepsilon}^{}_+$, and ${\varepsilon}_3 = {\varepsilon}^{}_-$ with $$\begin{aligned} {\varepsilon}^{}_\pm = \frac{\omega^{}_0 + \omega^{}_Q + 2\Delta_Q \pm \sqrt{12 \Delta_{0Q}^2 + \left( \omega^{}_0 - \omega^{}_Q - 2\Delta_Q \right)^2}}{2}.\end{aligned}$$ $\{\mathbf{A}_\mu\} \to \{\mathbf{\Phi}_{\kappa}\}$ is the associated orthogonal transformation, $$\begin{aligned} \begin{pmatrix} \mathbf{\Phi}_0 \\[7pt] \mathbf{\Phi}_1 \\[7pt] \mathbf{\Phi}_2 \\[7pt] \mathbf{\Phi}_3 \end{pmatrix} = \begin{pmatrix} 0 & \frac{2}{\sqrt{6}} & -\frac{1}{\sqrt{6}} & -\frac{1}{\sqrt{6}} \\[7pt] 0 & 0 & \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\[7pt] \frac{c^{}_+}{\sqrt{c_+^2 + 3}} & \frac{1}{\sqrt{c_+^2 + 3}} & \frac{1}{\sqrt{c_+^2 + 3}} & \frac{1}{\sqrt{c_+^2 + 3}} \\[7pt] \frac{c^{}_-}{\sqrt{c_-^2 + 3}} & \frac{1}{\sqrt{c_-^2 + 3}} & \frac{1}{\sqrt{c_-^2 + 3}} & \frac{1}{\sqrt{c_-^2 + 3}} \end{pmatrix} \begin{pmatrix} \mathbf{A}_0 \\[7pt] \mathbf{A}_1 \\[7pt] \mathbf{A}_2 \\[7pt] \mathbf{A}_3 \end{pmatrix}, \label{eq:orth}\end{aligned}$$ with $$\begin{aligned} c^{}_{\pm} = \frac{{\varepsilon}^{}_\pm - \omega^{}_Q - 2\Delta_Q}{\Delta_{0Q}}.\end{aligned}$$ By adding the Zeeman contribution, we have $$\begin{aligned} E_\text{var} &= \sum_{0 \le \kappa \le 3}{\varepsilon}_\kappa {\left\lvert{ \mathbf{\Phi}_\kappa }\right\rvert_{}^2} + 2 (J_Q - \lambda) \sum_{1\le\mu\le3} {\left\lvert{\mathbf{B}_\mu}\right\rvert_{}^2} \notag\\[-2pt] &\hspace{10pt} - \frac{(1 - \rho^{}_\text{imp})c^{}_+ - 6\rho^{}_\text{imp}}{\sqrt{c_+^2 + 3}} H {\Phi}^z_2 \notag\\ &\hspace{10pt} - \frac{(1 - \rho^{}_\text{imp})c^{}_- - 6\rho^{}_\text{imp}}{\sqrt{c_-^2 + 3}} H {\Phi}^z_3.\end{aligned}$$ Thus, by taking derivatives of $E_\text{var}$ with respect to $\Phi_\kappa^{x,y,z}$ and $B_\mu^{x,y,z}$, we can see that the magnetic field coupling to the triple-$Q$ modes ${\Phi}^z_{2,3}$ \[see Eq. \] leads to a field-induced triple-$Q$ state within the Luttinger-Tisza approximation. However, the resulting triple-$Q$ state is collinear, implying a rather strong violation of the fixed-length constraint. Real-space variational analysis ------------------------------- The above Luttinger-Tisza analysis suggests that a more strict treatment of the constraint is crucial, and the corresponding nonlinear effect is expected to drive the collinear triple-${\mathbf Q}$ state into a noncoplanar multi-${\mathbf Q}$ state. To proceed, we note that the commensurate ${\mathbf Q}$ vectors allow to work on a real-space variational calculation based on the following simple ansatz, $${\bm S}_j = \frac{1}{N_j} \left[ \mathbf{M}_{0} + \sum_{1 \le \mu \le 3} \left( \mathbf{M}_{{\mu}} e^{i\mathbf{Q}_\mu \cdot \mathbf{r}_j} + \text{c.c.}\right) \right], \label{var}$$ where $N_j$ is a normalization factor which enforces the ${\bm S} \cdot {\bm S} =1$ constraint exactly and $\mathbf{M}_{\mu}=\mathbf{M}^*_{-\mu}$. The variational parameters are seven: $M_0$ and three pairs of (Re$\mathbf{M}_\mu$, Im$\mathbf{M}_\mu$) ($\mu=1$-$3$). Figure \[Fig:variational\] shows the $H$ dependence of (a) the energy density, (b) magnetization, (c) the $xy$ component of the spin structure factor, and (d) the $z$ component of the spin structure factor, which are obtained by variational calculations at $T/J_3=0.00$ and Monte Carlo simulations at $T/J_3=0.05$. The Monte Carlo calculations are consistent with the variational results except for the region $1.6 \lesssim H/J_3 \lesssim 2.1$. The slight deviation around this region is due to a finite-temperature effect in the Monte Carlo simulations. In fact, another phase transition occurs at $T/J_3 \sim 0.03$ for $H/J_3=2.0$. Thus, most of the low-temperature phases obtained from the finite temperature Monte Carlo simulations shown in Fig. \[Fig:phase-diagram\_a8\](a) remain stable down to $T=0$. Towards more realistic considerations {#sec: more realistic} ===================================== So far, we have assumed a periodic array of impurities, which is commensurate with the spin texture. Below, we discuss the stability of the chiral phases upon relaxing this condition. Different impurity concentrations {#sec:Deviation from simple commensurability} --------------------------------- First we investigate the stability of the chiral phases shown in Fig. \[Fig:phase-diagram\_a8\] upon changing the impurity concentration. Our numerical results and the stability analysis have shown that the multiple-${\mathbf Q}$ structures arise from the fact that $\mathbf{Q}_\nu$ and $\mathbf{Q}_{\nu'}$ ($\nu \neq \nu'$) are connected by $\mathbf{K}_{\pm}$. Below we demonstrate that the multiple-${\mathbf Q}$ structure is suppressed upon changing the periodicity of impurity array, i.e., upon reducing the commensurability effect. Figure \[Fig:diffperiodic\_chi2\] shows the $H$-dependence of the square of the scalar chirality at a low enough $T$ for several different impurity concentrations and $J_3 / \lvert{J_1}\rvert \approx 0.394$ (corresponding to $Q = 2\pi/6 < Q_c$). In addition to the case without impurities, we show the results for $a_\text{imp} = 11$ and $a_\text{imp} = 12$. The case with $a_\text{imp} = 12$ is similar to the case we discussed in Sec. \[sec:a8\], in the sense that $a_\text{imp} = 12$ is twice as large as $2\pi / Q = 6$ and $\mathbf{K}_{+}+\mathbf{K}_{-} = \mathbf{Q}_1$ holds. In fact, the $H$-dependence of $(\chi^\text{tot})^2$ is qualitatively very similar to the previous case shown in Fig. \[Fig:phase-diagram\_a8\](e). Strictly speaking, the other case with $a_\text{imp} = 11$ is also commensurate with the optimal magnetic order. However, the minimal period of the combined structures is 44 lattice spacings, which is significantly large and thus almost incommensurate. Interestingly enough, we find that the net chirality is finite in the intermediate magnetic field range, although the functional form is rather different. The spin configuration is triple-$Q^M$ with vortices located around each impurity (Fig. \[Fig:a11\]). The corresponding chirality wave has a multiple-$Q^\chi$ ferrichiral structure with slightly different profiles in $S_\chi^u(\mathbf{q})$ and $S_\chi^d(\mathbf{q})$. Our results suggest that chiral states resulting from nonmagnetic impurities are rather robust against changing the impurity concentration when ${\mathbf Q}$ is small. Skyrmion crystal induced by small randomness {#subsec:Skyrmion state by slight charge fluctuations} -------------------------------------------- Finally, we introduce small quenched randomness into the array of impurities. In the example of a CDW, nonmagnetic ions (holes or doubly-occupied sites) can be frozen at positions slightly away from the perfect superlattice sites. Thus, it is natural to ask what would the consequence of such small quenched randomness in terms of the magnetic ordering. To simplify our discussion, we consider the following three cases where, as illustrated in Fig. \[Fig:randomness\], each nonmagnetic ion is uniformly and randomly placed at (i) one of seven sites comprising a site of the perfect superlattice and its six NNs of the underlying lattice, (ii) one of 13 sites comprising a perfect superlattice site, the six NNs, and the six second NNs or (iii) one of 19 sites that include up to the third NN sites. For each case, we take statistical averages by generating $32$–$144$ different impurity configurations. First, we show the impurity structure factor $$\begin{aligned} \label{Eq:impstructure} S_{\rm imp}(\mathbf{q})=\frac{1}{N} \sum_{j,l} \langle (1-p_j)(1-p_l) \rangle e^{i \mathbf{q}\cdot(\mathbf{r}_j - \mathbf{r}_l)}, \end{aligned}$$ for the different randomness (i)–(iii) in Fig. \[Fig:impurity\_structure\]. In the case of an ideal impurity array, the Bragg peaks appear at $c_1 \mathbf{K}_+ + c_2 \mathbf{K}_-$ ($c_1$ and $c_2$ are integer) with the same intensity, as shown in Fig. \[Fig:impurity\_structure\](a). With increasing the degree of randomness from the case (i) in Fig. \[Fig:impurity\_structure\](b) through the case (iii) in Fig. \[Fig:impurity\_structure\](d), the amplitudes of the Bragg peaks for large $\mathbf{q}$ diminish. In particular, the amplitude at $\mathbf{q} = \mathbf{Q}_\nu$ remains finite (almost disappears) for the cases (i) and (ii) \[the case (iii)\]. This difference is crucial for the emergence of the skyrmion crystal (see Fig. \[Fig:skyrmion\]) that we discuss below. As shown in Figs. \[Fig:randomness\_chi2\](a)–\[Fig:randomness\_chi2\](c), we compute $(\chi^\text{tot})^2$ at a low $T$ for $J_3 / \lvert{J_1}\rvert \approx 0.854$ ($Q = 2\pi / 4$). While there is no net chirality in the low and high magnetic field regions, we find that the scalar chirality is drastically enhanced in the intermediate field regime $1.2 \lesssim H / J_3 \lesssim 1.6$ for the moderate impurity randomness, i.e., the cases of (i) and (ii), as shown in Fig. \[Fig:randomness\_chi2\](c). Such a drastic enhancement indicates the emergence of a new phase induced by the quenched randomness. The spin configuration corresponds to a triple-$Q^M$ hexagonal skyrmion crystal (Fig. \[Fig:skyrmion\]), similar to the state found by Okubo [*et al.* ]{}in a different parameter regime of the same model without impurities [@Okubo_PhysRevLett.108.017206]. A similar phase is also obtained for the same model with easy-axis anisotropy. [@leonov2015multiply; @Shizeng_PhysRevB.93.064430; @Hayami_PhysRevB.93.184413] We note that, unlike the ferrochiral 3$Q^M$ spiral state in Fig. \[Fig:phase5-8\_a8\](f), the chirality structure factor of this state shows six peaks at $\mathbf{q}=\pm\mathbf{Q}_{1\le\nu\le3}$, in addition to the $\mathbf{q}=\mathbf{0}$ peak (see also Appendix \[sec:Finite-size scaling of each phase\]). The local spin reorientation induced by impurities is central to explain why the skyrmion crystal appears only for small charge randomness in the range of $1.2 \lesssim H /J_3 \lesssim 1.6$. Without the randomness, the ferrichiral 3$Q^M$-2$Q^\chi$ spiral I phase is realized in this region, where the spin texture creates the vortex configuration with $S^z \approx 0$ around each impurity. This state is more stable than the skyrmion crystal in the absence of randomness, because impurities would be near the center of the skyrmions (commensurability effect). Such a situation is energetically unfavorable due to the large $|S^z|$ value of the spins near the skyrmion core. (According to our preliminary considerations, the energy is minimized by increasing the $xy$ components of the spins near the nonmagnetic impurities relatively to the other spins.) By introducing the small quenched randomness, as shown in the left panel of Fig. \[Fig:skyrmion\], the impurities can escape from the skyrmion core towards the perimeter region where $S^z \approx 0$ and nucleate antivortices around themselves. While the chirality structure is to a large extent characterized by the $\mathbf{q} = 0$ component, as shown in the middle panel in Fig. \[Fig:skyrmion\], there are spots with the opposite sign around the impurities because of the induced antivortices. In contrast, the randomness increases the energy of the ferrichiral 3$Q^M$-2$Q^\chi$ spiral I phase because it pushes the impurities away from their “comfortable” $S^z \approx 0$ zone. This is also the reason why the skyrmion phase is destabilized when the randomness becomes too strong, i.e., in the case (iii) (see Fig. \[Fig:randomness\_chi2\]). Once the typical deviation exceeds the skyrmion radius, the impurities are pushed into the large $|S^z|$ region again. This result points to a new mechanism of stabilizing skyrmion crystals, namely, moderate randomness in the array of nonmagnetic impurities can induce a chiral phase when combined with frustrated exchange interactions. Summary and discussion {#sec: summary} ====================== As we discussed above, CDW ordering of a strongly-coupled single-band model away from half-filling can provide a natural realization of our periodic array of non-magnetic impurities. Moreover, it is natural to expect that charge orderings which are commensurate with the magnetic ordering wave-vectors will be naturally selected at the corresponding filling fractions of the Hubbard model. Our previous analysis of the effect of small randomness in the periodic array of impurities suggests the interesting possibility of having skyrmion crystals stabilized by the zero point fluctuations of the CDW. What are the alternative realizations of the model studied in this work? Here we present three additional proposals for realizing periodic arrays of nonmagnetic impurities. The first realization involves surface science technology. [@eigler1990positioning] For instance, a selective atom substitution based on the scanning tunneling microscope technique enables manipulate of atoms on the surface of Mott insulators with spiral order, such as Fe$_x$Ni$_{1-x}$Br$_2$ [@moore1985magnetic] and Zn$_x$Ni$_{1-x}$Br$_2$. [@day1981neutron] The second possibility is through Kondo lattice systems with long-range Coulomb interaction between conduction electrons. Even not taking into account Coulomb interaction explicitly (i.e., in the usual Kondo lattice model) charge ordering can be induced by magnetic ordering [@Sahinur_PhysRevB.91.140403; @Misawa_PhysRevLett.110.246401; @Hayami:PhysRevB.89.085124; @hayami2013charge; @ishizuka2012magnetic] and produce a periodic potential for the spin degrees of freedom similar to the one induced by the periodic array of nonmagnetic impurities. Coulomb interactions can enhance this tendency producing an even stronger CDW ordering. The third proposal is based on selective Kondo screening: some heavy fermion compounds are known to exhibit partially ordered magnetic states. For instance, the partially-disordered compound, UNi$_4$B, exhibits a magnetic vortex structure [@Mentink1994; @Oyamada2007; @Hayami_1742-6596-592-1-012101]. A possible scenario is that the partial disorder is produced by a site-selective formation of Kondo singlet states. [@Lacroix; @Motome2010; @Hayami2011] The site-selective Kondo screening is then an alternative mechanism for producing a nonmagnetic superlattice. To summarize, by taking the classical $J_1$-$J_3$ Heisenberg model on the triangular lattice as an example, we have shown that exotic multiple-$\mathbf{Q}$ states can be induced by periodically distributed nonmagnetic impurities. The interplay between the spin configuration and the underlying impurity superlattice renders most of the states chiral (i.e., with net scalar chirality). We have also shown that weak randomness in the impurity positions, relative to the periodic array, induces a skyrmion crystal phase for intermediate magnetic field values. Our results suggest that a variety of magnetically ordered states with nonzero net scalar chirality can be realized by changing the concentration of nonmagnetic impurities, magnetic field, and temperature. Computer resources for numerical calculations were supported by the Institutional Computing Program at LANL. This work was carried out under the auspices of the U.S. DOE contract No. DE-AC52-06NA25396 through the LDRD program. Y.K. acknowledges the financial supports from the RIKEN iTHES project. Characterization of each phase {#sec:Finite-size scaling of each phase} ============================== Figures \[Fig:sizedep\_opt1-4\], \[Fig:sizedep\_opt5-8\], and \[Fig:sizedep\_opt\_2times\] show the $1/L$ dependence of the $xy$ and $z$ components of the spin structure factor and the chirality structure factor normalized by the system size $N$. They should scale $S(\mathbf{q}) \sim {\mathcal{O}\bigl({1}\bigr)}$, $L^{2-\eta}$, and $N$, respectively, when the corresponding mode is disordered, critical, and long-range ordered. Note that the Mermin-Wagner theorem precludes the long-range order in the $xy$ component at finite $T$. [^1]: In our convention, we count the number of pairs $\pm Q^M$ and $\pm Q^\chi$ corresponding to the (quasi-)Bragg peaks in the structure factor.

--- abstract: | \[Section:Abstract\] The spectra of repeating fast radio bursts (FRBs) are complex and time-variable, sometimes peaking within the observing band and showing a fractional emission bandwidth of about 10–30%. These spectral features may provide insight into the emission mechanism of repeating fast radio bursts, or they could possibly be explained by extrinsic propagation effects in the local environment. Broadband observations can better quantify this behavior and help to distinguish between intrinsic and extrinsic effects. We present results from a simultaneous 2.25 and 8.36GHz observation of the repeating FRB 121102 using the 70m Deep Space Network (DSN) radio telescope, DSS-43. During the 5.7hr continuous observing session, we detected 6 bursts from FRB 121102, which were visible in the 2.25GHz frequency band. However, none of these bursts were detected in the 8.36GHz band, despite the larger bandwidth and greater sensitivity in the higher-frequency band. This effect is not explainable by Galactic scintillation and, along with previous multi-band experiments, clearly demonstrates that apparent burst activity depends strongly on the radio frequency band that is being observed. address: | $^{\text{1}}$ [Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA; ]{}\ $^{\text{2}}$ [Division of Physics, Mathematics, and Astronomy, California Institute of Technology, Pasadena, CA 91125, USA]{}\ $^{\text{3}}$ [Anton Pannekoek Institute for Astronomy, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands]{}\ $^{\text{4}}$ [ASTRON, Netherlands Institute for Radio Astronomy, Oude Hoogeveensedijk 4, 7991 PD Dwingeloo, The Netherlands]{}\ $^{\text{5}}$ [CSIRO Astronomy and Space Science, Canberra Deep Space Communications Complex, P.O. Box 1035, Tuggeranong, ACT 2901, Australia]{} author: - '[[](https:/orcid.org/0000-0002-4694-4221)]{}, [[](https:/orcid.org/0000-0002-8912-0732)]{}, [[](https://orcid.org/0000-0003-0510-0740)]{}, [[](https://orcid.org/0000-0003-2317-1446)]{}, [[](https:/orcid.org/0000-0002-8850-3627)]{}, [[](https://orcid.org/0000-0001-6898-0533)]{}, [[](https:/orcid.org/0000-0003-0249-7586)]{}, and [Shinji Horiuchi]{}' bibliography: - 'references.bib' title: | A Dual-band Radio Observation of FRB 121102 with the Deep Space Network\ and the Detection of Multiple Bursts --- [^1] [^2] Introduction {#Section:Introduction} ============ Fast radio bursts (FRBs) are bright (fluence$\sim$Jyms), short duration ($\sim$$\mu$s–ms) radio pulses with dispersion measures (DMs) that are well in excess of the expected Galactic contribution along their line of sights (see, e.g. @Petroff+2019 [@Cordes+2019] for recent reviews). The DMs, which are derived from frequency-dependent delays in the arrival times of the bursts due to the passage of the radio waves through the cold plasma between the source and the observer, are used as a proxy for the distances of these bursts. The high DM values have long suggested that the sources of FRBs are located at extragalactic distances. The localization of a subset of FRBs to host galaxies at redshifts of 0.034–0.66 has confirmed the extragalactic nature of FRBs [@Chatterjee+2017; @Bannister+2019; @Ravi+2019; @Prochaska+2019; @Marcote+2020]. FRBs have peak flux densities that are similar to those of radio pulsars, and their extragalactic distances imply total burst energies that are $\sim$10$^{\text{10}}$–10$^{\text{14}}$ times those of pulsars, if similar beaming fractions are assumed. There is currently no well-established progenitor theory that can explain this phenomenon, though dozens of hypotheses have been proposed (e.g., see @Platts+2019[^3] for a catalog of theories). Since the initial FRB discovery by [@Lorimer+2007], over a hundred distinct sources have been reported (e.g., see @Petroff+2016[^4] for a catalog). Interestingly, a subset of these sources have shown repeat bursts, which has provided an opportunity to study this enigmatic phenomenon in more detail through post-facto localization of the sources to a host galaxy (e.g., @Marcote+2020), studies of burst properties (e.g., @Gourdji+2019), and multi-wavelength searches for potential counterparts (e.g., @Scholz+2017 [@Scholz+2020]). Whether or not all FRBs are capable of repeating remains an active debate, though it has been argued that the high overall event rate requires that a large fraction of the population are repeaters [@Ravi2019]. FRBs are now also being localized precisely using the initial burst discovery data [@Bannister+2019]. This will help greatly in determining whether FRBs that have only been detected once come from a physically distinct progenitor type. FRB 121102 is the first known repeating FRB [@Spitler+2014; @Spitler+2016] and has been localized to a faint dwarf galaxy at a redshift of $z$$=$0.19 [@Chatterjee+2017; @Marcote+2017; @Tendulkar+2017]. Since the discovery of FRB 121102, hundreds of bursts have been detected by the Arecibo telescope and other instruments (e.g., @Gourdji+2019). Many of these detections were made at  (GHz), but FRB 121102 has also been detected at a wide range of radio frequencies using various radio telescopes (e.g., with CHIME/FRB at GHz, @Josephy+2019; the Green Bank Telescope (GBT) at GHz, @Scholz+2016 [@Scholz+2017]; the NASA Deep Space Network (DSN) 70m radio telescope, DSS-43, at 2.25GHz, @Pearlman+2019a; the Very Large Array (VLA) at GHz, @Chatterjee+2017 [@Law+2017]; the Arecibo telescope at GHz, @Michilli+2018a; the Effelsberg 100m telescope at GHz, @Spitler+2018; and the GBT at GHz, @Gajjar+2018 [@Zhang+2018]). Since the progenitor population of FRBs is still unknown, broadband and high-frequency radio observations of FRBs are important for understanding the underlying emission mechanism(s). In particular, simultaneous measurements across wide bandwidths are more robust against temporal evolution of scintillation and scattering as the interference patterns change over time because of the relative motion between the source, the scattering screen, and the observer. In this Letter, we present results from a simultaneous observation of FRB 121102 at 2.25 and 8.36GHz with the NASA DSN 70m telescope, DSS-43, and expand upon the initial results reported in @Pearlman+2019a. The observation and data analysis procedures are described in Section \[Section:Observation\]. In Section \[Section:Results\], we provide measurements of the detected bursts, including the DM, width, flux density, and fluence of each burst. In Section \[Section:Discussion\], we discuss our measurements of the burst spectra, previous multi-frequency measurements of FRB 121102, the impact of intrinsic and extrinsic effects on the burst properties, and the morphologies of the brightest bursts detected during this observation. Observation and Data Analysis {#Section:Observation} ============================= We observed FRB 121102 continuously for 5.7hr on 2019 Sep. 06, 17:27:54 UTC (MJD 58732.727708) using DSS-43, the NASA DSN 70m radio telescope located at the Canberra Deep Space Communication Complex (CDSCC) in Tidbinbilla, Australia. This observation was carried out as part of a recently initiated monitoring program of repeating FRBs at high frequencies with the DSN’s large 70m radio telescopes. DSS-43 is equipped with cryogenically-cooled, dual circular polarization receivers, which are capable of recording data simultaneously at $S$-band and $X$-band. The center frequencies of the recorded $S$-band and $X$-band data were 2.25 and 8.36GHz, respectively. The $S$-band system has a bandwidth of 115MHz, with an effective bandwidth of $\sim$100MHz after masking bad channels contaminated by radio frequency interference (RFI). The $X$-band receivers provide 450MHz of bandwidth, with $\sim$430MHz of usable bandwidth. Data from both polarization channels were simultaneously received and recorded at each frequency band with two different recorders at the site’s Signal Processing Center. The primary recorder is the pulsar machine, described previously in @Majid+2017, which provides channelized power spectral densities in filterbank format with a frequency resolution of 0.98MHz and a time resolution of 64.5$\mu$s. Data in both circular polarizations were also recorded at $S$-band using the stations’s very-long-baseline interferometry (VLBI) baseband recorder in six non-contiguous sub-bands. Each sub-band spanned 8MHz in bandwidth and provided a total bandwidth of 48MHz. The center frequency of the data was 2.24GHz. A detailed analysis of the baseband data will be presented in an upcoming publication. Most of the results in this Letter are derived from data obtained using the ultra-wideband pulsar machine, with the exception of the autocorrelation analysis (see Section \[Section:Results\]). The data were flux calibrated by measuring the system temperature, $T_{\text{sys}}$ at both frequency bands using a noise diode modulation scheme at the start of the observation, while the antenna was pointed at zenith. The $T_{\text{sys}}$ values were corrected for elevation effects, which are minimal for elevations greater than 20 degrees. The data processing procedures were similar to those described in previous single pulse studies of pulsars and magnetars with the DSN (e.g., @Majid+2017 [@Pearlman+2018; @Pearlman+2019b]). In each data set, we corrected for the bandpass slope across the frequency band and masked bad channels corrupted by RFI, which were identified using the PSRCHIVE software package [@Hotan+2004]. We also subtracted the moving average from each data value using 0.5s around each time sample in order to remove low frequency temporal variability. Next, the cleaned data were dedispersed with trial DMs between 500 and 700pccm$^{\text{--3}}$. A list of FRB candidates with detection signal-to-noise (S/N) ratios above 6.0 were generated using a matched filtering algorithm, where each dedispersed time-series was convolved with boxcar functions with logarithmically spaced widths between $\sim$64.5$\mu$s and $\sim$19.4ms. We used a GPU-accelerated machine learning pipeline based on the `FETCH`[^5] (Fast Extragalactic Transient Candidate Hunter) software package to determine whether or not each of these FRB candidates were astrophysical [@Agarwal+2019]. The same FRB candidates were also searched for astrophysical bursts using an automated classifier [@Michilli+2018c; @Michilli+2018d], after independently filtering each candidate for RFI. Both of these classification pipelines identified the bursts presented in Section \[Section:Results\] as genuine FRBs. In addition, we extracted raw voltages using 4.0s of data centered on the arrival times of each of the two brightest bursts for the autocorrelation analysis presented in Section \[Section:Results\]. The data were coherently dedispersed using a DM value of 563.6pccm$^{\text{--3}}$, the structure-optimized DM associated with the brightest burst. We then used the coherently dedispersed baseband data to form filterbanks comprised of channelized power spectral densities with temporal and spectral resolutions of 32$\mu$s and 31.25kHz, respectively. The resulting burst spectra were used to calculate autocorrelation functions (ACFs) for each burst. Results {#Section:Results} ======= Six bursts were detected at $S$-band with a DM value near the nominal DM of FRB 121102. In Table \[Table:Table1\], we list the peak time, peak S/N, DM value that maximized the peak S/N, burst width, peak flux density, spectral energy density, and fluence for each burst. We show the flux-calibrated, frequency-averaged burst profiles, dynamic spectra, and flux-calibrated, time-averaged spectra for all of these bursts in Figure \[Figure:Figure1\], after dedispersing each burst with a DM value of 563.6pccm$^{\text{--3}}$. For the brightest bursts (B1 and B6), the structure-optimized DM value was consistent with the DM value that maximized the peak S/N. However, the algorithm[^6] [@Seymour+2019] used to determined the structure-optimized DM performs poorly on low S/N bursts. Therefore, we have chosen to dedisperse all of the burst spectra shown in Figure \[Figure:Figure1\] using the structure-optimized DM associated with the brightest burst, B6. In the left diagram in Figure \[Figure:Figure2\], we show the dedispersed $S$-band and $X$-band dynamic spectra of the brightest burst, B6, after correcting for the dispersive delay between the two frequency bands. The frequency-averaged burst profiles are shown in the upper panel. Although the burst was detected with high S/N at $S$-band, there was no detectable signal during the same time at $X$-band. We also show the peak flux densities of the six detected $S$-band bursts as a function of time during our observation in the right diagram in Figure \[Figure:Figure2\]. The $X$-band and $S$-band 7-$\sigma$ detection thresholds are indicated with cyan and orange lines, respectively. Since no bursts were detected at $X$-band, we place a 7-$\sigma$ upper limit of 0.20Jy on the flux density of the emission at 8.36GHz during this observation, assuming a nominal pulse width of 1ms. If we further assume that the flux density scales as a power-law (i.e., $S(\nu)$$\propto$$\nu^{\alpha}$, where $S(\nu)$ denotes the flux density at an observing frequency $\nu$ and $\alpha$ is the spectral index), which is typical of most pulsar radio spectra, then we can place an upper limit of $\alpha$$<$–2.6 on the spectral index of the emission process using burst B6. However, we note that previous observations of bursts from FRB 121102 show that they may not be well-modeled by a power-law (e.g., @Spitler+2016 [@Scholz+2016; @Law+2017]). The two brightest bursts, B1 and B6, show remarkably similar temporal profiles, with two prominent central components and a precursor component. In addition, B1 shows evidence of an additional component towards the tail of the main burst envelope. Both bursts also show spectral-temporal features that are reminiscent of other FRBs (e.g., FRB 170827; @Farah+2018) and other bursts from FRB 121102 (e.g., @Hessels+2019). The diffractive interstellar scintillation (DISS) bandwidth roughly scales with frequency as: $$\Delta \nu_{\text{DISS}} \propto \nu^{4}. \label{Equation:DISS}$$ The scintillation bandwidth of FRB 121102 was previously measured to be 58.1$\pm$2.3kHz at 1.65GHz [@Hessels+2019]. The burst dynamic spectra in Figure \[Figure:Figure1\] show narrowband frequency structure. If we attribute the frequency structure in the burst spectra to DISS, then based on the scintillation bandwidth measured at 1.65GHz, Equation \[Equation:DISS\] would predict a scintillation bandwidth of $\Delta\nu_{\text{DISS}}$$\approx$200kHz at $\nu$$=$2.24GHz. We note that the data recorded from the pulsar machine is insufficient to resolve the predicted scintillation bandwidth due to its 1MHz spectral resolution. Therefore, to study the frequency-dependent brightness variations that arise due to scintillation, we used the baseband data to perform an ACF analysis on the burst spectra from B1 and B6. The procedure used to carry out the ACF analysis is described in detail in @Marcote+2020. We measure the scintillation bandwidth of B1 to be 177$\pm$17kHz and that of B6 to be 280$\pm$13kHz, both at a center frequency of 2.24GHz. We were unable to compute ACFs for the other four bursts because they did not have sufficient S/N. The ACFs of both B1 and B6 are shown in Figure \[Figure:Figure3\] up to frequency lags of 8MHz. We also show Lorentzian fits to the central bump in the ACFs, which corresponds to frequency lags up to 0.84MHz, after removing the zero lag noise spike. The scintillation bandwidth, defined as the half-width at half-maximum (HWHM) of the Lorentzian fit [@Cordes+1985], is labeled in the figure for each burst. The baseband data used in this analysis contained 8MHz frequency gaps between each of the six 8MHz-wide frequency sub-bands. This introduced noise spikes into the ACF at frequency lags that were close to integer multiples of the sub-band width. This is apparent in the ACFs shown in Figure \[Figure:Figure3\] toward frequency lags of 8MHz. We also include ACFs of the off-burst data in Figure \[Figure:Figure3\] to emphasize that the frequency structure is produced by scintillation, rather than instrumental effects. B6 shows a feature in the ACF at a frequency lag of approximately 1.7MHz, which we highlight using a black arrow in Figure \[Figure:Figure3\]. Similar behavior is not observed at the same frequency lag in the ACF of B1. We discuss this further in Section \[Section:Discussion\]. [cccccccc]{} B1 & 58732.82134550821 & 13.31 & 564.1$\pm$0.1 & 2.94$\pm$0.06 & 2.6$\pm$0.5 & 7.5$\pm$1.5 & 6.7$\pm$1.3\ B2 & 58732.85229671322 & 4.02 & 564.2$\pm$0.1 & 1.05$\pm$0.18 & 0.8$\pm$0.2 & 0.5$\pm$0.1 & 0.4$\pm$0.1\ B3 & 58732.86396118454 & 7.26 & 565.0$\pm$0.1 & 1.65$\pm$0.09 & 1.4$\pm$0.3 & 2.1$\pm$0.4 & 1.8$\pm$0.4\ B4 & 58732.86552034988 & 4.98 & 564.2$\pm$0.1 & 0.63$\pm$0.07 & 1.0$\pm$0.2 & 0.6$\pm$0.1 & 0.5$\pm$0.1\ B5 & 58732.86815249499 & 3.64 & 564.2$\pm$0.1 & 1.23$\pm$0.19 & 0.7$\pm$0.1 & 0.6$\pm$0.1 & 0.5$\pm$0.1\ B6 & 58732.93175395272 & 27.58 & 563.6$\pm$0.1 & 2.10$\pm$0.03 & 5.9$\pm$1.2 & 10$\pm$2.0 & 8.8$\pm$1.8 \[Table:Table1\] Discussion and Conclusions {#Section:Discussion} ========================== To date, FRB 121102 has been detected at radio frequencies from 600MHz [@Josephy+2019] up to 8GHz [@Gajjar+2018]. Early observations of FRB 121102 by @Spitler+2016 and @Scholz+2016 demonstrated that the bursts have variable spectra that sometimes peak within the observing band and are often not well-modeled by a power-law. This also clarifies the strange inverted spectrum of the discovery detection of FRB 121102 [@Spitler+2014], though the detection of that burst in the coma lobe of the receiver likely also affected the apparent spectrum. Broader-band observations (1.15–1.73GHz) by @Hessels+2019 demonstrated that the characteristic bandwidth of emission is roughly 250MHz at 1.4GHz and the bursts are sometimes composed of sub-bursts with characteristic peak emission frequencies that decrease during the burst envelope at a rate of $\sim$200MHzms$^{\text{--1}}$ in this frequency band. This “sad trombone” effect appears to be a characteristic feature of repeating FRBs [@CHIME+2019], and may be an important clue as to their emission mechanism. The available bandwidth used to detect the 2.25GHz bursts presented here is insufficient to resolve sub-burst drifts of this type. Similar narrowband, 100–200MHz brightness envelopes were also found by @Gourdji+2019 in a sample of 41 bursts detected using the Arecibo telescope during two $\sim$2hr observing sessions conducted on consecutive days. They also found tentative evidence for preferred frequencies of emission during those epochs, suggesting that FRB 121102’s detectability depends strongly on the radio frequency that is being utilized. In addition, recent simultaneous, multi-frequency observations of another repeating FRB, FRB 180916.J0158+65, demonstrated that its apparent activity may also be related to the observing frequency [@CHIME+2020]. CHIME/FRB detected two bursts from this source (with fluences of $\sim$2Jyms) in the 400–800MHz band within a 12min transit. However, no bursts (above a fluence threshold of 0.17Jyms) were detected from FRB 180916.J0158+65 with the Effelsberg telescope at $\sim$1.4GHz during 17.6hr of observations on the same day, which overlapped the times of the two CHIME/FRB detections. Clearly, the radio emission from repeating FRBs is not instantaneously broadband, which we further demonstrate with our simultaneous 2.25 and 8.36GHz observations of FRB 121102. Our results show that there was a period of burst activity from FRB 121102, lasting at least 2.6hr, where radio emission was detected at 2.25GHz but not at 8.36GHz. There are only a few multi-band radio observations of FRB 121102 in the literature. [@Law+2017] present results from a multi-telescope campaign of FRB 121102 using the VLA at 3GHz and 6GHz, the Arecibo telescope at 1.4GHz, the Effelsberg telescope at 4.85GHz, the first station of the Long Wavelength Array (LWA1) at 70MHz, and the Arcminute Microkelvin Imager Large Array (AMI-LA) at 15.5GHz. Nine bursts were detected with the VLA, and four of these bursts had simultaneous observing coverage at different frequencies. Only one of these bursts was detected simultaneously at two different observing frequencies with Arecibo (1.15–1.73GHz) and the VLA (2.5–3.5GHz). The remaining three bursts were detected solely with the VLA, despite the instantaneous sensitivity of Arecibo being $\sim$5 times better than the VLA. None of the four bursts were detected during simultaneous LWA1, Effelsberg, or AMI-LA observations, though we note that only Effelsberg’s sensitivity is comparable to the VLA’s. [@Gourdji+2019] describe 41 bursts detected with Arecibo at 1.4GHz, and no bursts were seen with the VLA during their simultaneous observations. They also report one VLA-detected burst that was not seen in their contemporaneous Arecibo data. [@Houben+2019] performed a search for bursts from FRB 121102 using both Effelsberg (1.4GHz) and the Low Frequency Array (LOFAR; 150MHz). In this search, they discovered nine bursts with Effelsberg, but there were no simultaneous detections with LOFAR. @Gajjar+2018 reported the detection of 21 bursts above 5.2GHz during a 6hr observation with the GBT. It is notable that all of these bursts were detected within a short 1hr time interval. The peak flux densities of these bursts ranged between $\sim$50 and $\sim$700mJy. These bursts also showed both large-scale ($\sim$1GHz-wide) and fine-scale frequency structures, none of which spanned the entire 4.5–8.0GHz frequency band. Assuming a flat spectral index, there are six bursts in @Gajjar+2018 with peak flux densities that are above our $X$-band sensitivity limit. Thus, similarly bright bursts would have been detected during our $X$-band observations, if they were present. Galactic scintillation cannot explain the observed narrowband spectrum of bursts from FRB 121102, which has a low Galactic latitude ($b$$=$–0.2$^{\circ}$). The bursts show evidence of narrowband scintillation that is consistent with the contribution expected from the Milky Way foreground ($\Delta\nu_{\text{DISS}}$$=$58.1$\pm$2.3kHz at 1.65GHz; @Hessels+2019). In this paper, we have measured $\Delta\nu_{\text{DISS,\,B1}}$$=$177$\pm$17kHz and $\Delta\nu_{\text{DISS,\,B6}}$$=$280$\pm$13kHz at 2.24GHz for the two brightest bursts, B1 and B6. Given the expected Galactic scintillation timescale of $\sim$4min at 2.24GHz [@Cordes+2002], it is not surprising that the measured scintillation bandwidths are different compared to their formal uncertainties. We are sampling a limited number of scintles in each case, and burst self-noise may also contribute to the difference. This likely also explains the other features in the on-burst ACFs, e.g. the 1.7MHz bump in B6. Previously, @Gajjar+2018 reported scintillation bandwidths of $\Delta\nu_{\text{DISS}}$$\sim$10–100MHz for bursts detected between 4.5–8.0GHz. Combining all available measurements, we estimate that the scintillation bandwidth is $\Delta\nu_{\text{DISS}}$$\approx$MHz at 2.25GHz and $\Delta\nu_{\text{DISS}}$$\approx$MHz at 8.36GHz, where the ranges correspond to assumed scalings of $\Delta\nu_{\text{DISS}}$$\propto$$\nu^{4}$ and $\Delta\nu_{\text{DISS}}$$\propto$$\nu^{4.4}$, respectively. Galactic scintillation therefore cannot explain the clear detections of the 2.25GHz bursts shown in Figure \[Figure:Figure1\] and the lack thereof in our simultaneous 8.36GHz data, where the bandwidth (430MHz) at $X$-band is many times larger than the scintillation bandwidth. [@Cordes+2017] discuss the possible role of plasma lensing on the burst spectra and apparent brightness of FRBs. They argue that FRBs may be boosted in brightness on short timescales through caustics, which can produce strong magnifications ($\lesssim$10$^{\text{2}}$). However, we note that larger spectral gains are possible since this depends strongly on various parameters, such as the geometry of the lens, the lens’ dispersion measure depth, and the scale size [@Cordes+2017; @Pearlman+2018], which are currently poorly constrained. It is therefore possible that the 2.25GHz detections shown in Figure \[Figure:Figure1\] may coincide with a caustic peak. The two brightest bursts (B1 and B6) in Figure \[Figure:Figure1\] display remarkably similar morphology: a weak precursor sub-burst, followed by a sharp rise and bright sub-burst (lasting for $\sim$0.5ms), and thereafter a broader component (lasting for a few milliseconds) perhaps composed of multiple unresolved sub-bursts, followed by a slow decay. The decaying tails of these bursts are far too long to be due to multipath propagation through the Galactic interstellar medium (ISM), which is expected to produce a scattering time of $\tau_{d}$$=$1.16/2$\pi\Delta\nu_{\text{DISS}}$$\approx$0.7$\mu$s at 2.25GHz [@Cordes+1998]. Rather, it appears that this structure may either be intrinsic to the burst emission mechanism or originate in FRB 121102’s host galaxy and/or local environment. Furthermore, many other bursts from FRB 121102 also show asymmetric burst morphologies, which cannot be explained by scattering (e.g., see @Hessels+2019). The burst tails observed in B1 and B6 may be caused by the same mechanism responsible for the sub-burst drift rate and the apparent “sad trombone” behavior [@Hessels+2019; @Josephy+2019]. Multi-frequency observations that densely cover the $\sim$GHz range can better clarify how the burst activity of FRB 121102 depends on the radio observing frequency. It is currently unclear whether there is an optimal frequency range for observing this source. While many bursts from FRB 121102 appear to span only a few hundred MHz of bandwidth, some appear to span at least $\sim$2GHz [@Law+2017]. Multi-frequency, broadband measurements can also better quantify the typical emission bandwidth and determine whether or not bursts show multiple brightness peaks at widely separate frequencies, both of which are important for disentangling propagation effects and studying the mechanism(s) responsible for the emission. Acknowledgments {#acknowledgments .unnumbered} =============== We thank Jim Cordes for useful discussions. A.B.P. acknowledges support by the Department of Defense (DoD) through the National Defense Science and Engineering Graduate (NDSEG) Fellowship Program and by the National Science Foundation (NSF) Graduate Research Fellowship under Grant No. . J.W.T.H. acknowledges funding from an NWO Vici fellowship. We thank the Jet Propulsion Laboratory’s Spontaneous Concept Research and Technology Development program for supporting this work. We also thank Charles Lawrence and Stephen Lichten for providing programmatic support. In addition, we are grateful to the DSN scheduling team (Hernan Diaz, George Martinez, Carleen Ward) and the Canberra Deep Space Communication Complex (CDSCC) staff for scheduling and carrying out these observations. A portion of this research was performed at the Jet Propulsion Laboratory, California Institute of Technology and the Caltech campus, under a Research and Technology Development Grant through a contract with the National Aeronautics and Space Administration. U.S. government sponsorship is acknowledged. ORCID iDs {#Section:OrcidIDs .unnumbered} ========= Walid A. Majid <https:/orcid.org/0000-0002-4694-4221>\ Aaron B. Pearlman <https:/orcid.org/0000-0002-8912-0732>\ Kenzie Nimmo <https://orcid.org/0000-0003-0510-0740>\ Jason W. T. Hessels <https://orcid.org/0000-0003-2317-1446>\ Thomas A. Prince <https:/orcid.org/0000-0002-8850-3627>\ Jonathon Kocz <https:/orcid.org/0000-0003-0249-7586>\ Charles J. Naudet <https://orcid.org/0000-0001-6898-0533>    [^1]: [$^{\text{6}}$ NDSEG Research Fellow.]{} [^2]: [$^{\text{7}}$ NSF Graduate Research Fellow.]{} [^3]: See [http://frbtheorycat.org.](http://frbtheorycat.org) [^4]: See [http://frbcat.org.](http://frbcat.org) [^5]: See [https://github.com/devanshkv/fetch.](https://github.com/devanshkv/fetch) [^6]: See [https://github.com/danielemichilli/DM\_phase.](https://github.com/danielemichilli/DM_phase)

--- abstract: 'We employ the density matrix renormalization group to construct the exact time-dependent exchange correlation potential for an impurity model with an applied transport voltage. Even for short-ranged interaction we find an infinitely long-ranged exchange correlation potential which is built up [instantly]{} after switching on the voltage. Our result demonstrates the fundamental difficulties of transport calculations based on time-dependent density functional theory. While formally the approach works, important information can be missing in the ground-state functionals and may be hidden in the usually unknown non-equilibrium functionals.' author: - 'P. Schmitteckert$^{1,2}$, M. Dzierzawa$^3$ and P. Schwab$^3$' title: 'Exact time-dependent density functional theory for impurity models ' --- Introduction ============ In recent years a combination of Kohn-Sham density functional theory (DFT) and the Landauer approach to transport has been developed that allows an ab-initio calculation of the current-voltage characteristics of molecules that are attached to reservoirs. Early comparisons between calculated and experimental conductances yielded discrepancies of several orders of magnitude, while in more recent comparisons a typical discrepancy of one order of magnitude was reported.[@lindsay2007] [Concerning experiments, the reproducibility of $I$-$V$ characteristics in molecular electronics poses a problem, as the contact configuration may change from sample to sample, which makes the comparison between theory and experiment difficult.]{} First DFT studies of lattice models like the Hubbard model or spinless fermions date already back to the late 1980s.[@gunnarsson1986; @schonhammer1987; @schonhammer1995] At that time ground state DFT was the main focus of interest. In the 2000s a lattice version of the local density approximation was constructed using the Bethe ansatz solution of the Hubbard model, and the accuracy of the method was tested in some detail.[@lima2003; @schenk2008; @dzierzawa2009] Due to the mentioned discrepancies between measured and calculated conductances through molecules, recently the focus turned to the DFT description of transport. In the conventional DFT approach to the two-terminal [linear]{} conductance of a molecular system the Kohn-Sham equations are solved in order to obtain the electronic structure, and the conductance of the Kohn-Sham system, $G_{\rm KS}$, is [used as a]{} theoretical estimate of the true conductance, $G$. There are thus two possible sources for errors, (i) the electronic structure calculation and (ii) the identification of $G$ with $G_{\rm KS}$. While in complex realistic structures it is not possible to quantify (i) and (ii) this is possible for lattice models, as demonstrated by Schmitteckert and Evers.[@schmitteckert2008] They observed that the Kohn-Sham conductance, $G_{\rm KS}$, based on the exact ground state exchange correlation potentials becomes accurate, i.e. $G_{\rm KS}= G$, close to (isolated) conductance resonances. Remarkably this holds even if the spectral properties of the Kohn-Sham system strongly differ from the true ones.[@schmitteckert2008; @troster2011] For example, the zero temperature conductance through a Kondo impurity is captured correctly by $G_{\rm KS}$. The reason for this coincidence is the Friedel sum rule which guarantees that $G$ and $G_{\rm KS}$ have identical functional dependencies on the charge density.[@schmitteckert2008; @mera2010; @mera2011] [However, as soon as the conductance is not given by a local Friedel sum rule there can be orders of magnitude between $G_{\rm KS}$ and the true conductance,[@troster2011; @bergfield2011; @stefanucci2011] and it can even be parametrically wrong.[@PS:DSB2012]]{} [It is important to note that this is not a general failure of DFT; rather the assumption that the conductance of the physical electrons is given by the conductance of the Kohn-Sham particles breaks down.]{} It has been stated several times in the literature that this discrepancy can be corrected by including dynamic contributions to the exchange-correlation potential that are not captured by ground state functionals.[@stefanucci2004; @koentopp2006; @schenk2011] For the linear current $I$ through an impurity this dynamic exchange-correlation potential $V^{\rm xc}$ renormalizes the voltage, such that $$\label{eq1} I= G V = G_{\rm KS}(V + V^{ \rm xc} ).$$ It is not clear whether a similar voltage renormalization is also present in the non-linear response. [The goal of our study is to investigate the nature of the dynamical response within a DFT framework beyond the linear transport regime.]{} To this end we construct the exact [time-dependent]{} exchange-correlation potential for an impurity model with an applied transport voltage. [Specifically we construct the time-dependent Kohn-Sham potentials for a two-terminal transport setup, which by construction yields the correct physical current within the time-dependent DFT description.]{} In this paper we will concentrate on a one-dimensional lattice model, where we can directly compare [time-dependent]{} DFT with accurate numerical and analytical results obtained by many-body techniques. [[@boulat2008]]{} In order to obtain meaningful results we have to solve much larger systems (here: $240$ lattice sites) than for instance in Ref. ($6$ to $12$ lattice sites). In addition the densities have to be calculated with an accuracy better than $ \sim 10^{-6}$; otherwise the reverse engineering of the time-dependent exchange-correlation potentials fails. We consider the interacting resonant level model, i.e. a one-dimensional model of spinless fermions, where a single interacting level is coupled to a left and a right lead, $$\begin{aligned} H &=& H_\mathrm{L} + H_\mathrm{LR}+ H_\mathrm{R} \label{eq2} \\ H_\mathrm{L}&=& - t \sum_{i=1}^{N_\mathrm{L}-1} c_{\mathrm{L} i}^+ c^{}_{\mathrm{L} i+1} + {\text{h.c}.} \\ H_\mathrm{R}&=& - t \sum_{i=1}^{N_\mathrm{R}-1} c_{\mathrm{R} i}^+ c^{}_{\mathrm{R} i+1} + {\text{h.c}.} \\ H_\mathrm{LR}&=& -t'(c^+ c_{\mathrm{ L},1} + c^+ c^{}_{\mathrm{R},1} + {\text{h.c.}} ) \nonumber \\ && + U\left(n-1/2\right)\left( n_{\mathrm{ L},1} + n_{\mathrm{R},1} - 1 \right) .\end{aligned}$$ Here $t=1$ is the hopping amplitude, $N_\mathrm{L}$ and $N_\mathrm{R}$ are the numbers of sites in the left and right lead, and $U$ is the interaction on the contact link. All the data presented in this article have been obtained for the half-filled lattice model. We assume that at time $T<0$ the system is in the ground state. At $T=0$ we include a voltage drop by applying a potential $eV/2$ ($-eV/2$) on the left (right) lead with a linear crossover on a scale of several sites left and right of the impurity. Then we follow the time evolution of the system. On time scales that are shorter than the transit time $T_t = L_{\rm lead}/v_F $ ($v_F = 2t$ is the Fermi velocity) the finite leads act as reservoirs, such that a time-dependent simulation allows to extract the current-voltage relation corresponding to infinite leads, see Refs.  and for details. [Note that in our model the linear conductance is given by a Friedel sum rule.[@Boulat_Saleur:PRB2008; @schmitteckert2008] Therefore, $V^{ \rm xc}$ vanishes in the linear regime.]{} We consider a Hamiltonian $H$ with parameters $U$ and $t'$ and a Kohn-Sham Hamiltonian $H_{\rm KS}$ with the same structure as Eq. (\[eq2\]) using different parameters $U_{\rm KS}=0$ and $t'_{\rm KS}$. The essence of DFT is a one-to-one correspondence between local densities and potentials: Starting with an uniquely defined initial state there exists an – up to a gauge transformation – unique set of potentials $ \{ v_{\rm KS}(T) \}$ such that the time-dependent densities $\{ n(T) \}$ are identical for $H$ and $H_{\rm KS}$. The numerical task is to calculate first the time-dependent local density for $H$ and in the second step the set of potentials $\{ v_{\rm KS} \}$ for $H_{\rm KS}$. We fix the gauge by imposing that the sum of the potentials is zero. To calculate the particle density we follow two strategies: (a) The particle density in an interacting system is obtained through time-dependent density matrix renormalization group (td-DMRG) as described in Refs. and . Since these calculations are extremely time-consuming we also (b) study a toy model of non-interacting fermions, where the time-dependent density can be obtained straightforwardly by exact diagonalization. As mentioned before, when calculating $\{ v_{\rm KS} \}$ from the densities a high accuracy is necessary. We use an iterative procedure that stops when the densities in $H$ and in $H_{\rm KS}$ agree within an error of $10^{-10}$. Noninteracting toy model ======================== We start by considering the non-interacting version of the Hamiltonian (\[eq2\]), i.e. we set the interaction $U$ as well as $U_{\rm KS}$ to zero. There remains only one free parameter, the hopping amplitude $t'$ between the impurity site and the leads. In our toy model we choose $t' = 0.5t$ and $t'_{\rm KS}=0.3t$. Typical results for the time-dependent Kohn-Sham potentials are shown in Fig. \[fig1\]. ![ \[fig1\] The Kohn-Sham potential $v_{\rm KS}$ as a function of time for the first ten lattice sites from the left edge of the chain. The two data sets are for a total chain length $N=200$ and $N=400$. For comparison the potential $v=0.2t$ in the toy model (corresponding to a voltage $eV=0.4t$) is also shown as dotted line. ](toy_V04_t03_dt02.pdf){width="40.00000%"} In the toy model a potential $eV=0.4t$ corresponding to a potential $v=\pm 0.2t$ in the left (right) lead is switched on at $T=0$. As a response a current flows through the resonant level that becomes stationary after $T \approx 30$ (the time is in units $\hbar/t$). By definition, the same current has to flow also in the Kohn-Sham system; here we find initially a time-dependent voltage which becomes stationary [after]{} $T \approx 30$. Remarkably the exchange-correlation potential (the difference between $v_{\rm KS}$ and $v$) is nearly position independent in the leads, and appears immediately after switching on the voltage. In the figure, data for a chain of 200 lattice sites and a chain of 400 lattice sites are presented. In both cases we plot the potentials for the first ten sites at the left edge of the chain. For the longer chain $v_{\rm KS}$ remains stationary until the end of the simulation, and only a single line is seen meaning that the potential is homogeneous in space. In the shorter chain the potential becomes position and time-dependent after $T \approx 50$. This happens since the simulation time is longer than the transit time $T_t = 100/2 =50$. The Kohn-Sham potential as a function of position is illustrated in Fig. \[fig2\], demonstrating that a major effect of the exchange-correlation potential is a time-independent renormalization of the [local potentials, which implies an additional voltage as anticipated in Eq. (\[eq1\]).]{} ![\[fig2\] Local potential $v_i$ in the toy model and in the Kohn-Sham system close to the impurity ($i = 0$) at time $T = 12$ (solid curve) and $T=20$ (symbols). ](toy_potentials.pdf){width="40.00000%"} A reasonable estimate for the voltage renormalization is obtained by comparing the $I$-$V$ characteristics for two noninteracting models with $t'=0.3t$ and $t'=0.5t$, see Fig. \[fig3\]. The current $I=0.3883$ (in units of $et/h$) corresponds to $eV=0.4t$ for $t'=0.5t$ and $eV=0.6061t$ for $t'=0.3t$ which is close [to]{} the voltage we found in the Kohn-Sham system, see Fig. \[fig1\]. Notice that the maximum current that can be achieved is larger in the case $t'=0.5t$ than for $t'=0.3t$. For DFT this implies that no stationary Kohn-Sham potential can generate such large currents. Theoretically, time-dependent Kohn-Sham potentials could generate a stationary current, however, the proofs of existence of time-dependent Kohn-Sham potentials [@runge1984; @leeuwen1999; @ruggenthaler2011] do not apply to the present situation, so that possibly the high voltages are not $v$-representable, compare Refs. and . A similar situation will also arise in the next section where we investigate the model with interaction. ![\[fig3\] Current (in units of $e t/h$) as a function of voltage for two non-interacting systems with $t'= 0.3t$ and $t'=0.5t$, respectively. In order to generate the same current in both systems (e.g. $I = 0.4$ as indicated by the dotted line) different potentials, $eV_{0.3}$ and $eV_{0.5}$, have to be applied. The voltage renormalization $V_{0.3}/V_{0.5}$ displayed in the inset gives a reasonable estimate of the voltage renormalization observed in the Kohn-Sham Hamiltonian for the toy model. ](toy_IV.pdf){width="40.00000%"} DMRG results for the interacting model ====================================== We now turn our attention to the interacting case and choose $H$ with $U= 2 t $ and $t' = 0.3t$, whereas for the Kohn-Sham Hamiltonian we set $U_{\rm KS}=0$ and $t'_{\rm KS}=0.3t$. The $I$-$V$ characteristics of this model is known analytically[@boulat2008] from field theoretical methods. For example, the current is given in closed form by[@PS:PRL2011] $$\label{eq6} I(V) = \frac{e^2 V }{2\pi \hbar } \;{}_3F_2 \left[ \left\{ \frac{1}{4},\frac{3}{4}, 1 \right\}, \left\{ \frac{5}{6},\frac{7}{6} \right\}, - \left( \frac{V}{V_{\mathrm c}}\right)^6 \right],$$ where $\;{}_3F_2$ is a hypergeometric function and $e V_{\mathrm c} = r {t'}^{4/3} $ sets the scale separating a charge $2e$ dominated low voltage regime, and a charge $e/2$ dominated high voltage regime.[@PS:PRL2010] The regularization $r\approx 3.2$ was determined from numerical simulation.[@boulat2008; @PS:PRL2010; @PS:PRL2011] The study of the toy model in the previous section shows that in order to obtain meaningful results the simulation time should be of the order $T \approx 30$ or longer, so that the length of each lead should be at least $ L_{\rm lead} \approx 60$ sites. [This means that the excitations originating form the charge quench at the impurity should not reach the boundary of the leads within the simulation time.]{} [In order to obtain the time-dependent densities we apply the full td-DMRG [@PS:PRB2004] as it has already proven to provide accurate results that agree perfectly with the analytic solution.]{} [The quench with a symmetric charge imbalance leads to oscillations with frequency $eV/2$ during the transient time and finite size induced Josephson like oscillations with frequency $eV$ in the steady state regime.[@PS:Ann2010] Therefore, when the effective voltage of the Kohn-Sham system is different from the voltage of the physical one, the time-dependent potentials have to compensate this mismatch. In order to reduce these finite time and finite size effects we ramp up the voltage linearly from $T=0\ldots5$, leading to reduced amplitudes of the transient oscillations.]{} ![\[fig4\] Local potentials on the first ten sites of the left lead as a function of time for an interacting system ($U=2t$, 240 lattice sites). Here we smoothly switched on the external voltage $V = 0.4t$ between $T=0$ and $T=5$. The number of states per block kept in the DMRG calculations varied between 2000 and 4000. ](frame.pdf){width="45.00000%"} Fig. \[fig4\] shows the Kohn-Sham potential in the left lead. As in the non-interacting toy model the main effect is a voltage renormalization. To obtain numerically converged data is a hard task. The figure shows results from three different DMRG runs varying the numerical accuracy. In all three runs the time-dependent particle density is almost identical, variations occur on the scale $10^{-6}$. The Kohn-Sham potentials [turn out to be]{} very sensitive functions of the particle density and show pronounced differences in the three cases. Only for a very large number of states per block in the DMRG calculations (highest accuracy) the final result becomes smooth. ![\[fig5\]Time-dependent Kohn-Sham potentials on the first ten sites of the left lead varying the voltage applied to the system [($t'=0.3t$ and $U=2t$)]{}. While for a small voltage ($eV =0.4t$) we are able to find a set of Kohn-Sham potentials for all times, for larger voltages ($eV = 0.6t$, $eV = 0.8t$) this is only possible for short times. The inset shows the $I$-$V$ characteristics for $U=0$ and $U=2t$. The curve for $U=2t$ is given by Eq. (\[eq6\]) and the symbols are the DMRG results. ](irlm_t03.pdf){width="40.00000%"} In Fig. \[fig5\] time-dependent Kohn-Sham potentials for different voltages are depicted. Whereas for a small voltage we are able to find Kohn-Sham potentials for all times, this is not the case for large voltage where the Kohn-Sham potential diverges at finite $T$. It will be interesting to see in future research, whether the time scale of the singularity has a deeper meaning. Since the $v$-representability is not guaranteed in the lattice model, we believe that Kohn-Sham potentials do not exist for all times in these cases, compare also the discussion relating to Fig. \[fig3\]. [In the inset of Fig. \[fig5\] we show the $I$-$V$ characteristics of the noninteracting ($U=0$) and interacting ($U=2t$) resonant level model. Again, there is a regime, where the current in the interacting case is higher than any current achievable in the noninteracting case. Since the noninteracting case is similar to the Kohn-Sham system, this is a hint that no stationary Kohn-Sham potential might exist in this regime. The cases where we find diverging Kohn-Sham potentials are indeed in the regime where the current for $U=2t$ exceeds the one for $U=0$.]{} Of course, the Kohn-Sham systems at finite voltage have additional potentials with a spatial structure which is not captured by a single number, $V^{\mathrm{xc}}$. It is also important to note, that the Kohn-Sham potentials should not create a current outside the “light cone” $v_F T$. The continuity equation then implies that any oscillations of the potentials outside this regime have to be instantaneous and constant in space including the reservoirs. [While one can gauge the resulting $V^{\mathrm{xc}}$ into a time-dependent phase $\exp( \mathrm{i}V^{\mathrm{xc}}T)$ of a hopping element at the impurity, one loses the property of an at least locally stationary system found in this work.]{} Summary ======= We calculated the exchange-correlation potential for a one-dimensional model system with an applied transport voltage. 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--- abstract: 'Through a sequence of large scale shell model calculations, total Gamow-Teller strengths ($S_+$ and $S_-$) in $^{54}$Fe and $^{56}$Fe are obtained. They reproduce the experimental values once the $\sigma\tau$ operator is quenched by the standard factor of $0.77$. Comparisons are made with recent Shell Model Monte Carlo calculations. Results are shown to depend critically on the interaction. From an analysis of the GT+ and GT$-$ strength functions it is concluded that experimental evidence is consistent with the $3(N-Z)$ sum rule.' address: - '$^{1}$Physique Théorique, Bât 40/1 CRN, IN2P3-CNRS/Université Louis Pasteur BP 28, F-67037 Strasbourg Cedex 2, France' - '$^{2}$Departamento de Física Teórica, Universidad Autónoma de Madrid, E-28049 Madrid, Spain' author: - 'E. Caurier$^{1}$[^1], A. Poves$^{2}$[^2], A. P. Zuker$^{1}$[^3] and G. Martínez-Pinedo$^{2}$[^4]' title: 'Gamow Teller strength in $^{54}$Fe and $^{56}$Fe' --- The charge exchange reactions $(p,n)$ and $(n,p)$ make it possible to observe, in principle, the total Gamow-Teller strength distribution in nuclei. The experimental information is particularly rich in $^{54}$Fe and $^{56}$Fe [@vett; @romq; @rapp; @ander; @elka] and the availability of both GT+ and GT$-$ makes it possible to study in detail the problem of renormalization of $\sigma \tau$ operators. Moreover, these nuclei are of particular astrophysical interest [@astro], and they have been the object of numerous theoretical studies. In this paper we present the results obtained with the largest shell model diagonalizations presently possible. First we concentrate on the study of the total strengths $S_+$ and $S_-$. After a brief review of existing calculations, we estimate the exact values in a full $0\hbar\omega$ space, stressing the need of ensuring the correct monopole behaviour for the interaction. The second part of the paper deals with the GT+ and GT$-$ stength functions. The analysis will confirm that the “standard” quenching factor of 0.77 is associated to suppression of strength in the $0\hbar\omega$ model space, but that little strength is actually “missing” (i.e., unobserved). [**I. Total Strengths $S_+$ and $S_-$**]{} The experimental situation is the following: - $^{54}$Fe$(n,p)^{54}$Mn,\ $S_+$ = 3.1$\pm$0.6 from [@vett], strength below 10 MeV.\ $S_+$ = 3.5$\pm$0.3$\pm$0.4 from [@romq], strength below 9 MeV. - $^{54}$Fe$ (p,n) ^{54}$Co,\ $S_-$ = 7.5$\pm$1.2 from [@vett], strength below 15 MeV.\ $S_-$ = 7.8$\pm$1.9 from [@rapp], strength below 13.5 MeV.\ $S_-$ = 7.5$\pm$0.7 from [@ander], strength below $\approx$ 24 MeV, (but see discussion in section II). - $^{56}$Fe$ (n,p) ^{56}$Mn,\ $S_+$ = 2.3$\pm$0.2$\pm$0.4 from [@romq], strength below 7 MeV.\ $S_+$ = 2.9$\pm$0.3 from [@elka], strength below 8.5 MeV. - $^{56}$Fe$ (p,n) ^{56}$Co,\ $S_-$ = 9.9$\pm$2.4 from [@rapp], strength below 15 MeV. The theoretical approaches include: shell model calculations in the $pf$ shell with different levels of truncation, RPA, quasiparticle RPA and Shell Model Monte Carlo (SMMC) extrapolations. Let us examine the results. ${\bf ^{54}Fe }$. [*Previous shell model calculations*]{}. Throughout the paper $f$ stands for 1$f_{7/2}$ and $r$ for any of the remaining orbits. Truncation level $t$ means that a maximum of $t$ particles are promoted from the $1f_{7/2}$ orbit to the higher ones, i.e., that the calculation includes the following configurations: $f^{n-n_0} r^{n_0}$, $f^{n-n_{0}-1} r^{n_{0}+1}$, $\cdots \cdots$ $f^{n-n_{0}-t} r^{n_{0}+t}$ with $n=A-40$. $n_0$ is different from zero when more than eight neutrons (or protons) are present and at $t=n-n_0$ we have the full space calculation. Given a choice of $t=t_p$ for a parent state having $n_0=n_{0p}$, to ensure respect of the $S_--S_+= 3(N-Z)$ sum rule, the truncation level for daughter states having $n_0=n_{0d}$, must be taken to be . In the simplest case we have $t=0$ i.e., 0p-2h configurations with respect to the $^{56}$Ni closed shell for the $^{54}$Fe ground state, and 1p-3h configurations for the $^{54}$Mn daughters. The result —$S_+$ =10.29 Gamow Teller units— is independent of the interaction. The calculation was extended to $t=1$ by Bloom and Fuller [@bloom], using the interaction of ref. [@bint], obtaining $S_+ =9.12$. A similar calculation by Aufderheide [*et al.*]{} [@auf] yields $S_+ =9.31$ (interaction from [@vanhe; @koops]). Muto [@muto], using the interaction [@muint], made a $t=$2-like calculation that did not respect the $3(N-Z)$ sum rule, but the author estimated the influence of this violation and proposed $S_+$ =7.4. Finally Auerbach [*et al.*]{} [@auer] have made a $t=$2 calculation using the interaction, MSOBEP, fitted in [@brint] (BR from now on), and obtain $S_+$ =7.05. [*QRPA*]{}. The calculation of Engel [*et al.*]{} [@engel] yields $S_+ =5.03$, to be compared with QRPA or RPA calculations of Auerbach [@auer] leading to $S_+ =6.70$. [*Shell Model Monte Carlo*]{}. The calculation of Alhassid [*et al.*]{} [@alha], in the full $pf$ shell, using the BR interaction extrapolates to $S_+ =4.32\pm0.24$. (here the error bar includes only the statistical uncertainties but not those associated to the extrapolation or to possible sistematic errors of the method). [*Large t Shell Model calculations*]{}. All the previous results point to a reduction of GT+ strength as correlations are introduced and to a rather large dispersion of the calculated values depending on the interaction and the approach used. Therefore, to obtain a reliable $S_+$ value, the method and the interaction must demonstrate their ability to cope with a large number of other properties of the region under study. Calculations in the $pf$ shell [@czpm] using the KB3 interaction —a minimally modified version of the Kuo Brown G-matrix [@kb]— fulfill this condition since they give an excellent description of most of the observables in the region up to A=50. The same interaction was used years ago in perturbation theory to describe nuclei up to $^{56}$Ni [@pz], with fair success. It should be mentioned that the monopole modifications in KB3 involve only the centroids $V_{ff}$ and $V_{fr}$. The $V_{rr}$ values were left untouched and may need similar changes. It is not yet possible to perform a full $pf$ shell calculation in $^{54}$Fe. However, we can come fairly close by following the evolution of the total strength as the valence space is increased. The shell model matrices are built and diagonalized and the GT strengths calculated with the code ANTOINE [@ant]. Full Lanczos iterations in spaces that reach maximum m-scheme dimension of $1.4 \cdot 10^{7}$ are necessary for the parent states. Acting on them with the $\sigma \tau$ operator to calculate the strength, leads to spaces of $m$-scheme dimension of $4.1 \cdot 10^{7}$. In addition to KB3, to compare with the results of the SMMC extrapolations of [@alha], we have used the BR interaction [@brint]. The results are collected in table I and we proceed to comment on them. 1. The $t=$5 calculation should approximate the exact ground state energy reasonably well, as can be gathered from the small gain of 270 KeV achieved when increasing the space from $t=$4 to $t=$5. 2. The SMMC result using the BR interaction, , is some 1 MeV above the exact energy since our $t=$5 result gives an upper bound. Consequently the SMMC error bars in [@alha] are underestimated. 3. The result of our $t=$2 calculation using BR differs slightly from the one in [@auer] (7.22 [*vs.*]{} 7.05). This is due to the readjustment of the single particle energies made in [@auer] with respect to the values of [@brint]. 4. Auerbach [*et al.*]{} proposed an extrapolation of their $t=2$ calculation to the full space, based on the behaviour of $S_+$ in $^{26}$Mg as a function of . Although it is true that there is a qualitative correlation between these two observables (the bigger the quadrupole collectivity the smaller the $S_+$ value), it is difficult to go further and to obtain a quantitative prediction. In figure 1 we have plotted the $S_+$ values [*vs.*]{} B(E2) for the BR interaction and several truncations. It is clear that no simple correlation pattern comes out. Notice also that the extrapolation in [@auer] gives $S_+ =6.4$ compared to $S_+=5.5$ in the $t=5$ calculation. Before we discuss the differences between the results of the BR and KB3 interactions and between shell model diagonalizations and Monte Carlo extrapolations we examine the situation in $^{56}$Fe. ${\bf ^{56}Fe }$. Bloom and Fuller made a $t=0$ calculation [@bloom] that yields $S_+ =10.0$ (interaction from [@bint]). Anantaraman [*et al.*]{} [@anan] (interaction from [@vanhe; @koops]) obtain $S_+ =9.25$ for $t=0$ and $S_+ =7.38$ for $t=1$. The SMMC result [@dean] is shown in table I together with the numbers coming out of several truncations for both KB3 and BR interactions. [**The influence of the interaction.**]{} The interactions KB3 and BR lead to different single particle spectra for $^{57}$Ni. The sequence of levels obtained in the calculations up to $t=4$ are compared with the experimental data in table II. It is apparent from the table that the BR interaction places the $1f_{5/2}$ orbit too low. As a consequence the dominant configuration in the $^{56}$Fe ground state predicted by BR is $(1f_{7/2})^{14} (1f_{5/2})^{2}$ instead of $(1f_{7/2})^{14} (2p_{3/2})^{2}$ as given by KB3. This explains the very large difference in $S_+$ values observed in table I, already at the $t=$0 level: For a pure $(1f_{7/2})^{14} (2p_{3/2})^{2}$ configuration the total strength is 10.3, while for a pure $(1f_{7/2})^{14} (1f_{5/2})^{2}$ it amounts to only 5.7. In $^{54}$Fe the situation is not so dramatic because the leading configuration, $(1f_{7/2})^{14}$, is the same in both cases. Still the BR value is 20$\%$ smaller than the KB3 one, due to an excess of $1f_{7/2}$ - $1f_{5/2}$ mixing in the ground state. From that we conclude that the BR interaction underestimates the $S_+$ values for nuclei with N or Z greater than 28. Table II also shows the valus of the “gaps” defined by: $\Delta= 2BE(^{56}$Ni$)- BE(^{57}$Ni,$\frac{3}{2}^{-})- BE(^{55}$Ni). The strong staggering between even and odd values of $t$ makes it difficult to obtain a reliable extrapolation. The overall trend for the gap is to decrease as $t$ increases. Nevertheless, it is probable that the exact value for KB3 will remain somewhat larger than the experimental one. In this case a slight revision of the monopole terms would be needed. [**Shell Model extrapolations.**]{} In figure 1 we show the evolution of the total GT strength with the level of truncation in $^{54}$Fe , $^{56}$Fe and several cases ($^{48}$Ti, $^{50}$Ti, $^{48}$Cr, and $^{50}$Cr) for which exact results are available. If we continue the $^{54}$Fe calculated values with lines paralel to the A=50 ones, we get the following extrapolated values: $^{54}$Fe ; $S_+$(KB3)= 6.0 ; $S_+$(BR)= 5.0 If we assume that the value of the difference between the $t=$4 to $t=$5 result and the exact one is the same in $^{54}$Fe and $^{56}$Fe, the corresponding extrapolation is $^{56}$Fe ; $S_+$(KB3)= 4.5 these values are fully consistent with the experimental results if we use the standard 0.77 renormalization of the Gamow-Teller operator ([@brwil; @oster; @cpz] and section II). For $^{54}$Fe we have $S_+$(exp)= 3.1$\pm$0.6 ; 3.5$\pm$0.7 [*vs.*]{} $S_+$(KB3)= 3.56 ; $S_+$(BR)= 2.96 the corresponding predictions for $S_-$ are compatible —again within the 0.77 renormalization— with the experimental results $S_-$(exp)= 7.5$\pm$1.2 ; 7.8$\pm$1.9 ;7.5$\pm$0.7 [*vs.*]{} $S_-$(KB3)= 7.11 ; $S_-$(BR)= 6.52. If we turn to $^{56}$Fe the corresponding numbers are: $S_+$(exp)= 2.3$\pm$0.6 ; 2.9$\pm$0.3 [*vs.*]{} $S_+$(KB3)= 2.7, and $S_-$(exp)= 9.9$\pm$2.4 [*vs.*]{} $S_-$(KB3)= 9.8. We have prefered to omit error bars in the extrapolated values which are simply reasonable visual guesses. However, they fall so confortably in the middle of the experimental intervals, that an estimate of computational uncertainties would leave the conclusions unchanged. [**Comparison of Monte Carlo and Shell Model extrapolations.**]{} The differences between our results and those of ref. [@dean] are mostly —but not only— due to the use of different forces. With the same force the shell model extrapolations yield values that are some 20$\%$ larger than the SMMC ones. The discrepancy is probably related to the lack of convergence of the SMMC energies detected in table I. [*Note* ]{} [@deanpc]. In the most recent SMMC calculations with finer $\Delta \beta$ steps of 1/32 (instead of 1/16) the binding energy goes down by 1 MeV thus eliminating the problem mentioned in point 2 above. Furthermore $S_+$ becomes 4.7$\pm$0.3 in full agreement with our extrapolated value . SMMC values have also become available for the KB3 interaction [@langa]. The values for $^{54}$Fe and $^{56}$Fe are 6.05$\pm$0.45 and 3.99$\pm$0.27, again in very good agreement with our values. [**II. Strength functions: standard quenching and missing intensity**]{} In fig. 3 we show the total $l=0$ cross sections obtained by Anderson [*et al.*]{} for $^{54}$Fe$(p,n)^{54}$Co. The individual peaks in table I of [@ander] have been associated to gaussians of $\sigma$=87 keV (the instrumental width) for the lowest and $\sigma$=141 keV for the others. The “background” (the area under the dashed lines) is obtained by converting the 2 MeV bins in table III of [@ander] into gaussians of $\sigma$=1.41 MeV. Since there is no direct experimental evidence to decide how much of this background is genuine strength, two extreme choices are possible to extract $S_-$ : either keep the whole area in the figure (i.e., $S_-=10.3\pm1.4$), or only the area over the dashed line (i.e., $S_-=6.0\pm0.4$). An intermediate alternative consists in keeping what is left of the background after subtracting from it a calculated contribution to quasi free scattering (QFS). The resulting profile (with $S_-=7.5\pm0.7$, the number adopted in I.ii) is shown in fig. 4 (tables I+II of [@ander]) and compared with the Lanczos strength function (see [@cpz] for instance) obtained after 60 iterations in a $t=3$ calculation for the parent state and $t=4$ for the daughters (the peaks are broadened by gaussians of $\sigma$=87 keV for the lowest, and $\sigma$=212 keV for the others). The areas under the measured and calculated curves are taken to be the same (we know that upon extrapolation to the exact results they coincide). To within an overall shift of some 2 MeV, the two profiles agree quite nicely. The discrepancy is easily traced to the (too large) value of the gap in table II at this level of truncation. Although a calculation closer to the exact one would be welcome, the elements we have point to a situation in all respects similar to that of the $^{48}$Ca$(p,n)^{48}$Sc reaction, analyzed in [@cpz]. What was shown in this reference can be summed up as follows: - The effective $\sigma\tau$ operator to be used in a $0\hbar\omega$ calculation is quenched by a factor close to the standard one ($\approx 0.77$) through a model independent mechanism associated to nuclear correlations. - The rest of the strength must be carried by “intruders” (i.e., non $0\hbar\omega$ excitations). Only a fraction of this strength is located under the resonance, but intruders are conspicuously present in this region and make their presence felt through mixing that “dilutes” the $0\hbar\omega$ peaks causing apparent “background”. In all probability, the long tail in fig. 3 corresponds to intruder strength and should be counted as such. What is achieved by subtracting the QFS contribution amounts —accidentally but conveniently— to isolate the $0\hbar\omega$ quenched strength. It is to this contribution that the notion of standard quenching applies but it should be kept in mind that the remaining strength —necessary to satisfy the $3(N-Z)$ sum rule— is not missing, but most probably present in the satellite structure beyond the resonance region as hinted in the very careful analysis of ref. [@ander]. In fig. 5, to make a meaningful comparison with the $^{54}$Fe$(n,p)^{54}$Mn data of [@romq] the spikes of a $t=3$ calculation have been replaced by gaussians with $\sigma=1.77$ MeV, chosen to locate some strength at $-$2 MeV, where the first experimental point is found. The resulting distribution is then transformed into a histogram with 1 MeV bins. The agreement is quite satisfactory. It should be pointed out that the measures of [@vett] are displaced to lower energies by some 700 keV with respect to those of[@romq]. Otherwise, the experiments are in good agreement, and both show satellite structure beyond the resonance (not included in fig. 5, but visible in figs. 10 and 7 in [@vett] and [@romq] respectively. Finally, in figs. 6 and 7 we show the corresponding results for $^{56}$Fe targets, for which a $t=2$ truncation level was chosen, going to $t=4$ for $(p,n)$, and $t=2$ for $(n,p)$. Though this numerical limitation is rather severe, the agreement with the data remains good enough to support the main conclusion of this paper: The GT strength functions for $^{54}$Fe and $^{56}$Fe, in the resonance region and below, are well described by $0\hbar\omega$ calculations that account for $(0.77)^2$ of the total strength. The remainder, due to intruder states, is likely to be present in the observed satellite structures, so that the $3(N-Z)$ sum rule is satisfied. 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C [**44**]{}, 398 (1991). D. J. Dean, P. B. Radha, K. Langanke, Y. Alhassid, S. E. Koonin and W. E. Ormand, Phys. Rev. Lett. [**72**]{}, 4066 (1994). B. A. Brown and B. H. Wildenthal, At. Data Nucl. Data Tables [**33**]{}, 347 (1985). F. Osterfeld, Rev. Mod. Phys. [**64**]{}, 491 (1992). E. Caurier, A. Poves and A. P. Zuker, Phys. Rev. Lett. [**74**]{} 1517 (1995). D. J. Dean, private communication. K. Langanke [*et al.*]{}, Caltech preprint, march 1995. NUC-TH/9504019 $^{54}$Fe KB3 BR E(BR) $^{56}$Fe KB3 BR ----------- ------- --------------- --------------- ----------- ------- --------------- $t=0$ 10.29 10.29 $-50.23$ 10.01 7.33 $t=1$ 9.30 9.34 $-51.38$ 7.73 5.70 $t=2$ 7.68 7.22 $-54.67$ 6.37 4.48 $t=3$ 7.24 6.66 $-55.30$ 5.61 3.75 $t=4$ 6.70 5.84 $-56.21$ 5.11 $t=5$ 6.53 5.62 $-56.48$ SMMC 4.32$\pm$0.24 $-55.5\pm$0.5 2.73$\pm$0.04 : $^{54}$Fe $\rightarrow ^{54}$Mn and $^{56}$Fe $\rightarrow ^{56}$Mn Gamow Teller strength $S_+$ in units of the GT sum rule. In column 4 nuclear two body energy of the ground state of $^{54}$Fe (in MeV). KB3 3/2 5/2 1/2 $\Delta$ BR 3/2 5/2 1/2 $\Delta$ ------- ----- ------ ------ ---------- ---- ------ ------ ------ ---------- $t=0$ 0.0 0.38 1.15 8.57 0.48 0.00 3.06 7.42 $t=1$ 0.0 0.47 1.14 7.33 0.07 0.00 2.11 5.80 $t=2$ 0.0 0.72 1.16 8.10 0.07 0.00 2.27 7.01 $t=3$ 0.0 0.76 1.14 7.74 0.00 0.08 1.89 6.41 $t=4$ 0.0 0.86 1.14 7.90 0.00 0.11 1.83 7.21 EXP 0.0 0.77 1.11 6.39 : Excitation energies of the low-lying states in $^{57}$Ni and the gap $\Delta$ in MeV (see text). KB3 and BR results for several truncations, compared with the experimental results. [^1]: E-mail: [email protected] [^2]: E-mail: [email protected] [^3]: E-mail: [email protected] [^4]: E-mail: [email protected]

--- abstract: 'In this paper, we apply the cut and paste procedure to charged black string for the construction of thin-shell wormhole. We consider the Darmois-Israel formalism to determine the surface stresses of the shell. We take Chaplygin gas to deal with the matter distribution on shell. The radial perturbation approach (preserving the symmetry) is used to investigate the stability of static solutions. We conclude that stable static solutions exist both for uncharged and charged black string thin-shell wormholes for particular values of the parameters.' author: - | M. Sharif$^1$ [^1] and M. Azam$^{1,2}$ [^2]\ $^1$ Department of Mathematics, University of the Punjab,\ Quaid-e-Azam Campus, Lahore-54590, Pakistan.\ $^2$ Division of Science and Technology, University of Education,\ Township Campus, Lahore-54590, Pakistan. title: '**Mechanical Stability of Cylindrical Thin-Shell Wormholes**' --- [**Keywords:**]{} Israel junction conditions; Stability; Black strings.\ [**PACS:**]{} 04.20.Gz; 04.40.Nr; 98.80.Jk. Introduction ============ The study of thin-shell wormholes is of great interest as it links two same or different universes by a tunnel (throat) [@1]. The throat is threaded by the unavoidable amount of exotic matter (which violates the null energy condition). It is necessary to minimize the violation of energy conditions for the physically viability of wormholes. For this purpose, cut and paste procedure was considered by various authors, for instance [@2]-[@4], to construct a theoretical wormhole (thin-shell). Moreover, the required exotic matter to support wormhole can be minimized with the appropriate choice of geometry [@5]. The key issue of thin-shell wormhole is to explored its mechanical stability under radial perturbations in order to understand its dynamical aspects. In this scenario, many people worked for the stability analysis of thin-shell wormholes. Poisson and Visser [@5a] have constructed the Schwarzschild thin-shell wormhole and explored its stability regions under linear perturbations. Visser [@6] analyzed stability of thin-shell wormholes with specific equations of state. It was found that charge [@10] and positive cosmological constant [@11] increase the stability regions of spherically symmetric thin-shell wormholes. Thibeault et al. [@12] explored stability of thin-shell wormhole in Einstein-Maxwell theory with a Gauss-Bonnet term. Cylindrical thin-shell wormholes associated with and without cosmic strings have been widely discussed in literature [@13]-[@16]. These cosmic strings have many astrophysical phenomena like structure formation in the early universe and gravitational lensing effects [@17]. Eiroa and Simeone [@18] have discussed the cylindrical thin-shell wormholes associated with local and global cosmic strings and found that the wormhole configurations are unstable under velocity perturbations. Bejarano et al. [@19] discussed stability of static configurations of cylindrical thin-shell wormholes under perturbations and found that the throat will expand or collapse depending upon the sign of the velocity perturbations. Richarte and Simeone [@20] found more configurations which are not stable under radial velocity perturbations, consistent with the conjecture discussed in a paper [@19]. Eiroa and Simeone [@21] summarized the above results and found that stable configurations are not possible for the cylindrical thin-shell wormholes. Recently, we have explored stability of spherical and cylindrical geometries at Newtonian and post-Newtonian approximations and also spherically symmetric thin-shell wormholes [@21a]. Several candidates have been proposed like quintessence, K-essence, phantom, quintom, tachyon, family of Chaplygin gas, holographic and new agegraphic DE [@29] for the explanation of accelerated universe. Besides all these candidates, Chaplygin gas (an exotic matter) has been proposed widely to explain the expansion of the universe. Kamenshchik et al. [@30] described the feature of Chaplygin gas and explored cosmology of FRW universe filled with a Chaplygin gas. The Chaplygin cosmological models support the observational evidence [@31; @32]. Models of exotic matter like phantom energy with equation of state $(p=\omega\sigma,~ \omega<-1)$ [@35] and Chaplygin gas $(p\sigma=-A)$ [@36], where $A$ is a positive constant, has been of interest in wormhole construction. Wormholes have also been studied in dilaton gravity [@37] and Einstein-Gauss-Bonnet theory [@38]. Eiroa [@39] found stable static solutions of spherically symmetric thin-shell wormholes with Chaplygin equation of state. In this paper, we construct black string thin-shell wormholes with and without charge supported by the Chaplygin gas. We study the mechanical stability of these constructed wormholes under radial perturbations preserving the cylindrical symmetry. The format of the paper is as follows. In section **2**, we formulate surface stresses of the matter localized on the shell through Darmois-Israel junction conditions. Section **3** provides the mechanical stability analysis of the static configuration of thin-shell wormholes. In the last section **4**, we conclude our results. Thin-Shell Wormhole Construction ================================ In this section, we build a thin-shell wormhole from charged black string through cut and paste technique and discuss its dynamics through the Darmois-Israel formalism. We consider the Einstein-Hilbert action with electromagnetic field as [@a] $$\begin{aligned} \label{a} \mathcal{A}+\mathcal{A}_{em}=\frac{1}{16\pi{G}}\int{\sqrt{-g}(R-2\Lambda)d^4x} -\frac{1}{16\pi}\int{\sqrt{-g}F^{\mu\nu}F_{\mu\nu}d^4x},\end{aligned}$$ where $\mathcal{A}$ is defined by $$\begin{aligned} \label{b} \mathcal{A}=\frac{1}{16\pi{G}}\int{\sqrt{-g}(R-2\Lambda)d^4x}.\end{aligned}$$ Here, $g,~R,~\Lambda,~F_{\mu\nu}$ are the metric determinant, the Ricci scalar, negative cosmological constant and the Maxwell field tensor ($F_{\mu\nu}=\partial_\mu{\phi_\nu}-\partial_\nu{\phi_\mu}$), respectively and $\phi_\nu=-\lambda(r)\delta^0_\nu$ is the electromagnetic four potential with an arbitrary function $\lambda(r)$. The Einstein-Maxwell equations from the above action yields a cylindrically symmetric vacuum solution, i.e., charged black string given as $$\label{1} ds^2=-g(r)dt^{2}+g^{-1}(r)dr^{2}+h(r)(d\phi^{2}+\alpha^2{dz^2}),$$ where $g(r)=\left(\alpha^2r^2-\frac{4M}{\alpha{r}}+\frac{4Q^2}{\alpha^2r^2}\right)$ is a positive function for the given radius and $h(r)=r^2$. We have following restrictions on coordinates to preserve the cylindrical geometry $$-\infty<t<\infty,\quad 0\leq{r}<\infty, \quad -\infty<{z}<{\infty},\quad 0\leq{\phi}\leq{2\pi}.$$ The parameters $M,~Q$ are the ADM mass and charge density, respectively and $\alpha^2=-\frac{\Lambda}{3}>0$. We remove the region with $r<a$ of the given cylindrical black string and take two identical $4D$ geometries $V^{\pm}$ with $r\geq{a}$ as $$\label{2} V^{\pm}=\{x^{\gamma}=(t,r,\phi,z)/r\geq{a}\},$$ where $``a"$ is the throat radius and glue these geometries at the timelike hypersurface $\Sigma=\Sigma^\pm=\{r-a=0\}$ to get a new manifold $V=V^{+}\cup{V^{-}}$. This manifold is geodesically complete exhibiting a wormhole with two regions connected by a throat satisfying the radial flare-out condition, i.e., $h'(a)=2a>0$ [@16]. If $`{r_h}$’ is the event horizon of the black string given in Eq.(\[1\]), then we assume $a>r_h$ in order to prevent the occurrence of horizons and singularities in wormhole configuration. We use the standard Darmois-Israel formalism [@b; @c] to analyze the dynamics of the wormhole. The two sides of the shell are matched through the extrinsic curvature defined on $\Sigma$ as $$\label{3} K^{\pm}_{ij}=-n^{\pm}_{\gamma}(\frac{{\partial}^2x^{\gamma}_{\pm}} {{\partial}{\xi}^i{\partial}{\xi}^j}+{\Gamma}^{\gamma}_{{\mu}{\nu}} \frac{{{\partial}x^{\mu}_{\pm}}{{\partial}x^{\nu}_{\pm}}} {{\partial}{\xi}^i{\partial}{\xi}^j}),\quad(i, j=0,2,3),$$ where $\xi^i=(\tau,\theta,\phi)$ are the coordinates on $\Sigma$ and $n^{\pm}_{\gamma}$ are the unit normals obtained to $\Sigma$ as $$\label{3a} n^{\pm}_{\gamma}=\left(-\dot{a},\frac{\sqrt{g(r)+\dot{a}^2}}{g(r)},0,0\right),$$ satisfying the relation $n^{\gamma}n_{\gamma}=1$. The induced metric on $\Sigma$ is defined as $$\label{4} ds^2=-d\tau^2+a^2(\tau)(d\phi^2+\alpha^2dz^2),$$ where $\tau$ is a the proper time on the hypersurface. The non-vanishing components of the extrinsic curvature turns out to be $$\label{5} K^{\pm}_{\tau\tau}=\mp\frac{g'(a)+2\ddot{a}}{2\sqrt{g(a)+\dot{a}^2}}, \quad K^{\pm}_{\phi\phi}= \pm{a}\sqrt{g(a)+\dot{a}^2},\quad K^{\pm}_{zz}=\alpha^2K^{\pm}_{\phi\phi}.$$ Here dot and prime mean derivative with respect to $\tau$ and $r$ respectively. The surface stress-energy tensor $S_{ij}=diag(-\sigma,~p,~p)$ provides surface energy density $\sigma$ and surface tensions $p$ of the shell. The Lanczos equations are defined on the shell as $$\label{6} S^i_{j}=-\frac{1}{8\pi}\left[\kappa^i_{j}-{\delta}^i_j{\kappa}^k_k\right],$$ where $\kappa_{ij}=K^{+}_{ij}-K^{-}_{ij}$ and due to the simplicity of cylindrical symmetry, we have $\kappa^i_j=diag(\kappa^\tau_{\tau},~\kappa^\phi_{\phi},~\kappa^\phi_{\phi})$. The Lanczos equations with surface stress-energy tensor provides $$\begin{aligned} \label{7} \sigma&=&-\frac{1}{4\pi}\kappa^\phi_\phi=-\frac{1}{2\pi{a}}\sqrt{g(a)+\dot{a}^2},\\\label{8} p&=&\frac{1}{8\pi}(\kappa^\tau_\tau+\kappa^\phi_\phi)=\frac{1}{8\pi{a}}\frac{2a\ddot{a}+2\dot{a}^2 +2g(a)+ag'(a)}{\sqrt{g(a)+\dot{a}^2}}.\end{aligned}$$ *Matter that violates the null energy condition is known as exotic matter*. We see from Eq.(\[7\]) that the surface energy density is negative indicating the existence of exotic matter at the throat. For the explanation of such matter, the Chaplygin equation of state is defined on the shell as $$\label{9} p=-\frac{A}{\sigma},$$ where $A>0$. Using Eqs.(\[7\]) and (\[8\]) in the above equation, we obtain a second order differential equation satisfied by the throat radius $`a$’ $$\label{10} 2a\ddot{a}+2\dot{a}^2-16\pi^2{A}a^2+2g(a)+ag'(a)=0.$$ Stability Analysis ================== In this section, we have adapted and applied the criteria introduced in Ref.[@39] for the stability analysis of static solutions. The existence of static solutions is subject to the condition $a_0>r_h$. For such solutions, we consider static configuration of Eq.(\[10\]) as $$\label{11} -16\pi^2{A}a^2_0+2g(a_0)+a_0g'(a_0)=0,$$ and the corresponding static configuration of surface energy density and pressure are $$\begin{aligned} \label{12} \sigma_0=-\frac{1}{2\pi{a_0}}\sqrt{g(a_0)},\quad p_0=\frac{2A\pi{a_0}}{\sqrt{g(a_0)}}.\end{aligned}$$ The perturbed form of the throat radius is given by $$\label{14} a(\tau)=a_0[1+\epsilon(\tau)],$$ where $\epsilon(\tau)\ll1$ is a small perturbation preserving the symmetry. Using the above equation, the corresponding perturbed configuration of Eq.(\[10\]) can be written as $$\label{15} (1+\epsilon)\ddot{\epsilon}+\dot{\epsilon}^2-8\pi^2{A}(2+\epsilon)\epsilon +D(a_0,\epsilon)=0,$$ where $$\label{16} D(a_0,\epsilon)=\frac{2g(a_0+a_0\epsilon)+a_0(1+\epsilon)g'(a_0+a_0\epsilon) -2g(a_0)-a_0g'(a_0)}{2a^2_0}.$$ Substituting $\upsilon(\tau)=\dot{\epsilon}(\tau)$ in Eq.(\[15\]), we have a first order differential equation in $\upsilon$ $$\label{17} \dot{\upsilon}=\frac{8\pi^2{A}(2+\epsilon)\epsilon-D(a_0,\epsilon)-\upsilon^2}{1+\epsilon}.$$ Using Taylor expansion to first order in $\epsilon$ and $\upsilon$, we obtain a set of equations $$\label{18} \dot{\epsilon}=\upsilon,\quad \dot{\upsilon}=\Delta{\epsilon},$$ where $$\begin{aligned} \label{19} \Delta&=&16\pi^2{A}-\frac{3g'(a_0)+a_0g''(a_0)}{2a_0}.\end{aligned}$$ This set of equations can be written in matrix form as $$\dot{\eta}=L\eta,$$ where $\eta$ and $L$ are $$\eta=\left[\begin{array}{ccccc} \epsilon \\ \upsilon \end{array} \right],\quad L=\left[\begin{array}{ccccc} 0&1 \\ \Delta&0 \end{array} \right].\\$$ The matrix $`L$’ has two eigenvalues $\pm{\sqrt{\Delta}}$. The stability analysis of static solutions on the basis of $\Delta$ can be formulated as follows [@39]:\ \ **i.** When $\Delta>0$, the given matrix has two real eigenvalues $\lambda_1=-\sqrt{\Delta}<0$ and $\lambda_2=\sqrt{\Delta}>0$. The negative eigenvalue has no physical significance, while the positive eigenvalue indicates the presence of unstable static solution.\ \ **ii.** When $\Delta=0$, the corresponding eigenvalues are $\lambda_1=\lambda_2=0$ and Eq.(\[18\]) leads to $\upsilon=\upsilon_0=constant$ and $\epsilon=\epsilon_0+\upsilon_0(\tau-\tau_0)$. In this case, the static solution is unstable.\ \ **iii.** When $\Delta<0$, we have two imaginary eigenvalues $\lambda_1=-\dot{\iota}\sqrt{\mid\Delta\mid}$ and $\lambda_2=\dot{\iota}\sqrt{\mid\Delta\mid}$. Eiroa [@39] discussed this case in detail and found that the only stable static solution with throat radius $a_0$ exists for $\Delta<0$ which is not asymptotically stable. Uncharged Black String Thin-Shell Wormhole ------------------------------------------ Here, we investigate the stability of thin-shell wormhole static solution constructed from the asymptotically anti-de Sitter uncharged black string [@a]. For the uncharged black string, we have $$\label{21} g(r)=\alpha^2r^2-\frac{4M}{\alpha{r}}.$$ The event horizon of the black string is given by $(g_{tt}=0)$ $$\label{21a} r_{h}=\frac{(4M)^{\frac{1}{3}}}{\alpha}.$$ The corresponding surface energy density and pressure of black string for static configuration leads to $$\begin{aligned} \label{22} \sigma_0=-\frac{\sqrt{\alpha^3a^3_0-4M}}{2\pi{a^{\frac{3}{2}}_0}\sqrt{\alpha}},\quad p_0=\frac{2A\pi{a^{\frac{3}{2}}_0}\sqrt{\alpha}}{\sqrt{\alpha^3a^3_0-4M}}.\end{aligned}$$ Using Eq.(\[21\]) in (\[11\]), we get a cubic equation satisfied by the throat radius with $A>0$ and $M>0$ $$\label{23} (4A\pi^2{\alpha}-\alpha^3)a^3_0+M=0.$$ The corresponding roots $a'^1_0,~a'^2_0,~a'^3_0,$ of the cubic equation are $$\begin{aligned} \label{24} a'^1_0&=&\frac{1}{\alpha}\left(\frac{M}{ 1-4A{\pi^2}\alpha^{-2}}\right)^\frac{1}{3},\\\label{25} a'^2_0&=&\frac{(-1+\dot{\iota}\sqrt{3})}{2\alpha}\left(\frac{M} {1-4A{\pi^2}\alpha^{-2}}\right)^\frac{1}{3},\\\label{26} a'^3_0&=&\frac{(-1-\dot{\iota}\sqrt{3})}{2\alpha}\left(\frac{M} {1-4A{\pi^2}\alpha^{-2}}\right)^\frac{1}{3}.\end{aligned}$$ Now, we analyze all the roots numerically and find which of them represent static solution. The condition for the existence of static solution is subject to $a'_0>\frac{(4M)^{\frac{1}{3}}}{\alpha}.$ For $(A\alpha^{-2})>(4\pi^2)^{-1},$ we have one negative real root $a'^3_0$ and two non-real roots $a'^1_0,~a'^2_0$. All the three roots have no physical significance. Thus, there are no static solutions corresponding to $(A\alpha^{-2})>(4\pi^2)^{-1}$. For $(A\alpha^{-2})<(4\pi^2)^{-1},$ there is one positive real root $a'^1_0$ and two non-real roots $a'^2_0,~a'^3_0$. The non-real roots are discarded being of no physical significance. We check numerically that for $(A\alpha^{-2})<(4\pi^2)^{-1},$ the only static solution corresponding to positive real root is $a'^1_0\geq{\frac{(4M)^{\frac{1}{3}}}{\alpha}}$. The critical value for which the throat radius greater than event horizon is $\beta=1.9\times10^{-2}<(4\pi^2)^{-1}$. It turns out that the static solution will exist if $\beta<(A\alpha^{-2})\leq(4\pi^2)^{-1}$. For the stability of $a'^1_0$, we work out $\Delta$ for the uncharged black string. From Eq.(\[19\]), we have $$\label{27} \Delta=16A\pi^2-4\alpha^2-\frac{2M}{\alpha{a^3_0}}.$$ The above equation with Eq.(\[23\]) leads to $$\label{28} \Delta=-\frac{6M}{\alpha{a^3_0}}.$$ This shows that $\Delta$ is always negative for $a'^1_0>\frac{(4M)^{\frac{1}{3}}}{\alpha}$. Thus, we have one stable solution for the uncharged black string corresponding to $(A\alpha^{-2})<(4\pi^2)^{-1}$ with throat radius $a'^1_0$ as shown in Figure **1**. There does not exist any other static solution corresponding to $(A\alpha^{-2})>(4\pi^2)^{-1}$. Charged Black String Thin-Shell Wormhole ---------------------------------------- For the case of charged black string, we have considered $g(r)$ from Eq.(\[1\]) $$\label{29} g(r)=\alpha^2r^2-\frac{4M}{\alpha{r}}+\frac{4Q^2}{\alpha^2r^2}.$$ The event horizons [@a] for the charged black string are obtained from the equation $$\label{29a} \alpha^2r^2-\frac{4M}{\alpha{r}}+\frac{4Q^2}{\alpha^2r^2}=0.$$ The above quartic equation has two real and two complex roots. We discard the complex roots being of unphysical and the real roots are taken as inner and outer event horizons of the charged black string $$\label{30} r_{\pm}=\frac{(4M)^\frac{1}{3}}{2\alpha}\left[\sqrt{s}\pm \sqrt{{2}\sqrt{s^2-Q^2\left(\frac{2}{M}\right)^\frac{4}{3}}-s}\right] =\frac{(4M)^\frac{1}{3}}{\alpha}\chi,$$ provided that the inequality $Q^2\leq\frac{3}{4}M^\frac{4}{3}$ holds, where $s$ and $\chi$ are given by $$\begin{aligned} \label{31} s&=&\left(\frac{1}{2}+\frac{1}{2}\sqrt{1-\frac{64Q^6}{27M^4}} \right)^\frac{1}{3} +\left(\frac{1}{2}-\frac{1}{2}\sqrt{1-\frac{64Q^6}{27M^4}} \right)^\frac{1}{3},\\\label{31a} \chi&=&\frac{1}{2}\left[\sqrt{s}\pm \sqrt{{2}\sqrt{s^2-Q^2\left(\frac{2}{M}\right)^\frac{4}{3}}-s}\right].\end{aligned}$$ For $Q^2>\frac{3}{4}M^\frac{4}{3}$, the given metric has no event horizon and represents a naked singularity. If $Q^2=\frac{3}{4}M^\frac{4}{3}$, the inner and outer horizons coincide, which corresponds to extremal black strings. For the static wormhole associated to the charged black string, the surface energy density and pressure turn out to be $$\begin{aligned} \label{32} {\sigma_0}&=&-\frac{\sqrt{\alpha^4a^4_0-4M\alpha{a_0+4Q^2}}}{2\pi{a^2}_0{\alpha}},\\\label{32a} p_0&=&\frac{2A\pi{a^2_0}{\alpha}}{\sqrt{\alpha^4a^4_0-4M\alpha{a_0+4Q^2}}}.\end{aligned}$$ We would like to find an equation satisfied by the throat radius. We substitute Eq.(\[29\]) in (\[11\]), it turns out that the charge terms cancel out and again we get the same equation (\[23\]) satisfied by the throat radius of the wormhole with the roots given in Eqs.(\[24\])-(\[26\]). For the existence of static solutions, the throat radius $a_0$ should be greater than $r_h=r_+$. It is noted that the event horizon (\[30\]) is a decreasing function of $Q$ for positive values of $M$ and $\alpha$ as shown in Figure **2**. Again, we check all the roots numerically and find that there exists only one static solution, $a'^1_0$, subject to large values of $Q$. Thus in the case of charged black string, we have one static solution for $(A\alpha^{-2})<(4\pi^2)^{-1}$ with $Q<M$ and large values of $Q$ and and no static solution for $(A\alpha^{-2})>(4\pi^2)^{-1}$. The stability of static solutions, whether they are stable or unstable, depends upon the sign of $\Delta$ which is again negative for $a'^1_0>r_h$. Hence, the obtained static solution is of stable type for the static wormhole associated with the charged black string. Summary and Discussion ====================== In this paper, we have constructed black string thin-shell wormholes and investigated their mechanical stability under radial perturbations preserving the cylindrical symmetry. We have formulated the Darmois Israel junction conditions and imposed Chaplygin equation of state on the matter shell. We have constructed wormholes using cut and paste procedure such that the throat radius should be greater than the event horizon of the given metric, i.e., $a_0>r_h$. A dynamical equation with the Chaplygin gas of the shell has been obtained by considering throat radius as a function of proper time. The proposed criteria for the stability analysis of static solutions has been extended to uncharged and charged constructed wormholes. It was shown in Refs.[@18]-[@21] that cylindrical thin-shell wormholes with equations of state depending upon the metric functions are always mechanically unstable. On the other hand, we have constructed cylindrical thin-shell wormholes from charged black string solution (whose causal structure is similar to Reissner-Nordstr$\ddot{o}$m solution) with Chaplygin equation of state. We have found stable configurations depending upon the parameters involving in the model. This shows that the choice of bulk solution as well as equation of state plays a significant role in the stability of wormhole configurations. For the case of static wormhole associated with uncharged black string, we have found one negative real root and two non-real roots corresponding to $(A\alpha^{-2})>(4\pi^2)^{-1}$, which implies no static solution. For $(A\alpha^{-2})<(4\pi^2)^{-1}$, we have two complex roots and one positive real root which is stable for $\beta<(A\alpha^{-2})\leq(4\pi^2)^{-1}$, where, $\beta=1.9\times10^{-2}<(4\pi^2)^{-1}$. In the case of static wormhole associated to charged black string, we see that event horizon $r_h$ is a decreasing function of $Q$. Also, we have found the same three roots as in the case of uncharged black string. Thus, there is one stable static solution corresponding to $(A\alpha^{-2})<(4\pi^2)^{-1}$ depending upon the values of the parameters $Q<M$ and $\alpha>0$. For $(A\alpha^{-2})>(4\pi^2)^{-1}$, there is no static solution. 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--- abstract: 'Diffuse star clusters (DSCs) are old and dynamically hot stellar systems that have lower surface brightness and more extended morphology than globular clusters (GCs). Using the images from HST/ACS Fornax Cluster Survey, we find that 12 out of 43 early-type galaxies (ETGs) in the Fornax cluster host significant numbers of DSCs. Together with literature data from the HST/ACS Virgo Cluster Survey, where 18 out of 100 ETGs were found to host DSCs, we systematically study the relationship of DSCs with GCs, and their host galaxy environment. Two DSC hosts are post-merger galaxies, with most of the other hosts either having low mass or showing clear disk components. We find that while the number ratio of DSCs to GCs is nearly constant in massive galaxies, the DSC-to-GC ratio becomes systematically higher in lower mass hosts. This suggests that DSCs may be more efficient at forming (or surviving) in low density environments. DSC hosts are not special either in their position in the cluster, or in the galactic color-magnitude diagram. Why some disk and low-mass galaxies host DSCs while others do not is still a puzzle, however. The mean ages of DSC hosts and non-hosts are similar at similar masses, implying that formation efficiency, rather than survival, is the reason behind different DSC number fractions in early-type galaxies.' author: - 'Yiqing Liu, Eric W. Peng, Sungsoon Lim, Andrés Jordán, John Blakeslee, Patrick C[ô]{}t[é]{}, Laura Ferrarese, Petchara Pattarakijwanich' bibliography: - 'ref\_DSC.bib' title: ' The ACS Fornax Cluster Survey. XII. Diffuse Star Clusters in Early-type Galaxies ' --- Introduction ============ Globular clusters (GCs) are relatively more massive and compact compared to other kinds of star clustersr. @Misgeld_Hilker_11 show that the surface density of GCs are well correlated with their masses, with the more massive GCs having higher surface densities, and the effective radii are distributed tightly around 3 pc. However, this view has been updated with the improvement of our detection ability. Using the Hubble Space Telescope (HST), @Larsen_Brodie_00 discovered a population of old star clusters that have GC-like luminosity but much larger sizes in a nearby S0 galaxy NGC 1023. Comparing with GCs, they are redder and mostly fainter than $M_V=-7$ with half-light radii ($r_h$) in the range of 7-15 pc, while the common GCs have a luminosity function peaked at $M_V=-7.4$ and a typical $r_h$ of 3 pc. On the other hand, they are significantly brighter and larger than the open clusters in the Milky Way. This discovery opened a new field, rapidly leading to more detections in other galaxies. Similar diffuse star clusters (DSCs) were detected in the nearby field galaxies NGC 3384, NGC 5195, NGC 5194 (M51), and NGC 6822 [@Larsen_01; @Lee_05; @Hwang_Lee_08; @Hwang_11], as well as 12 Early Type Galaxies (ETGs) in the Virgo Cluster [@P06 hereafter P06]. They are also detected in the outer halo of our Milky Way and M31, our satellite galaxies, and the dwarf elliptical galaxy Scl-dE1 [@vdBergh_Mackey_04; @Huxor_05; @daCosta_09]. These DSCs tend to have larger $r_h$ (20$-$40 pc), but are still smaller and brighter than the ultra-faint galaxies at similar magnitude. Nonetheless, there are galaxies with no DSCs detected. This naturally raises questions: Why are DSCs only detected in certain galaxies, instead of others? Do these galaxies have special physical conditions for DSC formation, or for their survival? Does the DSC formation follow the general picture of star cluster formation? The last question is the most fundamental one. Besides the typical way of star cluster formation, tidal stripping of galaxies and mergers of cluster complexes are two candidate mechanisms. In the former case, although the galactic cores left from stripping always have large sizes, they usually have relatively high surface brightness, which are more similar to ultra-compact dwarf galaxies (UCDs). The merger origin [@Fellhauer_Kroupa_02; @Burkert_05; @Bruns_09; @Bruns_11] is disfavored. @Assmann_11 found that the velocity dispersions of merger-produced DSCs would be too high. Furthermore, they also excluded the scenario in which DSCs formed by expanding normal star clusters due to the gas expulsion or stellar mass loss during their early evolution, as the observed star formation efficiency is not high enough. Therefore, DSCs probably form in a way similar to other star clusters. Then the question remains as to why they only exist in certain galaxies. Possibly, DSC formation may require special environmental conditions. @Harris_Pudritz_94 argue that the supergiant molecular clouds that form massive star clusters are pressure-confined by the interstellar medium (ISM) of their parent galaxies. Furthermore, @Mclaughlin_00 noted a relation between the binding energy and the galactocentric distance of the Milky Way globular clusters. These all imply that the more extended star clusters prefer to form in lower density regions. A more directed study is from @Elmegreen_08, who suggested that the difference between star formation in bound clusters and in loose groupings is attributed to the difference in cloud pressure. High-pressure regions place higher fraction of stars in bound clusters, while low-pressure regions prefer to make unbound stellar groupings; and the regions with moderately low density and moderately high Mach number would produce low-density bound clusters like DSCs. Low-mass galaxies provide such environments. Based on the evidence that extended star clusters are found in dwarf galaxies NGC 6822 and Scl-dE1, the low-mass halo origin is plausible, and the DSCs which are observed in the outer halo of massive galaxies can be explained by accretion from low-mass satellite galaxies. Moreover, @Masters_10 showed a trend between GC size and host galaxy mass, with the fainter galaxies have larger GCs. Galactic disks are another such low density environment. For example, @Pellerin_10 suggested that in a collisional ring galaxy NGC 922, the highly shocked low density ring which contains a number of star forming complexes and young massive clusters is a possible place for forming DSCs. Among all the previously founded DSC host galaxies, most are either dwarf or disky galaxies. In addition, DSCs in NGC 1023 have systematic rotation curve similar to the host galaxy [@Larsen_Brodie_02]. However, not all low-mass or disk galaxies are associated with DSCs. So the question that naturally follows is: are those DSC host galaxies different from their counterparts, or they are just in a stage of evolution when DSCs have not been entirely disrupted? Using N-body simulations, @Hurley_Mackey_10 found that DSCs can form naturally within weak tidal fields, which provides a possible scenario that the detected DSCs are just the ones that have not been tidally disrupted, because the disruption timescale is small when star clusters have larger radii [@Gnedin_99]. To further investigate these questions, a large and complete sample is necessary. Because of their low luminosities, DSC studies are limited to the nearby universe, and the sample from the literature is not big because the frequency of their appearance is relatively low. Moreover, except for the Virgo Cluster, which is the nearest galaxy cluster (16.5 Mpc away) that has been examined by P06, no other cluster environment has been used for DSC studies. Therefore, in order to build a larger sample for DSC study, we turn to the Fornax cluster, which is the second nearest cluster located 20 Mpc away. Space-based imaging is a powerful technique to detect these small low surface brightness DSCs. Previous work by P06 used the data from the ACS Virgo Cluster Survey (ACSVCS; @Cote_04) to study the DSCs in that cluster. This work uses the data from the ACS Fornax Cluster Survey (ACSFCS; @Jordan_07), which is a complementary program to the ACSVCS that imaged 43 galaxies in the Fornax cluster, to perform similar studies. We compare DSCs and GCs using this larger sample, and look for their dependence on galactic properties such as type, mass, and environment. A special advantage of this work is that these two surveys have the same instrument setups and data reduction processes, which aids in our comparison. The paper is structured as follow: Our data are introduced in $\S$\[obs\]. The selection and basic properties of DSCs are described in $\S$\[selection\]. Then we investigate the properties of DSC host galaxies in $\S$\[galaxy\], and compare the color, spacial distribution, and formation efficiency of DSCs and GCs in $\S$\[D\_G\]. Possible DSC formation scenarios are discussed in $\S$\[discussion\]. Conclusions are summarized in $\S$\[conclusion\].\ Observations {#obs} ============ The ACSFCS [@Jordan_07] is a program that has imaged 43 ETGs in the Fornax Cluster with the HST/ACS. This is a complete sample of Fornax galaxies brighter than $B_T\sim15.5$ ($M_B\sim-16$) mag, covering the morphological types of E, S0, SB0, dE, dE,N, dS0, or dS0,N. It includes 41 galaxies from the Fornax Cluster Catalog (FCC; @Ferguson_89a), as well as 2 outlying elliptical galaxies NGC 1340 and IC 2006. This survey took $202\arcsec \times 202\arcsec$ field of view (FOV) images for each galaxy in F475W and F850LP filters, with a pixel scale of $0.049\arcsec$. These two filters are roughly the same as the SDSS $g$ and $z$ bands (hereafter referred to as $g$ and $z$ band), and they are sensitive to metallicity and age of stellar populations. Because a primary science goal of the program is to study extragalactic globular clusters, the images are sufficiently deep that $\sim90\%$ of the GCs can be detected at a high level of completeness [@Cote_04] with a high spatial resolution. Moreover, the contaminants of background galaxies have been simulated by using 16 blank high-latitude control field images from HST archive, as in P06. We also use data from the ACSVCS, with the identical instrument setup. The ACSVCS sample contains 100 ETGs with $B_T<16$, but is only complete to $B_T<12.15$ ($M_B<-18.94$). In the luminosity range where the sample is incomplete, 63 low-mass galaxies were removed from the sample. The data reduction of both surveys was performed in the same way, following P06. One exception is on the star cluster candidates larger than 10 pc. For these objects from ACSVCS, their structural parameters were measured precisely by preforming a new model of profile fitting. However, it was not applied for ACSFCS, and we limit our sample to the objects smaller than 10 pc in this study.\ DSC Selection {#selection} ============= The data reduction process is described in @Jordan_04, for both image analysis and point source selection. Among the output of GC candidates, @Jordan_09 evaluated the probability $p_{GC}$ that a given object is a GC, according to its position in the size-magnitude parameter space. All the basic parameters of the GC candidates from ACSFCS are listed in @Jordan_15. In previous ACSVCS and ACSFCS studies, $p_{GC} \ge 0.5$ is used to select GCs, and we use the same criterion in this work. Usually, those objects with $p_{GC} < 0.5$ are not as concentrated as GCs and mainly consisted of background galaxies. However, since the expected number of background contaminants has been estimated from control fields, if the number of diffuse objects in a galaxy field significantly exceeds the expectation, we can infer that this galaxy hosts some DSCs. Following P06, we select those extended, background-liked DSCs using the criteria $p_{GC} \le 0.2$ and projected half-light radius $r_h \ge 4$ pc (typical GCs have median $r_h\sim3$ pc), avoiding most traditional GCs. This selection would leave a fraction of star clusters that are classified into neither GCs nor DSCs, but it is reasonable in this study. Because our primary goal is making a sample of star clusters that are significantly more diffuse than traditional GCs, instead of counting their absolute numbers. Figure \[rhMz\] shows our selection in the parameter spaces. All the GC candidates with $p_{GC} > 0$ from the DSC-excess galaxies (13 from Fornax and 19 from Virgo, which will be described below) are displayed in the $r_h$-$M_z$ diagrams. The left and right columns are for program and randomly selected control fields respectively. From top to bottom, the samples are from the star cluster systems of FCC 21 (NGC 1316; Fornax A), the combination of the rest of Fornax galaxies with DSC excess, VCC 798 (NGC 4382; M85), and the combination of the rest of Virgo galaxies with DSC excess. FCC 21 and VCC 798 have the highest number of DSCs in Fornax and Virgo respectively (Figure \[findDSC\] and Table \[gal\_DSC\] in both this paper and P06). We show these galaxies separately to show the distribution clearly, especially because they might dominate the total distribution by large numbers. The candidates that agreed with the criteria of GC and DSC are shown in blue and red respectively, and the rest are plotted in grey. The constant $z$-band mean surface brightness ($\mu_z$) of 18.0, 19.5, 21.0 mag/arcsec$^2$ are marked by diagonal dash lines. From these diagrams, all the star clusters distributed continuously in $r_h$-$M_z$ space, and our criteria are as good at separating them as using surface brightness in all host galaxies. The DSC candidates are located at the faint end of GC luminosity distributions, but this may be just a selection effect, because we select DSCs with faint surface brightness in a limited range of sizes. The parameter that fundamentally makes DSCs special is the surface brightness $\mu$, which is a combination of luminosity and size. Figure \[histSB\] displays the surface brightness distributions of star clusters in our sample. We divide the DSC host galaxies into two groups. One consists of the two merger remnants with the most massive DSC systems, FCC 21 and VCC 798 (the upper panel), and the other is made up by the rest galaxies (the bottom panel). In both panels, the black, blue, and red histograms represent the distributions of the entire star cluster systems, GCs, and DSCs in DSC host galaxies that normalized by the bin with the highest number of all star clusters respectively. In the bottom panel, the grey dash line shows the distribution of all star cluster candidates from the DSC non-excess galaxies, and normalized by the highest bin. The background contaminants are subtracted in each bin. In the merger remnants, the surface brightness distribution of their all star cluster candidates possibly peaks at a magnitude fainter than our detection limit. Because DSCs occupy the faint end of this distribution, it is hard to infer their substantial behavior in this work. From the bottom panel, the distribution of the star clusters in DSC host galaxies is more extended than that of the DSC non-hosts at the fainter end, while they are similar at the bright end and have peaks at similar magnitude. In addition, the distribution of the DSC non-host is symmetric and the faint excess of DSC host galaxies is mainly contributed by the DSC candidates. It indicates that DSCs are essentially a distinct population of star clusters. Figure \[im\_zoom\] shows how GC and DSC candidates look like on the image. GC and DSC candidates are circled in yellow and magenta respectively. DSCs are less compact than GCs, and some are not well separated from background galaxies. [llcccccl]{} 21 & 1316 & $03:22:42.09$ & $-37:12:31.63$ & $-24.45$ & $1.37$ & $179.6\pm 14.5$ & S0(pec)\ 213 & 1399 & $03:38:29.14$ & $-35:27:02.30$ & $-23.50$ & $1.41$ & $13.5 \pm 6.1 $ & E0\ – & 1340 & $03:28:19.70$ & $-31:04:05.00$ & $-22.22$ & $1.33$ & $43.3 \pm 10.2$ & E5\ 167 & 1380 & $03:36:27.45$ & $-34:58:31.09$ & $-22.46$ & $1.31$ & $59.9 \pm 10.8$ & S0/a\ 83 & 1351 & $03:30:35.04$ & $-34:51:14.51$ & $-21.18$ & $1.39$ & $20.9 \pm 9.2 $ & E5\ 184 & 1387 & $03:36:56.84$ & $-35:30:23.85$ & $-21.75$ & $1.59$ & $23.2 \pm 9.2 $ & SB0\ 47 & 1336 & $03:26:31.97$ & $-35:42:44.59$ & $-20.14$ & $1.28$ & $26.1 \pm 9.6 $ & E4\ 43 & IC 1919 & $03:26:02.30$ & $-32:53:36.80$ & $-19.51$ & $1.15$ & $31.7 \pm 10.7$ & dS0/2(5),N\ 190 & 1380B & $03:37:08.86$ & $-35:11:37.54$ & $-19.62$ & $1.37$ & $19.7 \pm 9.4 $ & SB0\ 148 & 1375 & $03:35:16.79$ & $-35:15:55.95$ & $-19.92$ & $1.21$ & $22.2 \pm 10.1$ & S0(cross)\ 335 & – & $03:50:36.64$ & $-35:54:29.27$ & $-18.36$ & $1.13$ & $21.7 \pm 10.2$ & E\ 182 & – & $03:36:54.24$ & $-35:22:22.69$ & $-18.38$ & $1.34$ & $21.0 \pm 10.2$ & S0 pec\ 202 & 1396 & $03:38:06.40$ & $-35:26:17.96$ & $-17.69$ & $1.19$ & $23.8 \pm 9.9 $ & dE6,N\ [llcccccl]{} 881 & 4406 & $12:26:11.74$ & $+12:56:46.4$ & $-23.53$ & $1.57$ & $32.3 \pm 7.79 $ & S0$_1$(3)/E3\ 798 & 4382 & $12:25:24.04$ & $+18:11:25.9$ & $-23.25$ & $1.38$ & $160.2\pm 14.05$ & S0$_1$(3)\ 1535 & 4526 & $12:34:03.10$ & $+07:41:59.0$ & $-22.40$ & – & $93.4 \pm 12.71$ & S0$_3$(6)\ 1903 & 4621 & $12:42:02.40$ & $+11:38:48.0$ & $-22.19$ & $1.53$ & $38.2 \pm 10.62$ & E4\ 1632 & 4552 & $12:35:39.82$ & $+12:33:22.6$ & $-22.33$ & $1.61$ & $19.5 \pm 8.98 $ & S0$_1$(0)\ 1231 & 4473 & $12:29:48.87$ & $+13:25:45.7$ & $-21.58$ & $1.53$ & $23.0 \pm 9.53 $ & E5\ 2095 & 4762 & $12:52:56.00$ & $+11:13:53.0$ & $-20.95$ & $1.44$ & $48.0 \pm 11.01$ & S0$_1$(9)\ 1154 & 4459 & $12:29:00.03$ & $+13:58:42.9$ & $-21.79$ & $1.44$ & $25.5 \pm 9.09 $ & S0$_3$(2)\ 1062 & 4442 & $12:28:03.90$ & $+09:48:14.0$ & $-21.34$ & $1.53$ & $44.7 \pm 11.04$ & SB0$_1$(6)\ 2092 & 4754 & $12:52:17.50$ & $+11:18:50.0$ & $-21.64$ & $1.50$ & $30.0 \pm 10.70$ & SB0$_1$(5)\ 369 & 4267 & $12:19:45.42$ & $+12:47:54.3$ & $-20.41$ & $1.57$ & $37.2 \pm 11.08$ & SB0$_1$\ 759 & 4371 & $12:24:55.50$ & $+11:42:15.0$ & $-21.41$ & $1.54$ & $65.0 \pm 12.22$ & SB0$_2$(r)(3)\ 1030 & 4435 & $12:27:40.49$ & $+13:04:44.2$ & $-21.38$ & – & $47.4 \pm 11.65$ & SB0$_1$(6)\ 1720 & 4578 & $12:37:30.61$ & $+09:33:18.8$ & $-20.68$ & $1.44$ & $37.8 \pm 11.43$ & S0$_{1/2}$(4)\ 355 & 4262 & $12:19:30.61$ & $+14:52:41.4$ & $-20.41$ & $1.52$ & $25.7 \pm 11.68$ & SB0$_{2/3}$\ 1883 & 4612 & $12:41:32.70$ & $+07:18:53.0$ & $-20.73$ & $1.32$ & $28.7 \pm 10.76$ & RSB0$_{1/2}$\ 9 & IC 3019 & $12:09:22.34$ & $+13:59:33.1$ & $-18.80$ & $1.15$ & $59.0 \pm 12.51$ & dE1,N\ 1192 & 4467 & $12:29:30.20$ & $+07:59:34.0$ & $-18.14$ & $1.52$ & $32.7 \pm 11.60$ & E3\ 1199 & IC 3602 & $12:29:34.97$ & $+08:03:31.4$ & $-16.94$ & $1.56$ & $44.4 \pm 12.26$ & E2\ Using such selection criteria, we claim that a galaxy hosts DSCs if the net number of diffuse objects (the number detected in the program field without completeness correction but subtracted by the mean number of contaminants from the 16 control fields) is 3$\sigma$ higher than the mean number of contaminants in the control fields, where the $\sigma$ is the standard deviation of the number counts from the 16 control fields. The upper panel of Figure \[findDSC\] shows that 13 galaxies in our sample have significant number of DSCs. However, because the star cluster system of FCC 202 belongs to the halo of the bright central galaxy (BCG) FCC 213 (NGC 1399), only 12 Fornax galaxies contain DSCs substantially. Basic parameters of these galaxies are listed in Table \[gal\_DSC\]. The errors are estimated as the 1$\sigma$ uncertainty from Poisson distributions, and the standard deviations of contaminants from the 16 control fields are considered. Because a large fraction of the low-mass host galaxies have ambiguous excess, we preform a test with the criteria of $p_{GC} < 0.5$ and $r_h \ge 7$ pc. This is similar to the cut used for “faint fuzzies” in other works (e.g. @Larsen_Brodie_00), which are essentially the same objects as the DSCs we are studying. Under the alternative criteria, the same 13 galaxies are selected out, as well as FCC 177, which is at the boundary of the cut. Therefore, we conclude all the 13 galaxies as DSC hosts in this work. Our selection of DSC host galaxies is different from P06. P06 also defined DSC hosts as 3$\sigma$ higher than the background, but the $\sigma$ was the errors of the number of DSC measurements instead of the scatter of background contaminants. Therefore, we use the new criterion to select DSC hosts from Virgo in this work, and the bottom panel of Figure \[findDSC\] shows that 20 ETGs in Virgo have DSC number excess. However, because the BCG VCC 1316 (M87) only have 3 DSC candidates with an expected background of $0.44\pm0.50$, we remove it from the DSC host galaxies. Besides, similar to the case of FCC 202 in Fornax, the DSC systems of VCC 1192 and VCC 1199 belong to the halo of VCC 1226 (M49), and only 18 ETGs from ACSVCS are real DSC hosts. Parallel to Table \[gal\_DSC\], we list their basic parameters in Table \[galV\_DSC\].\ Galaxies with DSCs {#galaxy} ================== Since not all galaxies contain DSCs, the natural question to ask is how these DSC host galaxies are special. First, in our sample, the DSC hosts include both low-mass and massive ETGs. A large fraction of the massive hosts are S0 galaxies, indicating that disk environment may be important for DSCs. In addition, although some galaxies are classified as elliptical galaxies in @Ferguson_89a, most of them look like containing disks from our images. However, in both clusters, not all disk galaxies have number excess of DSC-like objects. Second, in both clusters, some giant elliptical galaxies are DSC hosts. Furthermore, three low-mass host galaxies, FCC 202 from Fornax, and VCC 1192 and VCC 1199 from Virgo, contain star cluster systems of their nearby massive galaxies NGC 1399 and M49 (VCC 1226) respectively, implying the existence of DSCs in the halos of massive ETGs. Third, the merger remnant in each galaxy cluster (FCC 21 and VCC 798) have the highest number of DSCs in their respective clusters, which indicates that galactic merger is an efficient DSC producer. Last but not least, six DSC host galaxies in Virgo and two in Fornax contain significant amounts of dust, showing a possible relation between DSC detection and recent star formation. Especially, the two dusty hosts in Fornax, FCC 21 and FCC 167 (NGC1380), are the galaxies with the richest DSC systems, and the third richest DSC system NGC 1340 also has wispy dust and shells. Nonetheless, not all dusty galaxies in these two galaxy clusters contain DSCs. Then we investigate whether the internal properties or external environments cause the uniqueness of DSC host galaxies. Figures  \[gal\_CMD\] and \[gal\_posi\] display their positions in the galactic color-magnitude diagram and spatial distributions, but they occupy the same region of parameter space as normal galaxies. Figure \[gal\_CMD\] shows the $g-z$ color vs. $z$-band absolute magnitude of ACSFCS and ACSVCS galaxies. Red and magenta squares and black and grey circles indicate the DSC hosts and non-hosts in the Fornax and Virgo clusters respectively. The DSC systems of the three faintest hosts with crosses belong to the halos of their massive neighbors, and we only focus on the data points without crosses in this plot. DSC host galaxies generally follow the same broad color-magnitude distributions as others. However, there are slight differences between the two clusters. While the hosts in Virgo are mostly massive galaxies and lie on the same relation as the non-hosts, the hosts in Fornax cluster spread over a large mass range and half of them are at the blue edge of the distribution. In addition, nearly all the bluest galaxies at fixed mass in Fornax are associated with DSCs. In Virgo, only two galaxies are significantly bluer than the distribution, and one of them is a low-mass galaxy. One caveat is that the ACSVCS sample is not complete at low-mass ($B_T>12$), and the potential DSC hosts we missed could have special properties. Figure \[gal\_posi\] shows the locations of 43 galaxies in the Fornax Cluster (*top*) and 100 galaxies in the Virgo Cluster (*bottom*). The scales of the Viral radii are displayed at top-left of two panels. The big and small yellow stars are the first and second BCGs in each cluster, and the galaxies with significant number of DSCs in two galaxy clusters (13 in Fornax and 19 in Virgo) are marked with red squares. The three satellite galaxies FCC 202, VCC 1192 and VCC 1199 are marked with crosses. These galaxies distribute evenly across the full range of cluster-centric distances in both clusters. While the DSC hosts distribution in Fornax are concentrated in the central region, unlike Virgo, it may be biased by the more centrally concentrated distribution of all galaxies in Fornax. We performed a K-S test and found that the non-similarity of the cumulative radial distributions of DSC hosts and our entire sample in the Fornax is only 0.21, with the p-value of rejecting a null hypothesis as high as 0.74. We also investigate the global influence from galaxy clusters. Comparing with the Virgo cluster, in which 19 or 18 (when replacing VCC 1192 and VCC 1199 by M49) out of 100 ETGs contain significant number of DSCs, the fraction of such galaxies is slightly higher in the Fornax cluster. However, this might be due to selection effects, as ACSFCS has a more complete sample than ACSVCS. In ACSVCS, 63 low-mass galaxies or S0s with evidence of recent star formation, which are possibly DSC host candidates, are missed. If we only focus on the brightest galaxies ($M_B<-18.94$) which are completed in both the Fornax and Virgo Clusters, the fractions of DSC hosts become 6 out of 9 and 13 or 14 out of 26 respectively, and the difference becomes smaller. The upper and bottom panels of Firgure \[N\_sgm\] display the normalized distributions of $N_{DSC}$ and $N_{DSC}/\sigma$ respectively. The red and blue represent the distributions of the galaxies from ACSFCS and ACSVCS samples. $N_{DSC}$ is the number of DSCs selected from program images subtracted by the mean number of contaminants from 16 control fields, and $\sigma$ is the standard deviations of background galaxies from the 16 control fields. In the upper and bottom panels, the K-S test of the distributions in two galaxy clusters shows high p-value of 0.99 and 0.96 at $\alpha$ of 0.166 and 0.096, indicating high similarity of them. Therefore, it is evidence that the frequency of DSC hosts is independent with the environment of their location.\ DSC and GC {#D_G} ========== In this section, we compare the properties of DSCs with GCs, and investigate how the internal galactic environment relates to DSC formation. Color ----- The even rows of Figure \[histCMD\] displays the color-magnitude diagrams of GC (blue) and DSC (red) candidates in the 13 Fornax galaxies, as well as those of the entire Virgo DSC host galaxies. Grey points are DSC-like contamination from a randomly chosen control field. Above each diagram, we plot the normalized histograms of their $g-z$ color distributions in the same color coding correspondingly, and the dash lines represent the color of their host galaxies. Most DSC systems in Fornax have mean color similar to or slightly redder than the GC’s, but bluer than the field stars of their hosts. Nonetheless, unlike Fornax, the DSCs in Virgo are significantly redder than GCs, and comparable with the field stars. From Figures 7 and 11 in P06, red DSCs in Virgo tend to be associated with galactic disks when dividing DSCs by $g-z=1.0$. However, when we preform the same tests on Fornax galaxies, color separation does not decouple their spatial distributions, even in FCC 335, which has a DSC system redder than GC’s. Spatial Distribution {#distribution} -------------------- If the formation and evolution of DSCs have connections with GCs, spatial association of these two kinds of star clusters is expected. Figure \[denP\] displays their radial number density profiles in 13 Fornax DSC hosts. Blue and red lines represent GCs and DSCs respectively. The density is calculated by the third-nearest neighbor method, with corrections on completeness. Each GC or DSC candidate is corrected by its detection probability, and we also estimate the non-detection fraction of GCs basing on their luminosity functions [@Villegas_10]. The detection probability of each source is a function of three parameters: the apparent magnitude ($m$), the size ($r_h$), and the flux of its local background ($I_b$). The detection probability is tabulated for different values of $m$, $r_h$ and $I_b$ using Monte Carlo simulations with 4,993,501 fake GCs across the full range magnitude, size, and background surface brightness. Specifically, for every DSC in a galaxy, we calculate the density using the third closest neighbor, corrected for detection probability. We then divide radius into 10 bins with equal logarithmic interval and calculate the mean density value in each bin. In the end, they are globally subtracted by the average density of background contaminants derived from control fields, and the data points with density lower than zero are not plotted. Because the detection probability of DSC-like objects is small and varies highly at different galaxy radius with different background brightness, and the number from control fields is not large enough to smear the random effects, we do not apply completeness correction on the control fields, and our contamination correction has no effect on the shape of radial profiles. For comparison, GC number density profiles are derived in the same way, except for the consideration of objects with zero detection probability. To avoid the non-detection, we do not select objects 1$\sigma$ fainter than the peak of GC luminosity function of this galaxy [@Villegas_10], and divided by 0.84 for correction. A special case is FCC 21, which has a significantly fainter GC luminosity distribution. Therefore, we only select the GCs brighter than the peak of its luminosity function, and use a correction factor of 0.5. The density profiles of DSCs are mostly flat and possibly implying disky distributions. Some profiles are slightly increasing towards larger radii, apparently indicating their stronger formation/survival ability in lower density environment. The density profiles of GCs have negative gradients for most galaxies, except for FCC 21, FCC 213 and FCC 202 which have flat profiles similar to DSCs. Especially, FCC 202 is a low-mass satellite galaxy of FCC 213, and the mean densities of GCs and DSCs of FCC 202 can be regarded as the density at the outer halo of FCC 213. Thus for FCC 213, from the central region to the halo as far as FCC 202, the density of DSC remains roughly constant, while that of GC drops significantly, which is similar to most of the others. For most galaxies, the GC number densities in the central regions are higher than that of DSC. However, this is also possible to be purely due to a higher fraction of DSC non-detection in the central and brighter regions, as the difference between GC and DSC densities is smaller in fainter galaxies and at larger radii. In low-mass galaxies FCC 43, FCC 148, FCC 335, and FCC 182, the density of DSCs are comparable or even higher than that of GCs. Therefore, DSCs may be associated with GCs spatially, but we cannot detect the rise of their densities toward bright galactic centers. Formation Efficiency {#CFE} -------------------- Because of the potentially higher non-detection fraction in the central and brighter regions of galaxies, the flat DSC profiles shown in Figure \[denP\] might actually rise in the central region and follow that of GCs. To further investigate the relationship between DSCs and GCs in their formation and evolution, we compare their formation efficiency. Figure \[Ngd\] displays the number ratio between DSCs and GCs within the ACS FOV of 32 host galaxies from Fornax (magenta) and Virgo (cyan). Three squares at low-mass end represents FCC 202, VCC 1192 and VCC 1199, in which the DSC systems may belong to the nearby giant ETGs NGC 1399 and M49, and representing the properties of their outer halos. The numbers of DSCs and GCs are calculated using a similar method to what used for Figure \[denP\] and Section \[distribution\]. These are the sum of all objects corrected by their detection probability and background contaminants, with additional consideration for non-detection for GCs. The left panel of Figure \[Ngd\] shows the relation between the number ratio and total galactic $z$-band absolute magnitude, which represents their total stellar mass and the potential well depth. Over the full mass range, the ratio decreases as the galactic luminosity increases. However, this trend is mainly driven by the low-mass galaxies and the ETGs at massive end. At the low-mass end, except for the three satellite galaxies that represent the halos of massive galaxies, the ratios are systematically higher. For some objects, the ratios are even larger than unity, indicating a more efficient formation for DSCs than GCs. For the intermediate mass galaxies ($-23<M_z<-19.6$), the Pearson correlation coefficient is 0.35, showing a weak correlation. For the most massive galaxies ($M_z<-23$), because of their brighter background luminosity, the ratios of them are expected to be higher than others. Specially, the ratios of three satellites are similar to those of the intermediate mass galaxies, indicating similar formation efficiency at their outer halos. Therefore, there may be connections between the formation and evolution of DSCs and GCs across a wide range in mass and galactic environments. The right panel shows their relation with environmental density. $\Sigma_{15}$ is an indicator of environment, which is defined as the number of galaxy per square degree within a region that includes the 15 closest neighbors [@Guerou_15]. As shown in $\S$ \[galaxy\], there is no dependence with the external environment. Furthermore, we investigate whether the number of GCs is systematically different in DSC host galaxies. Figure \[NgcMz\] shows the relation between the galactic $z$-band absolute magnitude and the number of GCs within their images. The black and grey circles represents the DSC non-hosts in Fornax and Virgo, and the red and magenta squares represents the DSC hosts in these two clusters respectively. The three outliers at low-mass with high numbers are the satellites of the nearby massive ETGs, and the GCs inside belong to their host galaxies NGC 1399 and M49. Except for these three low-mass galaxies, the GC numbers of the DSC hosts and non-hosts at similar magnitude do not show systematical offsets. It implies that the formation of GCs and DSCs are independent and do not have direct effects on each other. Besides, the number of GCs increases with the galactic luminosity, even if only taking into account the GCs within the FOV for those massive galaxies. From the tables and Figure \[findDSC\], the number of DSCs does not have large scatter among the galaxies fainter than $M_z\sim22$, and the variation of the number ratios between DSCs and GCs shown in Figure \[NgcMz\] are mainly driven by the number of GCs.\ Discussion: The Origin of DSCs {#discussion} ============================== Low-density Environment: Formation or Survival? {#low-density} ----------------------------------------------- From literature, DSCs are detected in three kinds of environment: disk (spiral or S0) galaxies, low-mass galaxies, and galactic halos, all of which have relatively low density. In our sample, although some DSC host galaxies are classified as elliptical galaxies from old studies of @Ferguson_89a, they show disk-like structures in the ACS images. Especially, their DSC systems have disk-like distributions. One possible scenario is that the peaks of the formation radius distributions of initially bound star clusters vary with environment, with the clusters being more bound in denser environments [@Elmegreen_08]. During galactic evolution, less-bound star clusters with larger $r_h$ are disrupted in higher density regions [@Gnedin_99], and the low-density bound DSCs are left in the moderately low density environment. Such a picture may explain why some low-density environments are associated with DSCs while others are not. A test for this scenario is to compare the ages of stellar disks in host galaxies with and without DSCs. If DSCs are created in all disk equally at the beginning, but we only detect the ones that have not been disrupted as time passes, then the disks containing DSCs are expected to be younger. Figure \[disk\_age\] presents the ages of the massive galaxies from the ACSVCS sample that overlap with the ATLAS$^{3D}$ sample [@Cappellari_11] and have stellar population measurements from @McDermid_15. Red and blue circles are galaxies with or without significant DSC number excess. The ages (y-axis) are measured within 1 R$_e$, and the $B$-band absolute magnitude (x-axis) is derived from @Mei_07. Except for two galaxies at bright end, DSC hosts and non-hosts have similar ages at similar mass. Therefore, we suggest that the mechanisms by which the DSCs are only detected in a fraction of low-mass environments are are related to their formation instead of survival. Galactic Mergers and DSCs {#merger} ------------------------- In our sample, two merger remnants FCC 21 and VCC 798 are both DSC hosts, indicating that a galactic merger can trigger DSC formation. M51 is another example, which is an interacting system that hosts a number of DSCs. At the same time, however, there are also DSC hosts containing thin disks and X-shape bulges (e.g. FCC 83, FCC 148, and VCC 2095), which are impossible to have experienced merger events. Therefore, DSCs may have multiple origins, either low-density environments or galactic merger events. Alternatively, these two environments might essentially have the same physical conditions for DSC formation. From Figure \[Ngd\], FCC 21 have a similar DSC to GC number ratio to the other host galaxies, which supports this assumption. Although VCC 798 has a lower ratio, this may be due to the higher non-detection fraction. On the other hand, because they are brighter and have higher non-detection fraction than others, their number ratios could be substantially higher. In this case, special DSC formation mechanisms may play a role during galactic mergers. Other Origins {#strip_expanding} ------------- Because DSCs have relatively large sizes and diffuse light distributions, GC expansion and stripping from galaxies are two other candidates of their origin. In the former case, @Assmann_11 tested whether a DSC similar to Scl-dE1 GC1 can form during the early evolution of a normal star cluster through gas expulsion or stellar mass loss. They found that without the embedded dark matter halos, this scenario requires the star formation efficiency of at least 0.33, which is significantly higher than what observed. Alternatively, tidal forces may extend GCs. However, the flat density profiles in Figure \[denP\] indicate no environmental dependence of DSCs inside a galaxy, which does not support this scenario. As for stripping, there are luminous and large star clusters that show evidence of being stripped remnants of larger systems, like UCDs in Virgo (e.g., @Zhang_15 [@Liu_15]). However, this mechanism is not likely for our sample, since the existence of DSCs has no preference to the outer halos of galaxies, nor the denser environment in galaxy clusters.\ Summary {#conclusion} ======= From the images taken by ACSFCS, we find 12 out of 43 ETGs in the Fornax Cluster containing DSC-like objects more than the typical background galaxies at 3$\sigma$ level. The Virgo Cluster is the only other cluster environment with DSC detection. P06 found 12 DSC host galaxies in Virgo using the ACSVCS images of 100 ETGs, and we select out 18 hosts using the same criteria as for Fornax. In this work, we combine these two samples of 143 cluster ETGs and systematically study how the properties of DSCs relate with their host environment and GCs, in order to constrain their formation mechanisms. The main conclusions are listed as follow: - The 30 DSC hosts in our sample consist of low-mass ETGs, S0s, post-starburst merger remnants, as well as elliptical galaxies. Most elliptical galaxies contain potential disk features, except for NGC 1399, the BCGs of the Fornax Cluster. Both galaxy disks and low-mass galaxies have relatively low-density environment, indicating that DSCs can form in merger processes or low-density environments. It is possible that the physical origin of DSCs is essentially the same in these two environments, if merging places also has small tidal field. - A significant fraction of massive DSC host galaxies contain dust or shell-like structures, implying that the DSC formation is related with merger and recent star formation process. - Though all the DSC systems in our sample show flat galactic radial number density profiles and do not follow the distribution of GCs, the potential relations between their formation are shown in their similar color-magnitude distributions and nearly constant number ratios among the massive galaxies. The number ratios in low-mass galaxies are systematically higher, indicating a more efficient formation of DSCs in lower density environment. - No evidence shows that DSC formation has any dependence on the environment of their host galaxy locations inside a galaxy cluster. - In the end, why DSCs are not detected in all disky or low-mass early-type galaxies is still a puzzle. The mean ages of DSC hosts and non-hosts are similar at similar luminosities, suggesting that the reasons lie with formation history, rather than in the survival fraction. YL thanks for the great academic environment at KIAA. She also thanks for the helpful discussions with Long Wang on star cluster dynamics, as well as useful comments from Weijia Sun and Jinyi Shangguan. YL and EWP acknowledge support from the National Natural Science Foundation of China under Grant Nos. 11573002, and from the Strategic Priority Research Program, “The Emergence of Cosmological Structures”, of the Chinese Academy of Sciences, Grant No. XDB09000105.\

--- author: - | \ Institute of Theoretical Physics, University of Wroclaw, 50204 Wrocław, Poland\ E-mail: - | Peter Petreczky\ Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA\ E-mail: title: Hadron gas with repulsive mean field --- Introduction ============ Below the chiral crossover at zero net baryon density QCD thermodynamics can be well approximated by the hadron resonance gas (HRG) model. This model is based on the idea that the effect of interactions in hadron gas can be mimicked by including resonances as additional free particles in hadron gas—an idea, which can be justified using the relativistic virial expansion based on the S-matrix approach [@Dashen:1969ep]. In most interactions, after summation over spin and isospin channels, the second virial coefficient is dominated by the resonance contribution, and thus the interacting gas of hadrons can be approximated as a gas of non-interacting hadrons and hadronic resonances [@Venugopalan:1992hy]. The validity of the HRG model is further corroborated by the equation of state obtained in lattice QCD calculations, which agrees well with the HRG equation of state (see e.g. Refs. [@Borsanyi:2010bp; @Borsanyi:2013bia; @Bazavov:2014pvz; @Bazavov:2017dsy]). The fluctuations and correlations of conserved charges, defined as derivatives of pressure with respect to chemical potentials, have also been studied in the HRG model (see e.g. Ref. [@Huovinen:2009yb]). However, the applicability of HRG must have its limits. The cancellation of non-resonant contributions to thermodynamics is somewhat accidental [@Fernandez-Ramirez:2018vzu] and does not happen for all thermodynamic quantities [@Lo:2017lym]. There are also hadronic interactions which are not dominated by resonances, e.g. there are no resonances in nucleon-nucleon scatterings. Last but not least, the HRG model is based on free particle properties, but close to the chiral transition there are in-medium modifications to hadron properties [@Kelly:2018hsi; @Aarts:2017rrl]. Therefore, it is important to check the HRG model by comparing it to recent lattice QCD calculations of different thermodynamic quantities. It is known that for higher order ($>2$) fluctuations the agreement between lattice and HRG is not good for temperatures close to the chiral transition. It has been argued that this is not a signal of approaching transition, but rather due to repulsive baryon-baryon interactions which the HRG model does not describe [@Albright:2015uua; @Vovchenko:2016rkn; @Vovchenko:2017zpj]. If that is the case, the repulsive baryon-baryon interactions must be included in the model when HRG is extended to non-zero baryon densities, since the larger the baryon density, the larger the effect of repulsive interactions. As a step towards finding a convenient description of baryon repulsion, we study the trace anomaly and fluctuations of baryon number and strangeness using HRG with repulsive mean field. Nucleon gas with repulsive mean field and virial expansion ========================================================== According to the virial expansion, pressure of the interacting nucleon gas can be written as [@Huovinen:2017ogf] $$p(T,\mu)=p_0(T)\cosh(\beta \mu)+2 b_2(T) T \cosh(2 \beta \mu),~\beta=1/T,$$ where $$p_0(T)=\frac{4 M^2 T^2}{\pi^2} K_2(\beta M)$$ is the pressure of free nucleon gas at zero chemical potential, and $b_2(T)$ is the second virial coefficient. The second virial coefficient can be written as $$b_2(T) = \frac{2 T}{\pi^3} \int_0^{\infty} dE (\frac{ME}{2}+M^2) K_2\left(2 \beta \sqrt{\frac{M E}{2}+M^2}\right) \frac{1}{4 i}{\rm Tr} \left[ S^{\dagger}\frac{dS}{dE}-\frac{dS^{\dagger}}{dE} S \right],$$ where $S$ is the scattering S-matrix and $E$ the kinetic energy in the lab frame. Furthermore, $M$ is the nucleon mass and $K_2(x)$ is the Bessel function of second kind. The virial coefficient $b_2(T)$ can be evaluated using the experimentally measured phase shifts to parametrise the $S$-matrix, and it turns out that $b_2(T)$ is negative [@Huovinen:2017ogf], see Fig. \[fig:b2\]. For the comparison with the mean-field approach it is convenient to write the pressure as $$p(T,\mu)=p_0(T) ( \cosh(\beta \mu)+\bar b_2(T) K_2(\beta M) \cosh(2 \beta \mu) ), \label{red_vir}$$ where $$\bar b_2(T)=\frac{2 T b_2(T)}{p_0(T) K_2(\beta M)}.$$ is the reduced virial coefficient. In the mean-field approach pressure can be written in Boltzmann approximation as [@Huovinen:2017ogf] $$p(T,\mu)=T(n_b+\bar n_b)+\frac{K}{2} (n_b^2+\bar n_b^2),$$ where $n_b$ and $\bar n_b$ are the densities of nucleons and anti-nucleons, respectively, defined by the following self-consistent relations: $$n_b=4 \int \frac{d^3 p}{(2 \pi)^3} e^{-\beta(E_p-\mu+U)},~~ \bar n_b=4 \int \frac{d^3 p}{(2 \pi)^3} e^{-\beta(E_p+\mu+\bar U)},~E_p^2=p^2+M^2. \label{densities}$$ Here $U=K n_b$ and $\bar U=K \bar n_b$ are the mean-field potentials for nucleons and anti-nucleons. The chemical potential corresponding to the net nucleon density is denoted by $\mu$. This form of the pressure ensures thermodynamic consistency, i.e. $\partial p/\partial \mu=n_b-\bar n_b$. Since we are mostly interested in the region of not too high baryon densities we expand the above expressions in $\beta U=\beta K n_b$ and $\bar U=K \bar n_b$, and keep only the leading order terms in $K$. This simplifies the pressure to $$p(T,\mu) = T(n_b^0+\bar n_b^0)-\frac{K}{2} \left(\left(n_b^0\right)^2 +\left(\bar n_b^0\right)^2 \right), \label{expand}$$ where $n_b^0$ ($\bar n_b^0$) is the free nucleon (anti-nucleon) density. Equation (\[expand\]) can also be written as $$p(T,\mu) = \frac{4 T^2 M^2}{\pi^2} K_2(\beta M) \cosh(\beta \mu) -4 K \frac{T^2 M^4}{\pi^4} K_2^2(\beta M)\cosh(2 \beta \mu),$$ which is very similar to the virial expansion pressure Eq. (\[red\_vir\]). By comparing these two results, one can determine the value of the mean field coefficient $K$ in a limited temperature range. Since $\bar b_2(T)$ turns out to be negative, $K>0$, and the mean field is repulsive. The temperature dependence of $\bar b_2$ is shown in Fig. \[fig:b2\] and it turns out to be relatively mild except maybe at the highest temperature. The value of $\bar b_2$ is consistent with $K=250\,{\rm MeV/fm}^3$. The largest value allowed for $K$ by the virial expansion is around $K=450\,{\rm MeV/fm}^3$. ![The difference between fourth and second order baryon number fluctuations (squares) and the the sixth and second order baryon number fluctuations (triangles). The dashed lines correspond to the exact mean-field calculations, while the solid line to the expanded mean field (see text). The filled symbols correspond to lattice results of Ref. [@Bazavov:2017dus]. The open symbols are the lattice data for $\chi_4^B$ from Ref. [@Borsanyi:2014ewa] and $\chi_6^B$ from Ref. [@DElia:2016jqh], respectively.[]{data-label="fig:old"}](2b2.pdf){width="75mm"} ![The difference between fourth and second order baryon number fluctuations (squares) and the the sixth and second order baryon number fluctuations (triangles). The dashed lines correspond to the exact mean-field calculations, while the solid line to the expanded mean field (see text). The filled symbols correspond to lattice results of Ref. [@Bazavov:2017dus]. The open symbols are the lattice data for $\chi_4^B$ from Ref. [@Borsanyi:2014ewa] and $\chi_6^B$ from Ref. [@DElia:2016jqh], respectively.[]{data-label="fig:old"}](meanfield.pdf){width="85mm"} It is straightforward to generalise the mean-field approach to a multicomponent system if one assumes that the repulsive mean-field is the same for all ground state baryons [@Huovinen:2017ogf], and the baryon resonances are not affected by the mean field. Within this approach we calculated baryon number fluctuations defined as derivatives of the pressure with respect to baryon chemical potential [@Huovinen:2017ogf] $$\chi_n^B = T^n \frac{\partial^n p/T^4}{\partial \mu_B^n}.$$ In Fig. \[fig:old\] we show the differences of the baryon number fluctuations, $\chi_4^B-\chi_2^B$ and $\chi_6^B-\chi_2^B$ obtained in HRG model with the mean field and the value of $K=450\,{\rm MeV/fm}^3$. As seen, the model can qualitatively explain these differences, but at quantitative level it underpredicts the lattice data even if the value of $K$ is the largest the experimental data allows. We note that at temperatures above $150$ MeV the expanded mean-field expression (\[expand\]) is no longer accurate and one should use the exact mean-field expressions with particle densities determined self-consistently, Eq. (\[densities\]) [@Huovinen:2017ogf]. The exact mean-field results are shown as dashed lines in the figure. Comparison with lattice QCD =========================== After fixing the value of the mean field parameter, we compare the HRG calculations with repulsive mean field with lattice QCD results on the trace anomaly, fluctuations of baryon number up to sixth order, and second order strangeness fluctuations. First we note that there are many baryon resonances which are not experimentally observed but predicted by quark models and lattice QCD, so-called missing states. It was found that inclusion of these states in the HRG description improves the description of the strangeness fluctuations and baryon-strangeness correlation [@Bazavov:2014xya]. Therefore, in our analysis we will include the missing baryons from the quark model calculations of Refs. [@Loring:2001kx; @Loring:2001ky], missing mesons from Ref. [@Ebert:2009ub], and label the corresponding results as QM-HRG, whereas the HRG based on the Particle Data Group’s (PDG) resonance list [@PDG] is labeled PDG-HRG. The difference in the mass spectrum of these two approaches is depicted in Fig. \[fig:states\]. In our previous analysis we assumed that the baryon resonances are not affected by the mean field [@Huovinen:2017ogf]. This is not realistic because if the density of ground state baryons is reduced there will be also fewer baryon resonances in the system. Therefore, in the present analysis we include the effect of the repulsive mean field on baryon resonances as well, but assume that baryon resonances do no contribute to the strength of the mean field[^1]. We also use a more realistic value for the mean-field parameter, $K=250\,{\rm MeV/fm}^3$, and show the results for an exact mean-field calculation only. In Fig. \[fig:trace\] we show our calculations for the trace anomaly, $\epsilon-3P$, compared with the lattice results [@Borsanyi:2013bia; @Bazavov:2014pvz] at zero baryon chemical potential, $\mu_B=0$. As expected the missing states have a clear effect on the trace anomaly improving the fit to the lattice data: These states lead to significant increase of the trace anomaly around temperatures of $150$ MeV and higher compared to the HRG calculations with only PDG states (PDG-HRG). The HRG calculations with repulsive mean field are shown as dashed lines. The effect of the repulsive interactions turns out to be relatively small, because the contribution of baryons to the equation of state is small too. The situation will be different at sufficiently large $\mu_B$, when the contributions of mesons and baryons to the equation of state are comparable. ![The trace anomaly in HRG model with the Particle Data Group (PDG-HRG) and Quark Model (QM-HRG) lists of resonances compared to the lattice results [@Borsanyi:2013bia; @Bazavov:2014pvz]. The dashed lines correspond to HRG model with repulsive mean field.[]{data-label="fig:trace"}](spektri3.pdf){width="75mm"} ![The trace anomaly in HRG model with the Particle Data Group (PDG-HRG) and Quark Model (QM-HRG) lists of resonances compared to the lattice results [@Borsanyi:2013bia; @Bazavov:2014pvz]. The dashed lines correspond to HRG model with repulsive mean field.[]{data-label="fig:trace"}](trace.pdf){width="80mm"} In the left panel of Fig. \[fig:chi2B\] we show our results for second order baryon number fluctuations, again for QM-HRG and PDG-HRG with and without the effect of the repulsive mean field. The effect of the missing states is clearly visible and improves the fit to lattice data around 140–150 MeV temperature, but overshoots the lattice results at higher temperatures. The repulsive mean field has an opposite effect and is comparable in size. Therefore, it restores the agreement with the lattice data. The second order strangeness fluctuations are shown in the right panel of Fig. \[fig:chi2B\]. The need for more resonance states to fit the data is now even clearer than in the case of baryon number fluctuations, and even the full quark model spectrum of states hardly reaches the lattice data. The mean field, on the other hand, affects the strangeness fluctuations less, since they are dominated by kaons and other strange mesons which are not affected by baryonic interactions. ![The second order baryon number (left) and strangeness (right) fluctuations in HRG model with the Particle Data Group (PDG-HRG) and Quark Model (QM-HRG) lists of resonances compared to the lattice results [@Borsanyi:2014ewa; @Bazavov:2017dus]. The dashed lines correspond to HRG model with repulsive mean field.[]{data-label="fig:chi2B"}](chi2B.pdf "fig:"){width="8cm"} ![The second order baryon number (left) and strangeness (right) fluctuations in HRG model with the Particle Data Group (PDG-HRG) and Quark Model (QM-HRG) lists of resonances compared to the lattice results [@Borsanyi:2014ewa; @Bazavov:2017dus]. The dashed lines correspond to HRG model with repulsive mean field.[]{data-label="fig:chi2B"}](chi2S.pdf "fig:"){width="8cm"} ![The fourth order (left) and sixth order (right) baryon number fluctuations in HRG-QM and HRG-PDG models. The lattice results for $\chi_4^B$ are from Refs. [@Borsanyi:2014ewa; @Bazavov:2017dus], while the lattice results for $\chi_6^B$ are from [@DElia:2016jqh; @Bazavov:2017dus]. The dashed lines correspond to HRG model with repulsive mean field.[]{data-label="fig:chi46B"}](chi4B.pdf "fig:"){width="8cm"} ![The fourth order (left) and sixth order (right) baryon number fluctuations in HRG-QM and HRG-PDG models. The lattice results for $\chi_4^B$ are from Refs. [@Borsanyi:2014ewa; @Bazavov:2017dus], while the lattice results for $\chi_6^B$ are from [@DElia:2016jqh; @Bazavov:2017dus]. The dashed lines correspond to HRG model with repulsive mean field.[]{data-label="fig:chi46B"}](chi6B.pdf "fig:"){width="8cm"} Higher order baryon number fluctuations are more sensitive to the effects of the repulsive interactions than the second order fluctuations as shown in Fig. \[fig:chi46B\]. The QM-HRG model again overpredicts the lattice results and lies significantly above the PDG-HRG result. The repulsive interactions reduce the HRG prediction and lead to reasonable agreement with the lattice data. Note that the effects of the repulsive mean field are now larger than in the previous analysis shown in Fig. \[fig:old\] even though we use smaller value of $K$. This is because the resonances are also affected by the repulsive mean field. Note also that when the effect of the mean field is included, the sixth order baryon number fluctuation is almost independent of the mass spectrum of resonances. Conclusion ========== We studied the equation of state and fluctuations of baryon number and strangeness within the HRG framework, where the effect of the repulsive baryon-baryon interactions are included using the mean-field approach. We also studied the effects of missing resonances on these quantities. We have found that the missing states lead to significant increase of thermodynamic quantities. In the case of baryon number fluctuations missing states cause the HRG model to overshoot the lattice results for $T \ge 150$ MeV. The repulsive mean field has the opposite effect. The effects of repulsive interactions are small for the trace anomaly and strangeness fluctuations, but are significant for baryon number fluctuations, where they are needed to bring the HRG calculations in agreement with the lattice results. This implies that when extending the HRG model to calculate the equation of state at non-zero baryon density the repulsive interactions have to be taken into account. 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--- abstract: 'After introducing the Szekeres and Lemaître–Tolman cosmological models, the real-time cosmology program is briefly mentioned. Then, a few widespread misconceptions about the cosmological models are pointed out and corrected. Investigation of null geodesic equations in the Szekeres models shows that observers in favourable positions would see galaxies drift across the sky at a rate of up to $10^{-6}$ arc seconds per year. Such a drift would be possible to measure using devices that are under construction; the required time of monitoring would be $\approx10$ years. This effect is zero in the FLRW models, so it provides a measure of inhomogeneity of the Universe. In the Szekeres models, the condition for zero drift is zero shear. But in the shearfree normal models, the condition for zero drift is that, in the comoving coordinates, the time dependence of the metric completely factors out.' author: - 'Andrzej Krasiński$^*$' - 'Krzysztof Bolejko$^*$' title: Exact inhomogeneous models and the drift of light rays induced by nonsymmetric flow of the cosmic medium --- Inhomogeneous models in astrophysics$^1$ ======================================== \[introduction\] Just as was the case with our earlier reviews [@Kras1997; @BKHC2010; @BCKr2011], we define inhomogeneous cosmological models as those exact solutions of Einstein’s equations that contain at least a subclass of nonvacuum and nonstatic Friedmann – Lemaître – Robertson – Walker (FLRW) solutions as a limit. The reason for this choice is that such FLRW models are generally considered to be a good first approximation to a description of our real Universe, so it makes sense to consider only those other models that have a chance to be a still better approximation. Models that do not include an FLRW limit would not easily fulfil this condition. This is a live topic, with new contributions appearing frequently, so any review intended to be complete would become obsolete rather soon. Ref. 1 is complete until 1994, Ref. 2 is a selective update until 2009 and Ref. 3 is a more selective update until the end of 2010. The criterion of the selection in the updates was the usefulness of the chosen papers for solving problems of observational cosmology. In the very brief overview given here we put emphasis on pointing out and correcting the erroneous results that exist in the literature and are being taken as proven truths. Some of them have evolved to become research paradigms, with many followers, some others proceed in this direction. We hope to stop this process, which disturbs and slows down the recognition of the inhomogeneous models as useful devices for understanding the observed Universe. We first present the two classes of inhomogeneous models that, so far, proved most fruitful in their astrophysical application: the Szekeres model [@Szek1975], and its spherically symmetric limit, the Lemaître [@Lema1933] – Tolman [@Tolm1934] (L–T) model. Then we briefly mention the real-time cosmology program, and we give an overview of the erroneous ideas. Finally, we present an effect newly calculated in a few classes of models that is possible to observe and can become a test of homogeneity of the Universe: the drift of light rays induced by nonsymmetric flow of the cosmic medium The Szekeres solution {#Szek} ===================== The (quasi-spherical) Szekeres solution [@Szek1975; @PlKr2006] is, in comoving coordinates $${\rm d} s^2 = {\rm d} t^2 - \frac {{\cal E}^2 {(\Phi / {\cal E}),_r}^2} {1 + 2E(r)} {\rm d} r^2 - \frac {\Phi^2} {{\cal E}^2} \left({\rm d} x^2 + {\rm d} y^2\right),$$ $${\cal E} {\ {\overset {\rm def} =}\ }\frac {(x - {P})^2} {2{S}} + \frac {(y - {Q})^2} {2{S}} + \frac {{S}} 2,\label{2.1}$$ where $E(r)$, $M(r)$, ${P}{(r)}$, ${Q}{(r)}$ and ${S}{(r)}$ are arbitrary functions and $\Phi(t,r)$ obeys $$\label{2.2} {\Phi,_t}^2 = 2E(r) + \frac {2 {M}{(r)}} {\Phi} + \frac 1 3 \Lambda \Phi^2.$$ The source in the Einstein equations is dust, whose mass density in energy units is $$\label{2.3} \kappa \rho = \frac {2 \left(M / {\cal E}^3\right),_r} {(\Phi / {\cal E})^2 \left(\Phi / {\cal E}\right),_r}.$$ Eq. (\[2.2\]) implies that the bang time is in general position-dependent: $$\label{2.4} \int\limits_0^{\Phi}\frac{{\rm d} \widetilde{\Phi}}{\sqrt{2E + 2M / \widetilde{\Phi} + \frac 1 3 \Lambda \widetilde{\Phi}^2}} = t - {t_B(r)}.$$ The general Szekeres metric has no symmetry. It contains the spherically symmetric Lemaître [@Lema1933] – Tolman [@Tolm1934] (L–T) model as the limit of $(P, Q, S)$ being all constant. The latter is usually used with a different parametrisation of the spheres of constant $(t, r)$, namely $$\label{2.5} {\rm d}s^2 = {\rm d}t^2 - \frac{R,_r^2}{1 + 2E} {\rm d}r^2 - R^2(t,r) \left({\rm d}\vartheta^2 + \sin^2 \vartheta {\rm d}\varphi^2 \right),$$ where $R \equiv \Phi$, still obeying (\[2.2\]), and (\[2.3\]) simplifies to $$\label{2.6} \kappa \rho = \frac {2 M,_r} {R^2 R,_r}.$$ The Friedmann limit follows when, in addition, $\Phi (t,r) = r S(t)$, $2E = - k r^2$ where $k =$ const is the FLRW curvature index, and $t_B$ is constant. Real-time cosmology =================== There are ways in which the expansion of the Universe might be directly observed. The authors of Refs. [@QQAm2009; @QABC2012] composed them into a paradigm termed [***real-time cosmology***]{}. One of them is the [***redshift drift***]{}: the change of redshift with time for a fixed light source, induced by the expansion of the Universe. Consider, as an example, an L–T model with $\Lambda$, given by (\[2.5\]) with $R$ obeying (\[2.2\]). Along a single radial null geodesic, directed toward the observer, $t = T(t_o,r)$ (where $t_o$ is the instant of observation) the redshift is [@Bond1947] $$\label{3.1} 1 + z(t_o,r) = {\rm exp} \left[\int_{r_{\rm em}}^{r_{\rm obs}} \frac {R,_{tr}(T(t_o,r), r)} {\sqrt{1 + 2E(r)}} {\rm d} r\right].$$ For any fixed source (i.e. constant $r$), eq. (\[2.2\]) defines a different expansion velocity $R,_t$ for $\Lambda = 0$ and for $\Lambda \neq 0$, and thus allows us to calculate the contribution of $\Lambda$ to $z$ via (\[3.1\]). With the evolution type known, the expansion velocity depends on $r$, and so allows us to infer the distribution of mass along the past light cone. According to the authors of [@QQAm2009; @QABC2012], the “European Extremely Large Telescope” (E-ELT)[^1] could detect the redshift drift during less than 10 years of monitoring a given light source. The Gaia observatory[^2] could achieve this during about 30 years. For more on redshift drift see the contribution by P. Mishra, M.-N. Célérier and T. Singh in these Proceedings. The drift of light rays described in Sec. \[Szredshift\] and following is another real-time cosmology effect. Erroneous ideas and paradigms {#errideas} ============================= The L–T model was noticed in the astrophysics community – but: 1\. Many astrophysicists treat it as an enemy to kill rather than as a useful new device. (Citation from Ref. [@QABC2012]: The Gaia or E-ELT projects could distinguish FLRW from L–T “possibly eliminating an [*exotic alternative explanation to dark energy*]{}”). 2\. Some astrophysicists practise a loose approach to mathematics. An extreme example is to take for granted every equation found in any paper, without attention being paid to the assumptions under which it was derived. Papers written in such a style planted errors in the literature, which then came to be taken as established facts. In this section a few characteristic errors are presented (marked by [**[$\bullet$]{}**]{}) together with their explanations (marked by [**[$*$]{}**]{}). [**[$\bullet$]{}**]{} The accelerating expansion of the Universe is an observationally established fact (many refs., the Nobel Committee among them). [**[$*$]{}**]{} The established fact is the [*smaller than expected observed luminosity of the SNIa supernovae*]{} (but even this is obtained assuming that FLRW is the right cosmological model). [***The accelerating expansion is an element of theoretical explanation of this observation.***]{} When the SNIa observations are interpreted against the background of a suitably adjusted L–T model, they can be explained by matter inhomogeneities along the line of sight, with decelerating expansion [@KHBC2010; @INNa2002]. [**[$\bullet$]{}**]{} Positive sign of the redshift drift is a direct confirmation of accelerated expansion of the space. [**[$*$]{}**]{} The sign of redshift drift is only related to acceleration when homogeneity is assumed[@YKN2011] (see also Mishra, Célérier, and Singh in these Proceedings). [**[$\bullet$]{}**]{} $H_0$, supernovae and the cosmic microwave background radiation (i.e. the size of the sound horizon and the location of the acoustic peaks) are sufficient to rule out inhomogeneous L–T models. [**[$*$]{}**]{} These observations only depend on $D(z)$ and $\rho(z)$, and thus can be accommodated by the L–T model, which is specified by 2 arbitrary functions. Examples of such constructions are given in [@CBKr2010; @INNa2002]. In fact, as follows from the Sachs equations, in the approximation of small null shear (which for most L–T models works quite well[@BoFe2012]), there is a relation between $D(z)$, $\rho(z)$ and $H(z)$, meaning that these 3 observables are not independent[@BoFe2012], and thus allow the L–T model to accommodate more data on the past null cone. [**[$\bullet$]{}**]{} The gravitational potential of a typical structure in the Universe is of the order of $10^{-5}$ and thus, by writing the metric in the conformal Newtonian gauge, one immediately shows that inhomogeneities can only introduce minute ($\approx 10^{-5}$) deviations from the RW geometry. This also means that the evolution of the Universe must be Friedmannian (common argument among cosmologists). [**[$*$]{}**]{} Even if the gravitational potential remains small, its spatial derivatives do not, and thus the model has completely different optical properties than the Friedmann models. This was shown by rewriting one of the L–T Gpc-scale inhomogeneous models in the conformal Newtonian coordinates [@EMR2009]. The gravitational potential of this model remains small yet the distance–redshift relation deviates strongly from that for the background model. The question remains whether small-scale fluctuations (of the order of tens of Mpc) could also modify optical properties and evolution of the Universe. The problem is complicated as it requires solving the Einstein equations and null geodesics for a general matter distribution, which, with current technology, is not possible to do numerically. Therefore, the problem has been addressed in a number of approximations and toy models. Recent studies showed that the optical properties along a single line of sight can be significantly different than in the Friedmann model. Yet, if averaged over all directions, the average distance–redshift relation closely follows that of the model, which describes the evolution of the average density and expansion rate[@BoFe2012]. Thus, the problem reduces to the following one: do the average density and expansion rate follow the evolution of the homogeneous model, i.e. is the evolution of the background affected by small-scale inhomogeneities and does it deviate from the Friedmannian evolution? Some authors claim that matter inhomogeneities cannot affect the background and that the Universe must have Friedmannian properties[@IsWa2006; @GrWa2011], while others argue for strong deviation from the Friedmannian evolution[@Wilt2011; @RBCO2011]. Studies of this problem within the exact models, like L–T, proved that under certain conditions the back-reaction can be large, while under others it remains quite small [@Suss2011], leaving the problem unsolved. For an informative description of the problem and techniques used to address it see Ref. [@BuRa2012; @CELU2011]. [**[$\bullet$]{}**]{} Fitting an L–T model to number counts or the $D_L(z)$ relation results in predicting a huge void, several hundred Mpc in radius, around the centre (too many papers to be cited, literature still growing). Measurements of the dipole component of the CMB radiation then imply that our Galaxy should be very close to the center of this void, which contradicts the “cosmological principle”. [**[$*$]{}**]{} , for example constant $t_B$. When the model is employed at full generality, the giant void is not implied [@CBKr2010]. [**[$*$]{}**]{} , it cannot say which model is “right” and which is “wrong”. [**[$\bullet$]{}**]{} The bang time function must be constant, otherwise the decaying mode of density perturbation is nonzero, which implies large inhomogeneities in the early universe (see, for example, Ref. [@ZMSc2008]). [**[$*$]{}**]{} The bang time function describes the differences in the age between different regions of the Universe. In the L–T and Szekeres models it is also related to the amplitude of the decaying mode. However, these models describe the evolution of dust and therefore cannot be extended to times before the recombination, when the Universe was in a turbulent state: rotation, plasma, pressure gradients all did affect the proper time of an observer (${\rm d} \tau = {\rm d} t~ \sqrt{g_{00} (t,x^i)}$). Eventually, even if the Universe started with a simultaneous big bang, by the time of recombination, due to the standard physical processes, the age of the Universe would have been different at different spatial positions, giving rise to non-constant $t_B(r)$ of the dust L–T or Szekeres model that takes over there. Moreover, the relation between the nonsimultaneous big bang and the decaying mode was established only for the L–T [@Silk1977; @PlKr2006] and Szekeres [@GoWa1982] models. For more general models, not yet explicitly known as solutions of Einstein’s equations, like the ones mentioned above, the connection may be more complicated and indirect. Thus, citing this relation for such a general situation is an illegitimate stretching of a theorem beyond the domain of its assumptions (see also the next entry below). [**[$\bullet$]{}**]{} The L–T models used to explain away dark energy must have their bang-time function constant, or else they “can be ruled out on the basis of the expected cosmic microwave background spectral distortion” [@Zibi2011]. [**[$*$]{}**]{} . This sets them on the wrong track from the beginning. [**[$*$]{}**]{} , [***one must apply it at every step of analysis***]{} of the observational data. To do so, would require a re-analysis of a huge pool of data. C. Hellaby with coworkers [@McHe2008] is working on such a program applied to the L–T model, but the work is far from being completed. Lacking any better chance, we currently use observations interpreted in the FLRW framework to infer about the $M(r)$ and $t_B(r)$ functions in the L–T model. This is justified as long as we intend to point out possibilities, under the tacit assumption that these results will be verified in the future within a complete revision of the observational material on the basis of the L–T model. However, putting “precise” bounds on the L–T model functions using the self-inconsistent mixture of FLRW/L–T data available today is a self-delusion. An example: the spatial distribution of galaxies and voids is inferred from the luminosity distance vs. redshift relation that applies [*only*]{} in the FLRW models. Without assuming the FLRW background, we know nothing about this distribution until we reconstruct it using the L–T model from the beginning. The L–T and Szekeres models cannot be treated as exact models of the Universe, to be taken literally in all their aspects. They are [*exact as solutions of Einstein’s equations*]{}, but when applied in cosmology, they are merely [*the next step of approximation after FLRW*]{}. If the FLRW approximation is good for some purposes, then a more detailed model, [*when applied in a situation, in which its assumptions are fulfilled*]{}, can only be better. The redshift equations in the Szekeres models {#Szredshift} ============================================= Consider two light rays, the second one following the first after a short time-interval $\tau$, both emitted by the same source and arriving at the same observer. The trajectory of the first ray is given by $$\label{5.1} (t, x, y) = (T(r), X(r), Y(r)),$$ the corresponding equation for the second ray is $$\label{5.2} (t, x, y) = (T(r) + \tau(r), X(r) + \zeta(r), Y(r) + \psi(r)).$$ This means that while the first ray intersects a hypersurface $r = r_0$ at $(t, x, y) = (T, X, Y)$, the second ray intersects the same hypersurface not only later, but, in general, at a different comoving location. [***$\Longrightarrow$ In general the two rays will intersect different sequences of intermediate matter worldlines***]{}. The same is true for nonradial rays in the L–T model. Consequently, the second ray is emitted in a different direction and is received from a different direction by the observer. Thus, a typical observer in a Szekeres spacetime should see each light source slowly [***drift across the sky***]{}. How slowly will be estimated further on. As will be seen from the following, [***the absence of this drift is a property of exceptionally simple geometries***]{} (or exceptional directions in more general geometries). We assume that $(\zeta, \psi)$ and $({{{\rm d} {}} / {{\rm d} {r}}}) (\tau, \zeta, \psi)$ are small of the same order as $\tau$, so we neglect all terms nonlinear in any of them and terms involving their products. For any function $f(t, r, x, y)$ the symbol $\Delta f$ will denote $$\label{5.3} f(t + \tau, r, x + \zeta, y + \psi) - f(t, r, x, y)$$ [***linearized in $(\tau, \zeta, \psi)$.***]{} Note: [***the difference is taken at the same value of $r$***]{}. Applying $\Delta$ to the null geodesic equations parametrised by $r$ we obtain the equations of propagation of $(\tau, \zeta, \psi)$ and $(\xi, \eta) {\ {\overset {\rm def} =}\ }({{{\rm d} {}} / {{\rm d} {r}}})$ $(\zeta, \psi)$ along a null geodesic – see both sets of equations fully displayed in Ref. [@KrBo2011]. Repeatable light paths {#repeat} ====================== There will be no drift when, for a given source–observer pair, each light ray will proceed through the same intermediate sequence of matter world lines. Rays having this property will be called [***repeatable light paths (RLP)***]{}. For a RLP we have $$\label{6.1} \zeta = \psi = \xi = \eta = 0$$ all along the ray. The equations of propagation of $(\tau, \zeta, \psi, \xi, \eta)$ become then overdetermined (3 equations to determine the propagation of $\tau$ along a null geodesic), and imply limitations on the metric components. They can be used in 2 ways: 1\. As the condition (on the metric) for [***all***]{} null geodesics to be RLPs. 2\. As the conditions under which special null geodesics are RLPs in subcases of the Szekeres spacetime. In the first interpretation, the equations of propagation should be identities in the components of ${{{\rm d} {x^{\alpha}}} / {{\rm d} {r}}}$, and this happens when $$\label{6.2} \Psi {\ {\overset {\rm def} =}\ }\Phi,_{tr} - \Phi,_t \Phi,_r/\Phi = 0.$$ This means zero shear, i.e. the Friedmann limit. Thus, we have the following [**Corollary:**]{} [***The only spacetimes in the Szekeres family in which [all]{} null geodesics have repeatable paths are the Friedmann models.***]{} In fact, we proved something stronger. Since the drift vanishes in the Friedmann models, we have: [**Corollary 2:**]{} [***The presence of the drift would be an observational evidence for the Universe to be inhomogeneous on large scales.***]{} In the second interpretation, there are only 2 nontrivial (i.e. non-Friedmannian) cases: A. When the Szekeres spacetime is axially symmetric ($P$ and $Q$ are constant). In this case, the RLPs are those null geodesics that stay on the axis of symmetry in each 3-space of constant $t$. B. When the Szekeres spacetime is spherically symmetric ($P, Q, S$ are all constant) – then it reduces to the L–T model. In this case, the radial null geodesics are [*the only*]{} RLPs that exist. A formal proof of this statement is highly complicated [@KrBo2011]. For non-radial rays in the L–T model, the non-RLP phenomenon was predicted, by a different method (and under the name of [***cosmic parallax***]{}), by Quercellini [*et al.*]{} [@QQAm2009; @QABC2012]. Their review contains a broad presentation of the real-time cosmology paradigm, oriented toward observational possibilities.  Examples of non-RLPs in the L–T model {#numex} ===================================== The examples will show the non-RLP effect for nonradial null geodesics in two configurations of the L–T model, shown in Fig. \[densprof\], for different positions of the observers with respect to the center of symmetry. In Example 1 (see Fig. \[example1\]) we use Profile 1, the observer and the light source at 3.5 Gpc from the center of the void, the directions to them at the angle 1.8 rad, and [@BoWy2008] $t_B =0$ – simultaneous Big Bang, $\rho(t_0,r) = \rho_0 \left[ 1 + \delta - \delta \exp \left( - {r^2}/{\sigma^2} \right) \right]$ – the density profile at the current instant, $r {\ {\overset {\rm def} =}\ }R(t_0,r)$ – the radial coordinate, $\rho_0 {\ {\overset {\rm def} =}\ }\rho(t_0,0) = 0.3 \times (3H_0^2)/(8 \pi G) \equiv 0.3 \rho_{\rm critical}$ – the present density at the center of the void, $H_0 = 72$ km s$^{-1}$ Mpc$^{-1}$ – the present value of the Hubble parameter, $\delta = 4.05$, $\sigma = 2.96$ Gpc. The source in Fig. \[example1\] sends three light rays to the same observer, the first of which was received by the observer $5 \times 10^9$ years ago, the second is being received right now, and the third one will be received $5 \times 10^9$ years in the future. Fig. \[fig1\] shows these rays projected on the space $t =$ now along the flow lines of the L–T dust. The source is at the upper right corner of the graph and the observer is at the upper left corner. ![The configuration of the light source and the observer in example 1.[]{data-label="example1"}](example1.ps)  The [***time-averaged***]{} rate of change of the position of the source in the sky, seen by the observer is $$\begin{aligned} \label{7.1} && \dot{\gamma} = \frac {\rm angle\ between\ the\ earlier\ and\ the\ later\ ray} {\rm time\ interval = 5 \times 10^9\ years} \nonumber \\ && \sim 10^{-7}\ \frac {\rm arc sec} {\rm year}.\end{aligned}$$ The rate of drift in the next figures is calculated in the same way. In Examples 2, 3 and 4 (Fig. \[fig2\]), the observer O is at $R_0$ from the center; the angle between the direction toward the galaxy ([$_*$]{}) and toward the origin is $\gamma$. For each $\gamma$ we calculated the rate of change $\dot \gamma$ by eq. (\[7.1\]), and the graphs in Fig. \[fig3\] show $\dot{\gamma}$ as a function of $\gamma$. All the examples have $d = 1$ Gly $\approx 306.6$ Mpc. Example 2 (solid line) has $R_0 = 3$ Gpc and Profile 1; Example 3 (dashed line) has $R_0 = 1$ Gpc and Profile 1; Example 4 (dotted line) has $R_0 = 1$ Gpc and Profile 2 (for which $\delta = 10.0$ instead of $\delta = 4.05$; this is a deeper void in a higher-density background). The amplitude is $\sim 10^{-7}$ for (3) and $\sim 10^{-6}$ for (2) and (4). With the Gaia accuracy of $5-20 \times 10^{-6}$ arcsec, we would need a few years to detect this effect. ![The configuration for Examples 2, 3, 4.[]{data-label="fig2"}](fig2.eps)  RLPs in shearfree normal models {#BarnesRLP} =============================== In the Szekeres models, the condition for all null geodesics to be RLPs was the vanishing of shear. This suggests that the cause of the non-RLP phenomenon might be shear in the cosmic flow. To test this supposition, the existence of RLPs was investigated in those cosmological models in which shear is zero [@Kras2011] – the shearfree normal models found by Barnes [@Barn1973]. They obey the Einstein equations with a perfect fluid source and contain, as the acceleration-free limit, the whole FLRW family. There are four classes of them: the Petrov type D metrics that are spherically, plane and hyperbolically symmetric, and the conformally flat metric found earlier by Stephani [@Step1967]. In the Petrov type D case, the metric in comoving coordinates is $${\rm d} s^2 = \left(\frac {F V,_t} V\right)^2 {\rm d} t^2 - \frac 1 {V^2} \left({\rm d} x^2 + {\rm d} y^2 + {\rm d} z^2\right), \label{8.1}$$ where $F(t)$ is an arbitrary function, related to the expansion scalar $\theta$ by $\theta = 3 / F$. The Einstein equations reduce to the single equation: $$w,_{uu} /w^{2} = f(u), \label{8.2}$$ where $f(u)$ is an arbitrary function, while $u$ and $w$ are related to $(x, y, z)$ and to $V(t,x,y,z)$ differently in each subfamily. We have $$(u, w) = \left\{ \begin{array}{ll} (r^{2}, V) & \mbox{with spherical symmetry}, \\ \ & \ r^2 {\ {\overset {\rm def} =}\ }x^2 + y^2 + z^2;\\ (z, V) & \mbox{with plane symmetry};\\ (x/y, V/y) & \mbox{with hyperbolic symmetry}.\end{array} \right. \label{8.3}$$ The FLRW limit follows when $f = 0$ and $V = R(t) g(x,y,z)$. The conformally flat Stephani solution [@Step1967; @Kras1997] has the metric given by (\[8.1\]), the coordinates are comoving, and $V(t, x, y, z)$ is given by $$\begin{aligned} &&V = \frac 1 R \left\{1 + \frac 1 4 k(t) \left[\left(x - x_{0}(t)\right)^{2} + \left(y - y_{0}(t)\right)^{2} \right.\right. \nonumber \\ &&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ + \left.\left.\left(z - z_{0}(t)\right)^{2}\right]\right\}, \label{8.4}\end{aligned}$$ where $(R, k, x_0, y_0, z_0)$ are arbitrary functions of $t$. This a generalisation of the whole FLRW class, which results when $(k, x_0, y_0, z_0)$ are all constant. In general, (\[8.4\]) has no symmetry. In these models, in the most general cases, generic null geodesics are not RLPs. Consequently, [***it is not shear that causes the non-RLP property***]{}.[^3] In the general type D shearfree normal models, the only RLPs are radial null geodesics in the spherical case and their analogues in the other two cases. In the most general Stephani spacetime, RLPs do not exist. In the axially symmetric subcase of the Stephani solution the RLPs are those geodesics that intersect the axis of symmetry in every space of constant time. In the spherically-, plane- and hyperbolically symmetric subcases, the RLPs are the radial geodesics. The completely drift-free subcases are conformally flat, but more general than FLRW. Their defining property is that their time-dependence in the comoving coordinates can be factored out, and the cofactor metric is static. The FLRW models have the same property. For example, in the drift-free spherically symmetric type D case: $$\begin{aligned} {\rm d} s^2 = \frac 1 {V^2} && \hspace{-3mm} \left\{\left[\left(A_1 + A_2 r^2\right) \left(F S,_t {\rm d} t\right)\right]^2 - {\rm d} r^2\right. \nonumber \\ &&- \left.r^2 \left({\rm d} \vartheta^2 + r^2 \sin^2 \vartheta {\rm d} \varphi^2\right)\right\}, \label{8.5}\end{aligned}$$ the whole non-staticity is contained in $V$: $$\label{8.6} V = B_1 + B_2 r^2 + \left(A_1 + A_2 r^2\right) S(t).$$ The $(A_1, A_2, B_1, B_2)$ are arbitrary constants and $S(t)$ is an arbitrary function. This model is more general than FLRW because the pressure in it is spatially inhomogeneous. The FLRW limit follows when $A_1 \neq 0$ and $B_2 = (A_2/A_1) B_1$. Dependence of RLPs on the observer congruence ============================================= The RLPs are defined relative to the congruence of worldlines of the observers and light sources. So far, we have considered observers and light sources attached to the particles of the cosmic medium, whose velocity field is defined by the spacetime geometry via the Einstein equations. But we could as well consider other timelike congruences, or spacetimes in which no preferred timelike congruence exists, for example Minkowski. It turns out that even in the Minkowski spacetime one can devise a timelike congruence that will display the non-RLP property [@Kras2012]. Take the Minkowski metric in the spherical coordinates $$\label{9.1} {\rm d} s^2 = {\rm d} {t'}^2 - {\rm d} {r'}^2 - {r'}^2 \left({\rm d} \vartheta^2 + \sin^2 \vartheta {\rm d} \varphi^2\right),$$ and carry out the following transformation on it: $$\label{9.2} t' = (r - t)^2 + 1 / (r + t)^2, \qquad r' = (r - t)^2 - 1 / (r + t)^2.$$ The result is the metric $$\begin{aligned} && {\rm d} s^2 = \frac 1 {(r + t)^4}\left\{16 u \left({\rm d} t^2 - {\rm d} r^2\right)\right. \\ && \ \ \left.- \left(u^2 - 1\right)^2 \left({\rm d} \vartheta^2 + \sin^2 \vartheta {\rm d} \varphi^2\right)\right\}, \qquad u {\ {\overset {\rm def} =}\ }r^2 - t^2. \nonumber \label{9.3}\end{aligned}$$ Now we assume that the curves with the unit tangent vector field $u^{\alpha} = \left[(r + t)^2 / \left(4 \sqrt{u}\right)\right] {\delta^{\alpha}}_0$ are world lines of test observers and test light sources. 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--- abstract: 'Reinforcement learning (RL) algorithms allow agents to learn skills and strategies to perform complex tasks without detailed instructions or expensive labelled training examples. That is, RL agents can learn, as we learn. Given the importance of learning in our intelligence, RL has been thought to be one of the key components to general artificial intelligence, and recent breakthroughs in deep reinforcement learning suggest that neural networks (NN) are natural platforms for RL agents. However, despite the efficiency and versatility of NN-based RL agents, their decision-making remains incomprehensible, reducing their utilities. To deploy RL into a wider range of applications, it is imperative to develop explainable NN-based RL agents. Here, we propose a method to derive a secondary comprehensible agent from a NN-based RL agent, whose decision-makings are based on simple rules. Our empirical evaluation of this secondary agent’s performance supports the possibility of building a comprehensible and transparent agent using a NN-based RL agent.' author: - | Jung Hoon Lee\ Allen Institute for Brain Science\ Seattle, WA 98109\ `[email protected]`\ title: Complementary reinforcement learning towards explainable agents --- Introduction ============ Reinforcement learning (RL), inspired by our brain’s reward-based learning, allows artificial agents to learn a wide range of tasks without detailed instructions or labeled training sets which are necessary for supervised learning [@Hertz1991; @Sutton2017]. Given that RL agents’ learning resembles our process of learning and that learning is essential to our intelligence, it seems natural to assume that RL is one of the key components to brain-like intelligent agents or general artificial intelligence. Recent breakthroughs [@Mnih2015; @Silver2016] from DeepMind team showed that RL could train neural network (NN)-based agents to outperform humans in video-games and even ’Go’, reigniting interests in RL and its applications in NN-based agents, and noticeable developments in NN-based RL agents have been reported since then; see [@Mirowski2017; @Mnih2016; @Oh2016; @Plappert2018; @Wang2016] for examples. However, despite rapid improvements in RL, NN-based RL agents’ decision-making process remains incomprehensible to us. For instance, the AlphaGo’s strategies employed during the match with Sedol Lee exhibited efficiency leading to victories, but the exact reasons behind its moves are unknown. AlphaGo demonstrated that incomprehensible decision-making can still be effective, but its effectiveness does not mean that it cannot be faulty. In ‘high stake problems’ [@Rudin2018], any mistakes can be critical and need to be avoided. A loss in one Go match out of 100 matches is insignificant, but crashing a car once out of 100 drives is perilous and unacceptable. If we comprehend the exact internal mechanisms of RL agents, we can correct their mistakes without negative impact on their performance. That is, ’transparent’ agents with comprehensible internal decision-making processes [@Lipton2016] are necessary to safely deploy RL agents into high stake problems. Two earlier studies [@Hayes2017; @Waa2017] showed that the decision-making processes of RL agents can be translated into human-readable descriptions. Their proposed algorithms, which fall into the ’post-hoc’ interpretation approach [@Rudin2018], do provide some insights into RL agents’ decision-making process, but they do not address how we correct (or fix) RL agents’ actions. Then, how do we build transparent RL agents? We propose a secondary agent as a potential solution, which utilizes simple rules to choose the best action. This proposal is based on two ideas. First, the secondary agent’s action can be analyzed because it utilizes simple rules. Second, the secondary agent can perform general tasks, if it takes advantage of trained RL agents. In this study, we propose a quasi-symbolic (QS) agent as a secondary agent and compare its performance to RL agents’ performance. Specifically, QS agents learn, from RL agents’ behaviors, the values of transitions of states. After learning the values of transitions, QS agents identify the most valuable state-transitions (‘hub states’) and search for a sequence of actions (action plans) to reach one of the hub states. If QS agent cannot find a proper action plan to reach a hub state, they choose the best action by comparing the values of immediate transitions. In this study, we tested QS agents’ performance using the ’lunar-lander’ benchmark problem available in ’OpenAI Gym’ environments [@Brockman2016]. Our results show that QS agents’ performance is comparable to RL agents’ performance. While our experiments are conducted in a simple environment, the results indicate that comprehensible agents with transparent decision-making process can be derived from RL agents. Related work ============ While deep learning (DL) spreads its influence, it remains unclear whether DL can safely be applied to high stake decision problems (e.g., medicine and criminal justice) [@Rudin2018]. DL agents rely on cascades of nonlinear interactions among computing nodes (i.e., neurons), but the interactions are too complex to be analyzed, which limits our mechanistic understanding of DL agents’ decision-making. This means that despite extensive testings we cannot fully predict their actions. Thus, deploying DL agents to high stake problems may lead to critical failures; such failures have already been reported [@Rudin2018]. To deploy DL agents into high stake decisions, it is imperative to understand the exact process of their decisions. If reasoning behind their decisions become clear, developers could improve DL agents’ reliability, policy-makers could introduce regulations to ensure fairness of DL agents, and users could use DL agents more effectively [@Preece2018]. While we note the diverse aspects of DL explainability [@Preece2018], Lipton [@Lipton2016] pointed out that explainable intelligent agents can be obtained by analyzing the internal mechanisms or the agents’ behaviors, which were referred to as transparency and post-hoc interpretability, respectively. If we could understand the internal mechanisms, we will be able to fully understand DL agents’ decision-making process; that is, their decisions will become ’transparent’. Several methods have been proposed to analyze the internal mechanisms of DL agents [@Zeiler2014; @Yosinski2015; @Fong2017; @Erhan2009; @olah2017feature], and Olah et al.[@olah2018the] presented an intriguing way to synergistically use them to gain insights into DL agents’ decision-making. Also, multiple studies sought the post-hoc interpretability of DL agents, which we could use to predict and prevent potential failures. Specifically, human interpretable descriptions can be automatically generated by secondary agents [@McAuley2015; @Lipton2016], and representative examples or image parts can be identified by algorithms such as sensitivity analysis [@Samek2017; @Montavon2018]. The majority of studies have focused on feedforward DL, but a few studies pursued the explainability of RL. Specifically, Hayes and Shah [@Hayes2017] showed that a secondary network could be trained to provide user-interpretable descriptions, and Waa et al. [@Waa2017] further showed that user-interpretable descriptions could be contrasive; that is, their methods can explain why RL agents would prefer one option to another [@Miller2019; @Mittelstadt2019]. Although such post-hoc interpretability can be used to evaluate the quality/reliability of RL agents, it does not provide a way to fix RL agents’ mistakes. However, with transparent agents, we can correct their mistakes selectively. In our study, we propose a potential approach, which can help us obtain transparent RL agents. Network Structure {#network structure} ================= QS agents interact with RL agents and make plans for future actions by utilizing the model of environment (Env network). All these three networks/agents are constructed using the ’Pytorch’, an open-source machine learning toolkit [@Paszke2017]. Below, we discuss QS, RL and Env network in details. The structure of reference RL agent ------------------------------------ We used an actor-critic model as a reference RL agent [@Grondman2012; @Konda2000]. The implementation was adopted from the official pytorch github repository [@Pytorch-team2018], which maximizes the expected discounted reward $E_{\pi_{\theta}}[R]$, where $R=\sum^T_{t=0} \gamma^{t}r_t$, given policy $\pi_{\theta}$; where $T$ and $\theta$ represent a trajectory and parameters of the policy. The gradient ascent of $E$ is estimated as shown in eq. (\[eq\_grad\]). $$\label{eq_grad} \begin{aligned} & \bigtriangledown_{\theta}[R]=E \big[ \sum^T_{t=0} \bigtriangledown_{\theta} \log \pi_{\theta} (a|s) \{r'(t)-V(s,t)\} \big] \\ & r'(t)=\frac{R(t)-<R>}{\sigma_R} \end{aligned}$$ , where $a$, $\theta$, $s$, $V=V(s,t)$ represent action, parameters, state and value function, respectively; where $<R>$ and $\sigma_R$ represent the mean and standard deviation of R in episodes. The actor consists of 8 input, 100 hidden and 4 output nodes, and the critic, 8 input, 100 hidden and 1 output nodes. The initial learning rate is 0.003 and is decreased by 10 times at every 1000 episode. 0.2in ![The schematics of models. (A), the structure of the QS agents consisting of matching and value networks. The matching nodes (i.e., the outputs of matching network) are connected to the value nodes via one-to-one connection. When the input vector is novel (Eq. \[eq1\]), new nodes are added to the matching and value networks and are connected via exclusive one-to-one connection. That is, the number of value nodes is the same as that of matching nodes. (B), the planning of future actions. At each time step, a QS agent invokes a RL agent to get a probable action and feeds it into Env network to predict the next state. By recursively using them, QS agents can make action plans.[]{data-label="Fig1"}](Figure1){width="\columnwidth"} -0.2in The structure of quasi-symbolic (QS) agent ------------------------------------------ QS agent consists of matching and value networks, which are both single (synaptic) layer networks. The matching and value networks are sequentially connected (Fig. \[Fig1\]A). That is, the output node of matching network is directly connected to the output node of value network. The number of matching nodes (output nodes in the matching network) is identical to the number of value nodes (output nodes in the value network), and the connections between matching and value networks are one-to-one. The matching network memorizes input vectors by imprinting normalized inputs to synaptic weights converging onto the output node $O_i$, as shown in Eq. \[eq1\]; $h_i$ the synaptic input to $O_i$. $$\label{eq1} h^k_{i}=\sum_{j}w_{ij}\frac{s^k_j}{\left\Vert\vec{s_k}\right\Vert^2}, \text{where } w_{mn}=\frac{s^m_n}{\left\Vert\vec{s_m}\right\Vert^2}$$ , where superscript indices represent the inputs (state-transition vectors in this study; see below), and subscript indices represent the components of inputs. With normalized inputs stored in synaptic weights $w_{ij}$, the outputs of matching nodes represent cosine similarities between the current input and stored inputs. If all synaptic inputs $h_i$ to all output nodes $O_i$ are smaller than the threshold value $\theta $, the input is considered novel and stored into the matching network by adding a new output node into the matching network; the default threshold is $\theta=0.97$, unless stated otherwise. Once a new node is added to the matching network, a new node is also added to the value network, to keep one-to-one mapping (Fig. \[Fig1\]A). The strength of the connection between these two newly established units is determined by the reward obtained by RL agents with the selected action. When the current input is one of the previously stored ones (i.e., one of synaptic input to output nodes is higher than the threshold $\theta $), the maximally activated node is identified, and the connection to the corresponding value network node is updated by adding the reward induced by the input vector (Eq. \[eq2\]). It should be noted that only one matching node, which receives maximum synaptic input $h$, is allowed to be active one at a time, which makes the matching network operate in linear regimes. $$\label{eq2} W_k \leftarrow W_k+r_t$$ , where $W_k$ is the connection between the matching node $m_k$ and value node $v_k$, and $r_t$ is the reward the RL agent obtain at time $t$. In this study, the matching network is designed to memorize the transitions between states $S$ and $S'$. That is, the input to the network is $\Delta S=S'-S$. Since there are 8 state variables in the lunar-lander benchmark problem, the number of input nodes of the matching network is 8. The sizes of the matching and value networks are not fixed. Instead, they are determined during the training of QS agents. The more diverse inputs are, the bigger matching networks become. For instance, if all inputs are identical, there is only one matching node. Env network ----------- The Env network models the environment. Thus, it receives the state vector $S$ and action $a$ as inputs and returns the next state (Fig. \[Fig1\]B). The inputs layer includes 12 nodes which represent 8 state variables and 4 possible actions in the lunar lander environment, and the output layers include 8 nodes (due to 8 state variables). In our experiments, we set the hidden layer of the Env network to have 300 nodes. The Env network is trained using the mean squared error (squared L2 norm). In each episode of RL training, the error is accumulated. After each episode, the Env network is updated using the accumulated error; that is, 1 episode is a single batch of backpropagation. The initial learning rate is 0.05 and is decreased to 10 percent at every 1000 episode. Results ======= In this section, we discuss the operating principles of QS agents including the interplay between QS and RL agents. Then, we present our experiments which were conducted to evaluate QS agents’ performance in solving lunar-lander problem compared to RL agents. 0.2in ![Training of RL agent and Env network. (A), the total reward in each episode. (B), the error observed during training of the Env network. []{data-label="Fig2"}](Figure2){width="\columnwidth"} -0.2in Training and operating rules of Quasi-Symbolic (QS) agents ---------------------------------------------------------- Unlike RL agents, QS agents do not work alone. Instead, their operations depend on RL agents in both learning and inference modes; that is, RL and QS agents are complementary with each other. The matching and value networks in QS agents are updated by using RL agents’ behaviors during training. Specifically, in each time step in the training period, the previous state $S$ and the current state $S'$ are used to generate the state transition vectors $\Delta S=S'-S$, and this transition vector is fed into the matching network as inputs. The matching network first determines the novelty of the current transition vector; this novelty detection is done by inspecting synaptic inputs to matching nodes (see section \[network structure\] and Eq. \[eq1\] for details). When a novel transition vector is introduced, a new output node is added to the matching and value networks (Fig. \[Fig1\]A), and the connection between matching and value networks (one-to-one connection $w_k$ between newly added nodes, $m_k$ and $v_k$) is established (section \[network structure\]). The strength of the connection $w_k$ is determined by the reward given to the RL agent after the transition from $S$ to $S'$. When the earlier input is introduced again, the maximally activated matching node is identified, and the connection between the identified matching node and the value node, which is exclusive and one-to-one (Fig. \[Fig1\]A), is updated by adding the reward obtained to the previous strength (Eq. \[eq2\]). It should be noted that only one matching node is allowed to be active and used to assess the value of the transition. In brief, the matching network memorizes transition vectors observed during training, and the value network stores the amount of rewards induced by the observed transitions. In the inference mode, in which QS agents choose the best action, QS agents utilize both the trained RL agents and Env networks to make action plans, whose lengths are variable. In doing so, QS agents identify the most valuable transition vectors $\Delta S_k$ by inspecting connections’ strength between matching and value networks. In this study, we identified synaptic weights higher than $\theta_{hub}$ defined in Eq. \[eq3\]. $$\label{eq3} \theta_{hub}=<w_k>+\alpha \times \sigma(w_k)$$ , where $<w_k>$ and $\sigma(w_k)$ are mean and standard deviations of synaptic weights, and $\alpha $ is the scale constant which determines the magnitude of $\theta_{hub}$. The default value of $\alpha$ is $0.1$. Due to the one-to-one connections between matching and value networks, the synaptic weights allow us to identify the most valuable transitions observed during the training period, which are referred to as hub states hereafter. 0.2in ![Comparison between QS and RL agents’ performances. (A), the average reward of QS and RL agents in 100 environments, after the RL agent is trained in 1000 episodes. 10 independent QS and RL agents are tested for the same environment, and the rewards averaged over 10 agents are shown. The red and blue lines represent QS and RL agents, respectively. (B), the same as (A), but the RL agent is trained during 5000 episodes. (C), the average reward of QS and RL agents depending on the length of RL agents training. The mean values of the rewards are estimated using 10 agents tested in 100 environments. That is, the mean values illustrated are rewards averaged over 1000 individual experiments. The error bars are standard errors from the same 1000 experiments.[]{data-label="Fig3"}](Figure3){width="0.55\columnwidth"} -0.2in After identifying the hub states, QS agents search for an action plan to reach one of the hub states. In doing so, QS agents utilize the Env network and the trained RL agent to predict future states (and thus transitions as well) to make an action plan. At each state, RL agent provides a possible action, and Env network returns the next state in response to the suggested action. Employing them recursively (Fig. \[Fig1\]B), QS agents can predict the future states. During this planning, at each time step, QS agents examine whether a transition vector is one of the hub states (i.e., the most valuable transitions) or not. If QS agents do expect to reach one of the hub states, they stop planning and execute the current plan. The maximum length of the action plan is 10 time-step, unless stated otherwise. If no hub state cannot be reached within the predefined maximal time-step, QS agents start over and make a new plan. For each state, QS agents are allowed to make a total of 5 different plans. If they cannot find a path to the hub states in all five plans, they inspect the values of the first transition in the five plans and choose the best immediate action according to the reward estimated by the value network. In this case, rather than taking a sequence of actions, QS agents execute a single action (i.e., the first action in the plan) only. QS agents’ performance compared to RL agents -------------------------------------------- To evaluate QS agents’ performance, we compared their performance to that of RL agents by using the lunar-lander benchmark task included in the OpenAI gym environment [@Brockman2016]. In this study, we constructed RL agents (actor-critic model), QS agents and Env network using Pytorch, an open-source machine learning library [@Paszke2017]; their schematics are illustrated in Fig. \[Fig1\]. We trained RL, QS agents and Env network during 5000 episodes; see section \[network structure\] for training details. Figures \[Fig2\]A and B show the total amount of rewards given to the RL agent and the error function of the Env network during 5000 episodes. As shown in the figure, while the reward in each trial fluctuates from one trial to another, the amount of rewards, on average, increases rapidly until 1000 episodes. After 1000 episodes, the speed of improvement is reduced. Similarly, the error of Env network is reduced most rapidly in the first 1000 episodes, and then the speed of error-reduction slows down. At every 1000 episode, we froze the learning and tested both QS and RL agents using the same 100 environments of lunar lander; that is, the environments are instantiated with the same random seeds. The RL agent in this study uses stochastic policy to choose actions (eq. \[eq\_grad\]), and thus its behaviors depend on the random seed forwarded to the Pytorch. Moreover, QS agents’ behaviors are also stochastic, as they rely on RL agents for their decisions. To avoid potential biases based on their stochastic behaviors, we constructed 10 independent QS and RL agents by forwarding distinct random seeds to pytorch and calculated the average reward for both agents. Figures \[Fig3\]A and B show the average amount of QS and RL agents for all 100 instantiations of environment. $x$-axis represents the identity of environment, and $y$-axis represents the reward averaged over 10 independently constructed agents. Rewards are estimated after training the RL agent in 1000 episodes (Fig. \[Fig3\]A) and 5000 episodes (Fig. \[Fig3\]B). As shown in the figure, the performance of QS agents (shown in red) with 10 time-step action plan are comparable to that of RL agents (shown in blue). Figure \[Fig3\]C shows the average reward calculated using 10 independent agents in 100 environments, after training the RL agent in 1000, 2000, 3000, 4000 and 5000 episodes, respectively; that is, the mean values and standard errors are estimated from 1000 individual experiments (10 independent agents $\times$ 100 environments). 0.2in ![Parameters’ effects on the QS agents’ performance. (A), the average rewards depending on the maximal length of action plans. The mean values and standard errors of the rewards are estimated using 10 agents tested in 100 environments. We estimated them with RL agents trained during 1000, 2000, 3000, 4000 and 5000 episodes. The color codes represent the length of RL agents training. (B), the average rewards with a bigger set of hub states. The threshold value for the hub states is lowered ($\alpha$ in Eq. \[eq3\]: $0.1 \rightarrow 0.05$) to increase the number of states.[]{data-label="Fig4"}](Figure4){width="0.8\columnwidth"} -0.2in Then, we varied the parameters to examine how they affect QS agents’ performance. First, we tested the effects of action plans’ lengths. As shown in Fig. \[Fig4\]A, QS agents’ performance improves, as the maximal length of action plans increases. Second, we increased the number of hub states by lowering $\theta_{hub}$. Specifically, we set $\alpha=0.05$ and estimated the amount of rewards. As shown in Fig. \[Fig4\]B, QS agents’ performance improved. We also increased $\theta_{hub}$ to further examine the effects of the number of hub states on QS agents’ performance and found that QS performance is negatively correlated with $\theta_{hub}$ (Fig. \[Fig5\]A). Finally, we perturbed the threshold value $\theta $ for the novelty detection (used in matching networks) to see how it affects QS agents’ performance. As shown in Fig. \[Fig5\]B, QS agents obtained less rewards. This may be explained by the fact that the accuracy of predictions on QS agents’ future states decreases, as the threshold $\theta $ becomes lower, 0.2in ![Parameters’ effects on the QS agents’ performance. (A), the average rewards depending on the threshold value for the hub states (see Eq. \[eq3\]). (B), the average rewards depending on the threshold value $\theta $ for novel input detection (section \[network structure\]).[]{data-label="Fig5"}](Figure5){width="0.8\columnwidth"} -0.2in Discussion ========== In this study, we propose QS agents to develop transparent RL agents. The newly proposed QS agents have two operating units, matching and value networks. With these two units, QS agents evaluate actions suggested by RL agents and choose the most probable choice. To select the most probable action, QS agents search for a path to reach one of the hub states by utilizing the Env network which models the environment. Our results (Fig. \[Fig4\]) suggest that this future plan ensures QS agents’ good performance. Then, do QS agents have transparent decision processes? QS agents’ decisions are transparent for two reasons. First, the two operating units of QS agents have simple structures, which can be analyzed easily. The value network simply accumulates rewards given after the transition of states into synaptic weights, and it returns these stored values depending on inputs ($\Delta S$). That is, the value network is, in principle, equivalent to conventional memory units. The matching network identifies the old input ($\Delta S$), which is the closest to the current input by using the cosine similarity (eq. \[eq1\]). In addition, as only a single matching node is allowed to be active, it is clear that the matching network’s operation can be easily analyzed. Second, QS agents rely on a simple set of rules to select the best actions based on suggestions made by RL. With both simple inner mechanisms and operating rules, QS agents have transparent decision-making process. Moreover, it should be noted that the output nodes of the matching network are working independently from each other. That is, the matching nodes can be removed and added without interfering with other matching nodes’ operations. This property makes manual modification of QS agents’ actions possible. If a new piece of evidence finds a particular state-transition to be unacceptable, it can be removed. On the other hand, if some state-transitions $\Delta S$, which have not been previously observed, are considered valuable, they can be added to QS agents. Similarly, the synaptic weights of value networks and hub states can also be changed, if necessary. Therefore, QS agents can be continuously and incrementally improved (or repaired) to avoid mistakes. Potential variations of QS agents --------------------------------- To address the possibility of deriving a comprehensible secondary agent from RL agents, we sought generic algorithms to be applied to general RL problems, but QS agents and their operations can be customized in domain-specific ways. Below, we list a few potential variations of our generic QS agents. First, current QS agents evaluate individual transitions $\Delta S$ using the rewards RL agents obtain immediately after completing the transition. However, the actual values of transitions can be estimated differently. For instance, instead of using the immediately obtained reward, we can use the total rewards from the transition to the end of an episode for evaluating state transitions. Second, in this study, QS agents treat all state variables equally and rely on a single-state vector $\Delta S$. However, state variables do not have equivalent values in agents’ behaviors. For example, the coordinates and velocities included in the state vectors of lunar-lander do have different importance in terms of agents dynamics. If we deal with coordinates and velocities distinctively, QS agents may evaluate the state-transitions more effectively. In this case, there will be two state vectors; $S \rightarrow S_{\dot{x}}$, $S_{x}$. Similarly, a single state-transition vector can be split into multiple ones; $S \rightarrow S_1, S_2, ..., S_n$. In doing so, QS agents need to utilize multiple matching and value networks, and the actual value can be estimated with a linear summation of outputs of multiple value nodes. Third, the state transition vectors $\Delta S$ can be coupled with state vectors $S$ to estimate the values of actions more precisely. For instance, an agent’s horizontal move can be either bad or good depending on the current state. If both state and transition vectors are used, QS agent may have better estimation of agents’ actions. Implications for the brain’ complementary system ------------------------------------------------ Prefrontal cortex (PFC) has been thought to be a hub for high-level cognitive functions such as decision-making, learning and working memory [@Lara2015; @Miller2001; @Tanji2008; @Wang2012]. However, PFC does not work alone and is known to be connected to other brain areas. The two main areas that have strong interactions with PFC are hippocampus [@Jin2015; @Peyrache2011] and anterior cingulate cortex (ACC) [@Gehring2000; @Paus2001]. Notably, complementary learning system theory suggests central roles of the interplay between PFC and hippocampus in our ability to learn continuously [@OReilly2014]. 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--- abstract: 'A vapor bubble collapsing near a solid boundary in a liquid produces a liquid jet that points toward the boundary. The direction of this jet has been studied for boundaries such as flat planes and parallel walls enclosing a channel. Extending these investigations to enclosed polygonal boundaries, we experimentally measure jet direction for collapsing bubbles inside a square and an equilateral triangular channel. Following the method of Tagawa and Peters (Phys. Rev. Fluids **3**, 081601, 2018) for predicting the jet direction in corners, we model the bubble as a sink in a potential flow and demonstrate by experiment that analytical solutions accurately predict jet direction within an equilateral triangle and square. We further use the method to develop predictions for several other polygons, specifically, a rectangle, an isosceles right triangle, and a $30^{\circ}$-$60^{\circ}$-$90^{\circ}$ right triangle.' author: - 'Lebo Molefe$^{1,2}$' - 'Ivo R. Peters$^1$' title: Jet direction in bubble collapse within rectangular and triangular channels --- Introduction ============ Cavitation is the sudden formation of bubbles in liquid that occurs when absolute pressure in the liquid decreases below its vapor pressure. Due to surface tension, absolute pressure must actually drop to a threshold far below the vapor pressure, to large negative pressures, for cavitation to occur. The negative threshold pressure observed in experiment varies depending on the amount of impurities in the liquid [@cavitation; @pressure; @in; @water]. Cavities can be formed in fast flows of liquid when pressure drops to such a low level that vapor bubbles are formed [@textbook; @-; @Fundamentals; @of; @Fluid; @Mechanics]. They can also be produced by raising the local temperature of the fluid, as in laser-induced cavitation [@laser]. These cavities formed in local low-pressure zones collapse rapidly in the bulk liquid. Formation of cavities in low pressure flows happens in natural as well as industrial settings. There is evidence that dolphins and fish experience cavitation when they swim fast and the snapping shrimp uses cavitation to stun its prey [@dolphins; @and; @fish; @shrimp; @1; @shrimp; @2]. In tracheids, trees’ natural microfluidic pump system, water pressure can drop so low below atmospheric pressure that cavitation occurs, blocking water transport in that channel [@trees]. Near solid boundaries, collapsing cavities are known to form liquid jets that point toward the boundary [@cavitation; @near; @horizontal; @plane]. Due to these powerful and damaging liquid jets, in many cases cavitation in industrial settings is a negative occurrence – it causes damage to ship propellers and pumps, and adds complications to flow in fuel injection systems [@propellers; @pumps; @fuel; @injection]. Cavitation has found use, however, in medical treatment, in ultrasonic cleaning, and for mixing or initiating chemical reactions in microfluidic systems [@medical; @treatment; @kidney; @stones; @ultrasonic; @cleaning; @1; @mixing; @1; @mixing; @2; @catalyzing; @chemical; @reactions]. Cavitation near solid boundaries of different shapes has been studied, with the complexity of the boundary shape gradually increasing over time from single horizontal planes to parallel or perpendicular walls and curved boundaries [@parallel; @walls; @cavitation; @near; @horizontal; @plane; @cavitation; @near; @perpendicular; @walls; @cavitation; @near; @wall; @and; @corners; @cavitation; @near; @semi-enclosed; @parallel; @walls; @cavitation; @near; @curved; @boundaries]. Various aspects of cavitation including bubble shape, jet strength, and jet direction have been studied [@scaling; @laws; @for; @jets; @cavitation; @in; @corners; @cavitation; @near; @boundary; @with; @attached; @bubble]. Jet direction has been studied near various complex boundaries [@cavitation; @near; @boundary; @with; @attached; @bubble; @sawtooth; @boundary], but exact analytic predictions of jet direction are lacking in all but the most symmetric boundary cases [@cavitation; @near; @curved; @boundaries]. In jet formation flows develop around the bubble surface with very high velocities in the direction of the liquid jet [@flow; @field; @microfluidic; @triangle; @and; @square]. Moving away from the high degree of symmetry in previous studies, Tagawa and Peters (2018) predicted jet direction for bubbles collapsing near solid corners of angle $\pi / n$ (for natural number $n$). Their equation relating jet direction to position within the corner is an analytic solution based on modeling the flow field at the time of collapse initiation using potential flow analysis and the method of images [@cavitation; @in; @corners]. Here, we demonstrate that the potential flow method can be used to predict jet direction within several types of polygons, specifically, any rectangle, equilateral triangle, isosceles right triangle, and $30^{\circ}$-$60^{\circ}$-$90^{\circ}$ right triangle. A bubble in a liquid region enclosed by solid walls is modeled as a sink in a potential flow. The boundary condition that fluid cannot flow through the walls is satisfied by introducing an infinitely tessellating pattern of image sinks. Experiments were conducted using laser-induced cavitation, and the models are found to agree with experimental data. Conceptually, this demonstrates that regardless of the complex dynamics of the bubble surface, asymmetry in the flow field in the initial instants of collapse is sufficient to predict the direction of the liquid jet of a collapsing bubble. The method of images is a mathematical result that is widely applicable, including to our potential flow problem and any situation in which Laplace’s equation is satisfied [@method; @of; @images]. Because of this, the analytic model presented has further significance because it demonstrates the applicability of method of images solutions to describe a physical situation with relatively complex, enclosed boundary conditions. Experiments =========== Bubbles were produced by a laser in a large ($145 \times 145 \times 145$ mm$^3$) acrylic tank filled with degassed water. Figure \[fig:experiment\_diagram\]a is a diagram of the experimental arrangement. Glass microscope slides of dimensions $25.7 \times 75.8 \times 1.4$ $\text{mm}^{3}$ were glued together to form geometric shapes as shown in Figure \[fig:experiment\_diagram\]b and \[fig:experiment\_diagram\]c. The slides formed a channel with the cross-section of a square or equilateral triangle of side length $l = 15 \text{mm}$, and were long enough to approximate an infinitely long tunnel with the desired cross-section. The channel is long enough (for channel length $L$ and bubble diameter $d$, we have roughly $L / d > 7$) to ignore any influence of the finite length, since, at large distances, changes in boundary condition have very weak effects on bubble jets [@cavitation; @near; @perpendicular; @walls]. In addition, bubbles were produced midway between the channel’s two open ends, to reduce any effect of asymmetry in the $z$-direction. The microscope slide assembly was mounted using an acrylic arm attached to a translation stage accurate within 5 $\mu \text{m}$ (Figure \[fig:experiment\_diagram\]a). The assembly was mounted vertically (open ends toward top and bottom) to allow remnants of bubbles from previous cavitation events to escape by buoyancy. The slide assembly was positioned roughly equidistant between the tank bottom and the water surface. A Q-switched Nd:YAG laser (Bernoulli PIV; Litron) produces enough concentrated energy to vaporize a small amount of water and produce a vapor bubble [@laser]. Laser pulse duration was 6 ns and wavelength was 532 nm. Laser energy was halved using a 50:50 beam splitter, with half the beam directed toward an energy meter and the other half directed into the water tank. Laser energy was kept at about $5$ mJ per pulse. Using a $10 \times$ microscope objective (Nikon Plan Fluor; NA = 0.30; working distance in air = 16 mm; working distance in water = 21 mm; Airy disk radius = 1 $\mu \text{m}$), laser light was focused to a point within the microscope slide assembly after passing through the clear acrylic tank and the glass slide. ![Experiment setup. (a) Bubbles are produced in a large water bath by laser-induced cavitation. The tank is transparent. The beam passes through a microscope objective, which focuses the laser to a point where the bubble is nucleated within a geometrically shaped microscope slide assembly. Images are taken of the $xy$-plane after being reflected by a mirror placed below the tank. (b) Square cross section. Inner side length: 15 mm. (c) Equilateral triangle cross section. Inner side length: 15 mm.[]{data-label="fig:experiment_diagram"}](figure01.pdf){width="84mm"} Images were taken of bubbles growing and collapsing within the geometric shapes using a high-speed camera (Photron FASTCAM SA-X2) with a 105 mm Nikon Micro-Nikkor lens, recording at 100 kHz with an image size of $364 \times 284$ pixels. The camera recorded a bottom view by using a mirror positioned below the water tank. A 550 nm longpass filter blocked laser light from entering the camera lens. The position and jet angle of the bubbles were measured from these images. We calibrated our images using a millimeters per pixel resolution measured by the difference in pixel position of the slide assembly between pairs of images where it was moved by a known millimeter displacement using the translation stage. The resolution was between 0.0295 mm per pixel (square experiment) and 0.0301 mm per pixel (triangle experiment). A typical set of bubble growth and collapse images from two movies is displayed in Figure \[fig:movie\_images\]. The bubble first expands spherically from the point of nucleation (Figure \[fig:movie\_images\]c and \[fig:movie\_images\]d), grows to its largest diameter (Figure \[fig:movie\_images\]e and \[fig:movie\_images\]f), and then collapses, at which point a liquid jet pierces through the bubble (Figure \[fig:movie\_images\]g and \[fig:movie\_images\]h). ![Bubble growth and collapse. Position of the camera frame in relation to the shapes is shown in (a, b). Bubbles are shown shortly after nucleation (c, d), at maximum area (e, f), and after a liquid jet has pierced the bubble (g, h). The diameters of the bubbles at maximum area are both roughly $d \approx 1.7 \text{mm}$. Scale bar is $1$ mm. Image brightness has been increased by 50%.[]{data-label="fig:movie_images"}](figure02.pdf){width="60mm"} The image analysis procedure involves measuring position and jet angle of each bubble. First a background image (without the bubble) is subtracted from each frame, and a binary threshold is applied to the image, defined so that the bubble appears in white and the rest of the image in black. A remaining hole on the bubble surface caused by the backlighting in the video is filled in. The bubble’s area is calculated in each frame and converted from pixels to mm$^2$. The bubble position can then be measured. The bubble position is defined to be the bubble’s centroid when it has reached its maximum size. The bubble is the most circular at this stage of expansion, so its center is best measured at this stage. The jet angle is determined by the direction of the bubble’s displacement between its initial expansion and first rebound, which was found to be accurate compared to directly measured jet angles within 0.2 rad by Tagawa and Peters [@cavitation; @in; @corners]. The rebound position is determined by the bubble’s centroid when it has reached its second maximum size. Although the bubble is not spherical at this point, it is symmetric around the jet axis (Figure \[fig:movie\_images\]g and \[fig:movie\_images\]h), so the centroid position still reflects the jet direction accurately. Experiments were done along linear axes through the square and equilateral triangle, chosen based on behavior of the mathematical prediction of bubble jet direction in different regions of these shapes. Two sets of experiments were done for the square and four were done for the equilateral triangle. Model ===== A collapsing bubble forms a jet when the velocity field around the bubble is not spherically symmetric, which occurs in the presence of solid boundaries [@microfluidic; @triangle; @and; @square; @flow; @field]. Local regions of high fluid velocity appear to deform and indent a bubble’s surface, leading to the eventual formation of a jet. Because of this, our model seeks to describe the fluid velocity at the bubble’s surface during the initial part of the collapse. We use potential flow theory to describe the fluid velocity field at the bubble surface. The flow field $\vec{U}$ of an inviscid, incompressible fluid such as water can be expressed as the gradient of a velocity potential $\phi$, such that $\vec{U}=\vec{\nabla}\phi$ [@textbook; @-; @Fundamentals; @of; @Fluid; @Mechanics] and such that $\phi$ satisfies Laplace’s equation $\nabla^2\phi=0$. We neglect viscosity and compressibility for the same reasons cited by Tagawa and Peters in similar experiments – viscous effects are relevant only within thin boundary layers in the timescale of our experiments and the bubble surface speeds are minimal just after it has reached its maximum size [@cavitation; @in; @corners]. We model the bubble as a three-dimensional point sink in the fluid and require the boundary condition that fluid cannot pass through any solid boundaries. A three-dimensional sink, as opposed to a two-dimensional sink is required because it correctly represents the radial fluid velocity produced by a spherically collapsing bubble [@cavitation; @in; @corners]. In Figure 3, the left column shows the different channel shapes we want to model. We can produce problems with identical boundary conditions using the method of images. By placing image sinks in symmetric, tessellating patterns, solid boundaries can be modeled. Figure \[fig:image\_sinks\]a displays the simplest method of images solution in which a bubble and solid boundary are replaced by two sinks equidistant from an imaginary boundary plane (dashed line). Because of symmetry, fluid does not flow perpendicular to the plane of symmetry between the two sinks – at any position on the plane, the sinks have equal and opposite pull in the direction perpendicular to the plane. In Figure \[fig:image\_sinks\], solid boundaries of several shapes are modeled by patterns of image sinks: (a) a single wall, (b) a corner, (c) parallel walls, (d) semi-enclosed parallel walls, (e) a rectangle, (f) an equilateral triangle, (g) an isosceles right triangle, and (h) a $30^{\circ}$-$60^{\circ}$-$90^{\circ}$ right triangle. Each subsequent pattern follows the principle of the simple example in Figure \[fig:image\_sinks\]a: patterns are completely symmetric around each plane made by a solid boundary (dashed lines), so along these boundary planes there is no flow perpendicular to the plane. From Figure \[fig:image\_sinks\]a through Figures \[fig:image\_sinks\]c-e, for example, we see a progression of reflections across each successive boundary added so that the pattern remains symmetric until it is reflected across all four boundaries of a rectangle. Thus, the boundary condition is satisfied by these patterns of image sinks. ![Modeling boundary conditions using method of image sinks. Image sinks are used to satisfy the solid boundary conditions for several different shapes: (a) flat wall, (b) $60^{\circ}$ corner, (c) parallel walls, (d) semi-enclosed parallel walls, (e) rectangle, (f) equilateral triangle, (g) isosceles right triangle, and (h) 30$^{\circ}$-60$^{\circ}$-90$^{\circ}$ right triangle. The patterns are symmetric around each of the boundaries (dashed lines), so there is no flow perpendicular to the boundary and the solid boundary condition is met.[]{data-label="fig:image_sinks"}](figure03.pdf){width="70mm"} Because of the uniqueness theorem, two physical setups that are identical within a region of interest and satisfy the same boundary conditions will produce the one and only unique fluid velocity field $\vec{U}$ within that region (for flow which is singly connected) [@method; @of; @images]. For example, in Figure \[fig:image\_sinks\]f, the region of interest – which contains a point sink within an equilateral triangle fluid region – is identical in both diagrams, and both physical setups satisfy a boundary condition that no fluid flows across the triangular boundary. For the specified solid boundaries in the left column, tessellating image sink patterns in the right column produce the exact same descriptions of the fluid velocity field that would occur, except they do so using point sinks, which are easier to describe analytically than a spherical bubble near complex solid boundaries. Although we are not the first to identify these method of images solutions [@periodic; @image; @cloud; @cavitation; @near; @wall; @and; @corners], here we show that we can apply this to the study of cavitation in polygonal channels. In the remainder of this section, we will determine the equations representing sink positions in each pattern (A), calculate the velocity field induced by the sinks (B), and find the jet angle as a function of bubble position (C). Image sink positions for the four polygons are given by the equations below, although different formulations may be devised for the same patterns. Sink position ------------- ### Rectangle In our model of a collapsing bubble within a rectangle having side lengths $l_x$ and $l_y$, sink positions are given by $$\begin{aligned} \vec{s}_{ij} &= x_{s_{ij}} \hat{x} + y_{s_{ij}} \hat{y} \notag \\ &= [(-1)^i \cdot x_b + i \cdot l_x] \hat{x} + [(-1)^j \cdot y_b + j \cdot l_y] \hat{y} \label{eq:sink_positions_sq}\end{aligned}$$ for $-\infty < i < \infty$ and $-\infty < j < \infty$. For a square, we use equal side lengths, $l_x = l_y = l$. The position given by $i = j = 0$ is the bubble position itself, $\vec{s}_{00} = x_b \hat{x} + y_b \hat{y}$, while the rest of the positions are image sinks introduced to fulfill the boundary conditions at the rectangle boundaries. ### Equilateral triangle In our model of a collapsing bubble within an equilateral triangle of side length $l$, sink positions are given by $$\begin{aligned} \vec{s}_{ijk} &= x_{s_{ijk}} \hat{x} + y_{s_{ijk}} \hat{y} \notag \\ &= [a_{ijk} \cos (\frac{k \pi}{3}) - b_{ijk} \sin(\frac{k \pi}{3}) + c_{ijk} \cos(\pi + \frac{k \pi}{3})] \hat{x} \notag \\ &+ [a_{ijk} \sin(\frac{k \pi}{3}) + b_{ijk} \cos(\frac{k \pi}{3}) + c_{ijk} \sin(\pi + \frac{k \pi}{3})] \hat{y}, \label{eq:sink_positions_tri_y}\end{aligned}$$ where $$\begin{aligned} a_{ijk} &= x_b + i \cdot \frac{3l}{2} - k^2 x_b \label{eq:velocity_field_tri_p} \\ b_{ijk} &= (-1)^j \cdot (y_b - \frac{h}{2}) + j \cdot h + \frac{-h + (-1)^i h}{2} + \frac{h}{2} \label{eq:velocity_field_tri_q} \\ c_{ijk} &= k^2 x_b \label{eq:velocity_field_tri_r}\end{aligned}$$ for $k \in \{-1, 0, 1 \}$, $-\infty < i < \infty$, and $-\infty < j < \infty$. The height $h$ is the triangle height $h = \sqrt{l^2 - (l/2)^2}$. The three values of $k$ represent rotations, each separated by $60^{\circ}$, of a base pattern used to construct a hexagonal pattern of image sinks. The position given by $i = j = k = 0$ is the bubble position itself, $\vec{s}_{000} = x_b \hat{x} + y_b \hat{y}$, while the rest of the positions are image sinks introduced to fulfill the boundary conditions at the triangle boundaries. ### Isosceles right triangle In our model of a collapsing bubble within an isosceles right triangle having side length $l$, sink positions are given by $$\begin{aligned} \vec{s}_{ijk} &= x_{s_{ijk}} \hat{x} + y_{s_{ijk}} \hat{y} \notag \\ &= \left[d_{ij} \cos(\frac{k \pi}{2}) - f_{ij} \sin(\frac{k \pi}{2}) + l \right] \hat{x} \notag \\ &+ \left[d_{ij} \sin(\frac{k \pi}{2}) + f_{ij} \cos(\frac{k \pi}{2}) \right] \hat{y} \label{eq:sink_positions_isosceles_right}\end{aligned}$$ where $$\begin{aligned} d_{ij} &= (-1)^i \cdot \left(x_b - \frac{l}{2} \right) + i \cdot l - \frac{l}{2} \label{eq:velocity_field_isosceles_right_x} \\ f_{ij} &= (-1)^j \cdot \left(y_b - \frac{l}{2} \right) + j \cdot l + \frac{l}{2} \label{eq:velocity_field_isosceles_right_y}\end{aligned}$$ for $k \in \{0, 1 \}$, $-\infty < i < \infty$, and $-\infty < j < \infty$. The two values of $k$ represent rotations, separated by $90^{\circ}$, of a base pattern used to construct the full pattern of image sinks. As with the equilateral triangle, the position given by $i = j = k = 0$ is the bubble position itself, $\vec{s}_{000} = x_b \hat{x} + y_b \hat{y}$. ### 30$^{\circ}$-60$^{\circ}$-90$^{\circ}$ triangle In our model of a collapsing bubble within a 30$^{\circ}$-60$^{\circ}$-90$^{\circ}$ triangle having hypotenuse of length $l$ (short side length $\frac{l}{2}$ and long side length $h = \sqrt{l^2 - (l/2)^2}$), sink positions are given by $$\begin{aligned} \vec{s}_{ijk} &= x_{s_{ijk}} \hat{x} + y_{s_{ijk}} \hat{y} \notag \\ &= \left[m_{ij} \cos(\frac{k \pi}{3}) - n_{ij} \sin(\frac{k \pi}{3}) \right] \hat{x} \notag \\ &+ \left[m_{ij} \sin(\frac{k \pi}{3}) + n_{ij} \cos(\frac{k \pi}{3}) + h \right] \hat{y}\end{aligned}$$ where $$\begin{aligned} m_{ij} = (-1)^i \cdot (x_b - \frac{3l}{4}) + i \cdot \frac{3l}{2} + (-1)^q \cdot \frac{3l}{4} \\ n_{ij} = (-1)^j \cdot (y_b - \frac{h}{2}) + j \cdot h - (-1)^q \cdot \frac{h}{2}\end{aligned}$$ for $k \in \{-1, 0, 1\}$, $q \in \{0, 1\}$, $-\infty < i < \infty$, and $-\infty < j < \infty$. The three values of $k$ represent rotations, separated by $60^{\circ}$, of a base pattern used to construct the full pattern of image sinks. Both positive and negative values of $\frac{3l}{4}$ and $\frac{h}{2}$ are required to produce an offset section that completes the pattern, hence the $(-1)^q$ coefficient. As with the previous two cases, the position given by $i = j = k = 0$ is the bubble position itself, $\vec{s}_{000} = x_b \hat{x} + y_b \hat{y}$. Velocity field -------------- The fluid velocity field $\vec{u}_s (x, y)$ induced by one 3D point sink at a position $\vec{s} = (x_s, y_s)$ is $$\vec{u}_s (x,y) = -\frac{Q}{4 \pi R^3}[(x - x_s) \hat{x} + (y - y_s) \hat{y}]$$ where $Q$ is the sink strength in terms of volume per unit time and $R = \sqrt{(x - x_s)^2 + (y - y_s)^2}$ is distance from the sink position. For a set of image sink positions $\{s_{ij}\}_{i, j \in \mathbb{Z}}$, the velocity field induced by all of these sinks is the sum of the velocity fields induced by each individual sink. This velocity field for the square is given by $$\begin{aligned} \vec{u} (x, y) &= \sum_{\substack{i, j = -\infty \\ (i, j) \neq (0, 0)}}^{\infty} \vec{u}_{s_{ij}} \label{eq:velocity_field_sq}\end{aligned}$$ for the isosceles right triangle is given by $$\begin{aligned} \vec{u} (x, y) &= \sum_{\substack{i, j = -\infty \\ k \in \{0, 1 \} \\ (i, j, k) \neq (0, 0, 0)}}^{\infty} \vec{u}_{s_{ijk}} \label{eq:velocity_field_isc}\end{aligned}$$ and for the equilateral triangle and $30^{\circ}$-$60^{\circ}$-$90^{\circ}$ triangle is given by $$\begin{aligned} \vec{u} (x, y) &= \sum_{\substack{i, j = -\infty \\ k \in \{-1, 0, 1 \} \\ (i, j, k) \neq (0, 0, 0)}}^{\infty} \vec{u}_{s_{ijk}} \label{eq:velocity_field_eq_306090}\end{aligned}$$ where sink positions $\vec{s}_{ij}$ for a square, or $\vec{s}_{ijk}$ for an equilateral triangle, isosceles right triangle, or $30^{\circ}$-$60^{\circ}$-$90^{\circ}$ triangle are as given in Equations $(1$-$11)$. We have excluded the velocity field produced by the bubble itself by the condition $(i, j) \neq (0, 0)$ or the condition $(i, j, k) \neq (0, 0, 0)$, as applicable. This is because the flow field induced by the bubble itself is radially symmetric around the bubble position, so it is the resultant velocity from the *image* sinks, not the bubble, that deforms its surface. Jet direction ------------- We assume that the resultant velocity induced at the bubble position by all image sinks gives the direction of the jet, meaning the jet direction $\theta_j$ is given by $$\tan{\theta_j} = \frac{u_y}{u_x} \label{eq:jet_angle}$$ where $u_x$ and $u_y$ are the velocity components at the bubble position; that is, they are the $x$ and $y$ components of $\vec{u}(x, y)|_{(x, y) = (x_b, y_b)}$ as given by Equations (13-15) [@Note1]. ![Model predictions of jet direction. Polygons are (a, b) a square, (c, d) an equilateral triangle, (e, f) an isosceles right triangle, and (g, h) a $30^{\circ}$-$60^{\circ}$-$90^{\circ}$ triangle. Each vector within the polygons represents the jet direction produced by a bubble nucleated at that position (a, c, e, g). Plotting predictions along continuous lines within the shapes produces prediction curves (b, d, f, h), which can be tested experimentally. Side length of each shape is chosen to be $1$, except for (g), which is plotted so it is half the equilateral triangle’s size (c). Prediction curves for the triangles (d, f, h) are plotted against a position variable $\widetilde{x}$ normalized by length of the line for ease of comparison.[]{data-label="fig:model_plots"}](figure04.pdf){width="84mm"} The jet direction $\theta_j$ for a bubble collapse at any point in the fluid can be calculated using Equation . Calculations of jet direction for bubble collapse at different points within the four polygons are displayed by vector fields in Figure \[fig:model\_plots\]a, \[fig:model\_plots\]c, \[fig:model\_plots\]e, and \[fig:model\_plots\]g. The vector fields in Figure \[fig:model\_plots\] are *not* fluid velocity fields, but display the jet direction for bubbles nucleated at different points within the shape. Thus, Figure \[fig:model\_plots\] represents the complete model of analytic predictions of jet direction in bubble collapse within a square, an equilateral triangle, an isosceles right triangle, and a $30^{\circ}$-$60^{\circ}$-$90^{\circ}$ triangle. From this analytic model, we can make general observations about bubble collapse within these four polygonal shapes. 3D sink model behavior ---------------------- The model confirms our expectation that bubble jets point perpendicular to solid boundaries when close to the boundaries. In other words, bubble jets generally point toward the closest wall or walls. We also observe from the jet direction arrows in Figure \[fig:model\_plots\]a that along axes of symmetry of the square, the jet points along the symmetry axis. This behavior can also be observed within the equilateral triangle (Figure \[fig:model\_plots\]c) and isosceles right triangle (Figure \[fig:model\_plots\]g). The 30$^{\circ}$-60$^{\circ}$-90$^{\circ}$ triangle does not have a symmetry axis. This bubble behavior along symmetry axes is similar to that observed along the symmetry axis of all corners of angle $\alpha = \frac{\pi}{n}$ [@cavitation; @in; @corners]. Model curves can be plotted along straight lines within the shapes. Representative curves for all shapes are plotted in Figure \[fig:model\_plots\]b, \[fig:model\_plots\]d, \[fig:model\_plots\]f, and \[fig:model\_plots\]h. Calculating predicted jet directions at different positions along straight lines within shapes is one way to see the change in the respective influence of the solid walls at varied positions. At the left and right edges of the graphs in Figure \[fig:model\_plots\]b, \[fig:model\_plots\]d, \[fig:model\_plots\]f, and \[fig:model\_plots\]h, for example, all predictions converge to the same angle such that jets are perpendicular to the leftmost or rightmost wall, demonstrating that the influence of a single wall dominates at positions very close to the wall. Similarly, along the top edge of the square (Figure \[fig:model\_plots\]a and \[fig:model\_plots\]b, $y = 0.4$) and equilateral triangle (Figure \[fig:model\_plots\]c and \[fig:model\_plots\]d, $y = 0.8$), the jet angle hovers around $\theta_j = 90^{\circ}$, except closer to the left and right edges, where the jet is influenced by the other boundaries. Closer to the center of the shapes (square – $y = 0.05$, triangle – $y = 0.55$ or $y = 0.6$), a sharp transition in jet angle occurs when the dominant influence switches from the left boundary to the right one. Additionally, the model predicts jet direction to be highly sensitive to changes in position in the middle of these shapes and, to a lesser degree, at the corners. These are areas where a large change in jet angle may occur with only a small change in position. We call this sensitivity to changes in position the ‘volatility’ in jet direction, which varies by location. Volatility is predicted by the model to be highest near the center and corners of the shapes, as depicted in Figure \[fig:volatility\]. Here we quantify volatility as the magnitude of the gradient of the jet angle (plotted as the logarithm). ![Volatility of jet direction angle. Within the (a) square, (b) equilateral triangle, (c) isosceles right triangle, and (d) $30^{\circ}$-$60^{\circ}$-$90^{\circ}$ triangle, a bubble’s jet angle changes faster over small distances in the center and corners of the shape. In these regions, a bubble’s jet direction is very sensitive to its position. Here we quantify volatility as the magnitude of the gradient of the jet angle, plotted as the logarithm.[]{data-label="fig:volatility"}](figure05.pdf){width="84mm"} These observations may be used to apply jets to specific directions within microfluidic channels of geometric cross sections, at least when the channel is large enough that the approximation of the bubble as a point sink is reasonable. Comparison with Experiment ========================== Results from experiments within the square channel and the equilateral triangle channel are plotted in Figures \[fig:square\_results\] and \[fig:triangle\_results\]. For each of the six experiments, we measured bubble jet direction at a series of positions along a single line within the shape. The positions $x$ and $y$ are non-dimensional lengths, meaning bubble positions (blue data points) are normalized by the side length of the shapes. A large number of trials were conducted at each position [@Note2]. Data points display the mean jet angle over these trials in the same position. Error bars are standard error of the mean. The model is plotted in a solid black line and is found to be in agreement with the data. Because bubbles are nucleated at positions that deviate slightly from the intended axis, the spread in position must be considered in how the model is displayed. Shaded regions in Figure \[fig:square\_results\] and Figure \[fig:triangle\_results\] represent the model prediction that results from considering the distribution of bubble $y$-positions and moving two standard deviations above and below the mean bubble $y$ position. In each experiment, the shaded region should thus represent the analytic prediction for 98% of bubbles produced. Deviations in $y$ position tend to be small, $\pm 2$-$3 \%$ of the shape dimension, except in the case of Figure \[fig:triangle\_results\]c, where the deviation in $y$ position is $\pm 6 \%$ of the shape dimension. ![Square channel results. The jet angle of bubbles collapsing at several points along the axes (a) $y = 0.3$ and (b) $y = 0.07$ within a square channel was measured. Data points are the mean of several trials. Error bars are standard error of the mean. The model is plotted in a solid black line. Because bubble nucleation did not occur exactly on the intended axis, this is taken into account in how the model is displayed. Shaded regions represent model predictions resulting from moving two standard deviations above and below the mean bubble $y$ position.[]{data-label="fig:square_results"}](figure06.pdf){width="84mm"} ![Equilateral triangular channel results. The jet angle of bubbles collapsing at several points along the axes (a) $y = 0.67$, (b) $y = 0.57$, (c) $y = 0.5$, and (d) $y = 0.42$ within an equilateral triangular channel was measured. Data points are the mean of several trials. Error bars are standard error of the mean. The model is plotted in a solid black line. Because bubble nucleation did not occur exactly on the intended axis, this is taken into account in how the model is displayed. Shaded regions represent model predictions resulting from moving two standard deviations above and below the mean bubble $y$ position.[]{data-label="fig:triangle_results"}](figure07.pdf){width="84mm"} Across all experiments bubbles had an average diameter of 1.4 mm. Bubbles within the equilateral triangle had a slightly smaller average diameter, 1.36 mm, compared to the average diameter of bubbles within the square, 1.48 mm, but the difference is within a standard deviation of  0.3 mm. A typical errorbar length is 2-4 degrees, with the exception of Figure \[fig:triangle\_results\]c, in which errorbars have an average length of 9 degrees (despite having a similar number of trials to the other experiments). This suggests that, as the model predicts (Figure \[fig:model\_plots\]c), jet angle is much more volatile closer to the middle of the equilateral triangle shape. We also note that near the center of the square shape (Figure \[fig:square\_results\]b, $y = 0.07$), data has a significant deviation from the model around $x = 0$. A visual inspection of the data compared to the model reveals a fair to good match. Quantitatively, the root-mean-square-deviations between the experimental points and the model are 11 degrees and 16 degrees in Figures \[fig:square\_results\]a and \[fig:square\_results\]b, respectively, and 9, 21, 5, and 5 degrees in Figures \[fig:triangle\_results\]a, \[fig:triangle\_results\]b, \[fig:triangle\_results\]c, \[fig:triangle\_results\]d, respectively. Discussion & Conclusion ======================= We measured the direction of liquid jets produced by collapsing vapor bubbles in fluid enclosed by channels with a square or equilateral triangular cross-section. Experiments demonstrated that the theoretical model based on potential flow analysis agrees well with data. In total we have developed analytic predictions of jet direction for bubble collapse within any rectangle (including squares as a special case), equilateral triangle, isosceles right triangle, and $30^{\circ}$-$60^{\circ}$-$90^{\circ}$ triangle. Previous research modeled jet direction of collapsing cavities within corners by satisfying solid boundary conditions using the method of images. Our solutions extend this method using infinitely repeating symmetric patterns, and allow for the prediction of jet direction within enclosed channels with polygonal cross-section. The model is a powerful tool for analytically predicting jet direction within certain polygonal channels. Our results reinforce Tagawa and Peters’ claim that the velocity field around a bubble in the very beginning of its collapse is responsible for its later migration and jet direction [@cavitation; @in; @corners]. In addition, we introduce an analytic solution for enclosed cavitation, which has previously been studied only by experiment [@microfluidic; @triangle; @and; @square; @bubble; @in; @a; @tube] or numerical analysis [@numerical; @calculation; @for; @a; @completely; @enclosed; @bubble]. It is a key physical insight that the model does not describe the motion of the entire bubble surface throughout the collapse. Despite the complex shapes a bubble takes during collapse, describing the direction of its jet does not require a detailed model of the bubble surface boundary throughout the time of the collapse process. The bubble’s jet direction can be accurately predicted simply by observing the asymmetric flow field at its surface at the beginning of its collapse, which is the only thing our model takes into account. Bubbles in our experiments had a typical diameter of $0.1l$, where $l$ is the side length of the shape. No atypical deformations are observed for bubbles this size besides the usual jetting expected near a solid boundary. It is expected that with larger bubbles interesting deformations will occur. For example, our experiments resemble the cavitation experiments in microfluidic systems performed by Zwaan et al. [@microfluidic; @triangle; @and; @square]. In their case, the bubble was of comparable size to the enclosing solid boundary shape and the bubble’s collapse was considered a two-dimensional process. Deformations as well as multiple jetting was observed. Their experimental data was compared to a numerical model. Here, the bubble’s collapse is three-dimensional and an analytic solution predicting jet direction is presented. A goal of this research was to use a simple method to predict the direction of bubble jets in increasingly complex physical situations closer to those that may be encountered in human-made devices or in nature. In the realm of human-made devices, this research improves understanding of a bubble’s collapse behavior within enclosed channels, and this is, for example, applicable to microfluidic channels with a polygonal cross-section. The results are a starting point for studying how cavitation may be used more effectively in mixing or cleaning in channels of different designs. Since the method of images is less computationally expensive than numerical solutions [@periodic; @image; @cloud], it is a practical tool for understanding these applications. It is of theoretical benefit to find the most complex situations for which the relatively simple method of images can work. The method of images provides a better theoretical understanding of the influence of the geometry of the solid walls on collapsing bubbles’ jet formation. It appears that with the patterns presented in Figure \[fig:image\_sinks\] we have reached or are close to the limit of the method of images solutions under the conditions that the pattern (i) uses some (possibly infinite) number of point sinks that (ii) have equal strength and (iii) are arranged in a two-dimensional planar pattern. The authors suspect that the rectangle, equilateral triangle, isosceles right triangle, and $30^{\circ}$-$60^{\circ}$-$90^{\circ}$ triangle are the only enclosed polygons that can be modeled this way. This is because it seems no more shapes satisfy conditions required for this method to be applicable. First, we know that we cannot model polygons with more than four sides – for example, regular pentagons, hexagons, heptagons, etc. – because they have obtuse corner angles. For obtuse corner angles (angle greater than $\frac{\pi}{2}$), proper reflection of image sinks across each boundary eventually means that image sinks are placed within the region of interest rather than the boundary region, which will not produce the solution of a single bubble within the fluid region enclosed by the shape. Second, each solid boundary must have a mirror reflection of the image sink pattern on each side of the boundary plane. In practice, we have found that this means the boundary shape must be able to fully tessellate the plane. Only a small set of *regular* shapes, the equilateral triangle, square, and hexagon, tessellate the plane. (We have not modeled the hexagon because of its obtuse corner angles.) Irregular polygons were also explored. However, they only satisfy the second condition (to tessellate the plane) in a few cases, which are related nicely to regular polygons: the rectangle, isosceles right triangle, and $30^{\circ}$-$60^{\circ}$-$90^{\circ}$ triangle (see Fig. \[fig:image\_sinks\]). Considering these conditions, it appears that we have identified the four special enclosed polygonal channel shapes for which the method of images works to predict jet directions for collapsing bubbles. From a theoretical standpoint, these results encourage finishing the catalog of method of images solutions. It appears that we may have reached the limit of two-dimensional shapes that can be modeled by patterns of equal strength image sinks. However, there remain extensions to three dimensional shapes such as tetrahedra and dodecahedra, which can tessellate in three-dimensional space, as well as simpler extensions of the present results such as modeling cavitation within half-enclosed and fully-enclosed rectangular and triangular prisms. From a computational standpoint, there are many avenues by which these results can be expanded. While the method of images provides geometrical limitations if we desire analytical solutions, there are no such geometrical limitations for numerical methods. For bubble collapse within any shape, the flow produced by a point sink in a fluid domain with the appropriate solid boundary conditions could be solved numerically and jet direction could be predicted by finding the resultant fluid velocity at the bubble position. 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--- address: | High Energy Physics Division\ Argonne National Laboratory\ Argonne IL 60439\ [E-mail: [email protected]]{} author: - MAURY GOODMAN title: NEW PROJECTS IN UNDERGROUND PHYSICS --- Introduction ============ Large numbers of particle physicists first went underground in the early 1980’s to search for nucleon decay. Atmospheric neutrinos were a background to those experiments, but the study of atmospheric neutrinos has spearheaded tremendous progress in our understanding of the neutrino. Since neutrino cross sections, and hence event rates are fairly small, and backgrounds from cosmic rays often need to be minimized to measure a signal, many more other neutrino experiments are found underground. This includes experiments to measure solar $\nu$’s, atmospheric $\nu$’s, reactor $\nu$’s, accelerator $\nu$’s and neutrinoless double beta decay. In the last few years we have seen remarkable progress in understanding the neutrino. Compelling evidence for the existence of neutrino mixing and oscillations has been presented by Super-Kamiokande[@bib:sk1] in 1998, based on the flavor ratio and zenith angle distribution of atmospheric neutrinos. That interpretation is supported by analyses of similar data[@bib:sim] from IMB, Kamiokande, Soudan 2 and MACRO. And 2002 was a miracle year for neutrinos, with the results from SNO[@bib:sno] and KamLAND[@bib:kamland] solving the long standing solar neutrino puzzle and providing evidence for neutrino oscillations using both neutrinos from the sun and neutrinos from nuclear reactors. Finally, the K2K long-baseline neutrino oscillation experiment[@bib:k2k] has the first indications that accelerator neutrinos also oscillate. In preparation for this meeting, I asked the previous speaker, Sandip Pakvasa from the University of Hawaii, “What are the outstanding issues in the neutrino sector for future experiments to address?" His list was: 1. See the dip in atmospheric $\nu$ L/E distribution 2. Measure the whole solar $\nu$ energy spectrum. 3. Determine the value of Ue3 (${\theta_{13}}$). 4. Are CP violation effects large? 5. If so, what is the CP phase $\delta$? 6. Can we see the $\tau$’s in ${\nu_\mu \rightarrow \nu_\tau}$ oscillation? 7. Can we settle the LSND question? 8. What are the absolute $\nu$ masses? 9. Is the neutrino Majorana or Dirac? 10. Is the mass hierarchy normal or inverse? 11. Are there astrophysical sources of $>$TeV $\nu$’s? Indeed, those questions are just the ones that have motivated neutrino physicists, and there are a plethora of new proposals for future projects at underground laboratories and accelerators. These include the study of solar, atmospheric, reactor, accelerator and astrophysical sources of neutrinos. There is also the direct search for neutrino mass in Tritium beta decay, and the search for Majorana masses in neutrinoless double beta decay. In this paper I will review some underground projects focusing on those not found elsewhere on the agenda. I will start by discussing major existing and proposed facilities where underground physics research takes place. In Sections 3 and 4 I will briefly comment on plans for new real time solar neutrino experiments and site issues for future off-axis long-baseline neutrino experiments to measure ${\theta_{13}}$.[@bib:harris] In Section 5 I will present my thoughts about new reactor neutrino experiments to measure ${\theta_{13}}$. In Section 6 I will return to nucleon decay and describe the UNO experiment. Underground Laboratories for Neutrino Research ============================================== The nicest facility for underground physics is located at the Gran Sasso Laboratory in a mountain road tunnel in Italy. It is operated there by INFN for experimentalists from throughout the world. There are three large halls in which a number of neutrino experiments have operated and are being built. These include the Gallex and GNO experiments which have measured the pp solar neutrinos; the MACRO experiment which measured the angular distribution of atmospheric neutrinos; Borexino which is being built to measure the solar neutrinos from $^7Be$; the Heidelberg-Moscow neutrino-less $\beta\beta$ decay experiment on Germanium, which is a the most sensitive search to date; the LVD detector which is searching for neutrinos from supernovae; and the OPERA and ICARUS projects which will measure neutrinos associated with the CERN Neutrinos to the Gran Sasso (CNGS) program starting in 2006. The Kamioka mine is located in the Japanese Alps on the western side of Japan. It is home to the 50 kiloton Super-Kamiokande water Cerenkov detector. In November 2002 it started running again, with half the photo-tube coverage that it had before its accident in December 2001. Nearby is the KamLAND detector which looks at reactor neutrino disappearance from 26 reactors throughout Japan. There are also early plans to put the one megaton Hyper-Kamiokande experiment in a cavern in the Tochibora mine, about 3 km south of the present Mozumi mine. That could pair with the present Super-Kamiokande facility as an off-axis site from the new JPARC accelerator at Tokai. The Baksan facility is located in the Caucasus mountains in southern Russia. The Gallium solar $\nu$ experiment SAGE is located in a deep section with a minimum overburden 4700 MWE. The Gallium experiment was crucial in confirming the solar neutrino deficit. Also located at Baksan is the 4 layer Baksan Underground Scintillation Telescope (BUST) located at an minimum overburden of 850 MWE. BUST has been studying the zenith angle distribution of upward atmospheric induced neutrino events, which is relevant to the atmospheric neutrino deficit. The Sudbury Neutrino Observatory (SNO) is taking data that has provided revolutionary insight into the properties of neutrinos and the core of the sun. The detector is in INCO’s Creighton mine near Sudbury, Ontario. SNO uses 1000 tons of heavy water, on loan from Atomic Energy of Canada Limited (AECL), contained in a 12 meter diameter acrylic vessel. Neutrinos which react with the heavy water (D2O) are detected by an array of 9600 photomultiplier tubes. The detector laboratory is extremely clean to reduce background signals from radioactive elements. Besides the heavy water detector, the Canadian government has recently funded a new international facility for underground science called SNOLAB which will provide a low background facility nearby. The Soudan Underground Physics Laboratory is located in a facility maintained by the State of Minnesota Department of Natural Resources and operated as a tourist attraction as part of northern Minnesota’s iron range. Two connected laboratory spaces are located on the 27th level with an overburden of 2100 MWE. The nucleon decay experiment Soudan 2[@bib:s2] operated in one hall from 1986-2001. Also in that hall the Cryogenic Dark Matter Search (CDMS) is being installed. The 5.6 kton MINOS iron toroid detector which will serve as the far detector for the Fermilab long-baseline neutrino experiment is 96% installed as of May 2003. The Frejus laboratory is in a tunnel between France and Italy. It was also the site of a proton decay experiment which ran from 1984-1990. The construction of a parallel safety tunnel has been motivated by the 2000 accident in the Mont Blanc tunnel. That provides the opportunity for considering construction of a large new scientific laboratory space that could be used for a large new detector such as UNO. Such a detector, 190 km from CERN, would be useful in conjunction with a new neutrino superbeam from CERN using the proposed Superconducting Proton LINAC (SPL) or a beta beam using the SPL injecting $^6$He or $^{18}$Ne from ISOLDE into the SPS and a new storage ring.[@bib:beta] The Waste Isolation Pilot Plant, or WIPP facility, in Carlsbad New Mexico was built to store low-level radioactive waste from the US military. It is the world’s first underground repository licensed to safely and permanently dispose of transuranic radioactive waste left from the research and production of nuclear weapons. After more than 20 years of scientific study, public input, and regulatory struggles, WIPP began operations on March 26, 1999. It is considered as a possible site for the supernova experiment OMNIS which will feature a measurement of the neutral current events from supernovae; the 400 kiloton proton decay experiment UNO; and EXO, a new neutrino-less $\beta\beta$ decay experiment which uses Xenon. Experimental facilities located in the salt are sufficiently removed from the radioactive waste that they present no backgrounds to any of these experiments. There is considerable interest in the United States in developing a National Underground Laboratory Facility (NUSEL) for future neutrino experiments. A panel was appointed in 2002 by the U.S. National Research Council to study future neutrino facilities, with an emphasis on NUSEL and also a neutrino telescope ICE-CUBE. In its conclusion section, they wrote[@bib:nufac]: “In summary, our assessment is that a deep underground laboratory in the US. can house a new generation of experiments that will advance our understanding of the fundamental properties of neutrinos and the forces that govern the elementary particles, as well as shedding light on the nature of the dark matter that holds the Universe together. Recent discoveries about neutrinos, as well as new ideas and technologies make possible a broad and rich experimental program. Considering the commitment of the U.S. community and the existing scientific leadership in this field, the time is ripe to build such a unique facility." The favored location for NUSEL is the Homestake South Dakota mine where the famous Davis experiment ran for many years. There is also a proposal to locate the laboratory with horizontal access in a new facility near San Jacinto California. India was home to one of the earliest and deepest underground facilities with the KGF mine and experiment from the 60’s through the early 90’s. Now a group from several institutes throughout India is studying the possibility of a new neutrino observatory to measure atmospheric neutrinos and be the possible site for a long-baseline neutrino program from a neutrino factory. Two sites which have hydro-electric projects near large hills are being considered. The PUSHEP site (Pykara Ultimate Stage Hydro Electric Project) near Ooty in Tamil Nadu is located in a vein of high quality rock, and is also close to a high altitude cosmic ray facility. The RAMMAM Hydro Electric Project site is located in the Himalayas near Darjeeling, and has the possibility of a laboratory with a much greater overburden, suitable for solar neutrino experiments. A project team has gotten a grant to study both sites and design a 30 kiloton Iron/RPC calorimeter for atmospheric neutrinos. Real Time Solar Neutrino Experiments ==================================== Ray Davis, a winner of the 2002 Nobel prize in physics, started underground physics in the 1960’s with his Homestake solar neutrino experiment. Solar neutrino experiments have the most stringent background requirements to date which has generally made those experiments the deepest, and driven the depth requirements for consideration of a multi-purpose underground laboratory. Previous solar neutrino experiments have involved chemical extraction of chlorine or gallium. The next solar neutrino experiments will be Borexino and KamLAND,[@bib:suzuki] which will measure solar neutrino experiments in real time. Ideas for future real-time solar neutrino experiments are listed in Table \[tab:solar\]. Experiment[@bib:noi] Status Feature ---------------------- -------------------- ---------------------- Borexino under construction scintillator KamLAND running scintillator HELLAZ R&D Liquid Helium XMASS R&D Liquid Xenon CLEAN R&D Liquid Neon HERON R&D cryogenic LENS R&D $^{176}$Yb MOON R&D $^{100}$Mo GENIUS R&D $\nu e$ in Germanium : Possible Future Real Time Solar Neutrino Experiments[]{data-label="tab:solar"} Can Off-axis Experiments be at the Surface? =========================================== The long-baseline programs that are or soon will be running are in Japan (K2K), Europe (CNGS) and the United States (NuMI/MINOS). The far detectors for all three projects are located deep enough underground that there are negligible cosmic rays mimicking beam induced neutrino interactions. There was a previous proposal at Brookhaven[@bib:889] for a long-baseline neutrino experiment on the surface using a water Cerenkov Detector. In their proposal, they performed the only study of long-baseline neutrino backgrounds in the relevant energy region of which I’m aware. A new long-baseline experiment is being proposed that would use a 50 kton detector to search for ${\nu_\mu \rightarrow \nu_e}$ in the NuMI beam[@bib:p929] as evidence for a non-zero value of ${\theta_{13}}$. The beam spill with one-turn extraction from the Fermilab Main Injector will be $10\mu s$ long. Each year, assuming $2\times 10^7$ pulses and an area of about 1000 m$^2$, there will be 1.2 $\times~10^7$ muons and 8 $\times~10^5$ neutrons ($E~>~0.1$ GeV) going through a detector on the surface. There are four possible handles to reduce these backgrounds: 1. overburden 2. active veto 3. pattern recognition 4. bunch timing The last option would drive up the cost of electronics and cannot reduce the time window substantially. Of course most muons do not look like contained events, but the existence of cracks or other dead regions in the detector will make muons a problem. BNL889 had qualitatively different pattern recognition issues than the FNAL off-axis experiment will experience, but it was going to be a homogeneous detector without cracks. I scale and compare the backgrounds for illustrative purposes only. NuMI off-axis will need a rejection of 2.1 $\times~10^6$ for $\mu$’s and $1.6~\times~10^5$ for neutrons, to obtain Signal/Background $>$ 1 for $\sin^2 2{\theta_{13}}= 0.01$. BNL P889 calculated that they would achieve a rejection of 7 $\times~10^5$ for $\mu$’s and 3.4 $\times~10^2$ for neutrons using an active veto and pattern recognition for electrons 0.3 to 4 GeV. I conclude that operating NuMI-off-axis on the surface without overburden will be challenging. Reactor Experiments ===================  From the discovery of the neutrinos by Reines and Cowan[@bib:reines] at Savannah River to the evidence for $\bar{\nu}_e$ disappearance at KamLAND[@bib:kamland], reactor neutrino experiments have studied neutrinos in the same way – observation of inverse beta decay with scintillator detectors. Since the signal from a reactor falls with distance L as 1/L$^2$, as detectors have moved further away from the reactors over the years, it has become more important to reduce backgrounds. That can be done by putting experiments underground; and experiments one kilometer or more away from reactors (Chooz, Palo Verde and KamLAND) have been underground. The KamLAND experiment measured a 40% disappearance of $\bar{\nu}_e$ presumably associated with the 2nd term in Equation \[eq:p\]: $$\label{eq:p} P(\bar{\nu}_e \rightarrow \bar{\nu}_e) \cong - \sin^2 2 \theta_{13} \sin^2(\Delta m^2_{atm} L/4E) - \cos^4 {\theta_{13}}\sin^2 2 \theta_{12} \sin^2(\Delta m^2_{12} L/4E) + 1$$ The Chooz and Palo Verde data put a limit on ${\theta_{13}}$ (through the first term in Equation \[eq:p\]) of $\sin^2 2 \theta_{13} < 0.1$. Those experiments could not have had greatly improved sensitivity to ${\theta_{13}}$ because of uncertainties related to knowledge of the flux of neutrinos from the reactors. They were designed to test whether the atmospheric neutrino anomaly might have been due to ${\nu_\mu \rightarrow \nu_e}$ oscillations, and hence were searching for large mixing. Any new experiment to look for non-zero values of ${\theta_{13}}$ would need the following properties: - two or more detectors to reduce uncertainties to the reactor flux - identical detectors to reduce systematic errors related to detector acceptance - carefully controlled energy calibration - low backgrounds and/or reactor-off data In Equation \[eq:p\], the values of $\theta_{12}$, $\Delta m^2_{12}$ and $\Delta m^2_{atmo}$ are approximately known. In Figure \[fig:P\], The probability of $\bar{\nu}_e$ disappearance as a function of L/E is plotted with ${\theta_{13}}$ assumed to be at its maximum allowed value. Note that CP violation does not affect a disappearance experiment, and that matter effects can be safely ignored in a reactor experiment. The large variation in P for L/E$>$ 10 km/MeV is the effect seen by KamLAND and solar $\nu$ experiments. The much smaller deviations from unity for L/E $<$ 1 km/MeV are the goal for an accurate new reactor experiment. The optimization of detector distances for such a new experiment is straightforward. The statistical power comes from measuring a deficit of $\bar{\nu}_e$ (up to a few percent) at the far detector, along with a change in the energy spectrum consistent with that deficit. Up to systematic errors in the rate and energy spectrum, one can construct a $\chi^2$ difference between a near detector and a far detector. In Figure \[fig:r1\], the detector location has been fixed for each curve and the statistical power of these tests calculated as the location of the second detector is varied. In Figure \[fig:r2\], one detector has been fixed at 1000 m while the location of the second detector is varied with a 1% systematic error folded in. The different curves in that plot reflect uncertainty in the parameter $\Delta m^2_{atm}$. The optimum locations for detectors is thus sensitive to eventual systematic errors as well as oscillation parameters. But a near detector around 100m and a far detector around 1000 m will be near the optimum. Since the civil construction of laboratories might contribute half or more to the cost of an experiment, finding existing labs at those locations may change the optimization. A list of possible sites for a new reactor experiment is included in Table \[tab:sites\], along with a tabulation of previous reactor experiment sites. Two specific proposals have been made. The KR2DET[@bib:kr2det] experiment would use the Krasnoyarsk reactor in Russia, and two detectors located at 115 and 1000 m. That site has two attractive features: 1) There is an entire city built at 600 MWE depth with possible locations for the detector sites, and 2) The fuel cycle for that reactor is better understood than for most reactors because the fuel is changed every few months rather than years. The other specific idea is the Kashiwazaki[@bib:kash] site in Japan, the location of 7 reactors. It is the most powerful reactor complex in the world, 24 Gw. The reactors come in one cluster of 4 and one cluster of 3, so there would need to be two near detectors and one far detector, possibly located in shafts created by a large drill. Other sites in the United States, France, Taiwan and Brazil are being considered. The site in Taiwan may be interesting because of an existing road tunnel 2 km from the reactor.[@bib:taiwan] A sensitivity of 0.02 in $\sin^2 2{\theta_{13}}$ can be achieved with as little as 250 ton-Gigawatt-years, while an exposure of 8000 ton-Gigawatt-years may be required to achieve a sensitivity of 0.01.[@bib:huber] A two or more detector reactor experiment seems to be an attractive option as part of the search for ${\theta_{13}}$. It can probably find a non-zero value for ${\theta_{13}}$ faster and less expensively than an off-axis experiment. It does not face the degeneracies regarding CP parameters and the sign of ${\Delta m^2}$, and hence cannot address those issues. But a measurement of ${\theta_{13}}$ by reactors followed by optimized off-axis experiments would together measure neutrino parameters with much less uncertainty due to degeneracies and correlations. Reactor Location L Power Overburden Detector Mass -------------- ------------ ---------- ---------- ------------ --------------- Chooz France 1100 m 8.5 Gw 300 MWE 5 ton Bugey France 49/95 5.6 16 1/0.5 Palo Verde Arizona 890 11.6 32 11.3 KamLAND Japan $<180>$ 200 (26) 2700 1000 Krasnoyarsk Russia 115/1000 1 600 46 Diablo Canon California $\sim$ 1 6.1 600 Wolf Creek Kansas $\sim$ 1 3.2 Boulby UK 25 3.1 2860 Heilbronn Germany 19.5 6.4 480 Kashiwazaki Japan 1.7 24.3 20 ton Texono Taiwan 2.0 4.1 Angra Brazil 4.0 IMB Ohio 10 1.2 1570 : Past Reactor Sites and Future Possibilities\[tab:sites\] UNO === One of the reasons for the tremendous progress in understanding the neutrino has been the fact that several detectors were built underground to search for nucleon decay. They haven’t found nucleon decay, but have made a number of other discoveries. Perhaps the lesson is that we should build yet another detector to look for nucleon decay. And maybe we will find that the nucleon decays! The UNO detector[@bib:uno] is proposed as a next generation underground water Cerenkov detector that probes nucleon decay beyond the sensitivities of the highly successful Super-Kamiokande (Super-K) detector utilizing a well-tested technology. \[1\] The baseline conceptual design of the detector calls for a “Multi-Cubical" design with outer dimensions of 60x60x180 m3. The detector has three optically independent cubical compartments with corresponding photo-cathode coverage of 10%, 40%, and 10%, respectively. The total (fiducial) mass of the detector is 650 (440) kton, which is about 13 (20) times larger than the Super-K detector. The discovery potential of the UNO detector is multi-fold. The probability of UNO discovering proton decay via vector boson mediated $e^+\pi^0$ mode is predicted to be quite high ($\sim$50%) in modern Grand Unification Theories (GUTs). Water Cerenkov technology is the only realistic detector technology available today to allow a search for this decay mode for proton lifetimes up to 10$^{35}$ years. If the current super-symmetric GUT predictions are correct, UNO can discover proton decay via $\nu K^+$ mode. The important design issue for any nucleon decay detector is to keep backgrounds low while maintaining high efficiency. UNO is able to take advantage of the understandings about these two modes that have come from extensive analysis in the Super-Kamiokande experiment. Modeling the backgrounds and efficiencies in an UNO sized detector, the sensitivity for these two modes is plotted versus exposure in Figure \[fig:uno\]. While further advances are possible, it is seen that substantial increase in proton decay sensitivity would be achieved. In addition to nucleon decay, UNO will be sensitive to a large variety of other interesting topics. UNO will be able to detect neutrinos from supernova explosions as far away as the Andromeda galaxy. In case of a galactic supernova explosion, UNO will collect $\sim$100k neutrino events from which the millisecond neutrino flux timing structure can be extracted. This could provide us with an observation of black hole formation in real-time as well as a wealth of information to precisely determine the core collapse mechanism. Discovery of supernova relic neutrinos (SRN) is within the reach of UNO. SRN could very well be the next astrophysical neutrinos to be discovered. The predicted values of the SRN flux by various theoretical models are only up to six times smaller than the current Super-K limit. Some models have been already excluded. With much larger fiducial mass and lower cosmogenic spallation background, UNO situated at a depth 4000 MWE can cover all of the predicted flux range. UNO is an ideal distant detector for a long-baseline neutrino oscillation experiment with neutrino beam energies below about 10 GeV providing a synergy between the accelerator and the non-accelerator physics. UNO provides other opportunities, such as the ability to observe oscillatory behavior and appearance in the atmospheric neutrinos; precision measurement of temporal changes in the solar neutrino fluxes; and searches for astrophysical point sources of neutrinos and dark matter in an energy range difficult for larger, more coarse-grained undersea and under-ice detectors to cover. \[fig:uno\] Summary ======= There is a tremendous diversity of future neutrino projects and a number of interesting measurements to be made, almost all of them involving detectors located at facilities underground. Even with the tremendous progress we have already seen in the neutrino field, we are well on our way to answering the questions that were posed in Section 1. Acknowledgments =============== Thanks to Sandip Pakvasa and Chang Kee Jung for providing material for this talk. Thanks to Mike Shaevitz, Jerry Busenitz and Dave Reyna for information on reactor neutrino projects. [99]{} Y. Fukuda et al., Phys.Rev.Lett. 81 (1998) 1562-1567. D. Casper et al., Phys. Rev. Lett. 66 (1991) 2561; R. Becker-Szendy et al., Phys. Rev. D46 (1992) 3720; K.S. Hirata et al., Phys. Lett. B205, (1988) 416; K.S. Hirata et al., Phys. Lett. B280, (1992) 146; M. Sanchez et al., to be published by the Soudan 2 collaboration; M. Abrosio et al, Phys. Lett. B434 (1998) 451. Ahmad et al., Phys. Rev. Lett. 89 (2002) 011301. Eguchi et al., Phys. Rev. Lett. 90 (2003) 021802. Ahn et al., Phys. Rev. Lett. 90 (2003) 041801. Debbie Harris, these proceedings. W.W.M Allison et al., Phys.Lett. B391 (1997) 491-500. Mauro Mezzeto, “Physics Reach of the Beta Beam", hep-ex/0302007 B. Barish et al., “Neutrinos and Beyond: New Windows on Nature“ available at http://www7.nationalacademies.org/bpa/Neutrinos\_Sum.pdf A. Suzuki, these proceedings The web pages of most of these projects can be found on the ”solar neutrino" page of the neutrino oscillation industry: http://www.neutrinooscillation.org/ Dave Ayres et al., (2002) hep-ex/0210005. P.Huber et al., hep-ph/0303232. D. Beavis et al., BNL proposal 889, Report No. BNL52459, April 1995. Reines and Cowan Y. Kozlov et al., Nucl.Phys.Proc.Suppl. 87 (2000) 514-516. H. Minakata et al., hep-ph/0211111. Harry Wong, personal communication. M. Goodman et al., “Physics Potential and Feasibility of UNO", June 2001; C.K. Jung and C. McGrew, “UNO (Underground Nucleon decay and Neutrino Observatory)", February 7, 2003 at http://ale.physics.sunysb.edu/uno/UNO\_Narrative.pdf

--- address: 'Department of Mathematics, Brandeis University, MA' author: - 'R. İnanç Baykur' date: '06.17.2008.' title: | Handlebody argument for\ modifying achiral Lefschetz singularities --- In [@B1] we gave a handlebody description of a broken Lefschetz fibration on ${{\mathbb{CP}}{}^{2}}$ as a counterexample to Gay and Kirby’s conjecture on the necessity of negative Lefschetz singularities for generalized fibrations on arbitrary $4$-manifolds, and pointed out how this picture could be used to modify any given broken achiral Lefschetz fibration to a genuine broken Lefschetz fibration. Our general argument makes use of the following handlebody picture of a broken Lefschetz fibration over a disk, which can replace a regular neighborhood of a fiber with negative Lefschetz singularity: ![The broken Lefschetz fibration over a disk to replace the neighborhood of a negative node.[]{data-label="AchiralModification"}](AchiralModification) Here the diagram is drawn from the ‘higher side’, where there are three Lefschetz handles and a round $2$-handle attached to a fiber of genus $g+1$; so over the boundary of the base disk we have fibers of genus $g$. The $2$-handles corresponding to Lefschetz handles have fiber framing minus one, and the $2$-handle of the round $2$-handle given in red has fiber framing zero. The reader can turn to [@B0] for the conventions we use to depict broken Lefschetz fibrations in this note. To show that both fibrations have the same total space we proceed as follows: Using the $0$-framed $2$-handle of the round $2$-handle we split the diagram as in Figure \[AchiralSimplification\], where we also switch to the dotted-circle notation to perform the rest of our handle calculus. We can now cancel the $2$-handle of the round $2$-handle against the $1$-handle it is linked with. Keeping the upper part of the diagram with $2g$ $1$-handles and the $0$-framed $2$-handle as it is, we will simplify the remaining part of the diagram where we have one $1$-handle, three $2$-handles, and a $3$-handle. We first slide the $(+1)$-framed $2$-handles over the $(-2)$-framed $2$-handle to separate them from the bottom left $1$-handle, and cancel this $1$-handle against this $(-2)$-framed $2$-handle. We then get two $(+1)$-framed unknots, linking once. One more handle slide separates an unknotted $2$-handle with framing $0$, and this $2$-handle can be canceled against the $3$-handle. The $(+1)$-framed $2$-handle we are left with still links with the $1$-handle contained in the upper part of the diagram, thus giving us a handlebody picture of a genus $g$ fibration over a disk and with one negative Lefschetz singularity. ![Simplification of the handle diagram.[]{data-label="AchiralSimplification"}](AchiralSimplification) Relying on the construction of Gay and Kirby in [@GK], the above argument yields a handlebody proof of the existence of broken Lefschetz fibrations on arbitrary closed smooth oriented $4$-manifolds. (Another handlebody proof of this existence result was later given by Akbulut and Karakurt, where the achirality is avoided in a rather different way; see [@AK].) The purpose of the current note is to reconstruct our picture locally on a twice punctured torus in the fiber and with a $3$-fold symmetry so as to provide a comparable picture with Lekili’s achiral modification argument that uses singularity theory, given in Section 6 of his paper [@Lek]. One can certainly localize the handlebody picture we had in Figure \[AchiralModification\] by throwing away the $0$-framed $2$-handle corresponding to the fiber and all the $1$-handles but the three $1$-handles the Lefschetz $2$-handles are linked with. However, in order to achieve the $3$-fold symmetry, we also need to rearrange the Lefschetz $2$-handles. This symmetric picture and many observations contained in the following paragraphs arose during the author’s conversations with David Gay. ![The handlebody picture with a $3$-fold symmetry, which is obtained by rotating the diagram $2 \pi / 3$ degrees clockwise while shifting the order of the Lefschetz $2$-handles (towards the page) by one. All the $2$-handles but the $2$-handle of the round $2$-handle given in red have fiber framing minus one.[]{data-label="HandlePicture"}](HandlePicture) In Figure \[HandlePicture\] we see the twice punctured torus fiber, along with the vanishing cycles of three (positive) Lefschetz handles and a red curve which is the $2$-handle of the round $2$-handle. The $3$-handle completes the round $2$-handle. To show that this picture indeed gives a broken Lefschetz fibration over a disk one only needs to check that the attaching sphere of the $2$-handle of the round $2$-handle is sent to itself when the three Dehn twists prescribed by the Lefschetz handles are applied to it. Let us label the curves on the fiber as in Figure \[Stabilization\]. After applying the first right-handed Dehn twist along $C_1$ one sees that the image of $C$ can be isotoped so that it is positioned with respect to the curve $C_2$ as $C$ was positioned with respect to $C_1$ in the first place (Figure \[Stabilization\]). From the obvious $3$-fold symmetry one can conclude that after applying the right-handed Dehn twists along $C_1$, $C_2$ and $C_3$, the curve $C$ gets mapped onto itself (with the same orientation). ![On the left: The vanishing cycles on the fundamental domain of a twice punctured torus. On the right: The image of $C$ under the right-handed Dehn twist along $C_1$, represented by $C_1(C)$, can be isotoped on the fiber to the dashed red curve.[]{data-label="Stabilization"}](Stabilization) It remains to verify two things: First is to see that the total space of this fibration (which in fact is the $4$-ball) is the same as that of a fibration with an annulus fiber and a single negative Lefschetz singularity attached along a separating curve on an annulus. The required calculus for this is similar to the one we have given above, and will be left to the reader. Secondly, we double check that the reduced monodromy of the fibration over the boundary of the base disk in the former fibration is equivalent to that of the latter. This allows us to interchange these pieces while matching the fibrations along the boundary, and therefore to extend the given fibrations on the rest. For this, recall that the mapping class group of the annulus is generated by a Dehn twist along a boundary parallel curve. So it suffices to understand the effect of the reduced monodromy on a simple arc $A$ that runs from one boundary component to another; see Figure \[ReducedMonodromy\]. The curve $C_1$ does not intersect $A$, so the first nontrivially acting curve is $C_2$. The image of $A$ under the Dehn twist along $C_2$ is given on the first picture in Figure \[ReducedMonodromy\] by the dashed blue curve. The blue curve $C_3 C_2 C_1 (A)$ in the second picture is the image of $A$ after all three Dehn twists are applied. This is where the round $2$-handle gets into action. We can slide the curve $C_3 C_2 C_1 (A)$ over the red curve (twice) and get the dashed blue curve given in the third picture in Figure \[ReducedMonodromy\]. However, the third picture describes the effect of a *left-handed* Dehn twist along the curve $\Gamma$, which is a boundary parallel curve on the annulus obtained after surgering the punctured torus along the red curve. Hence the two monodromies are the same. ![Calculation of the reduced monodromy. []{data-label="ReducedMonodromy"}](ReducedMonodromy) We would like to finish with a few observations. The Figure \[FullSymmetry\] drawn in the most symmetric fashion presents a different choice of three vanishing cycles $D_1$, $D_2$ and $D_3$ on the twice punctured torus, and through similar arguments as above one can see that this picture stands for a broken Lefschetz fibration that can be used to replace a *positive* Lefschetz singularity locally. This contains the nontrivial part of Perutz’s example of a broken Lefschetz fibration (Example 1.3 in [@P]; also see Example 3.2 in [@B0]), and corresponds to the broken Lefschetz fibration that Lekili obtains after perturbing a positive Lefschetz singularity in his paper. One can then draw the curves $C_1$, $C_2$ and $C_3$ by joining the vertices of the hexagon formed by $D_1$, $D_2$, $D_3$ in the center as in Figure \[FullSymmetry\], and get the picture we had above (Figure \[Stabilization\], on the left) up to isotopy. From the very symmetry of the picture it is now easy to generalize our constructions to twice punctured $(4n+2)$-gons, for any $n \geq 1$, and thus to obtain various broken Lefschetz fibrations with higher fiber genera over disks. ![“Full symmetry.” Vanishing cycles used in the modification around both positive and negative Lefschetz singularities are given.[]{data-label="FullSymmetry"}](FullSymmetry) [99999]{} S. Akbulut and C. Karakurt, [*“Every $4$-manifold is BLF”* ]{}, J. of Gokova Geom. Topol. 2 (2008) 40-82. R.I. Baykur, [“Topology of broken Lefschetz fibrations and near-symplectic four-manifolds”,]{} Pacific J. Math. 240 (2009), no. 2, 201–230; http://arxiv.org/abs/0801.0192. R.I. Baykur, [*“Existence of broken Lefschetz fibrations”,*]{} Int. Math. Res. Not. IMRN 2008, Art. ID rnn 101, 15 pp. D.T. Gay and R. Kirby, [*“Constructing Lefschetz-type fibrations on four-manifolds”,*]{} Geom. Topol. [**11**]{} (2007), 2075–-2115. Y. Lekili, [*“Wrinkled Fibrations on near-symplectic manifolds”,*]{} Geom. Topol. 13 (2009), 277-318. T. Perutz, [*“Lagrangian matching invariants for fibred four-manifolds: I”*]{}, Geom. Topol. [**11**]{} (2007), 759–828.

--- address: - 'Department of Mathematics, Kerr Hall, One Shields Ave., University of California, Davis, CA 95616, USA' - 'Department of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA, The Fields Institute, 222 College Street, Toronto, Ontario, M5T 3J1, Canada' author: - Alexander Woo - Alexander Yong date: 'November 20, 2005' title: 'When is a Schubert Variety Gorenstein?' --- Introduction ============ The main goal of this paper is to give an explicit combinatorial characterization of which Schubert varieties in the complete flag variety are Gorenstein. Let $\mathrm{Flags}({\mathbb C}^{n})$ denote the variety of complete flags $F_{\bullet}:\langle 0\rangle \subseteq F_1\subseteq\ldots\subseteq F_n={\mathbb C}^n$. Fix a basis $e_1,e_2,\ldots,e_n$ of $\mathbb{C}^n$ and let $E_{\bullet}$ be the [*anti*]{}-canonical reference flag $E_{\bullet}$, that is, the flag where $E_{i}=\langle e_{n-i+1},e_{n-i+2},\ldots, e_n\rangle$. For every permutation $w$ in the symmetric group $S_n$, there is the [**Schubert variety**]{} $$X_{w}=\big\{F_{\bullet}\mid {\rm dim}(E_i\cap F_j)\geq \#\{k\geq n-i+1, w(k)\leq j\}\big\}.$$ These conventions have been arranged so that the codimension of $X_{w}$ is $\ell(w)$, that is, the length of any expression for $w$ as a product of simple reflections $s_i = (i\leftrightarrow i+1)$. The Gorenstein property gives a well-known measurement of how far an algebraic variety is from being smooth; all smooth varieties are Gorenstein, while all Gorenstein varieties are Cohen-Macaulay. In general, a variety is [**Gorenstein**]{} if it is Cohen-Macaulay and its canonical sheaf is a line bundle. (Throughout this paper we freely identify vector bundles and their sheaves of sections for convenience.) Recall that on a [*smooth*]{} variety $X$, the [**canonical sheaf**]{}, denoted $\omega_{X}$, is $\bigwedge^{{\rm dim}({X})}\Omega_{{X}}$, where $\Omega_{{X}}$ is the cotangent bundle of ${X}$. For a possibly singular but normal variety ${X}$, the canonical sheaf is the pushforward of the canonical sheaf $\omega_{{X}_{\rm smooth}}$ on the smooth part $X_{\rm smooth}$ of ${X}$ under the inclusion map. Since every Schubert variety is normal [@DeCon-Lak; @RR] and Cohen-Macaulay [@Ram85], the remarks above suffice to define Gorensteinness in the context of this paper. In Section 2.1, we will give the more commonly seen local definition of Gorensteinness. However, combining the above definition together with the results of Ramanathan [@Ram85; @Ram87] is what provides our starting point for determining which Schubert varieties are Gorenstein. Smoothness and Cohen-Macaulayness of Schubert varieties have been extensively studied in the literature; see, for example, [@BL00; @Ram85] and the references therein. While all Schubert varieties are Cohen-Macaulay, very few Schubert varieties are smooth. (See the table at the end of this Introduction.) Explicitly, $X_w$ is smooth if and only if $w$ is “$1324$-pattern avoiding” and “$2143$-pattern avoiding” [@LS90]; we give more details on pattern avoidance below. Our main result (Theorem \[thm:main\]) gives an explicit combinatorial characterization of which Schubert varieties are Gorenstein similar to the above smoothness criteria. This answers a question raised by M. Brion and S. Kumar and passed along to us by A. Knutson; see also [@Ram87 p. 88]. Our answer uses a generalized notion of pattern avoidance that we introduce. To describe our ideas in a simpler case, we first compare the classical smoothness criterion with a characterization of which Schubert varieties in the Grassmannian $\mathrm{Gr}(\ell,n)$ of $\ell$-planes in ${\mathbb C}^n$ are Gorenstein. (This is a special case of our main result, as we will explain in Section \[subsection:partial\].) Schubert varieties ${X}_{\lambda}$ of $\mathrm{Gr}(\ell,n)$ are indexed by partitions $\lambda$ sitting inside an $\ell\times (n-\ell)$ rectangle.[^1] The smooth Schubert varieties are those indexed by partitions $\lambda$ whose complement in $\ell\times (n-\ell)$ is a rectangle, as explained in, for example, [@BL00] and the references therein. For example, $\lambda=(7,7,2,2,2)$ indexes a smooth Schubert variety in ${\rm Gr}(5,12)$. $$\begin{picture}(340,100) \put(0,0){\framebox(140,100)} \thicklines \put(0,0){\line(1,0){40}} \put(40,0){\line(0,1){60}} \put(40,60){\line(1,0){100}} \put(140,60){\line(0,1){40}} \put(-30,50){$\lambda=$} \put(40,60){\circle*{4}} \put(0,0){\line(0,1){100}} \put(0,100){\line(1,0){140}} \thinlines \put(0,20){\line(1,0){40}} \put(0,40){\line(1,0){40}} \put(0,60){\line(1,0){40}} \put(0,80){\line(1,0){140}} \put(20,0){\line(0,1){100}} \put(40,60){\line(0,1){40}} \put(60,60){\line(0,1){40}} \put(80,60){\line(0,1){40}} \put(100,60){\line(0,1){40}} \put(120,60){\line(0,1){40}} \put(200,0){\framebox(140,100)} \thicklines \put(200,0){\line(1,0){40}} \put(240,0){\line(0,1){20}} \put(240,20){\line(1,0){20}} \put(260,20){\line(0,1){20}} \put(260,40){\line(1,0){40}} \put(300,40){\line(0,1){40}} \put(300,80){\line(1,0){20}} \put(320,80){\line(0,1){20}} \put(320,100){\circle*{4}} \put(170,50){$\mu=$} \put(240,20){\circle*{4}} \put(260,40){\circle*{4}} \put(300,80){\circle*{4}} \put(200,0){\line(0,1){100}} \put(200,100){\line(1,0){140}} \thinlines \put(220,0){\line(0,1){100}} \put(240,20){\line(0,1){80}} \put(260,40){\line(0,1){60}} \put(280,40){\line(0,1){60}} \put(300,80){\line(0,1){20}} \put(200,20){\line(1,0){40}} \put(200,40){\line(1,0){60}} \put(200,60){\line(1,0){100}} \put(200,80){\line(1,0){100}} \end{picture}$$ Alternatively, smooth Schubert varieties are those with at most one inner corner. View the lower border of partition as a lattice path from the lower left-hand corner to the upper right-hand corner of $\ell\times (n-\ell)$; an [**inner corner**]{} is then a lattice point on this path with lattice points of the path both directly below and directly to the right of it. The inner corners for the partitions $\lambda$ and $\mu$ above are marked by “dots”. Therefore, the partition $\mu=(6,5,5,3,2)$ above does not index a smooth Schubert variety. However, it does index a Gorenstein Schubert variety; in general, a Grassmannian Schubert variety ${X}_{\mu}$ is Gorenstein if and only if all of the inner corners of $\mu$ lie on the same antidiagonal. We mention that this condition can also be derived from [@svanes (5.5.5)]. In order to state our main result for $\mathrm{Flags}({\mathbb C}^{n})$, we will need some preliminary definitions. First we associate a Grassmannian permutation to each descent of a permutation $w$. Let $d$ be a [**descent**]{} of $w$, which is an index such that $w(d)>w(d+1)$. Now write $w$ in one-line notation as $w(1)w(2)\cdots w(n)$, and construct a subword $v_{d}(w)$ of $w$ by concatenating the right-to-left minima of the segment strictly to the left of $d+1$ with the left-to-right maxima of the segment strictly to the right of $d$. In particular, $v_{d}(w)$ will necessarily include $w(d)$ and $w(d+1)$. Let ${\widetilde v}_{d}(w)$ denote the [**flattening**]{} of $v_{d}(w)$, which is defined to be the unique permutation whose entries are in the same relative position as those of $v_{d}(w)$. Let $w=314972658\in S_{9}$. This permutation has descents at positions 1, 4, 5 and 7. We see that $v_{1}(w)=3149$, $v_{4}(w)=14978$, $v_{5}(w)=147268$, and $v_{7}(w)=12658$, so therefore ${\widetilde v}_{1}(w)=2134$, ${\widetilde v}_{4}(w)=12534$, ${\widetilde v}_{5}(w)=135246$, and ${\widetilde v}_{7}(w)=12435$. By construction, ${\widetilde v}_{d}(w)\in S_m$ is a [**Grassmannian permutation**]{}, meaning that it has a unique descent at some position we denote $e$. For any Grassmannian permutation $w\in S_m$ with its unique descent at $e$, let $\lambda(w)\subseteq e\times (m-e)$ denote the associated partition. The partition $\lambda(w)$ is the one whose lower border is obtained by drawing a lattice path which starts at the lower left corner of $e\times(m-e)$ and continues by a unit horizontal line segment at step $i$ (for some $i\in\{1,\ldots,m\}$) if $i$ appears strictly after position $e$ (or, in other words, if $w^{-1}(i)>e$), and a unit vertical line segment otherwise. For example, the Grassmannian permutation $w= 3589\ 11 \ \mid\ 12467\ 10\ 12$ corresponds to the partition $\lambda(w)=\mu=(6,5,5,3,2)$ depicted above. Now, given an inner corner of a partition $\lambda(w)$, let its [**inner corner distance**]{} be the sum of the distances from the top and left edges of the rectangle $e\times (m-e)$ to the inner corner. For example, in $\mu$ above, all the inner corner distances equal 6. Furthermore, suppose that $\lambda(w)$ has all its inner corners on the same antidiagonal; this is equivalent to requiring that the inner corner distance be the same for all inner corners. In this case we call this common inner corner distance ${\mathfrak I}(w)$; if there are no inner corners, we set $\mathfrak{I}(w)=0$ by convention. For our example permutation $w$, $\mathfrak{I}(w)=6$. Next we proceed to define [**Bruhat-restricted pattern avoidance**]{}. Recall that, classically, for $v\in S_\ell$ and $w\in S_n$, with $\ell\leq n$, an [**embedding of $v$ into $w$**]{} is a sequence of indices $i_1<i_2<\cdots<i_{\ell}$ such that, for all $1\leq a<b\leq\ell$, $w(i_a)>w(i_b)$ if and only if $v(a)>v(b)$. Then the classical definition of pattern avoidance is that [**$w$ pattern avoids $v$**]{} if there are no embeddings of $v$ into $w$. Now recall the [**Bruhat order**]{} $\succ$ on $S_n$. First we say that $w(i\leftrightarrow j)$ [**covers**]{} $w$ if $i<j$, $w(i)<w(j)$, and, for each $k$ with $i<k<j$, either $w(k)<w(i)$ or $w(k)>w(j)$; then the Bruhat order is the transitive closure of this covering relation. The Bruhat order is graded by the length of a permutation, and one can check that $v$ can cover $w$ only if $\ell(v)=\ell(w)+1$. Given a permutation $v\in S_\ell$, let ${\mathcal T}_{v}=\{(m_1 \leftrightarrow n_1), \ldots, (m_k \leftrightarrow n_k)\}$ be a set of [**Bruhat transpositions**]{} in $v$, by which we mean a subset of transpositions such that $v\cdot(m_j \leftrightarrow n_j)$ covers $v$ in the Bruhat order. We define a [**${\mathcal T}_{v}$-restricted embedding**]{} of $v$ into $w$ to be an embedding of $v$ into $w$ such that $w\cdot(i_{m_j} \leftrightarrow i_{n_j})$ covers $w$ for all $(m_j \leftrightarrow n_j)\in {\mathcal T}_{v}$. Then we say that [**$w$ pattern avoids $v$ with Bruhat restrictions ${\mathcal T}_{v}$**]{} if there are no ${\mathcal T}_{v}$-restricted embeddings of $v$ into $w$. Now we are ready to state our combinatorial characterization of which Schubert varieties in $\mathrm{Flags}({\mathbb C}^{n})$ are Gorenstein: \[thm:main\] Let $w\in S_n$. The Schubert variety $X_w$ is Gorenstein if and only if - for each descent $d$ of $w$, $\lambda({\widetilde v}_{d}(w))$ has all of its inner corners on the same antidiagonal, and - the permutation $w$ pattern avoids both $31524$ and $24153$ with Bruhat restrictions $\{(1\leftrightarrow 5),(2\leftrightarrow 3)\}$ and $\{(1\leftrightarrow 5),(3\leftrightarrow 4)\}$ respectively. In comparing the smoothness characterization of [@LS90] with Theorem \[thm:main\], considering our description of the Grassmannian case allows one to check that the $1324$-pattern avoidance condition of the former implies the “inner corner condition” of the latter. It is also easy to see that the $2143$-pattern avoidance condition of the former implies both of the Bruhat-restricted pattern avoidance conditions of the latter. We mention that Fulton [@Fulton:Duke92] has characterized $2143$-pattern avoidance in terms of the essential set of a permutation. A similar characterization can be given for the Bruhat-restricted pattern avoidance conditions of Theorem \[thm:main\]. The permutation $w=\underline{3} 7 \underline{1} 4 \underline{8} \underline{2}6\underline{5}\in S_8$ has descents at positions $2$, $5$ and $7$ and we have $${\widetilde v}_{2}(w)=24135, {\widetilde v}_{5}(w)=13524, \mbox{ and } {\widetilde v}_{7}(w)=1243.$$ Hence one checks that $w$ satisfies the inner corner condition with $${\mathfrak I}({\widetilde v}_{2}(w))=2, \ {\mathfrak I}({\widetilde v}_{5}(w))=2, \mbox{ and } {\mathfrak I}({\widetilde v}_{7}(w))=1.$$ The Schubert variety $X_{w}$ is Gorenstein, since there are no forbidden $31524$ and $24153$ patterns with Bruhat restrictions $\{(1\leftrightarrow 5),(2\leftrightarrow 3)\}$ or $\{(1\leftrightarrow 5), (3\leftrightarrow 4)\}$ respectively. Note that the underlined subword of $w$ is a $31524$-pattern, but since $w(1\leftrightarrow 8)$ does not cover $w$, it does not prevent $X_w$ from being Gorenstein. By combining Theorem \[thm:main\] with the descriptions of the singularities along the “maximal singular locus” of a Schubert variety $X_w$ given in [@Cortez; @manivel2], we obtain the following geometric corollary. \[cor:maxsingloc\] A Schubert variety $X_w$ is Gorenstein if and only if it is Gorenstein along its maximal singular locus. In other words, Corollary \[cor:maxsingloc\] states that a Schubert variety is Gorenstein if and only if its “smoothest” singularities (those at the generic points of the irreducible components of the singular locus) are Gorenstein. We now describe the canonical sheaf of a Gorenstein Schubert variety in terms of the Borel-Weil construction of line bundles. Let $T\cong(\mathbb{C}^*)^{n-1}$ be the subgroup of invertible diagonal matrices of determinant $1$ in $\mathrm{SL}_n(\mathbb{C})$; the Borel-Weil construction associates to each integral weight $\alpha\in\mathrm{Hom}(T,\mathbb{C}^*)$ a line bundle $\mathcal{L}_\alpha$. Let ${\mathcal L}_{\alpha}\big|_{X_{w}}$ denote the restriction of this line bundle to $X_w$. We will write weights additively in terms of the $\mathbb{Z}$-basis of fundamental weights $\Lambda_r$, defined by $\Lambda_r\left(\left[\begin{array}{ccc} t_1 & & 0 \\ & \ddots & \\ 0 & & t_n\end{array}\right]\right)=t_1\cdots t_r$. \[thm:second\_main\] If $X_{w}$ is Gorenstein, then $\omega_{X_{w}}\cong {\mathcal L}_{\alpha}\big|_{X_w}$ where $\alpha=\sum_{r=1}^{n-1} \widetilde{\alpha}_r\Lambda_{n-r}$ and $$\label{eqn:the_soln'} \widetilde{\alpha}_r = \left\{ \begin{array}{cc} -2+{\mathfrak I}({\widetilde v}_{r}(w)) & \mbox{ if $r$ is a descent} \\ -2 & \mbox{ otherwise.} \end{array} \right.$$ The proofs of Theorems \[thm:main\] and \[thm:second\_main\] as well as Corollary \[cor:maxsingloc\] will be given in Section 2. In Section 3, we end with a number of remarks and applications. Further study of the relationship between the geometry of Gorensteinness of Schubert varieties and related combinatorics should have potential. We conclude this introduction with some open problems and suggestions for further work. The most natural is: \[problem:1\] Give analogues of Theorems \[thm:main\] and \[thm:second\_main\] for generalized flag varieties corresponding to Lie groups other than ${\rm GL}_{n}({\mathbb C})$. We expect that the methods given in this paper will extend to solve Problem \[problem:1\]. It is not difficult to use Theorem \[thm:main\] to derive an analogue of Theorem \[thm:main\] for the case of the odd orthogonal groups ${\rm SO}(2n+1,{\mathbb C})$. However, we have found the combinatorial analysis required to be more intricate in general. Consequently, in the interest of brevity, we plan to discuss our investigations for the other Lie types in a subsequent paper. It should also be interesting to determine the “maximal non-Gorenstein locus” of a non-Gorenstein Schubert variety: Let $X$ be a variety that is Cohen-Macaulay but not Gorenstein; since the rank of any coherent sheaf on $X$ is upper semicontinuous (see, for example, [@Hartshorne III.12.7.2]), the canonical sheaf has rank strictly greater than 1 at some non-trivial closed subvariety. This subvariety then consists of all points of $X$ at which $X$ is not Gorenstein by the local definition. Since the canonical sheaf of a Schubert variety is $B_{-}$-equivariant for the subgroup $B_{-}\subseteq {\rm GL}_{n}({\mathbb C})$ of lower triangular matrices, this subvariety is a union of Schubert varieties contained in $X_w$. Therefore we ask: Give a combinatorial characterization for the minimal $v$ in the Bruhat order for which $X_{w}$ is not Gorenstein at $X_v$. In view of Corollary \[cor:maxsingloc\], it is natural to propose the following answer: \[conj:nongorloc\] The maximal non-Gorenstein locus of $X_w$ is the union of those Schubert varieties $X_v$ in the maximal singular locus of $X_w$ for which the generic point is not Gorenstein in $X_w$. One can give a combinatorial rule characterizing the set of $X_v$ appearing in Conjecture \[conj:nongorloc\] using the explicit description of the singular locus of Schubert varieties [@billey.warrington; @Cortez; @Gasharov; @KLR; @LS90; @manivel1; @manivel2] and facts mentioned in the proof of Corollary \[cor:maxsingloc\]. A geometric explanation was recently given in [@BilBrad] for the appearance of pattern avoidance in characterizations of smooth Schubert varieties. However, this explanation does not have an obvious modification to take into account Bruhat-restrictions. This leads to the following: Give a geometric explanation of Bruhat-restricted pattern avoidance which explains its appearance in Theorem \[thm:main\]. Lastly, for those interested in combinatorial enumeration: Give a combinatorial formula (for example, a generating series) computing the number of Gorenstein Schubert varieties in ${\rm Flags}({\mathbb C}^n)$. Using the methods of this paper, we computed the number of Gorenstein Schubert varieties in ${\rm Flags}({\mathbb C}^{n})$ for some small values of $n$ (see below). We compare this to the number of smooth Schubert varieties computed using the result of [@LS90] (by the recursive formulas found in [@Bona; @Stankova]). $n$ $n! = \#\mbox{ Cohen-Macaulay $X_w$}$ $\#\mbox{ Gorenstein $X_w$}$ $\#\mbox{ Smooth $X_w$}$ ----- --------------------------------------- ------------------------------ -------------------------- 1 1 1 1 2 2 2 2 3 6 6 6 4 24 24 22 5 120 116 88 6 720 636 366 7 5040 3807 1552 8 40320 24314 6652 9 362880 163311 28696 We are very grateful to M. Brion, A. Knutson and S. Kumar for bringing the problem addressed by Theorem \[thm:main\] to our attention, for outlining the argument used in Section 2.1, and for many other suggestions. We also thank A. Bertram, S. Billey, A. Buch, A. Cortez, R. Donagi, S. Fomin, M. Haiman, R. MacPherson, E. Miller, R. Stanley, B. Sturmfels, J. Tymoczko, and an anonymous referee for discussion and remarks on earlier drafts. This work was partially completed while the two authors were in residence at the Park City Mathematics Institute program on “Geometric Combinatorics” during July 2004. Proof of Theorems \[thm:main\] and \[thm:second\_main\] ======================================================= Geometry to combinatorics ------------------------- First we explain the algebraic definition of Gorensteinness and reduce the algebro-geometric problem of determining when a Schubert variety is Gorenstein to a problem in linear algebra; we will then solve this linear algebra problem combinatorially. This reduction to linear algebra appears to be folklore (and was told to us by M. Brion, A. Knutson and S. Kumar); we could not locate an explicit reference for it in the literature. Therefore, we include an argument for the sake of completeness. While we treat only $\mathrm{Flags}(\mathbb{C}^n)$ explicitly, the arguments of this section generalize easily to all semi-simple Lie groups with the substitution of the appropriate Monk-Chevalley formula [@Chevalley]. We found [@brion:book] an excellent resource for facts about the geometry of Schubert varieties. A local ring $(R,\mathfrak{m}, \Bbbk)$ is said to be [**Cohen-Macaulay**]{} if $\mathrm{Ext}_R^i(\Bbbk, R)=0$ for $i\leq\mathrm{dim} R$; it is [**Gorenstein**]{} if, in addition, $\mathrm{dim}_{\Bbbk} \mathrm{Ext}_R^{\mathrm{dim} R}(\Bbbk, R)=1$. A variety is Cohen-Macaulay (respectively Gorenstein) if the local ring at every point is Cohen-Macaulay (respectively Gorenstein). Using the Kozsul complex on a regular sequence, one can show that every regular local ring is Gorenstein; hence smooth varieties are Gorenstein. See [@Bruns-Herzog] for details. One might naively expect that, in order to check if a Schubert variety is Gorenstein, one would need to check if it is Gorenstein at all, or at least some, of its points. However, the alternative equivalent definition of the Gorenstein property alluded to in the introduction, which is based on Grothendieck duality theory (see [@HartsRes] or [@Altman-Kleiman]), allows for a different approach using the global geometry of Schubert varieties. Each projective variety has a dualizing complex (of sheaves) which plays a role analogous to that of the canonical bundle $\omega_X$, defined as the top exterior power of the cotangent bundle $\bigwedge^{{\rm dim}(X)}\Omega_{X}$, of a smooth variety in Serre duality. A connected projective variety is Cohen-Macaulay if and only if the dualizing complex is a sheaf, and Gorenstein if and only if the dualizing sheaf is locally free of rank one. For a normal, Cohen-Macaulay variety, one can realize the dualizing sheaf as the pushforward of the canonical sheaf $\omega_{X_\mathrm{smooth}}$ of the smooth part $X_{\mathrm{smooth}}$ under the inclusion map. As mentioned in the introduction, all Schubert varieties are known to be normal [@DeCon-Lak; @RR] and Cohen-Macaulay [@Ram85], so we can then use the calculation of the canonical sheaf of Schubert varieties by Ramanathan [@Ram85; @Ram87] to determine which Schubert varieties are Gorenstein. We now need some standard definitions which can be found in [@Hartshorne II.6]. Let $\mathrm{Cl}(X_w)$ denote the Weil divisor class group of $X_w$; its elements are linear equivalence classes $\left[ Z\right]$ of formal sums of codimension 1 subvarieties $Z$ of $X_w$. There is a natural group homomorphism $\mathrm{div}: \mathrm{Pic}(X_w) \rightarrow \mathrm{Cl}(X_w)$, where $\mathrm{Pic}(X_w)$ is the group of isomorphism classes of line bundles under tensor product. On a Schubert variety $X_w$ (or, in general, any normal irreducible variety over a field), $\mathrm{div}$ is injective and its image in $\mathrm{Cl}(X_w)$ is the [**Cartier class group**]{} $\mathrm{CaCl}(X_{w})$. (This is an unorthodox definition of the Cartier class group, but for convenience we have identified it with its isomorphic image in the Weil class group.) For smooth varieties, $\mathrm{div}$ is an isomorphism, so $\mathrm{CaCl}=\mathrm{Cl}$. We now proceed to describe explicitly $\mathrm{Cl}(X_w)$ and $\mathrm{CaCl}(X_w)$. The Schubert variety $X_w$ is the disjoint union of the open Schubert cell $X^\circ_w$ (which is isomorphic to the affine space ${\mathbb C}^{\binom{n}{2}-\ell(w)}$) together with the codimension 1 subvarieties $X_v$ for $v$ covering $w$ in the Bruhat order. Therefore, by repeatedly applying [@Hartshorne Prop. II.6.5], we see that $\mathrm{Cl}(X_w)$ is freely generated (as an abelian group) by $\left[ X_v \right]$ for $v$ covering $w$. To describe $\mathrm{CaCl}(X_w)$, we will need the Chow group $A_*(\mathrm{Flags}(\mathbb{C}^{n}))$ of the flag variety, whose elements are rational equivalence classes $\left[Z\right]$ of subvarieties $Z$ of $\mathrm{Flags}(\mathbb{C}^{n})$; see for example [@FultonIT Ch. 1]. Since $\mathrm{Flags}(\mathbb{C}^{n})$ is smooth, the Chow ring $A^*(\mathrm{Flags}(\mathbb{C}^{n}))$ is by definition equal as abelian groups to $A_*(\mathrm{Flags}(\mathbb{C}^{n}))$ [@FultonIT 8.3]. The graded pieces $A_d(\mathrm{Flags}(\mathbb{C}^n))=A^{\binom{n}{2}-d}(\mathrm{Flags}(\mathbb{C}^{n}))$ are freely generated by the classes $\left[X_v\right]$ of the Schubert varieties $X_v$ of dimension $d$, which are precisely those for which $d=\binom{n}{2}-l(v)$. Therefore, the natural map $\iota_*: \mathrm{Cl}(X_w) \rightarrow A_{\binom{n}{2}-\ell(w)-1}(\mathrm{Flags}(\mathbb{C}^n))$ induced by the inclusion $\iota: X_w \rightarrow \mathrm{Flags}(\mathbb{C}^n)$ is injective. Note that, by definition, $A_{\binom{n}{2}-1}(\mathrm{Flags}(\mathbb{C}^{n}))=\mathrm{Cl}(\mathrm{Flags}(\mathbb{C}^{n}))$. It is known [@Mathieu Prop. 6] that every line bundle on a Schubert variety is the restriction of a line bundle on $\mathrm{Flags}(\mathbb{C}^{n})$. Furthermore, for a line bundle $\mathcal{L}$ on $\mathrm{Flags}(\mathbb{C}^{n})$, general facts of intersection theory [@FultonIT Ch. 2] tell us that $\iota_*(\mathrm{div}(\mathcal{L}\big|_{X_w}))=\mathrm{div}(\mathcal{L})\cdot\left[X_w\right]$, where the right hand side is a product in $A^*(\mathrm{Flags}(\mathbb{C}^{n}))$. Therefore, since $\mathrm{CaCl}(\mathrm{Flags}(\mathbb{C}^{n}))=\mathrm{Cl}(\mathrm{Flags}(\mathbb{C}^{n}))$ is generated by $\left\{\left[X_{(r\leftrightarrow r+1)}\right]\right\}_{r=1}^{n-1}$, $\iota_*(\mathrm{CaCl}(X_w))\subseteq {\rm A}^{*}({\rm Flags}({\mathbb C}^n))$ is generated by $\left\{\left[X_{(r\leftrightarrow r+1)}\right]\cdot\big[X_w\big]\right\}_{r=1}^{n-1}$. By Monk’s formula [@Monk], $$\left[X_{(r\leftrightarrow r+1)}\right]\cdot\big[X_w\big] =\sum_{{\genfrac{}{}{0pt}{}{a\leq r<b}{\ell(w(a\leftrightarrow b))=\ell(w)+1}}} \left[ X_{w(a\leftrightarrow b)}\right],$$ so $\mathrm{CaCl}(X_w)$ is generated by these classes for $1\leq r\leq n-1$. (We can drop the $\iota_*$ since it is an injection.) Since Schubert varieties are Cohen-Macaulay [@Mus-Ses; @DeCon-Lak; @Ram85], a Schubert variety $X_w$ is Gorenstein if and only if its canonical sheaf $\omega_{X_w}$ is a line bundle. By results of Ramanathan [@Ram87 Thm. 4.2], the canonical sheaf of $X_w$ is $$\omega_{X_w}=\mathcal{L}_{-\rho}\mid_{X_w} \otimes \ \mathcal{I}(\partial X_w),$$ where $\mathcal{L}_{-\rho}\mid_{X_w}$ is the restriction to $X_w$ of the line bundle associated to the weight $-\rho=-\sum_{r=1}^{n-1}\Lambda_{r}$ by the Borel-Weil construction, and $\mathcal{I}(\partial X_w)$ is the ideal sheaf of the complement of $X^\circ_w$, or equivalently, the ideal sheaf of the reduced subscheme $\bigcup_v X_v$ where $v$ ranges over all permutations covering $w$ in the Bruhat order. Since $\mathcal{L}_{-\rho}\big|_{X_w}$ is a line bundle and $\mathrm{Pic}$ is a group, $\omega_{X_w}$ is a line bundle if and only if $\mathcal{I}(\partial X_w)$ is a line bundle. However, the ideal sheaf of a reduced codimension 1 subscheme $Y$ is a line bundle if and only if $\left[Y\right]$ is a Cartier divisor, in which case $\mathrm{div}(\mathcal{I}(Y))=-\left[Y\right]$; see, for example, [@Hartshorne II.6]. Therefore, $X_w$ is Gorenstein if and only if $$\left[\partial X_w\right]=\sum_{{\genfrac{}{}{0pt}{}{v\succ w}{\ell(v)=\ell(w)+1}}} \left[{X}_v\right]\in \mathrm{CaCl}(X_{w}).$$ Hence, by our previous calculation of $\mathrm{CaCl}(X_w)$ as a subgroup of $\mathrm{Cl}(X_w)$, we obtain the following: The Schubert variety $X_w$ is Gorenstein if and only if there exists an integral solution $(\alpha_1,\ldots,\alpha_{n-1})$ to $$\label{eqn:final_system} \sum_{r=1}^{n-1}\alpha_r \left(\sum_{{\genfrac{}{}{0pt}{}{a\leq r<b}{\ell(w(a\leftrightarrow b))=\ell(w)+1}}}\left[ X_{w(a\leftrightarrow b)}\right] \right)= \sum_{{\genfrac{}{}{0pt}{}{v=w(a\leftrightarrow b)}{\ell(v)=\ell(w)+1}}} \left[ X_v\right]\in {\rm CaCl}(X_w).$$ As an aside, a variety is said to be [**locally factorial**]{} if the local ring at every point is a unique factorization domain. It is well known (see [@Hartshorne Prop. II.6.11] or [@FultonIT 2.1]) that a normal variety is factorial if and only if $\mathrm{div}$ is an isomorphism. Therefore, factorial Schubert varieties can be characterized using the following proposition. The Schubert variety $X_w$ is factorial if and only if the classes $$\left\{\sum_{{\genfrac{}{}{0pt}{}{a\leq r<b}{\ell(w(a\leftrightarrow b))=\ell(w)+1}}}\left[ X_{w(a\leftrightarrow b)}\right]\right\}_{r=1}^{n-1}$$ span the free abelian group generated by $$\left\{\left[X_v\right] \mid v=w(a\leftrightarrow b), \ell(v)=\ell(w)+1\right\}.$$ Recently, M. Bosquet-Mélou and S. Butler [@BM-Butler] have used this proposition to give a characterization of locally factorial Schubert varieties in terms of Bruhat-resticted pattern avoidance. This solves a conjecture that we had distributed during the preparation of this article. Interlude: a diagrammatic formulation and two sample problems ------------------------------------------------------------- Although it is not used in our proof below, let us give a diagrammatic formulation of the above linear algebra problem (\[eqn:final\_system\]) that the reader may find useful. Label $n$ columns by the values $w(1), w(2), \ldots, w(n)$ of a permutation $w\in S_n$. Draw horizontal bars between the midpoints of columns $i$ and $j$ if and only if $w(i\leftrightarrow j)$ covers $w$ in the Bruhat order. Now draw vertical bars between columns $i$ and $i+1$ for $1\leq i\leq n-1$. Then a solution to (\[eqn:final\_system\]) is equivalent to an assignment $(\alpha_1,\ldots,\alpha_{n-1})\in {\mathbb Z}^{n-1}$ of integers to the vertical bars (from left to right respectively) such that, for each horizontal bar, the sum of the assignments to the vertical bars that it crosses equals 1. We encourage the reader to try out the following two sample problems; answers are at the bottom of the page[^2]: $$\begin{picture}(320,200) \put(0,190){$6$} \put(20,190){$3$} \put(40,190){$1$} \put(60,190){$4$} \put(80,190){$7$} \put(100,190){$2$} \put(120,190){$5$} \thicklines \put(2,180){\line(1,0){80}} \put(22,160){\line(1,0){40}} \put(42,140){\line(1,0){20}} \put(42,120){\line(1,0){60}} \put(62,100){\line(1,0){20}} \put(62,80){\line(1,0){60}} \put(102,60){\line(1,0){20}} \thinlines \put(12,200){\line(0,-1){180}} \put(32,200){\line(0,-1){180}} \put(52,200){\line(0,-1){180}} \put(72,200){\line(0,-1){180}} \put(92,200){\line(0,-1){180}} \put(112,200){\line(0,-1){180}} \put(6,0){$\alpha_1$} \put(26,0){$\alpha_2$} \put(46,0){$\alpha_3$} \put(66,0){$\alpha_4$} \put(86,0){$\alpha_5$} \put(106,0){$\alpha_6$} \put(200,190){$5$} \put(220,190){$3$} \put(240,190){$1$} \put(260,190){$7$} \put(280,190){$4$} \put(300,190){$2$} \put(320,190){$6$} \thicklines \put(202,180){\line(1,0){60}} \put(202,160){\line(1,0){120}} \put(222,140){\line(1,0){40}} \put(222,120){\line(1,0){60}} \put(242,100){\line(1,0){20}} \put(242,80){\line(1,0){40}} \put(242,60){\line(1,0){60}} \put(282,40){\line(1,0){40}} \put(302,20){\line(1,0){20}} \thinlines \put(212,200){\line(0,-1){180}} \put(232,200){\line(0,-1){180}} \put(252,200){\line(0,-1){180}} \put(272,200){\line(0,-1){180}} \put(292,200){\line(0,-1){180}} \put(312,200){\line(0,-1){180}} \put(206,0){$\alpha_1$} \put(226,0){$\alpha_2$} \put(246,0){$\alpha_3$} \put(266,0){$\alpha_4$} \put(286,0){$\alpha_5$} \put(306,0){$\alpha_6$} \end{picture}$$ Necessity of the combinatorial conditions in Theorem \[thm:main\] ----------------------------------------------------------------- It is possible to prove necessity by appealing to the geometric description of the singularities along the maximal singular locus found in [@Cortez; @manivel2]; however we will give a simple, purely combinatorial proof. We will need the following two lemmas, the first of which is immediate: \[lemma:zero\] The vector $(\alpha_1,\ldots, \alpha_{n-1})\in {\mathbb Z}^{n-1}$ is a solution to (\[eqn:final\_system\]) if and only if $\sum_{r=i}^{j-1}\alpha_r =1$ for all $(i\leftrightarrow j)$ such that $w(i\leftrightarrow j)$ covers $w$ in the Bruhat order. \[lemma:seq\_bars\] If there exists a solution $(\alpha_1,\ldots,\alpha_{n-1})\in {\mathbb Z}^{n-1}$ to (\[eqn:final\_system\]), and $i<j$ with $w(i)<w(j)$, then $\sum_{r=i}^{j-1}\alpha_r \geq 1$. Equality holds if and only if $w(i\leftrightarrow j)$ covers $w$. If $w(i\leftrightarrow j)$ covers $w$ then the claim holds by Lemma \[lemma:zero\]. Otherwise, it follows from the observation that there are indices $$i_0 = i <i_1<i_2 <\ldots < i_{t-1}<j=i_t$$ such that $w(i_s \leftrightarrow i_{s+1})$ covers $w$ for $0\leq s\leq t-1$. Now suppose that there is an embedding $i_1<i_2<i_3<i_4<i_5$ of a $31524$ pattern with Bruhat restrictions $\{(1\leftrightarrow 5),(2\leftrightarrow 3)\}$. Then by Lemma \[lemma:seq\_bars\], any solution would satisfy $$\sum_{r=i_1}^{i_{3}-1}\alpha_r \geq 1, \sum_{r=i_2}^{i_4 -1}\alpha_r\geq 1, \sum_{r=i_4}^{i_5 -1}\alpha_{r}\geq 1, \mbox{ and } \sum_{r=i_2}^{i_3 -1} \alpha_r =1.$$ Therefore, $$\label{eqn:askedforit} \sum_{r=i_1}^{i_5 -1}\alpha_r =\sum_{r=i_1}^{i_3 -1}\alpha_r + \sum_{r=i_2}^{i_4 -1}\alpha_r + \sum_{r=i_4}^{i_5 -1}\alpha_r -\sum_{r=i_2}^{i_3 -1}\alpha_r\geq 2.$$ Thus (\[eqn:askedforit\]) is a contradiction of Lemma \[lemma:zero\] (or Lemma \[lemma:seq\_bars\]) since $w(i_1 \leftrightarrow i_5)$ covers $w$. Therefore such an embedding cannot exist. A similar argument shows that there cannot exist an embedding into $w$ of a $24153$ pattern with Bruhat restrictions $\{(1\leftrightarrow 5),(3\leftrightarrow 4)\}$. It remains to show that for each descent $d$ of $w$, $\lambda({\widetilde v}_{d}(w))$ has all of its inner corners on the same antidiagonal. For this purpose, we need: \[lemma:bij\_inner\] Let $v$ be a Grassmannian permutation with descent at position $d$. Then the transpositions $(i\leftrightarrow j)$ with $i\leq d<j$ such that $v(i\leftrightarrow j)$ covers $v$ are in bijection with the inner corners of $\lambda(v)$. Moreover, if $(i\leftrightarrow j)$ corresponds to an inner corner of $\lambda(v)$ under this bijection, then the corresponding inner corner distance equals $j-i-1$. In terms of the lattice path description of $\lambda(v)$ given on page 2, an inner corner of $\lambda(v)$ occurs exactly when there is an “up step” at time $a$, followed by a “right step” at time $a+1$. In terms of $v$, this means $a$ and $a+1$ appear in positions $i$ and $j$ satisfying the hypotheses. Conversely, if $i\leq d < j$ and $v(i \leftrightarrow j)$ covers $v$, then $v(j)=v(i)+1$. The claims then follow. The next lemma is clear from the definition of $v_{d}(w)$: \[lemma:same\_both\] Let $d$ be a descent of $w$ and suppose $(i, j)$ is a pair $1\leq i<j\leq n$ that indexes two entries of $w$ included in the subword $v_d (w)$ of $w$. Let $(i',j')$ be the corresponding indices in ${\widetilde v}_{d}(w)$. Then $w(i\leftrightarrow j)$ covers $w$ if and only if ${\widetilde v}_{d}(w)(i'\leftrightarrow j')$ covers ${\widetilde v}_{d}(w)$. Let $d$ be a descent of $w$ and suppose that $$i_1<i_2<\ldots <i_f =a <\ldots <i_s =d <i_{s+1}=d+1 <i_{s+2}<\ldots <i_{g}=b <\ldots < i_t$$ are the indices of the subword $v_{d}(w)$ of $w$, where $w(a\leftrightarrow b)$ covers $w$. By Lemmas \[lemma:seq\_bars\] and \[lemma:same\_both\] combined, any solution satisfies $$1=\sum_{r=a}^{b-1}\alpha_r = (s-f) + (g-s-1) + \alpha_{d}=g-f-1+\alpha_{d}$$ Now, $g-f-1$ is the inner corner distance of the corresponding inner corner of $\lambda({\widetilde v}_{d}(w))$ under the bijection of Lemma \[lemma:bij\_inner\]. Since $\alpha_{d}$ is fixed, $g-f-1$ is independent of our choice of $a$ and $b$. Hence, all of the inner corners of $\lambda({\widetilde v}_{d}(w))$ have the same inner corner distance, and therefore they must all lie on the same antidiagonal. Sufficiency of the combinatorial conditions of Theorem \[thm:main\] ------------------------------------------------------------------- Assume that the combinatorial conditions of Theorem 1 hold. We will show that in fact $$\label{eqn:the_soln} \alpha_r = \left\{ \begin{array}{cc} 1-{\mathfrak I}(\widetilde{v}_{r}(w)) & \mbox{ if $r$ is a descent} \\ 1 & \mbox{ otherwise.} \end{array} \right.$$ for $1\leq r\leq n-1$ solves (\[eqn:final\_system\]). It suffices to show that $\sum_{r=i}^{j-1} \alpha_r =1$ whenever $w(i\leftrightarrow j)$ covers $w$. We prove this by induction on $j-i\geq 1$. The base case $j-i=1$ of the induction holds by our definition of $\alpha_r$, since in this case, $w$ does not have a descent at position $i$. Now suppose that $j-i>1$. Let $k$ be chosen (if possible) so that $i<k<j$ and $w(k)$ is minimal such that $w(k)>w(j)$. Similarly, let $\ell$ be chosen (if possible) so that $i<\ell<j$ and $w(\ell)$ is maximal such that $w(\ell)<w(i)$. Notice that since $w(i\leftrightarrow j)$ covers $w$, at least one of $k$ or $\ell$ must exist. We now separately examine the possible cases: First suppose $k$ exists but not $\ell$. Observe that $w$ has a descent at position $j-1$, since, in fact $w(j)<w(m)$ for all $i\leq m\leq j-1$. So we may consider the subword $v_{j-1}(w)$ of $w$. Notice that this necessarily includes $w(i)$, $w(j-1)$, and $w(j)$. By Lemma \[lemma:same\_both\], if $f$ and $g$ are indices between $i$ and $j-1$ in $w$ which correspond to successive entries of $v_{j-1}(w)$, then $w(f\leftrightarrow g)$ covers $w$. So by induction, $$\label{eqn:used_again} \sum_{r=f}^{g-1}\alpha_{r} =1.$$ Since by assumption, the inner corner distances of ${\widetilde v}_{j-1}(w)$ are all the same, by (\[eqn:the\_soln\]): $$\sum_{r=i}^{j-1}\alpha_{r} = \sum_{r=i}^{j-2}\alpha_{r} + \alpha_{j-1} = {\mathfrak I}({\widetilde v}_{j-1}(w))+\alpha_{j-1} =1$$ as desired. A similar argument works in the case that $\ell$ exists but not $k$, except that $v_{i}(w)$ must be used instead. Next suppose that both $k$ and $\ell$ exist. First consider the situation where $k>\ell$. Then by construction, $$w(i\leftrightarrow k), \ w(\ell\leftrightarrow j), \mbox{ and $w(\ell\leftrightarrow k)$ each cover $w$}.$$ Therefore, by the induction hypothesis, we have $$\sum_{r=i}^{k-1}\alpha_r = 1, \ \sum_{r=\ell}^{j-1} \alpha_r = 1, \mbox{ and }\sum_{r=\ell}^{k-1}\alpha_r = 1.$$ Hence, $$\sum_{r=i}^{j-1}\alpha_r = \sum_{r=i}^{k-1}\alpha_r + \sum_{r=\ell-1}^{j-1}\alpha_r - \sum_{r=\ell}^{k-1}\alpha_r=1$$ as desired. Finally, we have the case where $k<\ell$. Observe that the values of $w$ between $k$ and $\ell$ must consist of numbers larger than $w(k)$ followed by numbers smaller than $w(\ell)$, since otherwise it is easy to see that there must exist a $\{(1\leftrightarrow 5), (2\leftrightarrow 3)\}$-restricted embedding of $31524$ or a $\{(1\leftrightarrow 5), (3\leftrightarrow 4)\}$-restricted embedding of $24153$, contradicting the assumptions. Similarly, the values of $w$ between $i$ and $k$ are necessarily smaller than $w(i)$. Let $q$ be the last index $k\leq q<\ell$ such that $w(q)\geq w(k)$; hence $w$ has a descent at $q$. Consider the subword $v_{q}(w)$ of $w$ and observe that $w(i)$ and $w(j)$ are in $v_{q}(w)$, as, otherwise, we would find a bad $31524$ or $24153$ pattern. We are now ready to employ a similar argument as above. By Lemma \[lemma:same\_both\], if $f$ and $g$ are indices of $w$, with either both $f$ and $g$ in the interval $[i,q]$ or both in the interval $[q+1,j]$, and $f$ and $g$ correspond to consecutive entries of $v_{q}(w)$, then $w(f\leftrightarrow g)$ covers $w$; now the induction hypothesis implies (\[eqn:used\_again\]) as before. Therefore, by (\[eqn:the\_soln\]) and our assumptions about $\lambda({\widetilde v}_{q}(w))$, we have $$\sum_{r=i}^{j-1}\alpha_r = {\mathfrak I}({\widetilde v}_{q}(w))+\alpha_q =1$$ as required. Theorem \[thm:main\] follows immediately from the discussion above. Conclusion of the proof of Theorem \[thm:second\_main\] ------------------------------------------------------- In order to complete the above arguments to prove Theorem \[thm:second\_main\], we need two facts about the Borel-Weil construction; see, for example, [@brion:book Section 1.4] and the references therein. First, we note that, if $\mathcal{L}_{\Lambda_{n-r}}$ denotes the line bundle associated to the fundamental weight $\Lambda_{n-r}$ by the Borel-Weil construction, then $\mathrm{div}(\mathcal{L}_{\Lambda_{n-r}})=\left[X_{(r\leftrightarrow r+1)}\right]\in {\rm CaCl}({\rm Flags}(C^n))$; therefore, $$\mathrm{div}(\mathcal{L}_{\Lambda_{n-r}}\big|_{X_w})=\sum_{{\genfrac{}{}{0pt}{}{a\leq r<b}{\ell(w(a\leftrightarrow b))=\ell(w)+1}}} \left[ X_{w(a\leftrightarrow b)}\right]\in {\rm CaCl}(X_w).$$ (The line bundle $\mathcal{L}_{\lambda_{n-r}}$ can be concretely constructed using the isomorphism $\mathcal{L}_{\Lambda_{n-r}}\cong \bigwedge^{n-r} \mathcal{Q}_r$, , where $\mathcal{Q}_r$ is the tautological quotient bundle whose fiber at a flag $F_{\bullet}=(\langle 0\rangle \subseteq F_1\subseteq\ldots\subseteq F_n={\mathbb C}^n)$ is $\mathbb{C}^n/F_r$.) Secondly, addition of weights corresponds to tensor product of line bundles, so that, for any weights $\alpha$ and $\beta$, the line bundle $\mathcal{L}_{\alpha+\beta}=\mathcal{L}_\alpha\otimes\mathcal{L}_\beta$. We have shown that, when $X_w$ is Gorenstein, $$\sum_{r=1}^{n-1} \alpha_r \ \mathrm{div}(\mathcal{L}_{\Lambda_{n-r}}\big|_{X_w}) = \sum_{{\genfrac{}{}{0pt}{}{v\succ w}{\ell(v)=\ell(w)+1}}} \left[X_v\right]=\mathrm{div}(\mathcal{I}(\partial X_w)).$$ Therefore, we have that $\mathcal{I}(\partial X_w)\cong \mathcal{L}_\alpha\big|_{X_w}$, where $\alpha=\sum_{r=1}^{n-1}-\alpha_r\ \Lambda_{n-r}$. Since $\rho=\sum_{r=1}^{n-1}\Lambda_r$, and we have set $\widetilde{\alpha}_r=-1-\alpha_r$ in (\[eqn:the\_soln’\]), this proves Theorem \[thm:second\_main\]. Proof of Corollary \[cor:maxsingloc\] ------------------------------------- We prove Corollary \[cor:maxsingloc\] by comparing Theorem \[thm:main\] with a description of the generic singularities of a Schubert variety given in [@Cortez; @manivel2]. [^3] Suppose a Schubert variety $X_w$ is not Gorenstein along its maximal singular locus. By the local definition of Gorensteinness given in Section 2.1, it is not Gorenstein. To prove the other direction, suppose $X_w$ is not Gorenstein. Then $w$ contains one of the two forbidden patterns, or violates the inner corner condition. If $w$ contains a forbiddern pattern, then, in the language of Cortez [@Cortez], $w$ has a configuration II with $r=0$ and $s+t\geq 1$, and therefore has a generic singularity whose neighborhood is isomorphic to the product of $\mathbb{C}^k$ for some $k$ and the variety of $(s+t+2)\times 2$ matrices of rank at most 1. It is well known that the variety of $p\times q$ matrices of rank at most 1 is Gorenstein if and only if $p=q$; see for example [@Bruns-Herzog Thm. 7.3.6]; this shows that $X_w$ is not Gorenstein at a generic singularity. If $w$ violates the inner corner condition, then $w$ has a configuration I with $s\neq t$, yielding a corresponding generic singularity, which, as it a neighborhood isomorphic to the product of $\mathbb{C}^k$ for some $k$ and the variety of $s\times t$ matrices of rank at most 1, is not Gorenstein. Remarks and Applications ======================== Extension to partial flag varieties {#subsection:partial} ----------------------------------- More generally, let ${\rm Flags}(i_1<i_2<\ldots<i_k,{\mathbb C}^n)$ denote the variety of partial flags $F_{\bullet}:\langle 0\rangle \subseteq F_{i_1} \subseteq F_{i_2}\subseteq \ldots\subseteq F_{i_{k}}\subseteq {\mathbb C}^{n}$ in ${\mathbb C}^{n}$ where here ${\rm dim}(F_{i_{k}})=i_{k}$. By convention let $i_0=0$ and $i_{k+1}=n$. Now let $S=S_{i_1-i_0}\times S_{i_2-i_1}\times\cdots\times S_{i_{k+1}-i_k}\subseteq S_n$ denote the Young subgroup where the $S_{i_j-i_{j-1}}$ factor is generated by the simple reflections $s_{i_{j-1}+1},\ldots,s_{i_j -1}$ for all $j$ such that $1\leq j\leq k$. The Schubert varieties of ${\rm Flags}(i_1<i_2<\ldots<i_k,{\mathbb C}^n)$ are indexed by cosets of $S$. The natural “forgetting subspaces” projection $\pi:{\rm Flags}({\mathbb C}^n)\twoheadrightarrow {\rm Flags}(i_1<i_2<\ldots<i_k,{\mathbb C}^n)$ is a smooth fiber bundle. It follows that a Schubert variety $X_{wS}$ in ${\rm Flags}(i_1<i_2<\ldots<i_k,{\mathbb C}^n)$ indexed by a coset $wS$ is Gorenstein if and only if the Schubert variety $X_{\widetilde w}=\pi^{-1}(X_{wS})$ in ${\rm Flags}({\mathbb C}^n)$ is Gorenstein, where ${\widetilde w}$ is the minimal length element of $wS$. In particular, our main result implies the Grassmannian case as presented in the introduction. Uniqueness of (\[eqn:the\_soln\]) --------------------------------- It is worthwhile to note that the induction in Section 2.4 implies that (\[eqn:the\_soln\]) is a solution to (\[eqn:final\_system\]) if and only if $X_w$ is Gorenstein. Moreover, this solution is essentially unique. The only exception to uniqueness arises for those $r$ where $$\left[X_{(r\leftrightarrow r+1)}\right]\left[X_w\right] =\sum_{{\genfrac{}{}{0pt}{}{a\leq r<b}{\ell(w(a\leftrightarrow b))=\ell(w)+1}}} \left[ X_{w(a\leftrightarrow b)}\right]=0$$ because the sum on the right hand side is vacuous. In these cases, we can arbitrarily assign a value to $\alpha_r$ in order to arrive at a solution. (This is also apparent from the bar diagrams of Section 2.2, as in these cases no horizontal bars cross the $r^{th}$ vertical bar.) Consequently, the expression for $\omega_{X_w}$ given in Theorem \[thm:second\_main\] is unique, up to tensoring by bundles which are trivial when restricted to $X_{w}$. Furthermore: *${\mathbb Q}$-Gorensteinness:* A variety is said to be [**${\mathbb Q}$-Gorenstein**]{} if it is Cohen-Macaulay and some multiple of the canonical divisor is Cartier. Consequently, a Schubert variety $X_w$ is ${\mathbb Q}$-Gorenstein if (\[eqn:final\_system\]) has a rational solution. However, since if any solution exists, an integral solution exists, Gorensteinness and $\mathbb{Q}$-Gorensteinness are equivalent. This will not hold in general for flag varieties of other Lie types. *Computational efficiency:* In order to check if a permutation $w$ corresponds to a Gorenstein Schubert variety, it is typically more computationally efficient solve for (\[eqn:final\_system\]) than to use Theorem \[thm:main\]. In particular, it is enough to check if (\[eqn:the\_soln\]) works. Is it pattern avoidance? ------------------------ In view of [@LS90], it is natural to wonder if it is possible to reformulate Theorem 1 in terms of “classical pattern avoidance”, that is, if there is a finite list of permutations $w_1, w_2, \ldots, w_n$ such that $X_{w}$ is Gorenstein if and only if $w$ pattern avoids these permutations. In fact, this is already impossible for Grassmannian permutations. For example, we know $1346\ \mid \ 25\in S_6$ does not correspond to a Gorenstein Schubert variety. But $w' = \underline{1}2\underline{5}\underline{6}\underline{9}\ \mid \ \underline{3}4\underline{7}8\in S_9$ does. Note that $w'$ contains $w$ as a subpattern, so if a classical pattern avoidance permutation reformulation of Theorem \[thm:main\] existed, it would imply that $X_{w'}$ is not Gorenstein, which is not true. A characterization of Fano Schubert varieties --------------------------------------------- A Gorenstein algebraic variety is [**Fano**]{} if its anticanonical divisor is ample. It follows from Theorem \[thm:second\_main\] that a Gorenstein Schubert variety $X_{w}$ in ${\rm Flags}({\mathbb C}^n)$ is Fano if and only if all of the inner corner distances of $w$ are at most 1. This appears to give new examples of Fano varieties. It seems to have been previously unknown whether or not all smooth Schubert varieties of the flag variety are Fano. By the above remark, it is easy to find examples of Schubert varieties that are smooth but not Fano, in contrast to the case for Grassmannians, for which all smooth Schubert varieties are Fano. Matrix Schubert varieties and ladder determinantal varieties ------------------------------------------------------------ Let $v\in S_n$ be a permutation, and $Y_v$ the associated matrix Schubert variety; this was defined in [@Fulton:Duke92] as the closure in $\mathbb{C}^{n^2}$, considered as the space of $n \times n$ matrices, of $p^{-1}(X_v)$, where $p: \mathrm{GL}_{n}({\mathbb C}) \rightarrow \mathrm{GL}_n(\mathbb{C})/B=\mathrm{Flags}(\mathbb{C}^{n})$ is the quotient map. Now let $w=v \times {\rm id}\in S_n \times S_n \subseteq S_{2n}$ be the permutation agreeing with $v$ on $1,\ldots, n$ and fixing $n+1,\ldots, 2n$. The intersection of $X_w$ with the opposite big cell of flags intersecting the canonical reference flag (whose $i$-th vector space is $\langle e_1,\ldots,e_i\rangle$) generically is then isomorphic to $Y_v \times \mathbb{C}^{n^2-n}$. Every singularity of $X_w$ is represented in this opposite big cell, so $Y_v$ is Gorenstein if and only if $X_w$ is. Identifying ladder determinantal varieties with the appropriate matrix Schubert varieties allows us to recover the characterizations of Gorenstein ladder determinantal varieties found in [@Conca] and [@GonMiller]. Theorem \[thm:second\_main\] and cohomology of line bundles on Gorenstein Schubert varieties -------------------------------------------------------------------------------------------- Theorem \[thm:second\_main\] can be applied to obtain information about the sheaf cohomology groups ${\rm H}^{i}(X_w, {\mathcal L}_{\alpha}\big|_{X_w}\big)$ of the line bundle ${\mathcal L}_{\alpha}\big|_{X_w}$ on a Gorenstein Schubert variety $X_w$. The groups are classically known in the case $X_{\rm id}\cong {\rm Flags}({\mathbb C}^n)$ and $\alpha\in {\rm Hom}(T,{\mathbb C}^n)$ is arbitrary (the classical Borel-Weil-Bott theorem [@Bott]), and for arbitrary $w\in S_n$ when $\alpha$ is dominant [@Demazure]; see, for example, [@Jantzen]. It is an open problem to compute these groups in most of the remaining cases; see [@transform] for some recent progress on this problem. Serre duality (see, for example, [@Hartshorne III.7]) states that, for any projective, equidimensional, $d$-dimensional, Cohen-Macaulay scheme $X$, and any coherent sheaf $\mathcal{F}$ on $X$, we have $$H^i(X,\mathcal{F}) \cong\mathrm{Ext}^{d-i}(\mathcal{F},\omega_X)^* .$$ Let $\alpha$ be the (non-dominant) weight defined in Theorem \[thm:second\_main\], and $\beta$ any weight. Then: $$\begin{aligned} H^i\big(X_w,\mathcal{L}_{\alpha-\beta}\big|_{X_w}\big) & \cong & \mathrm{Ext}^{n-\ell(w)-i}(\mathcal{L}_{\alpha-\beta}\big|_{X_w},\omega_X)^* \\ & = & \mathrm{Ext}^{n-\ell(w)-i}(\mathcal{L}_{\alpha-\beta}\big|_{X_w}, \mathcal{L}_{\alpha}\big|_{X_w})^* \\ & \cong& \mathrm{Ext}^{n-\ell(w)-i}(\mathcal{O}_{X_w},\mathcal{L}_{\beta}\big|_{X_w})^* \\ & \cong & H^{n-\ell(w)-i}\big(X_w,\mathcal{L}_{\beta}\big|_{X_w}\big)^*.\end{aligned}$$ When $\beta$ is dominant, this relates the cohomology groups $H^i\big(X_w,\mathcal{L}_{\alpha-\beta}\big|_{X_w}\big)$ to the cohomology groups known by Demazure’s theorem. For example, it follows that, when $\beta$ is dominant, $H^i\big(X_w,\mathcal{L}_{\alpha-\beta}\big|_{X_w}\big)\cong 0$ for $i\neq n-\ell(w)$. [99]{} A. Altman and S. Kleiman, [*Introduction to Grothendieck duality theory*]{}, Lecture Notes in Math., 146, Springer, Berlin, 1970. 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W. Bruns and J. Herzog, *Cohen-Macaulay rings*, Cambridge Studies in Advanced Mathematics, 39. Cambridge University Press, Cambridge, 1993. C. Chevalley, *Sur les décompositions cellulaires des espaces $G/B$*, in *Algebraic Groups and their Generalizations: Classical Methods*, Proc. Sympos. Pure Math. [**56**]{} (1994), Amer. Math. Soc., 1–23. A. Conca *Gorenstein ladder determinantal rings*, J. London Math. Soc. (2) [**54**]{} (1996), 453–474. A. Cortez, *Singularités génériques et quasi-résolutions des variétés de Schubert pour le groupe linéaire*, Adv. Math. [**178**]{} (2003), 396–445. C. DeConcini and V. Lakshmibai, *Arithmetic Cohen-Macaulayness and arithmetic normality for Schubert varieties*, Am. J. Math. [**103**]{} (1981), 835–850. M. Demazure, *Désingularisation des variétés de Schubert généralisées*, Ann. Sci. Ècole Norm. Sup. (4) [**7**]{} (1974), 53–88. W. Fulton, *Flags, Schubert polynomials, degeneracy loci, and determinantal formulas*, Duke Math. J. [**65**]{} (1992), no. 3, 381–420. , *Intersection theory*, second edition, Springer-Verlag (1998). V. Gasharov, *Sufficiency of Lakshmibai–Sandhya singularity conditions for Schubert varieties*, Compositio Math. [**126**]{} (2001), 47–56 N. Gonciulea and C. Miller, *Mixed ladder determinantal varieties*, J. Algebra [**231**]{} (2000), 104–137. R. Hartshorne, *Algebraic geometry*, Graduate Texts in Mathematics, No. 52. Springer-Verlag, New York-Heidelberg, 1977. R. Hartshorne, *Residues and duality*, Lecture Notes in Mathematics, No. 20. Springer-Verlag, Berlin-New York, 1966. J. C. Jantzen, [*Representations of Algebraic groups*]{}, Pure and Appl. Math., Academic Press, 1987. C. Kassel, A. Lascoux and C. Reutenauer, *The singular locus of a Schubert variety*, J. Algebra [**269**]{} (2003), 74–108. V. Lakshmibai and B. Sandhya, *Criterion for smoothness of Schubert varieties in ${\rm SL}(n)/B$*, Proc. Indian Acad. Sci. Math. Sci. [**100**]{} (1990), no. 1, 45–52. L. Manivel, *Le lieu singulier des variétés de Schubert*, Internat. Math. Res. Notices [**16**]{} (2001), 849–871. , *Generic singularities of Schubert varieties*, arXiv:math.AG/0105239. O. Mathieu, *Formules de caractères pour les algèbres de Kac-Moody générales*, Astérisque [**159-160**]{} (1988). D. Monk, *The geometry of flag manifolds*, Proc. London Math. Soc., [**9**]{} (1959), 253–286. C. Musili and C.S. Seshadri, *Schubert varieties and the variety of complexes*, in *Arithmetic and Geometry, Vol. II*, Progr. Math. [**36**]{} (1983), Birkhäuser, Boston, 329–359. S. Ramanan and A. Ramanathan, [*Projective normality of flag varieties and Schubert varieties*]{}, Invent. Math. [**79**]{} (1985), no. 2, 217–224. A. Ramanathan, *Schubert varieties are arithmetically Cohen-Macaulay*, Invent. Math., [**80**]{} (1985), 283–294. , *Equations defining Schubert varieties and Frobenius splitting of diagonals*, Publ. Math. IHES [**65**]{} (1987), 61–90. Z. Stankova, *Forbidden subsequences*, Discrete Math. [**132**]{} (1994), 291–316. T. Svanes, *Coherent cohomology on Schubert subschemes of flag schemes and applications*, Adv. Math. [**14**]{} (1974), 369–453. [^1]: Consistent with our convention on Schubert varieties in $\mathrm{Flags}(\mathbb{C}^n)$, we index these Schubert varieties so that $\left|\lambda\right|$ is the codimension of $X_\lambda$. [^2]: The problem on the left is solved by $(\alpha_1,\alpha_2,\alpha_3,\alpha_4, \alpha_5,\alpha_6)=(-1,0,1,1,-1,1)$ while the problem on the right has no solution. [^3]: Note that our notation differs from the notation in these papers by right multiplication of a permutation $w$ by $w_0$.

--- abstract: 'High resolution observations of solar filaments suggest the presence of groups of prominence threads, i.e. the fine-structures of prominences, which oscillate coherently (in phase). In addition, mass flows along threads have been often observed. Here, we investigate the effect of mass flows on the collective fast and slow nonadiabatic magnetoacoustic wave modes supported by systems of prominence threads. Prominence fine-structures are modeled as parallel, homogeneous and infinite cylinders embedded in a coronal environment. The magnetic field is uniform and parallel to the axis of threads. Configurations of identical and nonidentical threads are both explored. We apply the $T$-matrix theory of acoustic scattering to obtain the oscillatory frequency and the eigenfunctions of linear magnetosonic disturbances. We find that the existence of wave modes with a collective dynamics, i.e. those that produce significant perturbations in all threads, is only possible when the Doppler-shifted individual frequencies of threads are very similar. This can be only achieved for very particular values of the plasma physical conditions and flow velocities within threads.' author: - 'R. Soler, R. Oliver, and J. L. Ballester' title: Propagation of nonadiabatic magnetoacoustic waves in a threaded prominence with mass flows --- Introduction ============ Prominences/filaments are fascinating coronal magnetic structures, whose dynamics and properties are not well-understood yet. The long life of the so-called quiescent prominences (several weeks) suggests that the cool and dense prominence material is maintained against gravity and thermally shielded from the much hotter and much rarer solar corona by means of some not well-known processes. However, it is believed that the magnetic field must play a crucial role in both the support and isolation of prominences. High-resolution observations of solar filaments reveal that they are formed by a myriad of horizontal structures called threads [e.g., @lin2005], which have been observed in the spines and barbs of both active region and quiescent filaments [@lin2008]. The width of these fine-structures is typically in the range 0.2 arcsec – 0.6 arcsec, which is close to the resolution of present-day telescopes, whereas their lengths are between 5 arcsec and 20 arcsec [@lin2004]. Threads are assumed to be the basic substructures of filaments and to be aligned along magnetic field lines. From the point of view of theoretical modeling, prominence threads are interpreted as large coronal magnetic flux tubes, with the denser and cooler (prominence) region located at magnetic field dips that correspond to the observed threads. Although some theoretical works have attempted to model such structures [e.g., @ballesterpriest; @schmitt; @rempel; @heinzel06], there are some concerns about their formation and stability that have not been resolved yet. Small amplitude oscillations, propagating waves and mass flows are some phenomena usually observed in prominences and prominence threads [see some recent reviews by @oliverballester02; @ballester; @banerjee]. Periods of small-amplitude prominence oscillations cover a wide range from less than a minute to several hours, and they are usually attenuated in a few periods [@molowny; @terradasobs]. Focusing on prominence threads, some works have detected oscillations and waves in such fine-structures [e.g., @yi1; @yi2; @lin2004; @okamoto; @lin2007]. In particular, @yi1 and @lin2007 suggested the presence of groups of near threads that moved in phase, which may be a signature of collective oscillations. On the other hand, mass flows along magnetic field lines have been also detected [@zirker94; @zirker; @lin2003; @lin2005], with typical flow velocities of less than 30 km s$^{-1}$ in quiescent prominences, although larger values have been detected in active region prominences [@okamoto]. Regarding the presence of flows, a phenomenon which deserves special attention is the existence of the so-called counter-streaming flows, i.e., opposite flows within adjacent threads [@zirker; @lin2003]. Motivated by the observational evidence, some authors have broached the theoretical investigation of prominence thread oscillations by means of the magnetohydrodynamic (MHD) theory in the $\beta = 0$ approximation. First, some works [@joarder; @diaz2001; @diaz2003] focused on the study of the ideal MHD oscillatory modes supported by individual nonuniform threads in Cartesian geometry. Later, @diaz2002 considered a more representative cylindrical thread and obtained more realistic results with respect to the spatial structure of perturbations and the behavior of trapped modes. Subsequently, the attention of authors turned to the study of collective oscillations of groups of threads, and the Cartesian geometry was adopted again for simplicity. Hence, @diaz2005 investigated the collective fast modes of systems of nonidentical threads and found that the only nonleaky mode corresponds to that in which all threads oscillate in spatial phase. Later, @diazroberts considered the limit of a periodic array of threads and obtained a similar conclusion. Therefore, these results seem to indicate that all threads within the prominence should oscillate coherently, even if they have different physical properties. However, one must bear in mind that the Cartesian geometry provides quite an unrealistic confinement of perturbations, and so systems of more realistic cylindrical threads might not show such a clear collective behavior. The next obvious step is therefore the investigation of oscillatory modes of systems of cylindrical threads. The first approach to a similar problem was done by [@luna1], who considered a system of two identical, homogeneous cylinders embedded in an unlimited corona. Although [@luna1] applied their results to coronal loops, they are also applicable to prominence threads. These authors numerically found that the system supports four trapped kink-like collective modes. These results have been analytically re-obtained by @tom, by considering the thin tube approximation and bicylindrical coordinates. Subsequently, @luna2 made use of the $T$-matrix theory of acoustic scattering to study the collective oscillations of arbitrary systems of non-identical cylinders. Although the scattering theory has been previously applied in the solar context [e.g., @bogdan87; @keppens], the first application to the study of normal modes of magnetic coronal structures has been performed by @luna2. They concluded that, contrary to the Cartesian case of @diaz2005, the collective behavior of the oscillations diminishes when cylinders with nonidentical densities are considered, the oscillatory modes behaving in practice like individual modes if cylinders with mildly different densities are assumed. The present study is based on @luna2 and applies their technique to the investigation of MHD waves in systems of cylindrical prominence threads. Moreover, we extend their model by considering some effects neglected by them. Here, the more general $\beta \neq 0$ case is considered, allowing us to describe both slow and fast magnetoacoustic modes. In addition, the adiabatic assumption is removed and, following previous papers [@soler1; @solerapj], the effect of radiative losses, thermal conduction and plasma heating is taken into account. The detection of mass flows in prominences has motivated us to include this effect in our study, and so the presence of flows along magnetic field lines is also considered here. Therefore, the present work extends our recent investigation [@solerapj hereafter Paper I], which was focused on individual thread oscillations, to the study of collective MHD modes in prominence multi-thread configurations with mass flows. On the other hand, the longitudinal structure of threads [e.g., @diaz2002] is neglected in the present investigation. For this reason, the effect of including a longitudinal variation of the plasma physical conditions within threads should be investigated in a future work. Finally, the prominence multi-thread model developed here could be an useful tool for future seismological applications [similar to that of @hinode]. This paper is organized as follows. The description of the model configuration and the mathematical method are given in § \[sec:math\]. Then, the results are presented in § \[sec:results\]. First, the case of two identical prominence threads is investigated in § \[sec:2treads\]. Later, this study is extended to a configuration of two different threads in § \[sec:ntreads\]. Finally, our conclusion is given in § \[sec:conclusions\]. Mathematical method {#sec:math} =================== Our equilibrium system is made of an arbitrary configuration of $N$ homogeneous and unlimited parallel cylinders, representing prominence threads, embedded in an also homogeneous and unbounded coronal medium. Each thread has its own radius, $a_j$, temperature, $T_j$, and density, $\rho_j$, where the subscript $j$ refers to a particular thread. On the other hand, the coronal temperature and density are $T_c$ and $\rho_c$, respectively. Cylinders are orientated along the $z$-direction, the $xy$-plane being perpendicular to their axis. The magnetic field is uniform and also orientated along the $z$-direction, ${\mathit {\bf B}}_j=B_j \hat{\bf e}_z$ being the magnetic field in the $j$-th thread, and ${\mathit {\bf B}}_c=B_c \hat{\bf e}_z$ in the coronal medium. In addition, steady mass flows are assumed along magnetic field lines, with flow velocities and directions that can be different within threads and in the corona. Thus, ${\mathit {\bf U}}_j=U_j \hat{\bf e}_z$ represents the mass flow in the $j$-th thread, whereas ${\mathit {\bf U}}_{\rm c}=U_{\rm c} \hat{\bf e}_z$ corresponds to the coronal flow. For simplicity, in all the following expressions a subscript 0 indicates local equilibrium values, while subscripts $j$ or $c$ denote quantities explicitly computed in the $j$-th thread or in the corona, respectively. Such as shown in Paper I, linear nonadiabatic magnetoacoustic perturbations are governed by the next equation for the divergence of the velocity perturbation, $\Delta = \nabla \cdot {\mathit {\bf v}}_1$, $$\Upsilon^2_0 \left[ \Upsilon^2_0 - \left( \tilde{\Lambda}_0^2 + {v_{\mathrm{A 0}}}^2 \right) \nabla^2 \right] \Delta + \tilde{\Lambda}_0^2 {v_{\mathrm{A 0}}}^2 \frac{{\partial}^2}{{\partial}z^2} \nabla^2 \Delta = 0, \label{eq:basic}$$ where $\Upsilon_0$ is the following operator, $$\Upsilon_0 = \frac{{\partial}}{{\partial}t} + U_0 \frac{{\partial}}{{\partial}z},$$ while ${v_{\mathrm{A 0}}}^2 = \frac{B_0^2}{\mu \rho_0}$ is the Alfvén speed squared and ${\tilde{\Lambda}_0}^2$ is the nonadiabatic sound speed squared, $$\tilde{\Lambda}^2_0 \equiv \frac{{c_{\mathrm{s 0}}}^2}{\gamma} \left[ \frac{\left( \gamma-1 \right) \left( \frac{T_0}{p_0} \kappa_{\parallel 0} k_z^2 + \omega_{T 0} - \omega_{\rho 0} \right) + i \gamma \Omega_0} {\left( \gamma -1 \right) \left( \frac{T_0}{p_0} \kappa_{\parallel 0} k_z^2 + \omega_{T 0} \right) + i \Omega_0} \right], \label{eq:lambda}$$ ${c_{\mathrm{s 0}}}^2 = \frac{\gamma p_0}{\rho_0}$ and $\gamma$ being the adiabatic sound speed squared and the adiabatic ratio, respectively. Terms with $\kappa_{\parallel 0}$, $\omega_{\rho 0}$, and $\omega_{T 0}$ are related to nonadiabatic mechanisms, i.e. radiative losses, thermal conduction, and heating (see Paper I for details). Finally, ${\Omega_0}$ is the Doppler-shifted frequency [@terra], $${\Omega_0}= \omega - k_z U_0,$$ where $\omega$ is the oscillatory frequency and $k_z$ is the longitudinal wavenumber. Considering cylindrical coordinates, namely $r$, $\varphi$, and $z$ for the radial, azimuthal, and longitudinal coordinates, respectively, we can write $\Delta$ in the following form, $$\Delta = \psi \left( r, \varphi \right) \exp \left( i \omega t - i k_z z \right), \label{eq:div}$$ where the function $ \psi \left( r, \varphi \right)$ contains the full radial and azimuthal dependence. By inserting this last expression into equation (\[eq:basic\]), the following Helmholtz equation is obtained, $$\nabla_{r \varphi}^2 \psi \left( r, \varphi \right) + m_0^2\, \psi \left( r, \varphi \right) = 0, \label{eq:hem}$$ where $ \nabla_{r \varphi}^2$ is the Laplacian operator for the $r$ and $\varphi$ coordinates, and $$m_0^2 = \frac{\left( \Omega_0^2 - k_z^2 {v_{\mathrm{A 0}}}^2 \right) \left( \Omega_0^2 - k_z^2 \tilde{\Lambda}^2_0 \right)} {\left( {v_{\mathrm{A 0}}}^2 + \tilde{\Lambda}^2_0 \right) \left( \Omega_0^2 - k_z^2 {\tilde{c}_{\mathrm{T 0}}}^2 \right)}, \label{eq:m0}$$ $${\tilde{c}_{\mathrm{T 0}}}^2 \equiv \frac{{v_{\mathrm{A 0}}}^2 \tilde{\Lambda}^2_0}{{v_{\mathrm{A 0}}}^2 + \tilde{\Lambda}_0^2}, \label{eq:ct}$$ are the radial wave number and the nonadiabatic tube speed squared, respectively. Moreover, due to the presence of nonideal terms $m_0^2$ is a complex quantity. Since nonadiabatic mechanisms produce a small correction to the adiabatic wave modes, $| \Re(m_0^2) | > | \Im(m_0^2) |$ and the dominant wave character depends on the sign of $\Re(m_0^2)$. Here, we investigate nonleaky modes, which are given by $\Re(m_c^2) < 0$. We impose no restriction on the wave character within threads. In order to solve equation (\[eq:hem\]), we consider the technique developed by @luna2 based on the study of wave modes of an arbitrary configuration of cylinders by means of the $T$-matrix theory of acoustic scattering. The novelty with respect to the work of @luna2 is that the method is applied here to solve a Helmholtz equation for the divergence of the velocity perturbation (our eq. \[\[eq:basic\]\]) whereas @luna2 considered an equation for the total pressure perturbation in the $\beta = 0$ approximation (their eq. \[1\]). The present approach allows us to study the more general $\beta \neq 0$ case, therefore slow modes are also described. In addition, nonadiabatic effects and mass flows are easily included in our formalism. However, the rest of the technique is absolutely equivalent to that of @luna2, and therefore the reader is refered to their work for an extensive explanation of the mathematical technique [see also an equivalent formalism in @bogdancatta]. We next give a brief summary of the method. The main difference between our application and that of @luna2 is in the definition of the $T$-matrix elements. These elements are obtained by imposing appropriate boundary conditions at the edge of threads, i.e., at $|{\bf r} - {\bf r}_j | = a_j$, where ${\bf r}_j$ is the radial vector corresponding to the position of the $j$-th thread center with respect to the origin of coordinates. In our case, these boundary conditions are the continuity of the total pressure perturbation, $p_{\rm T}$, and the Lagrangian radial displacement, $\xi_r =- i v_r / {\Omega_0}$. Expressions for these quantities as functions of $\Delta$ and its derivative are, $$p_{\rm T} = i \rho_0 \frac{\left( {\Omega_0}^2 - k_z^2 {\tilde{\Lambda}_0}^2 \right) \left( {\Omega_0}^2 - k_z^2 {v_{\mathrm{A 0}}}^2 \right)}{{\Omega_0}^3 m_0^2} \Delta,$$ $$\xi_r = i \frac{\left( {\Omega_0}^2 - k_z^2 {\tilde{\Lambda}_0}^2 \right)}{{\Omega_0}^3 m_0^2} \frac{\partial \Delta}{\partial r}.$$ Expressions for the rest of perturbations are given in Appendix A of Paper I. Thus, in our case the $T$-matrix elements have the following form, $$T_{mm}^j = \frac{m_c \rho_j \left( \Omega_j^2 - k_z^2 v_{{\rm A}_j}^2 \right) J_m \left( m_j a_j \right) J_m' \left( m_c a_j \right) - m_j \rho_c \left( \Omega_c^2 - k_z^2 v_{{\rm A}_c}^2 \right) J_m \left( m_c a_j \right) J_m' \left( m_j a_j \right)} {m_c \rho_j \left( \Omega_j^2 - k_z^2 v_{{\rm A}_j}^2 \right) J_m \left( m_j a_j \right) {H'}_m^{(1)} \left( m_c a_j \right) - m_j \rho_c \left( \Omega_c^2 - k_z^2 v_{{\rm A}_c}^2 \right) H_m^{(1)} \left( m_c a_j \right) J_m' \left( m_j a_j \right)},$$ where $H_m^{(1)}$ and $J_m$ are the Hankel function of the first kind and the Bessel function of order $m$, respectively, while the prime denotes the derivative taken with respect to $r$. Note that the denominator of $T_{mm}^j$ vanishes at the normal mode frequencies of an individual thread. This can be easily checked by comparing it to the dispersion relation of a single thread, see equation (19) of Paper I, in which Bessel $K_m$ functions are used instead of Hankel functions. The equivalence between both kinds of functions is given in @abram. Thus, following @luna2, the internal $\psi \left( r, \varphi \right)$ field of the $j$-th thread is, $$\psi^j_{\rm int} (r,\varphi)= \sum_{m = -\infty}^{\infty} A_m^j J_m \left( m_j |{\bf r} - {\bf r}_j | \right)e^{i m \varphi_j}, \label{eq:int}$$ whereas the external net field is, $$\psi_{\rm ext}(r,\varphi) = - \sum_j \sum_{m=-\infty}^{\infty} 2 \alpha_{2,m}^j T_{mm}^j H_m^{(1)} \left( m_c |{\bf r} - {\bf r}_j | \right) e^{i m \varphi_j}, \label{eq:ext}$$ where $\varphi_j$ is the azimuthal angle corresponding to the position of the $j$-th thread center with respect to the origin of coordinates, and $\alpha_{2,m}^j$ and $A_m^j$ are constants. Equations (\[eq:int\]) and (\[eq:ext\]) allow us to construct the spatial distribution of $\Delta$. Subsequently, the rest of perturbations can be obtained. Finally, the constants $\alpha_{2,m}^j$ form a homogeneous system of linear algebraic equations, $$\alpha_{2,m}^j + \sum_{k \neq j} \sum_{n=-\infty}^{\infty} T_{nn}^k \alpha_{2,n}^k H_{n-m}^{(1)} \left( m_c |{\bf r}_j - {\bf r}_k | \right) e^{i \left( n -m \right) \varphi_{j k}} = 0, \label{eq:system}$$ for $-\infty < m < \infty$. Once both integers $m$ and $n$ are truncated to a finite number of terms, the non-trivial (i.e., non-zero) solution of system (\[eq:system\]) gives us a dispersion relation for the oscillatory frequency, $\omega$, which is enclosed in the definitions of $m_j$ and $m_c$. In the next Sections, we apply the method to obtain the oscillatory frequency and the spatial distribution of perturbations of the wave modes supported by prominence thread configurations. We assume that $k_z$ is real, so a complex frequency is obtained, namely $\omega = \omega_{\rm R} + i \omega_{\rm I}$. The imaginary part of the frequency appears due to the presence of nonadiabatic mechanisms. The oscillatory period, $P$, and the damping time, ${\tau_{\mathrm{D}}}$, are related to the frequency as follows, $$P = \frac{2 \pi}{| \omega_{\rm R} |}, \qquad \tau_{\rm D} = \frac{1}{\omega_{\rm I}}.$$ Results {#sec:results} ======= Configuration of two identical threads {#sec:2treads} -------------------------------------- Now, we consider a configuration of two identical threads (see Fig. \[fig:model\]). Their physical conditions are typical of prominences ($T_1 = T_2 =8000$ K, $\rho_1 = \rho_2 = 5 \times 10^{-11}$ kg m$^{-3}$) while the coronal temperature and density are $T_{\rm c} = 10^6$ K and $\rho_{\rm c} = 2.5 \times 10^{-13}$ kg m$^{-3}$, respectively. Their radii are $a_1 = a_2 = a =30$ km, and the distance between centers is $d= 4 a = 120$ km. The magnetic field strength is 5 G everywhere. The flow velocity inside the cylinders is denoted by $U_{\rm 1}$ and $U_{\rm 2}$, respectively, whereas the flow velocity in the coronal medium is $U_{\rm c}$. Unless otherwise stated, these physical conditions are used in all calculations. ### Wave modes in the absence of flow First, we consider no flow in the equilibrium, i.e. $U_{\rm 1} = U_{\rm 2} = U_{\rm c} = 0$. We fix the longitudinal wavenumber to $k_z a = 10^{-2}$, which corresponds to a wavelength within the typically observed range. In addition to the four kink modes described by @luna1, i.e. the $S_x$, $A_x$, $S_y$, and $A_y$ modes, where $S$ or $A$ denote symmetry or antisymmetry of the total pressure perturbation with respect to the $yz$-plane, and the subscripts refer to the main direction of polarization of motions, we also find two more fundamental collective wave modes (one symmetric and one antisymmetric) mainly polarized along the $z$-direction, which we call $S_z$ and $A_z$ modes following the notation of @luna1. These new solutions correspond to slow modes which are absent in the investigation of @luna1 due to their $\beta = 0$ approximation. As was stated by @luna2, a collective wave mode is the result of a coupling between individual modes. So the reader must be aware that in the present work we indistinctly use both expressions, i.e., collective modes and coupled modes, to refer to wave solutions whose perturbations have significant amplitudes in both threads. The total pressure perturbation field, $p_{\rm T}$, and the transverse Lagrangian displacement vector-field, $\bf{\xi}_\perp$, corresponding to the six fundamental modes are displayed in Figure \[fig:eigen\]. On the other hand, Figure \[fig:cut\] displays a cut of the Cartesian components of the Lagrangian displacement ($\xi_x$, $\xi_y$, and $\xi_z$) at $y=0$, again for these six solutions. For simplicity, only the real part of these quantities are plotted in both Figures, since their imaginary parts are equivalent. One can see in Figure \[fig:cut\] that the amplitude of the longitudinal (magnetic field aligned) Lagrangian displacement, $\xi_z$, of the $S_z$ and $A_z$ modes is much larger than the amplitude of transverse displacements, $\xi_x$ and $\xi_y$, such as corresponds to slow modes in $\beta < 1$ homogeneous media, while the contrary occurs for the $S_x$, $A_x$, $S_y$, and $A_y$ fast kink solutions. Next, Figure \[fig:distkink\]a displays the ratio of the real part of the frequency of the four kink solutions to the frequency of the individual kink mode, $\omega_k$ (from Paper I), as a function of the distance between the center of cylinders, $d$. This Figure is equivalent to Figure 3 of @luna1 and, in agreement with them, one can see that the smaller the distance between centers, the larger the interaction between threads and so the larger the separation between frequencies. On the other hand, Figure \[fig:distkink\]b shows the ratio of the damping time to the period of the four kink modes as a function of $d$. We see that the damping times are between 4 and 7 orders of magnitude larger than their corresponding periods. Therefore, dissipation by non-adiabatic mechanisms cannot be responsible for the observed damping times of transverse thread oscillations, as was pointed out in Paper I. Recently, @arregui found that the mechanism of resonant absorption can provide kink mode damping times compatible with those observed. Regarding slow modes, Figure \[fig:distslow\]a displays the ratio of the real part of the frequency of the $S_z$ and $A_z$ solutions to the frequency of the individual slow mode, $\omega_s$ (from Paper I). One can see that the frequencies of the $S_z$ and $A_z$ modes are almost identical to the individual slow mode frequency, and so the strength of the interaction is almost independent of the distance between cylinders. This is consistent with the fact that transverse motions (responsible for the interaction between threads) are not significant for slow-like modes in comparison with their longitudinal motions. Therefore, the $S_z$ and $A_z$ modes essentially behave as individual slow modes, contrary to kink modes, which display a more significant collective behavior. Finally, Figure \[fig:distslow\]b shows ${\tau_{\mathrm{D}}}/ P$ corresponding to the $S_z$ and $A_z$ solutions versus $d$. One sees that both slow modes are efficiently attenuated by non-adiabatic mechanisms, with ${\tau_{\mathrm{D}}}/ P \approx 5$, which is in agreement with previous studies [@soler1; @solerapj] and consistent with observations. ### Effect of steady mass flows on the collective behavior of wave modes The aim of the present section is to assess the effect of flows on the behavior of collective modes. With no loss of generality, we assume no flow in the corona, i.e. $U_{\rm c} = 0$. On the other hand, the flow velocities in both cylinders, namely $U_1$ and $U_2$, are free parameters. We vary these flow velocities between -30 km s$^{-1}$ and 30 km s$^{-1}$, which correspond to the range of typically observed flow velocities in filament threads [e.g., @lin2003]. These flow velocities are below the critical value that determines the apparition of the Kelvin-Helmholtz instability [see details in @holzwarth]. In our configuration, a positive flow velocity means that the mass is flowing towards the positive $z$-direction, whereas the contrary is for negative flow velocities. From Paper I [see also @terra] we know that the symmetry between waves whose propagation is parallel ($\omega_{\rm R} > 0$) or anti-parallel ($\omega_{\rm R} < 0$) with respect to magnetic field lines is broken by the presence of flows. Hence, we must take into account the direction of wave propagation in order to perform a correct description of the wave behavior. Following Paper I, we call parallel waves those solutions with $\omega_{\rm R} > 0$, while anti-parallel waves are solutions with $\omega_{\rm R} < 0$. We begin this investigation with transverse modes. First, we assume $U_1 =$ 20 km s$^{-1}$ and study the behavior of the oscillatory frequency when $U_2$ varies (see Fig. \[fig:phase\]). Since frequencies are almost degenerate and, therefore, almost indiscernible if they are plotted together, we use the notation of @tom and call low-frequency modes the $S_x$ and $A_y$ solutions, while high-frequency modes refer to $A_x$ and $S_y$ solutions. In addition, we restrict ourselves to parallel propagation because the argumentation can be easily extended to anti-parallel waves. To understand the asymptotic behavior of frequencies in Figure \[fig:phase\], we define the following Doppler-shifted individual kink frequencies: $$\Omega_{k 1} = \omega_k + U_1 k_z, \label{eq:wkleft}$$ $$\Omega_{k 2} = \omega_k + U_2 k_z. \label{eq:wkright}$$ Since $U_1$ is fixed, $\Omega_{k 1}$ is a horizontal line in Figure \[fig:phase\], whereas $\Omega_{k 2}$ is linear with $U_2$. Three interesting situations have been pointed by means of small letters from $a$ to $c$ in Figure \[fig:phase\]. Each of these letters also corresponds to a panel of Figure \[fig:eigenampl\] in which the total pressure perturbation field of the $S_x$ mode is plotted. The three different situations are commented in detail next (remember that in all cases $U_1 = 20$ km s$^{-1}$): - $a)$ $U_2 = -10$ km s$^{-1}$ ($U_2 < U_1$). This corresponds to a situation of counter-streaming flows. From Figure \[fig:phase\] we see that the frequency of low-frequency modes is close to $\Omega_{k 2}$, whereas that of high-frequency solutions is near $\Omega_{k 1}$. Thus, these solutions do not interact with each other and low-frequency (high-frequency) solutions are related to individual oscillations of the second (first) thread. This is verified by looking at the total pressure perturbation field in Figure \[fig:eigenampl\]a, that shows that only the second thread is significantly perturbed. Therefore, for an external observer this situation corresponds in practice to an individual thread oscillation. - $b)$ $U_2 = 20$ km s$^{-1}$ ($U_2 = U_1$). The flow velocities and their directions are equal in both threads. In such a situation, low- and high-frequency modes couple. At the coupling, an avoided crossing of the solid and dashed lines is seen in Figure \[fig:phase\]. Because of this coupling solutions are related no more to oscillations of an individual thread but they are now collective and, for this reason, Figure \[fig:eigenampl\]b shows a significant pressure perturbation in both threads. - $c)$ $U_2 = 27$ km s$^{-1}$ ($U_2 > U_1$). This case is the opposite to situation $a)$. Therefore, in practice the present situation corresponds again to an individual thread oscillation. Our argumentation is supported by Figure \[fig:ampli\], which displays the amplitude of the transverse Lagrangian displacement, $\xi_\perp$, at the center of the second thread as a function of $U_2$, for parallel and anti-parallel kinklike waves. The displacement amplitude at the center of the first thread is always set equal to unity. The three previously commented situations have been pointed again in Figure \[fig:ampli\]. We clearly see that the displacement amplitude is only comparable in both threads, and so their dynamics is collective, when their flow velocities are similar. Next we turn our attention to slow modes. The behavior of the $S_z$ and $A_z$ modes is like that of low- and high-frequency kinklike solutions, so we comment them in short for the sake of simplicity. $S_z$ and $A_z$ solutions can only be considered collective when the flow velocity is the same in both threads because, in such a case, the $S_z$ and $A_z$ modes couple. If different flows within the threads are considered, the $S_z$ and $A_z$ slow modes lose their collective aspect and their frequencies are very close to the Doppler-shifted individual slow frequencies, $$\Omega_{s 1} = \omega_s + U_1 k_z, \label{eq:wsleft}$$ $$\Omega_{s 2} = \omega_s + U_2 k_z. \label{eq:wsright}$$ Then, the $S_z$ and $A_z$ solutions behave like individual slow modes. It is worth to mention that the coupling between slow modes is much more sensible to the flow velocities in comparison with fast modes, and the $S_z$ and $A_z$ solutions quickly decouple if $U_1$ and $U_2$ slightly differ. An example of this behavior is seen in Figure \[fig:eigenamplslow\], which displays the total pressure perturbation field of the $A_z$ mode for $U_1 - U_2=10^{-3}$ km s$^{-1}$. Although the difference of the flow velocities is insignificant, one can see that in this situation the $A_z$ mode essentially behaves as the individual slow mode of the second thread. The main idea behind these results is that fast or slow wave modes with a collective appearance (i.e., modes with a similar displacement amplitude within all threads) are only possible when the Doppler-shifted individual kink (eqs. \[\[eq:wkleft\]\] and \[\[eq:wkright\]\]) or slow (eqs. \[\[eq:wsleft\]\] and \[\[eq:wsright\]\]) frequencies are similar in both threads. In a system of identical threads, this can only be achieved by considering the same flow velocities within all threads, since all of them have the same individual kink and slow frequencies. However, if threads with different physical properties are considered (i.e., different individual frequencies), it is possible that the coupling may occur for different flow velocities. This is explored in the next section. Configuration of two threads with different physical properties {#sec:ntreads} --------------------------------------------------------------- Now we consider a system of two nonidentical threads and focus first on kink modes. From § \[sec:2treads\] we expect that collective kink motions occur when the Doppler-shifted individual kink frequencies of both threads coincide. The relation between flow velocities $U_1$ and $U_2$ for which the coupling takes place can be easily estimated from the equation, $$\omega_{k 1} + U_1 k_z \approx \omega_{k 2} + U_2 k_z.$$ Note that the individual kink frequency of each thread is different, i.e., $\omega_{k 1} \neq \omega_{k 2}$. Then, the relation between flow velocities at the coupling is, $$U_1 - U_2 \approx \frac{\omega_{k 2} - \omega_{k 1}}{k_z}.$$ This last expression can be simplified by considering the approximate expression for the kink frequency in the long-wavelength limit, $$\omega_{k i} \approx \pm \sqrt{\frac{2}{1+\rho_c / \rho_i}} v_{{\rm A} i} k_z \approx \pm \sqrt{2}\, v_{{\rm A} i} k_z,$$ for $i=1,2$, where the $+$ sign is for parallel waves and the $-$ sign is for anti-parallel propagation. Then, one finally obtains, $$U_1 - U_2 \approx \pm \sqrt{2} \left( v_{{\rm A} 2} - v_{{\rm A} 1} \right), \label{eq:velrelation}$$ where the meaning of the $+$ and $-$ signs is the same as before. In the case of identical threads, $v_{{\rm A} 1} = v_{{\rm A} 2}$ and so $U_1 - U_2 = 0$. Thus the flow velocity must be the same in both threads to obtain collective motions, as we concluded in § \[sec:2treads\]. An equivalent analysis can be performed for slow modes and one obtains, $$U_1 - U_2 \approx \pm \left( c_{s 2} - c_{s 1} \right). \label{eq:velrelationslow}$$ In general, the coupling between slow modes occur at different flow velocities than the coupling between kink modes. This makes difficult the simultaneous existence of collective slow and kink solutions in systems of non-identical threads. Next, we assume a particular configuration of two non-identical threads in order to verify the later argumentation. Thread radii are $a_1=30$ km and $a_2 = 45$ km, whereas their physical properties are $T_1=8000$ K, $\rho_1 = 5 \times 10^{-11}$ kg m$^{-3}$, and $T_2=12000$ K, $\rho_2 = 3.33 \times 10^{-11}$ kg m$^{-3}$. Coronal conditions are those taken in § \[sec:2treads\] (i.e., $T_{\rm c} = 10^6$ K, $\rho_{\rm c} = 2.5 \times 10^{-13}$ kg m$^{-3}$). The magnetic field strength is 5 G everywhere and the distance between the thread centers is $d= 120$ km. We assume $U_1 = 10$ km s$^{-1}$. In the present case, also four kinklike solutions are present, which are grouped in two almost degenerate couples. For this reason, we again refer to them as low- and high-frequency kink solutions. Their frequency as a function of $U_2$ is displayed in Figure \[fig:phasediff\]. At first sight, we see that solutions couple for a particular value of $U_2$, as expected. Applying equation (\[eq:velrelation\]), and considering that $v_{{\rm A} 1} = 63.08$ km s$^{-1}$ and $v_{{\rm A} 2} = 77.29$ km s$^{-1}$, we obtain $ U_1- U_2 \approx \pm 20.10\,\rm{km}\,\rm{s}^{-1}$, and since $U_1=10$ km s$^{-1}$, we get $U_2 \approx - 10.10$ km s$^{-1}$ for parallel propagation. We see that the approximate value of $U_2$ obtained from equation (\[eq:velrelation\]) is in good agreement with Figure \[fig:phasediff\]. In addition, we obtain that for parallel propagation collective dynamics appear in a situation of counter-streaming (opposite) flows. This result is of special relevance because counter-streaming flows have been detected in prominences [@zirker; @lin2003] and might play a crucial role in the collective behavior of oscillations. On the contrary, in the anti-parallel propagation case we obtain $U_2 \approx 30.10$ km s$^{-1}$ from equation (\[eq:velrelation\]), and so both flows are in the same direction and large a quite value of $U_2$ is obtained in comparison with the parallel propagation case. Regarding slow modes, considering that $c_{s 1} = 11.76$ km s$^{-1}$ and $c_{s 2} = 14.40$ km s$^{-1}$, equation (\[eq:velrelationslow\]) gives $U_2 \approx 7.36$ km s$^{-1}$ for parallel propagation and $U_2 \approx 12.64$ km s$^{-1}$ for anti-parallel waves. Note that in our particular example the flow velocities needed for the coupling situation are realistic and within the range of typically observed velocities. However, if threads with very different physical properties and, therefore, with very different Alfvén and sound speeds are considered, the coupling flow velocities could be larger than the observed values. This means that the conditions necessary for collective oscillations of systems of threads with very different temperatures and/or densities may not be realistic in the context of solar prominences. Conclusion {#sec:conclusions} ========== In this work, we have assessed the effect of mass flows on the collective behavior of slow and fast kink magnetosonic wave modes in systems of prominence threads. We have seen that the relation between the individual Alfvén (sound) speed of threads is the relevant parameter which determines whether the behavior of kink (slow) modes is collective or individual. In the absence of flows and when the Alfvén speeds of threads are similar, kink modes are of collective type. On the contrary, perturbations are confined within an individual thread if the Alfvén speeds differ. In the case of slow modes, the conclusion is equivalent but replacing the Alfvén speeds by the sound speeds of threads. On the other hand, when flows are present in the equilibrium, one can find again collective motions even in systems of non-identical threads by considering appropriate flow velocities. These velocities are within the observed values if threads with not too different temperatures and densities are assumed. However, since the flow velocities required for collective oscillations must take very particular values, such a special situation may rarely occur in real prominences. Therefore, if coherent oscillations of groups of threads are observed in prominences [e.g., @lin2007], we conclude that either the physical properties and flow velocities of all oscillating threads are quite similar or, if they have different properties, the flow velocities within threads are the appropriate ones to allow collective motions. From our point of view, the first option is the most probable one since the flow velocities required in the second case correspond to a very peculiar situation. This conclusion has important repercussions for future prominence seismological applications, in the sense that if collective oscillations are observed in large areas of a prominence, threads in such regions should possess very similar temperatures, densities, and magnetic field strengths Here, we have only considered two-thread systems, but the method can be applied to an arbitrary multi-thread configuration. 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--- abstract: 'We explore the phenomenology of new R-parity violating operators that can occur in $E_6$ models. The set of allowed operators is found to depend sensitively on the nature of the extension of the standard model gauge group. These new interactions lead to additional production processes for the exotic particles in such models and allow the LSP to decay but with a highly suppressed rate. The implications of these new interations are examined for the Tevatron, SSC, LHC, HERA, and $\sqrt{s} = 0.5$ and $1$ TeV $e^+e^-$ colliders.' --- -0.5in 6.0in 8.00in 4.0=0.5in -0.5in \#1[$^{#1}$ ]{} \#1 \#1 \#1\#2\#3[[**\#1**]{}, \#2 (19\#3)]{} \#1[\[\#1\]]{} 9.0in 6.5in [**Phenomenology of R-parity Breaking in $E_6$ Models**]{}[^1] Thomas G. Rizzo\ High Energy Physics Division\ Argonne National Laboratory\ Argonne, IL 60439\ and\ Ames Laboratory and Department of Physics\ Iowa State University, Ames, IA 50011\ 8.0in 6.0in It is by now well known that in the minimal SUSY standard model gauge symmetries alone do not forbid the existence of baryon ($B$) and/or lepton $(L)$ number violating term in the superpotential which break R-parity and lead to a rich phenomenology.$^1$ Of course, if rapid proton decay is to be avoided, not all of the terms allowed by gauge invariance can occur simultaneously so that additional symmetries must be present forbidding either the $\Delta B \neq 0\ \rm {or}\ \Delta L \neq 0$ terms. In $E_6$ models, the enlargement of the particle spectrum from the usual 15 to 27 2-component fields per generation already leads to a fairly complicated structure for the superpotential$^2$ when R-parity is conserved. In fact, even in this case not all of the usual 11 terms in the superpotential can be allowed simultaneously if we wish to ensure proton stability. Another way to think about this is that one finds that the $B$ and $L$ assignments for all of the ‘exotic’ fields in the  representation of $E_6$ will not be uniquely defined if all 11 terms of the superpotential are present. For example, we choose whether the color-triplet, isoinglet, $Q = -1/3$ field is a leptoquark, diquark, or an ordinary quark by which terms we allow in the superpotential. If we are willing to give up R-parity in these extended models this kind of complexity will be only significantly compounded. Of course in general $E_6$ models, the low energy gauge symmetry is larger than that of the SM by at least an additional U(1) factor. This additional symmetry may disallow several if not all of the new terms in the superpotential which violate R-parity. The structure of the new terms, whether they are allowed in models with particular extended gauge symmetries, and some of the possible phenomenological implications will be the main points we wish to address in this paper. We will restrict our attention to the existence of explicit, renormalizable R-parity violating terms in the superpotential in our discussion below. The possibility of introducing non-renormalizable higher dimension terms induced by loops will be discussed elsewhere. We know in the MSSM that requiring only $SU(3)_c \times SU(2)_L \times U(1)_Y$ gauge invariance and renormalizability leads not only to superpotential ($W$) terms which generate fermion masses and Higgs self-couplings $$W_1 = Qu^c\ H^c + Qd^c\ H + Le^c\ H + HH^c \auto$$ but also to $\Delta L,\ \Delta B \neq 0$ terms as well: $$W_2 = QLd^c + LLe^c + u^cd^cd^c + LH^c \auto$$ where coupling constants and generation indices are suppressed. The first, second, and fourth terms in $W_2$ violate $L$ whereas the third term violates $B$; the $LH^c$ terms can in principle be eliminated by a suitable rotation among the superfields.$^1$ As mentioned above, [*both*]{} $\Delta L \neq 0$ and $\Delta B \neq 0$ terms cannot occur simultaneously and this is usually insured by the existence of a new discrete symmetry. In $E_6$ models imposing [*only*]{} $SU(3)_C \times SU(2)_L \times U(1)_Y$ gauge symmetry leads to a large increase in the number of terms in $W$. In order to fix the notation for the exotic fields in the representation we follow the nomenclature used in Hewett and Rizzo$^2$ and shown in Table 1. One finds that there are now 25 possible trilinear terms (shown in Table 2) and 7 possible bilinear terms (shown in Table 3). Of these 32 terms, 11 are those usually associated with $W$ in $E_6$ models, 4 are the R-parity violating terms in (2) and one is the $HH^c$ term in (1); thus 16 new terms are generated which contain ‘exotic’ fields, , those present in the  of $E_6$ but not in the MSSM. We note that all of the new trilinear terms contain [*at least two*]{} exotic superfields. In what follows, we will generally ignore any new physics associated with the terms violating R-parity in the MSSM which appear in (2) since they have already been examined in the literature$^1$ and will already be familiar to the reader. Before discussing the phenomenology of the new terms in $W$ we remind the reader of the influence of the terms which already appear in the standard R-parity conserving version of $E_6$ models. For example, terms 4, 5, 8, 23, and 24 in Table 2 are responsible for generating the masses of $u-$ and $d-$type quarks, the charged leptons and the exotic fermions $H$ and $h$, respectively. If any of the terms 3, 13, and 15 in Table 2 are present, then $B(h) = 1/3\ \rm {and}\ $L$(h= -1)$, , $h$ is a leptoquark; whereas if terms 1 or 12 in Table 2 are present, then $B(h) = -2/3$ and $L(h) = 0$, , $h$ is a diquark. (If none of the terms are present, we can identify $h$ as an ordinary quark.) As is well known, these two sets of terms in $W$ cannot exist simultaneously since then $h$ would have ill-defined $B$ and $L$ quantum numbers and can thus mediate proton decay at a substantial rate unless $h$ has a mass of order the unification scale $\approx 10^{14}$ GeV or greater. We will consider the effect of the new terms in $W$ within the context of these two possible quantum number assignments. Lastly, we note that $B(H) = L(H) = 0 = B(S^c) = L(S^c)$ since the corresponding neutral scalar field components act as Higgs and obtain vacuum expectation values (vevs); in addition $B(\nu^c) = 0$ whereas $L(\nu^c) = 0\ \rm {or}\ -1$ depending on the model, , whether $\tilde{\nu}^c$ must have a large vev in order to break any additional gauge symmetries down to the SM. Which of these terms in $W$ would survive if extended gauge symmetries are present in the limit that such symmetries are unbroken? The answer, of course, depends on the nature of the additional symmetry. As an example of an additional U(1) symmetry, consider the breaking $$E_6 \rightarrow SO(10) \times U(1)_\psi \rightarrow SU(5) \times U(1)_\chi \times U(1)_\psi \rightarrow {\rm SM} \times U(1)_\theta \auto$$ where $U(1)_\theta \equiv U(1)_\psi \cos \theta - U(1)_\chi \sin \theta$ is the additional gauge symmetry with $-90^\circ \leq \theta \leq 90^\circ$. Clearly, as $\theta$ is varied and the couplings of the superfields change different terms in $W$ will be allowed. Of course, the 11 trilinear terms in $W$ present in R-parity conserving models will occur for all $\theta$ values. Tables 2 and 3 show which terms are allowed for various values of the parameter $\theta$. For example, model $\eta (\theta = 37.76^\circ)$ allows the most number of new terms including the all of the usual ones of the MSSM with R-parity breaking, whereas model $\chi (\theta = -90^\circ) $ only allows the $(S^c)^3$ term among the trilinear couplings and terms 5–7 among the bilinears. For [*arbitrary*]{} values of $\theta$, [*no*]{} R-parity violating terms are found to be allowed by the additional gauge symmetry and, thus, all bilinear couplings vanish. Thus, in general, the additional $U(1)_\theta$ symmetry removes the possibility of explicit R-parity violation [*except*]{} for specific values of the $\theta$ parameter. What about other gauge extensions? Let us consider the extension to the $SU(3)_c \times SU(2)_L \times SU(2)_R \times U(1)$ model; there are two versions of this scenario within the $E_6$ context. The first, which is the conventional left-right model (LRM)$^3$ places the usual right-handed quarks and leptons in right-handed doublets and, , $h_L, h_L^c$ are both $SU(2)_L \times SU(2)_R$ singlets. There is, however, a second possible quantum number assignment due to Ma and collaborators$^4$ which is an alternative version of the LRM (ALRM). Basically, the quantum number assignments of the pairs $L \leftrightarrow H, d^c \leftrightarrow h^c$, and $\nu^c \leftrightarrow S^c$ are interchanged in this scheme. Imposing this extended gauge symmetry, the only new trilinear term to survive is $(S^c)^3 [(\nu^c)^3]$ for the ${\rm LRM} [{\rm ALRM}]$ case while three bilinears are found to be allowed for each of the two scenarios. From this exercise we conclude that the existence of additional gauge symmetries greatly reduces the chance that explicit R-parity violating terms will be present in the superpotential of $E_6$ models. We now turn to a brief discussion of the phenomenology of the various new R-parity violating terms that can appear in $W$. Hattori $^5$ have already addressed the issues of chargino masses and generating neutrino mass hierarchies in $E_6$ models with R-parity violation so we will not deal with those subjects here, although further work in both areas clearly still needs to be done. Thus we will, for now, avoid operators which are most important for dealing with the problems of neutrino mass generation. We note, of course,that although additional gauge symmetries may forbid the existence of renormalizable tree-level operators in the superpotential which break R-parity,such terms can be generated at the one- or two-loop level thus resulting in rather small effective Yukawa couplings. In the LRM,for example, the existence of a vev for the scalar partner of $\nu^c$ implies that one can write down effective dimension-5 operators which are gauge invariant and R-parity conserving but lead to the conventional R-parity violating terms in the MSSM due to a spontaneous breaking of R-parity. This point has been stressed by Valle and collaborators $^1$. The first trilinear R-parity violating operator not occurring in the MSSM is $QHh^c$. Since $L(H) = B(H) = 0$ this operator is $|\Delta B| = 1$ if $h$ is a diquark whereas, if $h$ is a leptoquark, this operator is $|\Delta B| = |\Delta L| = 1$ which could allow for rapid proton decay in conjunction with the usual R-parity conserving operators in $W$. (We note that this operator, like all of the new R-parity violating trilinears, contains two exotic superfields.) With $H^T = (N,E)$, the operator decomposes into $dNh^c + uEh^c$, where $Q(N,E) = (0, -1)$, following the notation of ref. 2. This operator would, , allow for the associated production of exotic particles at hadron colliders which were not particle-antiparticle pairs. Conventionally,$^2\ h$’s are pair produced via $qq, gg \rightarrow h\overline{h}(\tilde{h}\bar{\tilde h})$ but we can now have both $gu \rightarrow \tilde{E}h(E\tilde{h})$ and $gd \rightarrow \tilde{N}h(N\tilde{h})$ reactions as well. Since $H$’s are color singlets, they can only be produced via purely electroweak interactions in the R-parity conserving case but the existence of this new operator, $QHh^c$, could allow for their production with much larger rates at hadron supercolliders as we will see below. Note that the operator $d^cS^ch$ leads to new physics similar to $QHh^c$ (except for the replacement $N \rightarrow S$) since it can lead to $gd \rightarrow \tilde{S}h(S\tilde{h})$ but there is no comparable $gu$ initiated process. The operator $LH^cS^c$ is $|\Delta L | = 1$ since $B (H,S) = L(H,S) = 0$ and decomposes into $\nu N^cS^c + eE^cS^c$. As in the previous case, this operator allows for the associated production of exotics, not at hadron supercolliders, but at $\epm$ colliders via the process $\gamma e \rightarrow E\tilde{S}(\tilde{E}S)$. The rates for this process can also be quite large as we will see below. The operator $e^cHH$ leads to similar physics since it generates the interaction $e^cNE$ but without a comparable term involving all neutral fields. The last trilinear operator is $u^ch^ch^c$. If $h$ is a diquark, then this term is $|\Delta B| = 1$ whereas if $h$ is a leptoquark, both $\Delta B$ and $\Delta L \neq 0$ interaction are induced. This type of operator can lead to $h\tilde{h}$ production in hadronic collisions via the process $gu \rightarrow h\tilde{h}$. We remind the reader that the conventional $gg$ and $q\overline{q}$ production mechanics lead to $h\overline{h}$ or $\tilde{h}\bar{\tilde h}$ and not to a mixed $h\tilde{h}$ final state. Among the bilinear operators almost all of them simply induce bare mass terms for $\nu^c,S^c,H$ and $h$. The $hd^c$ term parallels the usual $LH^c$ of the MSSM which induces mixing between the two superfields; in the $hd^c$ case, identifying $h$ as a diquark (leptoquark) this operator is $|\Delta B| = 1$ ($\Delta B, \Delta L \neq 0$). As in the $LH^c$ case, too, this term can be rotated away by a suitable redefinition of the fields. (Note that such a rotation may not always possible if additional gauge symmetries are present.) Thus, except for mass generation, there is not much additional physics associated with the new bilinear terms. One of the ‘hallmarks’ of R-parity violating models is the instability of the $LSP$, the lightest SUSY partner. In the MSSM (assuming the $LSP$ is the lightest neutralino, $\chi^0_1$) this particle can undergo a 3-body decay, via, , the $LLe^c$ term in Eq. (2), , $\chi^0_1 \rightarrow \overline{e}\tilde{e}^ *\rightarrow \overline{e} e \nu$. Depending on the size of the $LLe^c$ coupling, $\chi^0_1$ may decay inside the detector$^1$ and invalidate the usual SUSY signal of missing energy. If we do [*not*]{} consider the MSSM R-parity violating terms in $W$ here, we find that the $\chi^0_1$ decay will be very highly suppressed in comparison to the MSSM case since it must proceed through virtual exotics. For example $\chi^0_1 \rightarrow h^* {\bar{\tilde h}}^*$, followed by ${\bar{\tilde h}}^* \rightarrow q\overline{q}$, $h^* \rightarrow (N,S)^*d$, and $(N,S)^* \rightarrow q \overline{q}$. Thus instead of an effective dimension-6 operator responsible for $\chi^0_1$ decay in the MSSM case with R-parity violation we find an effective dimension-9 operator for $E_6$ models. So, in the MSSM case one obtains $\tau_\chi \sim m^5_\chi$ whereas in $E_6$ models $\tau_\chi \sim m^{11}_\chi$! In fact, scaling unknown Yukawa couplings to the electromagnetic strength and taking the effective scale of the dim-9 operator to be $\Lambda$ we estimate $$\Gamma_\chi = \lambda\ {{\alpha^4} \over {2048\ \pi^3}}\ m_\chi \ \left( {{m_\chi} \over {\Lambda}}\right)^{10} \auto$$ where $\lambda$ accounts for our ignorance of the Yukawas and the deviation of 5-body phase space integration of the matrix element from uniformity. The $\chi^0_1$ decay length is then given by $$d = 4.42.10^{-3} cm\ \gamma\beta\lambda^{-1} \left({{m_\chi} \over {100\ {\rm GeV}}}\right)^{11} \left({{\Lambda} \over {100\ {\rm GeV}}}\right)^{10} \auto$$ Fig. 1 shows $d$ as a function of $m_\chi$ assuming $\Lambda = 100$ GeV for $\chi$’s produced with an energy of $100$ GeV and different choices of $\lambda$. We see that if $m_\chi \leq 40$ GeV, even with $\lambda = 1$, the LSP will escape the detector before decaying. For larger masses, whether $\chi^0_1$ will have an observable decay depends critically on $\lambda, \Lambda,\ {\rm and}\ m_\chi$. Let us now turn to the various production processes for exotics discussed above that are now possible given an R-parity violating $W$ arising from $E_6$ models. We first consider the case $\gamma e^\pm \rightarrow E^\pm \tilde{S}$ at a high-energy $\epm$ collider; the corresponding $S\tilde{E}^\pm$ cross section is then obtainable by interchanging the roles of the two masses, $m_E$ and $m_S$, in the expressions below. (Similarly $\gamma e^\pm \rightarrow E^\pm \tilde{N}$ can be trivially obtained.) Of course, the real process we are studying is $e^+e^- \rightarrow E^\pm \tilde{S} e^\pm$ where the $e^\pm$ is unobserved and the initial state photon comes from one of the original $e^\pm$. The subprocess differential cross section for this reaction can be obtained with $(\hat{s} \equiv xs)$ where $x$ being the fraction of the initial $e^\pm$ energy carried by the photon: $${{d\hat{\sigma}} \over {dz}} = {{\pi \alpha^2} \over {\hat{s}^2}}\ \kappa \beta\ \left[ A_0\, \tilde{u}^{-2} + A_1 \tilde{u}^{-1} + A_2 - \tilde{u}\right] \auto$$ where $\kappa \equiv (\lambda/e)^2,\, z = \cos \theta,\, \tilde{u} \equiv \hat{u}-m^2_E = - \sqrt{\hat{s}} \, (E_E-p_Ez)$, and $$\eqalign{ A_0 &= -2\hat{s} m^2_E\, (m_E^2 - m^2_S)\cr A_1 &= - \left[ \hat{s}^2 + 2\hat{s} (m^2_E - m^2_S) + 2(m^2_E - m^2_S)^2 \right]\cr A_2 &= -2(\hat{s} + m^2_E - m^2_S)\cr \beta &= \left[ \left(1- {{m^2_E + m^2_S} \over {\hat{s}}} \right)^2 - {{4m^2_Em^2_S} \over {\hat{s}^2}}\right]^{{1} \over {2}}\cr } \auto$$ Here, $\lambda$ is the [*a priori*]{} unknown $e^cE^cS^c$ coupling. Eq. 6 can be integrated to yield the sub-process total cross section. $$\hat{\sigma} = {{2\pi\alpha^2} \over {\hat{s}^3}}\ \kappa \ \left[ A_0(u^{-1}_l-u^{-1}_u)+ A_1 \ell n {{u_u} \over {u_l}} + A_2 (u_u-u_l)- {{1} \over {2}} (u^2_u-u^2_l)\right] \auto$$ where $u_{u,l} \equiv - \sqrt{\hat{s}} (E_E \pm p_E z_0)$ with $-z_0 \leq z \leq z_0; P_E = \sqrt {E^2_E-m^2_E}$ and $$E_E = {{\hat{s} + m^2_E-m^2_S} \over {2\sqrt{\hat{s}}}} \auto$$ (For numerical purposes we take $z_0 =1$ in our calculations) The total cross section is then, summing over both $e^+$ and $e^-$ initial states, $$\sigma = 2\int^1_{x_L}\ dx\ f_{\gamma/e}(x)\ \hat{\sigma}(x) \auto$$ with $f_{\gamma/e}(x)$ being the photon flux and $x_L \equiv (m_E+m_S)^2/s$. In what follows we will calculate $\sigma$ for $\epm$ colliders with $\sqrt{s} = 0.5$ or $1$ TeV and using either the photon flux of the Weizsacker-Williams Effective Photon Approximation (EPA)$^6$ or the beamsstrahlung (BS) photon$^7$ flux. Figs. 2a–d show $\sigma$ as a function of $m_E$ for different choices of $m_S$ assuming $\kappa = 1$. Note that for smaller values of $m_{E,S}$ the BS spectrum produces a larger value of $\sigma$ in comparison to EPA whereas the reverse is true for large masses. As an example of the rates one should expect consider a $\sqrt{s} = 500$ GeV (1 TeV) collider with an integrated luminosity of $25 (100) fb^{-1}$ and take $M_E=M_S= 200(400)$ GeV. For the EPA spectrum we obtain 42 (141) events whereas the BS spectrum gives 91(99) events emplying a sizeable signal in either case. Similar calculations can be performed in the case where the itial photon arises from backscattered laser light. We note in passing that a parallel process might occur at the HERA $ep$ collider. Firstly, if the initial $q$ or $\overline{q}$ emits a photon such that $\gamma e$ scattering occurs we could have the reaction $\gamma e \rightarrow E \tilde{S}$ as above. On the other hand, if the initial $e$ emits the $\gamma$, then the parallel process $\gamma u \rightarrow h\tilde{E}$ or $\gamma d \rightarrow h\tilde{S}$ can occur via the above operators in $W$. Numerically, however, for $m_{E,h,S} \gsim 50$ GeV or so the cross sections for these processes are quite miniscule at HERA energies ($\sqrt{s} = 314$ GeV). Let us now turn to the reactions $gd \rightarrow \tilde{S} h$ and $gu \rightarrow \tilde{E} h$ which can arise from the above operators. The subprocess differential cross section is directly obtainable from Eq. (6) with the replacements $\alpha^2 \rightarrow {{1} \over {6}} \alpha\alpha_s(\hat{s})$ and $m_E \rightarrow m_h$. Placing a rapidity cut, $Y$, on the outgoing pair implies as usual $$z_0 = min \left[ \beta^{-1}_h tanh (Y -|y|), 1\right] \auto$$ where $\beta_h \equiv \beta \left[ 1 + (m^2_h - m^2_S)/\hat{s}\right]^{-1}$ with $\beta$ given in Eq. (7). Then $$\sigma = \int^1_{\tau_0}\, d\tau\, \int^Y_{-Y}\, dy\ \left[ g(x_a,\hat{s}) q(x_b,\hat{s}) + g(x_b, \hat{s}) q (x_a, \hat{s}) + q \rightarrow \overline{q} \right] \int^{z_0}_{-z_0}\, {{d\sigma} \over {dz_0}} \auto$$ where $\tau_0 \equiv (m_h + m_S)^2/s$, $x_{a,b} = \sqrt{\tau}\, e^{\pm y}$ and $q$ is either $u$ or $d$; we take $Y = 2.5$ in our calculations below. (Note that we sum over both $gq$ and $g\overline{q}$ subprocesses in the equation above.) $\tau$ is simply the scaled invariant mass of the h,S pair. Figures 3a–d show the integrated cross section, $\sigma$, assuming $\kappa=1$ as a function of $m_h$ at both the SSC ($\sqrt{s} = 40$ TeV) and LHC ($\sqrt{s} = 15.4$ TeV) for both $gd$- and $gu$-induced processes. As a comparison case, let us examine the values of $\sigma$ for $m_h$ = 3 TeV and $m_S$ or $m_E = 1$ TeV. For the $gd$ case we obtain $\sigma = 26.7 (1.21)fb$ at the SSC (LHC) and for the $gu$ case we obtain $\sigma = 5.84 (3.95) fb^{-1}$ at the SSC (LHC). (These numerical results were obtained using the NLO Morfin-Tung$^8$ structure functions ,set S1.) Note that the $h\tilde{E}$ cross section is larger than $h\tilde{S}$ due to the larger $gu$ luminosity in comparison to that available for $gd$. For the SSC (LHC) with an integrated luminosity of 10(400)fb$^{-1}$ these cross sections translate into a substantial event sample. If we demand that 100 events be produced at either machine to observe the associated production of exotic particles then, with the above luminosities,we find the following search ranges in the case that $\kappa = 1$: $$\eqalign{ h\tilde{S}\left\{ \eqalign{& SSC \hspace{0.65in} m_h + m_S \lsim 4.5\ {\rm TeV}\cr & LHC \hspace{0.65in} m_n + m_S \lsim 3.7\ {\rm TeV}\cr} \right.}$$ (13) $$\eqalign{ h\tilde{E}\left\{ \eqalign{& SSC \hspace{0.65in} m_h + m_E \lsim 5.2\ {\rm TeV}\cr & LHC \hspace{0.65in} m_h + m_E \lsim 4.3\ {\rm TeV}\cr} \right. }$$ The signature for $h,\ E$, and $S$ production themselves at supercolliders have been discussed elsewhere.$^2$ The rates for these types of processes at the Tevatron are, of course, substantially smaller. For example, for $m_h = m_{S(E)} = 50$ GeV we find the total cross section to be (now for $Y = 1.5$) $371\ pb\ (564\ pb)$ for $\kappa=1$ which is sizable for current integrated luminosities but falls fast as both $m_h$ and $m_{S(E)}$ increase. Clearly, however, if $\kappa$ is near unity such small masses for the exotics must already be disfavored by the existing data. Figs. 4a and 4b show the cross sections for $gd \rightarrow h\tilde{S}$ and $gu \rightarrow h\tilde{E}$ at the Tevatron ($\sqrt{s} = 1.8$ TeV) as functions of $m_h$ for different values of $m_{S(E)}$ and $\kappa=1$. Assuming as above that 100 events is necessary for a clear discovery of exotic particles and an integrated luminosity of $100\ pb^{-1}$ we see that the Tevatron search range for $h\tilde{E}$ and $h\tilde{S}$ final states are approximately $m_h + m_E \lsim 330$ GeV and $m_h +m_S \lsim 290$ GeV, respectively. These search ranges can each be extended by $\simeq 100$ GeV if the integrated luminosity were to reach $500\ pb^{-1}$. Clearly, though the search ranges are relatively restricted in comparison to the SSC/LHC, the Tevatron can do an excellent job in the low mass range of exotics if $\kappa$ is not too small. In this paper we have begun to explore the collider phenomenology associated with explicit R-parity violating operators in $E_6$ models. The main points of our discussions are as follows: - We have found that the enriched particle spectrum of $E_6$ models can lead to as many as 25 trilinear and 7 bilinear operators in the superpotential, $W$, when only $SU(3)_c \times SU(2)_L \times U(1)_Y$ symmetry is required. This is 16 more than is present in the standard $E_6$ model [*beyond*]{} the usual R-parity violating terms of MSSM. Demanding only $|\Delta B| \neq 0$ [*or*]{} $|\Delta L| \neq 0$ terms in $W$ was found to restrict the exotic particle quantum number assignments. Our discussion was, however, restricted to terms which explicitly break R-parity and are renormalizable. - Enlarging the gauge symmetry of the SM can greatly reduce the number of new operators in $W$. The set of allowed operators was found to be quite sensitive to the nature of this extension. We noted that in extended gauge models spontaneous violation of R-parity could result in the usual R-parity breaking terms of the conventional MSSM via loop corrections even when the extended gauge symmetries forbid them at tree level. - As in the MSSM with R-parity violation, the LSP was found to be unstable but in our case could only decay via virtual exotic particle exchange if the R-parity violating operators of the MSSM were absent. Since its lifetime was found to scale as the eleventh power of its mass, the possibility of using the missing energy signature for SUSY was found to be quite sensitive to the model parameters. - The new operators explicitly breaking R-parity in $E_6$ models allow for new ways to produce exotic particles via gluon-quark fusion at hadron colliders or $\gamma-e$ fusion at $e^+e^-$ colliders. Cross sections were found to be quite substantial over a wide range of particle masses provided the Yukawa couplings of these new operators were not too small. It is clear from the above that the phenomenology of $E_6$ models with R-parity violation is particularly rich and is just beginning to be explored. The author would like to thank J. L. Hewett for discussions related to this work. This research has been supported by the U.S. Department of Energy, Division of High Energy Physics, Contracts W-31-109-38 and W-7405-Eng-82. 1. The phenomenology of R-parity violation in the MSSM is particularly rich; see for example L. J. Hall and M. Suzuki, Nuc. Phys. [**B231**]{}, 419 (1984); S. Dawson, Nucl. Phys. [**B261**]{}, 297 (1985); S. Dimopoulos and L. S. Hall, Phys. Lett. [**B207**]{}, 210 (1987); V. Barger, E. G. Guidice, and T. Han, Phys. Rev. [**D40**]{}, 2987 (1989); , Phys. Rev. [**D41**]{}, 2988 (1990); H. Dreiner and G. Ross, Oxford preprint OUTP-91-15P (1991); R. Mohapatra, Phys. Rev. [**D34**]{}, 3457 (1986); R. Barbieri and A. Masiero, Nucl. Phys. [**B267**]{}, 679 (1986), S. Lola and J. McCurry, Oxford preprint OUTP-91-31P (1991); M. C. Gonzalez-Garcia, J. C. Romao, and J. W. F. Valle, Univ. of Wisconsin preprint MAD/PH/678 (1991); E. Ma, and D. Ng, Phys. Rev. [**D41**]{}, 1005 (1990); M. Doncheski and J. Hewett, Argonne preprint (1992). 2. For a review of $E_6$ phenomenology, see J. L. Hewett and T. G. Rizzo, Phys. Rev. [**183**]{}, 193 (1989). 3. For a review and original references, R. N. Mohapatra, [*Unification and Supersymmetry*]{}, (Springer, New York, 1986). 4. E. Ma, Phys. Rev. [**D36**]{}, 274 (1987); Mod. Phys. Lett. [**A3**]{}, 319 (1988); K. S. Babu , Phys. Rev. [**D36**]{}, 878 (1987); V. Barger and K. Whisnant, Int. J. Mod. Phys. [**A3**]{}, 879 (1988); J. F. Gunion , Int. J. Mod. Phys. [**A2**]{}, 118 (1987); T. G. Rizzo, Phys. Lett. [**B206**]{}, 133 (1988). 5. C. Hattori , Nagoya University preprints DPNU-91-21 (1991) and DPNU-91-32 (1991). 6. For a review of the EPA and original references, see P. W. Johnson, F. Olness, and W.-K. Tung in [*Proceedings of the 1986 Summer Study on the Physics of the Superconducting Supercollider*]{}, Edited by R. Donaldson and J. Marx (1986). 7. R. Blankenbecler and S. Drell, Phys. Rev. Lett. [**G1**]{}, 2324 (1988), Phys. Rev. [**D36**]{}, 277 (1987) and [**D37**]{}, 3308 (1988), D. V. Schroeder, SLAC-Report-371 (1990); F. Halzen, C. S. Kim and M. Stong, Univ. Of Wisconsin preprint MAD/PH/673 (1991). 8. J. C. Morfin and W.-K. Tung, Z. Phys. [**C52**]{}, 13 (1992). - Nomenclature and $SU(3)_c \times SU(2)_L \times U(1)_Y$ quantum numbers for the fields in the  representation of $E_6$. - The 25 trilinear terms allowed in the $E_6$ superpotential $W$ by $SU(3)_c \times SU(2)_L \times U(1)_Y$ gauge invariance following the notation of Table 1. HR labels the term as it appears in $W$ from the review of Hewett and Rizzo (ref. 2). $\theta$ labels the value of the mixing angle between $Z_\psi$ and $Z_\chi$ described in the text which allows this particular term: $\chi(\theta = -90^\circ)$, $\eta(\theta \simeq 37.76^\circ)$, $\nu_R(\theta \simeq -14.48^\circ)$, $\nu^I_R(\theta \simeq -66.72)$, $I(\theta \simeq -52.24^\circ)$, and $-\eta(\theta = -37.76^\circ)$. We also show which terms generate particle masses, induce leptoquark ($L$) or diquark ($D$) quantum numbers for $h$ and which terms appear in the R-parity violating version of the MSSM. Also shown is whether the term is allowed in either the LRM or ALRM case. - Same as Table 2 but for the bilinear terms in $W$ allowed by $SU(3)_c \times SU(2)_L \times U(1)_Y$ gauge invariance. Also shown is whether these terms are allowed in the LRM or ALRM. Table 1 -------- ---------------- ----------------------------------------------- ---------------- -------------------------------------------------- ------ --------------------------------------------- SO(10) SU(5) [ ]{} Color $T_{3L}$ Y/2 Q 16 10 $Q\equiv \left( {{u} \atop {d}} \right)_L$ 3 $\left( {{\phantom{-}1/2} \atop {-1/2}} \right)$ 1/6 $\left( {{2/3}\atop {1/3}}\right)$ $u^c_L$ $\overline{3}$ 0 -2/3 -2/3 $e^c_L$ 1 0 1 1 $\overline{5}$ $L\equiv \left( {{\nu} \atop {e}}\right)_L$ 1 $\left( {{\phantom{-}1/2} \atop {-1/2}}\right)$ -1.2 $\left( {{\phantom{-}0} \atop {-1}}\right)$ $d^c_L$ $\overline{3}$ 0 1/3 1/3 1 $\nu^c_L$ 1 0 0 0 10 $\overline{5}$ $H \equiv \left( {{N} \atop {E}}\right)_L$ 1 $\left( {{\phantom{-}1/2} \atop {-1/2}}\right)$ -1/2 $\left( {{\phantom{-}0} \atop {-1}}\right)$ $h^c_L$ $\overline{3}$ 0 1/3 1/3 5 $H^c\equiv \left( {{E} \atop {N}}\right)^c_L$ 1 $\left( {{\phantom{-}1/2} \atop {-1/2}}\right)$ 1/2 $\left( {{1} \atop {0}}\right)$ $h_L$ 3 0 -1/3 -1/3 1 1 $S^c_L$ 1 0 0 0 -------- ---------------- ----------------------------------------------- ---------------- -------------------------------------------------- ------ --------------------------------------------- 9.0in Table 2 Term HR $\theta$ Comment ---- ----------------- ---- ----------- -------------------------------------- 1 $QQh$ 9 D 2 $QLD^c$ $\eta$ ${\rm MSSM}\ \st{R}\ |\Delta L| = 1$ 3 $QLh^c$ 7 L 4 $Qh^cH^c$ 1 $u\ mass$ 5 $Qd^cH$ 2 $d\ mass$ 6 $Qh^cH$ $\eta$ 7 $LLe^c$ $\eta$ ${\rm MSSM}\ \st{R}\ |\Delta L| = 1$ 8 $LHe^c$ 3 $e\,mass$ 9 $LH^c\nu^c$ 11 $Dirac\ \nu\ mass$ 10 $LH^cS^c$ $\eta$ 11 $u^cd^cd^c$ $\eta$ ${\rm MSSM}\ \st{R}\ |\Delta B| = 1$ 12 $u^cd^ch^c$ 10 $D$ 13 $u^ce^ch$ 6 $L$ 14 $u^ch^ch^c$ $\eta$ 15 $d^c\nu^ch$ 8 $L$ 16 $d^cS^ch^c$ $\eta$ 17 $e^cHH$ $\eta$ 18 $(\nu^c)^3$ $\nu_R$ $ALRM$ 19 $\nu^c\nu^cS^c$ $-\eta$ 20 $HH^c\nu^c$ $\eta$ 21 $hh^c\nu^c$ $\eta$ 22 $\nu^cS^cS^c$ $\nu^I_R$ 23 $HH^cS^c$ 4 $H\ mass$ 24 $hh^cS^c$ 5 $h\ mass$ 25 $(S^c)^3$ $\chi$ $LRM$ 8.0in Table 3 Term $\theta$ Comment --- ------------- ---------- ------------------------------ -------- 1 $LH^c$ $\nu_R$ ${\rm MSSM}\ |\Delta L| = 1$ $ALRM$ 2 $hd^c$ $\nu_R$ $ALRM$ 3 $(\nu^c)^2$ $\nu_R$ $ALRM$ 4 $\nu^cS^c$ I 5 $HH^c$ $\chi$ MSSM $LRM$ 6 $hh^c$ $\chi$ $LRM$ 7 $(S^c)^2$ $\chi$ $LRM$ - Decay length in $cm$ for the $LSP(\chi)$ as a function of $m_\chi$ assuming $\Lambda = 100$ GeV and that $\chi$ is produced with an energy of 100 GeV. The bottom curve corresponds to $\lambda =1$ with subsequently higher curves corresponding to a decrease in $\lambda$ by a power of 10. - Cross section for $\tilde{S}E^\pm$ production in high energy $\epm$ collisions as a function of $m_E$ with $\kappa=1$ and different $m_S$ choices. (a) $\sqrt{s} = 500$ GeV using the EPA, (b) $\sqrt{s} = 500$ GeV using BS. The top curve in both cases corresponds to $m_S = 50$ GeV and each subsequently lower curve corresponds to an increase in $m_S$ by 50 GeV. (c) and (d) same as (a) and (b) but for a $\sqrt{s} = 1$ TeV collider. The top curve corresponds to $m_S = 100$ GeV and each subsequently lower curve corresponds to an increase in $m_S$ by 100 GeV. - Cross section after rapidity cuts for $h\tilde{S}$ \[(a) and (b)\] or $h\tilde{E}$ \[(c) and (d)\] production at hadron supercolliders for $\kappa=1$ as a function of $m_h$ assuming different values for $m_S$ or $m_E$. (a) $h\tilde{S}$ production at the SSC; (b) $h\tilde{S}$ at the LHC, (c) $h\tilde{E}$ production at the SSC; (d) $h\tilde{E}$ production at the LHC. The top curve corresponds to $m_{S(E)} = 0.1$ TeV with each successive curve corresponding to an increase in $m_{S(E)}$ by 0.1 TeV. - The $h\tilde{E}$ (a) and $h\tilde{S}$ (b) production cross section as a function of $m_h$ with $\kappa=1$ at the Tevatron. The top solid curve corresponds to $m_{E,S} = 50$ GeV with lower subsequent curves corresponding to $m_{E,S} = 100,\ 200,\ 300,$ etc. GeV. [^1]: Work supported by the U.S. Department of Energy, Division of High Energy Physics, ContractsW-31-109-ENG-38 and W-7405-Eng-82.

--- abstract: 'Semantic layouts based Image synthesizing, which has benefited from the success of Generative Adversarial Network (GAN), has drawn much attention in these days. How to enhance the synthesis image equality while keeping the stochasticity of the GAN is still a challenge. We propose a novel denoising framework to handle this problem. The overlapped objects generation is another challenging task when synthesizing images from a semantic layout to a realistic RGB photo. To overcome this deficiency, we include a one-hot semantic label map to force the generator paying more attention on the overlapped objects generation. Furthermore, we improve the loss function of the discriminator by considering perturb loss and cascade layer loss to guide the generation process. We applied our methods on the Cityscapes, Facades and NYU datasets and demonstrate the image generation ability of our model.' author: - | \ \*Corresponding author: Zhibin Yu Email: [email protected] bibliography: - 'acess.bib' title: Instance Map based Image Synthesis with a Denoising Generative Adversarial Network --- Generative Adversarial network (GAN), Denoising, Image-to-image translation Introduction ============ Image-to-image translation has been made much progress due to the successful of image generation and synthesizing. As a sub research topic of image translation, image-to-image translation task need the model not only to understand the image contents but also to convert the source images to the target images under some certain rules. Many image processing problems such as semantic segmentation, color restoration, etc. can be considered as image-to-image work. Since Generative Adversarial Network (GAN) was presented by Goodfellow in 2014[@goodfellow2014generative], GAN is proved to be very efficient in image generation and synthesizing tasks in many research fields[@zhu2017unpaired][@zhao2016energy][@mirza2014conditional][@press2017language]. Soon researchers show that the extension of GAN can not only generate images from random noise [@nguyen2016plug], but also achieve the image-to-image translation between different domains [@isola2016image]. It’s highly desirable if we could build a model which is able to create realistic images based on some simple graphics (e.g. semantic layouts). Actually, lots of works about GAN prove that GAN is a good candidate to handle the image-to-image translation work. But GAN based models are still will weak on precise generation tasks. How to organize the translated image contents precisely and enhance the generated image equality is still a challenging work. Take Fig.\[fig:demo\] as an example, it would be a challenging task if we want to create realistic overlapped cars only based on an adhesive semantic layout. In order to achieve this goal, the image-to-image translation model should be smart enough to understand the relationship and the position of each car in the space domain. ![Some image-to-image translating examples from semantic layouts to photorealistic images. (c) and (d) are local enlarged images from (a) and (b), respectively.[]{data-label="fig:demo"}](demo.pdf){width="\linewidth"} In this paper, we proposed a novel idea for generating realistic images from the semantic layout precisely. We take advantages from pix2pix, Cascade Refinement Network (CRN) and Inception-ResNet to design our model [@isola2016image] [@chen2017photographic][@szegedy2017inception]. The cascade layer loss from CRN can efficiently control the contents organizing while the adversarial structure can generate realistic sharp samples. It’s believed that the residual blocks can be used to build very deep neural networks [@He2015]. It can also help alleviate the gradient vanish and gradient explode. We also consider the skip connection for the encoder-decoder network, which is derived from the U-net[@ronneberger2015u]. Specially, we add skip connections between the convolution layers before residual blocks and deconvolution layers after the residual blocks. In order to generate the complex objects precisely, we compute the instance map of complex objects (such as the cars in the Cityscapes[@Cordts2016Cityscapes] and the furniture in the NYU [@silberman2012indoor] datasets) from the semantic layouts. And we concatenate the instance car map with the raw semantic layout to provide additional information to our model. Inspired by InfoGAN[@chen2016infogan], we develop a denoising framework by adding some auxiliary random noise to the input images. Our model will try to remove the noise during the generation process and synthesis a clear image. This operation can enhance the model stochasticity and increase the robustness in image generation stage. {width="\linewidth"} To further improve the generation, we follow the perturbed loss defined by [@denton2016semi]. The perturbed loss can improve the ability of the discriminator and push the generator to create more realistic images. The detail of our framework is displayed in Fig.  \[fig:architecture\]. In this paper we make the following contributions: - We develop a new framework to including residual blocks, cascade layer loss and perturbed loss to synthesis images from a semantic layout to a realistic RGB image. - We develop a denoising process enhance the model stochasticity and increase the robustness on image-to-image translation stage. - In order to generate complex objects precisely, we include an instance map to force the generator focus on the confusion parts and improve the final generation performance. Related work ============ Residual networks ----------------- Residual networks, which are developed by [@He2015], is proved to be efficient on image classification tasks. Some computer vision tasks such as image translation and image editing also use residual blocks as a strong feature representation architecture[@he2016identity][@he2016deep]. Residual networks are neural networks in which each layer consists of a residual block $f_{i}$ and a skip connection bypassing $f_{i}$. Layers in residual networks include multiple convolution layers. The residual blocks build connections between different layers $$y_{i}=f_{i}(y_{i-1})+y_{i-1}$$ Let $y_{i-1}$ and $y_{i}$ denote input and output of the $i$th residual block separately. Where $f_{i}$ is some sequence of convolutions, batch normalization[@ioffe2015batch], and Rectified Linear Units (ReLU) as nonlinearities. The shortcut connections can transmit gradients and propagated errors in very deep neural networks. Image-to-image translation -------------------------- There exists a large body of work on supervised representation learning. Early methods were based on the stacked auto-encoders or restricted Boltzman machines[@hinton2006fast]. Researchers use pairwise supervision to achieve the image feature representation using the auto-encoders[@zhu2014multi][@kan2014stacked]. And the Image-to-image translation problems are often regarded as feature representation and feature matching work. The researchers have already taken significant steps in the computer vision tasks, while deep convolution neural networks(CNNs) becoming the common workhorse behind a wide variety of image processing problems. For the style translation[@gatys2015neural], Gatys et al applied a VGG-16 network to extract features. Other methods based on CNNs use deep neural networks to capture feature representations of images and applied them to other images[@johnson2016perceptual][@simonyan2014very]. They try to capture the feature spaces of scattering networks and proposed the use of features extracted from a pre-trained VGG network instead of low-level pixel-wise error measures[@bruna2015super]. GANs ---- Generative adversarial networks (GANs)[@goodfellow2014generative] aim to map the real image distribution by forcing the synthesized samples to be indistinguishable from natural images. The discriminator try to differentiate synthesized image from real images while the generator aims to generate plausible images as possible to fool the discriminator. The adversarial loss has been a popular choice for many image-to-image tasks[@zhang2017age][@wang2016generative][@sangkloy2016scribbler][@pathak2016context]. Recent research related with GANs are mostly based on the work of DCGAN (deep convolutional generative adversarial network)[@radford2015unsupervised]. DCGAN has been proved to learn good feature representation from image pixels in many research. And the deep architecture has shown fantastic effectiveness of synthesizing plausible images in the adversarial networks. And conditional generative adversarial networks (cGAN) which developed by [@mirza2014conditional] offers a solution to generate images with auxiliary information. Previous works have conditioned GANs on discrete labels, text[@reed2016generative] and images. The application of conditional GANs on images processing contains the inpainting [@yeh2017semantic], image manipulation guided by user constrains [@zhu2016generative], future frame prediction [@mathieu2015deep], and so on. Researchers applied the adversarial training to the image-to-image translation, in which we use the images from one domain as inputs to get translated images of another domain [@karacan2016learning][@kaneko2017generative][@dong2017semantic][@isola2016image]. The recent famous method for image-to-image translation about GAN is pix2pix[@isola2016image]. It consists of a generator $G$ and a discriminator $D$. For image translation task, the objective of the generator $G$ is to translate input images to plausible images in another domain, while the discriminator $D$ aims to distinguish real images from the translated images. The supervised framework can build the mapping function between different domains. Towards stable training of GAN, Wasserstein generative adversarial network (WGAN)[@arjovsky2017wasserstein] replace Jensen-Shannon divergence by Wasserstein distance as optimization metric, recently some other researchers improved the training using gradient penalty method.[@gulrajani2017improved]. Proposed methods ================ In this paper, we use the residual blocks and convolution neural networks as the generators. Inspired by the U-net, we use the skip connections between feature maps in the generator to build the deep generator network. The residual block can help alleviate the gradient vanishing and gradient explosion problems. In order to generate more photorealistic images and balance the training, we have added some noise to the input images and found the noise can help improve the ability of generator. Architecture of our models -------------------------- As described in Fig. \[fig:architecture\], we use three convolution-Batchnorm-ReLu[@ioffe2015batch] to get smaller size feature map, then we use nine residual blocks to capture feature representations. Following these residual blocks, we use two deconvolution layers to get high resolution images. We use the Batchnorm layers and Leaky ReLU after both the convolution operation and deconvolution layers. The final outputs have been normalized from -1 to 1 using the Tanh activation to generate a $256 \times 256$ image outputs. We build the skip connections between the convolution layers and deconvolution layers to allow low-level information transmit. Such a network requires that all information flow pass through all the layers, including the bottleneck. All the filters in convolutions have kernel size 4 and stride 2. The residual blocks are using kernel size 3 and stride 1. To learn the difference between fake and real components, we add the perturbed loss as a regularization scheme. The Fig.  \[fig:architecture\] shows the architecture of our model. Let $k4n64s2$ denotes the convolution layer have 64 convolution filters with kernel size 4 and stride 2. We use the Adam optimizer of learning rate 0.0002 for both discriminator and generator. All the experiments have taken 200 epochs. And following the pix2pix method, we have applied random jittering by resizing the $256 \times 256$ input images to $286 \times 286$ and randomly cropped back to size $256 \times 256$. And the number of our model parameter is about one fourth as pix2pix and one eighth as Cascade Refinement Network (CRN) . We use less parameter to get better results. Perturbed loss -------------- The original GAN loss is expressed as $$\begin{split} \mathcal{L}(G,D)&=\mathbb{E}_{x,y\sim P_{data}(x,y)}\left[\log D(x,y)\right]+ \\ &\!\!\!\!\!\!\!\!\!\!\!\! \mathbb{E}_{x\sim P_{data}(x)}\left[\log (1-D(x,G(x,z)))\right]. \end{split} \label{eq:cgan}$$ Let $x$ denotes the semantic input and $z$ denotes the noise. As a result of concatenating the input image and generated images in the Fig.  \[fig:architecture\], $D(x,G(x,z)))$ shows the predict output of the discriminator. Ideally $D(x,y)=1$ and $D(x,G(x,z)))=0$ when G(x) denotes the generated samples. To make the discriminator more competitive, we use the perturbed training to improve the generations. Following Gulrajani and Denton et al.’s work [@denton2016semi] [@gulrajani2017improved], we define the perturbed loss $\mathcal{L}_{p}$ as: $$\begin{aligned} \mathcal{L}_{p} &= {E}_{x\sim P_{data}(x)}\left[\log (1-D(x,G(\hat{x})))\right]\\ \hat{x} &= \alpha G(x,z) +(1-\alpha) y\end{aligned}$$ where $\hat{x}$ represents the mixture of the synthesized fake image $G(x,z)$ and the target image $y$. Only a portion of $\hat{x}$ is from target image. And $\alpha$ is a random number from 0 to 1 following an uniform distribution. Here we can regard $\hat{x}$ as perturbed sample. The object of an perturbed sample $\hat{x}$ indicates which we changes the output of the generated sample in a desired way. We add this perturbed loss to improve the training of GANs and the quality of synthesized samples. We compute the perturbed loss $\mathcal{L}_{p}$. The $\mathcal{L}_{p}$ computes the distance between fake images and target images and encourage the discriminator to distinguish different components of the mixed images. It can help improve the ability of discriminator to distinguish the fake images. Noisy adding and one-hot instance map ------------------------------------- Gaussian noise $z$ are usually used in the traditional GANs as an input to the generator [@goodfellow2014generative] [@radford2015unsupervised]. As mentioned in Isola et al.’s work [@isola2016image], how to enhance the stochasticity of their pix2pix method is an important question. We increase the stochasticity of our model by adding the Gaussian noise to the input images. And we found the additional noise can actually help the generator synthesize images stably. And we have taken experiments using various standard deviations to try to found out the influence the noise. ![The procedure of adding noise and creating an instance map.[]{data-label="fig:water"}](watershed.pdf){width="\linewidth"} As mentioned in Fig. \[fig:demo\], it’s a challenging work when we want to create a realistic image only based on a semantic label map while the cars are overlapped. We compute the instance map of cars from the semantic input and concatenate them as the final input. We hope this operation can force our generator to focus on the boundary and structure information of overlapped objects. First we extract the car segmentation scene and make the background black. And the Fig.  \[fig:water\] shows how we got the instance map from the semantic labeled images and how we added the noise. Please note that we also concatenate the instance map and generated images for the discriminator. Cascade layer loss ------------------ In order to make the synthesized images more clear and perform better, we consider Chen et al.’s work and include an extra pre-trained VGG-19 network called CRN to provide the cascade loss in our hybrid discriminator[@chen2017photographic]. Fig. \[fig:architecture\] shows the detail of cascade loss. The feature maps of convolutional networks can express some important information of images. The CRN can provide extra criteria for our model to distinguish real images from fake samples. The cascade loss is considered as a measurement of similarity between the target and the output images. We initialize the network with the weights and bias of the pre-trained VGG network on the Imagenet dataset[@krizhevsky2012imagenet]. Each layer of the network will provide cascade loss between the real target images and the synthesized images. We use the first five convolutional layers as a component of our discriminator. $$\label{eq5} \mathcal{L}_{c}(\theta)=\sum^{N}_{n}{\lambda}_{n}||{\Phi}_{n}(x)-{\Phi}_{n}(G((x,z);\theta))||_{1}$$ Following the definition mentioned in Eq. \[eq5\], here $y$ denotes the target image and $(G(x,z))$ is the image produced by the generator $G$; $\theta$ is the parameters of the generator $G$; $ {\Phi}_{n}$ is the cascade response in the $n_{th}$ level in the CRN. We choose the first 5 convolutional layers in VGG-19 to calculate the cascade loss. So we have $N=5$. Please note the loss ${L}_{cascade}(\theta)$ mentioned in Eq. \[eq5\] is only used to train the parameter $\theta$ of the generator G. The CRN is a pre-trained network. The weights of CRN will not be changed during we train the generator $G$. The parameter ${\lambda}_{n}$ controls the influence of cascade loss in the $n_{th}$ layer of CRN. The cascade loss can provide the ability to measure the similarity between the output and the target images under $N$ different scales and enhance the ability of discriminator. It’s believed that the cascade loss from the higher layers control the global structure; the loss from the lower layers control the local detail during generation. Thus, the generator should provide better synthesized images to cheat the hybrid discriminator and finally improve the synthesized image quality. Objective --------- The goal is to learn a generator distribution over data $y$ that matches the real data distribution $P_{data}$ by transforming an observed input semantic image $x\sim P_{data}(x)$ and a random Gaussian noise $z\sim P(z)$ into a sample $G(x,z)$. This generator is trained by playing against an adversarial discriminator $D$ that aims to distinguish between generated samples from the true data distribution (real image pair $\{x,y\}$) and the generator’s distribution (synthesized image pair $\{x,G(x,z)\}$). As mentioned in Isola et al.’s work, location information is really important while synthesizing images. So we still use the L1 loss to the objective in our network to force the generator to synthesize images with considering the details of small parts. $$\mathcal{L}_{1}=\left\| y -G(x,z) \right\|_{1}.$$ The L1 loss is the fundamental loss of our generator which guarantee the image-to-image generation abilities. Our final objective is $$G^{*} = \arg\underset{G}{\min}\;\underset{D}{\max}\mathcal{L}(G,D)+ \gamma \mathcal{L}_{l1} + \theta \mathcal{L}_{p} + \sigma \mathcal{L}_{cl}$$ where $ \mathcal{L}_{l1}$ means the real-fake L1 loss, $\gamma$, $\sigma$ and $\theta$ are hyper parameters to balance our training. We use greedy search to optimize the hyper parameters with $\gamma=100, \theta=1, \sigma=1$ in our experiments. Experiments =========== To validate our methods, we conduct extensive quantitative and qualitative evaluations on Cityscapes, Facades [@tylevcek2013spatial] and NYU datasets [@silberman2012indoor], which contain enough object variations and are widely used for image-to-image translation analysis. We choose the contemporaneous approach (pix2pix) as the baseline [@isola2016image], and we also compare our model with CRN [@chen2017photographic], which is also a state of art method on image-to-image generation task. Results by the two compared methods are generated using the code and models released by their authors. Following their method, we use 400 images of Facades dataset for training and 100 for testing. On the image-to-image tasks, we use some image quality scores (R-MSE, SSIM, MSE and P-NSR) to evaluate the generations of used methods. For these supervised translation method, output images with lower R-MSE and MSE are better while images with higher P-NSR and SSIM higher are better. NYU dataset and Facades dataset ------------------------------- Method P-SNR MSE R-MSE SSIM --------- --------- --------- -------- -------- Ours 10.8562 0.08753 0.2911 6.4610 CRN 8.4967 0.15183 0.3827 7.5927 Pix2pix 10.8545 0.08801 0.2916 6.3742 : The image equality evaluation results of different methods on the Nyu2 dataset.[]{data-label="table:nyu2"} {width="\linewidth"} ![The evaluation results of the perturbed loss on the NYU dataset. []{data-label="fig:lp-nyu"}](perturb-nyu2.pdf){width="\linewidth"} In order to test the generative ability of our model, we use the RGB images from NYU dataset [@silberman2012indoor] and get the semantic layouts from Ivaneck’s work [@ivaneckydepth]. One challenge on this dataset is that sematic labels are quite limited. In this dataset, the semantic layouts only contain five different classes (furniture, structure, prop, floor and boardlines). Note different type of objects can be labeled as the same class (e.g. Beds and bookshelves are both labelled as ”furniture”). Through this way, we got 795 five-category semantic images from the training dataset of raw Nyu dataset[@silberman2012indoor], and 700 for training, 95 for testing. In this case, it is difficult for the generator to organize the contents and generate plausible results. We regard it as a challenging task for this one-to-many translation. So we compute the instance map of these objects and concatenate the instance map with the semantic layouts, which can provide our generator more information and force the model to pay attention to the overlapped objects. The Fig. \[fig:nyu2-compare\] shows one generation example on NYU dataset. We can find that our result is much better than the other two methods. The relationship between books and shelves are well modeled. On the other hand, we evaluated the effectiveness of the perturbed loss $\mathcal{L}_{p}$. The Fig.  \[fig:lp-nyu\] shows the comparison results, which illustrate that the perturbed loss can help generate better results. The Fig.  \[fig:nyu2-compare\] shows the comparing results using the three methods on NYU dataset. The picture generated by pix2pix performs well on generating some simple contents such as floor and the wooden furniture. But pix2pix fail to generate books on the bookshelf. The CRN method fail to generate images with enough details. Although the image-to-image tasks can be considered as a one to many problem, we still measure four kinds of image quality scores on these three methods for reference. The Table. \[table:nyu2\] shows the evaluation results of the three methods. The CRN is better at generating images without structure errors. That’s why CRN has higher SSIM score. Comparing with the pix2pix method, we have higher SSIM and lower MSE error. Our methods have better results than the other two methods. ![The evaluation results of the perturbed loss on the Facades dataset. []{data-label="fig:lp-facades"}](lp-facades.pdf){width="\linewidth"} We evaluate the effectiveness of the perturbed loss $\mathcal{L}_{p}$ on both NYU and Facades datasets. As the Fig.  \[fig:lp-facades\] shows, the perturbed loss can actually reduce the blur parts and help the generator create more plausible images with more detail information. On the other hand, we evaluate the performance of denoising process and compare our methods with the pix2pix method in the Fig.  \[fig:facades-noise\]. The result of pix2pix has some meaningless color blocks as shown in the red boxes. The results in Fig.  \[fig:facades-noise\] show that the proper noise can help alleviate such phenomenon. We also evaluate the effectiveness of noise in the NYU dataset in the Fig.  \[fig:noise-nyu\] and get similar performance of Fig.  \[fig:facades-noise\]. Proper noisy inputs make the generation process more robust. ![Different scale of noise induces different quality of results on Facades dataset. We get best results with deviation 0.1[]{data-label="fig:facades-noise"}](facades-noise.pdf){width="\linewidth"} ![The evaluation results of noisy inputs on the NYU dataset. []{data-label="fig:noise-nyu"}](noise-nyu.pdf){width="\linewidth"} ![The evaluation results of instance map on the NYU dataset. []{data-label="fig:facades-noise"}](ins-nyu2.pdf){width="\linewidth"} ![The evaluation of the cascade layer loss on the NYU dataset.[]{data-label="fig:layer-nyu"}](layer-nyu.pdf){width="\linewidth"} ![The evaluation of the cascade layer loss on the Facades dataset.[]{data-label="fig:layer-facades"}](layer-facades.pdf){width="\linewidth"} We also evaluate the cascade layer loss for our model. The Fig.  \[fig:layer-nyu\] shows the comparing results on NYU dataset. The generations without cascade layer loss are blurry and the color blocks are dirty and wrong. And we also evaluate this loss in the Facades dataset, we can see the results in the Fig.  \[fig:layer-facades\]. And we didn’t compute the instance map for the Facades dataset as the result of that this dataset is easy and doesn’t have the complex objects. Cityscapes dataset ------------------ {width="\linewidth"} Method P-SNR MSE R-MSE SSIM --------- --------- --------- -------- --------- Ours 18.3344 0.01677 0.1250 26.0268 CRN 11.9927 0.06791 0.2551 16.9266 Pix2pix 15.7161 0.03025 0.1686 16.9876 : The image equality evaluation results of different methods on the Cityscapes dataset.[]{data-label="table:city"} ![More detail of cars can be generated after using pertubed loss .[]{data-label="fig:lp-city"}](lp-city.pdf){width="\linewidth"} The cityscapes dataset contain 20 different kinds of objects and have various scenes. And the overlapped objects such as cars in the semantic layouts are difficult for our generator to synthesize. As described in Fig. \[fig:city\_compare\], although models like pix2pix can synthesize a clear image, such generated images often contain mistakes especially near by the boundary of two different objects (e.g. two neighboring cars). The results generated from CRN are basically correct but lack of the details on small objects and the generated objects are blurry. With the help of cascade layer loss, perturbed loss and the instance map, images generated by our model contain correct and clear objects. We also evaluate the image equality scores on the cityscape dataset. The results in Table. \[table:city\] show that our method performs significantly better than two other methods. And this dataset contains 2975 images and use 2675 for training and others for testing. Method P-SNR MSE R-MSE SSIM ------------------------------------------ --------- --------- -------- --------- Z0.1 16.1530 0.02823 0.1617 18.1796 Z0.04 15.7261 0.03098 0.1695 16.7215 Z0.4 15.6197 0.03101 0.1706 16.8597 $\mathcal{L}_{p}$ 16.6002 0.02443 0.1518 21.5974 $\mathcal{L}_{c}$ 16.4890 0.02642 0.1558 20.0253 $\mathcal{L}_{c}+ \mathcal{L}_{p}$ 16.7169 0.02386 0.1499 21.4486 $\mathcal{L}_{c}$+ Ins. 17.7136 0.02244 0.1451 21.3573 $\mathcal{L}_{p}$+ Ins. 16.3767 0.02349 0.1496 22.2922 $\mathcal{L}_{c}+ \mathcal{L}_{p}$+ Ins. 17.4317 0.01902 0.1236 25.4498 Ours 18.3344 0.01677 0.1250 26.0268 : The image equality evaluation results of different methods on the Cityscapes dataset.[]{data-label="table:comparison"} The perturbed loss $\mathcal{L}_{p}$ can improve the image generation in some way. The Fig.  \[fig:lp-city\] shows the results we explore the effectiveness of the $\mathcal{L}_{p}$.We see that using the perturbed loss can generate images with more details and the generation got sharper than cases without using perturbed loss. The Table. \[table:comparison\] also shows the evaluation results with the four evaluation metric. The perturbed loss allow the adversarial network to train longer efficiently. That’s why model with perturbed loss lead to lower MSE error and get better generations. The $\mathcal{L}_{p}$ can help improve the ability of discriminator to distinguish the fake component from the real sample distribution. So it can generate more plausible results. And we also consider to check the extra cascade layer loss $\mathcal{L}_{c}$. So we design the contrast experiments to show the effectiveness of this loss. The Fig.  \[fig:layer-city\] shows the experimental results. As this figure shows, the results using the cascade layer loss could be clearer. This loss could provide hierarchy constrains for the generator. The higher-layer feature maps can carry on interpretation information for this image-to-image translation. From the Table. \[table:comparison\], the $\mathcal{L}_{c}$ can help model to generate images with higher SSIM and lower MSE error. This loss can force the generator to generate images with considering small parts as well as matching the representations of fake images and real images in high-level space. The effect of instance map Ins. is also evaluated in our experiment in Table.  \[table:comparison\]. To further validate our car instance map Ins. can improve the generation results, we show some examples in the Fig.  \[fig:waterconcate\]. The additional car instance map can provide our generator more information and force the discriminator to concentrate on the car objects. It’s easy to find that inputs with car instance map lead to more plausible results. Results from Table. \[table:comparison\] also demonstrate this point. In order to explore the effectiveness of noise input, we evaluate how the denoising process can improve the generation results. The Fig.  \[fig:noise\] shows some experimenting results. We find that models trained with appropriate Gaussian noise (e.g. Gaussian noise with deviation $0.1$) generate more plausible images. But if the noise is larger (e.g. Gaussian noise with deviation $0.4$), the outputs get worse. We can conclude that an appropriate amount of Gaussian noise can improve the image generation process. But when the noise are too large, the results got worse than results without adding noise. The Table.  \[table:comparison\] shows some image equality scores under different condition on the Cityscapes dataset. With a proper noisy input, the denoising process can improve the robustness of our model as well as generating better images. Considering that we used multiple improvements, we evaluate the some combinations of theses methods on Cityscapes dataset. The Table. \[table:comparison\] shows the evaluation results. We find that our framework performs well when we mix Gaussian random noise with standard deviation 0.1 rather than 0.04 and 0.4. The image generation results can be further improved when we consider the perturbed loss $\mathcal{L}_{p}$, the cascade loss $\mathcal{L}_{c}$ and the instance map. We find that we can get better results by hybrid these methods all. We evaluate our final method on the Cityscapes dataset with $\mathcal{L}_{c}$, $\mathcal{L}_{p}$, car instance map and the Gaussian noise with a standard deviation 0.1 and achieved the highest P-SNR and SSIM scores as well as the lowest MSE and R-MSE scores. ![The car contents can be well modeled after using cascade loss.[]{data-label="fig:layer-city"}](layer-city.pdf){width="50.00000%"} ![The instance map can reduce the risk of wrong contents generation.[]{data-label="fig:waterconcate"}](waterconcate.pdf){width="50.00000%"} ![Different scale of Gaussian noise induce different generation results on the Cityscape dataset.[]{data-label="fig:noise"}](noise.pdf){width="50.00000%"} Conclusion ========== In this paper, we propose a new architecture based on GAN to achieve image translation between two different domains. We introduce a denoising image-to-image framework to improve the image generation process and increase the robustness of our model. We use the instance map to force the generator to concentrate on the overlapped objects to achieve better generation performance. The perturbed loss and the cascade loss can improve the contents organizing ability of the generator. Experimental results show that our method outperforms other state-of-the-art image-to-image translation methods. Although our model can achieve plausible image-to-image translation tasks among contemporaneous methods, the gap still exists between the real images and the synthesized ones. Generally, the key of image-to-image translation is how to teach the model comprehend the relationship among various objects. On this point, we use cascade layer loss to control the image structure information in different scales. We also use a denoising process and the perturbed loss to improve the robustness and model the invariability in the sample distribution. But they are still not enough to obtain a realistic synthesized image. It’s still a challenge to achieve a pixel-level translation. We need the discriminator to be smarter to guide the generation process. It requires the model achieve not only the feature matching, but also the global information extracting. We leave them to our future work. Acknowledgment {#acknowledgment .unnumbered} ============== This work was supported by the National Natural Science Foundation of China under Grant no. 61701463, the China Postdoctoral Science Foundation under Grant no. 2017M622277, Natural Science Foundation of Shandong Province of China under Grant no. ZR2017BF011, the Fundamental Research Funds for the Central Universities under Grant nos. 201713019 and the Qingdao Postdoctoral Science Foundation of China. We gratefully acknowledge the support of NVIDIA Corporation with the donation of the Titan X Pascal GPU used for this research.

--- abstract: 'We characterize non-decreasing weight functions for which the associated one-dimensional vertex reinforced random walk (VRRW) localizes on $4$ sites. A phase transition appears for weights of order $n\log \log n$: for weights growing faster than this rate, the VRRW localizes almost surely on at most $4$ sites whereas for weights growing slower, the VRRW cannot localize on less than $5$ sites. When $w$ is of order $n\log \log n$, the VRRW localizes almost surely on either $4$ or $5$ sites, both events happening with positive probability.' address: - 'Laboratoire Modal’X, Université Paris Ouest Nanterre La Défense, Bâtiment G, 200 avenue de la République, 92000 Nanterre, France.' - 'Département de Mathématiques, Bât. 425, Université Paris-Sud 11, F-91405 Orsay, cedex, France. ' author: - 'Anne-Laure BASDEVANT' - Bruno SCHAPIRA - Arvind SINGH title: 'Localization on $4$ sites for Vertex-reinforced random walks on ${\mathbb{Z}}$.' --- Introduction ============ The model of the *vertex reinforced random walk* (VRRW) was first introduced by Pemantle [@P] in 1992. It describes a discrete random walk $X = (X_n,\; n\geq 0)$ on a graph $G$, which jumps, at each unit of time $n$, from its actual position towards a neighboring site $y$ with probability proportional to $w(Z_n(y))$, where $w : {\mathbb{N}}\to {\mathbb{R}}_+^*$ is some deterministic weight sequence and where $Z_n(y)$ is the local time of the walk at site $y$ and time $n$. Thus, when $w$ is non-decreasing, the walk tends to favor sites it has already visited many times in the past. A striking feature of the model is that, depending on the reinforcement scheme $w$, it is possible for the walk to get “trapped” and visits only finitely many sites, even on an infinite graph. In this case, we say that the walk *localizes*. This unusual behaviour was first observed by Pemantle and Volkov [@PV] who proved that, with positive probability, the VRRW on the integer lattice ${\mathbb{Z}}$ with linear weight $w(n) = cn+1$ visits only $5$ sites infinitely often. This result was later completed by Tarrès [@T1] (see also [@T2] for a more recent and concise proof) who showed that localization of the walk on $5$ sites occurs almost surely. More generally, Volkov [@V1] and more recently Benaïm and Tarrès [@BT] proved that the linearly reinforced VRRW localizes with positive probability on any graph with bounded degree. It is conjectured that this localization happens, in fact, with probability $1$. However, this seems a very challenging question as it is usually difficult to prove almost sure asymptotics for VRRW (let us note that, even in the one-dimensional case, Tarrès’s proof of almost sure localization is quite elaborate). A seemingly closely related model is the so-called *edge reinforced random walk* (ERRW), introduced by Coppersmith and Diaconis [@CD] in 1987. The difference between VRRW and ERRW is only that the transition probabilities for ERRW depend on the *edge* local time of the walk instead of the *site* local time. In the one-dimensional case, Davis [@Dav] proved that the ERRW with non-decreasing reinforcement weight function $w$ is recurrent (i.e. the walk visits all sites infinitely often almost surely) i.f.f. $$\label{twosite} \sum_{i=0}^{\infty}\frac{1}{w(i)} = \infty.$$ Otherwise, the walk ultimately localizes on two consecutive sites almost surely. It may seem natural to expect a similar simple criterion for VRRW. However, the picture turns out to be much more complicated than for ERRW because the walk may localize on subgraphs of cardinality larger than $2$. Only partial results are currently available. For instance, when condition fails and the sequence $(w(n),n\ge 0)$ is non-decreasing, the VRRW also gets stuck on two consecutive sites[^1]. However, when holds, the walk may or may not localize depending on the weight function $w$. In particular, it is conjectured that for reinforcements $w(n) \sim n^\alpha$, the walk is recurrent for $\alpha<1$ and localizes on $5$ sites for $\alpha=1$. In this direction, it is proved in [@V2] that when $w(n)$ is of order $n^\alpha$ with $\alpha<1$, the VRRW cannot localize. When $\alpha<1/2$, this result was slightly refined by the second author in [@Sch], who proved that the process is either a.s. recurrent or a.s. transient. Yet, a proof of the recurrence of the walk in this seemingly simple setting is still missing. On the other hand (apart from the linear case) not much is known about the cardinality of the set of sites visited infinitely often when localization occurs and holds. The aim of this paper is to partially answer this question by investigating under which conditions the VRRW ultimately localizes on less than $5$ sites. In order to do so, we shall associate to each weight function $w$ a number $\alpha_c(w) \in [0,\infty]$ (the precise definition of $\alpha_c(w)$ is given in the next section). The main result of the paper states that: \[4probapositive\] Assume that $w$ is non-decreasing and that holds. Denote by $R'$ the set of sites which are visited infinitely often by the VRRW and by $|R'|$ its cardinality. Then, defining $\alpha_c(w)$ as in , we have $$\begin{aligned} |R'|=4 \textrm{ with positive probability} &\Longleftrightarrow & \alpha_c(w)<\infty.\\ |R'|=4 \textrm{ almost surely} &\Longleftrightarrow & \alpha_c(w)=0.\\ \begin{array}{c} |R'| \textrm{ equals $4$ or $5$ a.s., both events }\\ \textrm{occurring with positive probability} \end{array} &\Longleftrightarrow & \alpha_c(w)\in (0,\infty).\end{aligned}$$ It is easy to see that a VRRW can never localize on $3$ sites (see Proposition \[R’3\]) therefore Theorem \[4probapositive\] combined with criterion for localization on two sites covers all possible cases where a VRRW with non-decreasing weight localizes with positive probability on less than $5$ sites. The last part of the theorem shows that the size of $R'$ can itself be random. Such a result was already observed for graphs like ${\mathbb{Z}}^d$, $d\ge 2$ (for linear reinforcement) where different non-trivial localization patterns may occur (see [@BT; @V1]). Yet we find this result more surprising in the one-dimensional setting since $R'$ is necessarily an interval. The parameter $\alpha_c(w)$ can be explicitly calculated for a large class of weights $w$. In particular, if $w(n)\sim n\log\log n$, then $\alpha_c(w) = 1$. Moreover, \[alphac\_cor\] For any non-decreasing weight function $w$ such that holds: $$\begin{aligned} w(n)\ \gg \footnotemark \ n\log\log n & \Longrightarrow & \alpha_c(w) = 0. \\ w(n)\ \asymp \footnotemark\ n\log\log n & \Longrightarrow & \alpha_c(w) \in (0,\infty). \\ w(n)\ \ll \ n\log\log n & \Longrightarrow & \alpha_c(w) = \infty.\end{aligned}$$ Let us mention that, when $\alpha_c(w) = \infty$, Theorem \[4probapositive\] simply states that, if the walk localizes, $|R'| \geq 5$ necessarily. In fact, it is proved in a forthcoming paper [@BSS] that there exist non-decreasing weight functions $w$ for which the VRRW localizes almost surely on finite sets of arbitrarily large cardinality (this result is, in a way, similar to those proved in [@ETW1; @ETW2] for another related model of self-interacting random walks). The proof of Theorem \[4probapositive\] is based on two main techniques. First we use martingales arguments which were introduced by Tarrès in [@T1; @T2]. These martingales have the advantage of taking into account the facts that, on each site, the process is roughly governed by an urn process, but also the fact that all these urns are strongly correlated. The second tool is a continuous time construction of the VRRW, called Rubin’s construction, which was already used by Davis [@Dav] for urn processes, and by Sellke in [@Sel] in the case of edge reinforcement. Tarrès introduced in [@T2] a variant of this construction, which allows for powerful couplings in the case of non-decreasing weights and which will be very useful in this study. The remainder of the paper is organized as follows. In the next section we give some simple results concerning $w$-urns processes which will play an important role in the proof of the theorem. In Section \[section3\], we recall some classical results concerning VRRW. The proof of Theorem \[4probapositive\] is provided in Section \[section4\]. Finally we prove Proposition \[alphac\_cor\] in the appendix along with other technical lemmas concerning properties of the critical parameter $\alpha_c(w)$. $w$-urn processes {#section2} ================= Weight function $w$ and the parameter $\alpha_c$ ------------------------------------------------ In the rest of the paper, we call *weight sequence* $w$ a sequence $(w(n),n\ge 0)$ of positive real numbers. It will be convenient to extend $w$ into a weight function $w: {\mathbb{R}}_+ \to {\mathbb{R}}_+^*$ by $w(t) = w(\lfloor t\rfloor)$ where $\lfloor t\rfloor$ stands for the integer part of $t$. Then, given $w$, we set $$W(t) := \int_0^t \frac 1{w(u)}\, du.$$ When condition (\[twosite\]) holds, we have $W(\infty) = \infty$ and $W$ is an homeomorphism of ${\mathbb{R}}_+$ whose inverse we denote by $W^{-1}$. Then, for $\alpha>0$, we define the integral $$\label{defI} I_\alpha(w):= \int_0^\infty \frac{dx}{w(W^{-1}(W(x)+\alpha))}=\int_0^\infty \frac{w(W^{-1}(y))}{w(W^{-1}(y+\alpha))}\, dy.$$ If furthermore we assume that $w$ is non-decreasing, then $\alpha \to I_\alpha(w)$ is non-increasing and we can define the critical parameter $\alpha_c(w)$ by $$\label{defAlphac} \alpha_c(w):=\inf \{\alpha\ge 0 \ :\ I_\alpha(w)<\infty\}\in [0,\infty]$$ with the convention that $\inf \emptyset = \infty$. $w$-urn processes {#w-urn-processes} ----------------- A $w$-urn is a process $(R_n,B_n)_{n\ge 0}$ defined on some probability space $(\Omega,\mathcal{F},{\mathbb{P}})$, such that for all $n\ge 0$, $R_n+B_n=n$, $R_{n+1}\in \{R_n,R_n+1\}$, and $${\mathbb{P}}\{R_{n+1}=R_n + 1\} = \frac {w(R_n)}{w(R_n)+w(B_n)}.$$ We call $R_n$ (resp. $B_n$) the number of red (resp. blue) balls in the urn after the $n$-th draw. Set $R_\infty = \lim_{n\to \infty}R_n$ and $B_\infty=\lim_{n\to \infty} B_n$. Our interest towards $w$-urn processes comes from fact that, if we consider a VRRW on the finite set $\{-1,0,1\}$ (*i.e.* the walk reflected at $1$ and $-1$, see Section \[section3.4\]), then joint local times of the walk at sites $1$ and $-1$ and at time $2n$ is exactly a $w$-urn process. The next proposition describes the asymptotic behavior of such an urn. Several arguments used during the proof of the result below will also play an important role when proving Theorem \[4probapositive\]. \[propurne\] For any weight sequence $w$ (not necessarily non-decreasing), we have $$\begin{aligned} \sum_{n\ge 0} \frac 1 {w(n)}< +\infty \quad \Longleftrightarrow \quad R_\infty<+\infty\ \textrm{or}\ B_\infty <+\infty\;\hbox{ a.s.}\end{aligned}$$ The process $\hat{M} = (\hat{M}_n,\, n\geq 0)$ defined by $\hat{M}_n :=W(R_n)-W(B_n)$ is a martingale. Moreover, 1. If $\sum_n 1/w(n)=\infty$ and $\sum_n 1/w(n)^2<+\infty$, then $\hat{M}_n$ converges a.s. to some random variable $\hat{M}_\infty$, which admits a symmetric density with unbounded support. 2. If $\sum_n 1/w(n)^2 = \infty$ and $\inf_n w(n) >0$, then $\liminf \hat{M}_n = -\infty$ and $\limsup \hat{M}_n=+\infty$ a.s. The first equivalence is well known (see, for instance [@P2]) and follows immediately from Rubin’s construction of the urn process which we will recall below. Let us note that we can rewrite $\hat{M}$ in the form $$\hat{M}_n = \sum_{k=0}^{n-1} \left(\frac {1_{\{\textrm{the $(k+1)$-th draw is Red}\}}}{w(R_k)} - \frac {1_{\{\textrm{the $(k+1)$-th draw is Blue}\}}}{w(B_k)}\right).$$ Therefore, $\hat{M}$ is clearly a martingale. Let $$\begin{aligned} V_n &:=& \sum_{k\le n} (\hat{M}_{k+1}-\hat{M}_k)^2\\ &=& \sum_{k\le n } \left(\frac {1_{\{\textrm{the $(k+1)$-th draw is Red}\}}}{w(R_k)^2} + \frac {1_{\{\textrm{the $(k+1)$-th draw is Blue}\}}}{w(B_k)^2}\right).\end{aligned}$$ We have $${\mathbb{E}}[\hat{M}_n^2] = {\mathbb{E}}[ V_{n-1} ] \leq 2\sum_{k = 0}^{\infty}\frac{1}{w(k)^2}.$$ Thus, when assumption (a) holds, $\hat{M}$ converges almost surely and in $L^2$ towards some random variable $\hat{M}_\infty$. We now use Rubin’s construction to identify this limit: let $(\xi_n,n\ge 0)$ and $(\xi'_n,n\ge 0)$ be two sequences of independent exponential random variables with mean $1$. Define the random times $t_k=\xi_0/w(0) + \dots +\xi_k/w(k)$ and $t'_k=\xi'_0/w(0)+\dots +\xi'_k/w(k)$. We can construct the $w$-urn process $(R_n,B_n)$ from these two sequences by adding a red ball in the urn at each instant $(t_k)_{k\geq 0}$ and a blue ball at each instant $(t'_k)_{k\geq 0}$ (see the appendix in [@Dav] for details). Using this construction, we can rewrite $\hat{M}_n$ in the form $$\hat{M}_n=\sum_{k=0}^{R_n-1} \frac {1-\xi_k} {w(k)}- \sum_{k=0}^{B_n-1} \frac {1-\xi_k'} {w(k)}+ \sum_{k=0}^{R_n-1} \frac {\xi_k} {w(k)}-\sum_{k=0}^{B_n-1} \frac {\xi'_k} {w(k)}.$$ Observe that, by construction, for any $n$, $$\left|\sum_{k=0}^{R_n-1} \frac {\xi_k} {w(k)}-\sum_{k=0}^{B_n-1} \frac {\xi'_k} {w(k)}\right| \le \max\left( \frac {\xi_{R_{n}}} {w(R_n)}, \frac {\xi'_{B_{n}}} {w(B_n)}\right).$$ Since a.s. $R_n\wedge B_n \to \infty$, we deduce that the r.h.s. above tends to $0$ a.s. hence $$\label{limitUrn} \hat{M}_\infty = \sum_{k=0}^{\infty} \frac {1-\xi_k} {w(k)}- \sum_{k=0}^{\infty} \frac {1-\xi_k'} {w(k)}.$$ Both sums in the r.h.s. of the previous equation converge because they have a finite second moment. Thus, $\hat{M}_\infty$ admits a symmetric density with unbounded support since the $\xi_n$’s and $\xi'_n$’s are independent and $\xi_0$ has a non-vanishing density on ${\mathbb{R}}_+$. It remains to prove $(b)$. Let us observe that, when $\inf_n w(n)>0$, the martingale $\hat{M}_n$ has bounded increments. Using Theorem 2.14 in [@HH], it follows that, a.s., either $\hat{M}_n$ converges or $\limsup \hat{M}_n = +\infty$ and $\liminf \hat{M}_n = -\infty$. Moreover, when $\sum_n 1/w(n)^2 = \infty$, we have $\lim_{n\to\infty} V_n = \infty$ a.s. Therefore, we can define $k_n:=\inf\{m\ :\ V_m\ge n\}$ and Theorem $3.2$ of [@HH] states that $\hat{M}_{k_n}/\sqrt n$ converges in law towards a standard normal variable. In particular, $\hat{M}$ cannot converge. This completes the proof of the proposition. The next result illustrates how the parameter $\alpha_c(w)$ of Theorem \[4probapositive\] naturally appears in connection with $w$-urns. \[Yurn\] Consider a $w$-urn $(R_n,B_n)$. Assume that $w$ is non-decreasing with $\sum_n 1/w(n)=\infty$ and $\sum_n 1/w(n)^2 <\infty$ and set $$Y^B := \sum_{k=0}^\infty \frac{1_{\{\textrm{the $(k+1)$-th draw is Blue}\}}}{w(k)},\qquad Y^R := \sum_{k=0}^\infty \frac{1_{\{\textrm{the $(k+1)$-th draw is Red}\}}}{w(k)}.$$ Then, we have - If $\alpha_c(w)=0$ then, a.s., $\min(Y^B,Y^R)<\infty$. - If $\alpha_c(w)\in (0,\infty)$ then ${\mathbb{P}}\{\min(Y^B,Y^R)<\infty\}\in (0,1)$. - If $\alpha_c(w)=\infty$ then, a.s., $\min(Y^B,Y^R)=\infty$. According to Proposition \[propurne\], $W(R_n)-W(B_n)$ converges to some random variable $\hat{M}_\infty$ with a symmetric density and unbounded support. Let $\delta=|\hat{M}_\infty|/2$. On the event $\{\hat{M}_\infty>0\}$, we have, for $n$ large enough, $$W(n)\ge W(R_n) \ge W(B_n)+ \delta$$ which yields, for some (random but finite) constant $c$ $$Y^B\le c\sum_{k=0}^\infty \frac{1_{\{\textrm{the $(k+1)$-th draw is Blue}\}}}{w(W^{-1}(W(B_k)+ \delta))}= \sum_{k=0}^\infty \frac{c}{w(W^{-1}(W(k)+ \delta))}.$$ Thus, by symmetry and using ${\mathbb{P}}\{\hat{M}_\infty =0\} = 0$, we get, a.s., $$\min(Y^B,Y^R) \leq \sum_{k=0}^\infty \frac{c}{w(W^{-1}(W(k)+ \delta))}.$$ This proves (i) and also that ${\mathbb{P}}\{\min(Y^B,Y^R)<\infty\}>0$ whenever $\alpha_c(w)\in (0,\infty)$. Conversely, set $\delta' = 2|\hat{M}_\infty|$. On the event $\{\hat{M}_\infty \geq 0\}$, we have, for $n$ large enough $$n=B_n+R_n \le B_n+ W^{-1}(W(B_n)+ \delta') \le R_n+ W^{-1}(W(R_n)+ \delta').$$ This gives $$Y^B\ge c\sum_{k=0}^\infty \frac{1_{\{\textrm{the $(k+1)$-th draw is Blue}\}}}{w(B_k+W^{-1}(W(B_k)+ \delta'))}= \sum_{k=0}^\infty \frac{c}{w(k+W^{-1}(W(k)+ \delta'))}.$$ and the same bound also holds for $Y^R$. Therefore, by symmetry, we get, a.s., $$\min(Y^B,Y^R) \geq \sum_{k=0}^\infty \frac{c}{w(k+W^{-1}(W(k)+ \delta'))}.$$ We conclude the proof using Lemma \[teclem\] of the appendix which insures that the sum above is infinite when $\delta' < \alpha_c(w)$. Vertex reinforced random walk {#section3} ============================= The VRRW -------- In the remainder of the paper, given the weight sequence $w$, $(X_n,n\ge 0)$ will denote a nearest neighbour random walk on the integer lattice ${\mathbb{Z}}$, starting from $X_0=0$ with transition probabilities given by $$\label{transiVRRW} {\mathbb{P}}\{X_{n+1}= x\pm 1 \mid {\mathcal{F}}_n\}\, = \, \frac{w(Z_n(x\pm 1))}{w(Z_n(x+1)) + w(Z_n(x-1))},$$ where $({\mathcal{F}}_n,n\ge 0)$ is the natural filtration $\sigma(X_0,\ldots,X_n)$ and $Z_n(y)$ stands, up to a constant, for the local time of $X$ at site $y$ and at time $n$: $$Z_n(y):=z_0(y)+\sum_{k=0}^n 1_{\{X_k=y\}}.$$ We call the sequence $\mathcal{C} := (z_0(y),\, y\in{\mathbb{Z}})$ the *initial local time configuration*. We say that $X$ is a VRRW when it starts from the trivial configuration $\mathcal{C}_0 := (0,0,\ldots)$. However, it will sometimes be convenient to consider the walk starting from some other configuration $\mathcal{C}$. In that case, we shall mention it explicitly and emphasize this fact by calling $X$ a $\mathcal{C}$-VRRW. In the rest of this section, we collect some important results concerning the VRRW which we will use during the proof of Theorem \[4probapositive\] in Section \[section4\]. For additional details, we refer the reader to [@Dav; @P2; @Sel; @T1; @T2] and the references therein. The martingales $M_n(x)$ {#section3.2} ------------------------ For $x\in {\mathbb{Z}}$, define $Z_\infty(x) := \lim_{n\to \infty} Z_n(x)$. Recall that $R'$ stands for the set of sites visited infinitely often by the walk: $$R' :=\{x\in {\mathbb{Z}}\ :\ Z_\infty(x)=\infty\}.$$ The following quantities will be of interest: $$\label{defY} Y_n^{\pm}(x) := \sum_{k=0}^{n-1} \frac{1_{\{X_k=x\textrm{ and }X_{k+1}=x\pm 1\}}}{w(Z_k(x\pm 1))},$$ and $$M_n(x) := Y_n^+(x)-Y_n^-(x).$$ It is a basic observation due to Tarrès [@T1; @T2] that $(M_n(x),n\ge 1)$ is a martingale for each $x\in {\mathbb{Z}}$. Moreover, if $$\label{wsquare} \sum_{n=0}^{\infty} \frac{1}{w(n)^2}<\infty,$$ then these martingales are bounded in $L^2$, and thus converge a.s. and in $L^2$ towards $$M_\infty(x) := \lim_{n\to\infty} M_n(x).$$ We will also consider the (possibly infinite) limits: $$Y_\infty^\pm(x):= \lim_{n\to \infty} Y_n^{\pm}(x).$$ From the definition of $Y^{\pm}$, we directly obtain the identity $$\begin{aligned} \label{Ynpm} Y_n^+(x-1) + Y_n^-(x+1) = W(Z_{n}(x))-W(1)1_{\{x=0\}},\end{aligned}$$ which holds for all $x\in {\mathbb{Z}}$ and all $n\ge 0$. In particular, we get $$\begin{gathered} W(Z_{n}(x+2))-W(Z_{n}(x)) = Y_n^-(x+3)-Y_n^+(x-1)+M_{n}(x+1)\\ +W(1)(1_{\{x=-2\}}-1_{\{x=0\}}).\end{gathered}$$ More generally, if we now consider a $\mathcal{C}$-VRRW starting from some arbitrary initial local time configuration $\mathcal{C}$, then $M_n(x)$ is still a martingale and the equation above takes the form $$\label{eqW} W(Z_{n}(x+2))-W(Z_{n}(x))=Y_n^-(x+3)-Y_n^+(x-1)+M_{n}(x+1)+ c(x,\mathcal{C}).$$ where $c(x,\mathcal{C})$ is some constant depending only on $x$ and on the configuration $\mathcal{C}$. Time-line construction of the VRRW ---------------------------------- We now describe a method to construct the VRRW for a collection of exponential random variables which is in a way similar to Rubin’s algorithm for $w$-urns. This construction was introduced by Tarrès in [@T2] and may be seen as a variant for the VRRW of the continuous time construction previously described by Sellke in [@Sel] for edge reinforced random walks. One of the main advantages of this construction is that it enables to create non-trivial coupling between VRRWs. {width="12.6cm"} Let us fix a sequence $$\xi := (\xi_n^\pm(y),n\ge 0, y\in{\mathbb{Z}}) \in {\mathbb{R}}_+^{\mathbb{N}}$$ of positive real numbers. The value $\xi^-_n(y)$ (resp. $\xi^+_n(y)$) will be related to the duration of a clock attached to the oriented edge $(y,y-1)$ (resp. $(y,y+1)$). Given this sequence, we create a deterministic, integer valued, continuous-time process $(\widetilde X(t),t\ge 0)$ in the following way: - Set $\widetilde X(0)=0$ and attach two clocks to the oriented edges $(0,-1)$ and $(0,1)$ ringing respectively at times $\xi_0^-(0)/w(0)$ and $\xi_0^+(0)/w(0)$. - When the first clock rings at time $\tau_1:=\xi_0^+(0)/w(0)\wedge \xi_0^-(0)/w(0)$, stop both clocks and set $\widetilde X(\tau_1)=\pm1$ depending on which clock rung first. (if both clocks ring at the same time, we decide that $\widetilde X$ stays at $0$ forever). Assume that we have constructed $\widetilde{X}$ up to some time $t>0$ at which time the process makes a right jump from some site $x-1$ to $x$. Denote by $k$ the number of jumps from $x$ to $x-1$ and by $m$ the number of visits to $x-1$ before time $t$. We follow the procedure below: - Start a new clock attached to the oriented edge $(x,x-1)$, which will ring after a time $\xi_k^-(x)/w(m)$. - If the process already visited $x$ some time in the past, restart the clock attached to the oriented edge $(x,x+1)$ which had previously been stopped when the process last left site $x$. Otherwise, start the first clock for this edge which will ring at time $\xi_0^+(x)/w(0)$. - As soon as one of these two clocks rings, stop both of them and let the process jump along the edge corresponding to the clock which rung first (if both clocks ring at the same time, we decide that $\widetilde X$ stays in $x$ forever). We use a similar rule when the process makes a left jump from some site $x$ to $x-1$. We say that this construction *fails* if at some time, two clocks ring simultaneously. Let now $\tau_i$ stand for the time of the $i$-th jump of $\widetilde{X}$ (with the convention $\tau_0=0$ and $\tau_{n+1}=\tau_{n}$ if $\widetilde X$ does not move after time $\tau_n$) and define the discrete time process $X = (X_n,\, n\geq 0)$ by $$X_n := \widetilde{X}(\tau_n).$$ It is an elementary observation that if we now choose the $\xi_n^\pm(y)$ to be independent exponential random variables with mean $1$, then the construction does not fail with probability $1$ and the resulting process $X$ is a VRRW with weight $w$. For the sake of clarity, we only describe the construction for the VRRW starting from the trivial configuration $\mathcal{C}_0$. However, it is clear that we can do a similar construction for any $\mathcal{C}$-VRRW by simply replacing the duration of the clocks $\xi_k^\pm(x)/w(m)$ with $\xi_k^\pm(x)/w(z_0(x\pm 1) + m)$. A remarkable feature of this construction comes from the fact that we can simultaneously create a family $(\widetilde{X}^{(u)},\; u\geq 0)$ of processes with nice monotonicity properties with respect to the $u$ parameter. To this end, define, for $x\in{\mathbb{Z}}$, $${\mathcal{H}}_x:= \left((\xi_n^\pm(y),n\ge 0, y\neq x),(\xi_n^-(x),n\ge 0),(\xi_n^+(x),n\ge 1)\right) \in {\mathbb{R}}_+^{\mathbb{N}}.$$ Then, given ${\mathcal{H}}_x$ together with a real number $u>0$, the pair $({\mathcal{H}}_x,u)$ defines a deterministic process $X^{(u)} = (X_n^{(u)},n\geq 0)$ using the construction above with $\xi_0^+(x) = u$. The following lemma is easily obtained by induction. \[tarres1\] Suppose that $w$ is non-decreasing. Fix ${\mathcal{H}}_x$ and $0 < u\le u'$ and assume that the construction for $X^{(u)}$ and $X^{(u')}$ both succeed. Given $y\in {\mathbb{Z}}$ and $k\ge 1$, denote by $\sigma$ (resp. $\sigma'$) the time when $X^{(u)}$ (resp. $X^{(u')}$) visits $y$ for the $k$-th time. If $\sigma$ and $\sigma'$ are both finite, then $$\begin{aligned} Z_\sigma^{(u)}(y+1) \ge Z_{\sigma'}^{(u')}(y+1) \quad &\textrm{and}& \quad Z_\sigma^{(u)}(y-1) \le Z_{\sigma'}^{(u')}(y-1)\\ N_\sigma^{(u)}(y,y+1) \ge N_{\sigma'}^{(u')}(y,y+1) \quad &\textrm{and}& \quad N_\sigma^{(u)}(y,y-1) \le N_{\sigma'}^{(u')}(y,y-1),\end{aligned}$$ where $Z_n^{(s)}$ stands the local time of $X^{(s)}$ and $N_n^{(s)}(y,y\pm 1)$ denotes the number of jumps from $y$ to $y\pm 1$ up to time $n$. Moreover, denote by $\theta^{\pm}$ (resp. ${\theta'}^{\pm}$) the time when $X^{(u)}$ (resp. $X^{(u')}$) jumps for the $k$-th time from $y$ to $y\pm1$. If these quantities are finite, then $$Y^{(u)+}_{\theta^{+}}(y) \le Y^{(u')+}_{{\theta'}^{+}}(y)\qquad\hbox{and}\qquad Y^{(u)-}_{\theta^{-}}(y) \ge Y^{(u')-}_{{\theta'}^{-}}(y),$$ where $Y^{(s)\pm}$ is defined as in for the process $X^{(s)}$. The combination of the time-line construction of the walk from i.i.d. exponential random variables together with Lemma \[tarres1\] yields a simple proof of the following key result concerning the localization of the VRRW: \[Yfini\] Assume that $w$ is non-decreasing and that $\sum_n 1/w(n)^2$ is finite. Then, for any $x\in {\mathbb{Z}}$, a.s., $$\{Y_\infty^+(x)<\infty\}=\{Y_\infty^-(x)<\infty\}\ = \ \{Z_\infty(x-1)<\infty\} \cup \{Z_\infty(x+1)<\infty\}.$$ This result is proved in [@T2] only for linear reinforcements $w$ but the same arguments apply, in fact, for any non-decreasing weight function. However, since some details are omitted in [@T2], for the sake of completeness, we provide here a detailed proof (differing in some aspects from the original one). The first equality $\{Y_\infty^+(x)<\infty\}=\{Y_\infty^-(x)<\infty\}$ follows from the fact that the martingale $M_n(x)$ converges a.s. to some finite limit when $\sum_n 1/w(n)^2$ is finite. Concerning the second equality, the inclusion $$\{Y_\infty^+(x)<\infty\}=\{Y_\infty^-(x)<\infty\}\ \supset \ \{Z_\infty(x-1)<\infty\} \cup \{Z_\infty(x+1)<\infty\}.$$ is straightforward (one of the sums $Y_\infty^{\pm}$ has only a finite number of terms). We use the time-line construction of the VRRW $X$ from the sequence $(\xi_n^\pm(y),n\ge 0,y\in {\mathbb{Z}})$ to prove the converse inclusion. Denote by $N_k(x, x\pm 1)$ the number of jumps of $X$ from $x$ to $x\pm 1$ before time $k$, and set $$T_x^\pm :=\sum_{k\ge 0} \frac{1_{\{X_k=x,\ X_{k+1}=x\pm 1\}}\xi_{N_k(x, x\pm 1)}^\pm(x)}{w(Z_k(x\pm 1))}.$$ Thus, $T_x^\pm$ represents the total time consumed by the clocks attached to oriented edge $(x,x\pm 1)$. We claim that $$\label{Tx} \{Y_\infty^+(x)<\infty\}\cap \{Z_\infty(x-1)=\infty\}\cap \{Z_\infty(x+1) =\infty\} \subset \{T_x^+=T_x^-<\infty\}.$$ We prove the result for $x<0$ (the proof for $x>0$ and $x=0$ are similar). Let $\theta_k$ denote the time of the $k$-th jump from site $x+1$ to $x$ and let $i_k$ be the local time at site $x+1$ and at time $\theta_k$. With this notation, on the event $\{Z_\infty(x+1) =\infty\}$, we can write $$T_x^+= \sum_{k\ge 1} \xi_{k-1}^+(x)\frac{1_{\{\theta_k < \infty\}}}{w(i_k)}\quad \mbox{ and } \quad Y_\infty^+(x)= \sum_{k\ge 1} \frac{1_{\{\theta_k < \infty\}}}{w(i_k)}.$$ Define now $$T_n^+(x):= \sum_{k=1}^{n-1} (\xi_{k-1}^+(x)-1)\frac{1_{\{\theta_k < \infty\}}}{w(i_k)}.$$ Recall that $({\mathcal{F}}_n,n\ge 0)$ stands for the natural filtration of $X$ and notice that $i_k$ is ${\mathcal{F}}_{\theta_k}$-measurable whereas $\xi_{k-1}^+(x)$ is independent of ${\mathcal{F}}_{\theta_k}$. Thus, $(T_n^+(x),n\ge 1)$ is a ${\mathcal{F}}_{\theta_n}$-martingale. Moreover, using that $w(i_k)\ge w(k)$ for all $k\ge 1$, it follows that the $L^2$-norm of this martingale is bounded by $\sum_{k\ge 0} {w(k)^{-2}}<\infty$. In particular, this implies that $Y_\infty^+(x)$ is finite if and only if $T_x^+$ is finite. It is also clear from the construction of the time-line process that, on the event $\{Z_\infty(x-1)=\infty\}\cap \{Z_\infty(x+1) =\infty\}$, we have $T_x^+=T_x^-$. Thus we have established . It remains to prove that the event $$\mathcal{E}_x:=\{T_x^+=T_x^- <\infty\}\cap \{Z_\infty(x-1)=\infty\}\cap \{Z_\infty(x+1) =\infty\}$$ has probability $0$. Recall the notation $${\mathcal{H}}_x:= \left((\xi_n^\pm(y),n\ge 0)_{y\neq x},(\xi_n^-(x),n\ge 0),(\xi_n^+(x),n\ge 1)\right) \in {\mathbb{R}}_+^{\mathbb{N}}.$$ and denote by $\mu$ the product measure on ${\mathbb{R}}_+^{\mathbb{N}}$ under which ${\mathcal{H}}_x$ is a collection of i.i.d. exponential random variables with mean $1$. Given $\xi_0^+(x)$, the pair $({\mathcal{H}}_x,\xi_0^+(x))$ defines the (deterministic) process $X = X({\mathcal{H}}_x,\xi_0^+(x))$ via the time-line construction and $X$ under the product law ${\mathbb{P}}:=\mu\times\hbox{Exp}(1)$ is a VRRW. Let us note that, for $\mu$-a.e. realization of ${\mathcal{H}}_x$, the set of values of $\xi_0^+(x)$ such that the time-line construction fails is countable hence has zero Lebesgue measure. Moreover, Lemma \[tarres1\] implies that, for any $({\mathcal{H}}_x,u)$ and $({\mathcal{H}}_x,u')$ in $\mathcal{E}_x$ with $u'>u$, we have $$T_x^+({\mathcal{H}}_x,u') > T_x^+({\mathcal{H}}_x,u)\quad\hbox{ and }\quad T_x^-({\mathcal{H}}_x,u') \leq T_x^-({\mathcal{H}}_x,u).$$ Thus, for any ${\mathcal{H}}_x$, there is at most one value of $\xi_0^+(x)$ such that $({\mathcal{H}}_x,\xi_0^+(x))\in \mathcal{E}_x$. This yields $${\mathbb{P}}\{\mathcal{E}_x\}=\mathbb{E}_\mu\left(\int_0^\infty e^{-u} 1_{\{({\mathcal{H}}_x,u)\in\mathcal{E}_x\}}du\right)=\mathbb{E}_\mu(0)=0.$$ A weaker statement can also be obtained when the assumptions of Lemma \[Yfini\] do not hold. \[Yfini2\] For any weight sequence $w$ and for any $x\in {\mathbb{Z}}$, we have, a.s. $$\{Z_\infty(x-1)<\infty\} \cup \{Z_\infty(x+1)<\infty\}\subset \{Y_\infty^+(x)<\infty\}\cap\{Y_\infty^-(x)<\infty\}.$$ By symmetry, we can assume without loss of generality that $Z_\infty(x-1)<\infty$. On the one hand, $Y_\infty^-(x)$ is finite since it is a sum with a finite number of terms. On the other hand, the conditional Borel-Cantelli lemma implies that $Y_\infty^+(x)<\infty$ (apply for instance the theorem of [@Ch] with the sequence $1_{\{X_k=x-1\}}$). VRRW restricted to a finite set. {#section3.4} -------------------------------- In the sequel, it will be convenient to consider the vertex reinforced random walk restricted to some interval ${\left[\kern-0.15em\left[}a,b{\right]\kern-0.15em\right]}:= \{x\in {\mathbb{Z}}\ :\ a\le x\le b\}$ for some $a \leq 0 \leq b$, *i.e.* a walk with the same transition probabilities as the VRRW $X$ on ${\mathbb{Z}}$ except at the boundary sites $a$ and $b$ where it is reflected. We shall use the notation $\bar{X}$ to denote this reflected process. We also add a bar to denote all the quantities $\bar{Z}$,$\bar{Y}^\pm$,$\bar{M}$,…related with the reflected process $\bar{X}$. Let us emphasize the fact that, for $x\in {\left]\kern-0.15em\left]}a,b {\right[\kern-0.15em\right[}$, the processes $\bar{M}_n(x):=\bar{Y}^{+}_n(x)-\bar{Y}^{-}_n(x)$ are still martingales, which are bounded in $L^2$ when holds. In particular, Lemma \[Yfini\] still holds for the reflected random walk for all site $x\in {\left]\kern-0.15em\left]}a,b {\right[\kern-0.15em\right[}$. However, $\bar{M}_n(a)$ and $\bar{M}_n(b)$ are not martingales anymore. In particular, $\bar{Y}^+_\infty(a)$ or $\bar{Y}^{-}_\infty(b)$ can be infinite whereas $\bar{Y}^-_\infty(a)$ and $\bar{Y}^{+}_\infty(b)$ are, by construction, always equal to $0$. We can construct the VRRW $\bar{X}$ restricted to the interval ${\left[\kern-0.15em\left[}a,b{\right]\kern-0.15em\right]}$ using the same time-line construction used for $X$, choosing again the random variables $\xi_k^\pm(x)$ independent and exponentially distributed except for the two boundary r.v. $\xi_0^-(a)$ and $\xi_0^+(b)$ which are now chosen equal to $\infty$ (this prevent the walk from ever jumping from $a$ to $a-1$ or from $b$ to $b+1$). Let us note that, this construction depends only upon $(\xi^\pm_n(x), n\geq 0, x\in {\left]\kern-0.15em\left]}a,b{\right[\kern-0.15em\right[})$. Let $\bar{X}'$ denote another VRRW restricted to ${\left[\kern-0.15em\left[}a,b'{\right]\kern-0.15em\right]}\supset {\left[\kern-0.15em\left[}a,b{\right]\kern-0.15em\right]}$ for some $b'\geq b$. Using the time-line construction for $\bar{X}$ and $\bar{X}'$ with the same random variables $\xi_k^\pm(x)$, except for $\xi_0^+(b)$, we directly deduce from Lemma \[tarres1\] a monotonicity result between the local time processes of $X$ and $X'$ : \[couplrest\] Assume that $w$ is non-decreasing. Fix $z\in {\left[\kern-0.15em\left[}a,b{\right]\kern-0.15em\right]}$ and $k\ge 1$, let $\sigma,\sigma'$ be the times when $\bar{X},\bar{X}'$ visit $z$ for the $k$-th times. On the event $\{\sigma <\infty \hbox{ and } \sigma'<\infty\}$, we have, a.s., $$\begin{aligned} \bar{Z}_\sigma(z+1) \le \bar{Z}'_{\sigma'}(z+1) \quad &\textrm{and}& \quad \bar{Z}_\sigma(z-1) \ge \bar{Z}'_{\sigma'}(z-1)\\ \bar{N}_\sigma(z,z+1) \le \bar{N}'_{\sigma'}(z,z+1) \quad &\textrm{and}& \quad \bar{N}_\sigma(z,z-1) \ge \bar{N}'_{\sigma'}(z,z-1),\end{aligned}$$ where $N$ and $N'$ are defined as in Lemma \[tarres1\] for $\bar{X}$ and $\bar{X}'$. Moreover, if we denote by $\theta^{\pm}$ (resp. ${\theta'}^{\pm}$) the time when $\bar{X}$ (resp. $\bar{X}'$) jump for the $k$-th time from $z$ to $z\pm1$, then, on the event of these quantities being finite, we have, a.s., $$\bar{Y}^{+}_{\theta^{+}}(z) \ge \bar{Y}'^{+}_{{\theta'}^{+}}(z)\qquad\hbox{and}\qquad \bar{Y}^{-}_{\theta^{-}}(z) \le \bar{Y}'^{-}_{{\theta'}^{-}}(z).$$ We conclude this section with a simple lemma we will repeatedly invoke to reduce the study of the localization properties of the VRRW $X$ on ${\mathbb{Z}}$ to those of the VRRW $\bar{X}$ restricted to a finite set. \[lemrest2\] For $N>0$, let $\bar{X}$ be a VRRW on ${\left[\kern-0.15em\left[}0,N{\right]\kern-0.15em\right]}$ and define the events $$\begin{aligned} \mathcal{E}&=&\{\bar{Y}^+_\infty(0)<\infty\}\cap\{\bar{Y}^{-}_\infty(N)<\infty\},\\ \mathcal{E}' &=& \mathcal{E} \cap \{\hbox{$\bar{X}$ visits all sites of ${\left[\kern-0.15em\left[}0,N{\right]\kern-0.15em\right]}$ i.o.}\}.\end{aligned}$$ - If $\mathcal{E}$ (resp. $\mathcal{E}'$) has positive probability, then the VRRW on ${\mathbb{Z}}$ has positive probability to localize on a subset of length at most (resp. equal to) $N+1$. - Reciprocally, if the VRRW on ${\mathbb{Z}}$ has positive probability to localize on a subset of length at most (resp. exactly) $N+1$, then there exists some initial local time configuration $\mathcal{C}$ such that, for the $\mathcal{C}$-VRRW on ${\left[\kern-0.15em\left[}0,N{\right]\kern-0.15em\right]}$, the event $\mathcal{E}$ (resp. $\mathcal{E}'$) has positive probability. Define a sequence $(\chi_n)_{n\ge 0}$ of random variables which are, conditionally on $\bar{X}$, independent with law $${\mathbb{P}}\{\chi_n=1\ |\ \bar{X}\}=1-{\mathbb{P}}\{\chi_n=0\ |\ \bar{X} \}=\frac{w(0)1_{\{\bar{X}_n=0\}}}{w(0)+w(\bar{Z}_n(1))}+\frac{w(0)1_{\{\bar{X}_n=N\}}}{w(0)+w(\bar{Z}_n(N-1))}.$$ The Borel-Cantelli Lemma applied to the sequence $(\chi_{n})$ yields $$\label{incl} \mathcal{E}\subset\left\{\sum \chi_n<\infty\right\}.$$ Let us also note that if ${\mathbb{P}}\{\sum \chi_n<\infty\}>0$, then necessarily ${\mathbb{P}}\{\sum \chi_n=0\}>0$ since we just need to change a finite number of $\chi_n$. Moreover, it is clear that we can construct a VRRW $X$ on ${\mathbb{Z}}$ and a reflected random walk $\bar{X}$ on ${\left[\kern-0.15em\left[}0, N{\right]\kern-0.15em\right]}$ on the same probability space in such way that $X$ and $\bar{X}$ coincide on the event $\{\sum \chi_n=0\}$. Therefore, if ${\mathbb{P}}\{\mathcal{E}\}>0$, it follows from that the VRRW $X$ on ${\mathbb{Z}}$ localizes on ${\left[\kern-0.15em\left[}0,N{\right]\kern-0.15em\right]}$ with positive probability. Moreover, if ${\mathbb{P}}\{\mathcal{E}'\}>0$, we find that $${\mathbb{P}}\left\{\{\sum \chi_n=0\}\cap\{\hbox{$\bar{X}$ visits all sites of ${\left[\kern-0.15em\left[}0,N{\right]\kern-0.15em\right]}$ i.o.}\}\right\} >0$$ which implies that, with positive probability, $X$ visits every site of ${\left[\kern-0.15em\left[}0,N{\right]\kern-0.15em\right]}$ i.o. without ever exiting the interval. Reciprocally, if the VRRW on ${\mathbb{Z}}$ has positive probability to localize on some interval ${\left[\kern-0.15em\left[}x,x+N{\right]\kern-0.15em\right]}$, then, clearly, there exists some initial local time configuration $\mathcal{C}$ on ${\mathbb{Z}}$ such that the $\mathcal{C}$-VRRW on ${\mathbb{Z}}$ has positive probability never to exit the interval ${\left[\kern-0.15em\left[}0,N{\right]\kern-0.15em\right]}$. On this event, the $\mathcal{C}$-VRRW on ${\mathbb{Z}}$ and the restricted $\mathcal{C}$-VRRW on ${\left[\kern-0.15em\left[}0,N{\right]\kern-0.15em\right]}$ coincide. We conclude the proof using Lemma \[Yfini2\] which implies that, on this event, $Y^+_\infty(0)$ and $Y^{-}_\infty(N)$ are both finite. Proof of Theorem \[4probapositive\] {#section4} =================================== We split the proof of the theorem into several propositions. We start with two elementary observations: \[proptwosite\] Let $w$ be a weight sequence. - If $\sum 1/w(k) = \infty$, then we have, a.s., $|R'|\neq 2$. - Conversely, if the sum above is finite and the weight sequence $w$ is non-decreasing, then, a.s., $|R'|= 2$. Assume that $\sum 1/w(k) = \infty$ and consider the reflected VRRW $\bar{X}$ on ${\left[\kern-0.15em\left[}0,1{\right]\kern-0.15em\right]}$ starting from some initial configuration $\mathcal{C}$. We have $$\bar{Y}_\infty^+(0)=\sum_{k\ge 0} \frac{1_{\{\bar{X}_{k+1}=1\}}}{w(\bar{Z}_k(1))}=\sum_{i=\bar{Z}_0(1)}^\infty \frac{1}{w(i)}=\infty.$$ Thus, Lemma \[lemrest2\] implies that ${\mathbb{P}}\{|R'| = 2\}=0$. Reciprocally, if $w$ is non-decreasing and $\sum 1/w(k) < \infty$, then $\sum 1/w(k)^2 <\infty$ and we can invoke Lemma \[Yfini\] to conclude. \[R’3\] For any weight sequence $w$, we have, a.s., $|R'|\neq 3$. Consider the reflected VRRW $\bar{X}$ on ${\left[\kern-0.15em\left[}0,2{\right]\kern-0.15em\right]}$ starting from some initial configuration $\mathcal{C}$. We distinguish two cases: - if $\sum 1/w(n)<\infty$, then Rubin’s construction at site 1 implies that either $0$ or $2$ is visited only finitely many times (notice that we do not require here $w$ to be monotonic). - if $\sum 1/w(n)=\infty$, then we have $$\bar{Y}_\infty^+(0)+\bar{Y}_\infty^-(2)=\sum_{k\ge 0} \frac{1_{\{\bar{X}_{k+1}=1\}}}{w(\bar{Z}_k(1))}=\sum_{i=\bar{Z}_0(1)}^\infty \frac{1}{w(i)}=\infty.$$ Thus, in both cases, Lemma \[lemrest2\] implies that ${\mathbb{P}}\{|R'|= 3\}=0$. *Let us note that the result above does not hold for edge-reinforced random walks: if, for instance, $w(n)=1$ when $n$ is even and $w(n)=n^2$ when $n$ is odd, then $R'=\{-1,0,1\}$ a.s., see Sellke [@Sel].* In the rest of the paper, given two sequences $(u_n)_{n\ge 1}$ and $(v_n)_{n\ge 1}$, we shall use Tarrès’s notation [@T1; @T2] and write $u_n\equiv v_n$, when $(u_n-v_n)_{n\ge 1}$ is a converging sequence. \[lem4\_0\] Assume that $w$ is non-decreasing. Then $$\begin{aligned} |R'|=4 \textrm{ with positive probability} \quad \Longrightarrow \quad \sum_{n\ge 0} \frac 1 {w(n)^2} < \infty.\end{aligned}$$ Assume that $\sum 1/w(n)^2=\infty$. Let $\bar{X}$ be a VRRW on ${\left[\kern-0.15em\left[}0,3{\right]\kern-0.15em\right]}$ starting from some initial configuration $\mathcal{C}$. Recall that (\[eqW\]) states that $$\begin{aligned} \nonumber W(\bar{Z}_{n}(2))-W(\bar{Z}_{n}(0))&=&\bar{Y}_n^-(3)-\bar{Y}_n^+(-1)+\bar{M}_{n}(1)+ c\\ \label{eqw2}& = & \bar{Y}_n^-(3) +\bar{M}_{n}(1) + c,\end{aligned}$$ where $c$ is some constant depending on the initial configuration $\mathcal{C}$. Assume now that $\bar{Y}_\infty^-(3)$ is finite and let us prove that necessarily $\bar{Y}_\infty^+(0) = \infty$. Equation becomes $$\label{eqW3} W(\bar{Z}_{n}(2))-W(\bar{Z}_{n}(0))\equiv \bar{M}_{n}(1).$$ According to Theorem $2.14$ of [@HH], either $\bar{M}_n(1)$ converges or $\limsup \bar{M}_n(1) =-\liminf \bar{M}_n(1) =\infty$. On one hand, remark that $$\sum_{n\ge 0}(\bar{M}_{n+1}(1)-\bar{M}_n(1))^2\ge \sum_{n\ge 0}\frac{1_{\{\bar{X}_n=0\}}}{w(\bar{Z}_n(0))^2}.$$ Hence, on the event $\{\bar{Z}_\infty(0)=\infty\}$, we have $\limsup \bar{M}_n =-\liminf \bar{M}_n =\infty$. On the other hand, by periodicity, $\bar{Z}_\infty(2) \vee \bar{Z}_\infty(0) = \infty$. Recalling that $\lim_{x\rightarrow \infty} W(x)=\infty$, we deduce, using that $\{\bar{M}_n(1)\hbox{ converges}\}\subset \{\bar{Z}_\infty(2)=\infty\}\cap \{\bar{Z}_\infty(0)=\infty\}$. Therefore, a.s., $$\liminf_n \bar{M}_n(1) = -\infty \quad\hbox{and}\quad \limsup_n \bar{M}_n(1) = +\infty.$$ In particular, there exists a.s. arbitrarily large integers $n$, such that $W(\bar{Z}_n(0)) \ge W(\bar{Z}_n(2))+1$. Pick such an $n$ and let $m$ be the largest integer smaller than $n$ such that $W(\bar{Z}_m(0)) \le W(\bar{Z}_m(2))$. For $k\in (m,n]$, we have $\bar{Z}_k(0)\ge \bar{Z}_k(2)$ hence $$\bar{Z}_k(1)\le \bar{Z}_k(0)+\bar{Z}_k(2)+\bar{Z}_0(1)\le 3\bar{Z}_k(0),$$ assuming that $n$ is large enough. Since $w$ is non-decreasing, we get $$\begin{aligned} \sum_{k\in (m,n]} \frac{1_{\{\bar{X}_k=0\}}}{w(\bar{Z}_k(1))} \ge \sum_{k\in (m,n]} \frac{1_{\{\bar{X}_k=0\}}}{w(3\bar{Z}_k(0))} & = & \sum_{i = \bar{Z}_{m}(0)+1}^{\bar{Z}_{n}(0)}\frac{1}{w(3i)}\\ &\ge & \frac{1}{3} \Big\{W(\bar{Z}_n(0))-W(\bar{Z}_m(0)+1)\Big\}\ge \frac{1}{4}.\end{aligned}$$ As this holds for infinitely many $n$, we deduce that, a.s. $$\bar{Y}_\infty^{+}(0)=\sum_k \frac{1_{\{\bar{X}_k=0\}}}{w(\bar{Z}_k(1))} = \infty.$$ We conclude by using Lemma \[lemrest2\]. Let us note that Lemma \[lem4\_0\] together with Lemma \[Halphaw\] of the appendix imply that, for any non-decreasing weight sequence $w$, we have $${\mathbb{P}}\{|R'| = 4\} >0 \hbox{ or } \alpha_c(w) < \infty \ \Longrightarrow \ \sum_{n}\frac{1}{w(n)^2} < \infty.$$ Hence, when proving Theorem \[4probapositive\], we can assume, without loss of generality, that $\sum_n 1/w(n)^2 < \infty$. In particular, the martingales introduced in Section \[section3.2\] converge a.s. and in $L^2$. \[prop4pos\] Assume that $w$ is non-decreasing and that holds. We have $$|R'|=4 \textrm{ with positive probability}\ \Longleftrightarrow \ \alpha_c(w)<\infty.\\$$ Let us first suppose that $|R'|= 4$ with positive probability. Thus, according to Lemma \[lemrest2\], there exists some initial local time configuration $\mathcal{C}$ such that, for the $\mathcal{C}$-VRRW $\bar{X}$ on ${\left[\kern-0.15em\left[}0,3{\right]\kern-0.15em\right]}$, the event $\mathcal{E}:=\{\bar{Y}^+_\infty(0)<\infty\}\cap\{\bar{Y}^{-}_\infty(3)<\infty\}$ has positive probability. Using , we find that $$\begin{aligned} W(\bar{Z}_{n}(2))-W(\bar{Z}_{n}(0))=\bar{Y}_n^-(3)-\bar{Y}_n^+(-1)+\bar{M}_{n}(1)+C\\ W(\bar{Z}_{n}(3))-W(\bar{Z}_{n}(1))=\bar{Y}_n^-(4)-\bar{Y}_n^+(0)+\bar{M}_{n}(2)+C'.\end{aligned}$$ As we already noticed, we can assume without loss of generality that $\sum 1/w(n)^2 < \infty$ so the martingales $\bar{M}_n(1)$ and $\bar{M}_n(2)$ converge. Hence, there exist finite random variables $\alpha,\beta$, such that, on the event $\mathcal{E}$, $$\begin{aligned} \label{eqalphabeta} W(\bar{Z}_n(1)) - W(\bar{Z}_n(3)) = \alpha + o(1),\\ \notag W(\bar{Z}_n(2)) - W(\bar{Z}_n(0)) = \beta + o(1).\end{aligned}$$ For $n$ large enough, this yields $$\max(\bar{Z}_n(1),\bar{Z}_n(2))\le W^{-1}(W(\max(\bar{Z}_n(0),\bar{Z}_n(3)))+\gamma)$$ with $\gamma:=|\alpha|+|\beta|+1$. Hence, we have $$\begin{aligned} \bar{Y}^+_\infty(0)+\bar{Y}^{-}_\infty(3)&=&\sum_{k\ge 0} \left(\frac {1_{\{\bar{X}_k=0\}}}{w(\bar{Z}_{k}(1))} + \frac {1_{\{\bar{X}_k=3\}}}{w(\bar{Z}_{k}(2))}\right)\\ &\ge & \sum_{k\ge 0} \frac {1_{\{\bar{X}_k\in \{0,3\}\}}}{w(\max(\bar{Z}_{k}(1),\bar{Z}_{k}(2)))} \\ &\ge & c\sum_{k\ge 0} \frac {1_{\{\bar{X}_k\in \{0,3\}\}}}{w(W^{-1}(W(\max(\bar{Z}_k(0),\bar{Z}_k(3)))+\gamma))}\\ &\ge & c \sum_{k\ge 0} \frac {1}{w(W^{-1}(W(k)+\gamma))}.\end{aligned}$$ Therefore, on the event $\mathcal{E}$, we have $I_{\gamma}(w)<\infty$. This shows that $\alpha_c(w)<\infty$. We now prove the converse implication. Let us assume that $I_\delta(w)<\infty$ for some $\delta>0$. In particular, $\sum_n 1/w(n)^2<\infty$ (*c.f.* Lemma \[Halphaw\]). In view of Lemma \[lemrest2\], we will show that, for the reflected VRRW $\bar{X}$ on ${\left[\kern-0.15em\left[}0,3 {\right]\kern-0.15em\right]}$, the event $\{\bar{Y}^+_\infty(0)<\infty\}\cap\{\bar{Y}^{-}_\infty(3)<\infty\}$ has positive probability. This will insure that the VRRW on ${\mathbb{Z}}$ localizes with positive probability on a subset of size less or equal to $4$ which will complete the proof of the proposition since localization on $2$ or $3$ sites is not possible with our assumptions on $w$. We use the time-line construction. As explained in the previous section we can construct $\bar{X}$ from a sequence $(\xi_n^\pm(y),\, n\ge 0,\, y\in\{1,2\})$ of independent exponential random variables with mean $1$. Observe that the sequences $(\xi_n^\pm(1)/w(n),n\ge 0)$ define a $w$-urn process via Rubin’s construction (choosing $+$ for the red balls). Let $\hat{M}_\infty(1)$ denote the limit of this urn defined as in Proposition \[propurne\]. Then, we have $\hat{M}_\infty(1)\ge \delta+1$ with positive probability. Recall the definition of $Y^R$ given in Corollary \[Yurn\] and note that, on the event $\{\hat{M}_\infty(1)\ge \delta+1\}$, the random variable $Y^R$ is finite. Besides, using Lemma \[couplrest\] to compare $\bar{X}$ with the walk restricted on ${\left[\kern-0.15em\left[}0,2{\right]\kern-0.15em\right]}$ (which correspond to the urn process above), we get $$\begin{aligned} \bar{Y}^+_\infty(0)\le Y^R<\infty \quad \mbox{on the event } \{\hat{M}_\infty(1)\ge \delta+1\}.\end{aligned}$$ By symmetry, considering the limit $\hat{M}_\infty(2)$ of the urn process $(\xi_n^\pm(2)/w(n),n\ge 0)$, we also find that $$\begin{aligned} \bar{Y}^-_\infty(3) < \infty \quad \mbox{on the event } \{\hat{M}_\infty(2)\leq -\delta-1\}.\end{aligned}$$ The random variables $\hat{M}_\infty(1)$ and $\hat{M}_\infty(2)$ being independent, we conclude that $\{\bar{Y}^+_\infty(0)<\infty\}\cap\{\bar{Y}^{-}_\infty(3)<\infty\}$ has positive probability. \[prop5ex\] Assume that $w$ is non-decreasing and that holds. We have $$\alpha_c(w)\in (0,\infty) \ \Longrightarrow \ |R'|=5 \textrm{ with positive probability}.\\$$ Assume that $\alpha_c(w)\in (0,\infty)$. In particular, we have $\sum_{n}1/w(n)^2 < \infty$. Since the walks associated with a weight $w$ and any non-zero multiple of $w$ have the same law, we will assume without loss of generality that $w(0)\ge 1$. Let $\bar{X}$ denote the VRRW reflected on ${\left[\kern-0.15em\left[}0,4{\right]\kern-0.15em\right]}$. Let us prove that, with positive probability, $\bar{Y}^+_\infty(0)$ and $\bar{Y}^-_\infty(4)$ are both finite and $\bar{X}$ visits all sites of ${\left[\kern-0.15em\left[}0,4 {\right]\kern-0.15em\right]}$ infinitely often. We use again the time-line representation explained in Section \[section3\] except that we will change the construction slightly for the transition at site $2$. Recall that according to the original construction, when the process jumps for the $k$-th time from $1$ (resp. $3$) to $2$ and has made $m$ visits to $1$ (resp. $3$) before time $t$, then we attached to the oriented edge $(2,1)$ (resp. $(2,3)$) a clock which rings after time $\xi_k^-(2)/w(m)$ (resp. $\xi_k^+(2)/w(m)$). In our new construction, we choose to attach instead a clock which rings after time $\xi_m^-(2)/w(m)$ (resp. $\xi_m^+(2)/w(m)$). The random variables $(\xi_k^\pm(2),\, k\geq 0)$ being i.i.d, this modification does not change the law of $\bar{X}$ (some random variables $\xi_k^\pm(2)$ are simply never used). Fix some $0<\varepsilon < 1$ and consider the two $w$-urn processes $u_1:=(\xi_n^\pm(1)/w(n),n\ge 0)$, and $u_3:=(\xi_n^\pm(3)/w(n),n\ge 0)$. Since $\bar{Y}^+_\infty(0)$ is stochastically smaller than it would be for the process reflected in ${\left[\kern-0.15em\left[}0,2{\right]\kern-0.15em\right]}$, using similar arguments as in the proof of Proposition \[prop4pos\], we see that there exists a set $E_1\subset ({\mathbb{R}}_+^2)^{\mathbb{N}}$, such that the event $\mathcal{E}_1:=\{u_1\in E_1\}$ has positive probability and on which $\bar{Y}^+_\infty(0)\le \varepsilon^3$. By symmetry, there exists a set $E_2$, such that $\mathcal{E}_2:=\{u_3\in E_2\}$ has positive probability and on which $\bar{Y}^-_\infty(4)\le \varepsilon^3$. By independence of the urns $u_1$ and $u_3$, the event $\mathcal{E}_1\cap \mathcal{E}_2$ also has positive probability. In view of Lemma \[lemrest2\], it remains to prove that, on this event, $\bar{X}$ visits all the sites of ${\left[\kern-0.15em\left[}0,4 {\right]\kern-0.15em\right]}$ infinitely often with positive probability. We now consider the urn process $u_2:=(\xi_n^\pm(2)/w(n),n\ge 0)$. Recall that, according to , we may express the limit $\hat{M}_\infty(2)$ of this urn in the form: $$\label{eqs1} \hat{M}_\infty(2)= \sum_{n \ge 0} \left( \frac {1-\xi_n^+(2)}{w(n)} \right) - \sum_{n\ge 0} \left(\frac{1-\xi_n^-(2)}{w(n)}\right).$$ Similarly, it is not difficult to check that we can also express the limit of the martingale $\bar{M}_\infty(2) := \lim_{n\to \infty} (\bar{Y}^+_n(2)-\bar{Y}^-_n(2))$ in the form $$\label{eqs2} \bar{M}_\infty(2)= \sum_{n \ge 0} \left( \frac {1-\xi_{c_n}^+(2)}{w(c_n)} \right) - \sum_{n\ge 0} \left(\frac{1-\xi_{d_n}^-(2)}{w(d_n)}\right).$$ where $(c_n,n\ge 0)$ and $(d_n,n\ge 0)$ are the increasing (random) sequences such that $\bar{Y}^+_n(2)= \sum_{c_k \leq n } 1/w(c_k)$ and $\bar{Y}^-_{n}(2)=\sum_{d_k\leq n} 1/w(d_k)$. The idea now is to compare $\bar{M}_\infty(2)$ and $\hat{M}_\infty(2)$ and prove that, on the event $\mathcal{E}_1\cap \mathcal{E}_2$ their values are close. Then we will use the fact that $\hat{M}_\infty(2)$ has a density to deduce that $\bar{M}_\infty(2)$ can be smaller than $\alpha_c(w)$. Subtracting from , we find that $$\hat{M}_\infty(2) - \bar{M}_\infty(2) = \sum_{n\ge 0} \left(\frac {1-\xi_{i_n}^+(2)}{w(i_n)}\right) + \sum_{n\ge 0} \left(\frac{1-\xi_{j_n}^-(2)}{w(j_n)}\right),$$ where $(i_n,n\ge 0)$ and $(j_n,n\ge 0)$ are the complementary sequences of $(c_n,n\ge 0)$ and $(d_n,n\ge 0)$. Moreover, using relation , we have $$\bar{Y}^+_\infty(0) = \sum_n \frac{1}{w(j_n)} \quad\hbox{ and }\quad \bar{Y}^-_\infty(4) = \sum_n \frac{1}{w(i_n)}.$$ But, using similar arguments as in the proof of , we obtain $$\begin{gathered} {\mathbb{E}}\left[\left(\sum_{n\ge 0} \frac {1-\xi_{i_n}^+(2)}{w(i_n)}\right)^2\ \Big|\ \mathcal{E}_1\cap \mathcal{E}_2\right] = {\mathbb{E}}\left[ \sum_{n\ge 0} \frac 1 {w(i_n)^2}\ \Big|\ \mathcal{E}_1\cap \mathcal{E}_2\right] \\ \le \frac{1}{w(0)}{\mathbb{E}}\left[\sum_{n\ge 0} \frac 1 {w(i_n)}\ \Big|\ \mathcal{E}_1\cap \mathcal{E}_2\right] \le {\mathbb{E}}[\bar{Y}_\infty^-(4)\mid \mathcal{E}_1\cap \mathcal{E}_2] \le \varepsilon^3.\end{gathered}$$ Using Tchebychev’s inequality, we deduce $${\mathbb{P}}\left\{|\hat{M}_\infty(2)-\bar{M}_\infty(2)|\ge 2\varepsilon\mid \mathcal{E}_1\cap \mathcal{E}_2 \right\} \le \varepsilon.$$ Recalling that $\hat{M}_\infty(2)$ has a density with support on the whole of ${\mathbb{R}}$ (*c.f.* Proposition \[propurne\]), we can pick $\eta>0$ such that ${\mathbb{P}}\{|\hat{M}_\infty(2)|\le \eta\} = 2\varepsilon$. This yields $${\mathbb{P}}\Big\{|\bar{M}_\infty(2)|\ge \eta + 2\varepsilon \mid \mathcal{E}_1\cap \mathcal{E}_2\Big\} \le 1-2\varepsilon + \varepsilon \le 1-\varepsilon,$$ so the set $\mathcal{E}_3:=\mathcal{E}_1\cap \mathcal{E}_2\cap \{|\bar{M}_\infty(2))| \le \eta + 2\epsilon\}$ has positive probability. Moreover, we have, $$W(\bar{Z}_n(1))-W(\bar{Z}_n(3))= \bar{Y}^+_n(0) - \bar{Y}^-_n(4) -\bar{M}_n(2),$$ and therefore, for all $n$ large enough, on $\mathcal{E}_3$, $$\bar{Z}_n(2) \le \bar{Z}_n(1) + \bar{Z}_n(3) \le \bar{Z}_n(1) + W^{-1}(W(\bar{Z}_n(1))+\eta +4\varepsilon) .$$ Choosing $\varepsilon$ small enough such that $\delta:=\eta +4\varepsilon <\alpha_c(w)$, we obtain, for $N$ large $$\begin{aligned} \bar{Y}^+_\infty(1)&\ge& \sum_{n\ge N} \frac{1_{\{\bar{X}_n=1,\bar{X}_{n+1}=2\}}}{w(\bar{Z}_n(1) + W^{-1}(W(\bar{Z}_n(1))+\delta))}\\ & \ge & \sum_{n\ge N} \frac{1}{w(n + W^{-1}(W(n)+\delta))}-\sum_{n\ge N} \frac{1_{\{\bar{X}_n=1,\bar{X}_{n+1}=0\}}}{w(\bar{Z}_n(1))}\\ & \ge & \sum_{n\ge N} \frac{1}{w(n + W^{-1}(W(n)+\delta))}-\bar{Y}^+_\infty(0).\\\end{aligned}$$ It follows from Lemma \[teclem\] of the Appendix that $\bar{Y}^+_\infty(1)$ is infinite on $\mathcal{E}_3$. By symmetry, we also have $\bar{Y}^-_\infty(3)=\infty$ on $\mathcal{E}_3$. Therefore, according to Lemma \[Yfini\], on $\mathcal{E}_3$, the VRRW on ${\left[\kern-0.15em\left[}0,4{\right]\kern-0.15em\right]}$ visits every site infinitely often. This concludes the proof of the proposition. \[Prop45as\] Assume that $w$ is non-decreasing and that holds. Then $$\begin{aligned} \alpha_c(w)<\infty \quad \Longrightarrow \quad |R'|\in \{4,5\} \textrm{ almost surely.}\end{aligned}$$ We first argue that, if $\alpha_c(w)<\infty$, then $R'$ is a.s. finite and non-empty. Indeed, recalling Lemma \[couplrest\], each time $X$ visits a new site, say $x>0$, as long as it does not visit $x+2$, the restriction of $X$ to the set $\{x-1,x,x+1\}$ can be coupled with a $w$-urn process in such a way that it always makes less jumps to $x+1$ than the urn process. Then, Corollary \[Yurn\] and Lemma \[Yfini\] insure that $X$ never visits $x+2$ with a positive probability uniformly bounded from below by a constant depending only on this urn process (and therefore which does not depend on the past trajectory of $X$ before its first visit to $x$). It follows that, a.s., $\limsup X_n < \infty$ and by symmetry $\liminf X_n > -\infty$. Hence the walk localizes on a finite set almost surely. Let us now assume, by contradiction, that $|R'|=N+1\ge 6$ with positive probability. Thus, according to Lemma \[lemrest2\], there exists some initial local time configuration $\mathcal{C}$ such that, for the $\mathcal{C}$-VRRW $\bar{X}$ on ${\left[\kern-0.15em\left[}0,N{\right]\kern-0.15em\right]}$, the event $$\mathcal{E}:=\{\bar{Y}_\infty^+(0)+\bar{Y}_\infty^-(N)<\infty\}\cap\{\bar{X} \mbox{ visits 0 and $N$ i.o.} \}$$ has positive probability. Moreover, using , we have $$\begin{aligned} W(\bar{Z}_{n}(2))-W(\bar{Z}_{n}(0))&\equiv &\bar{Y}_n^-(3)\\ W(\bar{Z}_{n}(3))-W(\bar{Z}_{n}(1))&\equiv &\bar{Y}_n^-(4)-\bar{Y}_n^+(0).\end{aligned}$$ On the event $\mathcal{E}$, the quantity $\bar{Y}_\infty^+(0)$ is finite whereas $\bar{Y}_\infty^-(3)$ and $\bar{Y}_\infty^-(4)$ are infinite according to Lemma \[Yfini\] since $N\geq 5$. Thus, for all $A>0$, the stopping time $$\begin{aligned} T_A:=\inf \left\{n \ge 0 \ : \ \begin{array}{l} \bar{X}_n=2\\ W( \bar{Z}_n(0)) \le W( \bar{Z}_n(2)) - A \\ W( \bar{Z}_n(1)) \le W( \bar{Z}_n(3)) - A \end{array} \right\}\end{aligned}$$ is finite on $\mathcal{E}$. We claim that $$\begin{aligned} \label{Yinftyfini} {\mathbb{P}}\{ \bar{Y}_\infty^+(1)=\infty\mid T_A<\infty \} \to 0 \quad \textrm{as }A\to \infty.\end{aligned}$$ For the time being, assume that holds. As before, Lemma \[Yfini\] states that, on $\mathcal{E}$, we have $\bar{Y}_\infty^+(1) = \infty $. Thus, for all $A>0$, we get $${\mathbb{P}}\{\mathcal{E}\}\le {\mathbb{P}}\left\{\{\bar{Y}_\infty^+(1)=\infty\} \cap \{T_A<\infty\}\right\}\le {\mathbb{P}}\{ \bar{Y}_\infty^+(1)=\infty\mid T_A<\infty \},$$ which yields ${\mathbb{P}}\{\mathcal{E}\}=0$ and contradicts the initial assumption that the walk localizes with positive probability on more than $5$ sites. It remains to prove . For $A>0$, consider a process $\bar{X}^A$ which is, up to time $T_A$, equal to the VRRW $\bar{X}$ on ${\left[\kern-0.15em\left[}0,N{\right]\kern-0.15em\right]}$ and which, after time $T_A$, has the transition of the VRRW restricted on ${\left[\kern-0.15em\left[}0,2{\right]\kern-0.15em\right]}$. In view of Lemma \[couplrest\], we can construct $\bar{X}^A$ together with $\bar{X}$ in such way that, with obvious notation, $$\bar{Y}_\infty^+(0)\le \bar{Y}^{A,+}_\infty(0) \quad \mbox{ and }\quad \bar{Y}_\infty^+(1)\le \bar{Y}^{A,+}_\infty(1).$$ After time $T_A$, the process $\bar{X}^A$ is simply an urn process. Hence, using the same arguments as in the proof of Proposition \[propurne\], for $n\geq T_A$, the process $$\hat{M}^A_n := W( \bar{Z}^A_n(0))- W( \bar{Z}^A_n(2))$$ is a martingale with quadratic variation bounded by $2\sum_n 1/w(n)^2$. Noticing that, by definition of $T_A$, we have $\hat{M}^A_{T_A}\le -A$, the maximal inequality for martingales shows that, for any $\varepsilon >0$, there exists a constant $C>0$, such that for all $A>0$, $$\begin{aligned} \label{TA} {\mathbb{P}}\left\{\sup_{n\ge T_A}\, W(\bar{Z}^A_n(0))- W( \bar{Z}^A_n(2))\ge -A+ C \;|\; \mathcal{F}_{T_A}\right\} \le \varepsilon.\end{aligned}$$ Moreover, for every odd integer $n \ge T_A$, we have $\bar{X}^A_n = 1$ from which we deduce that, for all $n \ge T_A$, $$\bar{Z}^A_n(1) \;\ge\; \bar{Z}^A_n(2)+\bar{Z}^A_n(0)+\bar{Z}^A_{T_A}(1)-\bar{Z}^A_{T_A}(0)-\bar{Z}^A_{T_A}(2)\;\ge\; \bar{Z}^A_n(2)-\bar{Z}^A_{T_A}(2).$$ On the event $\{\sup_{n\ge T_A}\, W(\bar{Z}^A_n(0))- W( \bar{Z}^A_n(2))< -A+ C\}$, we get, for $n\ge T_A$, $$\bar{Z}^A_n(1)\ge W^{-1}(W(\bar{Z}^A_n(0)+A-C) -\bar{Z}^A_{T_A}(2).$$ This yields $$\begin{aligned} \bar{Y}^{A,+}_\infty(0) &=&\bar{Y}^{A,+}_{T_A}(0) + \sum_{n > T_A} \frac{1_{\{\bar{X}_n^A = 0\}}}{w(\bar{Z}_n^A(1))}\\ &\leq&\bar{Y}^{A,+}_{T_A}(0) +\sum_{n\ge 0} \frac{1}{w(W^{-1}(W(n)+A-C)-\bar{Z}^A_{T_A}(2))}\end{aligned}$$ with the convention $w(x)=w(0)$ for $x\le 0$. Thus, according to Lemma \[lemaa\] of the appendix, for $A >\alpha_c(w)+ C$, we have $\bar{Y}^{A,+}_\infty(0)<\infty$ on the event $\{T_A<\infty\}\cap\{\sup_{n\ge T_A}\, W(\bar{Z}^A_n(0))- W( Z^A_n(2))< -A+ C\}$. Using , we obtain $$\begin{aligned} \label{eka} {\mathbb{P}}\left\{\bar{Y}^{A,+}_\infty(0)=\infty\,|\, T_A<\infty\right\}\le \varepsilon.\end{aligned}$$ We can now choose $A_0 > \alpha_c(w) + C$ and $K>0$ such that $${\mathbb{P}}\{\bar{Y}^{A_0,+}_\infty(0)\ge K\, | \, T_{A_0}<\infty \} \le 2\varepsilon.$$ Notice that for $A>A_0$, the random variable $\bar{Y}^{A,+}_\infty(0)$ is stochastically dominated by $\bar{Y}^{A_0,+}_\infty(0)$ (this is again a consequence of Lemma \[couplrest\] using the same time-line construction for $\bar{X}^A$ and $\bar{X}^{A_0}$). Moreover, by hypothesis, $${\mathbb{P}}\{T_A<\infty \}\ge {\mathbb{P}}\{\mathcal{E}\}:= c >0,$$ hence $$\label{Y((A))+} \forall A>A_0, \quad {\mathbb{P}}\{\bar{Y}^{A,+}_\infty(0)\ge K\, |\, T_{A}<\infty \} \le 2\varepsilon/c.$$ Finally, we consider a third process $\tilde{X}^A$ which coincides up to time $T_A$ with $\bar{X}$ and $\bar{X}^A$, and which, after time $T_A$, has the transition of the VRRW restricted on ${\left[\kern-0.15em\left[}0,3{\right]\kern-0.15em\right]}$. Again, we can construct these processes in such way that $\bar{Y}_\infty^{A,+}(0)$ stochastically dominates $\tilde{Y}_\infty^{A,+}(0)$. This domination implies, that also holds with $\tilde{Y}^{A,+}_\infty(0)$ in place of $\bar{Y}^{A,+}_\infty(0)$. Moreover, $$\tilde{M}^A_n(2):=W(\tilde{Z}^A_n(3))-W(\tilde{Z}^A_n(1))+\tilde{Y}^{A,+}_n(0) \qquad n\ge T_A,$$ is a martingale with bounded quadratic variation. As before, we deduce from the maximal inequality for martingales, that, for some constant $C'>0$ depending only on $\varepsilon$ and the weight function $w$, $${\mathbb{P}}\left\{\inf_{n\ge T_A}\, \tilde{M}^A_n(2)-\tilde{M}^A_{T_A}(2)\le -C'\ \Big| \ {\mathcal{F}}_{T_A}\right\} \le \varepsilon.$$ Using the facts that $\tilde{M}^A_{T_A}(2)\ge A$ and that $\tilde{Y}^{A,+}_n(0)\ge K$ with probability smaller than $2\varepsilon/c$ on the event $\{T_{A}<\infty \}$, we obtain $${\mathbb{P}}\left\{\inf_{n\ge T_A}\, W(\tilde{Z}^A_n(3))-W(\tilde{Z}^A_n(1))\le -C'-K+A\ \Big| \ T_A<\infty\right\} \le \varepsilon',$$ with $\varepsilon'=\varepsilon(1+2/c)$. We now fix $A$ large enough such that $A-C'-K>\alpha_c(w)$. Using the trivial relation $\tilde{Z}_n^{A}(2)\ge \tilde{Z}_n^{A}(3)-\tilde{Z}_{T_A}^{A}(3)$ for $n\ge T_A$, we deduce, in the same way as for the proof of , that $${\mathbb{P}}\left\{ \tilde{Y}^{A,+}_\infty(1)=\infty \ \Big|\ T_A<\infty \right\} \le \varepsilon'.$$ We conclude the proof of by noticing that $\tilde{Y}^{A,+}_\infty(1)$ stochastically dominates $\bar{Y}_\infty^{+}(1)$. \[sansatome\] Assume that $w$ is non-decreasing and that $\sum_{n} 1/w(n)^2 < \infty$. Fix $\infty\le a<0<b\le \infty$ and let $\bar{X}$ be a $\mathcal{C}$-VRRW on ${\left[\kern-0.15em\left[}a,b{\right]\kern-0.15em\right]}$ for some initial local time configuration $\mathcal{C}$. Set $$\delta:=\lim_{n\rightarrow\infty} W(\bar{Z}_n(1))-W(\bar{Z}_n(-1)) \quad \mbox{ when the limit exists.}$$ Then, for any $\delta_0\in {\mathbb{R}}$, we have $${\mathbb{P}}\big\{\{\delta \hbox{ exists and equals }\delta_0\} \cap \{\bar{Z}_\infty(0)=\infty\}\big\}=0.$$ Since the weights $w$ and $\lambda w$ (for $\lambda>0$) define the same VRRW, we assume, without loss of generality that $w(Z_0(1))=1$. Recalling the time-line construction described in Section \[section3\], we create the $\mathcal{C}$-VRRW $\bar{X}$ on ${\left[\kern-0.15em\left[}a,b{\right]\kern-0.15em\right]}$ from a collection $((\xi_n^\pm(y),n\ge 0),y\in {\left]\kern-0.15em\left]}a,b{\right[\kern-0.15em\right[})$. Set $${\mathcal{H}}:=\left((\xi_n^\pm(y),n\ge 0, y\neq 0),(\xi_n^-(0),n\ge 0),(\xi_n^+(0),n\ge 1)\right)\in {\mathbb{R}}_+^{\mathbb{N}}$$ and let $\mu$ denote the product measure on ${\mathbb{R}}_+^{\mathbb{N}}$ under which ${\mathcal{H}}$ is a collection of i.i.d. exponential random variables with mean $1$. Then, given ${\mathcal{H}}$ and some other variable $\xi_0^+(0)$, the pair $({\mathcal{H}},\xi_0^+(0))$ defines a process $\bar{X} = \bar{X}({\mathcal{H}},\xi_0^+(0))$ via the time-line construction which is a VRRW under the product probability ${\mathbb{P}}:=\mu\times\hbox{Exp}(1)$. For $u>0$, define $$\mathcal{B}_{u}=\{{\mathcal{H}}\in {\mathbb{R}}_+^{\mathbb{N}},\; \delta({\mathcal{H}},u) \hbox{ exists and equals }\delta_0 \hbox{ and } \bar{Z}_\infty(0)=\infty\}$$ and $$\mathcal{B} =\{({\mathcal{H}},u) \in {\mathbb{R}}_+^{\mathbb{N}}\times{\mathbb{R}}_+,\; {\mathcal{H}}\in \mathcal{B}_u\}.$$ We will prove that, for almost every $u>0$ and $h>0$, $$\label{intersecB} \mu\{\mathcal{B}_u\cap \mathcal{B}_{u+h}\}=0.$$ This equation implies that $\{u\in {\mathbb{R}}^+,\mu\{\mathcal{B}_u\}>0\}$ has zero Lebesgue measure. Hence $${\mathbb{P}}\{\{\delta=\delta_0\}\cap\{\bar{Z}_\infty(0)=\infty\}\}= {\mathbb{P}}\{\mathcal{B}\} =\int_0^\infty e^{-u}\mu\{\mathcal{B}_u\}du=0.$$ It remains to prove . Given $({\mathcal{H}},u)$ such that $\bar{Z}_\infty(1)= Z_\infty(-1)=\infty$, we can define, as in the proof of Lemma \[Yfini\], the increasing sequences $(i_k^\pm,k\ge 0)$ such that, for all $n\ge 0$, $$Y_n^\pm(0)= \sum_{i_k^\pm\le n} \frac{1}{w(i_k^\pm)}.$$ For $t>0$, define also $$\label{defzip} z^\pm_t:=\inf\left\{n\ :\ \sum_{i_k^\pm \le n} \frac {\xi_k^\pm(0)}{w(i_k^\pm)}>t\right\}.$$ Thus, $z^\pm_t$ represents the local time at site $\pm 1$ when the clock process attached to site $0$ has consumed a time $t$. Hence, another way to define $z^\pm_t$ is to consider the continuous-time process $(\tilde{X}(s),s\ge 0)$ associated with $({\mathcal{H}},u)$ via the time-line construction (recall that $\bar{X}$ is deduced from $\tilde{X}$ by a change of time). Defining $$\tau_t:=\inf\Big\{s>0, \int_{0}^s1_{\{\tilde{X}_s=0\}}ds > t\Big\},$$ we get that $z^\pm_t=\tilde{Z}_{\tau_t}(\pm 1)$. Let us notice that, on $\mathcal{B}$, ${\mathbb{P}}$-a.s., we have $Z_\infty(-1)=Z_\infty(1)=\infty$ so the sequences $(i_k^{\pm})$ are well defined for all $k\ge 0$. Moreover, on the event $\{Z_\infty(-1)=Z_\infty(1)=\infty\}$, the total time consumed by the clock process at site $0$ is infinite ${\mathbb{P}}$-a.s. (see the proof of Lemma \[Yfini\]). Hence, on $\mathcal{B}$, the random variables $z^\pm_t$ are finite for all $t>0$, ${\mathbb{P}}$-a.s. Define $$\delta_{t} := W(z^+_t)-W(z^-_t)=W(\tilde{Z}_{\tau_t}(1))-W(\tilde{Z}_{\tau_t}(-1)).$$ By definition of $\delta$, we get $$\lim_{t \rightarrow \infty} \delta_t=\delta.\qquad {\mathbb{P}}\mbox{-a.s. on $\mathcal{B}$}.$$ Thus, for almost any $u>0$ (with respect to the Lebesgue measure), we have $$\label{dtmuas} \lim_{t \rightarrow \infty} \delta_t({\mathcal{H}},u)=\delta({\mathcal{H}},u)\qquad \mbox{for $\mu$-a.e. ${\mathcal{H}}\in\mathcal{B}_u$}.$$ Now let $u, h > 0$ be fixed and such that holds for $u$ and $u+h$. Pick ${\mathcal{H}}\in \mathcal{B}_u\cap\mathcal{B}_{u+h}$. Lemma \[tarres1\] implies that, for all $k\ge 0$, we have $$i_k^+({\mathcal{H}},u+h)\le i_k^+({\mathcal{H}},u) \quad\hbox{ and }\quad i_k^-({\mathcal{H}},u+h)\ge i_k^-({\mathcal{H}},u).$$ Recalling that $w(\bar{Z}_0(1))=w(i_0^+)=1$, we deduce from that, for $t>0$, $$z^+_t({\mathcal{H}},u+h)\le z^+_{t-h}({\mathcal{H}},u) \quad\hbox{ and }\quad z^-_t({\mathcal{H}},u+h)\ge z^-_t({\mathcal{H}},u).$$ This yields $$\begin{aligned} \delta_t({\mathcal{H}},u)-\delta_t({\mathcal{H}},u+h)&\ge& W(z^+_{t}({\mathcal{H}},u))- W(z^+_{t-h}({\mathcal{H}},u))\\ & \ge & \sum_{z^+_{t-h}\le k< z^+_{t}}\ \frac{1}{w(k)} \\ & \ge & \sum_{z^+_{t-h}\le i_k^+< z^+_{t}}\ \frac{1}{w(i_k^+)} := \Delta_{u,h}^+(t),\end{aligned}$$ where $z^+_{t-h}, z^+_{t}$ and $i_k^+$ stand for $z^+_{t-h}({\mathcal{H}},u), z^+_{t}({\mathcal{H}},u)$ and $i_k^+({\mathcal{H}},u)$. In view of , we deduce that, for almost every $u,h>0$, we have $$\mu\{\mathcal{B}_u\cap \mathcal{B}_{u+h}\} \le \mu\big\{\mathcal{B}_u\cap\{\limsup_{t\to\infty}\Delta_{u,h}^+(t)=0 \}\big\}.$$ It remains to prove that the r.h.s. in the previous inequality is equal to zero. For ${\mathcal{H}}\in\mathcal{B}_u$, the quantity $$h_t^*({\mathcal{H}},u) := \sum_{z^+_{t-h}\le i_k^+< z^+_t}\ \frac{\xi^+_k(0)}{w(i_k^+)}$$ is well defined. Moreover, it is clear that, $$\label{zr1} \lim_{t \rightarrow \infty} h^*_t=h\quad\hbox{ $\mu$-a.s. on $\mathcal{B}_u$.}$$ On the other hand, we have, $$|h^*_t-\Delta_{u,h}^+(t)|^21_{\mathcal{B}_u}\le 2 \left(\sum_{i_k^+\ge z^+_{t-h}} \!\!\!\!\frac{1-\xi^+_k(0)}{w(i_k^+)}1_{\{\theta_k<\infty\}}\right)^{\!\!\! 2}+ 2\left(\sum_{i_k^+\ge z^+_{t}}\!\!\!\! \frac{1-\xi^+_k(0)}{w(i_k^+)}1_{\{\theta_k<\infty\}}\right)^{\!\!\! 2},$$ where $\theta_k$ denotes the time of the $k$-th jump of $\bar{X}$ from $1$ to $0$. Let $(\tilde{\mathcal{F}}_t,t>0)$ denote the natural filtration of the continuous time process $\tilde{X}_t(\cdot,u)$. Using the same argument as in the proof of , we find that $$\begin{aligned} {\mathbb{E}}_\mu\left[(h^*_t-\Delta_{u,h}^+(t))^21_{\mathcal{B}_u}\ \Big| \ \tilde{\mathcal{F}}_{\tau_{t-h}}\right] &\le & 4{\mathbb{E}}_\mu\left[\sum_{i_k^+\ge z^+_{t-h}}\ \frac{1_{\{\theta_k<\infty\}}}{w(i_k^+)^2}\ \Big| \ \tilde{\mathcal{F}}_{\tau_{t-h}} \right] \\ &\le & \sum_{k\ge z^+_{t-h}} \frac 4 {w(k)^2}.\end{aligned}$$ This yields $$\mu\left\{|h^*_t-\Delta_{u,h}^+(t)|1_{\mathcal{B}_u}\ge h/2\ \Big| \ \ \tilde{\mathcal{F}}_{\tau_{t-h}}\right\} \le \frac{16}{h^2}\, \sum_{k\ge z^+_{t-h}} \frac 1 {w(k)^2}.$$ Hence, by monotone convergence, $$\label{zr2} \lim_{t\to\infty}\mu\left\{|h^*_t-\Delta_{u,h}^+(t)|1_{\mathcal{B}_u}\ge h/2\right\} = 0$$ Combining and , we conclude that $$\begin{gathered} \lim_{t\to\infty}\mu\left\{ \mathcal{B}_u\cap\{\Delta_{u,h}^+(t) \leq h/4\} \right\} \\ \leq \lim_{t\to\infty}\mu\left\{ \mathcal{B}_u\cap\{|h^*_t-\Delta_{u,h}^+(t)|\geq h/2\} \right\}+ \lim_{t\to\infty}\mu\Big\{ \mathcal{B}_u\cap\{h^*_t\leq 3h/4\} \Big\} = 0,\end{gathered}$$ which implies $\mu\big\{\mathcal{B}_u\cap\{\limsup_{t\to\infty}\Delta_{u,h}^+(t)=0 \}\big\} = 0$. We can now prove the last part of Theorem \[4probapositive\]. \[propnot5\] Assume that $w$ is non-decreasing and that holds. Then $$\begin{aligned} \alpha_c(w)=0 \quad \Longrightarrow \quad |R'|\neq 5 \textrm{ almost surely.}\end{aligned}$$ Assume by contradiction that $\alpha_c(w)=0$ and that $|R'|=5$ holds with positive probability. Thus there exists an initial local time configuration $\mathcal{C}$ such that, for the $\mathcal{C}$-VRRW $\bar{X}$ on ${\left[\kern-0.15em\left[}0,4{\right]\kern-0.15em\right]}$, the event $$\mathcal{E}:=\{\bar{Y}_\infty^+(0)+\bar{Y}_\infty^-(4)<\infty\}\cap\{\bar{X} \mbox{ visits 0 and $4$ i.o.} \}$$ has positive probability. Moreover, Equation yields, for $n\ge 0$, $$W(\bar{Z}_n(1))- W(\bar{Z}_n(3))= \bar{Y}_n^+(0) -\bar{M}_n(2) -\bar{Y}_n^-(4)+c,$$ for some constant $c$ depending on the initial configuration. On the event $\mathcal{E}$, each term on the r.h.s. of this equation converges to a limit, thus $$\lim_{n\to\infty} W(\bar{Z}_n(1))- W(\bar{Z}_n(3))=\bar{Y}_\infty^+(0) -\bar{M}_\infty(2) -\bar{Y}_\infty^-(4)+c =: \delta$$ exists and is finite. Moreover, Lemma \[sansatome\] implies that ${\mathbb{P}}\{\{\delta=0\} \cap \mathcal{E}\}=0$. Let us now prove that the event $ \mathcal{E}\cap \{\delta>0\}$ has probability 0 (the same result holds for $\delta<0$ by symmetry). On this event, for $n$ large enough, we get $$\begin{aligned} \label{x1x3} W(\bar{Z}_n(1))\ge W(\bar{Z}_n(3))+\delta',\end{aligned}$$ with $\delta'=\delta/2$. Besides, we have $$\begin{aligned} W(\bar{Z}_n(2))-W(\bar{Z}_n(0))\equiv \bar{Y}_n^-(3)\\ W(\bar{Z}_n(2))-W(\bar{Z}_n(4))\equiv \bar{Y}_n^+(1).\end{aligned}$$ Since, on $\mathcal{E}$, the quantities $\bar{Y}_\infty^-(3)$ and $\bar{Y}_\infty^+(1)$ are infinite (*c.f.* Lemma \[Yfini\]), we deduce that $\bar{Z}_n(2)$ is larger than $\bar{Z}_n(0)$ and $\bar{Z}_n(4)$ for $n$ large enough. Moreover, by periodicity, the sum of these three quantities is, up to a constant, equal to $n/2$. Thus, for $n$ large enough, we obtain $$3\bar{Z}_n(2)\ge \frac{n}{2} \ge \bar{Z}_n(1).$$ Using , we get, for $n$ large enough, on the event $ \mathcal{E}\cap \{\delta>0\},$ $$\begin{aligned} \frac{1_{\{\bar{X}_n=3,\,\bar{X}_{n+1}=2 \}}}{w(\bar{Z}_n(2))} &\le & \frac{1_{\{\bar{X}_n=3\}}}{w(\bar{Z}_n(1)/3)} \le \frac{1_{\{\bar{X}_n=3\}}}{w(W^{-1}(W(\bar{Z}_n(3))+\delta' )/3)}.\end{aligned}$$ Since $\alpha_c(w)=0$, it follows from Lemma \[lemaa\] of the Appendix that $\bar{Y}_\infty^-(3)$ is finite which contradicts the fact that $\bar{X}$ visits all the sites of ${\left[\kern-0.15em\left[}0,4 {\right]\kern-0.15em\right]}$ infinitely often. Theorem \[4probapositive\] is now a consequence of Propositions \[prop4pos\], \[prop5ex\], \[Prop45as\] and \[propnot5\]. Let us conclude this section by remarking that we can also describe the shape of the asymptotic local time configuration when $\alpha_c(w) <\infty$. Indeed, collecting the results obtained during the proof of the theorem, it is not difficult to check (the details being left out for the reader) that the asymptotic local time profile of the walk on $R'$ at time $n$ takes the form: {width="12.6cm"} In particular, when the walk localizes on $4$ sites, only the two central sites are visited a non-negligible proportion of time (this follows from of Lemma \[lemaa\] of the appendix). When the walk localizes on $5$ sites, a more unusual behaviour may happen. If the weight function is regularly varying (for example $w(n)\sim n\log \log n$), then, again, the walk spends asymptotically all its time on two consecutive vertices: $$\lim_{n\rightarrow\infty } \frac{L_n\wedge R_n}{n}=0 \qquad \hbox{ and }\qquad \lim_{n\rightarrow\infty } \frac{L_n\vee R_n}{n}=\frac{1}{2}.$$ However, this result is not true for general weight functions. In fact, the ratio $Z_n(y)/n$ of time spent at site $y$ may not converge. For instance, considering the weight sequence $w_0$ of Remark \[remn!\] of the appendix, we find that when the walk localizes on $5$ sites: $$\liminf_{n\rightarrow\infty } \frac{L_n\wedge R_n}{n}=0 \qquad \hbox{ but }\qquad \limsup_{n\rightarrow\infty } \frac{L_n\wedge R_n}{n}>0.$$ Finally, let us mention that the functions $\varepsilon(n)$ and $\varepsilon'(n)$ can also be explicitly computed for particular weights sequences. For example, for $w(n)=n\log \log n$, using and similar arguments as in the proof of , we find that $$W(\varepsilon(n))\equiv Y_n^+(x+1) \equiv \sum_{k=0}^{n/2}\frac{p_k}{w(k)} \quad \hbox{ and }\quad W(\varepsilon'(n))\equiv W(n/2)- W(\varepsilon(n))$$ where $p_k$ denotes the probability of the walk to jump to site $x+1$ at its $k$-th visit to $x+2$ (*i.e.* $p_k\sim L_{2k}/k$). Thus, after some (rather tedious) calculations, we deduce that, on the event $\delta:=\lim_{n\rightarrow \infty} W(R_n)-W(L_n) \in (0,1)$, the asymptotic local time profile on $R'$ takes the form: {width="9.6cm"} Appendix {#section5} ======== \[alphac\_prop\] Let $w$ and $\widetilde{w}$ denote two non-decreasing weight functions. 1. For any $\lambda>0$, we have $\alpha_c(w) = \lambda \alpha_c(\lambda w)$ **(scaling)**. 2. If $w \leq \widetilde w$, then $\alpha_c(w) \geq \alpha_c(\widetilde w)$ **(monotonicity)**. 3. If $w\sim \widetilde w$, then $\alpha_c(w)=\alpha_c(\widetilde w)$ **(asymptotic equivalence)**. The scaling property (a) follows directly from the relation $\lambda I_\alpha(\lambda w) = I_{\lambda \alpha}(w)$. We now prove (b). For $x\ge 0$ and $\alpha>0$, set $u(x,\alpha) :=W^{-1}(W(x)+\alpha)$ and define $\tilde{u}(x,\alpha)$ similarly for $\tilde{w}$. We have, $$\label{equ} \int_x^{u(x,\alpha)} \frac 1 {w(t)}\, dt = \int_x^{\widetilde u(x,\alpha)} \frac 1 {\widetilde w(t)}\, dt =\alpha.$$ When $w\le \widetilde w$, the equality above implies that $u(x,\alpha)\le \widetilde u(x,\alpha)$ for all $x$ and all $\alpha>0$. Since $w$ is non-decreasing we get $w( u(x,\alpha)) \le \widetilde w(\widetilde u(x,\alpha))$, hence $I_\alpha(w) \ge I_\alpha(\widetilde w)$. This establishes (b). Suppose now that $w$ and $\tilde{w}$ are two weight functions such that $w(x) = \tilde{w}(x)$ for all $x$ larger than some $x_0$. Then, in view of , we see that $u(x,\alpha)=\tilde{u}(x,\alpha)$ for all $\alpha >0$ and all $x\geq x_0$. Hence $\alpha_c(w) = \alpha_c(\tilde{w})$. This shows that $\alpha_c(w)$ does not depend upon the values taken by $w$ on any compact interval $[0,x_0]$. Thus (c) follows directly from (a) and (b). Theorem \[4probapositive\] states that when $\alpha_c(w)$ is finite and non-zero, the walk localizes on either $4$ or $5$ sites. It would certainly be interesting to estimate the probability of each of these events. This seems a difficult question. Let us remark that these probabilities are not (directly) related to $\alpha_c(w)$. Indeed, for any $\lambda>1$, the weight functions $w$ and $\lambda w$ define the same VRRW yet, $\alpha_c(w) \neq \alpha_c(\lambda w)$. We just need to check that $\alpha_c(x\log\log x) = 1$. For $x\geq 1$, set $$w(x): = x(1+L(\log x)),$$ where $L$ denotes the Lambert function defined as the solution of $L(x)e^{L(x)} = x$. Then it follows from elementary calculation that $$\frac{w(W^{-1}(x))}{w(W^{-1}(x+\alpha))} = \frac{x^x(1+\log x)}{(x+\alpha)^{x+\alpha}(1+\log(x+\alpha))} \underset{x\to\infty}{\sim} \frac{e^{-\alpha}}{x^\alpha}.$$ Therefore $\alpha_c(w) = 1$. Using now the well known equivalence $L(x) \sim \log(x)$, we conclude using (c) of Proposition \[alphac\_prop\] that $\alpha_c(x\log\log x) = 1$. \[Halphaw\] Assume that $w$ is non-decreasing. We have $$\liminf_{x\to\infty} \frac{w(x)}{x} \geq \frac{1}{\alpha_c(w)}.$$ In particular, when $\alpha_c(w) < \infty$ then $\sum 1/w(n)^2 < \infty$ and if $\alpha_c(w) = 0$ then $w$ has super-linear growth. In view of the scaling property $\lambda I_\alpha(\lambda w) = I_{\lambda \alpha}(w)$, we just need to prove that $\liminf w(x)/x < 1$ implies $I_{1}(w) = \infty$. Thus, let us assume that for some $\varepsilon>0$, there exist arbitrarily large $x$ such that $w(x)/x \leq 1-\varepsilon$. Then, for such an $x$ and for $y \leq \varepsilon x$, we have, since $w$ is non-decreasing, $$W(y + (1-\varepsilon)x) - W(y) = \int_{y}^{y+(1-\varepsilon)x}\frac{dz}{w(z)} \geq \frac{(1-\varepsilon)x}{w(x)} \geq 1,$$ which we can rewrite as $$\frac{1}{W^{-1}(W(y) + 1)} \geq \frac{1}{y+(1-\varepsilon)x}.$$ Thus, $$\label{apen1} \int_{\varepsilon x /2}^{\varepsilon x}\frac{1}{w(W^{-1}(W(y) + 1))} \geq \frac{\varepsilon}{2-\varepsilon}.$$ Since there exist arbitrarily large $x$ such that holds, we conclude that $I_{1}(w) = \infty$. \[remn!\] The previous lemma cannot be improved without additional assumptions on $w$. Indeed, consider the weight function $w_0$ defined by $w_0(x) = (n!)^{2}$ for $x \in [((n-1)!)^2,(n!)^2)$, $n\in {\mathbb{N}}^*$. It is easily seen that $$\liminf_{x\to\infty} \frac{w_0(x)}{x} = 1 = \frac{1}{\alpha_c(w_0)}.$$ This provides an example of a weight function which does not uniformly grow faster than linearly and yet for which the VRRW localizes on $4$ sites with positive probability. On the other hand, if $w$ is assumed to be regularly varying, then using similar arguments to those in the proof above, one can check that the finiteness of $\alpha_c(w)$ implies $\lim_\infty w(x)/x = \infty$. \[lemaa\] Assume that $w$ is non-decreasing and $\alpha_c(w) < \infty$, for any $0 < \delta < \delta'$, we have $$\label{almostlast} \liminf_{x\to\infty}\frac{W^{-1}(W(x) + \delta')}{W^{-1}(W(x) + \delta)} \geq e^{\frac{\delta'-\delta}{\alpha_c(w)}}.$$ Furthermore, for $\delta > \alpha_c(w)$, $$\label{andthelast} \lim_{x\to\infty}\frac{x}{W^{-1}(W(x)+\delta)} = 0.$$ As a consequence: - For any $\delta > \alpha_c(w)$ and any $c\in{\mathbb{R}}$, $$\sum_n^\infty \frac{1}{w(W^{-1}(W(n)+\delta)- c)} < \infty.$$ - If $\alpha_c(w) = 0$, then, for any $\delta , \gamma > 0$, $$\sum_n^\infty \frac{1}{w(\gamma W^{-1}(W(n)+\delta))} < \infty.$$ Recall the notation $u(x,\delta):= W^{-1}(W(x)+\delta)$. We have $$\int_{u(x,\delta)}^{u(x,\delta')}\frac{ds}{w(s)} = \delta'-\delta.$$ Using Lemma \[Halphaw\] and the fact that $u(x,\delta)$ tends to infinity as $x$ goes to infinity. we get, for any $\alpha>\alpha_c(w)$, $$\liminf_{x\to \infty}\log\left(\frac{u(x,\delta')}{u(x,\delta)}\right) = \liminf_{x\to \infty} \int_{u(x,\delta)}^{u(x,\delta')}\frac{ds}{s} \geq \lim_{x\to \infty} \int_{u(x,\delta)}^{u(x,\delta')}\frac{ds}{\alpha w(s)} = \frac{\delta'-\delta}{\alpha}$$ which yields . Assertion (a) now follows from noticing that, for $\alpha_c(w) < \gamma < \delta$, we have $W^{-1}(W(n)+\delta)- c \geq W^{-1}(W(n)+\gamma)$ for all $n$ large enough. The proof of Assertion (b) is similar. It remains to prove . Let $\delta > \alpha_c(w)$ and pick $\varepsilon >0$ small enough such that $\alpha_c(w) < \delta - \varepsilon$. Assume by contradiction that, for some $A>0$, we can find $x_0$ arbitrarily large such that $u(x_0,\delta) \leq A x_0$. Then, for all $x<x_0$, $$\varepsilon = \int_{u(x,\delta-\varepsilon)}^{u(x,\delta)}\frac{dy}{w(y)} \leq \int_0^{A x_0}\frac{dy}{w(u(x,\delta-\varepsilon))} = \frac{A x_0}{w(u(x,\delta-\varepsilon))}$$ which, in turn, implies $$\int_{x_0/2}^{x_0}\frac{dx}{w(u(x,\delta-\varepsilon))} \geq \frac{\varepsilon}{2A}$$ and contradicts the fact that $I_{\delta-\varepsilon}(w) < \infty$. \[teclem\] Assume that $w$ is non-decreasing. For any $\beta<\alpha_c(w)$, we have $$\sum_{n}^\infty \frac 1{w(n+W^{-1}(W(n)+\beta))} = \infty.$$ Choose $\alpha\in (\beta,\alpha_c(w))$. Since $w$ is non-decreasing, for any $t\ge 0$ and any $m\le n$, we have $$\label{lasteq} u(n,t)-u(m,t)\ge n-m.$$ Assume now that, for some large $n$, we have $n+u(n,\beta)\ge u(n,\alpha)$. Then, necessarily, there exists $k\in [u(n,\beta),u(n,\alpha)]$, such that $w(k)\le n/(\alpha-\beta)$. In particular, since $w$ is non-decreasing, $w(u(n,\beta))\le n/(\alpha-\beta)$. Moreover, using , we get $m+u(m,\beta)\le u(n,\beta)$ for all $m\le n/2$. Thus we also have $w(m+u(m,\beta))\le n/(\alpha-\beta)$, for all $m\le n/2$. It follows that $$\sum_{m=n/4}^{n/2} \frac 1{w(m+u(m,\beta))} \ge \frac{\alpha-\beta}{4}.$$ Therefore if $n+u(n,\beta)\ge u(n,\alpha)$, for infinitely many $n$, the desired result follows. Conversely, if the inequality above holds only for finitely many $n$, then because $w$ is non-decreasing, the result follows as well from the fact that $I_\alpha(w)= \infty$. *Acknowledgments: This work started following a talk given by Vlada Limic at the conference “Marches aléatoires, milieux aléatoires, renforcement” in Roscoff (2011) who shared some questions addressed to her by Bálint Tóth. We thank them both for having motivated this work.* [99]{} A.-L. Basdevant, B. Schapira and A. Singh. Localization results for one-dimensional vertex reinforced random walk with sub-linear reinforcement. *In preparation*. M. Benaïm and P. Tarrès. 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Vertex-reinforced random walk on ${\mathbb{Z}}$ has finite range. *Ann. Probab.* 27(3):1368–1388, 1999. B. Schapira. A $0-1$ law for Vertex Reinforced Random Walk on ${\mathbb{Z}}$ with weight of order $k^\alpha$, $\alpha<1/2$. Preprint, arXiv:1107.2505. T. Sellke. Reinforced random walk on the $d$-dimensional integer lattice. *Markov Process. Related Fields* 14(2):291–308, 2008. P. Tarrès. Vertex-reinforced random walk on ${\mathbb{Z}}$ eventually gets stuck on five points. *Ann. Probab.* 32(3B):2650–2701, 2004. P. Tarrès. Localization of reinforced random walks. Preprint, arXiv:1103.5536. S. Volkov. Vertex reinforced random walk on arbitrary graphs, *Ann. Probab.* 29(1):66–91, 2001. S. Volkov. Phase transition in vertex-reinforced random walks on ${\mathbb{Z}}$ with non-linear reinforcement. *J. Theoret. Probab.* 19(3): 691–700, 2006. [^1]: This result first appears at the end of [@V2]. However, there is a mistake in the original argument. For the sake of completness, we give an other proof of this result (for non-decreasing weight sequences) in Proposition \[proptwosite\].

--- abstract: 'We consider the 0–1 Incremental Knapsack Problem (IKP) where the capacity grows over time periods and if an item is placed in the knapsack in a certain period, it cannot be removed afterwards. The contribution of a packed item in each time period depends on its profit as well as on a time factor which reflects the importance of the period in the objective function. The problem calls for maximizing the weighted sum of the profits over the whole time horizon. In this work, we provide approximation results for IKP and its restricted variants. In some results, we rely on Linear Programming (LP) to derive approximation bounds and show how the proposed LP–based analysis can be seen as a valid alternative to more formal proof systems. We first manage to prove the tightness of some approximation ratios of a general purpose algorithm currently available in the literature and originally applied to a time-invariant version of the problem. We also devise a Polynomial Time Approximation Scheme (PTAS) when the input value indicating the number of periods is considered as a constant. Then, we add the mild and natural assumption that each item can be packed in the first time period. For this variant, we discuss different approximation algorithms suited for any number of time periods and for the special case with two periods.' address: - | Dipartimento di Ingegneria Gestionale e della Produzione, Politecnico di Torino,\ Corso Duca degli Abruzzi 24, 10129 Torino, Italy,\ [{federico.dellacroce, rosario.scatamacchia}@polito.it ]{} - 'CNR, IEIIT, Torino, Italy' - | Department of Statistics and Operations Research, University of Graz, Universitaetsstrasse 15, 8010 Graz, Austria,\ [[email protected]]{} author: - Federico Della Croce - Ulrich Pferschy - Rosario Scatamacchia bibliography: - 'ref8.bib' title: Approximating the Incremental Knapsack Problem --- Incremental Knapsack problem ,Approximation scheme ,Linear Programming Introduction {#sec:introIKP} ============ The 0–1 Knapsack Problem (KP) is one of the paradigmatic problems in combinatorial optimization and has been object of numerous publications and two monographs [@MarTot90] and [@KePfPi04]. In KP a set of items with given profits and weights is available and the aim is to select a subset of the items in order to maximize the total profit without exceeding a known knapsack capacity. KP is weakly NP-hard, although in practice fairly large instances can be solved to optimality within limited running time. Various generalizations of KP have been considered over the years. We mention among others the most recent contributions we are aware of: the collapsing KP (where the capacity of the constraint is inversely related to the number of items placed inside the knapsack) [@CDSS17]; the discounted KP (where it is required to select a set of item groups where each group includes three items and at most one of the three items can be selected) [@CXYSW16]; the parametric KP (where the profits of the items are affine-linear functions of a real-valued parameter and the task is to compute a solution for all values of the parameter) [@GHRT17]; the KP with setups (where items belong to disjoint families and can be selected only if the corresponding family is activated) [@SDSS17; @FUMOTR18; @PS17]; the penalized KP (where each item has a profit, a weight, a penalty and the goal is to maximize the sum of the profits minus the greatest penalty value of the selected items) [@DCPFSC17] and the temporal KP (where a time horizon is considered, and each item consumes the knapsack capacity during a limited time interval only) [@CFMT16]. We consider here a very meaningful and natural generalization of KP, namely the 0–1 Incremental Knapsack Problem ($\mbox{IKP}$) as introduced in [@BiSeYe13] where the constraining capacity grows over $T$ time periods. If an item is placed in the knapsack in a certain period, it cannot be removed afterwards. Each packed item contributes its profit for each time period in which it is included in the knapsack, with different impacts in the objective function depending on multiplicative time factors. These reflect the different importance of the periods or allow a discounting over time. The problem calls for maximizing the weighted sum of the profits accumulated over the whole time horizon.\ $\mbox{IKP}$ generalizes the time-invariant Incremental Knapsack Problem, namely with unit time multipliers, considered in [@hart08; @HaSh06; @sha07] and which we will denote here as [$\mbox{IIKP}$]{}. $\mbox{IKP}$ has many real-life applications since, from a practical perspective, it is often required in resource allocation problems to deal with changes in the input conditions and/or in a multi–period optimization framework. In manufacturing, for instance, a producer may activate contracts with customers for a periodic (e.g. monthly) supply of its products within an expanding production plan. The production capacity is increased in each time period and the goal is to decide which contracts (and when) should be set in order to maximize the overall profits over a given time horizon. Here, the increasing capacity values represent the available production resources in each time period while the items are the orders to satisfy with corresponding profits and resource consumption. Other applications of an online [$\mbox{IIKP}$]{}variant in the contexts of renewable resources and of trading operations are provided in [@Thtiwe16]. Related literature ------------------ From a general point of view, [@HaSh06] introduced incremental versions of maximum flow, bipartite matching, and also knapsack problems. The authors in [@HaSh06] discuss the complexity of these problems and show how the incremental version even of a polynomial time solvable problem, like the max flow problem, turns out to be NP–hard. General techniques to adapt the algorithms for the considered optimization problems to their respective incremental versions are discussed. Also, a general purpose approximation algorithm is introduced. In [@BiSeYe13], it is shown that even [$\mbox{IIKP}$]{}is strongly NP–hard. A constant factor algorithm is also provided under mild restrictions on the growth rate of the knapsack capacity. In addition, a PTAS is derived for the special variant [$\mbox{IIKP}$]{}under the assumption that $T$ is in $O(\sqrt{\log n})$, where $n$ is the number of items. In a recent contribution [@FaMa17], an improved PTAS for [$\mbox{IIKP}$]{}is lined out. The PTAS in [@FaMa17] polynomially depends on both $n$ and $T$ without any further restriction on the input data. Both algorithms introduced in [@BiSeYe13] and [@FaMa17] cannot be adapted to tackle arbitrary $\mbox{IKP}$ instances. The PTAS derived in [@BiSeYe13] could be extended (for fixed $T$) to $\mbox{IKP}$ instances with monotonically non-increasing time factors. Likewise, the PTAS proposed in [@FaMa17] can be extended to $\mbox{IKP}$ instances with monotonically increasing time multipliers. A number of problems closely related to $\mbox{IKP}$ were tackled in the literature. In [@faa81; @duwa85], a multiperiod variant of the knapsack problem is considered. The capacity increases over periods as in $\mbox{IKP}$ but each item becomes available for packing only at a certain time period. Thus, the set of items to choose from increases over time. In [@faa81], a branch and bound algorithm is proposed. In [@duwa85], an efficient algorithm for solving the linear programming relaxation of the problem and a procedure for reducing the number of variables are presented. In [@Thtiwe16], an online knapsack problem with incremental capacity is considered. In each period a set of items is revealed without knowledge of future items and the goal is to maximize the overall value of the accepted items. The authors in [@Thtiwe16] introduce deterministic and randomized algorithms for this online knapsack problem and provide an analysis of their performance. Our contribution ---------------- In this work, we provide approximation results for $\mbox{IKP}$ and its restricted variants. We employ Linear Programming (LP) to analyze the worst case performance of different algorithms for deriving approximation bounds. Interestingly, the proposed LP–based analysis can be seen as a valid alternative to formal proof systems based on analytical derivation. Indeed, recently a growing attention has been centered on the use of LP modeling for the derivation of formal proofs (see [@CW16; @ACCEHMW16]) and we also show here a successful application of this technique. In our contribution, in Section \[sec:ApproxIKP\] we first generalize for $\mbox{IKP}$ the performance analysis of the generic algorithmic approach derived in [@HaSh06] and originally applied to [$\mbox{IIKP}$]{}. Moreover, we show the tightness of some approximation ratios provided by the approach, including approximation bounds derived in [@HaSh06]. Then, we devise a PTAS in Section \[IKP\_PTAS\] when the number of time periods $T$ is a constant. While this is a stronger assumption than the one made for the PTAS for [$\mbox{IIKP}$]{}in [@BiSeYe13] and no assumption on input $T$ is needed in [@FaMa17], our algorithm is, to the authors’ knowledge, the first PTAS that applies to arbitrary $\mbox{IKP}$ instances. Also, the proposed approach is much simpler and does not require a huge number of complicated LP models as in [@BiSeYe13; @FaMa17]. In Section \[sec:RestrIKP\] we introduce the natural assumption that each item can be packed at any point in time, also in the first period. For this reasonable variant of $\mbox{IKP}$ we discuss two approximation algorithms suited for any $T$. Finally, in Section \[sec:RestrIKP23\] we focus on an $\mbox{IKP}$ variant with $T=2$ and show an algorithm with an approximation ratio bounded by $\frac{1}{2} + \frac{\sqrt{2}}{4} = 0.853\ldots$ and by $\frac{6}{7}=0.857\ldots$ for [$\mbox{IIKP}$]{}. A preliminary conference version containing some of the results of this paper derived for [$\mbox{IIKP}$]{}is presented in [@DEPFSC17]. Notation and problem formulation {#sec:themodelIKP} ================================ In $\mbox{IKP}$ a set of $n$ items is given together with a knapsack with capacity values $c_t$ for each time period $t=1,\ldots,T$ and increasing over time, i.e. $c_{t-1} \leq c_t$ for $t=2,\ldots,T$. Each item $i$ has a profit $p_{i} > 0$ and a weight $w_{i} > 0$. If an item is placed in the knapsack at time $t$, it cannot be removed at a later time. The contribution of an item $i$ at time $t$ is equal to $\Delta_t \, p_{i}$, where $\Delta_t > 0$ denotes the time multiplier of period $t$. The problem calls for maximizing the total profit of the selected items without exceeding the knapsack capacity summing up the profits attained for each time period. In order to derive an ILP-formulation, we associate with each item $i$ a $0-1$ variable $x_{it}$ such that $x_{it}=1$ iff item $i$ is contained in the knapsack in period $t$. $\mbox{IKP}$ can be formulated by the following ILP model: $$\begin{aligned} (\mbox{IKP}) \qquad \text{maximize}\quad & \sum\limits_{t=1}^T \sum\limits_{i=1}^{n} \Delta_t p_{i}x_{it}\label{eq:ObjIKP}\\ \text{subject to}\quad & \sum\limits_{i=1}^{n} w_{i}x_{it} \leq c_t \quad t= 1,\dots,T; \label{eq:capIKP}\\ & x_{i(t-1)} \leq x_{it} \qquad i= 1,\dots,n, \quad t= 2,\dots,T; \label{eq:PredIKP}\\ & x_{it} \in\{0,1\} \qquad i= 1,\dots,n, \quad t= 1,\dots,T. \label{eq:varDefIKP}\end{aligned}$$ The cost function (\[eq:ObjIKP\]) maximizes the sum of the profits over the time horizon. Constraints (\[eq:capIKP\]) guarantee that the items weights sum does not exceed capacity $c_t$ in each period $t$. Constraints (\[eq:PredIKP\]) ensure that an item chosen at time $t$ cannot be removed afterwards. Constraints (\[eq:varDefIKP\]) indicate that all variables are binary. In [$\mbox{IIKP}$]{}, we have $\Delta_t =1$ for each $t=1,\dots,T$.\ In the following, we denote by $x^*$ (resp. $z^*$) the optimal solution (resp. solution value) of $\mbox{IKP}$. The contribution of time period $t$ to $z^*$ is denoted by $z^*_t = \sum_{i=1}^{n} \Delta_t p_{i} x^*_{it}$. We also denote by $z^{Y}$ the solution value yielded by a generic algorithm $Y$ and by $pc_j = \sum_{t=1}^T \Delta_t p_{j} x_{jt}$ the [*profit contribution*]{} of an item $j$ of a given feasible solution $x$. For each period $t$ and the related capacity value $c_t$, we define the corresponding standard knapsack problem as $KP_t$. This means that in $KP_t$ we consider only one of the $T$ constraints (\[eq:capIKP\]). The optimal solution value and related item set of each $KP_t$ is denoted by $z_t$ and [$Sol(KP_t)$]{}respectively. Finally, we define for a generic item set $S$ the canonical weight sum $w(S) := \sum_{j \in S} w_j$ and profit sum $p(S) :=\sum_{j \in S} p_j$. We say that an approximation algorithm $Y$ has [*approximation ratio*]{} $r$, for $r\in [0,1)$, if $z^Y \geq r\cdot z^*$ for every instance of $\mbox{IKP}$. IKP Linear Relaxation --------------------- The linear relaxation of $\mbox{IKP}$, where fractions of items can be packed, i.e.  where constraints (\[eq:varDefIKP\]) are replaced by the inclusion in the interval $[0,1]$, can be easily computed. In fact, it suffices to order the items by non–increasing efficiencies $\frac{p_i}{w_i}$ in $O(n \log n)$ and to fill the capacity of the knapsack in each period according to this ordering in $O(n)$. Thus, the execution time for solving the linear relaxation of $\mbox{IKP}$, hereafter denoted by $\mbox{IKP}_{LP}$, can be bounded by $O(n \log n+T)$. An alternative approach would consider each time period $t$ separately and solve the LP-relaxation for each knapsack problem $KP_t$. Using the linear-time median algorithm to find the split item, this can be done in $O(n)$ time for every $t=1,\ldots, T$, thus requiring $O(n\cdot T)$ time in total. Clearly, both approaches deliver exactly the same solution structure. For small or even constant values of $T$ the latter approach will dominate the former. Taking a closer look at the $T$ iterations of this approach one can exploit the fact that the split item (and thus the LP-relaxation) is always computed on the same item set, only the capacities change. Avoiding repetitions one can even get a total running time of $O(n \log T)$. Since $T$ can be expected to be of moderate size, this is close to a linear time algorithm. The technical details of this approach can be briefly described as follows: The classical computation of the split item in linear time requires a sequence of $\log n$ iterations (see [@KePfPi04 ch 3.1]). In iteration $i$ for $i=1,\ldots, \log n$, there are $n/2^{i-1}$ items considered, their median w.r.t. the efficiencies is computed in $O(n/2^{i-1})$ time, and the considered items are partitioned into two subsets with efficiencies lower and higher than the median, respectively. Then, it is determined (in constant time) which of the two subsets contains the split item and the next iteration is performed on the selected subset with $n/2^i$ items. Applying this procedure several times for different capacity values on the same set of items does not change the positions of medians and the resulting subset structure. Of course, the selections of subsets in each iteration may change for different capacities. Nevertheless, once a certain subset was considered, its median computed and the partitioning into two halves performed, this process does not have to be repeated for the same subset again for a different capacity value. Thus, each subset of size $n/2^i$ has to be processed (in linear time) only once. We split the analysis of the total effort required for all $T$ capacity values into two phases. We first consider the iterations $i=1, \ldots, \log T$. In each such iteration a subset of size $n/2^{i-1}$ is considered. However, over all $T$ iterations each item can occur only once in a subset of size $n/2^{i-1}$. Thus, the total effort for the $T$ executions of an iteration indexed by $i$ is bounded by $O(n)$. The time for this first phase is thus $O(n \log T)$. The running time for the second phase can be simply summed up by $$T \cdot \sum_{i=\log T+1}^{\log n} \frac{n}{2^{i-1}} \leq T \cdot 2\: \frac{n}{2^{\log T}} = 2n\,.$$ We summarize our considerations in the following statement. \[th:lprelax\] The LP-relaxation of $\mbox{IKP}$ can be computed in\ $O(\min\{n \log T, n \log n+T\})$. Approximating IKP {#sec:ApproxIKP} ================= Approximation ratios of a general purpose algorithm {#sec:HarmNumRatios} --------------------------------------------------- In [@HaSh06], a general framework for deriving approximation algorithms is provided. Following the scheme in [@HaSh06] which was originally applied to [$\mbox{IIKP}$]{}, we consider the following algorithm $A$. The algorithm employs an $\eps$–approximation scheme to obtain a feasible solution for each knapsack problem $KP_t$ with solution value denoted by $z_{t}^{A}$ for $t=1, \ldots, T$. Each such solution is also a feasible solution for $\mbox{IKP}$ where the items constituting $z_t^A$ are present in all successive time periods (and no other items). The algorithm chooses as a solution value $z^A$ the maximum among all these candidates, i.e. $$\label{eq:z^a} z^A = \max_{t=1,\dots,T}\left\{\sum_{\tau=t}^T \Delta_\tau \, z_{t}^{A}\right\}. $$ The obvious extension of $A$, which solves each knapsack problem $KP_t$ to optimality with value $z_t$ (instead of $z_{t}^{A}$) and then proceeds as in (\[eq:z\^a\]), will be denoted by $A^*$ with solution values $z^{A^*}$. Clearly, $A^*$ is not polynomial. Let us define quantity $\Theta = {{‎‎\sum}}_{t=1}^T \frac{\Delta_t}{\sum_{\tau=t}^T \Delta_\tau}$ , with $\Theta \leq T$. The following generalization of Theorem 3 in [@HaSh06] holds. \[ShHaGEN\] Algorithm $A$ is an approximation algorithm for $\mbox{IKP}$ with approximation ratio at least $\frac{1-\eps}{\Theta} \geq \frac{1-\eps}{T}$. For each $t=1,\dots,T$ we have $\frac{\Delta_t z_{t}^{A}}{1 -\eps} \geq z_t^{*}$ and also $z_{t}^{A} \leq \frac{z^A}{\sum\limits_{\tau=t}^T \Delta_\tau}$ from . Then, the following series of inequalities holds: $$z^* = \sum\limits_{t=1}^T z_t^{*} \leq \sum\limits_{t=1}^T \frac{\Delta_t z_{t}^{A}}{(1 -\eps)} \leq \sum\limits_{t=1}^T \frac{\Delta_t z^{A}}{(1 -\eps)\sum\limits_{\tau=t}^T \Delta_\tau} = \frac{\Theta \, z^{A}}{1-\eps} \implies \frac{z^{A}}{z^*} \geq \frac{1-\eps}{\Theta}$$ For [$\mbox{IIKP}$]{}considered in [@HaSh06], we have $\Theta=\mathcal{H}_T$ with $\mathcal{H}_T\approx \ln T$ being the $T$-th harmonic number. At the present state of the art, the tightness of this bounds is an open question even for [$\mbox{IIKP}$]{}. Considering the variant $A^*$, where each $KP_t$ is solved to optimality (i.e. algorithm $A$ for $\eps = 0$), we can state the following result. \[GenTight\] For any value of $T$ and any choice of time multipliers $\Delta_t$, algorithm $A^*$ has a tight approximation ratio $\frac{1}{\Theta}$ for $\mbox{IKP}$. We can evaluate the performance of algorithm $A^*$ by an alternative analysis based on a Linear Programming model. More precisely, we consider an LP formulation with non-negative variables $h^A$ and $h_t$ associated with $z^{A^*}$ and $z_t$ respectively and a positive parameter $OPT > 0$ associated with $z^*$. The corresponding LP model for evaluating the worst case performance of algorithm $A^*$ is as follows: $$\begin{aligned} \text{minimize}\quad & h^A \label{eq:ObjAlgoA}\\ \text{subject to}\quad & h^A - \sum\limits_{\tau=t}^T \Delta_\tau \, h_t \geq 0 \qquad t= 1,\dots,T; \label{eq:MaxValueA}\\ & \sum\limits_{t=1}^T \Delta_t h_t \geq OPT \label{eq:UBIKP}\\ & h^A \geq 0 \label{eq:hAdef}\\ & h_t \geq 0 \qquad t= 1,\dots,T. \label{eq:hTdef}\end{aligned}$$ The value of the objective function (\[eq:ObjAlgoA\]) provides a lower bound on the worst case performance of algorithm $A^*$. Constraints (\[eq:MaxValueA\]) guarantee that the contribution of each optimal knapsack solution $KP_t$ as a solution of $\mbox{IKP}$ will be taken into account according to (\[eq:z\^a\]). Constraint (\[eq:UBIKP\]) indicates that the sum of the knapsack solutions $z_t$, multiplied by the associated $\Delta_t$, constitute a trivial upper bound on $z^*$. Constraints (\[eq:hAdef\]), (\[eq:hTdef\]) indicate that variables are non-negative. Solving model (\[eq:ObjAlgoA\])–(\[eq:hTdef\]) to optimality provides lower bounds on the performance ratio of algorithm $A^*$ equal to $\frac{h^A}{OPT}$ for any $T$. We will derive the optimal solutions of model – by considering the related dual problem. Let us introduce $T +1$ dual variables $\lambda_i ~ (i=1,\dots,T+1)$ associated with constraints – respectively. The dual formulation of model – is as follows: $$\begin{aligned} \text{maximize}\quad & OPT \, \lambda_{(T+1)} \label{eq:ObjDUALAlgoA}\\ \text{subject to}\quad & \sum\limits_{t=1}^T \Delta_t\leq 1 \label{eq:hAA}\\ & -\sum\limits_{\tau=t}^T \Delta_\tau \, \lambda_t + \Delta_t \lambda_{(T+1)} \leq 0 \qquad t= 1,\dots,T; \label{eq:htt}\\ & \lambda_t \geq 0 \qquad t= 1,\dots,T+1. \label{eq:lambdadef}\end{aligned}$$ Constraints – correspond to primal variables $h^A$, $h_t ~ (t= 1,\dots,T)$ respectively. Feasible solutions of primal and dual models are: $$\begin{aligned} & h^{A} = \frac{OPT}{\Theta}; \, h_t = \frac{OPT}{\sum\limits_{\tau=t}^T\Delta_\tau \, \Theta} \qquad t= 1,\dots,T; \label{PrimalGen}\\ &\lambda_{(T + 1)} = \frac{1}{\Theta}; \, \lambda_t = \frac{\Delta_t}{\sum\limits_{\tau=t}^T\Delta_\tau \, \Theta} \qquad t= 1,\dots,T. \label{DualGen}\end{aligned}$$ It is easy to verify that such solutions are feasible and satisfy all corresponding constraints to equality for any distribution of time multipliers and for any value of $T$. Since $h^{A} = OPT \, \lambda_{(T+1)}$, by strong duality both primal and dual solutions are also optimal. Hence, the bound $\frac{h^A}{OPT}$ is equal to $\frac{1}{\Theta}$ for any $T$. It was shown in Theorem \[ShHaGEN\] (for $\eps=0$ ) that the approximation ratio of $A^*$ is at least $\frac{1}{\Theta}$. Now we will derive instances where this bound can be reached. Therefore, we construct instances where the optimal solution values of $KP_t$ in each period $t$ are equal to the values of $h_t$ for $t=1,\dots, T$ in and where the optimal solution value for $\mbox{IKP}$ is equal to the weighted sum of all these solutions, namely $z^* = \sum\limits_{t=1}^T \Delta_t h_t$. Such target instances can be generated by the following procedure: 1. We first represent quantity $\Theta$ as a fraction, i.e. $\Theta = \frac{a}{b}$ where b is the smallest common multiple of the denominators of fractions $\frac{\Delta_1}{\Delta_1 + \dots + \Delta_T} + \dots + \frac{\Delta_{T}}{\Delta_T}$. Then, we set $OPT = a$ so as to get integer values $h_t$. 2. We generate an $\mbox{IKP}$ instance as follows: $$n = \frac{b}{\Delta_T},\ p_j = w_j = 1\ (j=1,\dots, n),\ c_t = h_t \, (t=1,\dots, T)$$ The optimal solution of each $KP_t$ will pack items until the corresponding capacity $c_t$ is filled and thus will yield a solution value equal to $h_t$. The number of items is $\frac{b}{\Delta_T}$ because the capacity in the last period $T$ is $c_T = h_T = \frac{OPT}{\Delta_T \Theta}= \frac{a}{\Delta_T \frac{a}{b}} = \frac{b}{\Delta_T}$. At the same time, the optimal solution for $\mbox{IKP}$ can be obtained by progressively packing all items over time periods while filling the capacities $c_t$, hence $z^* = \sum\limits_{t=1}^T \Delta_t c_t = \sum\limits_{t=1}^T \Delta_t h_t$. As an example of the outlined procedure, consider the [$\mbox{IIKP}$]{}variant with $T=3$ for which $\mathcal{H}_T = \frac{11}{6}$. Correspondingly, the following instance with $n=6$, $p_j = w_j = 1$ ($j=1,\dots, 6$), $c_1 = 2$, $c_2 = 3$, $c_3 = 6$ is generated. The optimal solution is given by greedily packing as many items as possible in each time period and thus filling the corresponding capacities ($z^* = 11$). The optimal solutions values of the KPs are equal to 2, 3, 6 respectively. Hence we have $z^{A^*} = \max \{3\cdot2, 2\cdot3, 6 \} = 6$ which proves the tightness of the approximation bound $\frac{1}{\mathcal{H}_3} = \frac{6}{11}$. We remark that the bound tightness cannot be straightforwardly generalized to algorithm $A$, where an $\eps$–approximation scheme is run for solving each KP. We could get the ratio in Theorem \[ShHaGEN\] by solving model (\[eq:ObjAlgoA\])–(\[eq:hTdef\]) where the term $\sum\limits_{t=1}^T \Delta_t h_t$ in constraint (\[eq:UBIKP\]) is divided by ($1 - \eps$) and the optimal values of primal variables are multiplied by $(1 - \eps)$. However, the generation of tight instances is strictly related to the choice of the approximation algorithm for KPs. A PTAS when T is a constant {#IKP_PTAS} --------------------------- In the following it will be convenient to consider a [*residual problem*]{} $\mbox{IKP}^R$ with optimal solution value $z_R^*$. It is derived from any given instance of $\mbox{IKP}$ by defining a subset of items, a residual capacity $c^R_t \leq c_t$ for every time period $t$ with $c^R_{t-1} \leq c^R_t$, and an [*earliest insertion time*]{} $t_j$, which means that item $j$ cannot be put into the knapsack in time periods before $t_j$. Formally, the conditions $x_{j \tau}=0$ for $\tau < t_j$ are added in the ILP model. Obviously, every $\mbox{IKP}$ instance is equivalent to a restricted instance $\mbox{IKP}^R$ with $t_j := \min \{t \mid c_t \geq w_j, t=1,\ldots, T\}$. Clearly, the solution value of the LP-relaxation of this $\mbox{IKP}^R$ can be smaller than the LP-relaxation of $\mbox{IKP}$. Concerning the structure of the LP-relaxation of $\mbox{IKP}^R$ we can observe the following. \[th:lpres\] The solution of the LP-relaxation of $\mbox{IKP}^R$ contains at most $T$ fractional items. We claim that considering the time periods from $t=1,\ldots, T$ at most one additional fractional item can appear in any period $t$ which implies the above statement. Assume that in some period $t$ there are two new fractional items $j_1$ and $j_2$ in the solution (which were not packed at all in periods before $t$) with efficiencies $p_{j_1}/w_{j_1} \geq p_{j_2}/w_{j_2}$. Similar to the standard KP (see [@KePfPi04 Theorem 2.2.1]) one can invoke an exchange argument and shift weight from $j_2$ to $j_1$ by setting $d:=\min\{w_{j_2}x_{j_2 t}, w_{j_1}(1-x_{j_1 t}) \}$ and defining a new solution with $x_{j_1 t}' := x_{j_1 t} + d/w_{j_1}$ and $x_{j_2 t}' := x_{j_2 t} - d/w_{j_2}$. Obviously, the total capacity consumed by items $j_1$ and $j_2$ does not change in period $t$ and neither in any later periods up to $T$. Hence, the new solution is feasible and gives a higher profit contribution in all periods $t, t+1,\ldots, T$. Moreover, in the new solution at most one of $j_1$ and $j_2$ has a fractional value in period $t$. Note that a fractional item introduced in some period $t$ may well be increased during a later period and possibly reach $1$. Moreover, different from the LP-relaxation for $\mbox{IKP}$, it may be the case that the capacity is not fully utilized in some periods. This can occur e.g. in some period $t$, if a highly efficient item with large weight has an earliest insertion time larger than $t$. In this case it can make sense to leave some empty space in period $t$ and thus allow this item to be fully packed when it becomes available. Taking a closer look at the solution structure of the LP-relaxation of $\mbox{IKP}^R$ we can also compute it by a combinatorial algorithm with polynomial running time. However, the details of this algorithm are beyond the scope of this paper. Computing the LP-relaxation of any restricted instance $\mbox{IKP}^R$ and rounding down all fractional values gives a feasible solution with value $z^{\prime}_R$. It follows from Lemma \[th:lpres\] that the maximum loss from rounding down is bounded by $T$ times the maximum profit contribution of a single item. Thus we have: $$\label{eq:zAprimeR} z_R^* \leq z^{\prime}_R + T \cdot \max_{j=1,\ldots,n} \sum_{\tau=t_j}^T \Delta_\tau p_j$$ For a residual problem $\mbox{IKP}^R$ we can apply the following approximation algorithm $A^{\prime}$, which is a variant of algorithm $A$ described in Section \[sec:HarmNumRatios\]. We run an FPTAS for each time period $t$ yielding $\eps$-approximations $z_{tR}^{A^\prime}$. Then we also consider an alternative solution derived by computing the optimal solution of the LP-relaxation of $\mbox{IKP}^R$ and rounding down all fractional variables to $0$ as described above reaching a solution value $z^{\prime}_R$. Finally, we take the maximum between these $T+1$ candidates obtaining a solution value $z_R^{A^{\prime}}$, namely $$\label{eq:z^B} z_R^{A^{\prime}} = \max\left\{ z^{\prime}_R,\: \max_{t=1,\ldots,T} \left\{ \sum_{\tau=t}^T \Delta_\tau z_{tR}^{A^\prime}\right\} \right\}.$$ With Theorem \[ShHaGEN\] the overall approximation ratio of algorithm $A^{\prime}$ for $\mbox{IKP}^R$ can be bounded by $\rho = \frac{1-\eps}{T}$. Note that the introduction of earliest insertion times for items in $\mbox{IKP}^R$ does not interfere with the analysis in Theorem \[ShHaGEN\]. Computing $z^{\prime}_R$ can be done in polynomial time $poly(n,T)$. The $T$ executions of a FPTAS for KP can be bounded by $O(T(n \log(\frac{1}{\eps}) + (\frac{1}{\eps})^3 \log^2(\frac{1}{\eps})))$, see [@KelPfe04]. Similarly to the line of reasoning for deriving PTAS’s for KP (see, e.g., [@Sahni75; @CaKePfPi00]), we propose an approximation scheme for $\mbox{IKP}$. It is based on guessing (by going through all possible choices) the set $S_k$ of $k$ items and the associated starting periods $st_j$ for each item $j \in S_k$ which give the largest profit contributions in an optimal solution. For convenience, the minimum profit contribution of an item in $S_k$ is denoted by $pcm = \min_{j\in S_k} \{\sum_{\tau=st_j}^T \Delta_\tau p_j\}$. For each such choice of $S_k$ and starting periods, if it is feasible, one can define the following residual instance $\mbox{IKP}^R$. It consists of the residual capacities $c^R_t = c_t - \sum_{j \in S_k, st_j \leq t} w_j$ in each time period $t$. To preserve the incremental capacity structure of $\mbox{IKP}$ we set $c^R_t := \min\{c^R_t, c^R_{t+1}\}$ for $t=T-1, T-2, \ldots, 1$, which is without loss of generality because of (\[eq:PredIKP\]). The item set of $\mbox{IKP}^R$ is restricted to items not in $S_k$ with profit contribution not exceeding $pcm$, which can be enforced by setting the earliest insertion time $t_j$ for all items $j \not \in S_k$ as $$t_j = \min \left\{t \mid c^R_t \geq w_j, \sum_{\tau=t}^T \Delta_\tau p_j \leq pcm,\, t=1,\ldots, T \right\}.$$ If the minimum is taken over the empty set, we remove $j$ from $\mbox{IKP}^R$. We can now state our approximation scheme, denoted by algorithm *Approx*. **Input:** $\mbox{IKP}$ instance, $\eps$. \[alg:1\] Set $k := \min \left \{ n, \Bigl\lceil \frac{T}{\eps} \Bigr\rceil \right \}$. \[alg:2\] Generate all subsets of $\{1,\ldots,n\}$ consisting of less than $k$ items.For each such subset generate all possible starting periods of items and check the feasibility for $\mbox{IKP}$.Let $z^\eps$ be the best objective value over all generated feasible solutions. \[alg:3\] For each subset $S_k \subseteq \{1,\ldots,n\}$ with $|S_k|=k$, generate all possible starting periods of items and check the feasibility for $\mbox{IKP}$. \[alg:4\] For every resulting feasible configuration, compute the overall profit contribution $PC(k)=\sum_{j\in S_k} pc_j$ and the minimum contribution $pcm$. Determine the corresponding residual $\mbox{IKP}$ instance $\mbox{IKP}^R$.Apply algorithm $A^{\prime}$ to $\mbox{IKP}^R$ yielding a solution value $z_R^{A^{\prime}}$.Update $z^\eps := \max\{z^\eps, z_R^{A^{\prime}} + PC(k)\}$ The following proposition characterizes the running time of *Approx*. \[ptas\_complex\] The running time complexity of *Approx* is polynomial in the size $n$ of the input. The running time mainly depends on the number of configurations generated in Step \[alg:3\]. There are $O(n^k)$ subsets $S_k$ and for each item $T$ possible starting periods, i.e. $O(T^k)$ possible choices for each $S_k$. For each such configuration algorithm *$A^{\prime}$* is performed in Step \[alg:4\]. This dominates the effort spent in Step \[alg:2\]. Plugging in the definition of $k$ the overall complexity is: $$O\left(n^{\lceil \frac{T}{\eps} \rceil} T^{\lceil \frac{T}{\eps} \rceil} \cdot \left( poly(n,T) + T\left(n \log(\frac{1}{\eps}) + (\frac{1}{\eps})^3 \log^2(\frac{1}{\eps})\right)\right)\right)$$ The following theorem describes the approximation ratio reached by *Approx*. Algorithm *Approx* is a PTAS for $\mbox{IKP}$ when $T$ is a constant. We will show in this proof that the algorithm is an $\eps$–approximation scheme. Then, it follows from the time complexity stated in Proposition \[ptas\_complex\] that *Approx* is a PTAS for $\mbox{IKP}$ when $T$ is a constant. If $k = n$, namely $ n \leq \Bigl\lceil \frac{T}{\eps} \Bigr\rceil$, Steps \[alg:2\] or \[alg:3\] of *Approx* would even compute an optimal solution since all subsets of the $n$ items are enumerated, and therefore the statement is trivial. Thus, it remains to consider the case where $k=\Bigl\lceil \frac{T}{\eps} \Bigr\rceil$. We consider two cases depending on a parameter $f \in (0,1)$ and analyze the iteration of Step \[alg:3\] where $S_k$ and the associated starting periods $st_j$ yield the largest profit contribution $PC(k)$ of any subset of $k$ items in the optimal solution. In the corresponding execution of Step \[alg:4\] we denote by $z_R^*$ and by $p_{max}^R$ the optimal solution value and the maximum profit of the items in the respective residual instance $R$. #### Case 1: $PC(k) \geq f\cdot z^*$ \ Since the $k$ items with maximal profit contribution in the optimal solution are considered, we have $z^* = z_R^* + PC(k)$. Using the approximation ratio $\rho$ of $A^{\prime}$, the following series of inequalities holds: $$\begin{aligned} \label{case1} &z^\eps \geq PC(k) + z_R^{A^{\prime}} \geq PC(k) + \rho\, z_R^* \\ & = PC(k) + \rho (z^* - PC(k)) = (1-\rho)PC(k) + \rho z^*\\ & \geq (1-\rho)fz^* + \rho z^* = ((1-\rho)f + \rho)\, z^* \end{aligned}$$ #### Case 2: $PC(k) < f\cdot z^*$ \ Since the profit contribution of any item in $\mbox{IKP}^R$ is bounded by $pcm$, we also for any item $j$ in $\mbox{IKP}^R$ $$\label{eq:pmaxR} pc_j \leq pcm \leq \frac{1}{k} PC(k) < \frac{1}{k} f\,z^*.$$ Then, the following series of inequality holds: $$\begin{aligned} \label{eq:case2} z^* &=& PC(k) + z_R^*\\ &\leq& PC(k) + z_R^{A^{\prime}} + T \cdot \max_{j \not\in S_k}\, pc_j \\ &\leq& PC(k) + z_R^{A^{\prime}} + T \cdot \frac{1}{k} f\,z^*\\ &\leq& z^\eps + T\frac{1}{k}f\,z^* \label{eq:lastline}\end{aligned}$$ The first inequality comes from (\[eq:zAprimeR\]), the second inequality from (\[eq:pmaxR\]). Now (\[eq:case2\])-(\[eq:lastline\]) is equivalent to $$\label{eq:FinalCase2} z^\eps \geq (1 - T\frac{1}{k}f)\,z^*.$$ Given $\eps$ and $\rho$, we now set $f$ as follows: $$f := 1 - \frac{\eps}{1-\rho} = \frac{1-\rho - \eps}{1-\rho} \label{eq:setf}$$ We easily note that $f < 1$ and we also have $f > 0$ since $\rho = \frac{1-\eps}{T}$ and thus $1-\eps > \rho$. For *Case 1*, plugging the value of $f$ in (\[case1\]) yields $$\begin{aligned} \label{eq:checkcase1} &z^\eps \geq ((1-\rho)f + \rho)\, z^* \\ &= ((1-\rho - \eps) + \rho)\, z^* = (1 - \eps)\,z^*. \end{aligned}$$ For *Case 2*, we plug in $k=\Bigl\lceil \frac{T}{\eps} \Bigr\rceil$ into (\[eq:FinalCase2\]) and get $$\label{eq:checkcase2} z^\eps \geq (1 - \eps f)\, z^* \geq (1 - \eps)\, z^*.$$ We remark that in [@BiSeYe13] a PTAS is introduced under the more general assumption that $T$ is in $O(\sqrt{\log n})$, but only for [$\mbox{IIKP}$]{}. For constant $T$, it can be extended to decreasing time multipliers $\Delta_t$. Furthermore, in [@FaMa17] a PTAS is given for [$\mbox{IIKP}$]{}for arbitrary $T$ and for $\mbox{IKP}$ with increasing time multipliers. All these PTASs rely on solving a huge number of non–trivial LP models. Our approach works for general multipliers $\Delta_t$ and does not require the solution of LP models. Approximation algorithms for a weight constrained IKP variant {#sec:RestrIKP} ============================================================= There are two essential decisions in $\mbox{IKP}$, namely whether to select an item at all, and – if yes – when to select it. Thus, it seems natural not to restrict the latter decision to a subset of time periods and allow each item to be selected at any point in time, also in period $1$. Therefore, in the reminder of the paper we will consider $\mbox{IKP}$ under the mild and natural assumption that each item can be packed in the first period, i.e. $w_i \leq c_1$, $i=1,\dots,n$. We refer to this weight constrained variant of the problem as $\mbox{IKP}^\prime$ and to the related version with unit time multipliers $\Delta_t=1$ for all $t$ as $\mbox{IIKP}^\prime$. Let us denote by $s_t$ the split items in the linear relaxation of each $KP_t$ for $t=1,\dots,T$, which also correspond to the fractional items in the optimal solution of $\mbox{IKP}^\prime_{LP}$. We state the following algorithm $H_1$, which is independent from the given multipliers $\Delta_t$. **Input:** $\mbox{IKP}^\prime$ instance. Sort items by decreasing $\frac{p_i}{w_i}$ $(i = 1,\dots,n)$ and solve $\mbox{IKP}^\prime_{LP}$. Let $\hat{t} = \min\{ t \mid c_t \geq \sum_{j=1}^{s_1} w_j,\, t=1,\ldots, T \}$. Let $p=\max \left \{\sum_{j=1}^{s_1 - 1} p_j, p_{s_1} \right \} $. Pack the item(s) yielding profit $p$ in time periods $1$ to $\hat{t}-1$.Pack all items $j=1,\dots, s_1$ in time periods $\hat{t}$ to $T$. Add the remaining items of the optimal solution of $\mbox{IKP}^\prime_{LP}$ ignoring all fractional values. The following theorem holds. Algorithm $H_1$ has a tight $\frac{1}{2}$-approximation ratio for $\mbox{IKP}^\prime$ for any choice of multipliers $\Delta_t$. Consider the optimal profit values $z_t$ of $KP_t$ for $t=1,\dots,T$ (without multipliers). From the properties of KP and since items are ordered by decreasing $\frac{p_i}{w_i}$, we have: $$\begin{aligned} \max \left \{\sum\limits_{j=1}^{s_t - 1} p_j,p_{s_t} \right \} \geq \frac{1}{2}\, z_t \qquad t=1,\dots,T \label{eq:genKP}\end{aligned}$$ $$\begin{aligned} \frac{\sum_{j=1}^{s_t - 1} p_j }{\sum_{j=1}^{s_t - 1} w_j} \geq \frac{p_{s_t}}{w_{s_t}} \qquad t=1,\dots,T \label{eq:orditems} \end{aligned}$$ Algorithm $H_1$ yields a solution with value $$\begin{aligned} \label{eq:H_1sol} z^{H_1} = \sum_{t=1}^{\hat{t} - 1} \Delta_t \cdot \max \left \{\sum\limits_{j=1}^{s_1 - 1} p_j,p_{s_1} \right \} + \sum_{t=\hat{t}}^{T} \Delta_t \cdot \sum\limits_{j=1}^{s_t - 1} p_j.\end{aligned}$$ Since inequalities $\sum_{j=1}^{s_{t} - 1} w_j \geq \sum_{j=1}^{s_1}w_j > c_1 \geq w_i$ hold for any item $i$ and $t \geq \hat{t}$, from (\[eq:orditems\]) we get $\sum_{j=1}^{s_t - 1} p_j > p_{s_t}$ for any $t \geq \hat{t}$ and thus $$\begin{aligned} \label{eq:afterC1} \max \left \{\sum\limits_{j=1}^{s_t - 1} p_j,p_{s_t} \right \} = \sum\limits_{j=1}^{s_t - 1} p_j \qquad t=\hat{t},\dots,T.\end{aligned}$$ Considering (\[eq:genKP\]), (\[eq:H\_1sol\]), (\[eq:afterC1\]) and that $\sum_{t=1}^{T} \Delta_t z_t$ is an upper bound on $z^*$, we get $$\begin{aligned} \label{eq:1over2} z^{H_1} \geq \frac{1}{2}\sum_{t=1}^{\hat{t} - 1} \Delta_t z_t + \frac{1}{2}\sum_{t=\hat{t}}^{T} \Delta_t z_t \geq \frac{1}{2}z^*\end{aligned}$$ which shows that algorithm $H_1$ has an approximation ratio of $\frac{1}{2}$. To prove the tightness of the bound, consider the following instance with $c_t = M + t$ for $t=1,\dots,T$ (with integer $M \gg T$) and 3 items with entries $$p_1 = \frac{M}{2} + \delta, w_1 = \frac{M}{2}; \; p_2 = \frac{M}{2} + T + 1, w_2 = \frac{M}{2} + T + 1; \; p_3 = \frac{M}{2} - \delta, w_3 = \frac{M}{2};$$ with $\delta > 0$ being an arbitrary small number. Algorithm $H_1$ will select only the second item over all time periods getting a solution with value $\sum_{t=1}^T \Delta_t (\frac{M}{2} + T + 1)$. The optimal $\mbox{IKP}^\prime$ solution will pack items 1 and 3 for all periods independently from the multipliers $\Delta_t$ reaching a value of $\sum_{t=1}^T \Delta_t \cdot M$. Hence, the approximation ratio of the algorithm is $$\frac{\sum_{t=1}^T \Delta_t(\frac{M}{2} + T + 1)}{\sum_{t=1}^T \Delta_t M} = \frac{1}{2} + \frac{T + 1}{M}$$ which reaches a value arbitrarily close to $\frac{1}{2}$ for large values of $M$ and arbitrary multipliers $\Delta_t$. Given the approximation ratio of this simple heuristic, it is worth to investigate to what extent computationally more involved iterative approaches could improve upon the worst case performance of algorithm $H_1$. We consider an algorithm, denoted as $H_2$, which also considers the periods from $t=1$ to $T$ but computes in each step an [*optimal*]{} knapsack solution instead of the rounded down LP-relaxation. **Input:** $\mbox{IKP}^\prime$ instance. Solve $KP_1$ to optimality and pack [$Sol(KP_1)$]{}in all periods $1,\ldots, T$. For all periods $t=2, \ldots, T$:Solve to optimality the knapsack problem induced by $KP_t$ with the condition that all items packed in previous periods remain packed in period $t$.Pack the additional item set in period $t$. Contrary to our expectation, we can show the following negative result for algorithm $H_2$. It turns out quite surprisingly that algorithm $H_2$, which locally dominates $H_1$ for each period, yields a worse performance ratio for the overall problem for almost any choice of multipliers. Only for extremely large $\Delta_1$, namely if $\Delta_1 > \sum_{t=3}^T \Delta_t (t-2) $, $H_2$ outperforms $H_1$ from a worst case perspective. Algorithm $H_2$ has a tight approximation ratio of $\frac{\sum_{t=1}^T \Delta_t}{\sum_{t=1}^T \Delta_t\, t} \geq \frac 1 T$ for $\mbox{IKP}^\prime$. Note that for $\mbox{IIKP}^\prime$ we get a ratio of $\frac{2}{T+1}$. We first show the lower bound on the performance ratio of $H_2$. For every period $t=1,\ldots, T$ we denote by $\pi^*_t$ (resp. $\pi_t$) the total profit of the items added in period $t$ in the optimal solution of $\mbox{IKP}^\prime$ (resp. added by the optimal solution of the residual knapsack problem solved in $H_2$). Note that $\pi_1$ is the optimal solution value of $KP_1$. For every period $t$ denote by $O_t$ all items selected in period $t$ in the optimal ILP solution and by $I_t$ all items packed by $H_2$ (independently from their starting period and without accounting for the multiplier $\Delta_t$). The total profit values contributed in period $t$ are thus $p(O_t) = \sum_{j=1}^t \pi^*_j$ and $p(I_t) = \sum_{j=1}^t \pi_j$ respectively. Now we compare the optimal solution with the outcome generated by $H_2$. By construction there is $\pi^*_1 \leq \pi_1$. We will prove the crucial claim that in each successive time period an additional profit deviation of at most $p_{\max}$ can be accrued, namely: $$\label{eq:indclaim} \pi^*_{t} - \pi_{t} \leq p_{\max} \quad t=2,\ldots, T$$ For any period $t \leq T - 1$, we partition the items added in period $t+1$ and contributing to the optimal profit value $\pi^*_{t+1}$ into a set $I^h$, which are items also in $I_t$, and a set $I^n$ consisting of items not yet packed by $H_2$. We now estimate the value of $\pi_{t+1}$ by constructing an auxiliary set $I'$ as follows. Let $I^p:=O_t \setminus I_t$. All items in $O_{t+1} \setminus I_t = I^p \cup I^n$ are clearly available for $\pi_{t+1}$ (possibly among many others).\ We fill items into $I'$ and stop as soon as $w(I') > c_{t+1} -c_t$. This is done by first considering the items in $I^n$ and then items in $I^p$, both in arbitrary order. In the unusual case that we do not reach the stopping criterion we would have $w(I') = w(I^n) + w(I^p) \leq c_{t+1} - c_t$ and thus $\pi_{t+1} \geq p(I')$. But then we have $O_{t+1} = I' \cup I_t$ and thus even $p(I_{t+1}) \geq p(O_{t+1})$.\ Otherwise, after removing again the item added most recently into $I'$, we have a solution at hand which can be packed by $H_2$ since its weight is at most $c_{t+1} -c_t$. We will prove the following inequality $$\label{eq:claim} p(I') \geq \pi^*_{t+1}.$$ From (\[eq:claim\]) our original claim (\[eq:indclaim\]) follows because removing an item from $I'$ gives a lower bound for $\pi_{t+1}$ and thus we would have $\pi_{t+1} \geq p(I') - p_{\max} \geq \pi^*_{t+1} - p_{\max}$. To prove (\[eq:claim\]), we first show that $I'$ contains all items of $I^n$. Assume otherwise that some item $i \in I^n$ is not in $I'$. Then we have $$c_{t+1} \geq w(O_t) + w(I^n) \geq w(O_t)+ w(I') +w_i > w(O_t) + c_{t+1} -c_t + w_i\, ,$$ which means that $w(O_t) + w_i < c_t$ and thus item $i$ could have been added already to $O_t$ in contradiction to optimality. Next we show that $w(I^h)$ cannot be too large. Denote as $I'':= I' \cap I^p$, i.e. the items added to $I'$ which were already included in $O_t$. Clearly $I' = I'' \cup I^n$. If $w(I^h) \geq c_t - w(O_t \setminus I'')$ then we have: $$\begin{aligned} w(O_{t+1}) &=& w(O_t \setminus I'') + w(I'') + w(I^h)+w(I^n) \\ &=& w(O_t \setminus I'') + w(I^h)+ w(I')\\ &>& w(O_t \setminus I'') + w(I^h) + c_{t+1} - c_t\\ &\geq & w(O_t \setminus I'') + c_t - w(O_t \setminus I'') + c_{t+1} - c_t = c_{t+1}\end{aligned}$$ which is a contradiction. Hence, we must have that $w(I^h) < c_t - w(O_t \setminus I'')$.\ Therefore, $I^h$ could be used in $O_t$ instead of $I''$. The fact that this was not done by the optimal solution implies that $p(I^h) \leq p(I'')$ which is equivalent to $\pi^*_{t+1} = p(I^h) + p(I^n) \leq p(I'') + p(I^n) = p(I')$ and (\[eq:claim\]) is shown. Thus, considering (\[eq:indclaim\]), for any period $t \geq 2$ we have $$\label{FinIneq} \pi^*_1 + \sum_{j=2}^t (\pi^*_j - \pi_j) \leq \pi_1 + (t-1)p_{\max} \implies p(O_t) \leq p(I_t) + (t-1)p_{\max}$$ Summing up (\[FinIneq\]) over all periods we get $$\begin{aligned} z^* &=& \sum_{t=1}^T \Delta_t \, p(O_t) \\ &\leq & \sum_{t=1}^T \Delta_t \left(p(I_t) + (t-1)p_{\max}\right) \\ &=& z^{H_2} + p_{\max} \sum_{t=1}^T \Delta_t(t-1) \\ & \leq& z^{H_2} + \frac{z^{H_2}}{ \sum_{t=1}^T \Delta_t} \cdot \sum_{t=1}^T \Delta_t(t-1) \label{AverZH2} = z^{H_2} \cdot \frac{\sum_{t=1}^T \Delta_t\, t}{ \sum_{t=1}^T \Delta_t}\end{aligned}$$ which shows the stated approximation ratio. The last inequality (\[AverZH2\]) derives from the trivial fact that $\pi_1 \geq p_{\max}$ and thus $z^{H_2} \geq p_{\max} \sum_{t=1}^T \Delta_t$. The minimum of this bound is reached for multipliers $\Delta_1 = \ldots = \Delta_{T-1} = \eps$ for small $\eps>0$ and $\Delta_T=1$ where the approximation ratio tends to $\frac 1 T$. We can prove the tightness of the bound for arbitrary multipliers $\Delta_t$ by considering the following instance with capacities $c_t = t\cdot M + (T-t)$ and $2\,T$ items with entries $$\begin{aligned} w_1 = p_1 &=& M + (T-1),\\ w_2 = w_3 = \cdots = w_T &=& M-1,\\ p_2 = p_3 = \cdots = p_T &=& 1,\\ w_{T+1} = w_{T+2} = \cdots = w_{2T} &=& M,\\ p_{T+1} = p_{T+2} = \cdots = p_{2T} &=& M.\end{aligned}$$ Note that $c_{t+1} - c_t = M-1$. The solution provided by algorithm $H_2$ starts with $KP_1$ and packs item $1$. For $KP_2$ the residual capacity is $c_2-c_1= M-1$, so only item $2$ with profit $1$ can be packed. This argument continues for each $KP_t$. Hence, we have as solution value $$z^{H_2} = \sum_{t=1}^T \Delta_t \cdot (M+(T-1)) + \sum_{t=2}^T \Delta_t (t-1).$$ The optimal $\mbox{IKP}^\prime$ solution packs $t$ copies of items with weight and profit $M$ in each time period $t$ which yields $$z^* = \sum_{t=1}^T \Delta_t \, t \cdot M.$$ This shows the tightness of the approximation bound for large $M$. Mirroring the iterative strategy of algorithm $H_2$, one may consider another natural heuristic approach which starts from the optimal solution of the last knapsack problem $KP_T$ and moves upwards until period $1$ by solving to optimality the knapsack problem induced by $KP_t$ $(t = T -1, \dots, 1)$ with item set restricted to those items packed in period $t+1$. Still, it turns out that this strategy cannot improve the approximation ratio of the polynomial time heuristic $H_1$. We show this observation by giving a class of instances where the above algorithm reaches only $z^*/3$. Consider the following $\mbox{IIKP}^\prime$ instance with capacities $c_t = 4 - \gamma$, ($t=1, \dots, T - 2$), $c_{(T - 1)} = 4 + \gamma$, $c_{T} = 5$ (with $\gamma > 0$ being an arbitrary small number) and four items with entries: $j$ 1 2 3 4 ------- -------------- -------------- --- -------------- $p_j$ $1 + \gamma$ $1 + \gamma$ 1 2 $w_j$ 2 2 1 $3 - \gamma$ The sketched alternative approach will select items 1, 2, 3 in the last period $T$, items 1 and 2 in period $T - 1$ and either item 1 or item 2 in the remaining periods. The corresponding profit is equal to $$(3+2\gamma) + (2+2\gamma) + (1+\gamma)(T-2) = (1+\gamma) T + 3 +2\gamma.$$ The optimal solution instead consists in packing items 3 and 4 in all periods, hence $z^{*} = 3T$. Consequently, the approximation ratio of the algorithm is strictly less than $\frac{1}{2}$ for large enough $T$ and cannot be greater than $\frac{1}{3}$ as the number of periods $T$ increases. An approximation algorithm for a weight constrained IKP variant with two periods \[sec:RestrIKP23\] =================================================================================================== In this section we consider $\mbox{IKP}^\prime$ for the special cases with two periods only, namely $T=2$. It turns out that a more careful analysis of the solution structures of the knapsack subproblems $KP_t$ by means of an LP model gives much better approximation ratios than the more general results presented in Section \[sec:RestrIKP\]. Let us define for the optimal solution sets [$Sol(KP_1)$]{}and [$Sol(KP_2)$]{}the following subsets illustrated in Figure \[FigureSi\]: - $S_{12}$: subset of items included in both optimal solutions [$Sol(KP_1)$]{}and [$Sol(KP_2)$]{}; - $S_1$: remaining subset of items in [$Sol(KP_1)$]{}; - $S_2^a$: remaining subset of items not exceeding capacity $c_1$ in [$Sol(KP_2)$]{}; - $S_2^\prime$: first item exceeding $c_1$ in [$Sol(KP_2)$]{}; - $S_2^b$: remaining subset of items in [$Sol(KP_2)$]{}; Each subset could as well be empty. The dashed lines in Figure \[FigureSi\] refer to the item $S_2^\prime$ which exceeds the first capacity value. \(c) at (0.6,0); ($(c)+(0,1)$) rectangle ($(c)+(2,1.5)$); at ($(c)+(1,1.25)$) [$S_{12}$]{}; ($(c)+(2,1)$) rectangle ($(c)+(5,1.5)$); at ($(c)+(3.5,1.25)$) [$S_1$]{}; ($(c)+(2,0)$) rectangle ($(c)+(4.5,0.5)$); at ($(c)+ (3.25,0.25)$) [$S_2^a$]{}; ($(c)+(4.5,0)$) rectangle ($(c)+(5.5,0.5)$); at ($(c)+(5,0.25)$) [$S_2^\prime$]{}; ($(c)+(5.5,0)$) rectangle ($(c)+(7,0.5)$); at ($(c)+(6.25,0.25)$) [$S_2^b$]{}; ($(c)+(0,0)$) rectangle ($(c)+(2,0.5)$); at ($(c)+(1,0.25)$) [$S_{12}$]{}; at (0,1.25) [$KP_1:$]{}; at (0,0.25) [$KP_2:$]{}; at ($(c)+(5.0,1.75)$) [$c_1$]{}; at ($(c)+(7.0,0.75)$) [$c_2$]{}; According to the above definitions, we have the following inequalities: $$\begin{aligned} & w(S_{12}) + w(S_1) \leq c_1 \label{CaseW1} \\ & w(S_{12}) + w(S_2^a) \leq c_1 \label{CaseW2} \\ & w(S_{12}) + w(S_2^a) + w(S_2^\prime) > c_1 \label{CaseW3}\\ & w(S_{12}) + w(S_2^a) + w(S_2^\prime) + w(S_2^b) \leq c_2 \label{CaseW4}\\ & w(S_{12}) + w(S_1) + w(S_2^b) < c_2 \label{CaseW5}\end{aligned}$$ Inequality (\[CaseW5\]) derives directly from inequalities (\[CaseW1\]), (\[CaseW3\]) and (\[CaseW4\]). The optimal solution values of $KP_1$ and $KP_2$ are $z_1 = p(S_{12}) + p(S_1)$ and $z_2 = p(S_{12}) + p(S_2^a) + p(S_2^\prime) + p(S_2^b)$ respectively. Now we can state three feasible solutions for $\mbox{IKP}^\prime$: 1. \[1Sol\] [$Sol(KP_1)$]{}in the two periods plus the additional packing of items in $S_2^b$ in the second period with total profit $$\Delta_1(p(S_{12}) + p(S_1)) + \Delta_2(p(S_{12}) + p(S_1) + p(S_2^b));$$ 2. \[2Sol\] [$Sol(KP_2)$]{}in the second period with the packing of items in subsets $S_{12}$ and $S_2^a$ in the first period and resulting profit $$\Delta_1(p(S_{12}) + p(S_2^a)) + \Delta_2(p(S_{12}) + p(S_2^a)+p(S_2^\prime) + p(S_2^b));$$ 3. \[3Sol\] [$Sol(KP_2)$]{}in the second period with item $S_2^\prime$ placed in the knapsack in the first period. The profit of this solution is $$\Delta_1p(S_2^\prime) + \Delta_2(p(S_{12}) + p(S_2^a)+ p(S_2^\prime) + p(S_2^b)).$$ We introduce an algorithm, hereafter denoted as $H_{T2}$, which simply returns the best of these solutions. **Input:** $\mbox{IKP}^\prime$ instance with $T=2$. Compute [$Sol(KP_1)$]{}and [$Sol(KP_2)$]{}. Identify subsets $S_{12}, S_1, S_2^a, S_2^\prime, S_2^b$ and compute solutions \[1Sol\], \[2Sol\], \[3Sol\]. Return the best solution found. \[Theo:T=2\] For $\mbox{IKP}^\prime$ with $T=2$, algorithm $H_{T2}$ has a tight approximation ratio of $\frac{1 + 3\Delta_r + 2\Delta_r^2}{1 + 4\Delta_r + 2\Delta_r^2}$, where $\Delta_r = \frac{\Delta_2}{\Delta_1}$. In order to evaluate the worst case performance of algorithm $H_{T2}$, we consider an LP formulation where we associate a non–negative variable $h$ with the solution value computed by the algorithm. In addition, the profits of the subsets $p(S_{(\cdot)})$ are associated with non–negative variables $s_{(\cdot)}$. As in Section \[sec:HarmNumRatios\], the positive parameter $OPT$ represents $z^*$. This implies the following parametric LP model with $\Delta_1$ and $\Delta_2$: $$\begin{aligned} \text{minimize}\quad & h \label{eq:ObjT2}\\ \text{subject to}\quad & h - (( \Delta_1 + \Delta_2)(s_{12} + s_1) + \Delta_2 s_2^b)\geq 0 \label{eq:AlgoH1}\\ & h - (( \Delta_1 + \Delta_2)(s_{12} + s_2^a) + \Delta_2(s_2^\prime + s_2^b)) \geq 0 \label{eq:AlgoH2}\\ & h - (( \Delta_1 + \Delta_2)s_2^\prime + \Delta_2( s_{12} + s_2^a + s_2^b)) \geq 0 \label{eq:AlgoH3}\\ & \Delta_1(s_{12} + s_1) + \Delta_2(s_{12} + s_2^a + s_2^\prime + s_2^b) \geq OPT \label{eq:UBAlgoH}\\ & h, s_{12}, s_1, s_2^a, s_2^\prime, s_2^b \geq 0 \label{eq:AlgoHprof} $$ The objective function value (\[eq:ObjT2\]) represents a lower bound on the worst case performance of algorithm $H_{T2}$. Constraints (\[eq:AlgoH1\])–(\[eq:AlgoH3\]) guarantee that $H_{T2}$ will select the best of the three feasible solutions. Constraint (\[eq:UBAlgoH\]) states that the sum $\Delta_1 z_1 + \Delta_2 z_2$ constitutes an upper bound on $z^*$. Constraints (\[eq:AlgoHprof\]) indicate that the variables are non-negative. A feasible solution of model (\[eq:ObjT2\])–(\[eq:AlgoHprof\]), for any $\Delta_1 > 0$ and $\Delta_2 > 0$, is: $$\begin{aligned} & h= \frac{(1 + 3\Delta_r + 2\Delta_r^2)OPT}{1 + 4\Delta_r + 2\Delta_r^2},\, s_1 = \frac{\Delta_r OPT}{\Delta_1(1 + 4\Delta_r + 2\Delta_r^2)} \nonumber \\ & s_{12} = s_2^\prime= \frac{(1 + \Delta_r) OPT}{\Delta_1(1 + 4\Delta_r + 2\Delta_r^2)},\, s_2^a = s_2^b = 0. \nonumber\end{aligned}$$ We will show by strong duality that such a solution is also optimal for any positive value of $\Delta_1$ and $\Delta_2$. If we denote by $\lambda_i$ with $i=1, \dots, 4$ the dual variables related to constraints (\[eq:AlgoH1\])–(\[eq:UBAlgoH\]), the dual linear problem corresponding to model (\[eq:ObjT2\])–(\[eq:AlgoHprof\]) is as follows: $$\begin{aligned} \text{maximize}\quad & OPT \cdot \lambda_4 \label{eq:ObjDualT2}\\ \text{subject to}\quad & \lambda_1 + \lambda_2 + \lambda_3 \leq 1 \label{eq:Dualh}\\ & \Delta_1\lambda_4 - (\Delta_1 + \Delta_2)\lambda_1 \leq 0 \label{eq:Duals1}\\ & -\Delta_2\lambda_3 - (\Delta_1 + \Delta_2)(\lambda_1 + \lambda_2 - \lambda_4) \leq 0 \label{eq:Duals12}\\ & \Delta_2(\lambda_4 - \lambda_3) - (\Delta_1 + \Delta_2)\lambda_2 \leq 0 \label{eq:Duals2a}\\ & \Delta_2(\lambda_4 - \lambda_2) - (\Delta_1 + \Delta_2)\lambda_3 \leq 0 \label{eq:Duals2prime}\\ & - \Delta_2(\lambda_1 + \lambda_2 + \lambda_3 - \lambda_4) \leq 0 \label{eq:Duals2b}\\ & \lambda_1, \lambda_2, \lambda_3, \lambda_4 \geq 0 \label{eq:DualvarT2}\end{aligned}$$ Constraints (\[eq:Dualh\])–(\[eq:Duals2b\]) correspond to primal variables $h, s_1, s_{12}, s_2^a, s_2^\prime, s_2^b$ respectively. A feasible dual solution (for any $\Delta_1 > 0$ and $\Delta_2 > 0$) reads $$\begin{aligned} & \lambda_1 = \frac{1 + 2\Delta_r}{1 + 4\Delta_r + 2\Delta_r^2},\, \lambda_2 = \lambda_3 = \frac{(1 + \Delta_r) \Delta_r}{1 + 4\Delta_r + 2\Delta_r^2},\, \lambda_4 = \frac{1 + 3\Delta_r + 2\Delta_r^2}{1 + 4\Delta_r + 2\Delta_r^2}. \nonumber \end{aligned}$$ Hence, the dual solution value $OPT\cdot \lambda_4 = h$ proves by strong duality the optimality of the primal solution for any value of the two time multipliers. The corresponding lower bound on the performance ratio provided by algorithm $H_{T2}$ is equal to $\frac{h}{OPT} = \frac{1 + 3\Delta_r + 2\Delta_r^2}{1 + 4\Delta_r + 2\Delta_r^2}$. We can show the tightness of the bound by considering two different instances for $\Delta_r \leq 1$ and $\Delta_r > 1$ respectively, with $n=5$, $c_1 = 3 + \gamma$, $c_2 = 4$ and following entries: $i$ 1 2 3 4 5 ------------------- ------- -------------------------------- ------------------------------- -------------------------------- ---------------------------------------- ---------------------- $\Delta_r \leq 1$ $p_i$ $\frac{1+\Delta_r}{\Delta_1}$ $\frac{1+\Delta_r}{\Delta_1}$ $\frac{\Delta_r}{\Delta_1}$ $\frac{1+2\Delta_r -\gamma}{\Delta_1}$ $\frac{1}{\Delta_1}$ $w_i$ $2$ $2$ $1 + \gamma$ $3 - 2\gamma$ $1 + 2\gamma$ $\Delta_r > 1$ $p_i$ $\frac{1+2\Delta_r}{\Delta_1}$ $\frac{1+\Delta_r}{\Delta_1}$ $\frac{1+ \Delta_r}{\Delta_1}$ $\frac{\Delta_r -\gamma}{\Delta_1}$ $\frac{1}{\Delta_1}$ $w_i$ $3 + \gamma$ $2$ $2$ $1$ $1$ First, consider the instance for the case $\Delta_r \leq 1$. The optimal solutions of $KP_1$ and $KP_2$ consist of items 1, 3 and items 1, 2 respectively. Correspondingly, we have $$S_{12} = \{ 1 \}, S_1 = \{ 3 \}, S_2^a = \{ \emptyset \}, S_2^\prime = \{ 2 \}, S_2^b = \{ \emptyset \}.$$ An easy computation reveals that all the solutions provided by algorithm $H_{T2}$ have the same profit value of $1+ 3\Delta_r + 2\Delta_r^2$. The optimal $\mbox{IKP}^\prime$ solution selects item 4 in the first period together with item 5 in the second one, hence $z^* = 1+ 4\Delta_r + 2\Delta_r^2 - (1 + \Delta_r)\gamma$.\ In the second instance with $\Delta_r > 1$, we have $Sol(KP_1) = \{ 1 \}, Sol(KP_2) = \{ 2, 3 \}$ and thus $$S_{12} = \{ \emptyset \}, S_1 = \{ 1 \}, S_2^a = \{ 2 \}, S_2^\prime = \{ 3 \}, S_2^b = \{ \emptyset \}.$$ Also in this case all solutions provided by the algorithm have a profit equal to $1+ 3\Delta_r + 2\Delta_r^2$. An optimal solution packs items 2 and 4 in the first period together with item 5 in the second one, hence $z^* = 1+ 4\Delta_r + 2\Delta_r^2 - (1 + \Delta_r)\gamma$.\ In both cases, the approximation ratio of algorithm $H_{T2}$ can be arbitrarily close to ratio $\frac{1 + 3\Delta_r + 2\Delta_r^2}{1 + 4\Delta_r + 2\Delta_r^2}$ as the value of $\gamma$ goes to 0. \[Coro:T2\_1\] The approximation ratio of algorithm $H_{T2}$ is bounded from below by $\frac{2\sqrt{2} + 3}{2\sqrt{2} + 4}= \frac{1}{2} + \frac{\sqrt{2}}{4}$. When $H_{T2}$ is applied to $\mbox{IIKP}^\prime$, the ratio is $\frac{6}{7}$. A straightforward computation reveals that function $\frac{1 + 3\Delta_r + 2\Delta_r^2}{1 + 4\Delta_r + 2\Delta_r^2}$ has a unique global minimum (for $\Delta_r > 0$) at $\Delta_r = \frac{\sqrt{2}}{2}$. Hence, we get that the ratio $\frac{h}{OPT}$ is bounded from below by $\frac{2\sqrt{2} + 3}{2\sqrt{2} + 4}$. For $\mbox{IIKP}^\prime$, we have $\Delta_r =1$. Correspondingly, $\frac{1+3\Delta_r+2\Delta^2_r}{1+4\Delta_r+2\Delta^2_r} =\frac{6}{7}$. Considering algorithm $H_{T2}$ the following improvement comes to mind: Solve to optimality $KP_2$ after packing the item set [$Sol(KP_1)$]{}in both periods (as in $H_2$) and, as an alternative, optimally solve $KP_1$ restricted to the item set [$Sol(KP_2)$]{}. However, the above tight instance would also apply to this computationally more demanding variation and thus no improvement of the approximation ratio can be gained. Algorithm $H_{T2}$ is not a polynomial time algorithm since it requires the optimal solutions of $KP_1$ and $KP_2$. We could solve these knapsack problems by an $\eps$–approximation scheme (FPTAS) to get a polynomial running time at the cost of a decrease of the approximation ratio. The following corollary shows that the ratio is decreased by a factor $(1 - \eps)$. \[Coro:T2\] If an $\eps$–approximation scheme is employed for solving the standard knapsack problems $KP_1$ and $KP_2$, the approximation ratio of algorithm $H_{T2}$ is at least $ \frac{1 + 3\Delta_r + 2\Delta_r^2}{1 + 4\Delta_r + 2\Delta_r^2}(1-\eps)$. We consider again model (\[eq:ObjT2\])–(\[eq:AlgoHprof\]) to evaluate the performance of the algorithm. If the subsets $S_{(\cdot)}$ correspond to the items of the $\eps$–approximations for $KP_1$ and $KP_2$, constraints (\[eq:AlgoH1\])–(\[eq:AlgoH3\]) straightforwardly hold. Then, we just replace constraint (\[eq:UBAlgoH\]) by constraint $$\label{eq:epsUBAlgoH} \Delta_1(s_{12} + s_1) + \Delta_2(s_{12} + s_2^a + s_2^\prime + s_2^b) \geq OPT(1-\eps) $$ which indicates that the weighted sum of the approximate solutions divided by $(1-\eps)$ provides an upper bound on the optimal solution value of $\mbox{IKP}$. In the related primal/dual LP models, the optimal values of primal variables are now multiplied by $(1-\eps)$ while the optimal dual solution remains the same with objective value $(1-\eps)OPT \cdot \lambda_4$. The corresponding value of the ratio $\frac{h}{OPT}$ shows the bound. Conclusions {#TheConcl} =========== We proposed for $\mbox{IKP}$ a series of results extending in different directions the contributions currently available in the literature. We proved the tightness of the approximation ratio of a general purpose algorithm presented in [@HaSh06] and originally applied to the time-invariant version of the problem. We also established a polynomial time approximation scheme (PTAS) when one of the problem inputs can be considered as a constant. We then focused on a restricted relevant variant of $\mbox{IKP}$ which plausibly assumes the possible packing of any item since the first period. We discussed the performance of different approximation algorithms for the general case as well as for the variation with two time periods. In future research, we will investigate extensions of our procedures to the design of improving approximation algorithms for variants involving more than two periods. Also, since to the authors’ knowledge no computational experience has been provided for $\mbox{IKP}$ so far, it would also be interesting to derive new solution approaches and test their performance after generating benchmarks and challenging to solve instances. Acknowledgments {#acknowledgments .unnumbered} --------------- Ulrich Pferschy was supported by the project “Choice-Selection-Decision" and by the COLIBRI Initiative of the University of Graz. \[sec:bib\]

--- abstract: 'Certain quantum codes allow logic operations to be performed on the encoded data, such that a multitude of errors introduced by faulty gates can be corrected. An important class of such operations are [*transversal*]{}, acting bitwise between corresponding qubits in each code block, thus allowing error propagation to be carefully limited. If any quantum operation could be implemented using a set of such gates, the set would be [*universal*]{}; codes with such a universal, transversal gate set have been widely desired for efficient fault-tolerant quantum computation. We study the structure of $GF(4)$-additive quantum codes and prove that no universal set of transversal logic operations exists for these codes. This result strongly supports the idea that additional primitive operations, based for example on quantum teleportation, are necessary to achieve universal fault-tolerant computation on additive codes.' author: - 'Bei Zeng, Andrew Cross, and Isaac L. Chuang[^1]' bibliography: - 'IEEEabrv.bib' - 'tn\_vs\_un.bib' title: | Transversality versus Universality\ for Additive Quantum Codes --- Introduction ============ he study of fault-tolerant quantum computation is essentially driven by the properties of quantum codes – specifically, what logic operations can be implemented on encoded data, without decoding, and while controlling error propagation [@shor:ftqc; @Preskill98b; @gottesman:HR; @nielsen:NC]. Quantum code automorphisms, and their close relatives, [*transversal*]{} gates, are among the most widely used and simplest fault-tolerant logic gates; uncorrelated faults before and during such gates result in uncorrelated errors in the multi-qubit blocks. Transversal gates, in particular, are gates that act bitwise, such that they may be represented by tensor product operators in which the $j$th term acts only on the $j$th qubit from each block [@gottesman:thesis]. Much like in classical computation, not all gate sets can be composed to realize an arbitrary operation, however. It would be very desirable to find a [*universal*]{} transversal gate set, from which any quantum operation could be composed, because this could dramatically simplify resource requirements for fault-tolerant quantum computation [@Metodi05a; @Oskin02]. In particular, the accuracy threshold would likely improve, if any quantum computation could be carried out with transversal gates alone[@aliferis:bs]. Many of the well-known $GF(4)$-additive codes (also known as [*stabilizer*]{} codes [@crss:gf4; @gottesman:thesis]) have been exhaustively studied, for their suitability for fault tolerant quantum computation. However, no quantum code has yet been discovered, which has automorphisms allowing a universal transversal gate set. Specifically, an important subset, the [*CSS*]{} codes [@crss:gf4; @steane:ec; @steane:fta], all admit a useful two-qubit transversal primitive, the controlled-[not]{} (“CNOT”) gate, but each [*CSS*]{} code seems to lack some important element that would fill a universal set. For example, the $[[n,k,d]] = [[7,1,3]]$ Steane code [@steane:ec], based on a Hamming code and its dual, has transversal gates generating the Clifford group. This group is the finite group of symmetries of the Pauli group[@nielsen:NC], and may be generated by the CNOT, the Hadamard, and the single-qubit phase gate. For the Steane code, a Clifford gate can be implemented by applying that gate (or its conjugate) to each coordinate [@shor:ftqc]. Moreover, encoding, decoding, and error correction circuits for [*CSS*]{} codes can be constructed entirely from Clifford operations, and thus Clifford group gates are highly desirable for efficient fault-tolerant circuits. Unfortunately, it is well known that gates in the Clifford group are not universal for quantum computation, as asserted by the Gottesman-Knill theorem [@nielsen:NC; @gottesman:HR]. In fact, Rains has shown that the automorphism group of any $GF(4)$-linear code (i.e. the [*CSS*]{} codes) lies in the Clifford group [@rains:d2]. Because of this, and also exhaustive searches, it is believed that the Steane code, which is a $GF(4)$-linear code, does not have a universal set of transversal gates. The set of Clifford group gates is not universal, but it is also well known that the addition of nearly any gate outside of this set (any “non-Clifford” gate) can complete a universal set [@nebe:clifford]. For example, the single-qubit $\pi/8$, or $T=\operatorname{diag}{(1,e^{i\pi/4})}$ gate, is one of the simplest non-Clifford gates which has widely been employed in fault-tolerant constructions. Codes have been sought which allow a transversal $T$ gate. Since additive codes have a simple structure, closely related to the abelian subgroup of Pauli groups, transversal Clifford gates for such codes may be constructed systematically [@gottesman:thesis]. However, how to find non-Clifford transversal gates for a given code is not generally known. Some intriguing examples have been discovered, however. Strikingly, the $[[15,1,3]]$ [*CSS*]{} code constructed from a punctured Reed-Muller code has a transversal $T$ gate [@knill:ta]. Rather frustratingly, however, this code does not admit a transversal Hadamard gate, thus leaving the Clifford gate set incomplete, and rendering the set of transversal gates on that code non-universal. In fact, all known examples of transversal gate sets on quantum codes have been deficient in one way or another, leading to non-universality. Some of the known $[[n,1,3]]$ code results are listed in Table \[table1\]. None of these codes listed, or known so far in the community, allows a universal set of transversal gates. [|c|c|c|]{} ------------------------------------------------------------------------ Code & Transversal gates & Gates not transversal\ ------------------------------------------------------------------------ $[[5,1,3]]$ & $PH$, M$_\text{3}$ & $H$, $P$, CNOT, $T$\ ------------------------------------------------------------------------ $[[7,1,3]]$ & $H$, $P$, CNOT & $T$\ ------------------------------------------------------------------------ $[[9,1,3]]$ & CNOT & $H$, $P$, $T$\ ------------------------------------------------------------------------ $[[15,1,3]]$ & $T$, CNOT & $H$\ ------------------------------------------------------------------------ $[[2^m-1,1,3]]$ &$T_\text{m}$, CNOT & $H$\ Considering the many unsuccessful attempts to construct a code with a universal set of transversal gates, it has been widely conjectured in the community that transversality and universality on quantum codes are incompatible; specifically, it is believed that *no universal set of transversal gates exists, for any quantum code $Q$*, even allowing for the possibility of additional qubit permutation operations inside code blocks. Our main result, given in Section \[sec:transversal\], proves a special case of this “$T$ versus $U$” incompatibility, where $Q$ is a $GF(4)$-additive code and coordinate permutations are not allowed. Our proof relies on earlier results by Rains [@rains:d2] and Van den Nest [@nest:lulc], generalized to multiple blocks encoded in additive quantum codes. In Section \[sec:permutation\], we prove $T$ vs. $U$ incompatibility for a single block of qubits encoded in a $GF(4)$-additive code, by clarifying the effect of coordinate permutations. In Section \[sec:applications\], we consider the allowable transversal gates on additive codes, using the proof technique we employ. We also present a simple construction based on classical divisible codes that yields many quantum codes with non-Clifford transversal gates on a single block. We begin in the next section with some preliminary definitions and terminology. Preliminaries ============= This section reviews definitions and preliminary results about additive codes [@crss:gf4], Clifford groups and universality, automorphism groups, and codes stabilized by minimal elements. Throughout the paper, we only consider $GF(4)$-additive codes, i.e. codes on qubits, leaving more general codes to future work. We use the stabilizer language to describe $GF(4)$-additive quantum codes, which are also called binary stabilizer codes. Stabilizers and stabilizer codes -------------------------------- The *$n$-qubit Pauli group* ${\mathcal G}_n$ consists of all $4\times 4^n$ operators of the form $R=\alpha_R R_1\otimes\dots\otimes R_n$, where $\alpha_R\in\{\pm 1,\pm i\}$ is a phase factor and each $R_i$ is either the $2\times 2$ identity matrix $I$ or one of the Pauli matrices $X$, $Y$, or $Z$. A *stabilizer* $\mathcal{S}$ is an abelian subgroup of the $n$-qubit Pauli group ${\mathcal G}_n$ which does not contain $-I$. A *support* is a subset of $[n]:=\{1,2,\dots,n\}$. The support $\operatorname{supp}{(R)}$ of an operator $R\in {\mathcal G}_n$ is the set of all $i\in [n]$ such that $R_i$ differs from the identity, and the *weight* $wt(R)$ equals the size $|supp(R)|$ of the support. The set of elements in $\mathcal{G}_n$ that commute with all elements of $\mathcal{S}$ is the *centralizer* $\mathcal{C}(\mathcal{S})$. We have the relation $[XXXX,ZZZZ]=0$ where $XXXX=X^{\otimes 4}$ represents a tensor product of Pauli operators. Consider the stabilizer $\mathcal{S}=\langle XXXX, ZZZZ\rangle$ where $\langle\cdot\rangle$ indicates a generating set, so $$\mathcal{S}=\{ IIII, XXXX, ZZZZ, YYYY\}.$$ We have $\operatorname{supp}{(XXXX)}=\{1,2,3,4\}$ and $\operatorname{wt}{(XXXX)}=4$, for example. Finally, the centralizer is $\mathcal{C}(\mathcal{S})=\langle\mathcal{S}, ZZII,ZIZI, XIXI, XXII\rangle$. A stabilizer consists of $2^m$ Pauli operators for some nonnegative integer $m\leq n$ and is generated by $m$ independent Pauli operators. As the operators in a stabilizer are Hermitian and mutually commuting, they can be diagonalized simultaneously. An $n$-qubit *stabilizer code* $Q$ is the joint eigenspace of a stabilizer $\mathcal{S}(Q)$, $$Q=\{ |\psi\rangle\in ({\mathbb C}^2)^{\otimes n}\ |\ R|\psi\rangle=|\psi\rangle,\forall R\in \mathcal{S}(Q)\}$$ where each state vector $|\psi\rangle$ is assumed to be normalized. $Q$ has dimension $2^{n-m}$ and is called an $[[n,k,d]]$ stabilizer code, where $k=n-m$ is the *number of logical qubits* and $d$ is the *minimum distance*, which is the weight of the minimum weight element in $\mathcal{C}(\mathcal{S})\setminus \mathcal{S}$. The code $Q$ can correct errors of weight $t\leq\lfloor\frac{d-1}{2}\rfloor$. Continuing, we have $$\begin{aligned} Q = \textrm{span}\{ & |0000\rangle + |1111\rangle, |0011\rangle+|1100\rangle, \\ & |1010\rangle+|0101\rangle, |1001\rangle+|0110\rangle\}\end{aligned}$$ so $n=4$, $m=2$, $\dim{Q}=4$, and $k=2$. From $C(\mathcal{S})\setminus \mathcal{S}$, we see that $d=2$. Therefore, $Q$ is a $[[4,2,2]]$ code. Each set of $n$ mutually commuting independent elements of $\mathcal{C}(\mathcal{S})$ stabilizes a quantum codeword and generates an abelian subgroup of the centralizer. This leads to the isomorphism $\mathcal{C}(\mathcal{S})/\mathcal{S}\cong \mathcal{G}_k$ that maps each element $X_i,Z_i\in\mathcal{G}_k$ to a coset representative $\bar{X}_i,\bar{Z}_i\in C(\mathcal{S})/\mathcal{S}$ [@gottesman:thesis]. The isomorphism associates the $k$ logical qubits to logical Pauli operators $\bar{X}_i,\bar{Z}_i$ for $i=1,\ldots,k$, and these operators obey the commutation relations of $\mathcal{G}_k$. One choice of logical Pauli operators for the $[[4,2,2]]$ code is $\bar{X}_1=XIXI$, $\bar{Z}_1=ZZII$, $\bar{X}_2=XXII$, and $\bar{Z}_2=ZIZI$. These satisfy the commutation relations of ${\mathcal G}_2$. Universality ------------ Stabilizer codes are stabilized by subgroups of the Pauli group, so some unitary operations that map the Pauli group to itself also map the stabilizer to itself, preserving the code space. The *$n$-qubit Clifford group* ${\mathcal L}_n$ is the group of unitary operations that map ${\mathcal G}_n$ to itself under conjugation. One way to specify a gate in ${\mathcal L}_n$ is to give the image of a generating set of ${\mathcal G}_n$ under that gate. ${\mathcal L}_n$ is generated by the single qubit Hadamard gate, $$H: (X,Z)\rightarrow (Z,X),$$ the single qubit Phase gate $$P: (X,Z)\rightarrow (-Y,Z),$$ and the two-qubit controlled-not gate $$\begin{aligned} \textrm{CNOT}: & (XI,IX,XX,ZI,IZ,ZZ)\rightarrow \\ & (XX,IX,XI,ZI,ZZ,IZ)\end{aligned}$$ by the Gottesman-Knill theorem [@nielsen:NC; @gottesman:HR]. \[def:universality\] A set of unitary gates $G$ is (quantum) *computationally universal* if for any $n$, any unitary operation $U\in SU(2^n)$ can be approximated to arbitrary accuracy $\epsilon$ in the sup operator norm $||\cdot||$ by a product of gates in $G$. In notation, $\forall \epsilon>0, \exists V=V_1V_2\dots V_{\eta(\epsilon)}\ \textrm{where each}\ V_i\in G\ \textrm{s.t.}\ ||V-U||<\epsilon.$ In this definition, gates in $G$ may be implicitly mapped to isometries on the appropriate $2^n$-dimensional Hilbert space. The Gottesman-Knill theorem asserts that any set of Clifford group gates can be classically simulated and is therefore not (quantum) computationally universal. Quantum teleportation is one technique for circumventing this limit and constructing computationally universal sets of gates using Clifford group gates and measurements of Pauli operators [@gottesman:teleport; @zhou:ft]. There is a large set of gates that arise in fault-tolerant quantum computing through quantum teleportation. The *${\mathcal C}_k^{(n)}$ hierarchy* is a set of gates that can be achieved through quantum teleportation and is defined recursively as follows: ${\mathcal C}_1^{(n)}={\mathcal G}_n$ and $${\mathcal C}_k^{(n)} = \{ U\in SU(2^n)\ |\ UgU^\dag\in {\mathcal C}_{k-1}^{(n)}\ \forall\ g\in {\mathcal C}_1^{(n)}\},$$ for $k>1$. ${\mathcal C}_k^{(n)}$ is a group only for $k=1$ and $k=2$ and ${\mathcal C}_2^{(n)}={\mathcal L}_n$.\[def:ck\] The Clifford group generators $\{H,P,CNOT\}$ plus any other gate outside of the Clifford group is computationally universal [@nebe:clifford]. For example, the gates $T=\operatorname{diag}{(1,e^{i\pi/4})}\in {\mathcal C}_3^{(1)}\setminus {\mathcal C}_2^{(1)}$ and $\textsc{Toffoli}\in {\mathcal C}_3^{(3)}\setminus {\mathcal C}_2^{(3)}$ are computationally universal when taken together with the Clifford group. Automorphisms of stabilizer codes --------------------------------- An automorphism is a one-to-one, onto map from some domain back to itself that preserves a particular structure of the domain. We are interested in quantum code automorphisms, unitary maps that preserve the code subspace and respect a fixed tensor product decomposition of the $n$-qubit Hilbert space. The weight distribution of an arbitrary operator with respect to the Pauli error basis ${\mathcal G}_n$ is invariant under these maps. With respect to the tensor product decomposition, we can assign each qubit a coordinate $j\in [n]$, in which case the quantum code automorphisms are those local operations and coordinate permutations that correspond to logical gates. In some cases, these automorphisms correspond to the permutation, monomial, and/or field automorphisms of classical codes [@huffman:HP]. This section formally defines logical gates and quantum code automorphisms on an encoded block. A unitary gate $U\in SU(2^n)$ acting on $n$ qubits is a *logical gate on $Q$* if $[U,P_Q]=0$ where $P_Q$ is the orthogonal projector onto $Q$ given by $$P_Q = \frac{1}{2^m}\sum_{R\in \mathcal{S}(Q)}R.$$ Let ${\mathbb V}(Q)$ denote the set of logical gates on $Q$. When $Q$ is understood, we simply say that the gate $U$ is a logical gate. The logical gates ${\mathbb V}(Q)$ are a group that is homomorphic to $SU(2^k)$ since it is possible to encode an arbitrary $k$-qubit state in the code. For the $[[4,2,2]]$ code, $P_Q=\frac{1}{4}(I^{\otimes 4}+X^{\otimes 4}+Y^{\otimes 4}+Z^{\otimes 4})$. Any unitary acting in the code manifold $$\begin{aligned} \alpha (|0000\rangle+|1111\rangle) + \beta(|0011\rangle+|1100\rangle) + \\ \gamma(|1010\rangle+|0101\rangle) + \delta(|1001\rangle+|0110\rangle)\end{aligned}$$ is a logical gate. The *full automorphism group* $\operatorname{Aut}{(Q)}$ of $Q$ is the collection of all logical operations on $Q$ of the form $P_\pi U$ where $P_\pi$ enacts the coordinate permutation $\pi$ and $U=U_1\otimes\dots\otimes U_n$ is a local unitary operation. The product of two such operations is a logical operation of the same form, and operations of this form are clearly invertible, so $\operatorname{Aut}{(Q)}$ is indeed a group. More formally, the full automorphism group $\operatorname{Aut}{(Q)}$ of $Q$ is sometimes defined as the subgroup of logical operations contained in the semidirect product $(S_n,SU(2)^{\otimes n},\nu)$, where $\nu:S_n\rightarrow\operatorname{Aut}{(SU(2)^{\otimes n})}$ is given by $$\nu(\pi)(U_1\otimes\dots\otimes U_n)=U_{\pi(1)}\otimes\dots\otimes U_{\pi(n)}$$ and $S_n$ is the symmetric group of permutations on $n$ items. The notation $S_n\ltimes SU(2)^{\otimes n}$ is sometimes used. When $\operatorname{Aut}{(Q)}$ is considered as a semidirect product group, an element $(\pi,U_1\otimes\dots\otimes U_n)\in\operatorname{Aut}{(Q)}$ acts on codewords as $U_1\otimes\dots\otimes U_n$ and on coordinate labels as $\pi$. The product of two automorphisms in $\operatorname{Aut}{(Q)}$ is $$(\pi_1,U)(\pi_2,V)=(\pi_1\pi_2,(U_{\pi_2(1)}V_1)\otimes\dots\otimes (U_{\pi_2(n)}V_n)),$$ by definition of the semidirect product. The full automorphism group contains several interesting subgroups. Consider the logical gates that are local $$\operatorname{LU}{(Q)} = \{ U\in {\mathbb V}(Q)\ |\ U=\otimes_{i=1}^n U_i,\ U_i\in SU(2)\}$$ and the logical gates that are implemented by permutations $$\operatorname{PAut}{(Q)} = \{ \pi\in S_n\ |\ P_\pi\in {\mathbb V}(Q)\}$$ where $P_\pi:S_n\rightarrow SU(2^n)$ is defined by $P_\pi|\psi_1\psi_2\dots\psi_n\rangle=|\psi_{\pi(1)}\psi_{\pi(2)}\dots\psi_{\pi(n)}\rangle$ on the computational basis states. The semidirect product of these groups is contained in the full automorphism group, i.e. $\operatorname{PAut}{(Q)}\ltimes\operatorname{LU}{(Q)}\subseteq \operatorname{Aut}{(Q)}$. In other words, the elements of this subgroup are products of automorphisms for which either $P_\pi=I$ or $U=I$, in the notation of the definition. In general, $\operatorname{Aut}{(Q)}$ may be strictly larger than $\operatorname{PAut}{(Q)}\ltimes\operatorname{LU}{(Q)}$, as happens with the family of Bacon-Shor codes [@aliferis:bs]. The automorphism group of $Q$ as a $GF(4)$-additive classical code is a subgroup of the full automorphism group, since classical automorphisms give rise to quantum automorphisms in the Clifford group. For the $[[4,2,2]]$, $\operatorname{LU}{(Q)}=\langle P^{\otimes 4},H^{\otimes 4}\rangle\cong S_3$ and $\operatorname{PAut}{(Q)}=S_4$. Furthermore, the full automorphism group $\operatorname{Aut}{(Q)}=S_4\times S_3$ equals the automorphism group of the $[[4,2,2]]$ as a $GF(4)$-additive code and $\operatorname{PAut}{(Q)}\ltimes\operatorname{LU}{(Q)}=\operatorname{Aut}{(Q)}$ [@rains:d2]. Fault-tolerance and multiple encoded blocks ------------------------------------------- As we alluded to earlier, the reason we find $\operatorname{Aut}{(Q)}$ interesting is because gates in $\operatorname{Aut}{(Q)}$ are “automatically” fault-tolerant. Fault-tolerant gate failure rates are at least quadratically suppressed after error-correction. Given some positive integer $t'\leq t$, two properties are sufficient (but not necessary) for a gate to be fault-tolerant. First, the gate must take a weight $w$ Pauli operator, $0\leq w\leq t'$, to a Pauli operator with weight no greater than $w$ under conjugation. Second, if $w$ unitaries in the tensor product decomposition of the gate are replaced by arbitrary quantum operations acting on the same qubits, then the output deviates from the ideal output by the action of an operator with weight no more than $w$. Gates in $\operatorname{Aut}{(Q)}$ have these properties for any $t'\in [n]$ if we consider the permutations to be applied to the qubit labels rather than the quantum state. We are also interested in applying logic gates between multiple encoded blocks so that it is possible to simulate a large logical computation using any stabilizer code we choose. In general, each block can be encoded in a different code. Logic gates between these blocks could take inputs encoded in one code to outputs encoded in another, as happens with some logical gates on the polynomial codes [@abo:ft] or with code teleportation [@gottesman:teleport]. In this paper, we only consider the simplest situation where blocks are encoded using the same code and gates do not map between codes. Our multiblock case with $r$ blocks has $rk$ qubits encoded in the code $Q^{\otimes r}$ for some positive integer $r$. The notion of a logical gate is unchanged for the multiblock case: $Q$ is replaced by $Q^{\otimes r}$ in the prior definitions. However, the fault-tolerance requirements become: (1) a Pauli operator with weight $w_i$ on input block $i$ and $\sum_i w_i\leq t'$ conjugates to a Pauli operator with weight no greater than $\sum_i w_i$ on *each* output block and (2) if $w\leq t'$ unitaries in the tensor product decomposition of the gate are replaced by arbitrary quantum operations acting on the same qubits, then *each* output block may deviate from the ideal output by no more than a weight $w$ operator. Gates in $\operatorname{Aut}{(Q^{\otimes r})}$ are also fault-tolerant, since the only new behavior comes from the fact that $\operatorname{PAut}{(Q)}^{\otimes r}$ is not generally equal to $\operatorname{PAut}{(Q^{\otimes r})}$. However, $\operatorname{Aut}{(Q^{\otimes r})}$ does not contain all of the fault-tolerant gates on $r$ blocks because we can interact qubits in different blocks and still satisfy the fault-tolerance properties. \[def:transversal\] A *transversal $r$-qubit gate* on $Q^{\otimes r}$ is a unitary gate $U\in{\mathbb V}(Q^{\otimes r})$ such that $$\label{eq:trans} U = \otimes_{j=1}^n U_j,$$ where $U_j\in SU(2^r)$ only acts on the $j$th qubit of each block. Let $\operatorname{Trans}{(Q^{\otimes r})}$ denote the $r$-qubit transversal gates. More generally, we could extend the definition of transversality to allow coordinate permutations, as in the case of code automorphisms, and still satisfy the fault-tolerance properties given above. However, we keep the usual definition of transversality and do not make this extension here. Codes stabilized by minimal elements and the minimal support condition {#subsec:minsupp} ---------------------------------------------------------------------- A *minimal support* of $\mathcal{S}(Q)$ is a nonempty set $\omega\subseteq [n]$ such that there exists an element in $\mathcal{S}(Q)$ with support $\omega$, but no elements exist with support strictly contained in $\omega$ (excluding the identity element, whose support is the empty set). An element in $\mathcal{S}(Q)$ with minimal support is called a *minimal element*. For each minimal support $\omega$, let $\mathcal{S}_{\omega}(Q)$ denote the stabilizer generated by minimal elements with support $\omega$ and let $Q_\omega$ denote the *minimal code associated to $\omega$*, stabilized by $\mathcal{S}_{\omega}(Q)$. Let $\mathcal{M}(Q)$ denote the *minimal support subgroup* generated by all minimal elements in $\mathcal{S}(Q)$. Consider the $[[5,1,3]]$ code $Q$ whose stabilizer is generated by $XZZXI$ and its cyclic shifts. Every set of 4 contiguous coordinates modulo the boundary is a minimal support: $\{1,2,3,4\}$, $\{2,3,4,5\}$, $\{3,4,5,1\}$, etc. The minimal elements with support $\omega=\{1,2,3,4\}$ are $XZZXI$, $YXXYI$, and $ZYYZI$. Therefore, the minimal code $Q_\omega$ is stabilized by $\mathcal{S}_{\omega}(Q) = \langle XZZXI, YXXYI \rangle$. This code is a $[[4,2,2]]\otimes [[1,1,1]]$ code, since this $[[4,2,2]]$ code is locally equivalent to the code stabilized by $\langle XXXX, ZZZZ\rangle$ by the equivalence $I\otimes C\otimes C\otimes I$, where $C:X\mapsto Y\mapsto Z\mapsto X$ by conjugation. The $[[5,1,3]]$ code is the intersection of its minimal codes, meaning $Q=\cap_\omega Q_\omega$ and $\mathcal{S}(Q)=\prod_\omega \mathcal{S}_\omega(Q)$ where the intersection and product run over the minimal supports. Furthermore, $\mathcal{M}(Q)=\mathcal{S}(Q)$. Given an arbitrary support $\omega$, the projector $\rho_{\omega}(Q)$ obtained by taking the partial trace of $P_Q$ over $\bar{\omega}:=[n]\setminus\omega$ is $$\label{omega} \rho_{\omega}(Q) = \frac{1}{B_{\omega}(Q)}\sum_{R\in \mathcal{S}(Q), supp(R)\subseteq\omega} \operatorname{Tr}_{\bar{\omega}} R,$$ where $B_\omega(Q)$ is the number of elements of $S$ with support contained in $\omega$ including the identity. The projector $\rho_{\omega}(Q)\otimes I_{\bar{\omega}}$ projects onto a subcode $Q_\omega$ of $Q$, $Q\subseteq Q_\omega$, that is stabilized by the subgroup $\mathcal{S}_{\omega}(Q)$ of $\mathcal{S}(Q)$. For the $[[5,1,3]]$, $\rho_{\{1,2,3,4\}}=\frac{1}{4}(IIII+XZZX+YXXY+ZYYZ)\cong P_{[[4,2,2]]}$. If $Q$, $Q'$ are stabilizer codes, a gate $U=U_1\otimes\dots\otimes U_n$ satisfying $U|\psi\rangle=|\psi'\rangle\in Q'$ for all $|\psi\rangle\in Q$ is a *local unitary (LU) equivalence from $Q$ to $Q'$* and $Q$ and $Q'$ are called *locally equivalent codes*. If each $U_i\in {\mathcal L}_1$ then $Q$ and $Q'$ are called *locally Clifford equivalent codes* and $U$ is a *local Clifford (LC) equivalence from $Q$ to $Q'$*. In this paper, we sometimes use these terms when referring to the projectors onto the codes as well. The following results are applied in Section \[sec:transversal\]. \[lem:rains\] Let $Q$ be a stabilizer code. If $U=U_1\otimes\dots\otimes U_n$ is a logical gate for $Q$ then $[U_\omega,\rho_\omega(Q)]=0$ for all $\omega$, where $U_\omega=\otimes_{i\in\omega} U_i$. More generally, if $Q'$ is another stabilizer code and $U$ is a local equivalence from $Q$ to $Q'$ then $$U_\omega\rho_\omega(Q)U_\omega^\dag=\rho_\omega(Q')\label{Uomega}$$ for all $\omega$. $U$ is a local gate, so $$\operatorname{Tr}_{\bar{\omega}} UP_QU^\dag = U_\omega(\operatorname{Tr}_{\bar{\omega}} P_Q)U_\omega^\dag = U_\omega\rho_\omega(Q)U_\omega^\dag.$$ Since $U$ maps from $Q$ to $Q'$, we obtain the result. By examining subcodes, we can determine if a given gate can be a logical gate using Lemma \[lem:rains\]. In particular, if $U$ is not a logical gate for each minimal code of $Q$, then $U$ cannot be a logical gate for $Q$. A stabilizer code is called *free of Bell pairs* if it cannot be written as a tensor product of a stabilizer code and a $[[2,0,2]]$ code (a Bell pair). A stabilizer code $\cal{S}$ is called *free of trivially encoded qubits* if for each $j\in[n]$ there exists an element $s\in{\cal S}$ such that the $j$th coordinate of $s$ is not the identity matrix, i.e. if $S$ cannot be written as a tensor product of a stabilizer code and a $[[1,1,1]]$ code (a trivially encoded qubit). Let ${\mathfrak m}(Q)$ be the union of the minimal supports of a stabilizer code $Q$. The following theorem is a major tool in the solution of our main problem. \[thm:d2lulc\] Let $Q$, $Q'$ be $[[n,k,d]]$ stabilizer codes, not necessarily distinct, that are free of Bell pairs and trivially encoded qubits, and let $j\in {\mathfrak m}(Q)$. Then any local equivalence $U$ from $Q$ to $Q'$ must have either $U_j\in {\mathcal L}_1$ or $U_j=Le^{i\theta R}$ for some $L\in {\mathcal L}_1$, some angle $\theta$, and some $R\in {\cal G}_1$. For completeness, we include a proof of this theorem here, though it can be found expressed using slightly different language in [@rains:d2], [@nest:lulc], and [@nest:thesis]. The proof requires several results about the minimal subcodes of a stabilizer code that we present as Lemmas within the proof body. The first of these results shows that each minimal subcode is either a quantum error-detecting code or a “classical” code with a single parity check, neglecting the $[[|\bar{\omega}|,|\bar{\omega}|,1]]$ part of the space. \[lem:Aw\] Let $A_\omega(Q)$ denote the cardinality of the set of elements $s\in {\cal S}$ with support $\omega$ and let $Q$ be a stabilizer code with stabilizer $\cal S$. If $\omega$ is a minimal support of $\cal S$, then exactly one of the following is true: 1. $A_\omega(Q)=1$ and $\rho_\omega(Q)$ is locally Clifford equivalent to $$\rho_{[[|\omega|,|\omega|-1,1]]}:=\frac{1}{2^{|\omega|}}(I^{\otimes|\omega|}+Z^{\otimes|\omega|}),$$ a projector onto a $[[|\omega|,|\omega|-1,1]]$ stabilizer code $Q_{[[|\omega|,|\omega|-1,1]]}$. 2. $A_\omega(Q)=3$, $|\omega|$ is even, and $\rho_\omega(Q)$ is locally Clifford equivalent to $$\begin{aligned} \rho_{[[2m,2m-2,2]]}:=\frac{1}{2^{|\omega|}} & (I^{\otimes|\omega|}+X^{\otimes|\omega|} \\ & +(-1)^{|\omega|/2}Y^{\otimes|\omega|}+Z^{\otimes|\omega|}),\end{aligned}$$ a projector onto a $[[2m,2m-2,2]]$ stabilizer code $Q_{[[2m,2m-2,2]]}$, $m=|\omega|/2$. For any minimal support $\omega$, $A_\omega(Q)\geq 1$. If $A_\omega(Q)=1$ then $\mathcal{S}_\omega(Q)$ is generated by a single element and we are done. If $A_\omega(Q)\geq 2$, let $M_1,M_2\in \mathcal{S}_\omega(Q)\setminus\{I\}$ be distinct elements. These elements must satisfy $I\neq (M_1)_j\neq (M_2)_j\neq I$ for all $j\in\omega$, otherwise $\operatorname{supp}(M_1M_2)$ is strictly contained in $\omega$, contradicting the fact that $\omega$ is a minimal support. It follows that $\operatorname{supp}(M_1M_2)=\omega$ and $\{(M_1)_j,(M_2)_j,(M_1M_2)_j\}$ equals $\{X,Y,Z\}$ up to phase for all $j\in\omega$. Therefore, $I$, $M_1$, $M_2$, and $M_1M_2$ are the only elements in ${\cal S}_\omega(Q)$. Indeed, suppose there exists a fourth element $N\in {\cal S}_\omega(Q)$. Fixing any $j_0\in\omega$, either $(M_1)_{j_0}$, $(M_2)_{j_0}$, or $(M_1M_2)_{j_0}$ equals $N_{j_0}$, say $(M_1)_{j_0}=N_{j_0}$. Then, $\operatorname{supp}(M_1N)$ is strictly contained in $\omega$, a contradiction. Therefore, if $A_\omega(Q)\geq 2$ then $A_\omega(Q)=3$. The number of coordinates in the support $|\omega|$ must be even since $M_1$ and $M_2$ commute. The next result shows that any local equivalence between two $[[2m,2m-2,2]]$ stabilizer codes with the same $m\geq 2$ must be a local Clifford equivalence. In the $m=1$ special case, we have a $[[2,0,2]]$ code, i.e. a Bell pair locally Clifford equivalent to the state $(|00\rangle+|11\rangle)/\sqrt{2}$, for which the result does not hold because $V\otimes V^\ast$ is a local equivalence of the $[[2,0,2]]$ for any $V\in SU(2)$. This special case is the reason for introducing the definition of a stabilizer code that is free of Bell pairs. \[lem:d2LC\] Fix $m\geq 2$ and let $Q$, $Q'$ be stabilizer codes that are LC equivalent to $Q_{[[2m,2m-2,2]]}$. If $U\in U(2)^{\otimes 2m}$ is a local equivalence from $Q$ to $Q'$ then $U\in {\mathcal L}_{2m}$. We must show that every $U\in U(2)^{\otimes 2m}$ satisfying $U\rho_{[[2m,2m-2,2]]}U^\dag=\rho_{[[2m,2m-2,2]]}$ is a local Clifford operator. Recall that any 1-qubit unitary operator $V\in U(2)$ acts on the Pauli matrices as $$\sigma_a \mapsto V\sigma_aV^\dag = o_{ax}X + o_{ay}Y + o_{az}Z,$$ for every $a\in \{x,y,z\}$ and where $(o_{ab})\in SO(3)$. In the standard basis $\{|0\rangle,|1\rangle,|2\rangle\}$ of ${\mathbb R}^3$, the matrix $$X^{\otimes 2m} + (-1)^m Y^{\otimes 2m}+Z^{\otimes 2m}$$ is associated to the vector $$v := |00\dots 0\rangle + (-1)^m |11\dots 1\rangle + |22\dots 2\rangle\in ({\mathbb R}^3)^{\otimes 2m}$$ acted on by $SO(3)^{\otimes 2m}$. We must show that every $O=O_1\otimes\dots\otimes O_{2m}\in SO(3)^{\otimes 2m}$ satisfying $Ov=v$ is such that each $O_i$ is a monomial matrix (see [@huffman:HP]; a matrix is monomial if it is the product of a permutation matrix and a diagonal matrix). Consider the single qutrit operator $$\label{eq:rainstr} \langle 0|_1 \operatorname{Tr}_{\{3,4,\dots,2m\}}(vv^T)|0\rangle_1,$$ acting on the second qutrit (second copy of ${\mathbb R}^3$). The matrix $vv^T$ has 9 nonzero elements, and the partial trace over the last $2m-2$ qutrits gives $$\operatorname{Tr}_{\{3,4,\dots,2m\}}(vv^T) = |00\rangle\langle 00| + |11\rangle\langle 11| + |22\rangle\langle 22|.$$ Hence the matrix in Eq. \[eq:rainstr\] equals the rank one projector $|0\rangle\langle 0|$. Therefore, if $Ov=v$ then the operator $$\langle 0|_1 \operatorname{Tr}_{\{3,4,\dots,2m\}} (Ovv^TO^T)|0\rangle_1$$ equals $|0\rangle\langle 0|$ as well. The operator is given by the matrix $$\begin{aligned} O_2\langle 0|_1(O_1\otimes I)\operatorname{Tr}_{\{3,4,\dots,2m\}}(vv^T)(O_1^T\otimes I)|0\rangle_1 O_2^T \\ = O_2\left(\begin{array}{ccc} (O_1)_{00}^2 & 0 & 0 \\ 0 & (O_1)_{01}^2 & 0 \\ 0 & 0 & (O_1)_{02}^2 \end{array}\right)O_2^T. \label{eq:monmatrix}\end{aligned}$$ where we have factored $O_2$ to the outside. The matrix within Eq. \[eq:monmatrix\] equals the rank one projector $O_2^T|0\rangle\langle 0|O_2$ if and only if exactly one of the elements $(O_1)_{00}$, $(O_1)_{01}$, or $(O_1)_{02}$ is nonzero. Repeating the argument for every row of $O_1$ by considering the operators $\langle i|_1\operatorname{Tr}_{\{3,4,\dots,2m\}}(Ovv^TO^T)|i\rangle_1$, $i\in\{0,1,2\}$, shows that every row of $O_1$ has exactly one nonzero entry. $O_1$ is nonsingular therefore $O_1$ is a monomial matrix. The vector $v$ is symmetric so repeating the analogous argument for each operator $O_i$, $i\in [2m]$, completes the proof. Now we can complete the proof of Theorem \[thm:d2lulc\]. Let $Q$, $Q'$ be stabilizer codes, let $U$ be a local equivalence from $Q$ to $Q'$, and take a coordinate $j\in {\mathfrak m}(Q)$. There is a least one element $M\in \mathcal{M}(Q)$ with $j\in\omega:=\operatorname{supp}(M)$. Either $A_\omega(Q)=1$ or $A_\omega(Q)=3$ by Lemma \[lem:Aw\]. If $A_\omega(Q)=3$ then $\rho_\omega(Q)$ is LC equivalent to $\rho_{[[|\omega|,|\omega|-2,2]]}$. Moreover, as $Q$ is locally equivalent to $Q'$, $\omega$ is also a minimal support of $\mathcal{S}(Q')$ with $A_\omega(Q')=3$. Therefore, $\rho_\omega(Q')$ is local Clifford equivalent to $\rho_{[[|\omega|,|\omega|-2,2]]}$. By Lemma \[lem:rains\], $U_\omega$ maps $\rho_\omega(Q)$ to $\rho_\omega(Q')$ under conjugation. Note that we must have $|\omega|>2$, otherwise $Q$ is not free of Bell pairs. Since $|\omega|$ is even, $|\omega|\geq 4$. By Lemma \[lem:d2LC\], $U_\omega\in {\mathcal L}_{|\omega|}$ so $U_j\in {\mathcal L}_1$. If $A_\omega(Q)=1$ and there are elements $R_1,R_2,R_3\in \mathcal{M}(Q)$ such that $(R_1)_j=X$, $(R_2)_j=Y$, and $(R_3)_j=Z$, then there exists another minimal element $N\in \mathcal{M}(Q)$ such that $j\in\mu:=\operatorname{supp}(N)$ and $M_j \neq N_j$. If $A_\mu(Q)=3$ then we can apply the previous argument to conclude that $U_j\in {\mathcal L}_1$. Otherwise, $A_\mu(Q)=1$ and $$\begin{aligned} \label{eq:Aw11} \rho_\omega(Q) & = \frac{1}{2^{|\omega|}}(I^{\otimes|\omega|}+M_\omega) \\ \rho_\mu(Q) & = \frac{1}{2^{|\mu|}}(I^{\otimes|\mu|}+N_\mu).\end{aligned}$$ Since $\omega$ and $\mu$ are also minimal supports of $\mathcal{S}(Q')$ with $A_\omega(Q')=1$ and $A_\mu(Q')=1$, there exist unique $M',N'\in S(Q')$ such that $$\begin{aligned} \rho_\omega(Q') & = \frac{1}{2^{|\omega|}}(I^{\otimes|\omega|}+M'_\omega) \\ \rho_\mu(Q') & = \frac{1}{2^{|\mu|}}(I^{\otimes|\mu|}+N'_\omega).\label{eq:Aw12}\end{aligned}$$ Applying Lemma \[lem:rains\] to $U_\mu$ and $U_\nu$, we have $$\begin{aligned} U_jM_jU_j^\dag & = \pm M_j' \\ U_jN_jU_j^\dag & = \pm N_j'\end{aligned}$$ from Eqs. \[eq:Aw11\]-\[eq:Aw12\]. These identities show that $U_j\in {\mathcal L}_1$. Finally, if $A_\omega(Q)=1$ and $R=(R_1)_j=(R_2)_j$ for any $R_1,R_2\in \mathcal{M}(Q)$ then [*any*]{} minimal support $\mu$ such that $j\in\mu$ satisfies $A_\mu(Q)=1$. Applying Lemma \[lem:rains\] to $U_\mu$, we have $$U_jRU_j^\dag = \pm R'$$ for some $R'\in\{X,Y,Z\}$. Choose $L\in {\mathcal L}_1$ such that $LRL^\dag=R'$. Then $U_j=Le^{i\theta R}$ and the proof of Theorem \[thm:d2lulc\] is complete. Coordinates not covered by minimal supports ------------------------------------------- It is not always the case the ${\mathfrak m}(Q)=[n]$, as the following example shows. \[em:nonminimal\] Consider a $[[6,2,2]]$ code with stabilizer $\mathcal{S}=\langle XXXXII,ZZIIZZ,IIIIXX,IIXXZZ\rangle$. For $j=1,2$, there is no minimal support $\omega$ of $\mathcal{S}$ such that $j\in\omega$ [@bravyi:thanks]. For coordinates which are not covered by minimal supports of $\mathcal{S}$, the results in Sec. \[subsec:minsupp\] tell us nothing about the allowable form of $U_j$ for a transversal gate $U=\bigotimes_{j=1}^n U_j$, so we need another approach for these coordinates. Let $\mathcal{S}_j=\{ R\ |\ R\in \mathcal{S}(Q),\ j\in\operatorname{supp}(R)\}$ and define the “minimal elements” of this set to be $\mathcal{M}(\mathcal{S}_j)=\{ R\in\mathcal{S}_j\ |\ \nexists R'\in\mathcal{S}_j\ \textrm{s.t.}\ \operatorname{supp}(R')\subsetneq\operatorname{supp}(R)\}$. Note that these sets do not define codes because they are not necessarily groups. \[lem:neqsupp\] If $j$ is not contained in any minimal support of $\mathcal{S}$, then for any $R,R'\in\mathcal{M}(\mathcal{S}_j)$ such that the $j$th coordinates satisfy $R|_j\neq R'|_j$, we must have $\operatorname{supp}(R)\neq\operatorname{supp}(R')$. We prove by contradiction. If there exist $R,R'\in\mathcal{M}(\mathcal{S}_j)$ such that $R|_j\neq R'|_j$ and $\operatorname{supp}(R)=\operatorname{supp}(R')=\omega$, then up to a local Clifford operation, we have $R=X^{\otimes|\omega|}$ and $R'=Z^{\otimes|\omega|}$. Without loss of generality, assume $j=1$. Since $\omega$ is minimal in $\mathcal{S}_j$ but not minimal in $\mathcal{S}$, there exists an element $F$ in $\mathcal{S}\setminus\mathcal{S}_j$ whose support $\operatorname{supp}(F)=\omega'$ is strictly contained in $\omega$, i.e. $\omega'\subsetneq\omega$. Since $F$ is not in $\mathcal{S}_j$, $RF$, $R'F$, $R'RF\in\mathcal{M}(\mathcal{S}_j)$. However, one of $RF$, $R'F$, $R'RF\in\mathcal{M}(\mathcal{S}_j)$ will have support that is strictly contained in $\omega$, contradicting the fact that $\omega$ is a minimal support of $\mathcal{S}_j$. \[lem:ujnsupp\] If $j$ is not contained in any minimal support of $\mathcal{S}$, then for any transversal gate $U=\bigotimes_{j=1}^n U_j$, one of the following three relations is true: $U_jX_jU_j=\pm X_j$, $U_jY_jU_j=\pm Y_j$, $U_jZ_jU_j=\pm Z_j$. In other words, $U_j=Le^{i\theta R}$ for some $L\in {\mathcal L}_1$, some angle $\theta$, and some $R\in {\cal G}_1$. For any element $R\in\mathcal{M}({S}_j)$ with a fixed support $\omega$, we have $R|_j=Z$ up to local Clifford operations by Lemma \[lem:neqsupp\]. Tracing out all the qubits in $\bar{\omega}$, we get a reduced density matrix $\rho_{\omega}$ with the form $$\label{eq:rhonsupp} \rho_{\omega}=\frac{1}{2^{|\omega|}}(I_j\otimes R_I+Z_j\otimes R_Z),$$ where $R_I$ and $R_Z$ are linear operators acting on the other $\omega\setminus\{j\}$ qubits. Since $U_{\omega}\rho_{\omega}U_{\omega}^{\dagger}=\rho_{\omega}$, we have $U_jZ_jU_j^{\dagger}=\pm Z_j$. The following corollary about the elements of the automorphism group of a stabilizer code is immediate from Lemma \[lem:ujnsupp\]. After this work was completed, we learned that the same statement was independently obtained by D. Gross and M. Van den Nest [@gross07] and that the theorem was first proved in the diploma thesis of D. Gross [@grossthesis]. \[cor:form\] $U\in\operatorname{Aut}{(Q)}$ for a stabilizer code $Q$ iff $$\label{eq:form} U=L\left( \bigotimes_{j=1}^n \operatorname{diag}{(1,e^{i\theta_j})}\right) R^\dag P_\pi\in {\mathbb V}(Q)$$ for some local Clifford unitaries $L=L_1\otimes\dots\otimes L_n$, $R=R_1\otimes\dots\otimes R_n$, product of swap unitaries $P_\pi$ enacting the coordinate permutation $\pi$, and angles $\{\theta_1,\dots,\theta_n\}$. Transversality versus Universality {#sec:transversal} ================================== In this section we prove that there is no universal set of transversal gates for binary stabilizer codes. \[def:encuniv\] A set $A\subseteq {\mathbb V}(Q^{\otimes n})$ is *encoded computationally universal* if, for any $n$, given $U\in {\mathbb V}(Q^{\otimes n})$, $$\forall \epsilon>0, \exists V_1,\dots,V_{\eta(\epsilon)}\in A,\ \textrm{s.t.}\ ||UP_Q^{\otimes n}-\left(\prod_i V_i\right)P_Q^{\otimes n}||<\epsilon.$$ Gates in $A$ may be implicitly mapped to isometries on the appropriate Hilbert space, as in Definition \[def:universality\]. \[thm:maintheorem\] For any stabilizer code $Q$ that is free of Bell pairs and trivially encoded qubits, and for all $r\geq 1$, $\operatorname{Trans}(Q^{\otimes r})$ is not an encoded computationally universal set of gates for even one of the logical qubits of $Q$. We prove this theorem by contradiction. We first assume that we can perform universal quantum computation on at least one of the qubits encoded into $Q$ using only transversal gates. Then, we pick an arbitrary minimum weight element $\alpha\in\mathcal{C}(\mathcal{S})\setminus\mathcal{S}$, and perform appropriate transversal logical Clifford operations on $\alpha$. Finally, we will identify an element in $\mathcal{C}(\mathcal{S})\setminus\mathcal{S}$ that has support strictly contained in $\operatorname{supp}(\alpha)$. This contradicts the fact that $\alpha$ is a minimal weight element in $\mathcal{C}(\mathcal{S})\setminus\mathcal{S}$, i.e. that the code has the given distance $d$. We first prove the theorem for A) the single block case and then generalize it to B) the multiblock case. The single block case ($r=1$) ----------------------------- The first problem we encounter is that general transversal gates, even those that implement logical Clifford gates, might not map logical Pauli operators back into the Pauli group. This behavior potentially takes us outside the stabilizer formalism. The *generalized stabilizer* ${\mathcal I}(Q)$ of a quantum code $Q$ is the group of all unitary operators that fix the code space, i.e. $${\mathcal I}(Q)=\{ U\in SU(2^n)\ |\ U|\psi\rangle = |\psi\rangle,\ \forall |\psi\rangle\in Q\}.$$\[GS\] The transversal $T$ gate on the 15-qubit Reed-Muller code is one example of this problem since it maps $\bar{X}=X^{\otimes 15}$ to an element $(\frac{1}{\sqrt{2}}(X-Y))^{\otimes 15}$. This element is a representative of $\frac{1}{\sqrt{2}}(\bar{X}+\bar{Y})$ but has many more terms in its expansion in the Pauli basis. These terms result from an operator in the generalized stabilizer $\mathcal{I}$. The 9-qubit Shor code gives another example. A basis for this code is $$|0/1\rangle \propto (|000\rangle+|111\rangle)^{\otimes 3}\pm (|000\rangle-|111\rangle)^{\otimes 3},$$ from which it is clear that $e^{i\theta Z_1}e^{-i\theta Z_2}\in \mathcal{I}(Q_{\textrm{Shor}})\setminus S$. This gate does not map $\bar{X}=X^{\otimes 9}$ back to the Clifford group, even though it is both transversal and logically an identity gate in the logical Clifford group. In spite of these possibilities, we will now see that we can avoid further complication and stay within the powerful stabilizer formalism. First, we review a well-known fact about stabilizer codes. \[lem:proj\] Let $\mathcal{S}=\langle M_1,\dots,M_{n-k}\rangle$ be the stabilizer of an $[[n,k,d]]$ code $Q$. For any $n$-qubit Pauli operator $R\notin \mathcal{C}(\mathcal{S})$, we have $P_QRP_Q=0$ where $P_Q$ is the projector onto the code subspace. We have $$\begin{aligned} P_Q & \propto\prod\limits_{i=1}^{n-k}(I+M_i)\ \textrm{and} \\ P_QR & \propto R\prod\limits_{i=1}^{n-k}(I+(-1)^{r(i)}M_i),\end{aligned}$$ where $r(i)=0$ if $R$ commutes with $M_i$ and $r(i)=1$ if $R$ anticommutes with $M_i$. $R\notin\mathcal{C}(\mathcal{S})$ so $r(i)=1$ for at least one $i$, and $(I-M_i)(I+M_i)=0$, which gives $P_QRP_Q=0$. \[lem:li\] Let $Q$ be a stabilizer code with stabilizer $\mathcal{S}$ and let $\alpha\in \mathcal{C}(\mathcal{S})\setminus\mathcal{S}$ be a minimum weight element in $\mathcal{C}(\mathcal{S})\setminus\mathcal{S}$. Without loss of generality, $\alpha\in\bar{X}_1\mathcal{S}$ (the $\bar{X}_1$ coset of $\mathcal{S}$), where the subscript indicates what logical qubit the logical operator acts on. If the logical Clifford operations $\bar{H}_1$ and $\bar{P}_1$ on the first encoded qubit are transversal, then there exists $\beta,\gamma\in\mathcal{C}(\mathcal{S})\setminus\mathcal{S}$ such that $\beta\in\bar{Z}_1\mathcal{S}$, $\gamma\in\bar{Y}_1\mathcal{S}$ and $\operatorname{supp}(\alpha)=\operatorname{supp}(\beta)=\operatorname{supp}(\gamma)$. $\bar{H}_1$ is transversal, so $\beta'':=\bar{H}_1\alpha\bar{H}_1^\dag\in\bar{Z}_1{\mathcal I}$ and $\xi:=\operatorname{supp}(\alpha)=\operatorname{supp}(\beta'')$. Expand $\beta''$ in the basis of Pauli operators $$\label{eq:expansion} \beta'' = \sum_{R\in C(\mathcal{S}), \operatorname{supp}(R)\subseteq\xi} b_R R + \sum_{R'\in G_n\setminus C(\mathcal{S})} b_{R'} R'$$ where $b_R,b_{R'}\in {\mathbb C}$. By Lemma \[lem:proj\], $b_R\neq 0$ for at least one $R\in C(\mathcal{S})$ in the first term of Eq. \[eq:expansion\]. The operator $\beta':=P_Q\beta''P_Q\in\bar{Z}_1{\mathcal I}$ is a linear combination of elements of $C(\mathcal{S})$, $$\beta' = \sum_{R\in C(\mathcal{S})\setminus \mathcal{S}, \operatorname{supp}(R)=\xi} b_R R + \sum_{R'\in \mathcal{S}, \operatorname{supp}(R')\subseteq\xi} b_{R'} R',$$ where the terms $R\in C(\mathcal{S})\setminus \mathcal{S}$ must have support $\xi$ since $\alpha$ has minimum weight in $C(\mathcal{S})\setminus \mathcal{S}$. Considering the action of $\beta'$ on a basis of $Q$, it is clear that there is a term $b_\beta \beta$ where $b_\beta\neq 0$, $\beta\in\bar{Z}_1\mathcal{S}$, and $\operatorname{supp}(\beta)=\xi$. Similarly, since $\bar{P}_1$ is transversal, there must exist $\gamma\in\bar{Z}_1\mathcal{S}$, and $\operatorname{supp}(\gamma)=\xi$. Note in the proof of the above lemma, we assume that $\bar{H}_1$ is exactly transversal, i.e. $\epsilon=0$ in Definition \[def:encuniv\]. However, the proof is also valid for an arbitrarily small $\epsilon>0$. Indeed, in this case $\beta''\notin\bar{Z}_1{\mathcal I}$, but $\beta''$ must have a non-negligible component in $\bar{Z}_1{\mathcal I}$ to approximate $\bar{H}_1$. Hence, when expanding $\beta''$ in the Pauli basis, there must exist a $\beta\in\bar{Z}_1{\mathcal S}$ such that $\operatorname{supp}(\beta)=\xi$, i.e. the same argument holds even for an arbitrarily small $\epsilon>0$. The choice of $\alpha\in\bar{X}_1\mathcal{S}$ is made without loss of generality, since for a given stabilizer code, we have the freedom to define logical Pauli operators, and this freedom can be viewed as a “choice of basis”. What is more, since we assume universal quantum computation can be performed transversally on the code, then no matter what basis (of the logical Pauli operators) we choose, $\bar{H}_1$ and $\bar{P}_1$ must be transversal. On the other hand, sometimes we would like to fix our choice of basis, as in the case of a subsystem code, to clearly distinguish some logical qubits (protected qubits) from other logical qubits (gauge qubits). In this case, we can choose $\alpha$ as a minimum weight element in $\{\bar{X}_s\mathcal{S},\bar{Y}_s\mathcal{S},\bar{Z}_s\mathcal{S}\}$, where $s$ is a distinguished logical qubit. Starting from this choice of $\alpha$, one can see that the arguments hold for subsystem codes as well as subspace codes, because the distance of the subsystem code is defined with respect to this subgroup. The procedure of identifying $\beta\in\bar{Z}_1{\mathcal S}$ from $\beta''\in\bar{Z}_1{\mathcal I}$ in the proof of Lemma \[lem:li\] is general in the following sense. We can begin with a minimum weight element of $\alpha\in \bar{X}_1\mathcal{S}\subset C(\mathcal{S})\setminus \mathcal{S}$ and apply any transversal logical Clifford gate to generate a representative $\beta\in C(\mathcal{S})\setminus \mathcal{S}$ of the corresponding logical Pauli operator such that $\operatorname{supp}(\alpha)=\operatorname{supp}(\beta)$. This procedure is used a few times in our proof, so we name this procedure the “$\mathcal{I}\rightarrow\mathcal{S}$ procedure”. Now we can begin the proof of Theorem \[thm:maintheorem\]. Assume that $\operatorname{Trans}{(Q)}$ is encoded computationally universal. Let $\alpha\in \bar{X}_1\mathcal{S}\subset C(\mathcal{S})\setminus\mathcal{S}$ be a minimum weight element in $C(\mathcal{S})\setminus\mathcal{S}$. Applying the “$\mathcal{I}\rightarrow\mathcal{S}$ procedure” to both $\bar{H}_1$ and $\bar{P}_1$, we obtain $\beta\in\bar{Z}_1\mathcal{S}$ and $\gamma\in\bar{Y}_1\mathcal{S}$ such that $\operatorname{supp}(\alpha)=\operatorname{supp}(\beta)=\operatorname{supp}(\gamma)=:\xi$ and $|\xi|=d$. The next lemma puts these logical operators into a simple form for convenience. \[lem:XYZ\] If $\alpha\in\bar{X}_1\mathcal{S}$, $\beta\in\bar{Z}_1\mathcal{S}$, and $\gamma\in\bar{Y}_1\mathcal{S}$, all have the same support $\xi$, and $|\xi|=d$ is the minimum distance of the code, then there exists a local Clifford operation that transforms $\alpha$, $\gamma$, and $\beta$ to $X^{\otimes |\xi|}$, $(-1)^{|\xi|/2}Y^{\otimes |\xi|}$, and $Z^{\otimes |\xi|}$, respectively. Let $\xi=\{i_1,i_2,\ldots,i_{|\xi|}\}$ and write $\alpha=\alpha_{i_1}\alpha_{i_2}\ldots\alpha_{i_{|\xi|}}$, $\beta=\beta_{i_1}\beta_{i_2}\ldots\beta_{i_{|\xi|}}$, where each $\alpha_{i_k}$ and $\beta_{i_k}$, $k\in[|\xi|]$, are one of the three Pauli matrices $X_{i_k}$,$Y_{i_k}$, or $Z_{i_k}$, neglecting phase factors $\pm i$ or $-1$. Apart from a phase factor, $\alpha_{i_k}\neq \beta_{i_k}$ for all $k\in [|\xi|]$. Indeed, if for some $k$, $\alpha_{i_k}=\beta_{i_k}$, then $\operatorname{supp}(\bar{Y})\neq\xi$, a contradiction. Therefore, for each $k\in [|\xi|]$ there exists a single qubit Clifford operation $L_{i_k}\in {\mathcal L}_1$ such that $L_{i_k}\alpha_{i_k}L_{i_k}^\dag=X_{i_k}$ and $L_{i_k}\beta_{i_k}L_{i_k}^\dag=Z_{i_k}$. The local Clifford operation $$L_{\xi}=\bigotimes_{k=1}^{j}L_{i_k}$$ applies the desired transformation. Applying Lemma \[lem:XYZ\], we obtain a local Clifford operation that we apply to $Q$. Now we have a locally Clifford equivalent code $Q'$ for which $\operatorname{supp}{\alpha}=\operatorname{supp}{\beta}=\operatorname{supp}{\gamma}=\xi$ and $\alpha=X^{\otimes |\xi|}\in\bar{X}_1\mathcal{S}$, $\gamma=(-1)^{|\xi|/2}Y^{\otimes |\xi|}\in\bar{Y}_1\mathcal{S}$ and $\beta=Z^{\otimes |\xi|}\in\bar{Z}_1\mathcal{S}$. Note that if $|\xi|=d$ is even, then the validity of Lemma \[lem:XYZ\] already leads to a contradiction since $\alpha, \beta, \gamma$ must anti-commute with each other. However, if $|\xi|=d$ is odd, we need to continue the proof. By Theorem \[thm:d2lulc\] and Lemma \[lem:ujnsupp\], if $U=\bigotimes_{j=1}^n U_j$ is a transversal gate on one block, then either $U_j\in {\mathcal L}_1$ for all $j\in\xi$ or $U_j=Le^{i\theta R}$ for one or more $j\in\xi$, where $L\in {\mathcal L}_1$, $\theta\in {\mathbb R}$, and $R\in {\cal G}_1$. If $U_j\in {\mathcal L}_1$ for all $j\in\xi$ then, for the first encoded qubit of $Q'$, the only transversal operations are logical Clifford operations. Therefore, there must exist a coordinate $j\in\xi$, such that $U_j=e^{i\theta Z}$ up to a local Clifford operation. Since $\bar{H}_1$ is transversal, when expanding $\bar{H}_1\beta\bar{H}_1^{\dagger}\in\bar{X}_1\mathcal{I}$ in the basis of Pauli operators, using the “$\mathcal{I}\rightarrow\mathcal{S}$ procedure”, we know that there exists $\alpha'\in\bar{X}_1\mathcal{S}$ and $\operatorname{supp}(\alpha')=\xi$. Furthermore, since $(\bar{H}_1)_jZ(\bar{H}_1)_j^{\dagger}=\pm Z$, we have $(\alpha')_j=Z_j$, i.e. $\alpha'$ restricted to the $j$th qubit is $Z_j$. We know $\gamma'=i\alpha'\beta\in\bar{Y}_1\mathcal{S}$, and $(\gamma')_j=I_j$. Therefore, $\operatorname{supp}(\gamma')$ is strictly contained in $\xi$. However, this contradicts the fact that $\alpha$ is a minimal weight element in $\mathcal{C}({\mathcal{S}})\setminus\mathcal{S}$. This concludes the proof of Theorem \[thm:maintheorem\] for the single block case. The multiblock case ($r>1$) {#subsec:multiblock} --------------------------- Now we consider the case with $r$ blocks. A superscript $(i)$, $i\in [r]$, denotes a particular block. For example, $U^{(i)}$ acts on the $i$th block. First, we generalize Theorem \[thm:d2lulc\] and Lemma \[lem:ujnsupp\] to the multiblock case. \[lem:bigblocklem\] Let $Q$ be an $[[n,k,d]]$ stabilizer code free of Bell pairs and trivially encoded qubits, and let $U$ be a transversal gate on $Q^{\otimes r}$. Then for each $j\in [n]$ either $U_j\in {\cal L}_r$ or $U_j=L_1VL_2$ where $L_1$,$L_2\in {\cal L}_1^{\otimes r}$ are local Clifford gates and $V$ either normalizes the group $\langle\pm Z_j^{(i)}, i\in [r]\rangle$, of Pauli $Z$ operators or keeps the linear span of its group elements invariant. Lemma \[lem:rains\] and Lemma \[lem:Aw\] can be generalized to the the multiblock case with almost the same proof, which we do not repeat here. In the multiblock case, the corresponding results of Lemma \[lem:Aw\] read $$\begin{aligned} A_{\omega}(Q)&=&1\ :\ S_{\omega}^{\otimes{r}}(Q)=\{I^{\otimes\omega}, \ Z^{\otimes\omega}\}^{\otimes{r}}\nonumber\\ A_{\omega}(Q)&=&3\ :\nonumber\\ S_{\omega}^{\otimes{r}}(Q)&=&\{I^{\otimes\omega},\ X^{\otimes\omega}, (-1)^{(|\omega|/2)}Y^{\otimes\omega}, Z^{\otimes\omega}\}^{\otimes r}\label{MEr},\end{aligned}$$ and the corresponding equation of Eq. \[Uomega\] in Lemma \[lem:rains\] is $$U_{\omega}\rho_{\omega}^{\otimes r}(Q')U_{\omega}^{\dagger}=\rho_{\omega}^{\otimes r}(Q).$$ When $A_{\omega}(Q)=3$, we need to generalize the result of Lemma \[lem:d2LC\]. In particular, if $U=\bigotimes_{i=1}^r U_j\in U(2^r)^{\otimes 2m}$ satisfies $U\rho_{[[2m,2m-2,2]]}^{\otimes r}U^\dag=\rho_{[[2m,2m-2,2]]}^{\otimes r}$, then for each $j\in\omega$, $U_j\in U(2^r)$ is a Clifford operator. Indeed, any $r$-qubit unitary operator $V\in U(2^r)$ acts on a Pauli operator $\sigma_{a_1}\sigma_{a_2}\dots\sigma_{a_r}$ as $$\begin{aligned} \sigma_{a_1}\sigma_{a_2}\dots\sigma_{a_r} & \mapsto V\sigma_{a_1}\sigma_{a_2}\dots\sigma_{a_r}V^\dag \\ & = \sum_{i_1,...i_r=0}^{3} o_{a_1\dots a_ri_1\dots i_r}\sigma_{i_1}\sigma_{i_2}\dots\sigma_{i_r},\end{aligned}$$ for every nonidentity Pauli string $a$, where $(o_{a,i_1,...i_r})\in SO(4^r-1)$ and $o_{a,0,\dots,0}=0$. We can rearrange the numbering of the coordinates in $\rho_{[[2m,2m-2,2]]}^{\otimes r}$ such that the coordinate $r(a-1)+b$ denotes the $a$th qubit of the $b$th block. In the standard basis $\{|0\rangle,...,|4^r-2\rangle\}$ of ${\mathbb R}^{4^r-1}$, $\rho_{[[2m,2m-2,2]]}^{\otimes r}$ is associated to the vector $$v := \sum_{j=0}^{4^r-2}{b_j}|jj\dots j\rangle\in ({\mathbb R}^{4^r-1})^{\otimes 2m}$$ where $b_j\in\{\pm 1\}$. The vector is acted on by orthogonal matrices in $SO(4^r-1)^{\otimes 2m}$. For a code free of Bell pairs, we have $|\omega|\geq 4$. By reasoning similar to the proof of Lemma \[lem:d2LC\], if $O=O_1\otimes\dots\otimes O_{2m}\in SO(4^r-1)^{\otimes 2m}$ satisfies $Ov=v$, then each $O_i$ is a monomial matrix. This implies that $U_j\in U(2^r)$ is a Clifford operator for each $j\in\omega$. When $A_{\omega}(Q)=1$, it is possible to follow reasoning similar to the proof of Theorem \[thm:d2lulc\]. Now the equations analogous to Eqs. \[eq:Aw11\] are $$\begin{aligned} \rho_\omega(Q)^{\otimes r} & = \left(\frac{1}{2^{|\omega|}}(I^{\otimes|\omega|}+M_\omega)\right)^{\otimes r}\label{eq:rhoromega} \\ \rho_\mu(Q)^{\otimes r} & = \left(\frac{1}{2^{|\mu|}}(I^{\otimes|\mu|}+N_\mu)\right)^{\otimes r}.\label{eq:rhormu}\end{aligned}$$ Up to local Clifford operations, we can choose $M_\omega=X^{\otimes|\omega|}$ and $N_\mu=Z^{\otimes|mu|}$ We again rearrange the numbering of the coordinates of $\rho_\omega(Q)^{\otimes r}$ such that the coordinate $r(a-1)+b$ denotes the $a$th qubit of the $b$th block. In the standard basis $\{|0\rangle,|1\rangle,...,|2^r-2\rangle\}$ of ${\mathbb R}^{2^r-1}$, the matrix $\rho_\omega(Q)^{\otimes r}$ is associated to the vector $$v := \sum_{j=0}^{2^r-2}|jj\dots j\rangle\in ({\mathbb R}^{2^r-1})^{\otimes |\omega|}\label{eq:vr}$$ acted on by $SO(2^r-1)^{\otimes |\omega|}$. Note for any coordinate $j\in[n]$, if there are elements $R_1,R_2,R_3\in\mathcal{M}(Q)$ such that $(R_1)_j=X$, $(R_2)_j=Y$, and $(R_3)_j=Z$, then we have both $|\omega|\geq 2$ and $|\mu|\geq 2$ [@zeng:lulc]. Following similar reasoning to the proof of Lemma \[lem:d2LC\], if $O=O_1\otimes\dots\otimes O_{|\omega|}\in SO(2^r-1)^{\otimes |\omega|}$ satisfies $Ov=v$, then each $O_i$ has a monomial subblock. Therefore, $U_j\in\mathcal{N}(\langle \pm X_j^{(i)}\rangle_{i=1}^{r})$ for each $j\in\omega$, where $\mathcal{N}(\langle\pm X_j^{(i)}\rangle_{i=1}^{r})$ is normalizer of the group $\langle\pm X_j^{(i)}\rangle_{i=1}^{r}$ of Pauli $X$ operators acting at the $j$th coordinate of the $i$th block. Similarly for $\rho_\mu(Q)^{\otimes r}$, $U_j\in\mathcal{N}(\langle\pm Z_j^{(i)}\rangle_{i=1}^{r})$ for each $j\in\mu$, where $\mathcal{N}(\langle\pm Z_j^{(i)}\rangle_{i=1}^{r})$ is normalizer of the group $\langle\pm Z_j^{(i)}\rangle_{i=1}^{r}$ of Pauli $Z$ operators. $\mathcal{N}(\langle\pm X_j^{(i)}\rangle_{i=1}^{r})\cap\mathcal{N}(\langle\pm Z_j^{(i)}\rangle_{i=1}^{r})$ is a subgroup of the Clifford group, therefore $U_j\in U(2^r)$ is a Clifford operator for all $j\in\omega\cap\mu$. If instead $(R_a)_j=(R_b)_j$ for all $R_a,R_b\in\mathcal{M}(Q)$ and if $|\omega|\geq 3$, then $U_j\in\mathcal{N}(\langle \pm Z_j^{(i)}\rangle_{i=1}^{r})$ up to local Clifford operations, but $U_j$ is not necessarily a Clifford operator. If $|\omega|=2$ and $A_{\omega}=1$, then the form of the vector in Eq. \[eq:vr\] leads to different behavior when $r>1$. When $r=1$, there is only one term in the summation, so $U_j\in\mathcal{N}(\langle\pm X_j\rangle)$. However, when $r>1$, generally $U_j\notin\mathcal{N}(\langle\pm X_j^{(i)}\rangle_{i=1}^{r})$. Nevertheless $U_j$ must keep $\text{span}(\langle\pm X_j^{(i)}\rangle_{i=1}^{r})$ invariant, where $\text{span}(\langle\pm X_j^{(i)}\rangle_{i=1}^{r}\rangle)$ is the space of linear operators spanned by the group $\langle\pm X_j^{(i)}\rangle_{i=1}^{r}$ with coefficients in $\mathbb C$. Indeed, consider Eq. \[eq:vr\] when $|\omega|=2$. For convenience, let $\omega=\{1,2\}$. We have $\text{Tr}_2(vv^T)=\sum_{j=1}^r|j\rangle\langle j|$. If $Ov=v$, then $\text{Tr}_2(Ovv^TO^T)=\sum_{j=1}^r|j\rangle\langle j|$. However, $$\begin{aligned} \text{Tr}_2(Ovv^TO^T)& =\text{Tr}_2(O_1vv^TO_1^T) \\ & =\sum_{j=1}^rO_1|j\rangle\langle j|O_1^T \\ & =\sum_{k,k'}(\sum\limits_{j=1}^r(O_1)_{jk}(O_1)_{jk'})|k\rangle\langle k'|. \end{aligned}$$ Therefore $\sum_{j=1}^r(O_1)_{jk}(O_1)_{jk'}=\delta_{kk'}$ for all $k,k'\leq r$ and $\sum_{j=1}^r(O_1)_{jk}(O_1)_{jk'}=0$ for any one of $k>r$ or $k'>r$, which means $(O_1)_{jk}=0$ for all $k>r$. Finally, we need to generalize Lemma \[lem:ujnsupp\] to the multiblock case. Recalling Eq. \[eq:rhonsupp\], we now have $$\rho_\omega=\left(\frac{1}{2^{|\omega|}}(I_j\otimes R_I+Z_j\otimes R_Z)\right)^{\otimes r}.$$ This projector can be associated with a vector $v=\sum_{j=1}^r|jj\rangle$. Using the same technique as for the case where $A_\omega=1$ and $|\omega|=2$, we conclude that $U_j$ must keep one of the three spaces of linear operators $\text{span}(\langle\pm X_j^{(i)}\rangle_{i=1}^{r})$, $\text{span}(\langle\pm Y_j^{(i)}\rangle_{i=1}^{r})$, or $\text{span}(\langle\pm Z_j^{(i)}\rangle_{i=1}^{r})$ invariant. Given these generalizations of Theorem \[thm:d2lulc\] and Lemma \[lem:ujnsupp\] to the multiblock case, we now show that all the arguments in the proof of the single block case can be naturally carried to the multiblock case. Most importantly, we show that the “$\mathcal{I}\rightarrow\mathcal{S}$” procedure is still valid. To specify the “$\mathcal{I}\rightarrow\mathcal{S}$” procedure for the multiblock case, we first need to generalize the concept of the generalized stabilizer defined in Definition \[GS\] to the multiblock case. The *generalized stabilizer* ${\mathcal I}(Q^{\otimes r})$ of an $r$-block quantum code $Q^{\otimes r}$ is the group of all unitary operators that fix the code space, i.e. $${\mathcal I}(Q^{\otimes r})=\{ U\in SU(2^{nr})\ |\ U|\psi\rangle = |\psi\rangle,\ \forall |\psi\rangle\in Q^{\otimes r}\}.$$ Similar to the single block case, we start by assuming that universal quantum computation can be performed using transversal gates. Then $\bar{H}_1^{(1)}$, the logical Hadamard operator acting on the first logical qubit of the first block, is transversal. Let $\alpha^{(1)}$ be a minimal weight element of $\mathcal{C}(\mathcal{S})\setminus\mathcal{S}$. Without loss of generality, we assume $\alpha^{(1)}\in\bar{X}_1\mathcal{S}$. Then $\bar{H}_1^{(1)}$ will transform $\alpha^{(1)}$ to some $\beta'\in\bar{Z}_1^{(1)}\mathcal{I}(Q^{\otimes r})$. This is to say, $\beta'$ acting on $P_Q^{\otimes r}$ is a logical $Z$ operation on the first logical qubit of the first block, and identity on the other $r-1$ blocks. However, this does not mean that $\beta'=\beta^{''(1)}\bigotimes\limits_{i=2}^{r}\delta^{'(i)}$, where $\beta''\in\bar{Z}_1\mathcal{S}$ and $\delta'^{(i)}\in\mathcal{I}(Q)$ for all $i=2,\dots,r$, because $$\bar{H}_1^{(1)}=\bigotimes_{j=1}^{n} U_j,$$ where each $U_j$ acts on $r$ qubits. Expanding $\beta'$ in the basis of $nr$ qubit Pauli operators. For the same reason shown in the proof of Lemma \[lem:li\], there must be at least one term in the expansion which has the form $$\beta^{(1)}\bigotimes\limits_{i=2}^{r}\delta^{(i)},$$ where $\beta\in\bar{Z}_1\mathcal{S}$, and $\delta^{(i)}\in\mathcal{S}$ for all $i=2,\dots,r$. So, the generalization of the “$\mathcal{I}\rightarrow\mathcal{S}$ procedure” to the multiblock case is clear: Pauli operators acting on a code are either logical Pauli operators (on any number of qubits and any number of blocks) or they map the code $P_Q^{\otimes r}$ to an orthogonal subspace. Nevertheless, this is an important observation. Now $\beta\in\mathcal{C}(\mathcal{S})\setminus\mathcal{S}$ is a logical $Z$ operation acting on the first logical qubit of a single block of the code. Due to Eq. \[eq:trans\], $\operatorname{supp}(\beta^{(1)})\subseteq\operatorname{supp}(\alpha^{(1)})$. However $\alpha^{(1)}$ is a minimal weight element of $\mathcal{C}(\mathcal{S})\setminus\mathcal{S}$, therefore we have $\operatorname{supp}(\beta^{(1)})=\operatorname{supp}(\alpha^{(1)}):=\xi$. For convenience, we now drop the superscript $(1)$ when referring to these logical operators. Since $\bar{P}_1^{(1)}$ is transversal, then there exists a $\gamma\in\bar{Y}_1\mathcal{S}$ which has the same support as $\alpha$. Now we have $\alpha\in\bar{X}_1\mathcal{S}$, $\beta\in\bar{Z}_1\mathcal{S}$ and $\gamma\in\bar{Y}_1\mathcal{S}$ such that $\operatorname{supp}(\gamma)=\operatorname{supp}(\beta)=\operatorname{supp}(\alpha)$. Like the single block case, by Lemma \[lem:XYZ\], there is a local Clifford operation such that $\alpha=X^{\otimes |\xi|}\in\bar{X}_1\mathcal{S}$, $\beta=(-1)^{|\xi|/2}Y^{\otimes |\xi|}\in\bar{Y}_1\mathcal{S}$ and $\gamma=Z^{\otimes |\xi|}\in\bar{Z}_1\mathcal{S}$. If $|\xi|=d$ is even there is a contradiction, since $\alpha$, $\beta$, $\gamma$ must anti-commute with each other. When $|\xi|=d$ is odd, we need the following arguments. If for all coordinates $j\in\xi$, there are elements $R_1,R_2,R_3\in\mathcal{M}(Q)$ such that $(R_1)_j=X$, $(R_2)_j=Y$ and $(R_3)_j=Z$, then $U_j\in U(2^r)$ is a Clifford operator for all $j\in\xi$ by Lemma \[lem:bigblocklem\]. Therefore, all the possible logical operations that are transversal on the first encoded qubit must be Clifford operations, contradicting the assumption that universality can be achieved by transversal gates. Therefore, there exists a coordinate $j\in\xi$ such that either (a) there is no minimal support containing $j$ or (b) $(R_1)_j=(R_2)_j\neq I$ for all $R_1,R_2\in \mathcal{M}(Q)$. Use the “$\mathcal{I}\rightarrow\mathcal{S}$ procedure” to expand $\bar{H}_1^{(1)}\beta(\bar{H}_1^{(1)})^{\dagger}$ in the basis of Pauli operators and extract $\alpha'\in\bar{X}_1\mathcal{S}$ with $\operatorname{supp}(\alpha')=\xi$. Since $\bar{H}_1$ is transversal, $U_j$ must keep $\text{span}(\langle\pm Z_j^{(i)}\rangle_{i=1}^{r})$ invariant, up to a local Clifford operation. This means that the $j$th coordinate of $\alpha'$ is either $I_j$ or $Z_j$. The former is not possible since $\operatorname{supp}(\alpha')=\xi$. For the later, $\gamma''=i\alpha'\beta\in\bar{Y}_1\mathcal{S}$, and the $j$th coordinate of $\gamma''$ is $I_j$. Therefore, $\operatorname{supp}(\gamma'')$ is strictly contained in $\xi$. However, this contradicts the fact that $\alpha$ is a minimal weight element in $\mathcal{C}({\mathcal{S}})\setminus\mathcal{S}$. The effect of coordinate permutations {#sec:permutation} ===================================== In this section we discuss the effect of coordinate permutations. \[thm:permutation\] For any stabilizer code $Q$ free of trivially encoded qubits, $\operatorname{Aut}(Q)$ is not an encoded computationally universal set of gates for any logical qubit. Choose a minimum weight element $\alpha\in\mathcal{C}(\mathcal{S})\setminus\mathcal{S}$. Without loss of generality, assume $\alpha\in\bar{X}_1\mathcal{S}$ and $\operatorname{supp}(\alpha)=\xi$. Define a single qubit non-Clifford gate $F$ by $$F:\ X\rightarrow X'=\frac{1}{\sqrt{3}}\left(X+Y+Z\right);\ Z\rightarrow Z'$$ where $Z'$ is any operator that is unitary, Hermitian and anticommuting with $X'$. We cannot use the idea of applying $\bar{H}_1$ and $\bar{P}_1$ from within $\operatorname{Aut}(Q)$ since they might involve different permutations. We instead assume the logical gate $\bar{F}_1$ can be approximated to an arbitrary accuracy by gates in $\operatorname{Aut}{(Q)}$. Then we have $$\bar{F}_1\alpha\bar{F}_1^{\dagger}=\eta\in\frac{1}{\sqrt{3}}\left(\bar{X}_1+\bar{Y}_1+\bar{Z}_1\right)\mathcal{I}(Q)$$ Applying the $\mathcal{I}\rightarrow\mathcal{S}$ procedure to $\eta$, we find $\alpha'\in{\bar{X}_1}\mathcal{S}$, $\beta'\in{\bar{Y}_1}\mathcal{S}$, and $\gamma'\in{\bar{Z}_1}\mathcal{S}$ such that $\operatorname{supp}{\alpha'}=\operatorname{supp}{\beta'}=\operatorname{supp}{\gamma'}=\xi'$ and $|\xi'|=|\xi|=d$. By Lemma \[lem:XYZ\], we can find a locally Clifford equivalent code such that $\alpha'=X^{\otimes |\xi'|}\in\bar{X}_1\mathcal{S}$, $\beta'=(-1)^{|\xi'|/2}Y^{\otimes |\xi'|}\in\bar{Y}_1\mathcal{S}$ and $\gamma'=Z^{\otimes |\xi|}\in\bar{Z}_1\mathcal{S}$. Again, $d$ must be odd. If for all coordinates $j\in\xi$, there are elements $R_1,R_2,R_3\in\mathcal{M}(Q)$ such that $(R_1)_j=X$, $(R_2)_j=Y$, and $(R_3)_j=Z$, then for $U\in\operatorname{Aut}(Q)$, $U_j\in U(2)$ is a Clifford operator for all $j\in\xi$ by Theorem \[thm:d2lulc\]. Permutations are Clifford operations as well, so all possible transversal logical operations on the first encoded qubit must be Clifford operations, contradicting the assumption that the transversal gates are a universal set. Therefore, there exists $j'\in\xi'$ such that either (a) only one of $\{X,Y,Z\}$ appears in $\mathcal{M}(Q)$ at coordinate $j'$ or (b) there is no minimal element with support at $j'$. Without loss of generality, we assume that $X$ appears at coordinate $j'$ in case (a). Since $\bar{F}_1$ can be performed via some transversal gate plus permutation, we have $$\bar{F}_1\alpha'\bar{F}_1^{\dagger}=\eta'\in\frac{1}{\sqrt{3}}\left(\bar{X}_1+\bar{Y}_1+\bar{Z}_1\right)\mathcal{I}(Q).$$ Again applying the $\mathcal{I}\rightarrow\mathcal{S}$ procedure to $\eta'$ we know there exist $\alpha''\in{\bar{X}_1}\mathcal{S}$, $\beta''\in{\bar{Y}_1}\mathcal{S}$, and $\gamma''\in{\bar{Z}_1}\mathcal{S}$ such that $\operatorname{supp}{\alpha''}=\operatorname{supp}{\beta''}=\operatorname{supp}{\gamma''}=\xi''$. And $|\xi''|=|\xi|=d$. The permutation maps $j'$ to $j''$. However, we know that $\eta'|_{j''}=X$, and this is also true in case (b) by similar reasoning to Lemma \[lem:ujnsupp\], hence $\alpha''|_{j''}=\beta''|_{j''}=\gamma''|_{j''}=X$. Then $\gamma'''=i\alpha''\beta''\in\bar{Z}_1\mathcal{S}$ such that $i\alpha''\beta''|_{j''}=I$. Therefore, $\operatorname{supp}(\gamma''')$ is strictly contained in $\xi$, which contradicts the fact that $\alpha$ is a minimal weight element in $\mathcal{C}(\mathcal{S})\setminus\mathcal{S}$. If $\operatorname{Aut}(Q)$ is replaced by $\operatorname{Aut}(Q^{\otimes r})$, the theorem still holds because we can view $Q^{\otimes r}$ as another stabilizer code. However, it is not a simple generalization to allow permutations between transversal gates acting on $r>1$ blocks. This is because permutations are permitted to be different on each block and may also be performed between blocks. Applications and Examples {#sec:applications} ========================= In this section, we apply the proof techniques we have used in previous sections to reveal more facts about the form of transversal non-Clifford gates. First, we describe the form of transversal non-Clifford gates on stabilizer codes. We explore further properties of allowable transversal gates in the single block case and discuss how the allowable transversal gates relate to the theory of classical divisible codes. Finally, we review a [*CSS*]{} code construction based on Reed-Muller codes that yields quantum codes with various minimum distances and transversal non-Clifford gates. Corollary \[cor:form\] gives a form for an arbitrary stabilizer code automorphism. Similarly, in the multiblock case, Lemma \[lem:bigblocklem\] provides possible forms of $U_j$ for any transversal gate $U=\bigotimes_{j=1}^n U_j$. These forms prevent certain kinds logical gates from being transversal on a stabilizer code. An $r$-qubit logical gate $U$ such that $U_j\notin {\mathcal L}_r$ for all $j$ is transversal on a stabilizer code only if $U$ keeps the operator space $\text{span}(\langle\pm\bar{Z}_i\rangle_{i=1}^{r})$ invariant up to a local Clifford operation\[cor:formr\]. Here, $\bar{Z}_i$ denotes the logical Pauli $Z$ operator on the $i$th encoded qubit. This is a direct corollary from Theorem \[thm:maintheorem\] in Sec. \[sec:transversal\] and Theorem \[thm:permutation\] in Sec. \[sec:permutation\]. Consider the three-qubit bit-flip code with stabilizer $\mathcal{S}=\{Z_1Z_2,Z_2Z_3\}$, and choose $|0\rangle_L=|000\rangle,|1\rangle_L=|111\rangle$. The Toffoli gate is transversal on this code and is given by $\text{Toffoli}=\bigotimes_{j=1}^3\text{Toffoli}_j$. The Toffoli gate up to a local Clifford is not in $\mathcal{N}(\langle\pm Z_i\rangle_{i=1}^{3})$; however, the Toffoli gate up to a local Clifford does keep $\text{span}(\langle\pm Z_i\rangle_{i=1}^{3})$ invariant. If $U_j$ up to a local Clifford keeps $\text{span}(\langle\pm Z_j^{(i)}\rangle_{i=1}^{r})$ invariant, i.e. $U_j$ transforms any diagonal matrix to a diagonal matrix, then $U_j$ is a monomial matrix. Similarly, if $U$ keeps $\text{span}(\langle\pm X_j^{(i)}\rangle_{i=1}^{r})$ $($or $\text{span}(\langle\pm Y_j^{(i)}\rangle_{i=1}^{r})$$)$ invariant, then $U_j$ is a monomial matrix in the $X_j$ (or $Y_j$) representation. This does not necessarily mean that $U=\bigotimes_{j=1}^n U_j$ is a monomial matrix (in one of the $X,Y,Z$ representations) in the $2^{nr}$ dimensional Hilbert space, since in general some of the $U_j$ might be Clifford operations. Corollary \[cor:formr\] also applies to a set of gates. A set of gates $V_i$, $i=1,\dots,k$, $(V_i)_j\notin {\mathcal L}_r$ for all $j\in [n]$, is transversal on a stabilizer code only if all of the $V_i$ up to the same local Clifford keep the operator space $\text{span}(\langle\pm Z_i\rangle_{i=1}^{r})$ invariant. The set of gates $\{\text{Hadamard}, \text{Toffoli}\}$ cannot both be transversal on any stabilizer code, since Hadamard keeps $\text{span}(\langle\pm Y_i\rangle_{i=1}^{r})$ invariant and Toffoli keeps $\text{span}(\langle\pm Z_i\rangle_{i=1}^{r})$ invariant. These observations imply that all transversal gates are Clifford, but Toffoli is not Clifford. Note $\{\text{Hadamard}, \text{Toffoli}\}$ is “universal" for quantum computation in a sense that all the real gates can be approximated to an arbitrary accuracy [@shi:littlehelp]. Now we restrict ourselves to the single block case. Up to local Clifford equivalence, Corollary \[cor:form\] and Corollary \[cor:formr\] say that the unitary part of a code automorphism is a diagonal gate. Therefore, we may restrict our discussion of the essential non-Clifford elements of $\operatorname{Aut}{(Q)}$ to diagonal gates, because we can imagine considering the diagonal automorphisms for all locally Clifford equivalent codes and their permutation equivalent codes to find all of the non-Clifford automorphisms. We further restrict ourselves to the case where the stabilizer code is [*CSS*]{} code. Let $Q$ be a [*CSS*]{} code $CSS(C_1,C_2)$ constructed from classical binary codes $C_2^\perp <C_1$. Then $$V = \bigotimes_{\ell=1}^n \operatorname{diag}{(1,e^{i\theta_\ell})}\in\operatorname{Aut}{(Q)}$$ iff $\forall c,c'\in C_2^\perp$ and $\forall a\in C_1/C_2^\perp$, $$\label{eq:diag} \sum_{\ell\in\operatorname{supp}{(a+c)}} \theta_\ell = \sum_{\ell\in\operatorname{supp}{(a+c')}} \theta_\ell\ \textrm{mod}\ 2\pi.$$ The states $$|\tilde{a}\rangle \propto \sum_{c\in C_2^\perp} |a+c\rangle, a\in C_1/C_2^\perp,$$ are a basis for $Q$. $V$ is diagonal, so $V|c\rangle=v(c)|c\rangle$ for $c\in C_1$ and a factor $v(c)\in {\mathbb C}$ that is a sum of angles. $V$ is a logical operation so $V|\tilde{a}\rangle\in Q$, which is possible for a diagonal gate iff $v(a+c)=v(a+c')$ for all $a\in C_1/C_2^\perp$ and all $c,c'\in C_2^\perp$. We now restrict to the case where the angles $\theta_\ell=\theta$ are all equal. Let $Q$ be a [*CSS*]{} code constructed from classical binary codes $C_2^\perp <C_1$. A gate $V\in\operatorname{Aut}{(Q)}$ is a tensor product of $n$ diagonal unitaries $V_\theta=\operatorname{diag}{(1,e^{i\theta})}$ iff $\forall c,c'\in C_2^\perp$ and $\forall a\in C_1/C_2^\perp$, $$\frac{\theta}{2\pi}(\operatorname{wt}{(a+c)}-\operatorname{wt}{(a+c')})\in {\mathbb Z},$$ where $\operatorname{wt}{c}$ denotes the Hamming weight of a classical codeword. The corollary’s condition can be satisfied if and only if the weight of all the codewords in $C_1$ are divisible by a common divisor. A classical linear code is said to be divisible by $\Delta$ if $\Delta$ divides the weight of each codeword. A classical linear code is divisible if it has a divisor larger than $1$. An $[n,k]$ classical code can be viewed as a pair $(V,\Lambda)$ where $V$ is a $k$-dimensional binary vector space and $\Lambda=\{\lambda_1,\dots,\lambda_n\}$ is a multiset of $n$ members of the dual space $V^\ast$ that serve to encode $v\in V$ as $c=(\lambda_1(v),\dots,\lambda_n(v))$ and the image of $V$ in $\{0,1\}^n$ is $k$-dimensional. The $b$-fold replication of $C$ is $(V,r\Lambda)$ where $r\Lambda$ is the multiset in which each member of $\Lambda$ appears $r$ times. The following theorem, which is less general than that proven in [@ward:divisible], gives evidence (though not a proof) that the allowable value $\theta$ might only be $\frac{\pi}{2^{(k+2)}}$, which implies $U\in\mathcal{C}_k^{(1)}$ (see Definition \[def:ck\]). It would be interesting if all of the transversal gates for stabilizer codes lie within the $\mathcal{C}_k$ hierarchy. Let $C$ be an $[n,k]$ classical binary code that is divisible by $\Delta$, and let $b=\Delta/\textrm{gcd}(\Delta,2^{k-1})$. Then $C$ is equivalent to a $b$-fold replicated code, possibly with some added $0$-coordinates. The Reed-Muller codes are well-known examples of divisible codes. Furthermore, they are nested in a suitable way and their dual codes are also Reed-Muller codes, which makes them amenable to the [*CSS*]{} construction. In particular: Let $RM(r,m)$ be the $r$th order Reed-Muller code with block size $n=2^m$ and $0\leq r\leq m$. Then - $RM(i,m)\subseteq RM(j,m)$, $0\leq i\leq j\leq m$ - $\dim{RM(r,m)} = \sum_{i=0}^r {m\choose i}$ - $d=2^{m-r}$ - $RM(m,m)^\perp=\{0\}$ and if $0\leq r<m$ then $RM(r,m)^\perp=RM(m-r-1,m)$. $RM(r,m)$ is divisible by $\Delta=2^{\lfloor m/r\rfloor - 1}$. Let $even(RM^\ast(r,m))=C_2^\perp <C_1=RM^\ast(r,m)$ where $0<r\leq \lfloor m/2\rfloor$. Then $CSS(C_1,C_2)$ is an $[[n=2^m-1,1,d=\textrm{min}(2^{m-r}-1,2^{r+1}-1)]]$ code with a transversal gate $G=\otimes_{j=1}^n \operatorname{diag}{(1,e^{i2\pi/\Delta})}$ enacting $\bar{G}=\operatorname{diag}{(1,e^{-i2\pi/\Delta})}\in \mathcal{C}_{\log_2 \Delta}^{(1)}$ where $\Delta=2^{\lfloor m/r\rfloor -1}$. For instance, the $[[2^m-1,1,3]]$ [*CSS*]{} codes constructed from the first-order punctured Reed-Muller code $R^\ast(1,m)$ and its even subcode $even(R^\ast(1,m))$ support the transversal gate $\exp(-i\frac{\pi}{2^{m-1}}\bar{Z})$ [@zeng:lulc; @steane:fta]. The smallest of these, a $[[15,1,3]]$ mentioned in the introduction, has found application in magic state distillation schemes [@bravyi:magic] and measurement-based fault-tolerance schemes [@raussendorf:oneway]. If we choose parameters $m=8$ and $r=2$ then we have a $[[255,1,7]]$ code with transversal $T$, but this is not competitive with the concatenated $[[15,1,3]]$ code. We leave open the possibility that other families of classical divisible codes give better [*CSS*]{} codes with $d>3$ or $k>1$ and transversal non-Clifford gates. Conclusion ========== We have proven that a binary stabilizer code with a quantum computationally universal set of transversal gates for even one of its encoded logical qubits cannot exist, even when those transversal gates act between any number of encoded blocks. Also proven is that even when coordinate permutations are allowed, universality cannot be achieved for any single block binary stabilizer code. To obtain the required contradiction, the proof weaves together results of Rains and Van den Nest that have been generalized to multiple encoded blocks. Along the way, we have understood the form of allowable transversal gates on stabilizer codes, which leads to the fact that the form of gates in the automorphism group of the code is essentially limited to diagonal gates conjugated by Clifford operations, together with coordinate permutations. This observation suggests a broad family of quantum [*CSS*]{} codes that can be derived from classical divisible codes and that exhibit the attainable non-Clifford single-block transversal gates. In general, it is not clear how to systematically find non-Clifford transversal gates, but the results in Section \[sec:applications\] take steps in this direction. It would be interesting to find more examples of codes with non-Clifford transversal gates. There remain some potential loopholes for achieving universal computation with transversal or almost-transversal gates on binary stabilizer codes. For example, we could relax the definition of transversality to allow coordinate permutations on all $nr$ qubits before and/or after the transversal gate. We could also permit each block to be encoded in a different stabilizer code, and even allow gates to take an input encoded in a code $Q_1$ to an output encoded in a code $Q_2$, provided the minimum distances of these codes are comparable. We could further relax the definitions of transversality and conditions for fault-tolerance so that each $U_i$ acts on a small number of qubits in each block. This latter method is fault-tolerant provided that each $U_i$ acts on fewer than $t$ qubits. Finally, the generalization to nonbinary stabilizer codes, and further to arbitrary quantum codes, remain open possibilities. Acknowledgments {#acknowledgments .unnumbered} =============== We thank Panos Aliferis, Sergey Bravyi, David DiVincenzo, Ben Reichardt, Graeme Smith, John Smolin, and Barbara Terhal for comments, criticisms, and corrections. AC is partially supported by a research internship at IBM. [^1]: The authors are with the Center for Ultracold Atoms, Department of Physics, Massachusetts Institute of Technology, Cambridge, MA, 02139 USA e-mail: [email protected].

--- abstract: 'We present two–dimensional slab–jet simulations of jets in inhomogeneous media consisting of a tenuous hot medium populated with a small filling factor by warm, dense clouds. The simulations are relevant to the structure and dynamics of sources such as Gigahertz Peak Spectrum and Compact Steep Spectrum radio galaxies, High Redshift Radio Galaxies and radio galaxies in cooling flows. The jets are disrupted to a degree depending upon the filling factor of the clouds. With a small filling factor, the jet retains some forward momentum but also forms a halo or bubble around the source. At larger filling factors channels are formed in the cloud distribution through which the jet plasma flows and a hierarchical structure consisting of nested lobes and an outer enclosing bubble results. We suggest that the CSS quasar 3C48 is an example of a low filling factor jet – interstellar medium interaction whilst M87 may be an example of the higher filling factor type of interaction. Jet disruption occurs primarily as a result of Kelvin–Helmholtz instabilities driven by turbulence in the radio cocoon not through direct jet–cloud interactions, although there are some examples of these. In all radio galaxies whose morphology may be the result of jet interactions with an inhomogeneous interstellar medium we expect that the dense clouds will be optically observable as a result of radiative shocks driven by the pressure of the radio cocoon. We also expect that the radio galaxies will possess faint haloes of radio emitting material well beyond the observable jet structure.' author: - | Curtis J. Saxton$^{1,2,3}$, Geoffrey V. Bicknell$^{3,4}$, Ralph S. Sutherland$^3$ & Stuart Midgley$^5$\ $^{1}$[ Mullard Space Science Laboratory, University College London, Holmbury St Mary, Dorking, Surrey RH5 6NT, UK ]{}\ $^{2}$[ Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany ]{}\ $^{3}$[ Research School of Astronomy & Astrophysics, Mt Stromlo Observatory, Australian National University, Weston ACT 2611, Australia ]{}\ $^{4}$[ Department of Physics & Theoretical Physics, Faculty of Science, Australian National University ACT 0200, Australia ]{}\ $^{5}$[ ANU Supercomputer Facility, Australian National University ACT 0200, Australia ]{} date: 'Accepted —-. Received —-; in original form —-' title: Interactions of Jets with Inhomogeneous Cloudy Media --- \[firstpage\] hydrodynamics – ISM: clouds – galaxies: active – galaxies: ISM – galaxies: jets – galaxies: individual (3C48) INTRODUCTION {#s:intro} ============ Previous models of jets propagating through the interstellar medium of radio galaxies have almost always assumed a uniform, usually hot ($T \sim 10^7 \> \mathrm{K}$) interstellar medium. These models have contributed substantially to our understanding of the propagation of jets in evolved radio galaxies wherein much of the dense matter that could have previously obstructed the jets has been cleared away. However, it is now clear that in many *young* radio galaxies, i.e. Gigahertz Peak Spectrum (GPS) and Compact Steep Spectrum (CSS) sources, substantial interaction between the expanding radio plasma and dense interstellar clouds occurs. This interaction has a number of effects: Two of the most important are that it significantly distorts the radio morphology and it also gives rise to shock excited line emission. Strong observational evidence for the latter effect has been obtained from combined Hubble Space Telescope (HST) and radio observations of a number of CSS sources [@devries1999a; @labiano03a; @odea2003a] showing a strong alignment effect between optical and radio emission and clear evidence for disturbance of the optically emitting gas by the expanding radio lobe. On the other hand, the dense gas cannot have a large filling factor in the case of GPS and CSS sources since the expansion speed of the radio plasma is usually quite high: $\sim 0.1 - 0.4~c$ [@conway2002; @murgia1999a; @murgia1999b; @murgia2002a; @murgia2003a]. Thus the picture that is conveyed by the combined radio and optical observations is of radio plasma propagating through a predominantly hot and tenuous interstellar medium in which dense clouds are embedded with a low filling factor. Nevertheless, as we show in this paper, even a low filling factor of dense gas can have a pronounced effect on the evolution of the radio lobes. High redshift radio galaxies provide another example in which radio plasma – cloud interactions are clearly important. For example, in the high redshift radio galaxy 4C41.17, interactions between the jets and large clouds in the Lyman-$\alpha$ emitting halo of the parent galaxy lead to shock-excited line emission and to accelerated rates of star formation [@bicknell98a]. Returning to the low-redshift Universe, radio plasma – clumpy ISM interactions may be responsible for the heating of the ISM that is necessary to counteract the otherwise large mass inflow rates that are inferred from X-ray observations of cooling flows. Thus, there is ample physical motivation for detailed investigations of the manner in which jets and the related radio lobes interact with realistic interstellar media. One of the most significant drivers of this research is that all of the scenarios discussed above represent examples of AGN-galaxy feedback which, it has been argued [e.g. @silk1998a; @kawakatu03a], is a significant determinant of the ultimate structure of galaxies and in particular, the relation between the mass of the central black hole and the mass of the galaxy bulge [@magorrian98a; @tremaine02a]. In this first paper in this series we explore the nature of the interaction between jets and their lobes through a series of two–dimensional slab jet simulations. The advantages of this approach is that two dimensional simulations offer the prospect of high resolution and an initial rapid investigation of parameter space, which capture some of the features of a more realistic three dimensional simulations. Fully three dimensional simulations will be presented in future papers. The choice of coordinate system is non-trivial when representing supersonic jets in two spatial dimensions. Many aspects of a jet burrowing through a uniform external medium have been adequately described in the past by axisymmetric simulations. However an axisymmetric simulation is less appropriate for physical systems where the jet is expected to be bent or deflected. When colliding directly with a dense cloud, an axisymmetric jet must either burrow through the cloud or cease its advance. However, a two-dimensional slab-jet can be deflected, although both jet and cloud are essentially infinite in the direction perpendicular to the computational grid. Our present work is concerned with systems where jet-cloud collisions and indirect interactions dominate. We therefore opt for a Cartesian grid and simulate transient slab jets. CHOICE OF JET AND ISM PARAMETERS ================================ Jet energy flux, cloud and ISM density -------------------------------------- The choice of jet parameters is based on a self-similar model for lobe expansion (@bicknell97a, based upon the model by @begelman96a). In this model, the lobe is driven by a relativistic jet which expands in such a way that $\zeta$ the ratio of hot spot to lobe pressure is constant. A value $\zeta \sim 2$ was inferred by Begelman from simulations by @lind89. Let $\rho_0$ be the atmospheric mass density at a fiducial distance $R_0$ from the core, let $\delta$ be the index of the power variation of atmospheric density with radius ($\rho = \rho_0 (R/R_0)^{-\delta}$) and let the jet energy flux be $F_E$. The temperature of the external medium is unimportant in this model since it is the inertia of the background medium that governs the expansion of the lobe. The advance speed, $v_\mathrm{b}$ of the head of the bowshock as a function of distance $R$ from the core is given by: $$\begin{aligned} \frac {v_{\rm b}}{c} &=& \frac {3}{\left[ 18 (8-\delta)\pi\right]^{\frac{1}{3}}} \, \zeta^{\frac{1}{6}} \, \left( \frac {F_E}{\rho_0 R_0^2} \right)^{1/3} \, \left( \frac {R}{R_0} \right)^{(\delta-2)/3} \nonumber \\ &\approx& 0.073 \, \left( \frac {F_{E,46}}{n(100 \> \rm pc)} \right)^{\frac {1}{3}} \, \left( \frac {R}{100 \> \rm pc} \right)^{-\frac{1}{6}} \label{e:vb}\end{aligned}$$ where the jet energy flux is $10^{46} F_{E,46} \> \rm erg \> s^{-1}$, $n(100 \> \rm pc)$ is the number density at 100 pc and the numerical values are for $\zeta=2$ and $\delta=1.5$. Typical expansion speeds of CSS and GPS sources are of order $(0.05 - 0.4)\> c$ with the largest velocities measured for the most compact sources [@conway2002; @murgia1999a]. @conway02a has reviewed estimates of number densities $\sim 1 \> \rm cm^{-3}$ for three GPS sources, and so a fiducial value of $n(100 \> \rm pc) = 1~\mathrm{cm}^{-3}$ is reasonable. Compatibility of the expansion speeds estimated from equation (\[e:vb\]) and the observations indicates jet energy fluxes in the range of $10^{46-47} \> \rm erg \> s^{-1}$. Jets of this power are also consistent with a conventional ratio $\sim 10^{-12} - 10^{-11} \> \rm Hz^{-1}$ of 1.4 GHz monochromatic radio power to jet energy flux and a median power at 5 GHz $\sim 10^{27.5} \> \rm W \> Hz^{-1}$. Nevertheless, in other sources which morphologically resemble CSS radio galaxies, such as the less radio powerful Luminous Infrared Galaxies studied by @drake2003, it is apparent that the jet powers are somewhat less. Thus, in our simulations, we have adopted jet powers of $10^{45}$ and $10^{46} \> \rm erg \> s^{-1}$. Hubble Space Telescope observations [@odea2002; @devries1997; @devries1999a] show abundant evidence for dense clouds of emission-line gas associated with CSS radio galaxies and quasars. For a temperature $10^4 T_{\rm c,4} \> \rm K$ of the dense clouds, a number density $n{_{\mathrm{x}}}\sim 1 \> \rm cm^{-3}$ and temperature $10^7 T_{\rm x,7}\> \rm K$ for the hot interstellar medium, the number density in the clouds is $n_{\rm c}\approx 10^3 \> n{_{\mathrm{x}}}T_{\rm x,7} /T_{\rm c,4} \> \rm cm^{-3}$, assuming that the initial, undisturbed clouds are in pressure equilibrium with their surroundings. If we adopt such a value for the density of the interstellar medium then equation (\[e:vb\]) implies that the velocity of advance would be $\sim 0.02 \> c$ — much lower than what is observed. Hence, as discussed in § \[s:intro\], the dense clouds must have a small filling factor. Accordingly, in the simulations described below, we create clouds of approximately the inferred density and investigate the effect of various filling factors. With the inferred radio source and cloud geometry, the line emission orginates from dense, relatively cool clouds that are shocked as they are overrun by the expanding radio source. A model of this type was proposed by @devries1999a and elaborated by @bicknell2003a. In both treatments, the excess pressure of the advancing radio lobe drives shocks into the dense clouds. In the simulations described below, this is seen to occur but some direct jet-cloud collisions are also apparent. For radio lobe driven shocks (the main source of shocks), the shock velocity is: $$v_{\rm sh} \approx 950 \, \left( \frac {v_{\rm b}}{0.1 \> c} \right) \, \left( \frac {n_{\rm c}/n_{\rm x}}{10^3} \right)^{\frac{1}{2}} \> \rm km \> s^{-1}$$ where $n_{\rm c}$ and $n_{\rm x}$ are the cloud and ISM number densities respectively. Shocks in the velocity range of $300-950 \> \rm km \> s^{-1}$ are produced for bow-shock advance speeds $\sim 0.1 \> c$ and density ratios $n_{\rm c}/n_{\rm x}\sim 10^4 - 10^3$. This is in accord with the observed velocity dispersions and velocity offsets inferred from HST spectra of a sample of CSS sources [@odea2002]. Radiative shocks do not develop instantaneously; the time required for a shock to become fully radiative is effectively the time, $t_4$, required for post-shock gas to cool to $\sim 10^4~\mathrm{K}$. In order that there be a display of optical emission lines, $t_4$ must be less than a fraction $f \sim 1$ of the dynamical time associated with the expansion of the lobe. In estimating $t_4$ we utilize a cooling function, which estimates the non–equilibrium cooling following a high velocity shock, and which is derived from the results of the one–dimensional MAPPINGS III code [@sutherland93c]. This cooling finction is also used in our simulations described below. The condition on $t_4$ becomes a condition on the bow-shock velocity $$\left( \frac {v_b}{c} \right) < 0.07 \> f^{0.2} \, \left(\frac{n_{\rm c}}{10^4 \> \rm cm^{-3}} \right)^{0.2} \, \left( \frac {n_{\rm c}/n_{\rm x}}{10^4} \right)^{0.4} \, \left( \frac {R}{\rm kpc} \right)^{0.2} \ , \label{e:vb_limit}$$ [@bicknell2003a]. This upper limit on $v_{\rm b}$ is insensitive to the various parameters, in particular, $f$. If condition (\[e:vb\_limit\]) is not satisfied near the head of the radio lobe it may be satisfied at the sides where the bow-shock expansion speed is about a factor of 2 lower. In this case one would expect to see emission from the sides but not the apex of the radio source. This condition gives a limiting bow–shock velocity $v_{\rm b}\sim 0.1 c$. Therefore, it would be interesting to see whether the more rapidly expanding sources have a lower ratio of emission line flux to radio power, and whether the emission line flux is distributed in a different manner to that in lower velocity sources. These conditions may also be related to the causes of the trailing optical emission in CSS sources [@devries1999a]. There are two possibilities: (a) The apex of the bow-shock is non-radiative (if $v_{\rm b}$ exceeds the above limit) or (b) The number of dense clouds per unit volume decreases at a few kpc from the core, i.e. beyond the normal extent of the narrow line region. We are inclined to favour option (b) but (a) should also be kept in mind. The simulations capture these characteristics that depend on the time of the effective onset of radiative shocks in the clouds. We see a range of shock conditions varying from almost pure adiabatic shocks to strongly radiative. When the cooling time of gas becomes shorter than the Courant time for the simulation, the integration of the energy equation ensures that the radiative shock occupies approximately one pixel in which the gas temperature is equal to the predefined equilibrium temperature ($10^4 \> \rm K$ in these simulations). **Overpressured jets — Mach number and density ratio** {#s:jet_pars} ------------------------------------------------------ In order to carry out realistic 2D simulations, we wish to establish jets with powers per unit transverse length that are relevant to the types of radio galaxies mentioned in the introduction, in combination with other reasonable values of parameters such as jet width and density. If the number density of the interstellar medium is $\sim 1 \> \rm cm^{-3}$ on 100 pc scales, then the jet must be overpressured. We first show this in the context of a sub-relativistic jet model and then support the proposition with a relativistic jet model. Let a nonrelativistic jet have a density $\eta$ times that of the external medium, an internal Mach number $M$, a specific heat ratio $\gamma$, a pressure $\xi$ times the external ISM pressure and a cross-sectional area $A_\mathrm{j}$. Further, let $\rho{_{\mathrm{x}}}$ and $v{_{\mathrm{x}}}$ be the density and isothermal sound speed of the ISM. Then the jet mass, momentum and energy fluxes are, collecting the dimensional quantities on the left: $$\begin{aligned} \dot{M}&=& (\rho{_{\mathrm{x}}}\, v{_{\mathrm{x}}}\, A_\mathrm{j})\ \ M\sqrt{\gamma\xi\eta} \label{mdot} \\ \Pi&=& (\rho{_{\mathrm{x}}}\, v{_{\mathrm{x}}}^2 \, A_\mathrm{j})\ \ \xi \, (1+\gamma M^2) \> \label{e:thrust} \\ F_E&=& (\rho{_{\mathrm{x}}}\, v{_{\mathrm{x}}}^3 \, A_\mathrm{j})\ \ M\left({\frac {1}{\gamma-1} + M^2}\right) \sqrt{{\gamma^3\xi^3}\over\eta} \label{e:power}\end{aligned}$$ Equation (\[e:power\]) clearly shows that the energy flux can be made arbitrarily large by decreasing the density ratio $\eta$. However, there is a natural lower limit on $\eta$ – the value at which an intially relativistic jet becomes transonic. The critical value is defined by the condition $\rho_{\rm j} c^2 / 4 p_{\rm j} = 1$ [@bicknell1994a], implying that $\eta_{\rm crit} = 4 \xi (v{_{\mathrm{x}}}/c)^2 \approx 5.9 \times 10^{-6} \xi T_{\rm x,7}$. With the overpressure factor, $\xi =1$, $\eta = 10^{-5}$, a jet diameter of 10 pc, $T{_{\mathrm{x}}}= 10^7 \> \rm K$ and $M=20$, the energy flux is only $\sim 10^{44} \> \rm erg \> s^{-1}$. One requires an overpressure factor of about 100 to achieve $10^{46} \> \rm erg \> s^{-1}$. This result is easily verified for a relativistic jet whose energy flux is given by $$F_E = (\rho{_{\mathrm{x}}}v{_{\mathrm{x}}}^2 c A_{\mathrm j})\ 4\,\xi\beta_{\rm jet} \,\Gamma_{\rm jet}^2 \> \left[1 + \frac {\Gamma_{\rm jet}-1}{\Gamma_{\rm jet}} \chi \right] \nonumber$$ where $\chi = \rho_{\rm j} c^2 /4 p_{\rm j}$, and $\rho_j$ and $p_j$ are the jet rest–mass density and pressure respectively. If $\xi = 1$, $\Gamma=10$, a jet diameter of 10 pc, $T{_{\mathrm{x}}}= 10^7 \mathrm{K}$ and $\chi = 0$ (the latter being appropriate for a jet dominated by relativistic plasma), then the energy flux is only $1.2 \times 10^{43} \> \rm erg \> s^{-1}$. Hence, for a relativistic jet, we require $\xi \sim 10^{2-3}$ in order to achieve the required jet power. Is an overpressured jet physical? The overpressure ratio refers to the pressure with respect to the undisturbed interstellar medium. However, the lobe itself is over-pressured as a result of the advancing bow shock so that it is possible for the jet to be in approximate equilibrium with the ambient lobe. In a real galaxy the over pressured lobe may not extend back to the core, in which case an overpressured jet would behave initially like a classical overpressured, underexpanded jet. Therefore, in our simulations, we have adopted low values of the density ratio, $\eta \sim 10^{-3}$, Mach numbers, $M$ of the order of 10 and a high overpressure ratio in order to obtain the necessarily high jet energy fluxes (or their equivalent in 2D). Such a low value of $\eta$ has the additional effect of producing superluminal velocities ($\beta \sim 5-6$) in some simulations, highlighting the shortcomings of simulating powerful jets with a non–relativistic code. However, the essential point is that it is mainly the jet power that is physically important in jet–cloud interactions. One could adopt the strategy of adopting non-relativistic parameters that correspond more closely to those of relativistic jets, for example by equating the velocity and pressure and matching either the power or thrust (e.g. see @komissarov96a and @rosen99a). We have not done this explicitly here since there is no transformation that perfectly maps a non-relativistic flow into a relativistic equivalent. Clearly, this initial investigation of parameter space needs to be followed by more comprehensive simulations utilising a fully relativistic code. Prescription for cloud generation {#s.clouds} --------------------------------- In our simulations we prescribe a uniform density for the hot interstellar medium. Within this medium we establish an initial field of dense clouds that mimics the distribution of clouds in the interstellar medium, with a random distribution of positions, shapes and internal density structures. We consider that this is more desirable than specifying, say circular clouds, albeit with random centres; in that case the subsequent evolution may be governed by the initial symmetry. An advantage of the approach that we describe here is that the distribution of the sizes, shapes and directions of propagation of radiative shocks is more realistic than that obtained from initially highly symmetric clouds. We prescribe the density and distribution of clouds using a Fourier approach. In the first step we establish a grid of complex numbers in Fourier space and assign a random phase to each point. The largest wave number is compatible with the spatial resolution of the simulations. An amplitude filter is applied, $\propto |\mathbf{k}|^{-5/3}$, in order to mimic the power spectrum of structures formed by turbulence. Taking the Fourier transform of the wavenumber data produces an array of density fluctuations in real space. These fluctuations, $\sigma(\mathbf{r})$, are normalised so that their mean is $0$ and their variance is $1$. The final number density of gas is then defined by the relation: $$n=\left\{{ \begin{array}{ll} n_\mathrm{min}+n_\sigma (\sigma-\sigma_\mathrm{min}) &\mbox{wherever $\sigma\ge\sigma_\mathrm{min}$} \\ n{_{\mathrm{x}}}&\mbox{elsewhere} \end{array} }\right. \ ,$$ where $n{_{\mathrm{x}}}$ is the number density of hot inter-cloud gas, $n_\mathrm{min}$ is the minimum number density of gas in a dense cloud, $n_\sigma$ is a scaling factor and $\sigma_\mathrm{min}$ is a threshold value determining whether a particular cell is part of a cloud or else hot inter-cloud medium. In practice we choose the mean density of cloud gas, $\bar{n}$ and set the value of $n_\sigma$ implicitly. We apply upper and lower limits to the density distribution in $\mathbf{k}$-space. Amplitudes are zeroed for $k<k_\mathrm{min}$, where $k_{\rm min}$ corresponds to the Jeans wave number. This prevents the insertion of clouds that would have collapsed to form stars and also avoids the generation of a cloud field dominated by a single large cloud-mass. Amplitudes are also zeroed for $k> k_\mathrm{max}$ where $k_\mathrm{max}$ corresponds to 3 or 4 pixels of spatial resolution. Fine features, at the 3-4 pixel level, can introduce excessive and artificial sensitivity to effects on the level of individual pixels. Finally, radial limits are applied to the cloud field as a whole. An outer limit, $r_\mathrm{max}$, (typically $\sim 0.5 - 5~{{\mathrm{kpc}}}$) limits the extent of the cloud field. The inner radius, $r_\mathrm{min}$, is defined by the Roche limit in the vicinity of a black hole: $$r_\mathrm{min} =\left({ {3\over{4\pi}} {{M_\mathrm{bh}}\over{\rho_\mathrm{c}}} }\right)^{1/3}$$ If the black hole mass $M_\mathrm{bh} = 10^9 M_\odot$ and the cloud density $\rho_\mathrm{c} \approx 15 M_\odot \> \mathrm{pc}^{-3}$ then $r_\mathrm{min} \approx 250 \> \mathrm{pc}$. Table \[table.cloudfield\] presents a summary of the properties of the clouds and intercloud medium, and the extent of cloud coverage in our simulations (which are each described in specific detail in §\[s.simulations\]). Figures \[fig.clouds.initial.1\] and \[fig.clouds.initial.2\] represent the initial distributions of cloud and intercloud gas in three of our simulations. Figures \[fig.cloudcdf\_s02\] and \[fig.cloudcdf\_e\] show how the masses of individual clouds are distributed in the initial conditions of four of our simulations. As the cumulative distribution functions show, most clouds have masses of less than a few thousand solar masses. The scatter plots show the characteristic depth of clouds: in mean terms the distance from an interior point to the surface, $r=DI/S$ where $D$ is the number of spatial dimensions, $I$ is the interior (area of a 2D cloud) and $S$ is the surface (perimeter of a 2D cloud). A small $r$ means that most of the cloud mass is well exposed to surface effects, such as ablation and the penetration of shocks from outside. If $r$ is large than the interior regions are typically deeper from the locally nearest surface, and correspondingly better insulated from effects at the surface. Large clouds, near the Jeans limit (e.g. thickness $2r \sim 105\ {{\mathrm{pc}}}$ in the [B]{} series) are rare in every simulation. The simulations with a greater cut-off $\sigma_{\mathrm min}$, and hence greater filling factor of clouds, also feature individual clouds of greater mass than the simulations with lower $\sigma_{\mathrm min}$. $\begin{array}{ccc} \ifthenelse{\isundefined{\rainbow}}{ \includegraphics[width=7cm]{f02a.eps} \\ \includegraphics[width=7cm]{f02b.eps} \\ \includegraphics[width=7cm]{f02c.eps} }{ \includegraphics[width=7cm]{f02a.hue.eps} \\ \includegraphics[width=7cm]{f02b.hue.eps} \\ \includegraphics[width=7cm]{f02c.hue.eps} } \end{array}$ $\begin{array}{cc} \includegraphics[width=7.5cm]{f03a.eps} &\includegraphics[width=7.5cm]{f03b.eps} \end{array}$ $\begin{array}{cc} \includegraphics[width=7.5cm]{f04a.eps} &\includegraphics[width=7.5cm]{f04b.eps} \\ \includegraphics[width=7.5cm]{f04c.eps} &\includegraphics[width=7.5cm]{f04d.eps} \\ \includegraphics[width=7.5cm]{f04e.eps} &\includegraphics[width=7.5cm]{f04f.eps} \\ \end{array}$ Code ---- In the simulations reported here, we have used a locally modified version of a Piecewise Parabolic Method [@colella1984] code, known as [VH-1]{}, published on the web by the computational astrophysics groups at the Universities of Virginia and South Carolina[^1]. Our modifications of the VH-1 code include restructuring of the code for efficient computation on the computers at the Australian National University (formerly a Fujitsu VP300; now a Compaq Alphaserver SC), addition of code for dynamically important cooling and other code for coping with a numerical shock instability [@sutherland2003a]. The code has already been used in several adiabatic applications (e.g. @saxton2002aa, @saxton2002bb) and most recently in two-dimensional simulations of fully radiative wall shocks [@sutherland2003b]. In the latter paper, cooling was dynamically important and the code for preventing numerically unstable shocks was implemented and proved to be important in elminating physically unrealistic features. We refer to this augmented code as [*ppmlr*]{}. We have recently ported the [*ppmlr*]{} code to the Message Passing Interface (MPI) environment (see Appendix \[app.mpi\]). This port enables us to take advantage of the Compaq Alphasever parallel processor architecture. The MPI port is described in the appendix. The simulations described here typically involve 16 processors. Future three-dimensional simulations will regularly use 256 processors and in exceptional circumstances, up to 512 processors. $ \begin{array}{ccccccccc} \hline \hline \mbox{Model} &\begin{array}{c}T_\mathrm{x}\\(\mathrm{K})\end{array} &\begin{array}{c}n_\mathrm{x}\\(\mathrm{cm}^{-3})\end{array} &n_\mathrm{c,min}/n_\mathrm{x} &\bar{n}_\mathrm{c}/n_\mathrm{x} &\sigma_\mathrm{min} &\begin{array}{c}A_\mathrm{c}\\(\mathrm{pc}^2)\end{array} &f_\mathrm{c} &\begin{array}{c}P_\mathrm{c}\\(\mathrm{pc})\end{array} \\ \hline {\tt A1} &10^7& 10^{-2} & 300 & 500 & 1.9 & 7.28 \times 10^5 &2.92\times10^{-2}&5.83 \times 10^4 \\ {\tt B1} & 10^7 &1 & 500 & 1000 & 2.6 & 1.55 \times 10^3 &4.11\times10^{-3}&6.16 \times 10^2 \\ {\tt B2} & 10^7 &1 & 500 & 1000 & 1.9 &1.15 \times 10^4 & 3.04\times10^{-2}&3.51 \times 10^3 \\ {\tt B3} & 10^7 &1 & 500 & 1000 & 1.2 & 4.43 \times 10^4 & 1.18\times10^{-1}& 9.74 \times 10^4 \\ {\tt B3a} & 10^7 &1 & 500 & 1000 & 1.2 & 4.43 \times 10^4 & 1.18\times10^{-1}& 9.74 \times 10^4 \\ \hline \end{array} $ $ \begin{array}{cccccccccccc} \hline \hline \mbox{Model}&M&\eta&\xi &\begin{array}{c}r_\mathrm{j}\\(\mathrm{pc})\end{array} &\begin{array}{c}\dot{M}\\(M_\odot / \mathrm{yr})\end{array} &\begin{array}{c}\Pi\\(10^{33} \mathrm{dyn})\end{array} &\begin{array}{c}F_E\\(\mathrm{erg} / \mathrm{s})\end{array} &{{ \dot{M} }\over{ A_\mathrm{j}\rho_\mathrm{x}c_\mathrm{x} }} &{{\Pi }\over{ A_\mathrm{j}\rho_\mathrm{x}c_\mathrm{x}^2 }} &{{F_E }\over{ A_\mathrm{j}\rho_\mathrm{x}c_\mathrm{x}^3 }} \\ \hline {\tt A1}& 12.7&0.384~~~& 947 & 25 & 3.5\times10^{-2} &5.43~\,&10^{45} &313&2.10 \times 10^4&2.10\times10^8 \\ {\tt B1}& 12.7& 1.16 \times 10^{-3}& 100 &10 & 1.0\times10^{-2} &9.17~\,&10^{46} &5.58&2.22 \times 10^3&1.31\times10^8 \\ {\tt B2}& 12.7& 1.16 \times 10^{-3}& 100 &10 & 1.0\times10^{-2} &9.17~\,&10^{46} &5.58&2.22\times 10^3&1.31\times10^8 \\ {\tt B3}& 12.7& 1.16 \times 10^{-3}& 100 &10 & 1.0\times10^{-2} &9.17~\,&10^{46} &5.58&2.22\times 10^3&1.31\times10^8 \\ {\tt B3a}& 12.7& 1.16 \times 10^{-3}& 100 &10 & 1.0\times10^{-2} &9.17~\,&10^{46} &5.58&2.22 \times 10^3&1.31\times10^8 \\ \hline \end{array} $ SIMULATIONS {#s.simulations} =========== In the following subsections we describe a number of simulations that exhibit a wide range of phenomena when we vary fundamental parameters such as the jet energy flux ($F_E$), the jet density ratio ($\eta$) and the filling factor of clouds ($f_\mathrm{c}$) obstructing the path of the jet. The jet parameters in the simulations are presented in Table \[table.jet\], and the characteristics of the clouds and intercloud media are given in Table \[table.cloudfield\]. In each simulation the X-ray emitting inter-cloud medium has temperature $T_\mathrm{x}=10^7\ \mathrm{K}$, and the corresponding isothermal sound speed is our unit of velocity in the simulations, $v_0=v{_{\mathrm{x}}}=365\ \mathrm{km}\ \mathrm{s}^{-1}$. Our unit for time measurement is $t_0=v_0 / 1\ {{\mathrm{pc}}}=2.68\times10^3\ \mathrm{yr}$. The jet is defined by a group of cells on the left boundary where the density, pressure and velocity are kept constant. Elsewhere on the left boundary a reflection condition is applied, representing the effect of bilateral symmetry of a galaxy with equally powerful jet and counter-jet. The boundary cells on other sides of the grid are subject to an open boundary condition: in each cycle of the hydrodynamic computation, the boundary cell contents are copied into the adjacent ghost cells. This one-way boundary allows hydrodynamic waves to propagate off the grid. In a journal paper, one is necessarily limited to showing illustrative snapshots from simulations. Movies of the simulations described in the following section, which show the detailed dynamical evolution, may be obtained from [http://macnab.anu.edu.au/radiojets/]{}. Model [A1]{}: high density, overpressured jet --------------------------------------------- This simulation exhibits both expected and unexpected features of jets interacting with a distribution of clouds in an active nucleus. The jet is initially unimpeded by clouds and generally behaves like a jet in a uniform medium showing the usual bow-shock, terminal jet-shock and backflow. Then two small clouds are overtaken by the bow-shock and the overpressure in the cocoon drives radiative shocks into them. Subsequently the jet backflow is distorted by the reverse shock emanating from the upper of the two clouds. The interaction with the cloud field becomes more dramatic when the apex of the bow shock strikes one of the larger clouds just below the jet axis. Again this drives a strong radiative shock into the cloud and the larger reverse shock deflects both the jet and backflow sideways. The bow-shock then intersects the larger of the two clouds and this produces an additional reverse shock that deflects the jet further sideways. The jet continues onwards past these two clouds whlist the dense clouds in the wake of the bow-shock (i.e. immersed within the jet cocoon) are gradually disrupted by the gas surrounding them. A large amount of turbulence is generated within the jet cocoon by the reverse shocks from the bow-shock impacted clouds and the jet flow is clearly affected but not disrupted. At late times many of the clouds immersed within the cocoon show trails resulting from the ablation of gas within the turbulent cocoon. However, the details of these features will be uncertain until we are able to resolve them fully and subdue the effects of numerical mass diffusion. Notable features of this interaction are: (I) The jet does not directly touch any of the clouds. The radiative shocks within the clouds are all produced as a result of the overpressure behind the bow-shock. (II) The importance of the reverse shocks within the cocoon in deflecting the jet and making the cocoon more turbulent. (III) The survival of the jet despite these interactions (aided by heaviness of the jet in this simulation, $\eta=0.3835$). The left panel of Figure \[fig.slab02b.1000.dens\] shows the jet interacting with clouds during an early stage of the simulation, $t=5t_0$. The bow shock and therefore the influence of the jet have only propagated about $1.7\ {{\mathrm{kpc}}}$ forward and $0.5\ {{\mathrm{kpc}}}$ laterally, and most of the clouds remain in their initial condition. The upper panels of Figure \[fig.slab02b.1000.dens.cut\] show details of the subregion immediately surrounding the jet at this time of the simulation. Within the high-pressure region bounded by the bow shock, all clouds are shocked. The cloud shocks are the intensely cooling regions on the cloud perimenters visible in the upper-right panel of Figure \[fig.slab02b.1000.cool.cut\]. The cloud density is sufficiently high that the shocks have penetrated only a few pc at this stage — a small depth compared to a cloud thickness. The other noteworthy cooling component visible in the cooling map (Figure \[fig.slab02b.1000.cool.cut\]) is the X-ray emission from the bow shock. This is more intense near the apex of the bow shock than at its sides. The lower panels of Figure \[fig.slab02b.2000.dens.cut\] show the density and cooling at a later time, $t=10t_0$, when the jet is more advanced and the clouds are more eroded. Some clouds that lay in the path of the jet at time $5t_0$ (located at $(0.85,0.15)\,{{\mathrm{kpc}}}$ and $(0.95,0.10)\,{{\mathrm{kpc}}}$) are now completely shocked and destroyed. Other clouds have survived closer to the nucleus but with less damage because they are out of the jet’s path, e.g. those at $(0.3,0.4)\,{{\mathrm{kpc}}}$ and $(0.3,0.25)\,{{\mathrm{kpc}}}$. The jet survives an oblique interaction with one group of eroded clouds near $(0.9,0.15)\,{{\mathrm{kpc}}}$, but its width increases to the right of this point. A second cloud collision near $(1.7,0.0)\,{{\mathrm{kpc}}}$ deflects the jet into the negative $y$ direction, and causes its decollimation into a turbulent bubble of hot plasma accumulating at the apex of the bow shock. The head of the jet is unsteady and changes direction completely within a few $t_0$. The density distribution at time $t=19t_0$, near the finish of the simulation, appears in the right panel of Figure \[fig.slab02b.3800.dens\]. Clouds are visible in all stages of shock compression. The two major cloud collisions, around $(1.00,0.05)\,{{\mathrm{kpc}}}$ and $(1.75,0.00)\,{{\mathrm{kpc}}}$, now leave the jet well collimated and relatively straight. The jet loses collimation at a third collision, around $(2.5,^-\!\!0.2)\,{{\mathrm{kpc}}}$. The turbulent bubbles around the head of the jet have grown, and spurt between the clouds in several different directions. As was the case at $t=5t_0$ and $10t_0$, the clouds situated off to the sides have survived much better than clouds that lay close to the direct path of the jet. In summary, whether a cloud behaves radiatively and contracts, or adiabatically and ablates is affected by its size, location and density. Small clouds are more effectively adiabatic and less durable than large clouds. This may be partly because their exposed surfaces are large compared to their internal volume, and partly because they are geometrically narrower and on average less dense than large clouds. Some of the smallest swell and develop ablative tails almost as soon as they enter the cocoon, e.g. those between $(1.3,^-\!\!0.7)\,{{\mathrm{kpc}}}$ and $(2.1,^-\!\!0.7)\,{{\mathrm{kpc}}}$ when $t=10t_0$, in Figure \[fig.slab02b.2000.dens.cut\]. A larger cloud near $(0.2,^-\!\!0.5)\,{{\mathrm{kpc}}}$ was within the cocoon for most of the simulation and yet it was barely affected, even until $t=19t_0$ (right panel, Figure \[fig.slab02b.3800.dens\]). Qualitatively, each cloud appears to suffer one of three possible fates. Small clouds are destroyed quickly regardless of their position. Large clouds near the path of the jet affect the jet through reflection shocks (even if the jet doesn’t strike directly) but they inflate and ablate away in the long term. Large bystander clouds situated well off the jet axis endure almost indefinitely Interaction with clouds alters the propagation and shape of the bow shock. The bow shock is severely retarded when it strikes a cloud and refracts through the gaps between clouds. Thus the bow shock surface is conspicuously indented (“dimpled”) for as long as it remains amidst the cloud field. We would expect this effect to manifest in 3D simulations also, although the extra degree of freedom in 3D is also likely to bring about some differences. The jet generally remains identifiable and collimated after deflection in two or at most three cloud interactions. These interactions need not be direct: it is sufficient for the jet to interact with the reverse shocks that reflect off a cloud lying within several times $r_\mathrm{j}$ off the jet’s path. These backwashing shocks spread continually in the turbulent cocoon; they appear to act like a “cushion” against collision between jet and cloud. More than three consecutive cloud interactions disrupts the jet. $\begin{array}{cc} \ifthenelse{\isundefined{\rainbow}}{ \includegraphics[width=9cm]{f05a.eps} &\includegraphics[width=9cm]{f05b.eps} }{ \includegraphics[width=9cm]{f05a.hue.eps} &\includegraphics[width=9cm]{f05b.hue.eps} } \end{array}$ \[fig.slab02b.3800.dens\] $ \begin{array}{cc} \ifthenelse{\isundefined{\rainbow}}{ \includegraphics[width=9cm]{f06a.eps} &\includegraphics[width=9cm]{f06bv.eps} \\ \includegraphics[width=9cm]{f06c.eps} &\includegraphics[width=9cm]{f06dv.eps} }{ \includegraphics[width=9cm]{f06a.hue.eps} &\includegraphics[width=9cm]{f06bv.hue.eps} \\ \includegraphics[width=9cm]{f06c.hue.eps} &\includegraphics[width=9cm]{f06dv.hue.eps} } \end{array} $ \[fig.slab02b.1000.cool.cut\] \[fig.slab02b.2000.dens.cut\] \[fig.slab02b.2000.cool.cut\] Low density, overpressured jets ------------------------------- ### Jet stability properties, model [B0]{}: {#results.eruptyy} Our main series of simulations depicts a more powerful jet ($F_E=10^{46}~\mathrm{erg}~\mathrm{s}^{-1}$) with a much lower density contrast ($\eta=1.16\times10^{-3}$) and a large overpressure ($\xi=100$). The initial internal Mach number of the jet is $M=12.72$ at the nucleus. The jet radius is $r_\mathrm{j}=10\ {{\mathrm{pc}}}$, the square grid is $4\ {{\mathrm{kpc}}}$ wide and the resolution is $2000\times2000$ cells. The background gas again has a temperature $T_\mathrm{x}=10^7~\mathrm{K}$ but the density is raised to $n_\mathrm{x}=1\ \mathrm{cm}^{-3}$. These parameters are suggested by the observations of GPS sources [e.g. in the summary by @conway02a]. One of the aims of this work is to distinguish the effects of an inhomogeneous medium from the phenomena that are intrinsic to the jet itself. Therefore we have performed a control simulation containing only the jet and an initially uniform intercloud medium. The main large-scale features at the end of this simulation (see Figure \[fig.eruptyy.0320.dens\]) are: an outer shell of highly overpressured gas processed through the bow shock, enclosing a turbulent cocoon where jet plasma mixes with external gas. The length of the jet is small compared to the diameter of the bubble surrounding it. The jet is highly unstable: it stays recognisably straight throughout its first $0.1\ {{\mathrm{kpc}}}$ but at greater distances from the origin it flaps transversely with increasing amplitude, losing collimation at positions beyond $\sim0.5\ {{\mathrm{kpc}}}$ from the origin. This highlights a feature of two–dimensional simulations that may not be present in three dimensions. The instability in this jet is driven by 2D turbulence in the cocoon that results from the mixing of gas at the edge, although the instability is not as rapid as in the ensuing cases with cloudy media. The turbulent driving is probably stronger here because of the persistence of vortices associated with the well–known inverse cascade in two dimensions. The instability evident here is seeded by a very small amount of numerical noise in the initial data. @hardee1988 performed a linear stability analysis of the instabilities of two-dimensional slab jets. Two qualitative kinds of instabilities occur. An unstable pinching mode is responsible for diamond shocks at quasi-periodic locations along the length of the jet. Sinusoidal instabilities are responsible for transverse oscillations, which disrupt and ultimately decollimate the jet. According to the analytical and numerical results of @hardee1988, the maximally growing fundamental mode has wavelength $$\lambda^*\approx {{0.70}\over{\alpha_\mathrm{sj}}} {{2\pi M r_\mathrm{j}}\over{0.66+\eta^{1/2}}} \, , \label{hardee.wave}$$ an $e$-folding length scale of $$l^*_e \approx {{1.65}\over{\alpha_\mathrm{sj}}} M r_\mathrm{j} \ , \label{hardee.growth}$$ and angular frequency $$\omega^*\approx 1.5\ \alpha_\mathrm{sj} a_\mathrm{e} / r_\mathrm{j} \ , \label{hardee.omega}$$ where $a_\mathrm{e}$ is the adiabatic sound speed in the cocoon immediately exterior to the jet surface, the parameter $\alpha_\mathrm{sj}\rightarrow 1$ for $M\gg1$, and $\eta$, in this case, is the density contrast between the jet and its immediate surroundings within the cocoon. In simulation [B0]{}, the internal Mach number within the jet is reduced to typically $M\approx6$, downstream of the first diamond shock (see Figure \[fig.eruptyy.0320.mach\]). The cocoon gas immediately surrounding the jet has a density typically $\sim4\rho_\mathrm{j}$ and adiabatic sound speed $a_\mathrm{e}\sim800v_0$, where $v_0\approx365\ \mathrm{km} \> \mathrm{s}^{-1}$ is the isothermal sound speed of the undisturbed medium outside the bow shock. Upon substitution into equations (\[hardee.wave\]) and (\[hardee.growth\]), these values imply exponential growth of sinusoidal instabilities on scales $l\sim100\ \mathrm{pc}$ and with wavelengths $\lambda\sim200\ \mathrm{pc}$. This is consistent with the growing oscillations that are apparent in individual frames of our simulations, e.g. in Figure \[fig.eruptyy.0320.mach\]. From equation (\[hardee.omega\]), the implied period of the instability is $T^*<0.052\ t_0$. The simulated jet thrashes back and forth transversely within this typical time. The sinusoidal transverse instability revealed in this homogeneous ISM simulation are similar in wavelength and growth time to instabilities manifest in the inhomogeneous simulations discussed below. However, the appearance of this instability in this “clean” simulation shows that the instability can be properly regarded as a Kelvin-Helmholtz instability of the jet itself — albeit, in the inhomogeneous cases, stimulated by the turbulence in the cloudy medium. Note that growth and wavelength of the transverse KH instability at first sight seems to be suggestive of a transonic jet. However, the comparison with the [@hardee1988] expressions for maximally growing wavelength and growth time shows that our simultions are consistent with the expected behavior of *supersonic* jets with the quoted Mach numbers. $\begin{array}{cc} \ifthenelse{\isundefined{\rainbow}}{ \includegraphics[width=6cm,trim=1.5cm 0.5cm 1.5cm 0.0cm]{f07b.eps} \\\includegraphics[width=6cm, trim=1.5cm 0.5cm 1.5cm 0.0cm]{f07c.eps} \\\includegraphics[width=6cm, trim=1.5cm 0.5cm 1.5cm 0.0cm]{f07av.eps} }{ \includegraphics[width=6cm,trim=1.5cm 0.5cm 1.5cm 0.0cm]{f07b.hue.eps} \\\includegraphics[width=6cm, trim=1.5cm 0.5cm 1.5cm 0.0cm]{f07c.hue.eps} \\\includegraphics[width=6cm, trim=1.5cm 0.5cm 1.5cm 0.0cm]{f07av.hue.eps} } \end{array}$ \[fig.eruptyy.0320.mach\] ### Small cloud filling factor: model [B1]{} This is our simulation with the fewest clouds: their filling factor is $f_\mathrm{c}=4.1\times10^{-3}$ ($\sigma_\mathrm{min}=2.6$) within a region spanning radially between $0.1{{\mathrm{kpc}}}$ and $0.5{{\mathrm{kpc}}}$ from the nucleus. All of the clouds are quickly overtaken by the expanding bow shock. As in model [A1]{}, each cloud is shocked by the overpressure of the cocoon (up to $5\times10^4$ times greater the external pressure), and contracts as it cools radiatively. Some clouds are accelerated away from their initial positions by the thrust of the jet. For some other clouds, radiative cooling eventually proves inadequate; the tendency to contract is overcome, and the cloud’s fate is expansion and ablation. Halfway through this simulation at time $t=24t_0$, (top panel, Figure \[fig.erupt03.0160.cut.dens\]), shows the destruction of one such cloud in the path of the jet, situated at $(0.6,0.0)\,{{\mathrm{kpc}}}$ but ablating away across about $0.2\ {{\mathrm{kpc}}}$ to the right. Meanwhile, a few extremely dense and compact clouds persist in apparent safety off to the sides of the jet, within a few hundred ${{\mathrm{pc}}}$ of the nucleus. They appear to survive indefinitely and appear scarcely changed by the time the simulation ends, as seen in the middle panel of Figure \[fig.erupt03.0320.dens\]. In comparison with Figure \[fig.eruptyy.0320.dens\] we see much more dense gas (derived from the ablated clouds) intermixed with jet plasma in the cocoon. The last panel in Figure \[fig.erupt03.0320.cut.glow\] represents the appearance of the jet in synchrotron emission at late stages of the simulation. Here we have used the approximation [e.g. @saxton2001; @saxton2002a] that the emissivity $j_\nu\propto \varphi p^{(3+\alpha)/2}$, where $p$ is the pressure, $\varphi$ is the relative concentration of mass originating from the jet (a scalar tracer that is passively advected in our code) and $\alpha$ is the spectral index that we take to be $0.6$. Several diamond shocks are visible in the well collimated section of the jet closest to the nucleus. The brighter, barred “fishbone” shaped pattern seen at distances $x>0.4\ {{\mathrm{kpc}}}$ is a transient and typical manifestation of the transverse instability of the jet (as in § \[results.eruptyy\]). $\begin{array}{cc} \ifthenelse{\isundefined{\rainbow}}{ \includegraphics[width=7.3cm]{f08av.eps} \\\includegraphics[width=7.3cm]{f08b.eps} \\\includegraphics[width=7.3cm]{f08c.eps} }{ \includegraphics[width=7.3cm]{f08av.hue.eps} \\\includegraphics[width=7.3cm]{f08b.hue.eps} \\\includegraphics[width=7.3cm]{f08c.hue.eps} } \end{array}$ \[fig.erupt03.0320.dens\] \[fig.erupt03.0320.cut.glow\] ### Intermediate cloud filling factor: model [B2]{} In this case the filling factor of clouds is higher with $f_\mathrm{c}=3.0 \times 10^{-2}$, corresponding to $\sigma_{\mathrm{min}}=1.9$. In the first few $t_0$ of simulation time the jet is completely disrupted within $< 0.1\ {{\mathrm{kpc}}}$ of the nucleus as a result of the KH instability induced by waves reflected off the innermost clouds. By $t\sim12t_0$ (see top panel, Figure \[fig.erupt06.0080.dens\]), the jet has ablated the nearest clouds and cleared an unobstructed region out to $\sim0.1\ {{\mathrm{kpc}}}$ on either side. Within this region the jet remains well collimated during the rest of the simulation. The clouds that define the sides of this sleeve are shocked but they survive until the last frame (middle panel, Figure \[fig.erupt06.0320.dens\]). The flow of plasma from the head of the jet splits into a fork on either side of a cloud group that sits nearly on the jet axis at about $x\approx0.5\ {{\mathrm{kpc}}}$. Throughout the simulation these clouds retain dense, efficiently cooling cores but they continually lose dense gas entraining into the stream of jet plasma passing to either side. Near the end of the simulation ($t=48t_0$, Figure \[fig.erupt06.0320.dens\]) these obstructive clouds are trailed by dense ablated gas extending throughout a triangular area up to $0.5\ {{\mathrm{kpc}}}$ further out in the positive $x$ direction. The radio emissivity (last panel, Figure \[fig.erupt06.0320.cut.glow\]) has some similarities to the appearance of model [B1]{}, with a bright jet remaining well collimated out to around $0.5$ kpc. However the turbulent plumes of jet-derived plasma beyond the jet’s head are more disrupted by the dense obstructions of the cloud field, Radio bright plasma divides in several directions, as Figure \[fig.erupt06.0320.cut.glow\] shows. The brightest areas of the jet and post-jet flows are typically concentrated closer to the nucleus than in the less cloudy model [B1]{}. $\begin{array}{cc} \ifthenelse{\isundefined{\rainbow}}{ \includegraphics[width=7.3cm]{f09av.eps} \\\includegraphics[width=7.3cm]{f09b.eps} \\\includegraphics[width=7.3cm]{f09c.eps} }{ \includegraphics[width=7.3cm]{f09av.hue.eps} \\\includegraphics[width=7.3cm]{f09b.hue.eps} \\\includegraphics[width=7.3cm]{f09c.hue.eps} } \end{array}$ \[fig.erupt06.0320.dens\] \[fig.erupt06.0320.cut.glow\] ### Large filling factor: model [B3]{} This is our simulation with largest filling factor of clouds, $f_\mathrm{c}=0.12$ and $\sigma_{\mathrm{min}}=1.2$. As a result of the high coverage of clouds, the jet is disrupted when it is less than $0.1\ {{\mathrm{kpc}}}$ from its origin. Jet plasma streams through the available gaps between the persistent clouds (see the density distribution in right panel of Figure \[fig.erupt05.0320.dens\]). These low-density, radio-bright channels evolve as the cloud distribution changes. A channel may close when nearby clouds expand ablatively or are blown into positions where they create a new obstruction. Turbulent eddies of jet plasma accumulate in pockets within the cloud field, most conspicuously in places where a channel has closed. The density image, Figure \[fig.erupt05.channels\], illustrates the initial development of intercloud channels. The left panel of Figure \[fig.erupt05.0320.cut.glow\] shows simulated radio emissivity in the inner regions of the galaxy at the end of the simulation. The radio-bright material is concentrated near the nucleus within an irregular, branched region that is not directly related to the alignment of the initial jet. This particular configuration of channels is ephemeral, occurring after the opening and closure of previous channels as the cloudfield evolves. Eventually, one or more channels pushes into the open region outside the cloudy zone. A spurt of turbulent jet plasma accumulates at that point. In later stages of our simulations, these eruptions eventually become large compared to the cloudy zone, and merge with adjacent eruptions to form a diffuse, turbulent, approximately isotropic outer cocoon composed of a mixture of jet plasma, ablated cloud gas and intercloud gas processed through the bow shock. $\begin{array}{cc} \ifthenelse{\isundefined{\rainbow}}{ \includegraphics[width=9cm]{f10a.eps} & \includegraphics[width=9cm]{f10b.eps} \\ \includegraphics[width=9cm]{f10c.eps} & \includegraphics[width=9cm]{f10d.eps} }{ \includegraphics[width=9cm]{f10a.hue.eps} & \includegraphics[width=9cm]{f10b.hue.eps} \\ \includegraphics[width=9cm]{f10c.hue.eps} & \includegraphics[width=9cm]{f10d.hue.eps} } \end{array}$ $\begin{array}{cc} \ifthenelse{\isundefined{\rainbow}}{ \includegraphics[width=9cm]{f11a.eps} }{ \includegraphics[width=9cm]{f11a.hue.eps} } & \ifthenelse{\isundefined{\rainbow}}{ \includegraphics[width=9cm]{f11b.eps} }{ \includegraphics[width=9cm]{f11b.hue.eps} } \end{array}$ \[fig.erupt05.0320.dens\] ### Effect of cooling: model [B3a]{} We ran a further simulation with the same cloud field as in [B3]{} but with radiative cooling switched off (see Figure \[fig.erupt00\]). In the absence of radiative cooling, the compression of clouds is insignificant. The outflow of jet plasma rapidly turns the clouds into ablating, flocculent banana shapes. Large clouds and small clouds ablate efficiently. The survival time of individual clouds is shorter than in the simulation including the effects of radiative cooling. Dense material is more readily ablated from the clouds, and this may block channels of jet plasma more readily or at an earlier time, but otherwise the radio morphology is qualitatively similar to that of [B3]{}. $\begin{array}{cc} \ifthenelse{\isundefined{\rainbow}}{ \includegraphics[width=9cm]{f12a.eps} &\includegraphics[width=9cm]{f12b.eps} }{ \includegraphics[width=9cm]{f12a.hue.eps} &\includegraphics[width=9cm]{f12b.hue.eps} } \end{array}$ Evolution of the outer part of the radio source ----------------------------------------------- In each case, as a consequence of the disruption of the jet, its momentum is distributed throughout the cocoon in an isotropic way. Hence, the long term evolution is similar to that of an energy-driven bubble. Figure \[f.2d\_bubble\] presents plots of the average radius of the furthermost extent of the radio plasma versus time, for represenative simulations. The plots also show the corresponding analytic solution for a two dimensional bubble with the same parameters of energy flux per unit length and density (grey line, see Appendix \[app.bubble\] for a derivation). It is evident that the analytic slope of the radius/time curves is replicated very well by the simulations. The offset may be attributed to the time taken for the isotropic phase to be established. This may be related to the finite minimum width of the jet at its nozzle, the size of the innermost nuclear region that is initially cloudless, and the intrinsic length scale of the jet’s instability (§ \[results.eruptyy\]). We expect that a similar power-law expansion will occur in three dimensions, although this of course is yet to be confirmed. However, given that jet momentum and power can be redistributed isotropically in three dimensions, it is reasonable to expect that the long term evolution of the outer bubble will be described by a three–dimensional energy–driven bubble. $\begin{array}{cc} \includegraphics[width=8cm]{f13a.eps} &\includegraphics[width=8cm]{f13b.eps} \end{array}$ Comparison of model [B1]{} with 3C48 {#s:3C48} ==================================== Whilst these simulations have been two-dimensional, some of their features should persist in three–dimensional simulations with similar parameters. In particular the trends with filling factor will probably persist, to some extent, so that comparisons with existing observationsla are of interest, even at this stage. We therefore make a tentative comparison of one of these simulations (model [B1]{}) with the CSS quasar 3C48. We can also draw some general conclusions from these simulations, concerning the likely overall morphology of young radio sources. Simulation [B1]{} has the lowest filling factor of dense clouds and impedes the jet the least. The jet is still disrupted but the flow associated with it continues with reduced collimation and with its forward momentum spread over a much larger region. In the disrupted section, transverse oscillations are produced, with shocks created at points where the jet bends significantly, leading to local enhancements in emissivity. A large scale bubble is produced outside the cloudy region but the jet also retains some coherency in the intermediate region. This particular simulation bears a strong qualitative resemblance to 3C48. In Figure \[f.3C48.1\] a snapshot of the simulated radio emissivity at a particular time is compared with the 1.6 GHz VLBI image of 3C48 [@wilkinson91a]. In two dimensions, the alternating direction of the jet results from the transverse instability that we have already discussed (§\[results.eruptyy\]). In three dimensions, the corresponding instability would be a helical $m=1$ mode. In both cases the alternating direction of the supersonic flow produces shocks. We see evidence for these, in both the snapshot of the emissivity from the 2D simulation and the image of the jet. This particular simulation predicts that there will be a diffuse halo of emission external to the jet region of the source. This was, in fact, detected in larger scale images of 3C 48. One such image presented in @wilkinson91a, shows diffuse emission on a 1 arcsecond scale in 3C48 extending well beyond the compact jet structure evident in the VLBI image. More recent higher quality images have been obtained by @ojha04a and one of their contour maps is reproduced in Figure \[f.3C48.2\]. These images clearly reveal the extended structure of 3C48. @wilkinson91a had already concluded that the morphology of 3C48 is likely the result of jet–cloud interactions and cited evidence for asymmetric emission line gas northwest of the core as well as the presence of dust revealed by the high IRAS flux. They interpreted the radio knot close to the core as the result of a jet–cloud interaction. Whilst this is possible, our simulation shows that the sort of jet disruption observed in 3C48 can occur without direct jet–cloud encounters. The general turbulence in the jet cocoon can excite jet instabilities, producing shocks in the supersonic flow either as a result of the jet being bent through an angle greater than the Mach angle, or though pinching–mode instabilities. (Both of these occur in the simulations.) An explanation for the first bright knot involving cocoon turbulence may therefore be more favorable since direct jet–cloud encounters are usually highly disruptive and [@wilkinson91a] appealed to a glancing deflection in order to explain the radio structure. Given that the sort of jet–ISM interaction revealed in our simulations, is occurring, we expect that the emission line gas would show evidence for shock excitation as a result of radiative shocks induced by the overpressured radio cocoon. [@chatzichristou01a] has suggested that the emission line spectrum of 3C48 may show evidence for a mixture of AGN and shock excitation. @wilkinson91a also noted out that 3C48, classified as a CSS quasar, is much less extended than other similar power radio–loud quasars. They speculated that the confinement results from the emission line gas and dust inferred to be present in the ISM. Qualitatively, we agree with this, except to point out that the dispersion of the momentum flux (by a small number of clouds) prevents a radio source from expanding rapidly. As we have shown, the source becomes an expanding bubble with a much slower rate of expansion than a jet with the same kinetic power. Hence the gas and dust is not required to physically confine the radio plasma. The slower expansion can be effected by a small filling factor of dense clouds that redistribute the jet momentum isotropically. $\begin{array}{ccc} \ifthenelse{\isundefined{\rainbow}}{ \includegraphics[height=8cm]{f14a.eps} &\includegraphics[height=8cm]{f14b.eps} &\includegraphics[height=8cm]{f14cc.eps} }{ \includegraphics[height=8cm]{f14a.eps} &\includegraphics[height=8cm]{f14b.hue.eps} &\includegraphics[height=8cm]{f14cc.hue.eps} } \end{array}$ \[f.3C48.2\] We also note a qualitative resemblance between the simulation [B3]{} and M87. The channeling of the flow in [B3]{} produces a hierarchy of structures resembling the hierarchy of radio structures in M87 apparent in the low frequency radio image [@owen00a]. However, we have not emphasized this with a detailed comparison since the scale of the source and the simulation are quite different. Nevertheless, the production of the structure observed in simulation [B3]{} depends mainly on the filling factor of the clouds and this mechanism should also work on larger scales. We propose to confirm this in future with a fully relativistic 3D simulation. Other sources with similar morphology to 3C48 and which may be being disrupted in a similar way include the TeV $\gamma$–ray blazars, MKN 501 and MKN 421. The most recent VLBI images of these sources [@giovannini00a; @piner99a] show similar structures in the form of disrupted jets and large scale plumes suggesting that these sources are disrupted on the sub–parsec scale by a cloudy medium. Again the scale of the simulation [B1]{} is such that it does not [*strictly*]{} apply. However, the physical principle of these jets being disrupted by a small filling factor of dense clouds is probably still valid with interesting repercussions for the production of the $\gamma$–ray emission [@bicknell04a]. DISCUSSION ========== We have presented a series of two–dimensional simulations of slab jets propagating through inhomogeneous media. Despite the limitations of the restrictions to two dimensions, and to non–relativistic flow, these simulations provide useful insight into the characteristics of jets in radio galaxies which have an inhomogeneous interstellar medium. These include GPS and CSS radio sources, high redshift radio galaxies and galaxies at the center of cooling flows. In all of these galaxies there is ample evidence for dense gas clouds that may interact with the radio source producing at least some of the features evident in our simulations. In the case of GPS and CSS sources the inhomogeneous medium may be the result of a fairly recent merger; in the case of high redshift radio galaxies an inhomogeneous medium is likely to be the result of the galaxy formation process at those epochs, as large galaxies are assembled from sub-galaxian fragments. As well as the limitations to two dimensions and non–relativistic flow, there is another feature of our simulations that requires some comment and that is the ablation of cloud material that is evident in most cases. There is likely to be a large contribution of numerical mass diffusion due to the very high density contrasts between jet and cloud gas. Although this has some effect on the evolution, the dominant processes in the case of a low filling factor are waves emanating from the clouds impacted by the high pressure cocoon and, in the high filling factor simulations, direct obstruction of the flow plus wave effects. Despite the above–mentioned caveats, we have been able to deduce these general features from our simulations: 1. [**Jet instability.**]{} When propagating through the type of inhomogeneous interstellar medium that we have imposed, the jets become unstable. However, this instability does not always result from direct jet–cloud collisions but occurs indirectly from the waves generated by the impact of the high–pressure jet cocoon on the dense clouds. This leads to a general level of turbulence in the medium that induces unstable Kelvin–Helmholtz oscillations in the jet. An unstable jet was also produced in the homogeneous simulation ([B0]{}) and this was related to turbulence caused by the entrainment of ISM into the cocoon as well as the higher level of large scale turbulence in two dimensions. However, the simulations with an inhomogeneous medium became unstable in a shorter distance and we attribute this to the effect of the clouds. 2. [**Production of halo/bubble.**]{} The instability of the jet is associated with the spreading of the jet’s momentum and a turbulent diffusion of the jet material into a halo or bubble. These are characteristics that we expect to be present in three dimensions. The radius-time curve for the outer part of the halo is well–described by the analytic expression for a two–dimensional bubble. In three dimensions a energy driven bubble model should also be appropriate. This is useful since it gives us a way of estimating the energy flux fed into a large scale radio source from the dimensions of the outer halo and the density of the surrounding interstellar medium. The production of a bubble means that the overall increase of size of the radio source with time is diminished with respect to a source that propagates unimpeded. However, this restriction on the size of the source is not due to direct physical confinement of the radio plasma by denser gas. Rather, it is the result of the isotropic spread of the jet momentum by the process of jet instability described above. 3. [**The effect of filling factor.**]{} The main parameter that we have varied in these simulations is the filling factor of the dense clouds and we can see that this makes a considerable difference to the morphology of the radio source. With only a small filling factor of clouds, the jet is readily disrupted, but retains some directed momentum and produces the 3C48–type morphology described in § \[s:3C48\]. As the filling factor increases, the jet propagates via a number of different channels which open and close leading to an hierarchical structure of radio plasma that we noted is similar to the large–scale structure of M87. 4. [**The effect of jet density ratio.**]{} We also varied the jet density ratio parameter, $\eta$, but only in one simulation. The simulation [A1]{} had $\eta = 0.38$ whereas in the other simulations $\eta \sim 10^{-3}$. The main noticeable effect is that the heavier jet is affected less by instabilities and direct jet–cloud collisions than the lighter jets. This points to an important difference between models with $\eta {\raisebox{-0.5ex}{$\,\stackrel{>}{\scriptstyle \sim}\,$}}0.01$ and models with $\eta {\raisebox{-0.5ex}{$\,\stackrel{<}{\scriptstyle \sim}\,$}}0.001$. Of course, such low density ratios are important to provide the relevant power for radio galaxy jets. However, reassuringly for physics but difficult for modelers, such low values of $\eta$ push models into the relativistic regime, as we have discussed in § \[s:jet\_pars\]. There are two important points to be made here. The first is that many jet simulations usually involve a modest value of $\eta \ga 0.01$ and these may be misleading. Second, the most realistic simulations will be those that directly include relativistic physics and which also incorporate the mixing of relativistic and thermal plasma. This paper complements other work on the physics of direct jet–cloud collisions (e.g. @fragile04a, @wiita02a) in which the focus has been on the effect of a jet on a single cloud. Direct jet–cloud collisions are clearly important in the context of sources such as Minkowksi’s object [@vanbreugel85a]. In this paper we have clearly been more concerned with the overall radio source morphology, the evolution of an ensemble of clouds, the effects of indirect jet–cloud collisions (i.e. the effect of the high pressured cocoon on dense clouds) and trends with cloud filling factor. However, we note an interesting connection between @wiita02a and this work: We have both noted that less jet disruption occurs in the case of slow dense jets. We also note that the morphology of radiative cloud shocks deduced by @fragile04a is also reproduced in some of the individual clouds in our simulations. Our simulations have clearly all been two–dimensional. What are the expected differences between two–dimensional and three–dimensional work? There are at least two (1) In two dimensions, vortices survive for a lot longer as a result of the inverse cascade of turbulent energy. These vortices play a significant role in disrupting the jet and clouds. We expect much less disruption from vortical motions in three dimensions. (2) The number of independent directions that the jet can take in three dimensions is greater so that we expect that the clouds will impede the jet less severely and that escape from the cloudy region surrounding the nucleus will be easier. These issues as well as the inclusion of special relativistic effects will be addressed in future papers. ACKNOWLEDGMENTS {#acknowledgments .unnumbered} =============== Our research has been supported by ARC Large Grant A699050341 and allocations of time by the ANU Supercomputer Facility. Professor Michael Dopita provided financial support from his ARC Federation Fellowship. We are very grateful to P.N. Wilkinson and R. Ojha for their permission to use their respective images of 3C48. We thank the anonymous referee for the several suggestions improving the format of the paper. GRID SUBDIVISION FOR MULTIPROCESSOR COMPUTATION {#app.mpi} =============================================== We confront two practical obstacles to the calculation of high resolution hydrodynamics simulations in higher dimensions: the duration of the run in real-time (which inhibits the exploration of a wide parameter space), and the finite computer memory (which directly limits the number of grid cells that can be computed). We circumvent these limitations by subdividing the spatial grid into $N = 2^n$ subgrids (with $n$ an integer) and assigning each to a program running in parallel on a separate processor. The processors communicate between themselves using functions of the [MPI]{} (Message Passing Interface) library[^2]. If the global grid has dimensions of $I\times J$ cells in the $x$ and $y$ directions, then we define a “long subgrid” as an area with $I \times (J/N)$ cells (see the left panel of Figure \[fig.mpi.1\]), and a “wide subgrid” as an area of $(I/N) \times J$ cells (see the right panel of Figure \[fig.mpi.1\]). Within the memory at each processor we maintain representations of one long subgrid or one short subgrid: see for example the shaded long and wide subgrids held by processor 3 in Figure \[fig.mpi.1\]. The total number of cells in a long subgrid is the same as that of a wide subgrid. In fact for each hydrodynamic variable on each processor, we identify the long and wide subgrid arrays with the same section of memory, which is feasible and reliable since the program uses only one of the subgrid representations at any time (as we shall detail below). We then conceptually subdivide each long or wide subgrid into $N$ blocks separated by transverse cuts (e.g. the dotted line divisions in processor 3’s subgrids depicted in Figure \[fig.mpi.1\]). For each processor $p$ the block numbered $p$ is the only area of overlap between the long and wide subgrids that it is assigned. At the start of the simulation each processor is assigned long and wide subgrids according to its identity number. These regions are filled according to the initial conditions which are defined in terms of global grid coordinates. The first hydrodynamic sweep [@colella1984] is in the $x$ direction, and is performed by each processor within its long subgrid arrays (see left panel of Figure \[fig.mpi.2\]). Thus the $x$–sweeps do not encounter any of the subgrid interfaces, obviating the need for any communication of flux information between processors at this stage. The one-dimensional array vectors associated with each sweeep are the maximum possible length, which makes optimal use of processor cache. After the $x$ sweep is complete, the evolved density, velocity, pressure and other zone data are communicated between the processors. By use of the [MPI\_ALLTOALL]{} function, the data of block $i$ of the long subgrid of processor $j$ is copied to block $j$ of the wide subgrid of processor $i$, for all $i$ and $j$. These are corresponding areas of the global computational grid. As the communication sequence proceeds, each processor receives data blocks and assembles them into complete, ordered arrays describing the wide subgrid that it is conceptually assigned. The wide subgrid arrays overwrite the memory areas that was previously used for long subgrid data. However the sequence of the [MPI\_ALLTOALL]{} communication provides for a complete redistribution of data across the set of processors without any collective loss of information. When the communication is complete, each processor performs a hydrodynamic sweep in the $y$ direction within its respective wide subgrid arrays. (See right panel of Figure \[fig.mpi.2\].) Again, as in the $x$ sweep, no processor boundaries are encountered during the computation, and the one-dimensional sweep arrays are long. At the end of the $y$ sweep, the processors communicate blocks of their evolved wide grid arrays in another [MPI\_ALLTOALL]{} operation, ensuring that the long subgrid arrays are up to date. At this point program control operations are performed, e.g. generating output files recording the hydrodynamic variable arrays of each processor. Care is needed to ensure that the processors evolve their subgrids synchronously in simulation time. Within each subgrid $g$ the maximum allowable timestep of integration for each $x$ or $y$ sweep is determined by a Courant condition: $\delta t_g$ must be less than the characteristic timescales of fluid motion or sound waves crossing a cell, e.g. $\delta t_g < 0.6 \min( \delta x / v_{x,i}, \delta x / c_{s,i})$, for all cells $i$ with side length $\delta x$. In order to synchronise the simulation across all the subgrids, the processors are made to inter-communicate their locally calculated values of $\delta t_g$, and then adopt the global minimum allowable timestep, $\delta t = \min(\delta t_0, \delta t_1, ... \delta t_{N-1})$. This straightforward scheme for multiprocessor [PPM]{} hydrodynamics simulations is readily generalised to a three-dimensional grid. The 3D analogues of long and wide subgrids are “flat slabs” (full size in the $x$ and $y$ directions, but subdivided into $N$ segments in the $z$ direction) and “tall slabs” (full size in the $x$ and $z$ directions, but subdivided into $N$ parts in the $y$ direction). Using 32, 64, 128 or more of the processors of the Australian National University’s Compaq Alphaserver SC, we have been able to perform 3D simulations with cubic dimensions of 256, 512 or more cells per side (work in preparation). SELF-SIMILAR SOLUTION FOR INFLATION OF A TWO-DIMENSIONAL BUBBLE {#app.bubble} =============================================================== Dimensional analysis {#app.dimensional} -------------------- This appendix provides an analytical solution that can be used for the comparison of slab-jet simulations with expected asymptotic behaviour. The two parameters that determine the evolution of the bubble are $A$, the energy flux per unit transverse length, and $\rho_0$ the density of the background medium. In terms of jet parameters, $$A = \left( \frac{1}{2} \rho v^2 + \frac{\gamma}{\gamma-1} p \right) \, v D$$ where $D$ is the width of the jet. Consider the parameter $A/\rho_0$. The dimensions of this parameter are given by: $$\left[ \frac{A}{\rho_0} \right] = L^4 T^{-3} \Rightarrow \left[ \frac{A}{\rho_0} \right]^{1/4} = L T^{-3/4}$$ Let $R(t)$ be the bubble radius at time $t$. The dimensions of $R t^{-3/4}$ are also $L T^{-3/4}$. Hence the self-similar evolution of the bubble will be given by: $$\frac{R}{t^{3/4}} = C \left[ \frac{A}{\rho_0} \right]^{1/4}$$ where $C$ is a numerical constant to be determined. The radius of the bubble is therefore given as a function of time by: $$R = C \, \left[ \frac{A}{\rho_0} \right]^{1/4} \, t^{3/4}$$ This derivation is justified and elaborated in the next section, where the constant $C$ is derived. Temporal evolution of bubble ---------------------------- ### Formulation of equations Consider a cylindrical bubble that is being fed by a jet with an energy density per unit length (transverse to the jet) equal to $A$. We consider the jet energy to be effectively thermalised so that the pressure derived from the shocked jet plasma drives cylindrical expansion. Let the energy density of the bubble be $\epsilon$ and the pressure of the bubble be $p$. We represent the bubble in two dimensions in the $x-y$ plane and take a length, $L$ of bubble in the $z-$direction. The total energy, $E= \epsilon \pi R^2 L$ of this length of bubble with volume $\pi R^2 L$ is governed by the equation: $$\frac{d}{dt} \left( \epsilon \pi R^2 L \right) + p \frac{d}{dt} \left( \pi R^2 L \right) = AL$$ The second term represents the work done by the pressure during the expansion; the term on the right represents the rate of energy fed into the bubble. Note that we are assumig that the dominant form of energy in this bubble is internal and that kinetic energy does not contribute significantly. Taking out the common factor of $L$, the evolution of the bubble is governed by the equation $$\frac{d}{dt} \left( \epsilon \pi R^2 \right) + p \frac{d}{dt} \left( \pi R^2 \right) = A \label{e:bubble}$$ We solve this equation by introducing the equation of state \[$\epsilon = p/(\gamma-1)$\] and by relating the pressure in the bubble to its rate of expansion. Assuming that the bubble is expanding supersonically with respect to the background medium, then its structure consists of a strong shock outside the bubble and a contact discontinuity separating the bubble material from the shocked ISM. In this region, the pressure of the shocked ISM and the bubble are equal. Suppose that the strong shock is moving with velocity $v_{\rm sh}$. Then the gas behind the shock is moving with speed $3v_{\rm sh}/4$ so that $$\dot R = \frac{3}{4} \, v_{\rm sh} \label{e:rdot}$$ Let the density of the background medium be $\rho_0$. Then the pressure behind the shock is given by $$p \approx \frac{3}{4} \rho_0 v_{\rm sh}^2 \approx \frac{4}{3} \rho_0 {\dot R}^2 \label{e:pressure}$$ Note that we have not specified the $\gamma$ for the bubble but we have assumed that $\gamma=5/3$ for the background ISM. Substituting the equation of state into (\[e:bubble\]), and expanding the first term, the bubble equation becomes: $$\frac{\gamma}{\gamma-1} \pi p \frac{dR^2}{dt} + \frac{1}{\gamma-1} \pi R^2 \frac{dp}{dt} = A$$ Then, using equation (\[e:pressure\]) for the pressure, we obtain, after some simplification: $$R {\dot R}^3 + \frac{1}{\gamma} R^2 \dot R \stackrel{..}{R} = \frac {\gamma-1}{\gamma} \frac{3A}{8 \pi \rho_0} \label{e:final_equation}$$ ### Solution It would be reasonably straightforward to develop a general solution of equation (\[e:final\_equation\]). However, if we settle for a self-similar solution, for which: $$R = a t^\alpha$$ where $a$ and $\alpha$ are constants. Subsitution into equation (\[e:final\_equation\]) gives: $$a^4 \alpha^2 \, \left[ \alpha + \frac{\alpha-1}{\gamma} \right]\, t^{4 \alpha -3} = \frac{\gamma-1}{\gamma} \, \frac{3 A}{8 \pi \rho_0} \ .$$ Equating the powers of $t$ on both sides: $$\alpha = \frac{3}{4}$$ as expected. Equating the constants: $$a = \left[ \frac{3}{8} \frac{\gamma-1}{\alpha^2 \left[ (\gamma+1) \alpha -1 \right]} \frac{A}{\pi \rho_0} \right]^{1/4} \ .$$ The form of the dependence on the parameter $A/\rho_0$ was anticipated in §\[app.dimensional\]. For $\gamma = 5/3$, as used in our simulations: $$a = \left[ \frac{4 A}{9 \pi \rho_0} \right]^{1/4}$$ and the solution for $R$ is: $$R = \left[ \frac{4 A}{9 \pi \rho_0} \right]^{1/4} \, t^{3/4} \ .$$ Radius of shock wave -------------------- In comparing the analytic solution with simulations it is easiest to locate the position of the discontunity in the pressure. This is the location of the outer shock wave, whose velocity is derived from equation (\[e:rdot\]), i.e. $$v_{\rm sh} = \frac{4}{3} \, \dot R$$ Hence the radius of the pressure discontinuity is given by $$R_{\rm p} = \frac{4}{3} a t^\alpha = \frac{4}{3} \left[ \frac{4 A}{9 \pi \rho_0} \right]^{1/4} \, t^{3/4}$$ for $\gamma = 5/3$. \[lastpage\] [^1]: See [http://wonka.physics.ncsu.edu/pub/VH-1/]{}. See also @blondin93a for the extension of the PPM method to curvilinear coordinates. [^2]: http://www-unix.mcs.anl.gov/mpi/

--- abstract: | The dynamics of power distribution between longitudinal modes of a multimode semiconductor laser subjected to external optical feedback is experimentally analyzed in the low-frequency fluctuation regime. Power dropouts in the total light intensity are invariably accompanied by sudden activations of several longitudinal modes. These activations are seen not to be simultaneous to the dropouts, but to occur after them. The phenomenon is statistically analysed in a systematic way, and the corresponding delay is estimated. [*OCIS codes:*]{} 140.1540, 140.5960 address: - 'Departament de Física i Enginyeria Nuclear, Universitat Politècnica de Catalunya, Colom 11, E-08222 Terrassa, Spain ' - ' Departament de Física, Universitat de les Illes Balears, E-07071 Palma de Mallorca, Spain' author: - '**J.M. Buldú, J. Trull, M.C. Torrent, J. García-Ojalvo**' - '**Claudio R. Mirasso**' title: Dynamics of modal power distribution in a multimode semiconductor laser with optical feedback --- [2]{} Semiconductor lasers are devices very susceptible to exhibiting unstable dynamical behavior. When subjected to reflections of its own emitted radiation, in particular, they easily enter complex dynamical regimes, exhibiting for instance low-frequency fluctuations (LFF) in the form of intensity dropouts [@risch], or fully-developed chaotic fluctuations leading to coherence collapse [@lenstra]. Most of the related theoretical and experimental studies undertaken so far have dealt with the dynamics of the total emitted intensity [@lff; @claudio]. However, the low-cost semiconductor lasers employed in technological applications operate usually in several longitudinal modes. Therefore, analysing the mode dynamics would be necessary, a need which has been recognized only recently [@ryan]. In particular, recent experiments have indeed shown the importance of multimode operation in the LFF regime [@huyet; @vaschenko]. Different dynamical [@mandel; @rogister] and statistical [@sukow] characteristics of this regime have been described in terms of a multimode extension of the well-known Lang-Kobayashi model [@lang]. In the course of the above-mentioned investigations, it was observed that when the feedback was frequency-selective (such as that provided by a diffraction grating), the intensity dropouts were accompanied by a sudden activation of other longitudinal modes of the laser [@huyet; @giudici]. These modes, located at the sides of the main mode in the gain curve, will be called longitudinal side modes, or simply side modes, in the rest of this Letter. The activation of these modes was heuristically interpreted as the mechanism producing the intensity dropouts [@giudici], and was numerically reproduced again by a multimode LK model [@mandel2]. In this Letter, we show experimentally that the side-mode activation also appears in the presence of non-frequency-selective feedback, and that it occurs neither simultaneously nor previously to the intensity dropout, but [*after*]{} it. Therefore, in this case this activation can not be the cause, but rather the effect, of the dropout. Our experimental setup is shown schematically in Fig. \[fig:setup\]. We use an index-guided single-transverse-mode AlGaInP semiconductor laser (Roithner RLT6505G), emitting at a nominal wavelength of 650 nm with a threshold current of 20.1 mA. Its temperature is set to $24.00\pm0.01\;^{\rm o}$C. The laser output is collimated by an antireflection-coated laser-diode objective. An external mirror is placed 60 cm away from the front facet of the solitary laser, which corresponds to a feedback time of 4 ns. The threshold reduction due to the feedback is 9.4%. Throughout the paper, the injection current is set to 1.09 times the solitary laser threshold. Part of the total output intensity is detected by a fast photodiode and sent to a 500 MHz-bandwidth HP 54720D digital oscilloscope. The rest passes through a $1/8m$ CVI monochromator with a resolution better than 0.2 nm, used to select the laser modes, and whose output is sent to a Hamamatsu PS325 photomultiplier. The photomultiplier signal is also recorded by the oscilloscope. The optical spectrum of our solitary laser shows at least ten active longitudinal modes, with its maximum located at $\sim$658.4 nm and a FWHM of $\sim$0.9 nm. When the feedback is turned on, the spectrum broadens (up to a FWHM of $\sim$1.3 nm), and its maximum becomes shifted $\sim$0.5 nm towards higher wavelengths [@claudio; @javier]. For the feedback parameters chosen, mentioned above, the laser emits in the low-frequency fluctuation regime. In this regime, we have compared the dynamical behavior of the total emitted intensity of the laser with that of the main mode of the laser with feedback (MMF), and with a longitudinal side mode corresponding to the original main mode of the solitary laser (MMS). The typical behavior is displayed in Fig. \[fig:delay\], which compares the total intensity evolution with either that of the MMF (traces a-b) or the MMS (traces c-d). It can be seen that a power dropout is associated to an abrupt decay of the former and a sudden activation of the latter. Note that the recovery of the MMF is much slower than that of the total intensity, a fact that has been already reported in the literature [@huyet; @wallace]. The activation is seen not to be symmetric, i.e. it does not occur in the other side of the spectrum. We have found similar behavior in other semiconductor lasers of similar quality, including nearly-single-mode lasers. We note that, even though the pairs of measurements (a,b) and (c,d) in Fig. \[fig:delay\] were acquired simultaneously, the time traces exhibit a systematic delay of $\sim$20 ns between the total-intensity dropouts and the corresponding modal powers (see vertical dashed lines in the figure). This delay is spurious, due to the electronic response time of the photomultiplier used in the mode-selecting path of the experimental setup (Fig. \[fig:setup\]), which is substantially larger than that of the photodiode used to measure the total intensity. However, as we will show in what follows, a closer inspection of these results reveals that this spurious delay is slightly [*larger*]{} for the MMS activation than for the MMF dropout. Since both of these signals are measured with the same detector, this observation leads to the conclusion that the side-mode activation does not occur simultaneously to (nor before) the dropout, but [*after*]{} it. In order to estimate the delay between each dropout and its MMS activation we proceed as follows. First, several (typically 40) time-trace pairs containing a single total-intensity dropout and its simultaneously measured modal event (either MMF dropout or MMS activation) are averaged using a predefined event (a given decay of the total intensity in our case) as a trigger. In this way, we average out fluctuations before the dropout event and during the subsequent build-up, and refer all the time traces to a common time origin (given by the predefined event mentioned above). The result of this procedure is shown in Fig. \[fig:average\](a). One can already see in this figure, which shows several averaged sets for each one of the three quantities measured (total intensity, MMF intensity, and MMS intensity), that the MMS activation occurs somewhat later ($\sim$1 ns) than the MMF dropout (see vertical dashed lines in the figure). In order to identify such a delay more clearly, we compare in Fig. \[fig:average\](b) the MMS signal to the inverted MMF one. The delay becomes now evident. Note also that the escape trajectories of the two modes (from the lasing state in the MMF case, and from the off state in the MMS case) are basically parallel, which indicates that the instability mechanisms are the same, and hence a direct comparison between them can be made. We estimate the delay between the dropout and the side-mode activation as the distance between the two corresponding parallel escaping trajectories, which can be clearly identified in Fig. \[fig:average\](b) as two distinct sets of straigth lines with the same positive slope. We perform a piecewise local linear fit of each one of the averaged MMF and MMS time series, and identify the time instants at which the slope takes its maximum value. Figure \[fig:histogram\] represents the distribution of these times, for both the MMS activation and the MMF dropout, computed from an statistics of 3000 dropout events. The distribution functions of these two quantities are clearly separated, with a time difference between their two mean values of 1.5$\pm$1.1 ns. In conclusion, we have experimentally shown that low-frequency fluctuations in a multimode semiconductor laser with global (i.e. non-frequency selective) optical feedback are associated to sudden activations of a longitudinal side mode corresponding to the main mode of the solitary laser. These activations are seen to occur after the dropouts of the main mode of the laser with feedback, and hence after the total intensity dropouts of the system. Therefore, in this case the side-mode activation cannot account for the destabilization giving rise to the low-frequency fluctuations. On the contrary, one can conjecture that the activations are a natural consequence of the loss of power in the main modes of the laser with feedback. Work directed to the theoretical modeling of these phenomena is in progress [@javier]. We thank J. Martorell and R. Vilaseca for lending us part of the experimental setup. We acknowledge financial support from Ministerio de Ciencia y Tecnologia (Spain), under projects PB98-0935 and BFM2000-0264, from the EU IST network OCCULT, and from the Generalitat de Catalunya, under project 1999SGR-00147. [99]{} Ch. Risch and C. Voumard, J. Appl. Phys. [**48**]{}, 2083 (1977). D. Lenstra, B. Verbeek, and A.J. de Boef, IEEE J. Quantum Electron. [**21**]{}, 764 (1985). J. Mork, J. Mark, B. Tromborg, Phys. Rev. Lett. [**65**]{}, 1999 (1990); T. Sano, Phys. Rev. A [**50**]{}, 2719 (1994); A. Hohl, H.J.C. van der Linden, and R. Roy, Opt. Lett. [**20**]{}, 2396 (1995); I. Fischer, G.H.M. van Tartwijk, A.M. Levine, W. Elsäßer, E. Göbel, and D. 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--- abstract: 'Supermassive black hole (BH) binaries would comprise the strongest sources of gravitational waves (GW) once they reach $\ll1$pc separations, for both pulsar timing arrays (PTAs) and space based (SB) detectors. While BH binaries coalescences constitute a natural outcome of the cosmological standard model and galaxy mergers, their dynamical evolution is still poorly understood and therefore their abundances at different stages. We use a dynamical model for the decay of BH binaries coupled with a cosmological simulation and semi-empirical approaches to the occupation of haloes by galaxies and BHs, in order to follow the evolution of the properties distribution of galaxies hosting BH binaries candidates to decay due to GWs emission. Our models allow us to relax simplifying hypothesis about the binaries occupation in galaxies and their mass, as well as redshift evolution. Following previously proposed electromagnetic (EM) signatures of binaries in the subparsec regime, that include spectral features and variability, we model possible distributions of such signatures and also set upper limits to their lifespan. We found a bimodal distribution of hosts properties, corresponding to BH binaries suitable to be detected by PTA and the ones detectable only from space missions, as eLISA. Although it has been discussed that the peak of eLISA sources may happen at high z, we show that there must be a population of such sources in the nearby Universe that might show detectable EM signatures, representing an important laboratory for multimessenger astrophysics. We found a weak dependence of galaxy host properties on the binaries occupation, that can be traced back to the BH origin. The combination of the host correlations reported here with the expected EM signal, may be helpful to verify the presence of nearby GW candidates, and to distinguish them from ’regular’ intrinsic AGN variability.' author: - | Eva Martínez Palafox$^{1}$[^1] Octavio Valenzuela$^{1}$, Pedro Colín$^{2}$, Stefan Gottlöber$^3$\ $^{1}$ Instituto de Astronomía, Universidad Nacional Autónoma de México, A.P. 70-264 Ciudad Universitaria, D.F., México\ $^2$ Centro de Radioastronomía y Astrofísica, Universidad Nacional Autónoma de México, A.P. 72-3 (Xangari), Morelia,\ Michoacán 58089, México.\ $^{3}$ Leibniz Institute for Astrophysics, An der Sternwarte 16, D-14482 Potsdam, Germany.\ date: 'Received ; in original form ' title: 'Supermassive Black Hole Binaries: Environment and Galaxy Host Properties of PTA and eLISA sources.' --- \[firstpage\] black hole physics $-$ binaries: general $-$ galaxies: nuclei Introduction {#Intro} ============ The current understanding of the cosmic large scale structure origin is included in the $\Lambda$CDM scenario (Planck Collaboration 2013). According to this cosmological model the self-gravitating systems at galactic scales have been formed following a hierarchical process of mass accretion and mergers, and galaxies grew up by the collapse of baryons into the bottom of dark matter haloes potential wells (White & Rees 1978; Rees & Ostriker 1977; Silk & Mamon 2012 and references therein). Therefore it is possible to find a relationship between dark matter haloes mass accretion/merging histories and the interactions between galaxies. The consistency between the halo mass function and the galaxy luminosity function predicts a relationship between the halo mass and the observed galaxy stellar mass, and it is at the core of the halo occupation distribution (HOD) techniques, widely used to test the consistency between galaxies population and $\Lambda$CDM cosmology. On the other hand, stellar kinematic studies of nearby galaxies (Kormendy & Richstone 1995; Richstone et al. 1998; Ferrarese & Ford 2005) suggest that a supermassive black hole reside at their centres. Currently, our Milky Way (MW) is the most striking example (Ghez et al. 2008; Gillessen et al. 2009). There is also a variety of investigations confirming that the mass of the BH is highly correlated to the mass and velocity dispersion of the host galaxy bulge (Magorrian et al. 1998; Ferrarese & Merritt 2000; Gebhardt et al. 2000; Marconi & Hunt 2003; Häring & Rix 2004; Graham et al. 2011; McConnell & Ma 2013). Additionally, still controversial correlations between total luminosity or mass and BH mass (Ferrarese 2002; Baldana 2009; Läsker 2013) suggest that BH and their host galaxies evolve in a symbiotic way (Cattaneo 2009; Kormendy & Ho 2013). A natural consequence is that from galaxy mergers, a large number of BH binaries must form along the cosmic history (Begelman, Blandford & Rees 1980; Volonteri, Haardt & Madau 2003). However, few spatially resolved subkpc binaries are known so far (Rodriguez et al. 2006; Fabbiano et al. 2011; Woo et al. 2014), and the few observations of sub parsec candidates reported (Tsalmanza 2011; Eracleous 2012; Liu 2014) could also be explained by a number of scenarios other than the presence of a binary (Dotti 2012). The search for and characterization of the corresponding binary BH population are of great importance to understand both galaxy formation and gravitational wave (GW) physics. The dynamical evolution of the binary system within the merged galaxy, and the interaction with the host both dynamically and possibly via feedback during baryonic accretion onto one or both BHs, encode crucial information about the assembly of the bulge and the central BH. If the binary eventually coalesces, it will produce a gravitational siren that could be detected with future low frequency GW experiments (e.g., Colpi & Dotti 2011). Detecting BH binaries is then crucial in understanding the coevolution of galaxies and BHs, and for testing fundamental physics. The BH binaries may produce a wide range of EM signals in their path to coalesce: Emission associated to a circumbinary disk (CBD) (Milosavljević 2005; Tanaka & Menou 2010; Tanaka 2012; Sesana 2012; McKernan 2013; Roedig 2014), variability due to the orbital motion (Gower 1982; Sillanpaa 1988; Kaastra & Roos 1992; Britzen 2010; Kudryavtseva 2011; Godfrey 2012), multiple accreting BHs spatially resolved (Komossa 2003; Rodriguez 2006; Fabbiano 2011; Koss 2011; Paggi 2013), enhanced tidal disruption events (Ivanov 2005; Chen 2009; Wegg & Nate Bode 2011) and BHs wandering as a result of three-body interactions (Governato 1994; Iwasawa 2006) or post-merger recoiling BHs (Komossa & Merritt 20008; Merritt 2009; O’Leary & Loeb 2009). The frequency window of GW from BH binaries corresponds to the low frequencies in the range ($10^{-2} - 10^{-9}$)Hz covered by two projects: eLISA and PTAs. The *Evolved Laser Interferometer Space Antenna[^2]* (eLISA) will be the first dedicated space based gravitational wave detector. It will measure gravitational waves at low frequencies, from about $0.1$ mHz to $1$ Hz (Amaro-Seoane 2013). The potential *eLISA* sources include BH close binaries in the process of coalescence, with total mass in the range $M_{\rm BH}=M_{\bullet,1}+M_{\bullet,2} \sim 10^4 - 10^7 \text{M}_{\odot}$ (where $M_{\bullet,1}$ and $M_{\bullet,2}$ are the primary and secondary BH masses respectively). The technique to detect GW of ultra-low frequencies ($\sim10^{-9}$ to $10^{-8}$ Hz) is called Pulsar Timing Arrays (PTAs), consisting of sets of millisecond pulsar whose joint pulse arrival time produce correlations that have encoded information of gravitational waves passing the Earth and/or the pulsar (Foster & Backer, 1990). PTAs will detect GW from the coalescence of BH binaries with masses $M_{\rm BH} \gtrsim10^8 M_\odot$ and mass ratio $10^{-3} \lesssim q = M_{\bullet,2}/M_{\bullet,1}\lesssim 1$ (Sesana 2012). Before we reach the direct GW detection era, the possibility of detecting EM counterparts revealing binary coalescence events unambiguously has valuable potential, either constraining the dynamics of the coalescence process (e.g., Sesana 2007; Trias & Sintes 2008; Amaro-Seoane et al. 2012), or the origin and evolution of BHs (e.g., Volonteri 2010, Kulkarni & Loeb 2012), and the environments where these events take place. Even though the richness offered by the EM signatures produced in binaries the aim is to reach the multimessenger astrophysics era of trans-spectral studies (GW+EM) to narrow the degeneracies of a possible GW detection (e.g., Bloom 2009). Recent studies have aimed to narrow this multimessenger search, Sesana (2012) used a semi-analytical galaxy formation model and a massive BH halo population algorithm in order to predict the frequency of different EM signals related with PTA sources. Later Rosado & Sesana (2013) study the environment of BH binaries suitable to be detected by PTAs, finding a bias to high density environments. Lastly Simon (2014) assuming that each galaxy has a binary BH of equal mass, evaluated probable locations of PTA sources, suggesting nearby clusters. In this work we intend to relax some of the assumptions adopted in previous studies, we build a more realistic binary population and we do not assume that all binaries in the Local Universe are in the same coalescence stage, however we do not try to model the detectability. By focusing the modelling of binaries since the last major merger (LMM) of their host galaxies and following their dynamical evolution, we expect to understand better those objects that are more likely to be reliably considered galaxy candidates to host binaries. In order to do so, we populate under different strategies the dark matter haloes in a cosmological simulation during the last major merger event and track down the binary evolution, including the interaction with stars and a gaseous circumbinary disk. We use halo occupation distribution (HOD) techniques and semiempirical halo/galaxies scaling relationships (Firmani & Ávila-Reese 2010) in order to assign a BH and galaxy properties to each halo, satisfying scale relationships such as luminosity function, in order to asses the binary BH galaxy host different properties with more accuracy. The assignment was performed at different epochs in order to track down binary galaxy host properties at different redshifts. We estimate the lifespan and variability of electromagnetic signals. This paper is organized as follows. In Section 2 we describe the parameters and analysis adopted for our cosmological simulation. In section 3 we describe the prescriptions implemented in order to populate dark matter haloes with stellar disks and BHs. In section 4 we describe the semi-analytical model implemented to follow the dynamical evolution of BH pairs from cosmological scales down to subparsec scales. In section 5 we discuss the properties found for galaxies hosting binaries with separations $~10^{-3}$pc, candidates to decay due to GW emission. In section 6 we classify binaries in the previous section into PTA and SB sources, then we present the bimodality found in host galaxy properties, as total mass, stellar mass, and environment. In section 7 we present the assumptions made in order to assign a possible EM signature, then we discuss the abundance and characteristics of the EM features that might be found in the binary populations resulting from our models. Finally, we summarize the main conclusions of this paper in Section 8. Cosmological simulation. ======================== {width="\linewidth"} In this work we use a cosmological N-body dark matter simulation which is part of the Constrained Local UniversE Simulations (CLUES) project,[^3] whose aim is to perform cosmological simulations that reproduce the main features that characterize the local large scale structure in the Universe (Gottlöeber 2010). Figure \[fig:mapatot\] shows the distribution in the CLUES simulation of haloes more massive than $M_h \geq 5\times 10^{10} \text{M}_{\odot} h^{-1}$, shown as blue dots. It also shows the projected positions of the halo identified with the Virgo cluster and the halo dubbed as *MW candidate* as black circle and black square, respectively. The Hoffman & Ribak (1991) algorithm was used to generate the initial conditions as constrained realizations of Gaussian random fields. Constrained simulations represent a powerful tool to understand the history of the best known part of the Universe. By using them it is possible to construct a constrained sample of possible merging galaxies corresponding to any epoch of the local Universe. Also, constrained simulations allow us to explore the assembly history of MW and M31-like galaxies (Forero-Romero 2011), considered as candidates for such systems due not only to theirs masses (as usually assumed in other dark matter simulations) but also because their environments and MAHs have been shaped along the cosmic history by the large scale structure in which the Local Universe actually dwells. Advantages we will use to explore the origin and evolution of the BHs in galaxies in the Local Group in an upcoming paper. We summarize the simulation main properties here for clarity. The simulation has $1024^3$ particles with mass $m_p= 1.63\times10^{7}h^{-1} \text{M}_{\odot}$ in a box with side length of $L_{box} = 64 h^{-1}$Mpc. It was run using the PMTree-SPH MPI code `Gadget`2 (Springel 2005). The simulation has standard $\Lambda$CDM initial conditions, assuming a WMAP3 cosmology (Spergel et. al. 2007), i.e. $\Omega_m=0.24$, $\Omega_{b}=0.042$, $\Omega_{\Lambda}=0.76$, a normalization of $\sigma_8=0.75$ and a spectral index of primordial density perturbations $n=0.95$. As a frame, our model uses mass aggregation histories (MAHs) from the simulation to identify the last major merger redshift (zLMM) and to estimate the *cosmic mass accretion rate* which corresponds to the mass accreted going to the very centre of the galaxy. The impact of cosmic variance on the CLUES simulation’s halo population has been studied before (Forero-Romero et. al. 2011). They conclude that constrained simulation MAHs are essentially unbiased when compared to unconstrained simulations with higher resolution within larger volumes. Halo identification and mass aggregation histories construction. ---------------------------------------------------------------- To identify haloes we use a friends-of-friends (FOF) algorithm with linking length of $b=0.17$ times the mean inter-particle separation. We have identified the haloes for 133 snapshots more or less equally spaced over the 13.4 Gyrs between redshifts $0 < z \lesssim 13.9$. All objects with 20 or more particles are kept in the halo catalogue and considered in the merger tree construction. This corresponds to a minimum halo mass of $M_{min}= 3.26\times 10^{8} h^{-1}\mathrm{M}_{\odot}$. Within this simulation a Milky Way dark matter halo of mass $10^{12} h^{-1}\mathrm{M}_{\odot}$ is resolved with $\sim6\times10^4$ particles. The MAHs do not include any information of the substructure in each halo. MAHs are constructed by comparing the particles in FOF groups in two consecutive snapshots. Starting at $z=0$ for every halo $\rm H_{f}$ in the catalogue, we find all the FOF groups in the previous snapshots that share at least 13 particles with the halo $\rm H_{f}$ at $z=0$, and label them as potential progenitors. Then, for each tentative progenitor, we find all the descendants sharing at least 13 particles. Since the smallest group has 20 particles, at least $2/3$ of the particles must be identified in provisional progenitors or descendants. Only the tentative progenitors whose main descendant is $\rm H_{f}$, are labelled as confirmed progenitors at that snapshot. We iterate this procedure for each confirmed progenitor, until reaching the last available snapshot at high redshift. By construction, each halo in the MAH can have only one descendant, but many progenitors. The time at which haloes mergers occur corresponds to the moment when their radii overlap for the first time. Major mergers (MM) are defined in terms of an increase of at least $20\%$ in the total mass of a descendant halo. The LMM time is identified in the MAHs for each halo with mass $M_{\rm H} \geq 1.64 \times 10^{10} M_\odot h^{-1}$ (at $z=0$). We verify the reliability of the merger event inspecting the neighbour snapshots. The zLMM provides a starting point for our model, and we consider it the first stage in the coalescence process. Populating haloes with galaxies and black holes =============================================== Our N-body simulation includes only dark matter, if we intend to constraint the properties of binary BH host galaxies we require a strategy to populate dark matter halos with galaxies, below we describe our galaxies/black holes assignment scheme. Seeding Galaxies ---------------- The galaxy/dark halo connection has received an important deal of attention in recent years. Such connection have been commonly studied using *ab initio* strategies: N-body simulations coupled with semi-analytic galaxy formation models or self consistent cosmological hydrodynamic plus N-body simulations. An alternative strategy with an empirical spirit is to assume a direct match between dark halos and observed galaxy properties, in particular stellar mass. Although this approach does not determine the physical processes behind the correspondence, it can be extended even to trace the galaxies halo/stellar mass evolution assuring a consistency with a set of observations. We use the latter option, a semi-empirical approach to define a galaxy population consistent with current observations, including: luminosity function and downsizing (Firmani & Ávila-Reese 2010). The implementation assumes that we can transform the dark halo MAHs into average stellar mass growth histories, $M_s(M_{\rm H}(z),z)$, following the semi-empirical relation given by Firmani & Ávila-Reese (2010), $$\begin{split} \mathrm{ log}(M_{\rm H}(M_s)) &= \mathrm{ log}(M_1) + \beta \mathrm{ log} \left( \frac{M_s}{M_{s,0}} \right) \\ & + \frac{\left( \frac{Ms}{M_{s,0}} \right)^{\delta} }{1+\left( \frac{M_{s,0}}{M_s} \right)^{\gamma} } - \frac{1}{2}, \label{vladi} \end{split}$$ where the dependence on $z$ is introduce in the parameters of equation \[vladi\] as $$\begin{array}{l c l} \mathrm{ log}(M_1(a)) &=& M_{1,0} + M_{1,a}(a-1), \\ \mathrm{ log}(M_{s,0}(a)) &=& M_{s,0,0} + M_{s,0,a}(a-1) + \chi(z), \\ \label{paravladi} \delta(a) &=& \delta_0 + \delta_a (a-1), \\ \chi(z) &=& -0.181z(1-0.378z(1-0.085z)), \\ \end{array}$$ whose values were chosen according to table \[vladt\]. $M_{\rm H}$ is the total mass of the host galaxy and $a=1/(1+z)$ is the scale factor. Parameter value --------------- -- ---------- $ M_{s,0,0}$ $10.70 $ $ M_{s,0,a}$ $-0.80 $ $ M_{1,0}$ $ 12.35$ $ M_{1,a}$ $ -0.80$ $ \beta$ $ 0.44 $ $ \delta_0$ $ 0.48 $ $ \delta_{a}$ $-0.15 $ $ \gamma$ $ 1.56 $ : Best parameters for the $M_{\rm H}(M_s, 0<z<4)$, total mass-stellar mass relation (Firmani & Ávila-Reese 2010).[]{data-label="vladt"} Now, provided the total mass of the galaxy, we have to associate a mass density distribution. This is a complicated and unfinished task in galaxy formation theory. Providing a prescription to choose between spiral and elliptical morphology and their structural parameters is an active field in semi-analytical models and hydrodynamic galaxy formation experiments. As a working simplification hypothesis we associated to each remnant halo a scale radius for late type galaxies as given by Shen (2003) $$R(kpc) = 0.1 \left(\frac{M_{\rm H}}{M_{\odot}}\right) ^{0.14} \left(1+\frac{M_{\rm H}}{3.98\times10^{10}M_{\odot}}\right)^{0.25}.$$ This assumption allowed us to model galaxies’ potentials for each remnant halo using a Miyamoto-Nagai disk (1975) and a NFW halo (1996) in order to investigate the binary BH decay due to angular momentum exchange with stars at kpc scales. The assumption is not accurate, however as we can see in figure 2, last major merger redshift for our binaries is clustered around 1, the typical gas richness at that epoch favours disk reformation in the remnant galaxy (Robertson et al. 2006). Assuming a disk/spheroid component is more realistic, however assigning relative parameters is highly uncertain from theoretical grounds (e.g. Ávila-Reese et al. 2014). The presence of a bulge might shorten the time scale determined by the angular momentum exchange with stars and some binaries stalled at this stage might reach the phase of decay due to GW emission. In this paper our main focus are the galaxy hosts properties of BH binaries at the GW regime and not the exact abundance, therefore we consider that the effect is not critical for our results. Seeding black holes ------------------- We used two different approaches in order to provide a BH mass appropriate to the characteristics that each progenitor had before the merger, and to investigate the BH-halo occupation effects in the final properties of the binaries populations. First, we used the empirical relation between black hole mass and the total gravitational mass of the host galaxy suggested by Ferrarese (2002), $$M_{\bullet} = 0.67 \left( \frac{M_{\rm H,i}}{10^{12} M_{\odot}} \right)^{1.82},$$ \[Fbhmass\] where $M_{\bullet}$ is the mass of the BH, in $10^8 M_{\odot}$ units and $M_{\rm H,i}$ the total mass of the progenitor galaxy at hand. Models with a BH seeded following the Ferrarese (2002) relationship will be called F. Second, an approach based on HOD calibrated by hydrodynamic cosmological simulations that follows BH growth (Di Matteo 2008). Despite the uncertainties in AGN physics and accretion mechanisms, these simulations reproduce both the observed $M_{\rm BH}-\sigma$ relation and total BH mass density $\rho_{\rm BH}$ (Di Matteo 2008), as well as the quasar luminosity function (Degraf 2010). In our model, the BH population seeded in dark matter haloes following this approach inherits such properties. The conditional BH mass function (CMF) gives the mean number of BHs per logarithmic BH mass bin found in haloes of mass $M_{\rm H}$ and is fitted by a Gaussian distribution (Degraf 2011), in $log_{10}(M_{\bullet})$. $$\frac{dN_{\rm BH}}{d logM_{\bullet}} = \frac{1}{\sqrt{2\pi \sigma^2}} e^{-\frac{(log(M_{\bullet})-\mu)^2}{2\sigma ^2}}$$ where the parameters $\sigma(M_{\rm H})$ and $\mu(M_{\rm H})$ corresponds to their best fit (Degraf 2011). Models with a BH seeded following the CMF in Degraf (2011) will be called O hereafter. Figure \[fig:IC\] shows the primary and secondary BH mass, primary (or main progenitor) halo mass, BH mass ratio, last major merger redshift and progenitors mass ratio initial distributions used in the 10 parameter sets explored (see table \[parametros\]). Blue dotted lines correspond to the $M_{\rm H}-M_{\rm BH}$ empirical BH seeding prescription F and consists of 24815 galaxy pairs, black solid lines correspond to the BH seeding prescription based on occupation models O and consists of 2680 galaxy pairs. The difference in the number of pairs is due to the fact that the O seeding prescription is defined only for haloes with masses between $10^{11}\: -\: 10^{14}\:M_{\odot}$ as can be seen in the third top panel. Also, given the restriction imposed to the halo progenitor’s mass ratio of $q_{p} \geq 0.3$, the F seeding prescription constrains the minimum possible binary BH mass ratio $q$ to be $0.11$, as seen in figure \[fig:IC\] first bottom panel, whilst O prescription allows all values of $q$ between $0$ and $1$ given its broader scatter. It is important to mention that not all populated halos will reach the decay regime dominated by GW emission. Given the differences between seeding approaches, they might be considered as the consequence of different BH formation scenarios. Then, by using two approaches we are able to investigate their effect in the distribution of galaxy hosts properties at the end of the binary evolution. If there is a significant effect, it may set an avenue to constrain BH formation scenarios that depends on the binary population. {width="\linewidth"} Dynamical Decay Model Description ================================= In our dynamical coalescence model, supermassive black hole binaries are considered to evolve via four stages, identified by the separation between BHs, from large to small separations, and the main interactions with the surrounding material. The four stages considered are (1) the last major merger epoch obtained from a constrained cosmological simulation, as mentioned in section 2; (2) time scale for decay of merging galaxies’ baryonic components within a remnant halo, (3) dynamical friction of the secondary BH $M_{\bullet,2}$ to reach the centre of the newly formed galaxy, and (4) binary-gaseous disk interaction until the GW driven decay regime is reached. Here we introduce the dominant dynamical mechanisms that we have considered drive the binaries evolution. Merger time scale for galaxies within the remnant halo ------------------------------------------------------ The halo merger time, obtained from dark matter only simulations of large cosmological boxes, is frequently underestimated. Typically, it is defined when the satellite halo has crossed the host’s virial radius. This definition only includes the beginning of the merger and it does not take into account the coalescence time needed to have a new remnant halo. Additionally, because of the large simulated volume, the mass and spatial resolution are not always enough to fully capture all the density modes related to dynamical friction. These inaccuracies have stimulated the inclusion of some corrections, for instance: those based on the analytical description of dynamical friction given by Chandrasekhar or modified versions due to finite halo size and density gradients, or angular momentum (see Zavala 2012 and references therein). Given this uncertainties, we have chosen all haloes with known progenitors at their LMM and kept only those with progenitors’ mass ratios between $0.3 \leq q_{p} =M_{\rm H,2}/M_{\rm H,1} \leq 1$. Since Boylan-Kolchin, Ma & Quataert (2008) in their N-body experiments showed that, for such mass ratios, the coalescence time scale for an extended satellite to sink from the host halo virial radius down to the centre, is similar to the host halo dynamical time scale. This time scale is much longer for more dissimilar mass ratios. Thus, the time scale for the second stage in our model for the coalescence process is given by the dynamical time scale at the virial radius of the main progenitor, $$\tau_{dyn} \equiv \frac{r_{vir}}{V_c(r_{vir})} = \left( \frac{r_{vir}^3}{GM_{\rm H,1}} \right)^{1/2},$$ where $r_{vir}$ is the virial radius, $V_c(r_{vir})$ is the circular velocity at $r_{vir}$, and $M_{\rm H,1}$ is the halo mass before the merger. Binary interaction with stars ----------------------------- In the third stage, the black holes sink into the potential well depth given by the remnant galaxy due to dynamical friction with stars and dark matter. At this point we will start studying the dynamical evolution of the BHs from the host and the satellite haloes. The deceleration experimented by the black hole embedded in stars and dark matter can be provided by the Chandrasekhar dynamical friction formula (Chandrasekhar 1943; Binney & Tremaine 1987), assuming that at the sub-kpc spatial scale, density gradients are not large. We have considered the secondary BH moving in the remnant disk plane and decaying slowly enough to be considered at any given moment on a circular orbit. The change in the orbit of the BH is, $$\frac{dr}{dt}= - \frac{rF_{df}}{M_{\bullet,2} \left[ V_c + r\frac{dV_c}{dr} \right] }.$$ where $F_{df}$ is the force experimented by the BH due to dynamical friction and $V_c$ is the circular velocity at radius $r$. Such a formula gives us a lower limit to the BH’s decay time scale due to its interaction with dark matter and stars, which corresponds to the third stage in the coalescence process. It is worth noting that most BH pairs ($\gtrsim 90\%$) for all the model parameters explored, do not reach the next stage where the interaction with a gaseous disk occurs, mostly because the time scale needed for the stellar component to bring together the BHs to a few tens of parsecs is larger than the time available since the LMM. Also, the binary-stellar medium interaction is not a finished topic, since there have been some recent results claiming that triaxiality or axisymmetry in stellar distributions is enough to overcome the final parsec problem in gas-free galaxies (Khan & Holley-Bockelmann 2013). Whilst other studies claim that, given the binaries hardening rate dependence on the simulation’s number of particles, there is not enough evidence to reach any conclusions yet (Vasiliev, Antonini & Merritt 2014). A number of recent studies have shown that the presence of a gaseous circumbinary disk is highly efficient to drain angular momentum out from a binary BH (e.g. Dotti et al. 2012, Hayasaki 2009). Given that it is still unknown whether the interaction with stars might be able to overcome the final parsec problem, and because galaxy mergers funnel copious amounts of gas towards the remnant galaxy centre, we include as a fourth and last stage in our models the interaction within a gaseous disk as the mechanism responsible for the binaries’ angular momentum loss. The interaction within gaseous environment have the added value of promising EM signatures as we will discuss in section 7. Evolution within a circumbinary disk. ------------------------------------- In the fourth and last stage of our model, the binary exchanges angular momentum with a gaseous circumbinary disk at a rate that can be approximated by the angular momentum exchange at resonances (Goldreich & Tremaine, 1979; Hayasaki 2009). The mass transfer from the CBD adds momentum to the binary and might form accretion disks around each BH. We will follow the analytical model calibrated using SPH simulations suggested by Hayasaki (2009). It is important to acknowledge that vigorous research work on the angular momentum exchange between the CBD and the binary BH is currently being undertaken by several groups, and details such as, gas equation of state, CBD fragmentation, and star formation are still uncertain (Roskar et al 2014, Colpi 2014). It is also worth noting that our model is flexible enough to extend its implementation to include the results of such studies when the discussion is settled. For now the efficiency of the angular momentum exchange will be locked in phenomenological parameters in our approach. The CBD initial mass is given by the CBD to total BHs mass ratio, we explore three values $M_{\rm CBD} / M_{\rm BH}=0.1$, $0.01$ and $0.001$. The $M_{\rm CBD}$ grows along the binary’s evolution at a rate given by the parameter $f_{mah}= 0.01$ and $0.001$ (see table \[parametros\]) that constrains the cosmic mass transfer. Given that typical time scales at this stage are much shorter than the time between the snapshots of the simulation, we interpolate the MAHs so the new time steps are at least half the characteristic time scale for the evolution of the binary given by $t_c \simeq 8\times10^{4} ( M_{\rm BH}^{1/2}/M_{\odot}) [yr]$ (Hayasaki 2009, eq. 26). The evolution of the semi-major axis can be calculated as (Hayasaki 2009) $$\frac{a}{a_0} = \left( \frac{\dot{M}_{c}}{\dot{M}_{crit}} \right)^{-2} \left[ 1- \left( 1- \frac{\dot{M}_{c}}{\dot{M}_{crit}}\right) exp\left( \frac{t}{t_c} \frac{\dot{M}_{c}}{\dot{M}_{crit}}\right) \right]^2.$$ Here $a_0$ is the initial semi major axis and $\dot{M}_{crit}$ is a critical mass transfer rate defined as: $$\dot{M}_{crit} = \frac{1.39}{f_{am}}\frac{q}{(1+q)^2} \frac{M_{\rm BH}}{t_c}.$$ The semi-major axis $a$ decays with time when $\dot{M}_{crit}$ is larger than the cosmic accretion rate, $\dot{M}_{c}$, while it increases when $\dot{M}_{crit} < \dot{M}_{c}$. The parameter $f_{am}$ determines how much the torque of the mass transfer is converted to the torque of the binary. As long as material keeps reaching the CBD, this will provide a considerably large amount of mass for the BHs to accrete. We will consider accretion triggered only from this stage on, with both BHs accreting mass and following a phenomenological approach where both BH accrete an amount of mass is given by: $$\label{mpunto} \begin{array}{c c c} \varDelta M_{\bullet,1} = \frac{1}{1+q} + (f_{mah} M_{mah}) + (f_{d} M_{\rm CBD}) \\ % M_{\bullet,1}. \\ \\ \varDelta M_{\bullet,2} = \frac{q}{1+q} + (f_{mah} M_{mah}) + (f_{d} M_{\rm CBD}) \\ % M_{\bullet,2}. \\ \end{array}$$ per time step, where $f_{mah}$ is the fraction of the mass $M_{mah}$ gained by the dark halo MAH in a time step, so the product $f_{mah} M_{mah}$ is the amount of mass that reaches the galaxy centre in a time step, which we have called *cosmic accretion rate*; and $f_{d}$ is the fraction of the circumbinary disk mass $M_{\rm CBD}$ accreted by the black holes per time step. It is clear that the accretion model oversimplifies the complexity of AGN activity, however it is beyond the scope of our work to accurately investigate AGN activity. We follow the dynamical evolution of the initial galaxy pairs population tracking their separation, for the two BH seeding prescriptions (O and F) and the 10 parameter sets shown in table \[parametros\], so we have 20 different models. Once the binaries reach separations of $~ 10^{-3}$pc, they are considered binaries likely to decay and coalesce due to the emission of gravitational waves or GW candidates. In our model the destiny of BH binaries is set by a competition between cosmic accretion that replenish gas and angular momentum into the CBD ($f_{mah}$) and the binary ($f_{d}, f_{am}$), versus the angular momentum drain out of the binary by the CBD torques ($M_{\rm CBD} / M_{\rm BH}$). The CBD viscosity may have either tidal or stochastic origin (Fiacconi 2013), ours is an effective model, therefore it does not capture the full detail of the gaseous disk physics. model $\text{M}_{\text{CBD}}/\text{M}_{\text{BH}}$ $f_{am}$ $f_{mah}$ $f_{d}$ ------- ---------------------------------------------- ---------- ----------- --------- 1 0.1 0.01 0.01 0.001 2 0.1 0.01 0.01 0.0001 3 0.1 0.01 0.001 0.001 4 0.1 0.01 0.001 0.0001 5 0.01 0.01 0.01 0.001 6 0.01 0.01 0.01 0.0001 7 0.01 0.01 0.001 0.001 8 0.01 0.01 0.001 0.0001 9 0.1 0.01 0.01 0.1 10 0.001 0.01 0.01 0.001 : Parameters used for both BH seeding prescriptions O and F. $M_{\rm CBD}/M_{\rm BH}$ is the initial circumbinary disk to total BH mass ratio, $f_{am}$ indicates the angular momentum fraction that reaches the BHs, $f_{mah}$ is the MAH mass fraction that goes to the galaxy centre and $f_{d}$ is the fraction of CBD mass that is accreted by the BHs.[]{data-label="parametros"} Binary BH host galaxy properties ================================ Characterizing the types of galaxies likely to host BH binaries can be useful to determine a low frequency GW source location, to search for peculiar EM signatures, and to elucidate whether such a peculiar EM signature is truly due to a coalescing binary as predicted by various scenarios. Furthermore, if we are able to identify the host galaxy, and therefore the GW source position; luminosity distance measurements based on GWs would be greatly improved (Arun 2009). It also reduces the parameter space of search templates, and correspondingly increases the observed signal-to-noise ratio. Identifying and studying plausible binary BHs host galaxies by their properties and EM signatures is possible now and will teach us about mergers rate, galaxy assembly histories, accretion and even might help us narrow uncertainties about the possibility of detecting individual GW sources. Here we present our predictions for the general properties of galaxies hosting BH binaries prone to decay due to the emission of GWs and with separations $<10^{-3}$pc. Redshift -------- ![Binaries redshift distributions for all sets of parameters and both BH seeding prescriptions represented by box and whiskers plots. The box show the interquartile range (IQR) where the central $50\%$ of the data falls. Each whisker or dashed line represents the lower and upper $25\%$. Darker blue and grey horizontal lines within each box denotes the median values and the notch displays the $95\%$ confidence interval around the median. Blue and grey open circles represent the outliers outside 1.5 times the interquartile range. The red crosses represent the mean values. The boxes width is proportional to the square-root of the number of GW candidates. We conclude that the redshift distribution is slightly asymmetric with high significance, and half of the systems distributed below z$\sim$0.21.[]{data-label="fig:boxz"}](boxz2.eps){width="0.9\linewidth"} Regardless of both the model parameters and the BH seeding prescription, the binary BHs redshift distributions, shown in figure \[fig:boxz\], peak around $z \sim 0.21$ with a $95\%$ confidence interval. The distributions reveal an asymmetric shape with $\sim75\%$ of the binaries highly concentrated between $0.1 < $ z $< 1$ and a tail towards low redshift including the remaining $25\%$. We note that $\text{M}_{\text{CBD}}/\text{M}_{\text{BH}}$ is the most important parameter determining the redshift distribution. The distribution peaks at slightly lower redshifts for less massive CBDs (models 5 to 10) due to a less efficient angular momentum loss mechanism. In figure \[fig:redshift\] we show the number of candidates per unit volume likely to reach the GW emission regime at different redshifts, for three different values of $M_{\rm CBD}$. Since most of the galaxies had their last major merger at $z\leq1$ and the large scale characteristics of the binaries are dominated by the cosmology, we see as a consequence that most ($\sim 75\%$) of the candidates to decay due to GW emission occur at $z\lesssim0.5$. Even though, most pairs ($>90\%$) remain at much larger separations, of the order of $\sim\text{kpc}$. ![Number of binaries per unit volume likely to reach the GW emission regime as function of redshift. The line colour represents the BH seeding prescription and each panel corresponds to a parameter set. The errors are Poissonian. Models O1 and F1 with CBDs initial masses $\text{M}_{\text{CBD}} = 0.1 \, \text{M}_{\text{BH}}$, O5 and F5 have $\text{M}_{\text{CBD}} = 0.01 \, \text{M}_{\text{BH}}$, and O10 and F10 with $\text{M}_{\text{CBD}} = 0.001 \, \text{M}_{\text{BH}}$. Regardless of the CBD mass and BH seeding, by observing $10^6$ galaxies one must find some tens of GW candidates and an even larger number of kpc scale BH pairs.[]{data-label="fig:redshift"}](GWzV.eps){width="0.9\linewidth"} Host galaxy halo mass. ---------------------- ![Host galaxy halo mass distributions. Green horizontal region denotes Milky Way halo mass $1.2^{+0.7}_{-0.4} \text{(stat.)} ^{+0.3}_{-0.3} \text{(sys.)} \times 10^{12} M_\odot$ (68% confidence) from Busha (2011). Distributions represented by box and whiskers plots as in figure \[fig:boxz\]. The distributions are asymmetric with more than $50\%$ of the systems with halo masses larger than the MW one, with the halo mass mean value depending mostly on the CBD mass, and the high mass tail depending on the BH seeding scheme.[]{data-label="fig:boxmhall"}](boxmh2.eps){width="0.9\linewidth"} ![Host galaxy halo mass distribution for three models showing the dependence on $\text{M}_{\text{CBD}} / \text{M}_{\text{BH}}$. The errors are Poissonian. It is evident from the figure that $\text{M}_{\text{CBD}}/\text{M}_{\text{BH}}$ is the parameter driving the host galaxy halo mass normalization and shape. However the BH seeding seems to reflect on the high mass tail.[]{data-label="fig:MH"}](GWMhV.eps){width="0.9\linewidth"} Determining the distribution of dark matter halo masses hosting binary BHs in the GW regime is a first step towards constraining its origin and abundance. When comparing the host galaxy halo mass distribution for all models, we see from figure \[fig:boxmhall\] that even though there is no agreement on the mass at which the distributions peak across parameter sets, most BH binaries are systematically found in host galaxies with total mass larger than the Milky Way halo mass. Models with more massive CBD (1-4 in table \[parametros\], with either O and F BH seeding prescription) result in more numerous binary populations, as indicated by the boxes width (figure \[fig:boxmhall\]) and the histogram normalization in figure \[fig:MH\]. It is clear from figures \[fig:boxmhall\] and \[fig:MH\] that systems with less massive initial CBD, $\text{M}_{\text{CBD}}/\text{M}_{\text{BH}} = 0.01$, tend to peak at lower halo mass, $\sim2.7 \times 10^{12} \text{M}_\odot$ for F seeding prescription, while for O seeding prescription the more typical host galaxy halo mass for less massive CBD is $\sim3.5 \times 10^{12} \text{M}_\odot$. Unfortunately, total mass estimation is a challenging quantity to measure with current observations, useful constraints have been set using stacked galaxies plus satellites systems (Yang et al. 2012) or weak lensing (Mandenbaum et al. 2006) but until the next generation of surveys, like LSST, the statistical significance will be limited. We can see, also from figure \[fig:MH\], that $\text{M}_{\text{CBD}}/\text{M}_{\text{BH}}$ is the the driving parameter. If $\text{M}_{\text{CBD}}$ is smaller the number of BH binaries will be smaller, in particular for those hosted in galaxies with higher total mass, given the fact their LMM happen at low redshifts (Fakhouri & Ma 2009) and the binaries are embedded in less massive CBD, they have less time available to reach the GW emission regime in a non-favourable angular momentum exchange with the CBD. The high mass tail shows a mild but interesting dependence with the BH seeding scheme, that can be tracked down to the fact that the random nature of the BH seeding prescription O, may populate high-mass halos with more massive BHs than the deterministic F seeding scheme. The last fact assign larger CBD masses and therefore more efficient angular momentum drain from the binary to the O seeding model, pushing some BH binaries into the GW regime and as a consequence populating the high mass end of the halo mass distributions. Host galaxy stellar mass. ------------------------- ![Stellar mass distributions represented by box and whiskers plots as in figure \[fig:boxz\]. The green line represent $M^*$ from Baldry (2008). The average stellar mass of galaxies hosting binary GW candidates is larger than $M^*$ and the distributions are asymmetric.[]{data-label="fig:boxms"}](boxms2.eps){width="0.9\linewidth"} ![Distributions of host galaxy stellar mass for three models showing the dependence on $\text{M}_{\text{CBD}} / \text{M}_{\text{BH}}$. GW candidates are found in galaxies with stellar mass in the range $10^{10} − 3 \times 10^{11} M_\odot $. The errors are Poissonian. The galaxy population hosting GW candidates is more massive in average than $M^*$, and the slope of the host stellar mass distributions at high values may contain information of BH seeding prescription.[]{data-label="fig:MS"}](GWMSV.eps){width="0.9\linewidth"} Measurement of galaxies stellar mass, although not free from biases, can be accurately achieved with current galaxy surveys (Baldry 2008). Thus, it is important and useful to investigate the potential host galaxies stellar mass distribution. Using the halo mass of the host galaxy, we assign stellar mass following Firmani & Ávila-Reese (2011) semi-empirical modelling, whose $M_s/M_{\rm H}$ ratio function has a bell shape around $M^*$. A consequence of the dependence on the $M_s/M_{\rm H}$ ratio is that the shape of the stellar mass distribution is skewed towards low masses. In figures \[fig:boxms\] and \[fig:MS\] we can see that BH binaries host galaxies have stellar masses distributions with a broad peak in the range $10^{10} - 3\times10^{11} M_\odot$ and a tail in the low mass end going up to $3\times10^{9} M_\odot$. Most GW candidates ($\sim 75\%$) are hosted in galaxies more massive than $M^* = 4.45 \times 10^{10} \text{M}_\odot$ (Baldry 2008), regardless the model parameters. However for model F10 (fig. \[fig:boxms\]) the effect of a low mass CBD is more dramatic, binaries in massive host galaxies get stuck at large separations, not close enough to enter the GW emission regime. Thus, for this model, the galaxy population hosting GW candidates is less massive in average with mean stellar mass equal to $M^*$. The scatter in the BH seeding prescription O, may assign more massive BHs, and as a consequence more massive CBDs, to the same galaxies than the F prescription, pushing some host galaxies towards the GW regime. Hence, the slope of the host galaxies stellar mass function at high values may contain information of the BH seeding. PTA and SB sources host galaxy properties. ========================================== Recently the possibility of using Pulsar Timing Arrays has been put forward by several groups (Jenet 2009; Manchester 2013; Ferdman 2010) stimulating studies constraining the binary BH galaxy host abundance in the sky (Simon et al. 2014), suggesting that the expected abundance of PTA systems is low (Rosado & Sesana 2013). We split the binary BHs population, and their respective hosts, in candidates to emit GWs in the PTA frequency range and consider the rest as candidates to emit in the SB gravitational wave observatories frequency range, such as eLISA, in order to explore consistency with previous PTA studies and to extend the study to SB systems, which will also increase the number of binary systems prone to show an anomalous EM signature. Typical PTA sources have $\text{M}_{\text{BH}} \geq 10^8 \text{M}_\odot$ and mass ratios in the range $10^{-3} < q < 1$ (Sesana 2012). Redshift, halo mass, and stellar mass. -------------------------------------- ![PTA and SB sources redshift distributions in box and whisker plots as in figure \[fig:boxz\]. As in the general binary BH host galaxy population both distributions are asymmetric, as a consequence the mean an median are different. An important number of systems will be low z SB sources.[]{data-label="fig:ptaboxZ"}](ptaboxZ.eps){width="0.8\linewidth"} The redshift distributions for our potential PTA and SB sources are depicted in figure \[fig:ptaboxZ\]. It shows that most of the closest PTA sources (given that they have been formed after the last major merger of their host halos/galaxies) peaked at redshift $0.15<z<0.6$ with a tail going up to $z\sim0.03$. We have found an important population of SB sources, peaking at $0.06<z<0.33$. For some models, even $50\%$ of the GW candidates might be SB sources that would represent an interesting case for multimessenger astrophysics in the Local Universe. ![Host galaxy halo mass distribution of PTA detectable GW candidates (blue lines) and space-based (SB) detectable GW candidates (red lines) for 4 different models. Normalized to the number of galaxies with stellar mass $\geq 1.5 \times 10^{8} M_{\odot}$. Means and confidence intervals are compare in fig. \[fig:ptabsbox\]. The expected typical dark halo mass for PTA and SB sources are segregated in a bimodal distribution.[]{data-label="fig:PTAtot"}](nPTAtot.eps){width="0.9\linewidth"} ![Host galaxy stellar mass distribution of PTA detectable GW candidates (blue lines) and space-based (SB) detectable GW candidates (red lines) for 4 different models. Normalized to the number of galaxies with stellar mass $\geq 1.5 \times 10^{8} M_{\odot}$. Means and confidence intervals are compared in fig. \[fig:ptabsbox\]. The stellar mass distributions also present a bimodal behaviour.[]{data-label="fig:PTAstar"}](nPTAstar.eps){width="0.9\linewidth"} The host dark halo and stellar masses distributions for SB and PTA binaries are shown in figures \[fig:PTAtot\], \[fig:PTAstar\] and \[fig:ptabsbox\]. We observe a bimodal distribution for both quantities. Given the nature of the F seeding prescription (equation \[Fbhmass\]) we expect a clear separation between galaxies hosting PTA and SB binaries. However, we observe a similar segregation for O seeding prescription. The distributions peak around the same mass range for PTA and SB candidates (figure \[fig:PTAtot\]), we conclude that such distributions are due not only to the BH seeding process but mostly to the galaxy pair dynamical evolution. The bimodal distribution shows the following characteristics: PTA candidates total mass host galaxy distributions have a broad peak between $5 \times 10^{12} - 3 \times 10^{13} \, \text{M}_\odot$, with a median in a $95\%$ confidence interval around $10^{13} \text{M}_\odot$ for all models and stellar mass distributions consistently peaking in the range $9 \times 10^{10} -\, 2\times 10^{11} \text{M}_\odot$ with a median $1.3 \times 10^{11} \text{M}_\odot$ within a $95\%$ confidence interval as seen in figures \[fig:PTAstar\] and \[fig:ptabsbox\]. SB detectable sources are found in galaxies with dark halo mass peaking in the range $9.6 \times 10^{11} -\,4.5 \times 10^{12}\,\text{M}_\odot$ with no agreement in the median across parameter sets and average stellar masses between $2\times 10^{10}$ and $8\times 10^{10} M_\odot$ Given that O seeding prescription allows for lighter BH masses in more massive galaxies which results in a larger spread in the host halo mass distributions, O models have a slightly higher median when compared to F models. As shown in the upper panel in figure \[fig:ptabsbox\] most galaxies containing SB sources (over $50\%$ and up to $75\%$, depending on the parameters and BH seeding prescription) have average stellar masses between $2 \times10^{10} $ and $8 \times10^{10} \text{M}_\odot$. ![Host galaxy properties for PTA and SB candidates. Stellar mass distributions for 4 models for SB and PTA candidates. The green line shows $M^*$ (Baldry 2008) (Upper panel). Total mass distributions for galaxies hosting PTA and SB detectable sources. The green horizontal region denotes Milky Way halo mass (Busha 2011) (Lower panel). In both panels, black lines represent medians with $95\%$ confidence intervals. Black crosses indicate the mean values. We have estimated that SB sources would be hosted by galaxies with $M^*$ in the Local Universe. []{data-label="fig:ptabsbox"}](ptaboxst2.eps){width="0.8\linewidth"} SB candidates are more likely to dwell in Milky Way-like haloes with stellar mass similar to $M^*$. Whilst, galaxies hosting PTA candidates are in average an order of magnitude more massive than galaxies containing SB candidates. Such distribution bimodality might be exploited in order to constraint the nature of a set of binary BHs candidates. Environment. ------------ Host galaxies’ environment is yet another key piece of information that may contribute to improve the search strategies of GW candidates, specifically for the ones with EM anomalous signatures. In order to characterize the host galaxy environment of each binary candidate to coalesce due to GW emission in the PTA or SB frequency window, we estimate environment overdensities on two different length scales for the number density of galaxies with masses larger or equal to $10^{10}\,\text{M}_{\odot}h^{-1}$. The overdensity estimator $\delta_r$ on spheres of radius $r = 5$ and $1.5$ Mpc $h^{-1}$ $$\delta_r = \frac{\rho_r - \rho_a}{\rho_a}$$ is used, where $\rho_a$ is the average number density of galaxies with mass larger or equal to $10^{10} h^{-1} M_{\odot}$ counted from the whole simulation volume, and $\rho_r$ is the number density in a sphere of radius $r$ centred in each candidate. $\rho_r$ does not take into account the galaxy where the sphere is centred, i.e. the host galaxy of the binary. A galaxy in an environment with cosmic mean density has $\delta_r =0$. If $\delta_r= -1$, the galaxy is isolated to the radius r. If $-1 < \delta_r < 0$ the galaxy is within an underdensity. {width="0.9\linewidth"} From the map of PTA and SB projected positions it is possible to investigate the typical environment in which BH coalescences take place. We conclude that there is a relationship between the environment and the BH seeding prescriptions (figure \[fig:PTA-mapa\]). Thus, binary BHs detections combined with environmental information might offer insight on the occupation of black holes and therefore on BH formation scenarios. The overdensity for spheres with radii $5$ and $1.5 h^{-1}$Mpc as a function of halo and BH mass, averaged over logarithmic bins with size $\text{log}_{10}(M_{\rm H}) = \text{log}_{10}(M_{\rm BH}) = 0.5$, is depicted in figure \[fig:overtot\] and shows that the overdensity as a function of BH mass shows a clear tendency for more massive BHs, PTA sources in our case, to inhabit more dense environments. However, the overdensity as function of halo mass (figure \[fig:overtot\], bottom panel) shows such a clear separation only when the BH-halo occupation is given by an increasing power law, as in the case of F models. For O models, whose BH-halo occupation is given by a broader distribution, we found potential SB and PTA sources in galaxies within similar environments making the environment an unreliable parameter to determine a binary host when used on its own. Despite the previous discussion, when inspecting the overdensity distribution for PTA and SB sources as shown in figure \[fig:overbox\] we have found that, regardless the differences introduced by the BH seeding prescriptions, PTA hosts dwell in denser surroundings with a median $\delta_{n,1.5} = 11.75$ consistent for all models in a $95\%$ confidence interval. SB have overdensity distributions peaking around $\delta_{n,1.5} = 5.2$. Even though the properties of host galaxies overlap in some cases depending mostly on the BHs demographics, there are galaxy populations with both, galaxy properties and environments, that might be separated in order to improve the likelihood of identifying the GW host sources. ![Average overdensity for spheres with radii $5$ and $1.5 h^{-1}$Mpc as a function of halo and BH mass, grouped in $\text{log}_{10}(M_{\rm H}) = \text{log}_{10}(M_{\rm BH}) = 0.5$ bins. Blue solid lines correspond to PTA sources and red dotted lines correspond to SB sources. Standard errors for each bin are indicated by the $1\sigma$ error bars.[]{data-label="fig:overtot"}](overTOT-3.eps){width="0.99\linewidth"} ![Overdensity of galaxies with mass $\geq 10^{10} \text{M}_\odot h^{-1}$ in a sphere of $1.5h^{-1}$Mpc for PTA and SB sources and 2 different models. Even though both kinds of binaries might be found in any environment there is a clear tendency for PTAs to be found in denser environments regardless of the model.[]{data-label="fig:overbox"}](over-box3.eps){width="0.9\linewidth"} Electromagnetic signature. Lifespan and variability =================================================== If the binary hardening happens in a gas rich environment, as we are assuming, circumbinary and individual accretion disks around the BHs may form. Whilst on the fourth stage of the model, we explore the possible electromagnetic (EM) signatures associated to the CBD and accretion disks. Following Sesana (2012), by calculating the angular momentum at the inner CBD radius and the size of the possible accretion disks at each time step for all binaries in our sample we can identify the possible EM emission scenarios. The change in angular momentum at the CBDs inner radius is given by Hayasaki (2009): $$\dot{J}_{cbd} \simeq 3 \left( \frac{m+1}{l}\right)^{1/3} \frac{(1+q)^2}{q} \frac{M_{\rm CBD}}{M_{\rm BH}} \frac{1}{\sqrt{1-e^2}} \frac{J_b}{\tau_{vis}^{cbd}}.$$ We numerically integrate at every time step in order to get the radii for the Keplerian orbit $r_i = J/GM_{\bullet,i}$ at which the infalling material is captured around each BH given by the angular momentum of the material in the inner orbit of the CBD. Following Sesana (2012) and Rees (1988) if $r_i<3R_S$, (with $R_S$ the Schwarzschild radius) the inflow is radial and there will be Bondi-like accretion. For larger values, the infalling material settles at radius $r_i$ around $M_{\bullet,i}$ and dissipates its angular momentum through viscous processes, eventually being accreted by the BHs. The time needed for an annulus of material at radius $r$ to be accreted for an $\alpha$ disk (Shakura & Sunyaev 1973) is $$t_{acc, i} = 0.837\: \text{yr}\: \left( \frac{\dot{M}_i}{\epsilon} \right)^{-2} \frac{ r^{7/2}_{10} M_{8,i}}{\alpha},$$ where $\alpha=0.1$ is the viscosity parameter, the radiative efficiency $\epsilon=0.1$, for all binaries, $r_{10}$ is the accretion disk size in $10\:R_S$ units of the $i$-BH. If $t_{acc}$ is longer than the binary orbital period $P$, then a lasting, periodically fed accretion disk will form around the BH. If $t_{acc}$ is shorter than $P$, then accretion is periodic without lasting accretion disks, which by itself has the distinctive signature of variable accretion associated with the binary period. The radius at which the transition between this two possibilities happens is found by equating both, the period $P$ and the accretion time $t_{acc}$. $$\label{eq:REM} r_{\text{trans}} = 5.3 \: R_S\: \alpha^{2/7} \left( \frac{\dot{M}_{i}}{\epsilon} \right)^{4/7} M^{-2/7}_{8,i} P^{2/7}$$ where $P$ is the binary period in years, $\dot{M}_{0.3,i}$ is the accretion mass rate as defined by equation \[mpunto\] and $M_{8,i}$ is the mass of the $i$-BH in units of $10^8 M_{\odot}$. It is possible to identify 3 scenarios for EM signatures for binaries that are still coupled to the CBD and a spectral feature in binaries that have decoupled. We explore the abundance of the following EM signatures in our complete binary population and its dependence on the BH seeding prescription. 1. If $r_i < r_{\text{trans}}$ or equivalently $t_{acc,i} < P$, there will be steady optical/IR emission from the CBD. Material falling from the CBD inner radius will form an accretion disk around the BH, whose accretion time will be short compared to the orbital period, thus presenting strong periodicity in the UV/soft X-ray. 2. If $r_i > r_{\text{trans}}$ or $t_{acc,i} > P$ there will be steady UV/soft X-ray emission due to the long lived accretion disk around the BH, which is likely to produce a relativistic Fe $\text{K}_{\alpha}$ line. Thus, in case this condition is met by the two BHs, the two Fe $\text{K}_{\alpha}$ superposed lines might be observable, in addition to the optical/IR emission from the CBD. 3. If $r_i < 3R_S$, there is radial accretion. No signatures of an accreting system are present, only the optical/IR emission from the CBD. 4. If $P < 1 $ yr and the binary is decoupled from the CBD, the combined spectrum from the CBD plus accretion disk might show a deficient soft X-ray continuum and UV emission compared with a single BH of mass $\sim M_{\rm BH}$. Such deficient emission being more evident for shorter periods. Although the aforementioned features have been proposed by Sesana (i,ii,iii; 2012) and Tanaka (iv, 2012) for PTA sources, we have extended their application to less massive binaries. We keep track of the time during which the conditions for one of the EM signatures proposed are met and consider this the lifespan of the associated signature. As the binary’s angular momentum and separation decreases due to the interaction within gaseous disks, the GW driven coalescence takes over, such mechanism is not included in the model current version. Therefore, the electromagnetic signature lifespan found here might be considered as an upper limit, since the orbital decay is much quicker once the GW emission drives the angular momentum loss (Peters, 1964; Heiman 2012). Almost all SB sources have a 3-disk configuration whilst on the last stage, i.e. EM signature type ii: A CBD and two long lived accretion disks around each BH with viscous time scale longer than the period, resulting in a EM signature given by an optical/IR continuum from the CBD and Double X-ray Fe $K_\alpha$ emission lines from the accretion disks. While accretion variability might not be associated with the period, shocks between the infalling material and the accretion disks may produce X-ray hot spots varying on the orbital period, most SB sources have periods of $1.25 - 18$ yr. In figure \[fig:period\] we see that SB sources in haloes populated with F seeding prescription (empirical with increasing power law behaviour) do not show binaries with $P < 1$ yr. For binaries with O seeding prescription there are a few binaries with period equal or shorter than a year, the more massive ones allowed by the broader dispersion in the seeding prescription. Thus, less than $\sim8\%$ binaries with different parameter values (O seeding prescription only) might show the EM signature type iv, abnormally weak soft X-ray continuum and UV emission as proposed by Tanaka (2012). We found that the lifespan of the EM signature due to a 3-disk configuration ranges between $10^5 - 10^9$ years (figure \[fig:lifespan\]) compatible with the theorized AGN duty-cycle. For less massive CBDs (models 5,6,7,8 & 10), the angular momentum loss decreases, producing less pairs reaching the GW regime. Those that are able to reach such regime will spend more time interacting with the CBD resulting in a longer lifespan for the EM signature. As in the case of SB sources, most PTA sources, regardless the model, show the dynamical conditions to have a 3-disk configuration with a EM signature type ii, with some differences. Models F10 and O10 (our models with the less massive CBD) are dominated by signature type i, same configuration suggested by Roedig (2011) simulations, which consists of two accretion disks with viscous time scale shorter than the period whose signature might be observed as accretion associated variability in the UV/Soft X-ray. For models $5-8$ (for both seeding prescriptions, O and F) $\sim 3-4\%$ of the binaries do not fulfill the conditions to form accretion disks, so a small fraction of binaries would not be detected as an accreting system. According with our model, PTA sources have periods within the range $0.4 \sim 6$ yr, figure \[fig:period\], with a tail going up to a few days for some models. Indicating that those binaries with a long lived accretion disks might show variability associated to shocks caused by streaming material into the accretion disk coming from the CBD or if the typical CBD in nature turn out to be light (as in the case of models O10 and F10) there might be an accretion associated emission with variability of a few years. ![Period of the binaries when they enter the GW decay stage. The blue distribution describes the binaries that emit GW in the PTA detection frequency window. The red distribution correspond to the binaries emitting in the space based detector frequency window. Normalized to the number of galaxies with stellar mass $\geq 1.5\times10^8 \text{M}_\odot$[]{data-label="fig:period"}](Ppta2.eps){width="0.9\linewidth"} ![Fraction of SB and PTA sources per number of galaxies with stellar mass larger or equal to $1.5\times 10 M_\odot$ as a function of the time spent within a CBD. This time scale might be considered an upper limit of the EM signature lifespan.[]{data-label="fig:lifespan"}](emLifeS.eps){width="0.9\linewidth"} Summary and conclusions ======================= We have developed a dynamical model for the decay of supermassive black hole binaries. It has been applied to create a population of binaries likely to decay due to GW emission within a cosmological volume similar to the Local Universe. Our model is coupled with a cosmological simulation, we retrieved the redshift of the last major merger for all haloes with masses $M_{\rm H} \geq 1.64 \times 10^{10} M_\odot h^{-1}$ (at $z=0$), this zLMM constitutes the starting point for the binary evolution. We used semi-empirical approaches to assign galaxies and BHs to the haloes. In the final stage, the interaction within a gaseous disk is responsible for the binaries’ coalescence. The binary’s destiny is set by a competition between cosmic accretion that replenish gas and angular momentum into the CBD and the binary, versus the angular momentum drain out of the binary by the CBD torques. For the binaries found we used a simple criteria based on BHs mass and separation to create a population of binaries prone to emit GW in the PTA frequency window and in that of the space-based detectors. We found a bimodal correlation between host galaxy properties, environment, and binary period. Such correlation defines the properties that the host galaxy populations of PTA or SB sources might show. We anticipate that individual cases are rather difficult to distinguish between binaries and anomalous AGNs, but at a demographic level it might be possible to identify a subset of AGNs powered by binaries in the subpc regime. We investigate the BH binaries host galaxies properties, environment characterization and likelihood to observe some of the EM signatures previously proposed. A short summary of our results follows here: - In our model for the binaries in the gaseous disk stage the most dominant parameter is the $ M_{\rm CBD}/M_{\rm BH}$ ratio. The CBD mass works as a regulator of the angular momentum exchange mechanism for the whole binary populations across models, overpowering even the cosmic accretion rate. - The expected number density of binaries (SB and PTA sources combined) in the Local Universe is $10^{-4} h^3\text{Mpc}^{-3}$. With a probability of finding a binary every 1000 galaxies with stellar masses larger or equal to $1.5\times10^{8} \text{M}_{\odot}$ in our best case scenario. - Most of the candidates to decay due to GW emission occur at redshifts between $0.1 -\, 0.5$, as seen in figure \[fig:redshift\], with $\sim 80\%$ occurring at $z\lesssim0.5$ (fig. \[fig:boxz\]). The distribution peaks at lower redshifts for less massive CBDs due to the lack of an efficient angular momentum loss mechanism, but the tendency towards low redshifts remains regardless of model parameters and seeding prescription. - GW candidates host galaxies have typical stellar masses between $10^{10}$ - $3\times10^{11}$ $\text{M}_{\odot}$ (fig.\[fig:MS\]), with $\sim75\%$ of the candidates having masses larger or equal to $\text{M}^* = 4.45 \times 10^{10} \text{M}_{\odot}$. - Most GW candidates are systematically found in host galaxies with total masses larger than the Milky Way halo mass $1.2^{+0.7}_{-0.4} \text{(stat.)} ^{+0.3}_{-0.3} \text{(sys.)} \times 10^{12} M_\odot$ (Busha 2011). - As expected, PTA detectable GW candidates occupy the most massive galaxies (in agreement with Rosado & Sesana, 2013; Simon 2014) with mean total mass $\sim 10^{13} \text{M}_{\odot}$ and mean stellar mass $\sim 1.4\times 10^{11} \text{M}_{\odot}$. They are located on average in overdensity regions of $\delta_{n,1.5} = 11.75$, about 10 times the average density of the Universe. - Most PTA sources might show EM signatures of type ii, long lived CBD plus accretions disks, with emission associated to shocks between the accretion disks and streaming material from the CBD with variabilities in the range $1.25 - 20$ yr peaking at $\sim3$ yr with a lifespan for the EM signature due to the 3 disk configuration between $10^6 - 10^9$ yrs. There is a good chance ($\sim 50 \%$) in particular for O seeding prescription, to show a deficient soft X-ray continuum and UV emission compared to ’normal’ AGNs powered by one BH as suggested by Tanaka (2012). - According to our model, SB sources might be found in galaxies with total mass peaking in the range $9.6\times10^{11} - \, 4.5\times10^{12} M_\odot $, with average stellar masses between $2 \times10^{10}$ and $8\times10^{10} M_\odot$. Dwelling in overdensities of $\delta_{n,1.5} = 5.2$, less dense than the typical environment of PTA sources. - Most SB sources might show EM signatures associated to shocks between the accretion disks and streaming material from the CBD with variabilities in the range $1.25 - 20$ yr peaking at $\sim3$ yr with a lifespan for the EM signature due to the 3-disk configuration (EM signature type ii) between $10^5 - 10^6$ yrs and $10^6 - 10^7$ yrs for less massive CBDs. The correlations found may serve as a strategy to distinguish binary local effects in AGNs using a combination of host galaxy properties and EM signals. The distributions of the binary properties depends on the BHs seeding prescription, thus a sample of binaries would represent an independent way to learn about BHs formation models and initial BHs population, a possibility that should be explored. A better understanding on the mechanisms driving the evolution and final coalescence of BH binaries might be reached when better observational constraints on the number density of BH binaries are available. This will help constrain the parameters used in this and similar models. Acknowledgements ================ We thank the CLUES collaboration (www.clues-project.org) for providing access to the simulation analysed in this paper. Our semi-analytical modelling and analyses were done using IA-UNAM cluster Atocatl. EMP thanks J. A. de Diego, V. Ávila-Reese, W. H. Lee, Y. Krongold and A. Batta for helpful conversations. This work was supported by UNAM-PAPIIT grant IN112313 and a graduate CONACyT fellowship. 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--- author: - | Simone Alioli\ Deutsches Elektronen-Synchrotron DESY\ Platanenallee 6, D-15738 Zeuthen, Germany\ and INFN, Sezione di Milano-Bicocca\ E-mail: - | Paolo Nason\ INFN, Sezione di Milano-Bicocca, Piazza della Scienza 3, 20126 Milan, Italy\ E-mail: - | Carlo Oleari\ Università di Milano-Bicocca and INFN, Sezione di Milano-Bicocca\ Piazza della Scienza 3, 20126 Milan, Italy\ E-mail: - | Emanuele Re\ Institute for Particle Physics Phenomenology, Department of Physics\ University of Durham, Durham, DH1 3LE, UK\ and INFN, Sezione di Milano-Bicocca\ E-mail: bibliography: - 'paper.bib' title: 'Vector boson plus one jet production in [[POWHEG]{}]{}' --- Introduction ============ The process of vector boson production in association with jets plays an important role at present hadron colliders. In the early LHC phase, $Z$ plus one jet production will play a major role in jet calibration. Furthermore, vector boson production in association with jets is an important background to new physics signals. In particular, when the $Z$ decays into neutrinos, it becomes a source of missing-energy signals. The analogous process of $W$ production, in association with jets, gives origin to an important background to the production of a lepton in association with missing energy and jets. $Z$ plus one jet production has a very distinctive signature, when the $Z$ decays into muons or electrons, and several studies have been performed at the Tevatron, both by the CDF [@Aaltonen:2007cp] and by the D0 Collaboration [@Abazov:2006gs; @Abazov:2008ez; @Abazov:2009av; @Abazov:2009pp]. They are all carried out by correcting the measured quantities to the particle level, according to the recommendations developed in the 2007 Les Houches workshop [@Buttar:2008jx], and then compared to theoretical calculations, computed mostly at the next-to-leading order (NLO) level, and corrected for showering, hadronization and underlying-event effects. These corrections, in turn, are extracted from shower Monte Carlo (SMC) programs. In the present work, we present a calculation for the NLO cross section for vector boson plus one jet that can be interfaced to a shower Monte Carlo program, within the [[POWHEG]{}]{} framework [@Nason:2004rx; @Frixione:2007vw]. This is the first time that such calculation has been performed. More specifically, we have implemented this process using the [[POWHEG BOX]{}]{}, a general computer code framework for embedding NLO calculations in shower Monte Carlo programs according to the [[POWHEG]{}]{} method. In fact, the [[POWHEG BOX]{}]{} framework was developed using the $Z/\gamma+1j$ process as its first testing example.[^1] It is clear that, by using the [[POWHEG]{}]{} implementation of $V+1j$ production, the comparison of the theoretical prediction with the experimental results is eased considerably, and is also made more precise. Rather than estimating shower and underlying-event corrections using a parton shower program, one interfaces directly the parton shower to the hard process in question, yielding an output that can be compared to the experimental results at the particle level. We will carry out this task for the $Z+1j$ case, and compare our results to the Tevatron findings. We remark, however, that a further improvement to this study could be carried out, by using the [[POWHEG]{}]{} program to generate the events that are fed through the detector simulation, and are directly compared to raw data. We are not, of course, in a position to perform such a task, that should instead be carried out by the experimental collaborations. The paper is organized as follows. In section \[sec:calculation\] we give a brief description of our calculation. Since we used the [[POWHEG BOX]{}]{} to implement our process, we refer the reader to the [[POWHEG BOX]{}]{} publication [@Alioli:2010xd], and report here only a few details that are particularly relevant to the process in question. In the following, we always consider the $Z+1j$ case, since the $W+1j$ is fully analogous. In section \[sec:validation\] we describe the generation of the $Z+1j$ sample, together with some consistency checks of our calculation, and new features added to the [[POWHEG BOX]{}]{}. In section \[sec:phenomenology\], we present results for the Tevatron and for the LHC at 14 TeV. Comparison with available data from the CDF and D0 Collaborations are carried out. Finally, in section \[sec:conclusions\], we give our final remarks. Description of calculation {#sec:calculation} ========================== We have considered the hadroproduction of a single vector boson plus one jet at NLO order, with all spin correlations from the decay taken into account. All fermions (quarks and leptons) have been treated in the massless limit (no top-quark contributions have been taken into account). In order to implement a new process at NLO into the [[POWHEG BOX]{}]{}, we have to provide the following ingredients: The list of all flavour structures of the Born processes. The list of all flavour structures of the real processes. The Born phase space. \[item:real\] The real matrix elements squared for all relevant partonic processes. \[item:virtual\] The finite part of the virtual corrections computed in dimensional regularization or in dimensional reduction. \[item:born\] The Born squared amplitudes ${\cal B}$, the colour correlated ones ${\cal B}_{ij}$ and spin correlated ones ${\cal B}_{\mu\nu}$. The Born colour structures in the limit of a large number of colours. For the case at hand, the list of processes is generated going through all possible massless quarks and gluons that are compatible with the production of the vector boson plus an extra parton. The Born phase space for this process poses no challenges: we generate the momentum of the vector boson distributed according to a Breit-Wigner function, plus one extra light particle. The vector boson momentum is then further decayed into two momenta, describing the final-state leptons. At this stage, the momentum fractions $x_1$ and $x_2$ are also generated and the momenta of the incoming partons are computed. =0.3 A sample of Feynman diagrams that contribute to the $Z+1j$ process at the Born level (${\cal B}$) is depicted in panels (a) and (b) of fig. \[fig:graphs\]. Together with the Born diagrams, we have to consider the one-loop corrections to the tree level graphs, and the diagrams with an extra radiated parton. A sample of virtual and real contributions is depicted in panels (c)–(f) of fig. \[fig:graphs\]. We have computed the Born and real contributions ourselves, using the helicity-amplitude technique of refs. [@Hagiwara:1985yu; @Hagiwara:1988pp]. The amplitudes are computed numerically in a [fortran]{} code, as complex numbers, and squared at the end. The finite part of the virtual corrections has been taken from the [[MCFM]{}]{} program [@Campbell:2002tg], that uses the virtual matrix elements given in ref. [@Bern:1997sc] and computed first in ref. [@Giele:1991vf]. The use of the helicity-amplitude technique to compute the amplitudes at the Born level facilitates the calculation of the spin-correlated matrix elements ${\cal B}_{\mu\nu}$. In fact, they are the Born amplitudes just before being numerically contracted with the polarization vector of the initial- or final-state gluon. Since we are dealing with three coloured partons at the Born level, the colour correlated Born amplitudes ${\cal B}_{ij}$ are all proportional to the Born one. The two independent colour-correlated matrix elements are given by B\_[q q’]{}=$2 {C_{{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}}\rm F}}-{C_{{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}}\rm A}}$ [B]{},B\_[q g]{}=, and, from colour conservation, they satisfy $$\label{eq:colcons} \sum_{i,i\ne j}{{\cal B}}_{ij}=C_{f_j}{{\cal B}}\,,$$ where $i$ runs over all coloured particles entering or exiting the process, and $C_{f_j}$ is the Casimir constant for the colour representation of particle $j$. Finally, the Born colour structures, in the limit of a large number of colours, are straightforward, due to the presence of a single gluon and two quarks at the Born level. For the two Born diagrams depicted in fig. \[fig:graphs\], the colour structures are shown in fig. \[fig:colors\]. Generation cut and Born suppression factor ------------------------------------------ The $V+1j$ process differs substantially from all processes previously implemented in [[POWHEG]{}]{}, in the fact that the Born contribution itself is collinear and infrared divergent. In all previous implementations, the Born diagrams were finite, and it was thus possible to generate an unweighted set of underlying Born configurations covering the whole phase space. In the present case, this is not possible, since they would all populate the very low transverse momentum region. Of course, this problem is also present in standard shower Monte Carlo programs, where it is dealt with by generating the Born configuration with a cut ${k_{\rm gen}}$ on the transverse momentum of the $V$ boson. After the shower, one must discard all events that fail some transverse momentum analysis cut ${k_{\rm an}}$ in order to get a realistic sample. The analysis cut ${k_{\rm an}}$ may be applied to the transverse momentum of the vector boson, or to the hardest jet. We assume here, for sake of discussion, that it is applied to the vector boson transverse momentum. Taking ${k_{\rm an}}\gtrsim {k_{\rm gen}}$ is not enough to get a realistic sample. In fact, in an event generated at the Born level with a given ${k_{{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}}\rm T}}< {k_{\rm gen}}$, the shower may increase the transverse momentum of the jet so that the final vector-boson transverse momentum ${k_{{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}}\rm T}}^V$ can be bigger than ${k_{\rm an}}$. Thus, even if the generation cut is below the analysis cut, it may reduce the number of events that pass the analysis cut. Of course, as we lower ${k_{\rm gen}}$ keeping ${k_{\rm an}}$ fixed, we will reach a point where very few events below ${k_{\rm gen}}$ will pass the analysis cut ${k_{\rm an}}$. In fact, generation of radiation with transverse momentum larger than ${k_{\rm gen}}$ is strongly suppressed in [[POWHEG]{}]{}, and, in turn, radiation from subsequent shower is required to be not harder than the hardest radiation of [[POWHEG]{}]{}. Thus, if we want to generate a sample with a given ${k_{\rm an}}$ cut, we should choose ${k_{\rm gen}}$ small enough, so that the final sample remains substantially the same if ${k_{\rm gen}}$ is lowered even further. Together with the option of generating events with a generation cut ${k_{\rm gen}}$, a second option for the implementation of processes with a divergent Born contribution is also available. It requires that we generate weighted events, rather than unweighted ones. This is done by using a suppressed cross section for the generation of the underlying Born configurations $$\label{eq:supp_fac} \bar{B}_{\rm supp}=\bar{B}\times F({k_{{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}}\rm T}})\,,$$ where $\bar{B}$ is the inclusive NLO cross section at fixed underlying Born variables, ${k_{{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}}\rm T}}$ is the transverse momentum of the vector boson in the underlying Born configuration and $F({k_{{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}}\rm T}})$ is a function that goes to zero in the ${k_{{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}}\rm T}}\to 0$ limit fast enough to make $\bar{B}\times F({k_{{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}}\rm T}})$ finite in this limit. In this way, $\bar{B}_{\rm supp}$ is integrable, and one can use it to generate underlying Born configurations according to its value. The generated events, however, should be given a weight $1/F({k_{{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}}\rm T}})$, rather than 1, in order to compensate for the initial $F({k_{{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}}\rm T}})$ suppression factor. With this method, events do not concentrate in the low ${k_{{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}}\rm T}}$ region, although their weight, in the low ${k_{{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}}\rm T}}$ region, becomes divergent. After shower, if one imposes the analysis cut, one gets a finite cross section, since it is unlikely that events with small transverse momentum at the Born level may pass the cut after shower. In recent [[POWHEG BOX]{}]{} revisions, both methods can be implemented at the same time. We wanted in fact to be able to implement the following three options: - Generate events using a transverse momentum generation cut ${k_{\rm gen}}$. - Generate events using a Born suppression factor, and a small transverse momentum cut, just enough to avoid unphysical values of the strong coupling constant and of the factorization scale that appears in the parton distribution functions. In this case, since we are also generating events with very small transverse momenta, radiative corrections may become larger than the Born term, and negative-weight events could be generated. We should then be prepared to track them. This issue will be further discussed in the next section. - Apply a Born suppression factor, and set the transverse momentum cut ${k_{\rm gen}}$ to zero. In this case the program cannot be used to generate events. It can be used, however, to produce NLO fixed order distributions, provided the renormalization and factorization scales are set in such a way that they remain large enough even at small transverse momentum ${k_{{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}}\rm T}}^V$. This feature is only used for the generation of fixed-order distributions. The generation cut is activated by setting the token [bornktmin]{} to the desired value in the [powheg.input]{} file. The Born suppression is activated by setting the token [[bornsuppfact]{}]{} to a positive real value. The process-specific subroutine [born\_suppression]{} sets the suppression factor of eq. (\[eq:supp\_fac\]) to $$F({k_{{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}}\rm T}}) = \frac{{k_{{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}}\rm T}}^2}{{k_{{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}}\rm T}}^2+{{\tt bornsuppfact}}^2} \,.$$ If [[bornsuppfact]{}]{} is negative, the suppression factor is set to 1. The need of a transverse momentum cut is not only a technical issue. The NLO calculation of $V+1j$ production holds only if the transverse momentum of the vector boson is not too small. In fact, as the ${k_{{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}}\rm T}}$ decreases, large Sudakov logarithms arise in the NLO correction, and the value of the running coupling increases, up to the point where the cross section, at fixed order, becomes totally unreliable. These large logarithms should all be resummed in order to get a sensible answer in this region. In the [[POWHEG]{}]{} implementation of single vector-boson production [@Alioli:2008gx], in fact, these logarithms are all resummed. Then it is clear that some sort of merging between the $V+1j$ and the $V$ production processes should be performed at relatively small transverse momentum, in order to properly deal with this problem. Here we will not attempt to perform such merging, that we leave for future work. We will simply recall, when looking at our results, that we expect to get unphysical distributions when the vector boson transverse momentum is too small. We will discuss this fact in a more quantitative way in section \[sec:validation\]. Negative-weight events ---------------------- In the [[POWHEG]{}]{} method, negative-weight events can only arise if one is approaching a region where the NLO computation is no longer feasible. In our study for the $V+1j$ process, we approach this region at small transverse momenta. In order to better understand what happens there, rather than neglecting negative weights (that is the default behaviour of the [[POWHEG BOX]{}]{}), we have introduced a new feature in the program, that allows one to track also the negative-weight events. This feature is activated by setting the token [[withnegweights]{}]{} to 1 (true). If [[withnegweights]{}]{} is set to 1, events with negative weight can thus appear in the Les Houches event file [@Boos:2001cv; @Alwall:2006yp]. While we normally set the [ IDWTUP]{} flag in the Les Houches interface to 3, in this case we set it to -4. With this flag, the SMC is supposed to simply process the event, without taking any other action. Furthermore, the [XWGTUP]{} (Les Houches) common block variable is set by the [[POWHEG BOX]{}]{} to the sign of the event times the integral of the absolute value of the cross section, in such a way that its average equals the true total cross section. We preferred not to use the option [IDWTUP]{}=-3 for signed events with constant absolute value. This option is advocated by the Les Houches interface precisely in such cases, and it requires that the event weight [ XWGTUP]{} assumes the values $\pm 1$. However, the Les Houches interface does not provide a standard way to store the integral of the absolute value of the cross section, that would be needed to compute correctly the weight of the event in this case. In fact, the [XSECUP]{} variable is reserved for the true total cross section $ \sigma_{\rm NLO}$. More specifically, suppose that the total NLO cross section receives contributions from regions in the phase space where the differential cross section is positive, and where it is negative, so that we can write $$\sigma_{\rm NLO} = \sigma_{(+)}-|\sigma_{(-)}|\,.$$ We then generate a sample of $N=N_+ + N_-$, with $N_+$ events with weight $W_i=+1$ (so that the sum of the $W_i$ on events with positive weights is $N_+$) and $N_-$ events with weight $W_i=-1$ (so that the sum of the $W_i$ on events with negative weights is $-N_-$) in such a way that $$N_+ = \frac{\sigma_{(+)}}{\sigma_{(+)}+|\sigma_{(-)}|} \, N\,, \qquad N_- = \frac{\sigma_{(-)}}{\sigma_{(+)}+|\sigma_{(-)}|} \,N\,.$$ In order to give the correct NLO cross section, the $N$ events should then be weighted with the sum of the positive plus the absolute value of the negative part of the cross section, in such a way that $$\frac{1}{N}\sum_{i=1}^N W_i \left(\sigma_{(+)}+|\sigma_{(-)}|\right)= \sigma_{(+)}-|\sigma_{(-)}| = \sigma_{\rm NLO}\,.$$ Summarizing, since there is no standard way in the Les Houches interface to pass this absolute value when [IDWTUP]{}=-3, we set it to -4, and set the weight [XWGTUP]{} of the $i$-th event to $W_i \times \left(\sigma_{(+)}+|\sigma_{(-)}|\right)$, so that, event by event, we have this information. In this case the average value of the [XWGTUP]{} variable is equal to the total cross section, as required by the Les Houches interface when [IDWTUP]{}=-4. Notice that, if [[withnegweights]{}]{} is set to true and a Born suppression factor is also present, the events will have variable [XWGTUP]{} of either signs. In this case, [XWGTUP]{} is set to the sign of the event, times the absolute value of the cross section, divided by the suppression factor (the output of the [born\_suppression]{} routine). Also in this case, the average value of [XWGTUP]{} coincides with the true total cross section. Weighted events are also useful if one wants to generate a homogeneous sample from relatively-low up to very-high transverse momenta. In this case, it is convenient to pick a very large [[bornsuppfact]{}]{} value, of the order of the maximum transverse momentum one is interested in. The large-momentum region will be more populated in this way. The form of the [born\_suppression]{} function can also be changed at will by the user. Validation of the generated samples {#sec:validation} =================================== In this section we discuss the output of the [[POWHEG BOX]{}]{} for $Z+1j$ production. We consider here $p\bar{p}$ collisions at $1.96$ TeV. We use the CTEQ6M pdf set [@Pumplin:2002vw]. The factorization and renormalization scales (in the computation of the $\bar{B}$ function) are fixed to the transverse momentum of the vector boson of the underlying Born configuration. For the generation of radiation, these scales are set by the [[POWHEG BOX]{}]{}, according to the prescriptions given in [@Alioli:2010xd]. We have first produced one sample with a generation cut ${k_{\rm gen}}=5$ GeV on the transverse momentum of the $Z$ boson in the underlying Born configuration. This sample, that we call sample U (for unweighted), was produced with positive weights. Then we produced a second sample W, where we used a Born suppression factor, with ${{\tt bornsuppfact}}=10$ GeV, and a generation cut of 1 GeV, in order to avoid unphysical values for the strong coupling and pdf’s (we will assume, in the following, that this tiny generation cut has no effects on distributions where the $Z$ boson or the hardest jet have transverse momenta of several GeV). We have also set [[withnegweights]{}]{} to 1, so that we are able to record negative weighted events that may arise in the region of very small transverse momenta. Sample W is thus weighted, with weights of either signs. We analyzed the events of the two samples at the [[POWHEG]{}]{} level, i.e. without feeding them to a shower Monte Carlo. We begin by comparing the transverse momentum distribution of the $Z$ of the two samples, and of the fixed order NLO QCD result (obtained as a byproduct of the event generation) in fig. \[fig:UW\_ptZ\]. The figure should be interpreted in the following way. The weighted sample gives the reference result, since it is unaffected by the generation cut. The U sample feels clearly the effect of the 5 GeV generation cut. However, for ${p_{{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}}T}^{{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}}Z}}\gtrsim 10$ GeV, the U and W results coincide, showing that the effect of the generation cut in this region is fully negligible. A second important observation has to do with results in the W sample. It becomes unphysical for ${p_{{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}}T}^{{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}}Z}}\lesssim 5$ GeV. This indicates that NLO corrections in this region are out of control, and that our program should not be used for ${p_{{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}}T}^{{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}}Z}}\lesssim 10$ GeV. The fixed order NLO result displays a similarly unphysical behaviour in the low transverse momentum region. The similarity is more apparent in the right panel of fig. \[fig:UW\_ptZ\], which is given in linear scale. There is a very large negative value of the NLO cross section in the bin between 1 and 1.5 GeV. The [[POWHEG]{}]{} result follows the NLO result up to a certain amount of smearing. Plots for the transverse momentum of the hardest jet are shown in fig. \[fig:UW\_pt1\]. They are similar to the plots in the $Z$ transverse momentum, except for a more modest value of the NLO distribution at small transverse momenta. Finally, in fig. \[fig:UW\_yZj1\_pt10\], the rapidity distribution of the $Z$ and of the hardest jet are plotted for the U and W sample and for the NLO contribution. A cut of 10 GeV is applied on either the $Z$ or the jet transverse momentum. We see, again, very good agreement between the U and W samples at this transverse momentum. We have also analyzed many other distributions for the U and W sample. They all lead to the conclusion that, when cuts on the transverse momenta of the order of 10 GeV (twice the generation cut) are applied, the U sample yields distributions that are equivalent to the W sample. More specifically, at 10 GeV, in the differential cross section of the jet or of the $Z$ transverse momentum, the U sample differs from the W sample by 6%, and the difference decreases with ${p_{{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}}\rm T}}$. Thus, we conclude that for ${p_{{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}}\rm T}}>10$ GeV, at Tevatron energies, the U sample is substantially independent of a generation cut less or equal than 5 GeV. Negative weights and folding {#sec:posnegfold} ---------------------------- In order to get a reasonably small fraction of negative-weight events in the [[POWHEG BOX]{}]{}, it is at times necessary to increase the folding parameters [foldcsi]{}, [foldy]{} and [foldphi]{}. These parameters were introduced in the framework of the [[POWHEG]{}]{} implementation of heavy-quark pair production [@Frixione:2007nw], and are mentioned in the corresponding manual [@Frixione:2007nu], and discussed extensively in ref. [@Alioli:2010xd]. We briefly recall here their genesis and use. In [[POWHEG]{}]{}, the underlying Born configuration for an event is generated according to the $\bar{B}$ function, that, in the simplest cases, can be represented by the equation (see ref. [@Frixione:2007vw]) $$\bar{B}(\Phi_B)=B(\Phi_B)+V(\Phi_B)+\int d\Phi_{\rm rad} \lq R\(\Phi_B,\Phi_{\rm rad}\)-C\(\Phi_B,\Phi_{\rm rad}\) \rq \,.$$ In order to generate $\Phi_B$ distributed according to this probability density, we actually parametrize the radiation variables in terms of three variables in the unit cube, $X_i$, and then perform the integral $$\int d \Phi_B \; d^3 X\; \tilde{B}(\Phi_B,X)\,,$$ where $$\label{eq:btilde} \tilde{B}(\Phi_B,X)=B(\Phi_B)+V(\Phi_B)+ \left|\frac{\partial \Phi_{\rm rad}}{\partial X}\right| [R(\Phi_B,\Phi_{\rm rad})-C(\Phi_B,\Phi_{\rm rad})]\,,$$ using the [[MINT]{}]{} integration program [@Nason:2007vt]. This program stores appropriate grids and upper bounds, so that, after the integration, it is possible to use it to generate points in the integration variables, i.e. in $(\Phi_B,X)$, distributed with a probability proportional to $\tilde{B}$. The $X$ values are ignored, which amounts to integrating over them, and only $\Phi_B$ is retained. This procedure is sufficient to generate positive weights if the process in question has a sufficiently-high scale, like in vector boson, Higgs or $t\bar{t}$ production. In these cases, ${\alpha_{{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}}\rm S}}$ (present in eq. (\[eq:btilde\]) in the virtual $V$, real $R$ and counterterms $C$ contributions) is small enough so that $B$ is always larger than the other terms, yielding a positive result for any value of $\Phi_B$ and $X$. As the scale of the process decreases (and ${\alpha_{{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}}\rm S}}$ increases), it becomes more likely that $\tilde{B}$ may become negative for some value of the $X$ parameters, even if $\bar{B}$ is positive for any value of $\Phi_B$. Of course, this depends upon the way that the $C$ counterterm is defined. A brute force remedy to this problem, is to fold the integration of the $(R-C)$ term as many time as necessary to yield a positive $\bar{B}$ (that is to say, to get a negligible fraction of negative-weight events). The details of the folding method are illustrated in the [[MINT]{}]{} manual. Here we only illustrate it with an example. If we want to fold the $X_1$ variable twice, we define $$\tilde{B}_{\rm folded}(\Phi_B,x_1,X_2,X_3)= \tilde{B}(\Phi_B,x_1,X_2,X_3)+\tilde{B}(\Phi_B,1/2+x_1,X_2,X_3),$$ and integrate in $x_1$ from 0 to $1/2$. This clearly yields the same result as without the folding. Furthermore, if we increase the number of foldings for all the three parameters $X_i$, the folded function becomes less and less dependent upon the $x_i$ values. With this technique, regions of integration where the function is positive and negative are combined together. It is clear that, if the $\bar{B}$ function is positive, with large enough folding numbers, we will achieve the positivity of the folded $\tilde{B}$ function. The [[MINT]{}]{} integrator and the [[POWHEG BOX]{}]{} can perform the folding automatically. The user needs only to specify how many times each variables should be folded. The [[MINT]{}]{} integrator divides each integration coordinates in 50 bins, such that each bin contributes equally to the integral of the absolute value of the $\tilde{B}$ function. Folding is achieved by overlapping these intervals. Thus, the folding number must be a divisor of 50: 2, 5, 10, 25 or 50. Three tokens in the [powheg.input]{} file can be set to the folding number: [ifoldcsi]{}, [ifoldy]{} and [ifoldphi]{}, that refer to the folding of the three radiation variables $\xi$, $y$ and $\phi$. In fig. \[fig:folding-effects\], we display the effect of folding on the amount of negative weights that enter the computation of a physical quantity, namely the $Z$ transverse momentum (by choosing similar variables, like the transverse momentum of the hardest jet, we get similar results). All the histograms in the figure were obtained with the [[withnegweights]{}]{} flag set to true. The solid line was obtained in the standard way, i.e. by adding positive and negative contributions. The dashed, dotted and dotdashed lines were obtained by plotting only the absolute value of the negative weights. The dashed line was obtained with no folding, the dotted line was obtained with a 2-5-1 folding (i.e. [ifoldcsi]{}$=2$, [ifoldy]{}$=5$ and [ifoldphi]{}$=1$), and the dot-dashes line was obtained with the 5-10-5 folding. From the figure, we see that the amount of negative weights increases as the transverse momentum becomes smaller. This can be understood as being due to the increase of ${\alpha_{{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}}\rm S}}$ at small scales. By increasing the folding numbers, the number of negative weights is strongly reduced, and, furthermore, it only affects the region of very small transverse momenta, where the calculation becomes however unreliable. Increasing the folding number has a cost on the execution time. As a rule of thumb, we expect the execution time to increase as the product of the three folding numbers. Actually, the increase is somewhat less than that, because the Born and virtual contribution are evaluated only once for each set of foldings, since they do not depend upon the integration variables. Thus, the folding should be chosen as a function of the generation cut, and, ultimately, as a function of the cutoff one is imposing on the hardest-jet transverse energy or on the $Z$ transverse momentum. $\xi$-$y$-$\phi$ folding $\Longrightarrow$ 1-1-1 2-5-2 5-5-5 5-10-5 10-10-10 -------------------------------------------- ------- ------- -------- --------- -------------------- -- LHC, 14 TeV, ${k_{\rm gen}}=5$ GeV 0.25 0.067 0.026 0.016 0.0074 LHC, 14 TeV, ${k_{\rm gen}}=10$ GeV 0.17 0.020 0.0029 0.00036 $3.4\times10^{-5}$ Tevatron, 1.96 TeV, ${k_{\rm gen}}=5$ GeV 0.23 0.029 0.0086 0.0043 0.0028 : \[table:LHC\] Negative-weight fractions $\sigma_{(-)}/(\sigma_{(+)}+|\sigma_{(-)}|)$ for given folding numbers. As an example, in table \[table:LHC\], we have collected the fraction of negative weights $\sigma_{(-)}/(\sigma_{(+)}+|\sigma_{(-)}|)$ for the LHC at 14 TeV, with two generation cuts, and for the Tevatron. As can be seen, the higher is the folding, the smaller is the fraction of negative-weight events. The typical performance of event generation is, without folding, roughly 1000 events per minute, on typical workstation cpu’s. It is clear that, in the worse case of very low transverse-momentum jets, assuming that we can tolerate around $2\%$ negative weights (that would populate, in all cases, the very small ${p_{{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}}\rm T}}$ region), by using a 5-5-5 folding we end up generating of the order of eight events per minute. It is also clear that we have a compelling reason to generate independently samples with different generation cuts. Besides benefiting of a more abundant generation in the high ${p_{{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}}\rm T}}$ region, we also benefit from the possibility of using smaller folding numbers. A clarifying consideration on negative weights in [[POWHEG]{}]{} can be made in comparison with [[MC@NLO]{}]{} [@Frixione:2002ik]. The negative weights that we get in [[POWHEG]{}]{} are fully analogous to the negative weights that one gets in the ${\cal S}$ events in [[MC@NLO]{}]{}. Also in that case, the number of negative weights could be reduced or eliminated by using a folding technique, as in our case. However, negative weights in the ${\cal H}$ event sample of [[MC@NLO]{}]{} cannot be reduced, and so there is no reason to attempt to reduce the negative weight fraction in the ${\cal S}$ events. Finally, let us stress again that by performing any analysis on a [[POWHEG]{}]{} event sample with signed events, or with weighted events, or with positive-weight events (using eventually a large enough folding number to get a negligible fraction of negative weights) yields the same result within statistics. This property can be easily verified by running the program with the same [powheg.input]{} files that differs only for the presence of the [[withnegweights]{}]{} flag set to 1, or for the presence of the [[bornsuppfact]{}]{} token, or, again, differing only in the folding numbers. In the current version of the [[POWHEG BOX]{}]{} we leave this choice to the user. The performance cost of getting positive weights may be well balanced if the events have to be run through costly detector simulators, or if it is undesirable to have negative weighted events in some complex, multivariate statistical analysis. The default [[POWHEG BOX]{}]{} setting is with no negative weights and no weighted events (i.e. $F({k_{{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}}\rm T}})=1$). Comparison with [[MCFM]{}]{} ---------------------------- As a final check of the code, we have compared the NLO output of the [[POWHEG BOX]{}]{} to the output produced by the NLO code from [[MCFM]{}]{} [@mcfm], and total agreement was found. In spite of the fact that we have taken the virtual correction formulae (that we have checked against the [[BlackHat]{}]{} [@Berger:2008sj] results too) from the [[MCFM]{}]{} program, this comparison is a highly non-trivial check, since the subtraction schemes used by the two programs are completely different. Phenomenology {#sec:phenomenology} ============= In this section we directly compare the output of our program to distributions that have been measured by the CDF and D0 Collaborations at the Tevatron. The only aim of this study is to validate, to some extent, our program. We believe that a more thorough analysis can only be performed by the experimental collaborations themselves, using our event generator. All the results displayed in this section have been obtained with a sample of roughly 1.3 million events, at the center-of-mass energy of $1.96$ TeV, in proton-antiproton collisions. The sample was generated with positive weights, with the folding parameters 5-10-5 (see section \[sec:posnegfold\]). The generation cut was set to $5$ GeV. The fraction of negative-weight events generated with this setting of parameters was 0.4%. As shown in section \[sec:posnegfold\], this fraction is concentrated in events with low transverse momenta. We have used the CTEQ6M pdf set, with the corresponding value of ${\Lambda_{\rm\scriptscriptstyle QCD}}$. The renormalization and factorization scales for the calculation of the $\bar{B}$ function are set equal to the $Z$ transverse momentum in the underlying Born configuration. The events were showered using [[PYTHIA]{}]{} 6.4.21 [@Sjostrand:2006za]. We have compared two different choices of tune for [[PYTHIA]{}]{}: the tune A ([ PYTUNE(100)]{}), that uses the old shower and underlying event model, and the Perugia 0 tune ([PYTUNE(320)]{}), that uses the new transverse-momentum ordered shower and the new underlying event model. Results for the Tune A model will be displayed as blue dashed histograms, while the Perugia 0 will be shown as red solid lines. The data will be displayed as simple points with error bars. We have switched off photon radiation off leptons ([mstj(41)=3]{}), so that, in the case of $Z\to {\ell^+ \ell^-}$, the lepton energy better represents what would be measured in an electromagnetic calorimeter. In addition we have set the $Z$ mass and width to the values 91.188 GeV and 2.486 GeV respectively, $\sin^2\theta_W^{\rm eff}=0.2312$ and $\alpha^{-1}_{\rm em}(M_Z)=128.930$. We applied the jet algorithm to all particles in the event, including all leptons, except for those coming from $Z$ decay. In other words, when comparing to experimental results, we assume that the jet energies are fully corrected to the particle level, including those particles that would not be visible in the detector. We have used the jet algorithms as implemented in the [FASTJET]{} package [@Cacciari:2005hq]. CDF results ----------- The CDF Collaboration provided and still provides results for $Z/\gamma\ (\to e^+ e^-) +1j$ and $Z/\gamma\ (\to \mu^+ \mu^-) +1j$ events. In order to perform an analysis as similar as possible to the one done by the CDF Collaboration, we used the midpoint algorithm [@Abulencia:2005yg] to combine hadrons (from [[POWHEG]{}]{} events showered by [[PYTHIA]{}]{}) into jets, with cone radius $R=0.7$ and a merging/splitting fraction of 0.75, starting from seed towers with transverse momenta above 1 GeV. ### $\boldsymbol{Z/\gamma \ (\to e^+ e^-)}$ + jets {#boldsymbolzgamma-to-e-e--jets .unnumbered} We begin by considering the $Z/\gamma\to e^+ e^-$ results of CDF. In figures \[fig:cdfepem1\], \[fig:cdfepem2\] and \[fig:cdfepem3\] we compare the [[POWHEG]{}]{} results showered by [[PYTHIA]{}]{} using Tune A (blue dashed lines) and Perugia 0 tuning (red solid lines), with the CDF data. \ The results in figs. \[fig:cdfepem1\] and \[fig:cdfepem2\] were published in [@Aaltonen:2007cp], while those in fig. \[fig:cdfepem3\] were extracted from the blessed results in [@cdf:webpage], at an integrated luminosity of 2.5 fb$^{-1}$. In figs. \[fig:cdfepem1\] and \[fig:cdfepem2\] we plot results for the inclusive total cross section and inclusive ${p_{{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}}\rm T}}$ distributions for the production of at least one and two jets. In fig. \[fig:cdfepem3\] we plot the ${p_{{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}}\rm T}}$ distribution of the hardest and next-to-hardest jet and the inclusive rapidity distributions for events with at least one and two jets. In order to compare with CDF data, we adopted the following cuts \[eq:CDF\_epem\_cuts\] &&66 [GeV]{} &lt; M\_[ee]{} &lt; 116 [GeV]{},p\_[T]{}\^e &gt; 25 [GeV]{}, |\^[e\_1]{}|&lt; 1.0, 1.2 &lt; |\^[e\_2]{}|&lt; 2.8,\ &&|y\^[jet]{}|&lt;2.1, p\_[T]{}\^[jet]{} &gt; 30 [GeV]{}, R\_[e, [jet]{}]{}&gt;0.7, where $y$ and $\eta$ represent the rapidity and pseudorapidity of the specified particles, and where $R$ is the distance in the azimuth-rapidity plane. We notice the good agreement between the [[POWHEG]{}]{} prediction and the data. It parallels the agreement between data and the NLO [[MCFM]{}]{} result displayed in refs. [@Aaltonen:2007cp; @cdf:webpage], despite the fact that, when more than two jets are considered, [[MCFM]{}]{} has NLO accuracy, while our generator is limited to leading order. However, we emphasize that the [[POWHEG]{}]{} results are directly compared to data, while the [[MCFM]{}]{} ones are first corrected by parton-to-hadron correction factors, as detailed in [@Aaltonen:2007cp]. Notice also the dependence of the results from the chosen tune of [[PYTHIA]{}]{}. The Perugia 0 tune seems to give a slightly better agreement with data. We point out that the differences between the [[POWHEG]{}]{} results and the data is of the same order of the differences between the two tunes, thus suggesting that, by directly tuning the [[POWHEG]{}]{} results to data, one may get an even better agreement. ### $\boldsymbol{Z/\gamma\ (\to \mu^+ \mu^-)}$ + jets {#boldsymbolzgamma-to-mu-mu--jets .unnumbered} \ Similar studies for the $Z/\gamma$ decaying in the $\mu^+ \mu^-$ channel were also performed by CDF. In figs. \[fig:cdfmupmum1\] and \[fig:cdfmupmum2\] we display the total cross section for inclusive jet production and the inclusive ${p_{{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}}\rm T}}$ and rapidity distributions for events with at least one and two jets. In order to perform an analysis as close as possible to the CDF experimental settings, we have applied the following cuts \[eq:CDF\_mupmum\_cuts\] &&66 [GeV]{} &lt; M\_ &lt; 116 [GeV]{},p\_[T]{}\^&gt; 25 [GeV]{}, |\^|&lt; 1.0,\ &&|y\^[jet]{}|&lt;2.1, p\_[T]{}\^[jet]{} &gt; 30 [GeV]{}, R\_[, [jet]{}]{}&gt;0.7. In this case, no clear conclusions can be drawn on which of the two chosen tunes better reproduces the data. D0 results ---------- The D0 Collaboration performed analyses similar to those done by the CDF Collaboration, focusing on more exclusive jet cross sections, and considering also some angular distributions. To perform an analysis as similar as possible to the one done by the D0 Collaboration, we used the D0 Run II iterative seed-based cone jet algorithm [@D0:runIIjetalgo] in order to recombine hadrons into jets, with a splitting/merging fraction of 0.5 and a cone radius $R=0.5$. ### $\boldsymbol{Z/\gamma \ (\to \mu^+ \mu^-)}$ + jets {#boldsymbolzgamma-to-mu-mu--jets-1 .unnumbered} In ref. [@Abazov:2008ez; @Abazov:2009pp], several distributions were studied by the D0 experiment using the set of cuts \[eq:D0\_mupmum\_cuts\] &&65 [GeV]{} &lt; M\_ &lt; 115 [GeV]{},p\_[T]{}\^&gt; 15 [GeV]{}, |\^|&lt; 1.7,\ &&|y\^[jet]{}|&lt;2.8, p\_[T]{}\^[jet]{} &gt; 20 [GeV]{}, R\_[, [jet]{}]{}&gt;0.5 . \ \ In fig. \[fig:D0mupmum1\], we plot distributions of the transverse momentum and rapidity of the hardest jet and of the $Z$ boson. In fig. \[fig:D0mupmum2\], we show the azimuthal separation and rapidity separation between the $Z$ boson and the leading jet, for events with $p_{{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}}T}^Z > 25$ GeV and $p_{{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}}T}^Z > 45$ GeV, while in fig. \[fig:D0mupmum3\] we plot the absolute value of the average rapidity of the $Z$ and of the leading jet, $|y_{\rm boost}(Z+{\rm jet})|$, for events with $p_{{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}}T}^Z > 25$ GeV and $p_{{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}}T}^Z > 45$ GeV, where y\_[ boost]{}(Z+[jet]{}) = $y^Z + y^{\rm jet}$, $y^{\rm jet}$ being the rapidity of the leading jet. In ref. [@Abazov:2009pp], the D0 Collaboration plotted the normalized differential cross sections, in order to reduce the associated error bars. We have, instead, preferred to plot the differential cross sections itself, in order to appreciate how well [[POWHEG]{}]{} reproduce not only the shapes but also the overall normalization. The total cross section they use to normalize is given in [@Abazov:2008ez] and is equal to \[eq:sigZ\] \_Z=1180.5([stat.]{})4([syst.]{})4([muon]{}) 7([lumi.]{}) [ pb]{}. In addition, we have added, in quadrature, a flat systematic luminosity error of 8% to every bin. We see a noticeable discrepancy between data and the [[POWHEG]{}]{} prediction in the shape of the transverse-momentum spectra of the jet and of the $Z$. This discrepancy also manifest itself in differences in the normalization of the angular distributions in figs. \[fig:D0mupmum2\] and \[fig:D0mupmum3\], since transverse momentum cuts are applied there. We do not wish to comment further on this problem, that, on the other hand, is not present in the CDF analyses, and in the D0 analysis of the $Z/\gamma\to e^+e^-$ channel. We remark that the problem may well be due to our failure to understand some features of the D0 analysis, rather than to problems in the data or in [[POWHEG]{}]{}. We also notice that the angular correlation between the $Z$ and the jet ${p_{{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}}\rm T}}$ is only qualitatively described by [[POWHEG]{}]{}. We remind the reader, however, that [[POWHEG]{}]{} has only LO accuracy for this quantity, that is determined by the emission of a second hard parton not included in the hardest jet, and that near the back-to-back region, Sudakov resummation effects, as well as non perturbative effects, become determinant. ### $\boldsymbol{Z/\gamma\ (\to e^+ e^-)}$ + jets {#boldsymbolzgamma-to-e-e--jets-1 .unnumbered} D0 has published studies for the $Z/\gamma\ (\to e^+ e^-)+1j$ channel in ref. [@Abazov:2009av]. \ In fig. \[fig:d0epem\] we have plotted the ${p_{{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}}\rm T}}$ distributions of the hardest, next-to-hardest and next-to-next-to-hardest jet. No information is given in the paper on the value of the total inclusive $Z$ cross section used to normalized the distributions. It is reasonable to use the same total cross section of eq. (\[eq:sigZ\]). The cuts that we have applied for this analysis were the following &&65 [GeV]{} &lt; M\_[ee]{} &lt; 115 [GeV]{},p\_[T]{}\^e &gt; 25 [GeV]{}, |\^[e]{}|&lt; 1.1  [or]{}  1.5 &lt; |\^[e]{}|&lt; 2.5,\ &&|y\^[jet]{}|&lt;2.5, p\_[T]{}\^[jet]{} &gt; 20 [GeV]{}. We can see quite a good agreement among the data and the two [[POWHEG]{}]{} distributions for the hardest and next-to-hardest jet. Unexpectedly, there is quite a good agreement for the next-to-next-to-hardest jet too: in fact, this jet is a shower jet, so we do not expect it to be correct away from the collinear limit. LHC results ----------- In this section, we show a few results for the LHC at 14 TeV. We have produced 1.400.000 events, generated with the [[withnegweights]{}]{} flag set to 1, with ${k_{\rm gen}}=5$ GeV and with a 5-10-5 folding. With this folding, the fraction of negative weights is around 1.6%. We have then showered the events generated by [[POWHEG]{}]{} with [[PYTHIA]{}]{}, using the tune A and the Perugia 0 tuning. \ \ In figs. \[fig:LHC\_CDF\_epem\] and \[fig:LHC\_D0\_mupmum\] we have plotted a few distributions obtained for the LHC, using the same cuts and the same jet algorithms used by the two Collaborations at the Tevatron: in fig. \[fig:LHC\_CDF\_epem\], we have applied the cuts of eq. (\[eq:CDF\_epem\_cuts\]) that CDF used for the $Z/\gamma \ (\to e^+ e^-) +1j$ analysis, while in fig. \[fig:LHC\_D0\_mupmum\], we have applied the cuts of eq. (\[eq:D0\_mupmum\_cuts\]), that D0 used for the $Z/\gamma \ (\to \mu^+\mu^-)+1j$ study. In fig. \[fig:LHC\_CDF\_epem\] we can see a pattern of the two tunes similar to the one depicted in the corresponding fig. \[fig:cdfepem3\]: small differences for the distribution of the hardest jet, some differences both at low and high ${p_{{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}}\rm T}}$ for the next-to-hardest jet, and increasing differences in their inclusive rapidity distributions, with peaks of $\sim\! 20$%. Similarly, fig. \[fig:LHC\_D0\_mupmum\] should be compared with the corresponding panels in figs. \[fig:D0mupmum1\] and \[fig:D0mupmum2\]. Tune A gives a higher cross section at low transverse momentum and central rapidity of the vector boson with respect to the Perugia 0 tuning, while they agree very well in the azimuthal separation and rapidity separation between the $Z$ boson and the leading jet, for events with $p_{{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}}T}^Z > 25$ GeV. Conclusions {#sec:conclusions} =========== In this paper, we have presented, for the first time, a calculation of vector boson plus one jet NLO cross section interfaced to a Shower Monte Carlo program, within the [[POWHEG]{}]{} framework. More specifically, we have implemented this process using the [[POWHEG BOX]{}]{}, a general computer code framework for embedding NLO calculations in shower Monte Carlo programs according to the [[POWHEG]{}]{} method. We have compared the [[POWHEG]{}]{} results for $Z/\gamma+1j$ with the data available from the CDF and the D0 Collaborations, using two different tunes for [[PYTHIA]{}]{}. We notice that differences between the [[POWHEG]{}]{} results and the data is of the same order of the differences between the two tunes, thus suggesting that, by directly tuning the [[POWHEG]{}]{} results to data, one may get an even better agreement. Finally, we have presented similar results for the LHC, running at 14 TeV, showing that, in some distributions, the difference between the two tunes is of the order of 20%. The code of our generator can be accessed in the [[POWHEG BOX]{}]{} svn repository:\ <svn://powhegbox.mib.infn.it/trunk/POWHEG-BOX>, with username [ anonymous]{} and password [anonymous]{}. We thank John Campbell for helping us to access the virtual corrections in the [[MCFM]{}]{} code. We also thank Gavin Hesketh, Sabine Lammers and Henrik Nilsen for useful discussions on the experimental side. E.R. thanks Daniel Maître for his help comparing [[BlackHat]{}]{} and [[MCFM]{}]{} results. The work of S.A. has been supported in part by the Deutsche Forschungsgemeinschaft in SFB/TR 9. S.A. and E.R. would like to thank DESY Zeuthen for the use of the computing facilities and the hospitality, and CERN, where part of this work was performed. [^1]: In the rest of the paper, for ease of notation, we will refer to the $Z/\gamma+1j$ process simply as “$Z+1j$ production”, without mentioning the presence of the photon, whose effects, together with all the spin correlations of the decay products, have been fully taken into account.

--- abstract: 'The simplest modeling of planar quantum waveguides is the Dirichlet eigenproblem for the Laplace operator in unbounded open sets which are uniformly thin in one direction. Here we consider V-shaped guides. Their spectral properties depend essentially on a sole parameter, the opening of the V. The free energy band is a semi-infinite interval bounded from below. As soon as the V is not flat, there are bound states below the free energy band. There are a finite number of them, depending on the opening. This number tends to infinity as the opening tends to $0$ (sharply bent V). In this situation, the eigenfunctions concentrate and become self-similar. In contrast, when the opening gets large (almost flat V), the eigenfunctions spread and enjoy a different self-similar structure. We explain all these facts and illustrate them by numerical simulations.' address: 'Labortatoire IRMAR, UMR 6625 du CNRS, Campus de Beaulieu 35042 Rennes cedex, France' author: - 'Monique Dauge,  Yvon Lafranche,  Nicolas Raymond' title: Quantum waveguides with corners --- fig4tex.tex (update=yes) \#1\#2\#3\#4\#5[ 102 = \#2 /\#1, \#3 / 111 \[\#2,\#1\] 112 \[111\] 104 = \#2 /\#4, 112/ 105 = 104 /\#1, \#3 / (color=\#5) (width=1.2) \#1 ; ]{} Introduction {#introduction .unnumbered} ============ A quantum waveguide refer to nanoscale electronic device with a wire or thin surface shape. In the first case, one speaks of a quantum wire. The electronic density is low enough to allow a modeling of the system by a simple one-body Schrödinger operator with potential $$\psi\ \longmapsto\ -\Delta\psi + V\psi\quad\mbox{in}\quad\xR^3.$$ The structure of the device causes the potential to be very large outside and very small inside the device. As a relevant approximation, we can consider that the potential is zero in the device and infinite outside ; this can be described by a Dirichlet operator $$\psi\ \longmapsto\ -\Delta\psi \ \ \mbox{in}\ \ \Omega \quad\mbox{and}\quad\psi=0 \ \ \mbox{on}\ \ \partial\Omega$$ where $\Omega$ is the open set filled by the device. We refer to [@BDPR] where we can see, at least on the numerical simulations, the analogy between a problem with a confining potential and a Dirichlet condition. These kinds of device are intended to drive electronic fluxes. But their shape may capture some bound states, eigenpairs of the Dirichlet problem: $$-\Delta\psi=\lambda\psi \ \ \mbox{in}\ \ \Omega \quad\mbox{and}\quad\psi=0 \ \ \mbox{on}\ \ \partial\Omega .$$ The topic of this paper is two-dimensional wire shaped structures, structures which coincide with strips of the form $\xR_+\times(0,\alpha)$ outside a ball of center ${\bf 0}$ and radius $R$ large enough. These structures can be called *planar waveguides*. More specifically, there are the *bent waveguides* and the *broken waveguides*: Bent waveguides have a constant width around some central smooth curve, see Figure \[F:0a\], and the central curve of a broken waveguide is a broken line, see Figure \[F:0b\]. 1:( 0, 0) 2:( -5, 10) 3:( 50, 0) 12 = 2 /1, 0.75/ 22 = 2 /1, 1.25/ 1:( 0, 0) 2:( -0.05, 0.1) 3:( 50, 0) 4:( 25, 0) 30 (1,0) 11 = 1,2,3 /6, 30/ 21 = 1,2,3 /12, 30/ Due to the semi-infinite strips contained in such waveguide, the spectrum of the Laplacian $-\Delta$ with Dirichlet conditions is not discrete: It contains a semi-infinite interval of the form $[\mu,+\infty)$ which is the energy band where electronic transport can occur. The presence of discrete spectrum at lower energy levels is not obvious, but nevertheless, frequent. A remarkable result by Duclos and Exner [@Duclos95] (and generalized in [@Duclos05]) tells us that if the mid-line of a planar waveguide is smooth and straight outside a compact set, then there exists bound states as soon as the line is not straight everywhere. For broken guides, a similar result holds, [@Exner89; @ABGM91]: There exist bound states as soon as the guide is not a straight strip. The question of the increasing number of such bound states for sharply bent broken waveguides has been answered in [@CLMM93]. Before stating the main results of this paper, let us mention a few other works involving broken waveguides such as [@FriSolo08; @FriSolo09]. It also turns out that our analysis leads to the investigation of triangles with a sharp angle as in [@Fre07] (see also [@BoFre10]). The generalization to dimension 3 of the broken strip arises in [@ExTa10] where a conical waveguide is studied, whereas a Born-Oppenheimer approach is in progress in [@Ourm11]. In this paper, we revisit the results of [@ABGM91; @CLMM93] and prove several other quantitative or qualitative properties of the eigenpairs of planar broken waveguides. Here are the contents of the present work: In section \[s:1\] we recall from the literature notions of unbounded self-adjoint operators, discrete and essential spectrum, Rayleigh quotients. After proving that the Rayleigh quotients are increasing functions of the opening angle $\theta$ of the guide (section \[s:3\]), we adapt the technique of [@Duclos05] to give a self-contained proof of the existence of bound states (section \[s:4\]). We prove that the number of bound states is always finite, though depending on the opening angle $\theta$ (section \[s:5\]), that this number tends to infinity like the inverse $\theta^{-1}$ of the opening angle when $\theta\to0$ (section \[s:7\]). Concerning eigenvectors we prove that they satisfy an even symmetry property with respect to the symmetry axis of the guide (section \[s:2\]). Their decay along the semi-infinite straight parts of the guide can be precisely evaluated (section \[s:6\]) and is stronger and stronger when $\theta$ decreases. We perform numerical computations by the finite element method, which clearly illustrate these decay properties. When the opening gets small, concentration and self-similarity appears, which can be explained by a semi-classical analysis: We give some overview of the asymptotic expansions established in our other paper [@DauRay11]. We end this work by evaluations of the numerical convergence of the algorithms used for our finite element computations (section \[s:9\]). [Notation.]{} The $\xLtwo$ norm on an open set $\mathcal{U}$ will be denoted by $\|\cdot\|_{\mathcal{U}}$. Unbounded self-adjoint operators {#s:1} ================================ In this section we recall from the literature some definitions and fundamental facts on unbounded self-adjoint operators and their spectrum. We quote the standard book of Reed and Simon [@ReSi1 Chapter VIII] and, for the readers who can read french, the book of Lévy-Bruhl [@L-B03] (see in particular Chapter 10 on unbounded operators). Let $H$ be a separable Hilbert space with scalar product $\langle\cdot,\cdot\rangle_H$. We will consider operators $A$ defined on a dense subspace ${\mathsf{Dom}}(A)$ of $H$ called the *domain of $A$*. The adjoint $A^*$ of $A$ is the operator defined as follows: 1. The domain ${\mathsf{Dom}}(A^*)$ is the space of the elements $u$ of $H$ such that the form $${\mathsf{Dom}}(A)\ni v\mapsto \langle u,Av\rangle_H$$ can be extended to a continuous form on $H$. 2. For any $u\in{\mathsf{Dom}}(A^*)$, $A^*u$ is the unique element of $H$ provided by the Riesz theorem such that $$\label{DomA*} \forall v\in{\mathsf{Dom}}(A),\quad \langle A^*u,v\rangle_H = \langle u,Av\rangle_H$$ The operator $A$ with domain ${\mathsf{Dom}}(A)$ is said *self-adjoint* if $A=A^*$, which means that $$\label{Sym} {\mathsf{Dom}}(A^*)={\mathsf{Dom}}(A) \quad\mbox{and}\quad \forall u,v\in{\mathsf{Dom}}(A),\quad \langle Au,v\rangle_H = \langle u,Av\rangle_H.$$ Operators in variational form ----------------------------- The operators that we will consider can be defined by variational formulation. Let us introduce the general framework first. Let be given two separable Hilbert spaces $H$ and $V$ with continuous embedding of $V$ into $H$ and such that $V$ is dense in $H$. Let $b$ be an hermitian sesquilinear form on $V$ $$b: V\times V \ni (u,v) \mapsto b(u,v)\in\xC$$ which is assumed to be continuous and coercive: This means that there exist three real numbers $c$, $C$ and $\Lambda$ such that $$\label{eq:1-3} \forall u\in V,\quad c\|u\|^2 _V\le b(u,u) + \Lambda\langle u,u\rangle_H \le C\|u\|^2_V\,.$$ Let ${\underline{A}}$ be the operator defined from $V$ into its dual $V'$ by the natural expression $$\forall v\in V,\quad \langle {\underline{A}}u,v\rangle_H = b(u,v).$$ In other words, for all $u\in V$, ${\underline{A}}u$ is the linear form $v\mapsto b(u,v)$. Note that the operator ${\underline{A}}+\Lambda\,{\mathrm{Id}}$ is associated in the same way to the sesquilinear form $$b + \Lambda \langle \cdot,\cdot\rangle_H : V\times V \ni (u,v) \mapsto b(u,v) + \Lambda\langle u,v\rangle_H\in\xC$$ which is strongly elliptic by . As a consequence of the Riesz theorem (or the more general Lax-Milgram theorem) there holds $$\label{eq:1-4} {\underline{A}}+\Lambda\,{\mathrm{Id}}\quad\mbox{is an isomorphism from}\ \ V \ \ \mbox{onto}\ \ V'.$$ This situation provides many examples of (unbounded) self-adjoint operators. The following lemma is related to the Friedrichs’s lemma [@L-B03 Section 10.7]. \[lem:1-1\] Let ${\underline{A}}$ be the operator associated with an hermitian sesquilinear form $b$ coercive on $V$. Let $A$ be the restriction of the operator ${\underline{A}}$ on the domain $${\mathsf{Dom}}(A) = \{u\in V:\quad {\underline{A}}u \in H\}.$$ Then $A$ is self-adjoint. The operator $A$ is symmetric because $b$ is hermitian. Moreover we check immediately that $$\forall u,v\in {\mathsf{Dom}}(A),\quad \langle A u,v\rangle_H = \langle u,Av\rangle_H = b(u,v).$$ In particular, we deduce ${\mathsf{Dom}}(A)\subset{\mathsf{Dom}}(A^*)$. Let us prove that ${\mathsf{Dom}}(A^*)\subset{\mathsf{Dom}}(A)$. Since the domain of $A$ and $A^*$ are unchanged by the addition of $\Lambda\,{\mathrm{Id}}$, and since $$(A+\Lambda\,{\mathrm{Id}})^*=A^*+\Lambda\,{\mathrm{Id}}$$ we can consider $A+\Lambda\,{\mathrm{Id}}$ instead of $A$, or, in other words, using , assume that ${\underline{A}}$ is bijective. Then we deduce that $A$ is an isomorphism from ${\mathsf{Dom}}(A)$ onto $H$: Indeed, $A$ is injective because ${\underline{A}}$ is injective; If $f\in H$, there exists $u\in V$ such that ${\underline{A}}u = f$, and $u\in{\mathsf{Dom}}(A)$ by the very definition of ${\mathsf{Dom}}(A)$. Let $w$ belong to ${\mathsf{Dom}}(A^*)$. This means, , that $w\in H$ and $A^*w=f\in H$. Let us prove that $w$ belongs to ${\mathsf{Dom}}(A)$. Since $A$ is bijective there exists $u\in {\mathsf{Dom}}(A)$ such that $A u = f$. We have $$\forall v\in V,\quad b(u,v) = \langle f,v\rangle_H$$ and therefore $$\forall v\in {\mathsf{Dom}}(A),\quad \langle u,Av\rangle_H = \langle w,Av\rangle_H.$$ Hence, as $A$ is bijective, for all $g\in H$, $\langle u,g\rangle_H = \langle w,g\rangle_H$. Finally $u=w$, which ends the proof. \[ex:1-1\] Let $\Omega$ be an open set in $\xR^n$ and $\partial_{\mathsf{Dir}}\Omega$ a part of its boundary. On $$V = \{\psi\in {\mathrm{H}}^1(\Omega):\quad \psi=0 \ \ \mbox{on}\ \ \partial_{\mathsf{Dir}}\Omega\}$$ we consider the bilinear form $$b(\psi,\psi') = \int_\Omega \nabla\psi({\mathbf{x}}) \cdot \nabla\psi'({\mathbf{x}})\, \xdif{\mathbf{x}}.$$ The operator $A$ is equal to $-\Delta$ and it is self-adjoint on $H=\xLtwo(\Omega)$ with domain $${\mathsf{Dom}}(A) = \{\psi\in V:\quad \Delta \psi\in \xLtwo(\Omega)\quad\mbox{and}\quad \partial_n\psi=0 \ \ \mbox{on}\ \ \partial\Omega\setminus\partial_{\mathsf{Dir}}\Omega\}.$$ Discrete and essential spectrum ------------------------------- Let $A$ be an unbounded self-adjoint operator on $H$ with domain ${\mathsf{Dom}}(A)$. We recall the following characterizations of its spectrum $\sigma(A)$, its essential spectrum $\sigma_{\mathsf{ess}}(A)$ and its discrete spectrum $\sigma_{\mathsf{dis}}(A)$: - Spectrum: $\lambda\in\sigma(A)$ if and only if $(A-\lambda \,{\mathrm{Id}})$ is not invertible from ${\mathsf{Dom}}(A)$ onto $H$, - Essential spectrum: $\lambda\in\sigma_{\mathsf{ess}}(A)$ if and only if $(A-\lambda \,{\mathrm{Id}})$ is not Fredholm[^1] from ${\mathsf{Dom}}(A)$ into $H$ (see [@ReSi1 Chapter VI] and [@L-B03 Chapter 3]), - Discrete spectrum: $\sigma_{\mathsf{dis}}(A) := \sigma(A)\setminus\sigma_{\mathsf{ess}}(A)$. We list now several fundamental properties of essential and discrete spectrum. \[Weyl\] We have ${\lambda}\in\sigma_{{\mathsf{ess}}}(A)$ if and only if there exists a sequence $(u_{n})\in {\mathsf{Dom}}(A)$ such that $\|u_{n}\|_{H}=1$, $(u_{n})$ has no subsequence converging in $H$ and $(A-{\lambda}\,{\mathrm{Id}})u_{n}\underset{n\to+\infty}{\to }0$ in $H$. From this lemma, one can deduce (see [@L-B03 Proposition 2.21 and Proposition 3.11]): The discrete spectrum is formed by isolated eigenvalues of finite multiplicity. \[stab-ess\] The essential spectrum is stable under any perturbation which is compact from ${\mathsf{Dom}}(A)$ into $H$. \[ex:1-5\] Let $A$ be the self-adjoint operator on $H$ associated with an hermitian sesquilinear form $b$ coercive on $V$, Lemma \[lem:1-1\]. Let us assume that $V$ is compactly embedded in $H$. Then the spectrum of $A$ is discrete and formed by a non-decreasing sequence $\nu_k$ of eigenvalues which tend to $+\infty$ as $k\to+\infty$ (see [@L-B03 Chapter 13]). Let $(v_k)_{k\ge1}$ be an associated orthonormal basis of eigenvectors: $$Av_k = \nu_k v_k,\quad \forall k\ge1.$$ Then we have the following identities $$\begin{gathered} \label{eq:nu0} \forall u\in H,\quad \|u\|^2_H = \sum_{k\ge1} \big|\big\langle u, v_k\big\rangle_H\big|^2, \\ \label{eq:nu1} \forall u\in V,\quad b(u,u) = \sum_{k\ge1} \nu_k \big|\big\langle u, v_k\big\rangle_H\big|^2, \\ \label{eq:nu2} \forall u\in {\mathsf{Dom}}(A),\quad \|Au\|^2_H = \sum_{k\ge1} \nu_k^2\big|\big\langle u, v_k\big\rangle_H\big|^2 .\end{gathered}$$ \[ex:1-2\] Let us define $\Delta^{\mathsf{Dir}}_\Omega$ as the Dirichlet problem for the Laplace operator $-\Delta$ on an open set $\Omega\subset\xR^n$, Example \[ex:1-1\] with $\partial_{\mathsf{Dir}}\Omega = \partial\Omega$. *(i)* If $\Omega$ is bounded, $\Delta^{\mathsf{Dir}}_\Omega$ has a purely discrete spectrum, which is an increasing sequence of positive numbers. *(ii)* Let us assume that there is a compact set $K$ such that $$\Omega\setminus K = \bigcup_{j\ \rm finite} \Omega_j \quad\mbox{(disjoint union)}$$ where $\Omega_j$ is isometrically affine to a half-tube $\Sigma_j = (0,+\infty)\times \omega_j$, with $\omega_j$ bounded open set in $\xR^{n-1}$. Let $\mu_j$ be the first eigenvalue of the Dirichlet problem for the Laplace operator $-\Delta$ on $\omega_j$. Then, we have: $$\sigma_{\mathsf{ess}}(\Delta^{\mathsf{Dir}}_\Omega) = \cup_j [\mu_j,+\infty) = [\min_j \mu_j, +\infty).$$ The proof can be organized in two main steps. Firstly, for each $j$ we construct Weyl sequences supported in $\Sigma_j$ associated with any $\lambda>\mu_{j}$, which proves that $\sigma_{{\mathsf{ess}}}(\Delta^{\mathsf{Dir}}_\Omega)\subset[\min_j \mu_j, +\infty)$. Secondly we apply Lemma \[stab-ess\] with $A=B-C$ where $A=\Delta^{\mathsf{Dir}}_\Omega$ and $B=\Delta^{\mathsf{Dir}}_{\Omega}+W$, where $W$ is a non negative and smooth potential which is compactly supported and such that $W\geq \min_{j}\mu_{j}$ on $K$. On one hand, since $C$ is compact, we get that $\sigma_{{\mathsf{ess}}}(A)=\sigma_{{\mathsf{ess}}}(B)$. On the other hand, we notice that: $$\int_{\Omega} \|\nabla\psi\|^2\, \xdif{\mathbf{x}}+\int_{\Omega}W |\psi|^2\, \xdif{\mathbf{x}}\geq \int_{\Omega} \|\nabla\psi\|^2\, \xdif{\mathbf{x}}+\min_{j}\mu_{j}\int_{K} |\psi|^2\, \xdif{\mathbf{x}}$$ and, using the Poincaré inequality with respect to the transversal variable in each strip $\Sigma_j$: $$\int_{\Omega} \|\nabla\psi\|^2\, \xdif{\mathbf{x}}\geq\sum_{j}\int_{\Sigma{j}} \|\nabla\psi\|^2\, \xdif{\mathbf{x}}\geq\sum_{j}\int_{\Sigma{j}} \mu_{j} |\psi|^2\, \xdif{\mathbf{x}}\geq\min_{j}\mu_{j}\int_{\Omega\setminus K}|\psi|^2.$$ We infer that: $$\int_{\Omega} \|\nabla\psi\|^2\, \xdif{\mathbf{x}}+\int_{\Omega}W |\psi|^2\, \xdif{\mathbf{x}}\geq \min_{j}\mu_{j}\int_{\Omega}|\psi|^2 \, \xdif{\mathbf{x}}.$$ The min-max principle provides that $\inf\sigma (B)\geq \min_{j}\mu_{j}$ so that $\inf\sigma_{{\mathsf{ess}}} (B)\geq \min_{j}\mu_{j}$, and finally we get $\displaystyle{\inf\sigma_{{\mathsf{ess}}} (A)\geq \min_{j}\mu_{j}}$. For an example of this technique, we refer for instance to [@Duclos05 Section 3.1]. Let us notice that the Persson’s theorem provides a direct proof (see [@Persson60] and [@FouHel10 Appendix B]). The same formula holds even if the boundary conditions on $\partial\Omega\cap K$ are mixed Dirichlet-Neumann. Rayleigh quotients ------------------ We recall now the definition of the Rayleigh quotients of a self-adjoint operator $A$ (see [@L-B03 Proposition 6.17 and 13.1]). The Rayleigh quotients associated to the self-adjoint operator $A$ on $H$ of domain ${\mathsf{Dom}}(A)$ are defined for all positive natural number $j$ by $$\lambda_j = {\mathop{\operatorname{\vphantom{p}inf}}}_{ \substack {u_1,\ldots,u_j\in {\mathsf{Dom}}(A)\\ \mbox{\footnotesize independent}}} \ \ \sup_{u\in[u_1,\ldots,u_j]} \frac{\langle Au,u\rangle_H} {\langle u,u\rangle_H} .$$ Here $[u_1,\ldots,u_j]$ denotes the subspace generated by the $j$ independent vectors $u_1,\ldots,u_j$. The following statement gives the relation between Rayleigh quotients and eigenvalues. \[th:1-1\] Let $A$ be a self-adjoint operator of domain ${\mathsf{Dom}}(A)$. We assume that $A$ is semi-bounded from below, i.e., there exists $\Lambda\in\xR$ such that $$\forall u\in{\mathsf{Dom}}(A),\quad \langle Au,u\rangle_H + \Lambda\langle u,u\rangle_H \ge 0.$$ We set $\gamma=\min\sigma_{\mathsf{ess}}(A)$. Then the Rayleigh quotients $\lambda_j$ of $A$ form a non-decreasing sequence and there holds 1. If $\lambda_j<\gamma$, it is an eigenvalue of $A$, 2. If $\lambda_j\ge\gamma$, then $\lambda_j=\gamma$, 3. The $j$-th eigenvalue $<\gamma$ of $A$ (if exists) coincides with $\lambda_j$. \[lem:1-2\] Let $A$ be the self-adjoint operator on $H$ associated with an hermitian sesquilinear form $b$ coercive on $V$, Lemma \[lem:1-1\]. Then the Rayleigh quotients of $A$ are equal to $$\lambda_j = {\mathop{\operatorname{\vphantom{p}inf}}}_{ \substack {u_1,\ldots,u_j\in V\\ \mbox{\footnotesize independent}}} \ \ \sup_{u\in[u_1,\ldots,u_j]} \frac{b(u,u)} {\langle u,u\rangle_H} .$$ \[co:lem1-2\] Let $A$ and $\hat A$ be the self-adjoint operators on $H$ and $\hat H$ associated with the hermitian sesquilinear forms $b$ and $\hat b$ coercive on $V$ and $\hat V$, respectively. We assume that $$\hat H\subset H,\quad \hat V\subset V,\quad \hat b(u,u) \ge b(u,u) \ \forall u\in\hat V.$$ Let $\lambda_j$ and $\hat \lambda_j$ be the Rayleigh quotients associated to $A$ and $\hat A$, respectively. Then $$\forall j\ge1,\quad \hat \lambda_j \ge \lambda_j \,.$$ \[ex:1-3\] This consists in choosing a finite dimensional subspace $\hat V$ of $V$, which also defines a subspace of $H$, and $\hat b =b$. The Rayleigh quotients $\hat \lambda_j$ are the eigenvalues of the discrete operator $\hat A$, and they are larger than the Rayleigh quotients $\lambda_j$ of $A$. \[ex:1-4\] Let $\Omega$ be an open subset of $\xR^n$ and $\widehat \Omega\subset\Omega$. The extension by $0$ from $\widehat\Omega$ to $\Omega$ realizes a natural embedding of ${\mathrm{H}}^1_0(\widehat\Omega)$ into ${\mathrm{H}}^1_0(\Omega)$, and of $\xLtwo(\widehat\Omega)$ into $\xLtwo(\Omega)$. Then, with obvious notations: $$\forall j\ge1,\quad \lambda_j(\Delta^{\mathsf{Dir}}_{\widehat\Omega}) \ge \lambda_j(\Delta^{\mathsf{Dir}}_{\Omega}) .$$ The Neumann problem consists in taking $H^1(\Omega)$ as variational space. The argument above does not work because there is no canonical embedding of $H^1(\widehat\Omega)$ into $H^1(\Omega)$. Moreover, the monotonicity with respect to the domain is wrong: *(i)* Let us choose $\Omega$ bounded and connected, and $\widehat \Omega$ a subset of $\Omega$ with two connected components. Then $$\lambda_1(\Delta^{\mathsf{Neu}}_{\widehat\Omega}) = \lambda_2(\Delta^{\mathsf{Neu}}_{\widehat\Omega}) = 0 \quad\mbox{and}\quad \lambda_1(\Delta^{\mathsf{Neu}}_{\Omega}) = 0,\ \lambda_2(\Delta^{\mathsf{Neu}}_{\Omega}) > 0.$$ *(ii)* If we take two embedded intervals for $\widehat\Omega$ and $\Omega$, then explicit calculations show that $\lambda_j(\Delta^{\mathsf{Neu}}_{\widehat\Omega}) \ge \lambda_j(\Delta^{\mathsf{Neu}}_{\Omega})$. Another nice and non trivial counter-example can be found with the de Gennes operator appearing in the superconductivity theory, [@DauHel]. The broken guide {#s:2} ================ Let us denote the Cartesian coordinates in $\xR^2$ by ${\mathbf{x}}=(x_1,x_2)$. The open sets $\Omega$ that we consider are unbounded plane V-shaped sets. The question of interest is the presence and the properties of bound states for the Laplace operator $\Delta=\partial^2_1+\partial^2_2$ with Dirichlet boundary conditions in such $\Omega$. We can assume without loss of generality that our set $\Omega$ is normalized so that - it has its non-convex corner at the origin ${\bf 0}=(0,0)$, - it is symmetric with respect to the $x_1$ axis, - its thickness is equal to $\pi$. The sole remaining parameter is the opening of the V: We denote by $\theta\in(0,\frac\pi2)$ the half-opening and by $\Omega_\theta$ the associated broken guide, see Figure \[F:1\]. We have $$\label{eq:Ot} \Omega_{\theta}=\left\{(x_{1},x_{2})\in\xR^2 : \quad x_{1}\tan\theta<|x_{2}|<\left(x_{1}+\frac{\pi}{\sin\theta}\right)\tan\theta\right\}.$$ Finally, like in Example \[ex:1-2\], we specify the *positive* Dirichlet Laplacian by the notation $\Delta^{\mathsf{Dir}}_{\Omega_{\theta}}$. This operator is an unbounded self-adjoint operator with domain $${\mathsf{Dom}}(\Delta^{\mathsf{Dir}}_{\Omega_{\theta}}) = \{\psi\in{\mathrm{H}}^1_0(\Omega_\theta):\quad \Delta\psi\in\xLtwo(\Omega_\theta)\}.$$ 1:( 0, 0) 2:( 45, 0) 3: = 2 /1, 30/ 4:(-30, 0) 5:(12, 0) 6: = 5 /1, 120/ 101 \[1,4\] 14 = 3/1, 101/ 23 = 3, 14 /1, 2/ 10:(60, 0) 10(9) 11 : 10(9) 1 ; 20 (0,30) arrowhead (fillmode=yes, length=2) (dash=4) 1 ; 6.5 (30,120) The boundary of $\Omega_\theta$ is not smooth, it is polygonal. The presence of the non-convex corner with vertex at the origin is the reason for the domain ${\mathsf{Dom}}(\Delta^{\mathsf{Dir}}_{\Omega_{\theta}})$ to be distinct from ${\mathrm{H}}^2\cap{\mathrm{H}}^1_0(\Omega_\theta)$. Nevertheless this domain can be precisely characterized as follows. Let us introduce polar coordinates $(\rho,\varphi)$ centered at the origin, with $\varphi=0$ coinciding with the upper part $x_2=x_1\tan\theta$ of the boundary of $\Omega_\theta$. Let $\chi$ be a smooth radial cutoff function with support in the region $x_{1}\tan\theta<|x_{2}|$ and $\chi\equiv1$ in a neighborhood of the origin. We introduce the explicit *singular function* $$\label{eq:sing} \psi_{\mathsf{sing}}(x_1,x_2) = \chi(\rho)\, \rho^{\pi/\omega} \sin\frac{\pi\varphi}{\omega},\quad \mbox{with}\quad \omega = 2(\pi-\theta).$$ Then there holds, the classical references [@Kondratev67; @Grisvard85]: $$\label{eq:dom} {\mathsf{Dom}}(\Delta^{\mathsf{Dir}}_{\Omega_{\theta}}) = {\mathrm{H}}^2\cap{\mathrm{H}}^1_0(\Omega_\theta) \oplus [\psi_{\mathsf{sing}}]$$ where $[\psi_{\mathsf{sing}}]$ denotes the space generated by $\psi_{\mathsf{sing}}$. Essential spectrum ------------------ \[P:ess\] For any $\theta\in(0,\frac\pi2)$ the essential spectrum of the operator $\Delta^{{\mathsf{Dir}}}_{\Omega_{\theta}}$ coincides with $[1,+\infty)$. This proposition is a consequence of Example \[ex:1-2\] *(ii)*: Outside a compact set, $\Omega_\theta$ is the union of two strips isometric to $(0,+\infty)\times(0,\pi)$. Since the first eigenvalue of $-\partial_y^2$ on ${\mathrm{H}}^1_0(0,\pi)$ is $1$, the proposition is proved. Symmetry -------- In our quest of bounded states $(\lambda,\psi)$ of $\Delta^{{\mathsf{Dir}}}_{\Omega_{\theta}}$, i.e. $$\label{eq:1} \begin{cases} \ -\Delta \psi = \lambda \psi &\quad \mbox{in}\quad \Omega_\theta, \\ \qquad \psi = 0 &\quad \mbox{on}\quad \partial\Omega_\theta \end{cases} \qquad\mbox{with}\quad \lambda<1,\quad\psi\neq 0,$$ we can reduce to the half-guide $\Omega^+_\theta$ defined as (see Figure \[F:2\]) $$\Omega^+_\theta = \{(x_{1},x_{2})\in\Omega_\theta : x_2>0\}, \qquad\mbox{with the Dirichlet part of its boundary}\ \ \partial_{\mathsf{Dir}}\Omega^+_{\theta} = \partial\Omega_{\theta} \cap \partial\Omega^+_{\theta},$$ as we are going to explain now. Let us introduce $\Delta^{\mathsf{Mix}}_{\Omega_{\theta}^+}$ as the positive Laplacian with mixed Dirichlet-Neumann conditions on $\Omega_{\theta}^+$ with domain, Example \[ex:1-1\]: $${\mathsf{Dom}}(\Delta^{\mathsf{Mix}}_{\Omega_{\theta}^+}) = \big\{ \psi\in H^1(\Omega_{\theta}^+) : \quad\Delta\psi\in L^2(\Omega_{\theta}^+),\quad \psi=0 \ \mbox{ on } \ \partial_{\mathsf{Dir}}\Omega^+_{\theta} \ \ \mbox{and}\ \ \partial_{2}\psi=0 \ \mbox{ on }\ x_{2}=0 \big\}.$$ Then $\sigma_{\mathsf{ess}}(\Delta^{\mathsf{Mix}}_{\Omega_{\theta}^+})$ coincides with $\sigma_{\mathsf{ess}}(\Delta^{\mathsf{Dir}}_{\Omega_{\theta}})$. Concerning the discrete spectrum we have: 1:( 0, 0) 2:( 50, 0) 3: = 2 /1, 30/ 4:(-14.1, 0) 10:(100, 0) 101 \[1,4\] 14 = 3/1, 101/ 23 = 3, 14 /1, 2/ 50 = 1, 3, 4, 14/-4, 101/ 10(9) 11 : 10(9) (dash=4) \[P:dis\] For any $\theta\in(0,\frac\pi2)$, $\sigma_{\mathsf{dis}}(\Delta^{{\mathsf{Dir}}}_{\Omega_{\theta}})$ coincides with $\sigma_{\mathsf{dis}}(\Delta^{\mathsf{Mix}}_{\Omega_{\theta}^+})$. The proof relies on the fact that $\Delta^{\mathsf{Dir}}_{\Omega_{\theta}}$ commutes with the symmetry $S : (x_1,x_2)\mapsto(x_1,-x_2)$. [*(i)*]{} If $({\lambda},\psi)$ is an eigenpair of $\Delta^{\mathsf{Mix}}_{\Omega_{\theta}^+}$, the even extension of $u_{\lambda}$ to $\Omega_\theta$ defines an eigenfunction of $\Delta^{\mathsf{Dir}}_{\Omega_{\theta}}$ associated with the same eigenvalue $\lambda$. Therefore we have the inclusion $$\sigma_{\mathsf{dis}}(\Delta^{\mathsf{Mix}}_{\Omega_{\theta}^+})\subset\sigma_{\mathsf{dis}}(\Delta^{\mathsf{Dir}}_{\Omega_{\theta}}).$$ [*(ii)*]{} Conversely, let $({\lambda},\psi)$ be an eigenpair of $\Delta^{\mathsf{Dir}}_{\Omega_{\theta}}$ with ${\lambda}<1$. Splitting $\psi$ into its odd and even parts $\psi^{{\mathsf{odd}}}$ and $\psi^{{\mathsf{even}}}$ with respect to $x_{2}$, we obtain: $$\psi = \psi^{{\mathsf{odd}}} + \psi^{{\mathsf{even}}},\quad \Delta^{\mathsf{Dir}}_{\Omega_{\theta}}\psi^{{\mathsf{odd}}}={\lambda}\psi^{{\mathsf{odd}}}\quad \mbox{and}\quad \Delta^{\mathsf{Dir}}_{\Omega_{\theta}}\psi^{{\mathsf{even}}}={\lambda}\psi^{{\mathsf{even}}}.$$ We note that $\psi^{{\mathsf{odd}}}$ satisfies the Dirichlet condition on the line $x_{2}=0$, and $\psi^{{\mathsf{even}}}$ the Neumann condition on the same line. Let us check that $\psi^{{\mathsf{odd}}}=0.$ If it is not the case, this would mean that ${\lambda}$ is an eigenvalue for the Dirichlet Laplacian on the half-waveguide $\Omega_{\theta}^+$. But, by monotonicity of the Dirichlet spectrum with the respect to the domain, Example \[ex:1-4\], we obtain that the spectrum on $\Omega_{\theta}^+$ is higher than the spectrum on the infinite strip which coincides with $\Omega_{\theta}^+$ when $x_1>0$. This latter spectrum is equal to $[1,+\infty)$. Therefore the Dirichlet Laplacian on the half-waveguide $\Omega_{\theta}^+$ cannot have an eigenvalue below $1$. Thus, we have necessarily: $\psi^{{\mathsf{odd}}}=0$ and $\psi=\psi^{{\mathsf{even}}}$ which is an eigenfunction of $\Delta^{\mathsf{Mix}}_{\Omega_{\theta}^+}$ associated with ${\lambda}$. We take advantage of Proposition \[P:dis\] for further proofs and for numerical simulations. Monotonicity of Rayleigh quotients with respect to the opening {#s:3} ============================================================== From now on we consider the Rayleigh quotients associated with the operator $\Delta^{\mathsf{Mix}}_{\Omega_{\theta}^+}$, which we write in the form, Lemma \[lem:1-2\] $$\label{eq:3-1} \lambda_j(\theta) = {\mathop{\operatorname{\vphantom{p}inf}}}_{ \substack {\psi_1,\ldots,\psi_j\ \mbox{\footnotesize independent in}\ {\mathrm{H}}^1(\Omega^+_\theta),\\ \psi_1,\ldots,\psi_j=0\ \mbox{\footnotesize on}\ \partial_{\mathsf{Dir}}\Omega^+_\theta }} \ \ \sup_{\psi\in[\psi_1,\ldots,\psi_j]} \frac{\|\nabla\psi\|^2_{\Omega^+_\theta}} {\|\psi\|^2_{\Omega^+_\theta}} .$$ Those $\lambda_j(\theta)$ which are $<1$ are all the eigenvalues of $\Delta^{\mathsf{Dir}}_{\Omega_\theta}$ sitting below its essential spectrum. \[prop:3-1\] For any integer $j\ge1$, the function $\theta\mapsto\lambda_j(\theta)$ defined in is non-decreasing from $(0,\frac\pi2)$ into $\xR_+$. We cannot use directly Corollary \[co:lem1-2\] because of the part of the boundary where Neumann conditions are prescribed. Instead we introduce the open set $\widetilde{\Omega}_{\theta}$ isometric to $\Omega_{\theta}^+$, see Figure \[F:3\], $$\widetilde{\Omega}_{\theta} = \left\{(\tilde x,\tilde y)\in \left(-\frac{\pi}{\tan\theta},+\infty\right)\times\left(0,\pi\right) : \quad \tilde y<\tilde x\tan\theta + \pi\ \mbox{ if }\ \tilde x\in\left(-\frac{\pi}{\tan\theta},0\right)\right\}.$$ 1:( 0, 0) 2:( 50, 0) 3: (50, 15) 4: (0, 15) 5:(-15, 0) 10:(-50, 0) 10(9) 11 : 10(9) (dash=4) The part $\partial_{\mathsf{Dir}}\widetilde{\Omega}_{\theta}$ of the boundary carrying the Dirichlet condition is the union of its horizontal parts. The numbers $\lambda_j(\theta)$ can be equivalently defined by the Rayleigh quotients on $\widetilde{\Omega}_{\theta}$. Let us now perform the change of variable: $$x=\tilde x\tan\theta, \quad y=\tilde y,$$ so that the new integration domain $\widetilde\Omega:=\widetilde{\Omega}_{\pi/4}$ is independent of $\theta$. The bilinear gradient form $b$ on $\widetilde{\Omega}_{\theta}$ is transformed into the anisotropic form $b_\theta$ on the fixed set $\widetilde\Omega$: $$\label{eq:3-2} b_\theta(\psi,\psi') = \int_{\widetilde\Omega} \tan^2\!\theta \,(\partial_x\psi\,\partial_x\psi') + (\partial_y\psi\,\partial_y\psi') \xdif x\xdif y,$$ with associated form domain $$\label{eq:3-3} V:=\{\psi\in {\mathrm{H}}^1(\widetilde\Omega) : \psi=0 \mbox{ on } \partial_{{\mathsf{Dir}}}\widetilde\Omega \}$$ independent of $\theta$. The function $\theta\mapsto\tan^2\theta$ being increasing on $(0,\frac\pi2)$, we have $$\forall\psi\in V,\quad \theta\mapsto b_\theta(\psi,\psi) \ \mbox{non-decreasing on $(0,\frac\pi2)$.}$$ We conclude thanks to Corollary \[co:lem1-2\]. Using the perturbation theory, we also know that, for all $j\geq 1$, the function $\theta\mapsto{\lambda}_{j}(\theta)$ is continuous with respect to $\theta\in\left(0,\frac{\pi}{2}\right)$. Moreover, ${\lambda}_{1}$ being simple (as the first eigenvalue of a Laplace-Dirichlet problem), it is analytic because we are in the situation of an analytic family of type $(B)$ (see [@Kato80 p. 387 and 395]). In fact the numerical simulations lead to think that all the eigenvalues below $1$ are simple and thus analytic. Existence of discrete spectrum {#s:4} ============================== We recall that the lower bound of the essential spectrum of the operator $\Delta^{\mathsf{Mix}}_{\Omega^+_\theta}$ is $1$. Its first Rayleigh quotient is given by, , $$\label{eq:4-1} \lambda_1(\theta) = {\mathop{\operatorname{\vphantom{p}inf}}}_{ \psi\in {\mathrm{H}}^1(\Omega^+_\theta),\ \psi=0\ \mbox{\footnotesize on}\ \partial_{\mathsf{Dir}}\Omega^+_\theta } \frac{\|\nabla\psi\|^2_{\Omega^+_\theta}} {\|\psi\|^2_{\Omega^+_\theta}} .$$ In this section, we prove the following proposition: \[prop:4-1\] For any $\theta\in(0,\frac\pi2)$, the first Rayleigh quotient $\lambda_1(\theta)$ is $<1$. This statement implies that $\lambda_1(\theta)$ is an eigenvalue of $\Delta^{\mathsf{Mix}}_{\Omega^+_\theta}$ (application of Theorem \[th:1-1\]), hence of the Dirichlet Laplacian $\Delta^{\mathsf{Dir}}_{\Omega_\theta}$ on the broken guide (Proposition \[P:dis\]). This statement was first established in [@ABGM91]. Here we present a distinct, more synthetic proof, using the method of [@Duclos05 p. 104-105]. For convenience, it is easier to work in the reference set $\widetilde\Omega=\widetilde{\Omega}_{\pi/4}$ introduced in the previous section, with the bilinear form $b_\theta$ and the form domain $V$ . We are going to work with the shifted bilinear form $$\tilde b_\theta(\psi,\psi') = b_\theta(\psi,\psi') - \int_{\widetilde\Omega} \psi\,\psi' \xdif x\xdif y$$ which is associated with the quadratic form: $$\widetilde Q_{\theta}(\psi) = \tilde b_\theta(\psi,\psi).$$ Then the first Rayleigh quotient of $\widetilde Q_\theta$ is equal to $\lambda_1(\theta) - 1$. To prove our statement, this is enough to construct a function $\psi\in V$ such that: $$\widetilde Q_{\theta}(\psi)<0.$$ This will be done by the construction of 1. A sequence $\psi_n\in V$ such that $\widetilde Q_{\theta}(\psi_n)\to 0$ as $n\to\infty$, 2. An element $\phi$ of $V$ such that $\tilde b_\theta(\psi_n,\phi)$ is nonzero and independent of $n$. The desired function will then be obtained as a suitable combination $\psi_n+\varepsilon\phi$. Let us give details now. [Step 1.]{} In order to do that, we consider the Weyl sequence defined as follows. Let $\chi$ be a smooth cutoff function equal to $1$ for $x\leq 0$ and $0$ for $x\geq 1$. We let, for $n\in\mathbb{N}\setminus\{0\}$: $$\chi_{n}(x)=\chi\left(\frac{x}{n}\right) \quad\mbox{and}\quad\psi_{n}(x,y)=\chi_{n}(x)\sin y.$$ Using the support of $\chi_{n}$, we find that $\widetilde Q_{\theta}(\psi_{n})$ is equal to $$\int_{-\pi}^0\int_{0}^{x+\pi} \!\! (\cos^2 y-\sin^2 y) \xdif y \xdif x + \int_0^\infty\!\!\int_{0}^\pi\! \big( \tan^2\theta(\chi'_{n})^2\sin^2 y+\chi_{n}^2(\cos^2 y-\sin^2 y)\big) \xdif y \xdif x.$$ Then, elementary computations provide: $$\int_{0}^\pi (\cos^2y-\sin^2y) \xdif y =0 \quad\mbox{and}\quad \int_{-\pi}^0\int_{0}^{x+\pi} (\cos^2y-\sin^2 y) \xdif y \xdif x=0.$$ Moreover, we have: $$\int_0^\infty\!\!\int_{0}^\pi\tan^2\theta(\chi'_{n})^2\sin^2 y \xdif y \xdif x\leq \left(\int_{0}^1|\chi'(u)|^2 \xdif u\right) \frac{\pi\tan^2 \theta}{2n}.$$ Hence we have proved that $\widetilde Q_{\theta}(\psi_{n})$ tends to $0$ as $n\to\infty$: $$\label{eq:4-2} \widetilde Q_{\theta}(\psi_{n}) \le \frac{K_\theta}{2n} \quad\mbox{with}\quad K_\theta = \left(\int_{0}^1|\chi'(u)|^2 \xdif u\right) \pi\tan^2 \theta\,.$$ [Step 2.]{} We introduce a smooth cutoff function $\eta$ of $x$ supported in $(-\pi,0)$. We consider a function $f$ of $y\in[0,\pi]$ to be determined later and satisfying $f(0)=0$. We define $\phi(x,y)=\eta(x)f(y)$. For ${\varepsilon}>0$ to be chosen small enough, we introduce: $$\psi_{n,{\varepsilon}}(x,y)=\psi_{n}(x,y)+{\varepsilon}\phi(x,y).$$ We have: $$\widetilde Q_{\theta}(\psi_{n,{\varepsilon}}) = \widetilde Q_{\theta}(\psi_{n})+2{\varepsilon}\,\tilde b_{\theta}(\psi_{n},\phi) +{\varepsilon}^2\widetilde Q_{\theta}(\phi).$$ Let us compute $\tilde b_{\theta}(\psi_{n},\phi)$. We can write, thanks to considerations of support: $$\tilde b_{\theta}(\psi_{n},\phi) = \int_{-\pi}^0\int_{0}^{x+\pi} \hskip-2.0ex\eta(x)\bigl(\cos y f'(y)-\sin y f(y)\big) \xdif y \xdif x= \int_{-\pi}^0\int_{0}^{x+\pi} \hskip-2.0ex\eta(x)\bigl(\cos y f(y)\big)' \xdif y \xdif x.$$ Using $f(0)=0$, this leads to: $$\tilde b_{\theta}(\psi_{n},\phi)=\int_{-\pi}^0 \eta(x)\cos(x+\pi)f(x+\pi) \xdif x.$$ We choose $f(y)=\eta(y-\pi)\cos(y-\pi)$ and we find: $$\tilde b_{\theta}(\psi_{n},\phi)= -\int_{-\pi}^0 \eta^2(x)\cos^2(x) \xdif x = -\Gamma<0.$$ This implies, using : $$\widetilde Q_{\theta}(\psi_{n,{\varepsilon}})\leq \frac{K_\theta}{2n}-2\Gamma{\varepsilon}+D{\varepsilon}^2,$$ where $D=\widetilde Q_{\theta}(\phi)$ is independent of $\varepsilon$ and $n$. There exists ${\varepsilon}>0$ such that: $$-2\Gamma{\varepsilon}+D{\varepsilon}^2\leq- \Gamma{\varepsilon}.$$ The angle $\theta$ being fixed, we can take $N$ large enough so that $$\frac{K_\theta}{2N}\leq \frac{\Gamma}{2}\,{\varepsilon},$$ from which we deduce that $\widetilde Q_{\theta}(\psi_{N,{\varepsilon}})\leq -\varepsilon\Gamma/2 <0$, which ends the proof of Proposition \[prop:4-1\]. Finite number of eigenvalues below the essential spectrum {#s:5} ========================================================= For a self-adjoint operator $A$ and a chosen real number $\lambda$ we denote by ${\mathcal{N}}(A,\lambda)$ the maximal index $j$ such that the $j$-th Rayleigh quotient of $A$ is $<\lambda$. By extension of notation, if the operator $A$ is defined by a coercive hermitian form $b$ on a form domain $V$, and if $Q$ denotes the associated quadratic form $Q(u)=b(u,u)$, we also denote by ${\mathcal{N}}(Q,\lambda)$ the number ${\mathcal{N}}(A,\lambda)$. This is coherent with the fact that in this case the Rayleigh quotients can be defined directly by $Q$, Lemma \[lem:1-2\]: $$\lambda_j = {\mathop{\operatorname{\vphantom{p}inf}}}_{ \substack {u_1,\ldots,u_j\in V\\ \mbox{\footnotesize independent}}} \ \ \sup_{u\in[u_1,\ldots,u_j]} \frac{Q(u)} {\langle u,u\rangle_H} \,.$$ This section is devoted to the proof of the following proposition: \[prop:5-1\] For any $\theta\in(0,\frac{\pi}{2})$, ${\mathcal{N}}(\Delta^{\mathsf{Dir}}_{\Omega_\theta},1)$ is finite. Thus in any case $\Delta^{\mathsf{Dir}}_{\Omega_\theta}$ has a nonzero finite number of eigenvalues under its essential spectrum. For the proof of Proposition \[prop:5-1\] we use a similar method as [@MoTr05 Theorem 2.1]. Like for the proof of Proposition \[prop:4-1\], it is easier to work in the reference set $\widetilde\Omega$ introduced in section \[s:3\], with the bilinear form $b_\theta$ and the form domain $V$ . The opening $\theta$ being fixed, we drop the index $\theta$ in the notation of quadratic forms and write simply as $Q$ the quadratic form associated with $b_\theta$: $$Q(\psi) = b_\theta(\psi,\psi) = \int_{\widetilde\Omega} \tan^2\!\theta\,|\partial_{x}\phi|^2+|\partial_{y}\phi|^2 \xdif x\xdif y.$$ We recall that the form domain $V$ is the subspace of $\psi\in H^1(\widetilde\Omega)$ which satisfy the Dirichlet condition on $\partial_{{\mathsf{Dir}}}\widetilde\Omega$. We want to prove that $${\mathcal{N}}(Q,1)\quad\mbox{is finite}.$$ We consider a $\xCone$ partition of unity $(\chi_{0},\chi_{1})$ such that $$\chi_{0}(x)^2+\chi_{1}(x)^2=1$$ with $\chi_{0}(x)=1$ for $x<1$ and $\chi_{0}(x)=0$ for $x>2$. For $R>0$ and $\ell\in\{0,1\}$, we introduce: $$\chi_{\ell,R}(x)=\chi_{\ell}(R^{-1}x).$$ Thanks to the IMS formula (see for instance [@Cycon]), we can split the quadratic form as: $$\label{eq:IMS} Q(\psi)=Q(\chi_{0,R}\psi)+Q(\chi_{1,R}\psi) -\|\chi'_{0,R}\psi\|^2_{\widetilde\Omega}-\|\chi'_{1,R}\psi\|^2_{\widetilde\Omega}\,.$$ We can write $$|\chi'_{0,R}(x)|^2+|\chi'_{1,R}(x)|^2 = R^{-2}W_R(x)\quad \mbox{with}\quad W_R(x) = |\chi'_{0}(R^{-1}x)|^2+|\chi'_{1}(R^{-1}x)|^2 \,.$$ Then $$\begin{aligned} \label{eq:5-1} \|\chi'_{0,R}\psi\|^2_{\widetilde\Omega}+\|\chi'_{1,R}\psi\|^2_{\widetilde\Omega} &= \int_{\widetilde\Omega} R^{-2} W_{R}(x)|\psi|^2 \xdif x\xdif y \nonumber\\ &= \int_{\widetilde\Omega} R^{-2} W_{R}(x) \big(|\chi_{0,R}\psi|^2 + |\chi_{1,R}\psi|^2 \big)\xdif x\xdif y.\end{aligned}$$ Let us introduce the subsets of $\widetilde\Omega$: $$\mathcal{O}_{0,R} = \{(x,y)\in\widetilde\Omega : x<2R\} \quad\mbox{ and }\quad \mathcal{O}_{1,R}=\{(x,y)\in\widetilde\Omega : x>R\}$$ and the associated form domains $$\begin{gathered} V_0 = \left\{\phi\in H^1(\mathcal{O}_{0,R}) : \quad \phi=0 \ \mbox{ on }\ \partial_{{\mathsf{Dir}}}\widetilde\Omega \cap \partial \mathcal{O}_{0,R} \ \mbox{ and on }\ \{2R\}\times(0,\pi)\right\} \\ V_1 = {\mathrm{H}}^1_0(\mathcal{O}_{1,R}). \end{gathered}$$ We define the two quadratic forms $Q_{0,R}$ and $Q_{1,R}$ by $$\label{eq:Qell} Q_{\ell,R}(\phi) = \int_{\mathcal{O}_{\ell,R}} \tan^2\theta|\partial_{x}\phi|^2+|\partial_{y}\phi|^2-R^{-2}W_{R}(x)|\phi|^2 \xdif x\xdif y \quad\mbox{for}\quad \psi\in V_\ell,\ \ \ell=0,1.$$ As a consequence of and we find $$\label{Q=Q0+Q1} Q(\psi) = Q_{0,R}(\chi_{0,R}\psi) + Q_{1,R}(\chi_{1,R}\psi)\quad \forall\psi\in V.$$ Let us prove \[lem:5-1\] We have: $${\mathcal{N}}(Q,1) \leq {\mathcal{N}}(Q_{0,R},1) + {\mathcal{N}}(Q_{1,R},1).$$ We recall the formula for the $j$-th Rayleigh quotient of $Q$: $$\lambda_j = {\mathop{\operatorname{\vphantom{p}inf}}}_{\substack{E\subset V\\ \dim E=j}} \ \sup_{\substack{ \psi\in E}} \frac{Q(\psi)}{\|\psi\|^2_{\widetilde\Omega}} \,.$$ The idea is now to give a lower bound for $\lambda_j$. Let us introduce: $$\mathcal{J} : \left\{\begin{array}{ccc} V &\to& V_0 \times V_1 \\ \psi&\mapsto&(\chi_{0,R}\psi\ ,\,\chi_{1,R}\psi) \,. \end{array}\right.$$ As $(\chi_{0,R},\chi_{1,R})$ is a partition of the unity, $\mathcal{J}$ is injective. In particular, we notice that $\mathcal{J} : V \to\mathcal{J}(V)$ is bijective so that we have: $$\begin{aligned} \lambda_j &= \inf_{\substack{F\subset \mathcal{J}(V)\\ \dim F=j}} \ \sup_{\substack{ \psi\in \mathcal{J}^{-1}(F)}} \frac{Q(\psi)}{\|\psi\|^2_{\widetilde\Omega}}\\ &= \inf_{\substack{F\subset \mathcal{J}(V)\\ \dim F=j}} \ \sup_{\substack{ \psi\in \mathcal{J}^{-1}(F)}} \frac{Q_{0,R}(\chi_{0,R}\psi)+Q_{1,R}(\chi_{1,R}\psi)} {\|\chi_{0,R}\psi\|^2_{\widetilde\Omega} + \|\chi_{1,R}\psi\|^2_{\widetilde\Omega}}\\ &= \inf_{\substack{F\subset \mathcal{J}(V)\\ \dim F=j}} \ \sup_{\substack{ (\psi_{0},\psi_{1})\in F}} \frac{Q_{0,R}(\psi_{0})+Q_{1,R}(\psi_{1})} {\|\psi_{0}\|^2_{\mathcal{O}_{0,R}}+\|\psi_{1}\|^2_{\mathcal{O}_{1,R}}}.\end{aligned}$$ As $\mathcal{J}(V) \subset V_0 \times V_1$, we deduce by an application of Corollary \[co:lem1-2\]: $$\begin{aligned} \label{eq:5-2} \lambda_j &\geq \inf_{\substack{F\subset V_0\times V_1\\ \dim F=j}} \ \sup_{\substack{ (\psi_{0},\psi_{1})\in F}} \frac{Q_{0,R}(\psi_{0})+Q_{1,R}(\psi_{1})} {\|\psi_{0}\|^2_{\mathcal{O}_{0,R}} + \|\psi_{1}\|^2_{\mathcal{O}_{1,R}}} =: \nu_{j},\end{aligned}$$ Let $A_{\ell,R}$ be the self-adjoint operator with domain ${\mathsf{Dom}}(A_{\ell,R})$ associated with the coercive bilinear form corresponding to the quadratic form $Q_{\ell,R}$ on $V_\ell$. We see that $\nu_{j}$ in is the $j$-th Rayleigh quotient of the diagonal self-adjoint operator $A_R$ $$\begin{pmatrix} A_{0,R} & 0\\ 0 & A_{1,R} \end{pmatrix} \quad\mbox{with domain}\quad {\mathsf{Dom}}(A_{0,R}) \times {\mathsf{Dom}}(A_{1,R})\,.$$ The Rayleigh quotients of $A_{\ell,R}$ are associated with the quadratic form $Q_{\ell,R}$ for $\ell=0,1$. Thus $\nu_j$ is the $j$-th element of the ordered set $$\{\lambda_k(Q_{0,R}), \ k\ge1\} \cup \{\lambda_k(Q_{1,R}), \ k\ge1\}.$$ Lemma \[lem:5-1\] follows. The operator $A_{0,R}$ is elliptic on a bounded open set, hence has a compact resolvent. Therefore we get: For all $R>0$, ${\mathcal{N}}(Q_{0,R},1)$ is finite. To achieve the proof of Proposition \[prop:5-1\], it remains to establish the following lemma: There exists $R_{0}>0$ such that, for $R\geq R_{0}$, ${\mathcal{N}}(Q_{1,R},1)$ is finite. For all $\phi\in V_1$, we write: $$\phi=\Pi_{0}\phi+\Pi_{1}\phi,$$ where $$\label{eq:Pi0} \Pi_{0}\phi(x,y) = \Phi(x) \sin y\quad\mbox{with}\quad \Phi(x) = \int_0^\pi \phi(x,y)\sin{y}\xdif y$$ is the projection on the first eigenvector of $-\partial^2_y$ on ${\mathrm{H}}^1_0(0,\pi)$, and $\Pi_{1}={\mathrm{Id}}-\Pi_{0}$. We have, for all ${\varepsilon}>0$: $$\begin{aligned} Q_{1,R}(\phi) &=Q_{1,R}(\Pi_{0}\phi)+Q_{1,R}(\Pi_{1}\phi) -2\int_{\mathcal{O}_{1,R}} R^{-2}W_{R}(x)\Pi_{0}\phi\, \Pi_{1}\phi\xdif x\xdif y \nonumber\\ &\geq Q_{1,R}(\Pi_{0}\phi)+Q_{1,R}(\Pi_{1}\phi) -{\varepsilon}^{-1}\int_{\mathcal{O}_{1,R}} R^{-2}W_{R}(x)|\Pi_{0}\phi|^2\xdif x\xdif y -{\varepsilon}\int_{\mathcal{O}_{1,R}} R^{-2}W_{R}(x)|\Pi_{1}\phi|^2\xdif x\xdif y \label{eq:Q1R} $$ Since the second eigenvalue of $-\partial^2_y$ on ${\mathrm{H}}^1_0(0,\pi)$ is $4$, we have: $$\int_{\mathcal{O}_{1,R}} |\partial_{y}\Pi_{1}\phi|^2 \xdif x\xdif y \ge 4 \|\Pi_{1}\phi\|^2_{\mathcal{O}_{1,R}}\,.$$ Denoting by $M$ the maximum of $W_R$ (which is independent of $R$), and using we deduce $$Q_{1,R}(\Pi_{1}\phi)\geq (4-MR^{-2})\|\Pi_{1}\phi\|^2_{\mathcal{O}_{1,R}}\,.$$ Combining this with where we take $\varepsilon=1$, and with the definition of $\Pi_0$, we find $$Q_{1,R}(\phi) \geq q_{R}(\Phi) +(4-2MR^{-2})\|\Pi_{1}\phi\|^2_{\mathcal{O}_{1,R}},$$ where $$\begin{aligned} q_{R}(\Phi) &= \int_{R}^\infty \tan^2\theta|\partial_{x}\Phi|^2+|\Phi|^2 -R^{-2}W_{R}(x)|\Phi|^2 \xdif x \\ &\ge \int_{R}^\infty \tan^2\theta|\partial_{x}\Phi|^2+|\Phi|^2 -R^{-2}M\mathds{1}_{[R,2R]}|\Phi|^2 \xdif x.\end{aligned}$$ We choose $R=\sqrt{M}$ so that $(4-2MR^{-2})=2$, and then $$\label{eq:5-3} Q_{1,R}(\phi) \geq \tilde q_{R}(\Phi) + 2\|\Pi_{1}\phi\|^2_{\mathcal{O}_{1,R}},$$ where now $$\label{eq:5-4} \tilde q_{R}(\Phi) = \int_{R}^\infty \tan^2\theta|\partial_{x}\Phi|^2+ (1-\mathds{1}_{[R,2R]})|\Phi|^2 \xdif x.$$ Let $\tilde a_R$ denote the 1D operator associated with the quadratic form $\tilde q_R$. From -, we deduce that the $j$-th Rayleigh quotient of $A_{1,R}$ admits as lower bound the $j$-th Rayleigh quotient of the diagonal operator: $$\begin{pmatrix} \tilde a_R & 0\\ 0 & 2\,{\mathrm{Id}}\end{pmatrix}$$ so that we find: $${\mathcal{N}}(Q_{1,R},1)\leq {\mathcal{N}}(\tilde q_{R},1).$$ Finally, the eigenvalues $<1$ of $\tilde a_R$ can be computed explicitly and this is an elementary exercise to deduce that ${\mathcal{N}}(\tilde q_{R},1)$ is finite. This concludes the proof of Proposition \[prop:5-1\]. Decay of eigenvectors at infinity – Computations for large angles {#s:6} ================================================================= Decay at infinity ----------------- In order to study theoretical properties of eigenvectors of the operator $\Delta^{\mathsf{Dir}}_{\Omega_\theta}$ corresponding to eigenvalues below $1$, we use the equivalent configuration on $\widetilde\Omega_\theta$ introduced in section \[s:3\], see also Figure \[F:4\]. The eigenvalues $<1$ of $\Delta^{\mathsf{Dir}}_{\Omega_\theta}$ are the same as those of $\Delta^{\mathsf{Mix}}_{\widetilde\Omega_\theta}$ (with Dirichlet conditions on the horizontal parts of the boundary of $\widetilde\Omega_\theta$) and the eigenvectors are isometric. The main result of this section is a quasi-optimal decay in the straight part $(0,+\infty)\times(0,\pi)$ of the set $\widetilde\Omega_\theta$ as $x\to\infty$. 1:( 0, 0) 2:( 50, 0) 3: (50, 15) 4: (0, 15) 5:(-26, 0) 10:(-60, 0) 10(9) 11 : 10(9) (dash=4) (dash=2) \[prop:6-1\] Let $\theta\in(0,\frac\pi2)$. Let $\psi$ be an eigenvector of $\Delta^{\mathsf{Mix}}_{\widetilde\Omega_\theta}$ associated with an eigenvalue $\lambda<1$. Then for all $\varepsilon>0$ the following integral is finite: $$\label{eq:6-1} \int_0^\infty \int_0^\pi {\mathrm{e}}^{2 \tilde x(\sqrt{1-\lambda}-\varepsilon)} \left( |\psi(\tilde x, \tilde y)|^2 + |\nabla\psi(\tilde x, \tilde y)|^2 \right) \xdif \tilde x\xdif \tilde y < \infty.$$ We give here an elementary proof based on the representation of $\psi$ as solution of the Dirichlet problem in the half-strip $\Sigma := \xR_+\times(0,\pi)$ $$\label{eq:6-2} \begin{cases} \ -\Delta \psi = \lambda \psi &\quad \mbox{in}\quad \Sigma, \\ \ \psi(\tilde x,0) = 0,\ \psi(\tilde x,\pi) = 0&\quad \forall \tilde x>0,\\ \ \psi(0,\tilde y) = g &\quad \forall \tilde y \in (0,\pi) \\ \end{cases}$$ where $g$ is the trace of $\psi$ on the segment $\Gamma$, see Figure \[F:4\]. Since $\psi$ belongs to $\xHone(\Sigma)$, its trace on $\partial\Sigma$ belongs to ${\mathrm{H}}^{1/2}$. Because of the zero trace on the lines $\tilde y=0$ and $\tilde y=\pi$, we find that $g$ belongs to the smaller space[^2] ${\mathrm{H}}^{1/2}_{00}(\Gamma)$, which is the interpolation space of index $\frac12$ between $\xHone_0(\Gamma)$ and $\xLtwo(\Gamma)$, [@LionsMagenes68]. We expand $\psi$ on the eigenvector basis of the operator $-\partial^2_y$, self-adjoint on ${\mathrm{H}}^2\cap{\mathrm{H}}^1_0(0,\pi)$. Its normalized eigenvectors are $$v_k(\tilde y) = \sqrt{\frac2\pi}\, \sin k\tilde y\quad \mbox{with eigenvalue}\quad \nu_k = k^2.$$ We expand $g$ in this basis: $$g(\tilde y)=\sum_{k\geq 1} g_{k}\, v_{k}(\tilde y),\quad \mbox{ where } g_k = \int_0^\pi g(\tilde y)v_k(\tilde y) \xdif\tilde y.$$ Interpolating between and , we find $$\|g\|^2_{{\mathrm{H}}^{1/2}_{00}(\Gamma)} \simeq \sum_{k\ge1} k \,g^2_k.$$ We can easily solve by separation of variables. We find $$\label{eq:6-3} \psi(\tilde x, \tilde y) = \sum_{k\ge 1}{\mathrm{e}}^{-\tilde x \sqrt{k^2-\lambda}} \, g_{k}\, v_k(\tilde y).$$ The estimate of $\psi$ in is then trivial. Let us prove now the estimate of $\partial_{\tilde x}\psi$. We use that $\sum_{k\ge1} k \,g^2_k$ is finite so that we can write: $$\partial_{\tilde x}\psi(\tilde x, \tilde y) = -\sum_{k\ge 1} \sqrt{k^2-\lambda}\ {\mathrm{e}}^{-\tilde x \sqrt{k^2-\lambda}} \, g_{k}\,v_k(\tilde y).$$ We have: $${\mathrm{e}}^{\tilde x(\sqrt{1-\lambda}-{\varepsilon})}\, \partial_{\tilde x}\psi(\tilde x, \tilde y) = -\sum_{k\geq 1} \sqrt{k^2-\lambda}\ {\mathrm{e}}^{\tilde x \left(\sqrt{1-\lambda}-\sqrt{k^2-\lambda}\right)}\ {\mathrm{e}}^{-{\varepsilon}\tilde x} \,g_{k}\, v_k(\tilde y)$$ leading to the $\xLtwo$ estimate: $$\begin{aligned} \int_0^\infty \int_0^\pi \big| {\mathrm{e}}^{\tilde x(\sqrt{1-\lambda}-{\varepsilon})}\,\partial_{\tilde x}\psi(\tilde x, \tilde y) \big|^2 \xdif \tilde x\xdif \tilde y &= \sum_{k\geq 1} \int_0^\infty (k^2-\lambda)\ {\mathrm{e}}^{2\tilde x \left(\sqrt{1-\lambda}-\sqrt{k^2-\lambda}\right)}\ {\mathrm{e}}^{-2{\varepsilon}\tilde x} \,g^2_{k} \xdif \tilde x\\ &\le \Big(\sum_{k\geq 1}k\,g^2_{k}\Big) \, \sup_{k\ge1} \int_0^\infty k \,{\mathrm{e}}^{-2\gamma\tilde x\sqrt{k^2-1}}\ {\mathrm{e}}^{-2{\varepsilon}\tilde x} \xdif \tilde x,\end{aligned}$$ where $\gamma=\gamma({\lambda})>0$ is a constant, uniform with respect to $k\geq 1$. Using the change of variables $\tilde x \mapsto \sqrt k\,\tilde x$, we can see that the integrals $$\int_0^{+\infty} k {\mathrm{e}}^{-2\tilde x(\varepsilon + \gamma\sqrt{k^2-1})}$$ are uniformly bounded as $k\to\infty$, which ends the proof of the estimate of $\partial_{\tilde x}\psi$ in . The estimate of $\partial_{\tilde y}\psi$ is similar. Under the conditions of Proposition \[prop:6-1\], we have also a sharp global $\xLinfty$ estimate $$\| {\mathrm{e}}^{\tilde x\sqrt{1-\lambda}} \, \psi \|_{\xLinfty(\widetilde\Omega_\theta)} < \infty.$$ To prove this we use again the representation and the fact that the trace $g$ is more regular than ${\mathrm{H}}^{1/2}_{00}(\Gamma)$. In fact, as a consequence of the characterization of the domain of the operator $\Delta^{\mathsf{Dir}}_{\Omega_\theta}$, the trace $g$ belongs to ${\mathrm{H}}^1_0(\Gamma)$ and, therefore, the sum $\sum_{k\ge1} k^2 \,g^2_k$ is finite by . Then, we have: $${\mathrm{e}}^{\tilde x\sqrt{1-{\lambda}}} \,\psi(\tilde x,\tilde y)= \sum_{k\geq 1}{\mathrm{e}}^{\tilde x \left(\sqrt{1-\lambda}-\sqrt{k^2-\lambda}\right)} \, g_{k}\,v_k(\tilde y).$$ Taking absolute values, we get: $${\mathrm{e}}^{\tilde x\sqrt{1-{\lambda}}}|\psi(\tilde x,\tilde y)|\leq \sqrt{\frac2\pi} \sum_{k\geq 1}|g_{k}|\leq \sqrt{\frac2\pi}\Big(\sum_{k\geq 1}k^{-2}\Big)^{1/2}\Big(\sum_{k\geq 1}k^2|g_{k}|^2\Big)^{1/2}.$$ The estimate can also be proved with a general method due to Agmon (see for instance [@Agmon82]). Computations for large angles {#ss:CompLA} ----------------------------- When $\theta\to\frac\pi2$, $\lambda_1(\theta)$ tends to $1$, see [@ABGM91]. The representation shows that in such a situation, the behavior of the associated eigenvector $\psi$ is dominated by its first term, proportional to $${\mathrm{e}}^{-\tilde x \sqrt{1-\lambda}} \, \sin(\tilde y).$$ 1:( 0, 0) 2:( 40, 0) 5:( 45.9, 0) 3: = 2 /1, 45/ 4:(-14.1, 0) 6: = 5 /4, 45/ 10:(100, 0) 101 \[1,4\] 50 = 1, 3, 4, 6/-4, 101/ 10(9) 11 : 10(9) (dash=4) (dash=9) In order to compute such an eigenpair by a finite element method, we have to be careful and take large enough domains — we simply put Dirichlet conditions on an artificial boundary far enough from the corners of the guide. Our computations are performed in the model half-guide $\Omega:=\Omega^+_{\pi/4}$ for the scaled operator $$\label{eq.Ltheta} {\mathscr{L}}_\theta := -2\sin^2\!\theta\,\partial_{{u}}^2-2\cos^2\!\theta\,\partial_{{v}}^2$$ equivalent to $-\Delta$ in $\Omega^+_\theta$ through the variable change $${u}= x_1\sqrt2\sin\theta\quad\mbox{and}\quad {v}= x_2\sqrt2\cos\theta.$$ ![Computations for moderately large angles. Plots of the first eigenvectors in the physical domain (rotated by $\frac\pi2$). Numerical value of the corresponding eigenvalue $\lambda_1(\theta)$.[]{data-label="F:5"}](h05000-VP01.eps) $\theta = 0.5000*\pi/2$$\lambda_1^{\mathsf{cpt}}(\theta)=0.92934$ ![Computations for moderately large angles. Plots of the first eigenvectors in the physical domain (rotated by $\frac\pi2$). Numerical value of the corresponding eigenvalue $\lambda_1(\theta)$.[]{data-label="F:5"}](h05983-VP01.eps) $\theta = 0.5983*\pi/2$$\lambda_1^{\mathsf{cpt}}(\theta)=0.96897$ ![Computations for moderately large angles. Plots of the first eigenvectors in the physical domain (rotated by $\frac\pi2$). Numerical value of the corresponding eigenvalue $\lambda_1(\theta)$.[]{data-label="F:5"}](h06889-VP01.eps) $\theta = 0.6889*\pi/2$$\lambda_1^{\mathsf{cpt}}(\theta)=0.98844$ We use a Galerkin discretization by finite elements in a truncated subset of $\Omega$ with Dirichlet condition on the artificial boundary, see Figure \[F:6\]. According to Corollary \[co:lem1-2\], also Examples \[ex:1-3\] and \[ex:1-4\], the eigenvalues $\lambda^{\mathsf{cpt}}_j(\theta)$ of the discretized problem are larger than the Rayleigh quotients $\lambda_j(\theta)$ of ${\mathscr{L}}_\theta$. When the discretization gets finer and the computational domain, larger, $\lambda^{\mathsf{cpt}}_j(\theta)$ tends to $\lambda_j(\theta)$ for $j=1,2,\ldots$ For the values of $\theta$ considered in this section ($\theta\ge\frac\pi4$), the numerical evidence is that the discrete spectrum of ${\mathscr{L}}_\theta$ has only one element $\lambda_1(\theta)$. The numerical effect of this is the convergence to $1$ of all other computational eigenvalues $\lambda^{\mathsf{cpt}}_j(\theta)$ for $j=2,3,\ldots$ The computations represented in Figure \[F:5\] are performed with the artificial boundary set at the abscissa ${u}=5\pi\sqrt2$. The plots are mapped back to the corresponding physical domain by a postprocessing of the numerical results. ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![Computations for very large angles with the mesh M4L (see Figure \[F:mesh4L\]). Plots in the computational domain $\Omega$.[]{data-label="F:7"}](h07022-VP01r.eps "fig:") ![Computations for very large angles with the mesh M4L (see Figure \[F:mesh4L\]). Plots in the computational domain $\Omega$.[]{data-label="F:7"}](h08538-VP01r.eps "fig:") ![Computations for very large angles with the mesh M4L (see Figure \[F:mesh4L\]). Plots in the computational domain $\Omega$.[]{data-label="F:7"}](h09702-VP01r.eps "fig:") $\theta = 0.7022*\pi/2$ $\theta = 0.8538*\pi/2$ $\theta = 0.9702*\pi/2$ $\lambda_1^{\mathsf{cpt}}(\theta)=0.9903037$ $\lambda_1^{\mathsf{cpt}}(\theta)=0.9994215$ $\lambda_1^{\mathsf{cpt}}(\theta)=0.9999998$ ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------- The computations in Figure \[F:7\] are performed with the artificial boundary set at the abscissa ${u}=10\pi\sqrt2$. The plots use the computational domain because the corresponding physical domains would be too much elongated to be represented. Accumulation of eigenpairs for small angles {#s:7} =========================================== When $\theta$ tends to $0$, there is more and more room for eigenvectors between the two corners of the guide. 1:( 0, 0) 2:( 80, 0) 3:( 80, 10) 4:( 100, 0) 5 = 1, 3 /2, 0.25/ 7 := 3 /1, 0.75/ 8 := 3 /1, 0.6/ 9 := 8 /5, 7/ 10 := 8 /5, 1/ 23 = 3, 4, 5, 6, 7, 8, 9, 10 /1, 2/ 4(9) 11 : 4(9) (dash=5) For any rectangular box ${\mathcal{B}}$ contained in $\Omega_{\theta}$ like in Figure \[F:8\], by the monotonicity of Dirichlet eigenvalues (Example \[ex:1-3\]), we know that for any $j$ $$\lambda_j(\theta) \le \lambda_j({\mathcal{B}})$$ where $\lambda_j(\theta)$ are the Rayleigh quotients of $\Delta^{\mathsf{Dir}}_{\Omega_\theta}$ and $\lambda_j({\mathcal{B}})$ the Dirichlet eigenvalues on ${\mathcal{B}}$. We choose ${\mathcal{B}}$ in the form of a rectangle bounded by the vertical lines $x_1=-\alpha\pi$ and $x_1=0$, and the horizontal lines $x_2=\pm\beta\pi$. Thus $\alpha$ and $\beta$ satisfy, $$\alpha\in(0,\frac1{\sin\theta}),\quad \beta\pi = (-\alpha\sin\theta + 1)\frac\pi{\cos\theta}\,.$$ The eigenvalues of the Dirichlet problem in ${\mathcal{B}}$ are $$\frac{k^2}{4\beta^2} + \frac{\ell^2}{\alpha^2},\quad k,\ell\in\{1,2,\ldots\}\,.$$ We look for the eigenvalues $\lambda_j({\mathcal{B}})$ less than $1$. Therefore $\alpha$ has to be chosen $>1$. Thus $\beta<(1-\sin\theta)/\cos\theta\le1$ for any $\theta$. As a consequence $k=1$ and the eigenvalues $\lambda_j({\mathcal{B}})$ less than $1$ are necessarily of the form $$\begin{aligned} \lambda_j({\mathcal{B}}) &= \frac{1}{4\beta^2} + \frac{j^2}{\alpha^2} \\ & = \frac{\cos^2\theta}{4(1-\alpha\sin\theta)^2} + \frac{j^2}{\alpha^2} \,.\end{aligned}$$ We optimize ${\mathcal{B}}$: The minimum of $\lambda_j({\mathcal{B}})$ is obtained for $\alpha$ such that $$\frac{\sin\theta\cos^2\theta}{2(1-\alpha\sin\theta)^3} - \frac{2j^2}{\alpha^3} = 0$$ ![Computations for $\theta=0.0226*\pi/2\sim2^\circ$ with the mesh M64S (see Figure \[F:mesh64S\]). Numerical values of the 10 eigenvalues $\lambda_j(\theta)<1$. Plots of the associated eigenvectors in the physical domain. []{data-label="F:11"}](VP01.eps) $\lambda_1^{\mathsf{cpt}}(\theta)= 0.32783 $ ![Computations for $\theta=0.0226*\pi/2\sim2^\circ$ with the mesh M64S (see Figure \[F:mesh64S\]). Numerical values of the 10 eigenvalues $\lambda_j(\theta)<1$. Plots of the associated eigenvectors in the physical domain. []{data-label="F:11"}](VP02.eps) $\lambda_2^{\mathsf{cpt}}(\theta)= 0.40217 $ ![Computations for $\theta=0.0226*\pi/2\sim2^\circ$ with the mesh M64S (see Figure \[F:mesh64S\]). Numerical values of the 10 eigenvalues $\lambda_j(\theta)<1$. Plots of the associated eigenvectors in the physical domain. []{data-label="F:11"}](VP03.eps) $\lambda_3^{\mathsf{cpt}}(\theta)= 0.47230 $ ![Computations for $\theta=0.0226*\pi/2\sim2^\circ$ with the mesh M64S (see Figure \[F:mesh64S\]). Numerical values of the 10 eigenvalues $\lambda_j(\theta)<1$. Plots of the associated eigenvectors in the physical domain. []{data-label="F:11"}](VP04.eps) $\lambda_4^{\mathsf{cpt}}(\theta)= 0.54181 $ ![Computations for $\theta=0.0226*\pi/2\sim2^\circ$ with the mesh M64S (see Figure \[F:mesh64S\]). Numerical values of the 10 eigenvalues $\lambda_j(\theta)<1$. Plots of the associated eigenvectors in the physical domain. []{data-label="F:11"}](VP05.eps) $\lambda_5^{\mathsf{cpt}}(\theta)= 0.61194 $ ![Computations for $\theta=0.0226*\pi/2\sim2^\circ$ with the mesh M64S (see Figure \[F:mesh64S\]). Numerical values of the 10 eigenvalues $\lambda_j(\theta)<1$. Plots of the associated eigenvectors in the physical domain. []{data-label="F:11"}](VP06.eps) $\lambda_6^{\mathsf{cpt}}(\theta)= 0.68328 $ ![Computations for $\theta=0.0226*\pi/2\sim2^\circ$ with the mesh M64S (see Figure \[F:mesh64S\]). Numerical values of the 10 eigenvalues $\lambda_j(\theta)<1$. Plots of the associated eigenvectors in the physical domain. []{data-label="F:11"}](VP07.eps) $\lambda_7^{\mathsf{cpt}}(\theta)= 0.75607 $ ![Computations for $\theta=0.0226*\pi/2\sim2^\circ$ with the mesh M64S (see Figure \[F:mesh64S\]). Numerical values of the 10 eigenvalues $\lambda_j(\theta)<1$. Plots of the associated eigenvectors in the physical domain. []{data-label="F:11"}](VP08.eps) $\lambda_8^{\mathsf{cpt}}(\theta)= 0.83040 $ ![Computations for $\theta=0.0226*\pi/2\sim2^\circ$ with the mesh M64S (see Figure \[F:mesh64S\]). Numerical values of the 10 eigenvalues $\lambda_j(\theta)<1$. Plots of the associated eigenvectors in the physical domain. []{data-label="F:11"}](VP09.eps) $\lambda_9^{\mathsf{cpt}}(\theta)= 0.90610 $ ![Computations for $\theta=0.0226*\pi/2\sim2^\circ$ with the mesh M64S (see Figure \[F:mesh64S\]). Numerical values of the 10 eigenvalues $\lambda_j(\theta)<1$. Plots of the associated eigenvectors in the physical domain. []{data-label="F:11"}](VP10.eps) $\lambda_{10}^{\mathsf{cpt}}(\theta)= 0.98195 $ Since we are interested in the behavior as $\theta\to0$, we take without asymptotic loss $$\alpha = 4^{1/3}j^{2/3}\sin^{-1/3}\theta$$ which provides $$\begin{aligned} \lambda_j({\mathcal{B}}) & = \frac14\Big( \frac{\cos^2\theta}{(1-4^{1/3}j^{2/3}\sin^{2/3}\theta)^2} + 4^{1/3}j^{2/3}\sin^{2/3}\theta \Big)\,.\end{aligned}$$ As a consequence, as soon as the quantity $Z:=4^{1/3}j^{2/3}\sin^{2/3}\theta$ is less that the first root of the equation $$\frac14\Big(\frac1{(1-Z)^2} + Z\Big) = 1,$$ for $Z\le 0.4679$, we have $\lambda_j({\mathcal{B}})<1$ and, hence, $\lambda_j(\theta)<1$. This implies that the maximal number $J$ such that $\lambda_J({\mathcal{B}})<1$ is greater than $$0.4679^{3/2}\cdot 0.5 \,\sin^{-1}\theta \simeq 0.1601\,\sin^{-1}\theta\,.$$ Therefore the number of eigenvalues of $\Delta^{\mathsf{Dir}}_{\Omega_\theta}$ less than $1$ tends to infinity (at least) like $\theta^{-1}$ as $\theta$ tends to $0$. We present in Figure \[F:11\] the computations of the all eigenpairs of $\Delta^{\mathsf{Dir}}_{\Omega_\theta}$ for the angle $\theta=0.0226*\pi/2$. We find 10 eigenvalues $<1$. Note that for this angle $\theta$, the numerical value of $0.1601\,\sin^{-1}\theta$ is $4.5103$. Asymptotic behavior of eigenpairs for small angles {#s:8} ================================================== We present in Figures \[F:9\] and \[F:10\] computations of the first eigenvector for smaller and smaller values of the angle $\theta$. We notice that the eigenvectors look similar, with the appearance of a short scale in the horizontal variable. ---------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------- ![Computations for small angles. Plots in the computational domain $\Omega$.[]{data-label="F:9"}](h01482-VP01r.eps "fig:") ![Computations for small angles. Plots in the computational domain $\Omega$.[]{data-label="F:9"}](h01032-VP01r.eps "fig:") ![Computations for small angles. Plots in the computational domain $\Omega$.[]{data-label="F:9"}](h00701-VP01r.eps "fig:") $\theta = 0.1482*\pi/2$ $\theta = 0.1032*\pi/2$ $\theta = 0.0701*\pi/2$ $\lambda_1^{\mathsf{cpt}}(\theta)= 0.56209 $ $\lambda_1^{\mathsf{cpt}}(\theta)= 0.48754 $ $\lambda_1^{\mathsf{cpt}}(\theta)= 0.42763 $ ---------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------- ![Computations for very small angles. Plots in the computational domain $\Omega$.[]{data-label="F:10"}](h00416-VP01r.eps "fig:") ![Computations for very small angles. Plots in the computational domain $\Omega$.[]{data-label="F:10"}](h00270-VP01r.eps "fig:") ![Computations for very small angles. Plots in the computational domain $\Omega$.[]{data-label="F:10"}](h00112-VP01r.eps "fig:") $\theta = 0.0416*\pi/2$ $\theta = 0.0270*\pi/2$ $\theta = 0.0112*\pi/2$ $\lambda_1^{\mathsf{cpt}}(\theta)= 0.37085 $ $\lambda_1^{\mathsf{cpt}}(\theta)= 0.33845 $ $\lambda_1^{\mathsf{cpt}}(\theta)= 0.29766 $ ---------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------- We perform in [@DauRay11] asymptotic expansions of the first eigenpairs at any order as $\theta\to0$, using techniques of semi-classical analysis (see for instance [@HelSj84; @Hel88] and also [@Martinez89]). We briefly describe now some of the results proved there. Let us recall that the eigenvalues $\lambda_j(\theta)<1$ of our operator $\Delta^{\mathsf{Dir}}_{\Omega_\theta}$ coincide with those of the scaled operator $${\mathscr{L}}_\theta := -2\sin^2\!\theta\,\partial_{{u}}^2-2\cos^2\!\theta\,\partial_{{v}}^2$$ in the model half-guide $\Omega$. The construction and validation of asymptotic expansions for the eigenpairs of ${\mathscr{L}}_\theta$ rely on a Born-Oppenheimer approximation. Roughly, this consists of four steps: 1. Definition of an operator ${\mathscr{L}}_\theta^{\mathsf{BO}}$ in the horizontal variable ${u}$ by replacing the operator in the vertical variable ${\mathscr{N}}_{u}:= -2\cos^2\!\theta\,\partial_{{v}}^2$ in each slice ${u}$ = const. by its first eigenvalue $\Lambda({u})$ 2. Semi-classical analysis of the eigenpairs of ${\mathscr{L}}_\theta^{\mathsf{BO}}$ as $\theta\to0$. 3. Determination of a change of variables $({u},{v})\mapsto(u,t)$ on $\Omega$ in order to exhibit a tensor product structure. Here appears the role of the limit operator ${\mathscr{N}}_{0-}$ as ${u}\nearrow 0$. Its first eigenvector is ${v}\mapsto\cos({v}/2)$. 4. Construction of expansions of eigenvectors in the new variables. [Step 1:]{} Definition of ${\mathscr{L}}_\theta^{\mathsf{BO}}$.\ Let ${\mathcal{I}}({u})$ be the intersection of $\Omega$ with the vertical line of abscissa ${u}$. - For ${u}\in(-\pi\sqrt2,0)$, ${\mathcal{I}}({u}) = (0,{u}+\pi\sqrt2)$. The operator ${\mathscr{N}}_{u}$ has Neumann condition at $0$ and Dirichlet at ${u}+\pi\sqrt2$. Thus $$\Lambda({u}) = 2\cos^2\!\theta\,\frac{\pi^2}{4({u}+\pi\sqrt2)^2}$$ - For ${u}\in(0,+\infty)$, ${\mathcal{I}}({u}) = ({u},{u}+\pi\sqrt2)$. The operator ${\mathscr{N}}_{u}$ has Dirichlet conditions at both ends. Thus $$\Lambda({u}) = \cos^2\!\theta.$$ The Born-Oppenheimer operator is $${\mathscr{L}}^{\mathsf{BO}}_\theta = -2\sin^2\!\theta\,\partial_{{u}}^2 + \Lambda({u}).$$ [Step 2:]{} Semi-classical analysis of ${\mathscr{L}}_\theta^{\mathsf{BO}}$.\ The operator ${\mathscr{L}}_\theta^{\mathsf{BO}}$ can be viewed as a 1D Schrödinger operator with potential $\Lambda$. The potential has a well with bottom at ${u}=0$. The well is not smooth and has a triangular shape, see Figure \[F:12\]. ![The Born-Oppenheimer potential $\Lambda$ and its left tangent at ${u}=0$ (here $\cos^2\!\theta$ is set to $1$).[]{data-label="F:12"}](Lambda.eps) The behavior of the eigenpairs of ${\mathscr{L}}_\theta^{\mathsf{BO}}$ as $\theta\to0$ is governed by the Taylor expansion of the potential $\Lambda$ at the well bottom ${u}=0$, by the tangent potential $V$ defined by $$V({u}) = \cos^2\!\theta \begin{cases} \,{\displaystyle}\frac14 -\frac{{u}}{2\pi\sqrt{2}}, &\mbox{if ${u}<0$,} \\ \,1, &\mbox{if ${u}>0$.} \end{cases}$$ The corresponding model problem is the problem of the behavior as $h\to0$ of the eigenpairs of the operator $${\mathscr{H}}_h = -h^2\partial^2_{u}+ \begin{cases} -{u}, &\mbox{if ${u}<0$,} \\ \,1, &\mbox{if ${u}>0$.} \end{cases}$$ The potential barrier on $(0,+\infty)$ produces a Dirichlet condition at ${u}=0$ for the leading terms of the asymptotics: We are led to the Airy-type eigenvalue problem $$\label{eq:airyh} -h^2\partial^2_{u}\psi_h - {u}\psi_h = E_h\psi_h \ \ \mbox{on}\ \ (-\infty,0),\quad \mbox{with}\quad \psi(0)=0.$$ The change of variables ${u}\mapsto X = {u}h^{-2/3}$ transforms the above equation into the reverse Airy equation $$-\partial^2_X \Psi - X\Psi = \mu\Psi \ \ \mbox{on}\ \ (0,+\infty),\quad \mbox{with}\quad\Psi(0)=0, \quad (\mbox{where } \mu=h^{-2/3}E_h)$$ whose solutions can be easily exhibited: As we will see, the eigenvalues $\mu$ are the zeros of the reverse Airy function ${\mathsf{A}}(X):={\mathsf{Ai}}(-X)$, where ${\mathsf{Ai}}$ is the standard Airy function. The zeros of ${\mathsf{A}}$ form an increasing sequence of positive numbers, which we denote by $z_{\mathsf{A}}(j)$, $j\ge1$. Since $-{\mathsf{A}}''(X)-X{\mathsf{A}}(X) = 0$ we have for any $E\in\xR$ $$-{\mathsf{A}}''(X)-(X-E){\mathsf{A}}(X) = E{\mathsf{A}}(X) \quad\mbox{ \ie}\quad -{\mathsf{A}}''(X+E)-X{\mathsf{A}}(X+E) = E{\mathsf{A}}(X+E)\,.$$ If $E=z_{\mathsf{A}}(j)$, then ${\mathsf{A}}(X+E)$ vanishes at $X=0$, hence $$\Big(z_{\mathsf{A}}(j), {\mathsf{A}}\big(X+z_{\mathsf{A}}(j)\big) \Big) \quad\mbox{is an eigenpair}.$$ Conversely, all eigenpairs are of this form. We deduce that the eigenvalues of problem are $E_h=h^{2/3}z_{\mathsf{A}}(j)$ and the associated eigenvectors are $\psi({u})={\mathsf{A}}({u}h^{-2/3} + z_{\mathsf{A}}(j))$. Coming back to our operator ${\mathscr{L}}^{\mathsf{BO}}_\theta$, we prove in[@DauRay11] that its eigenvalues have asymptotic expansions of the form $$\label{eq:8-1} \lambda^{\mathsf{BO}}_j(\theta) \underset{\theta\to0}{\simeq} \frac14 + \frac{ 2\theta^{2/3} z_{\mathsf{A}}(j)} {(4\pi\sqrt{2})^{2/3}} + \sum_{n\ge3} \theta^{n/3}\gamma^{\mathsf{BO}}_{j,n}\,,$$ where $\gamma^{\mathsf{BO}}_{j,n}$ are some real coefficients. The eigenvectors have expansions in powers of $\theta^{1/3}$, using the scale ${u}h^{-2/3}$ for ${u}<0$ and ${u}h^{-1}$ for ${u}>0$. [Step 3:]{} Tensorial structure of ${\mathscr{L}}_\theta$.\ On the left part of $\Omega$, its triangular part ${\mathsf{Tri}}$ in the half-plane ${u}<0$, we perform a change of variables to transform ${\mathsf{Tri}}$ into a square: $$\label{eq:Eut-} {\mathsf{Tri}}\ni({u},{v}) \mapsto ({u},t)\in(-\pi\sqrt{2},0)\times(0,\pi\sqrt{2}) , \quad\mbox{with}\quad t=\frac{{v}\,\pi\sqrt{2}}{{u}+\pi\sqrt{2}}.$$ On the right part of $\Omega$, its strip part contained in the half-plane ${u}>0$, we perform a change of variables to transform it into a horizontal strip: $$\label{eq:Eut+} ({u},{v}) \mapsto ({u},\tau)\in(0,+\infty)\times(0,\pi\sqrt{2}) , \quad\mbox{with}\quad \tau={v}-{u}.$$ 1:( 0, 0) 2:( 40, 0) 3:( 40, 40) 4:( 60, 0) 8:( -30, 0) 5 = 1, 3 /2, 0.67/ 100 \[6,3\] 7 = 5/1, 100/ 21 = 1, 5, 2, 4, 8/-0.5, 100/ 31 = 21, 22, 23, 24, 25/-1, 100/ 4(9) 11 : 4(9) 34(9) 13 : 34(9) 35(9) 15 : 35(9) (dash=4) (dash=9) This allows to work with the Taylor expansions of the transformed operator around ${u}=0$, on the left and on the right. The operator ${\mathscr{N}}_{0-} = -2\partial^2_t$ appears on the left with Dirichlet condition at $t=1$ and Neumann at $t=0$. Its first eigenvector is $\cos\frac t 2$, associated with the eigenvalue $\frac14$. The operator ${\mathscr{N}}_{0+} = -2\partial^2_\tau$ appears on the right with Dirichlet condition at $t=0,1$. [Step 4:]{} Asymptotics of the eigenpairs of ${\mathscr{L}}_\theta$.\ The projection on the eigenvector $t\mapsto\cos\frac t 2$ appears naturally at the first step of the expansion (this projection is sometimes called Feshbach or Grushin projection) and lets appear ${\mathscr{L}}^{\mathsf{BO}}_\theta$. A complete asymptotics for eigenpairs can be constructed, and in a further step, validated. As a result $\lambda_j(\theta)$ has an expansion similar to $\lambda^{\mathsf{BO}}_j(\theta)$, with different coefficients for $n\ge3$: $$\label{eq:8-2} \lambda_j(\theta) \underset{\theta\to0}{\simeq} \frac14 + \frac{ 2\theta^{2/3} z_{\mathsf{A}}(j)} {(4\pi\sqrt{2})^{2/3}} + \sum_{n\ge3} \theta^{n/3}\gamma_{j,n} \,$$ where $\gamma_{j,n}$ are some real coefficients. In Figure \[F:EVtg\] we display the functions $(\theta*2/\pi)^{2/3}\mapsto \lambda_j(\theta)$ for $j=1,2$ (computed by finite elements) and their linear approximation as $\theta\to0$, corresponding to the two-term approximation in , $(\theta*2/\pi)^{2/3}\mapsto \frac14 + \frac{ 2\theta^{2/3} z_{\mathsf{A}}(j)} {(4\pi\sqrt{2})^{2/3}} $. ![The eigenvalues $\lambda_1$ and $\lambda_2$ as functions of $(\theta*2/\pi)^{2/3}$.[]{data-label="F:EVtg"}](EVtg.eps) -3.5ex In the reference domain $\Omega$, the eigenvectors of ${\mathscr{L}}_\theta$ have a multiscale expansion in powers of $\theta^{1/3}$. On the left side, there are two scales, the ultra-short scale $({u}\theta^{-1},{v})$ and the short scale $({u}\theta^{-2/3},{v})$, on the right side there is only the ultra-short scale. The asymptotics is dominated by its first term, which is $${\mathsf{A}}\Big(\frac{u}{\theta^{2/3}}-z_{\mathsf{A}}(1)\Big)\cos\frac{t}2\ .$$ This structure explains the results seen in Figures \[F:9\]-\[F:10\]. Convergence of the finite element method {#s:9} ======================================== We now present some aspects of the computation process that led to the numerical results shown in the previous sections. As described in the section \[ss:CompLA\], the eigenvalue problem writes ${\mathscr{L}}_\theta \psi = \lambda \psi$ in the domain $\Omega$ with homogeneous Dirichlet conditions on its boundary, except on the horizontal segment where Neumann conditions are set, see identity and Figure \[F:6\]. The associate bilinear form is $$b(\psi,\psi') = \int_\Omega 2\sin^2\theta (\partial_u\psi\,\partial_u\psi' ) + 2\cos^2\theta (\partial_v\psi\,\partial_v\psi') \xdif u\xdif v$$ defined on the corresponding form domain $$V = \{\psi\in{\mathrm{H}}^1(\Omega):\ \psi=0\ \mbox{on}\ \partial_{\mathsf{Dir}}\Omega\}.$$ The eigenvalue problem writes in variational form: find non-zero $\psi\in V$ and $\lambda$ such that $$\forall\psi'\in V,\quad b(\psi,\psi') = \lambda\big\langle\psi,\psi'\big\rangle_{\xLtwo(\Omega)}.$$ By Galerkin projection on a finite dimension subspace $V_{\sf fds}$ of $V$, this problem can be rewritten as the generalized eigenvalue problem: find the eigenpairs $(\lambda, w)$ such that $Sw = \lambda Mw$, where $S$ and $M$ are the stiffness and mass matrices associated with a basis $(\Psi_1,\ldots,\Psi_N)$ of $V_{\sf fds}$: $$S = \Big( b(\Psi_j,\Psi_k) \Big)_{1\le j,k\le N} \quad\mbox{and}\quad M = \Big( \big\langle\Psi_j,\Psi_k\big\rangle_{\xLtwo(\Omega)} \Big)_{1\le j,k\le N} .$$ The computation process consists of two main steps: first, using a finite element method that leads to the two matrices $S$ and $M$, and second, using an algorithm to compute the eigenpairs. In the following two sections, we focus on the algorithm for the computation of the eigenpairs and then on the influence of the choice of meshes and polynomial degrees in the finite element method. All the computations have been done in double precision arithmetic, on a iMac computer (4 GB memory, 3.06 GHz Intel Core 2 Duo processor). Computation of the eigenpairs ----------------------------- We have first written a program using the Fortran 77 finite element library “Melina" [@Melina], which also provides a routine to compute eigenpairs of real symmetric eigenvalue problems. This routine uses an algorithm based on subspaces iterations. We call [Mel]{} this method. We have also written another program using the C++ version of the previous library, called “Melina++". The computation of the eigenpairs have been made using the well-known package ARPACK++ [@ARPACK++]. We call [Arp]{} this method. We thus were able to reproduce the results obtained with the [Mel]{} method, which is a cross validation of both methods. Here, we compare the computation of the eigenpairs with the two methods. For this purpose, in both cases, we fix the parameters governing the finite element part: interpolation degree 6 at Gauss-Lobatto points, quadrature rule of degree 13. We have selected two test configurations: - “large angle" configuration: $\theta = 0.9702*\pi/2$,  4 eigenvalues computed, mesh M4L with 656 triangles (see Figure \[F:mesh4L\]), leading to matrices of size 12325$\times$12325; - “small angle" configuration: $\theta = 0.0226*\pi/2$,  12 eigenvalues computed, mesh M64S with 6144 triangles (see Figure \[F:mesh64S\]), leading to matrices of size 111265$\times$111265. ![Mesh M4L for large angle, 656 triangles.[]{data-label="F:mesh4L"}](M4-80-4.eps "fig:") -4ex ![Mesh M64S for small angle, 6144 triangles.[]{data-label="F:mesh64S"}](M64-16-64.eps "fig:") -4ex Although the internal algorithms of the two methods are different, the parameters governing the computation of the eigenvalues are the same: a tolerance $\varepsilon$ that controls the end of the iteration process, and the dimension $N_{\sf sub}$ of the subspaces involved in the subspace iteration — $N_{\sf sub}$ is at least equal to $N_{\sf val}+1$, where $N_{\sf val}$ is the number of desired eigenvalues. We let $N_{\sf sub}$ vary between $10$ and $70$ and ran the programs for $\varepsilon=10^{-n}$, $n=4,5,6,7$, while recording the number of iterations and the CPU time needed. The CPU time of the finite element part, i.e. the computation of the matrices $S$ and $M$, does not depend on these parameters. It appears on the graphs as [CPUef]{}. The results are gathered on Figure \[F:CalGdAng\] for the “large angle" configuration and on Figure \[F:CalPtAng\] for the “small angle" configuration. We can observe that the number of iterations decreases as $N_{\sf sub}$ increases. A horizontal line at the beginning of the first two graphs indicates a failed computation (no convergence after the chosen maximal number of iterations). For the values tested, the parameter $\varepsilon$ does not play a significant role on the number of iterations, nor really on the CPU time (although it does on the residual). Since the algorithms used in the two methods are not the same, the number of iterations cannot be compared directly. They are mainly a good indicator of the computation process behavior. It is also worth to notice that the memory requirements increase as the subspace dimension $N_{\sf sub}$ increases. This parameter is difficult to handle since it is problem dependent. These graphs suggest that there is no critical value: it can be chosen large enough to ensure computation to succeed, but not too large, mainly because of memory considerations. The CPU graphs of the [Arp]{} method seem a bit chaotic: both algorithms need an initial vector which is chosen as a random vector by default. From our experience, we can say that this method is more sensible to this initial vector than the [Mel]{} method. Finally, these graphs show clearly that the [Arp]{} method is more efficient than the [Mel]{} method. ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- ![Comparison of the behavior of the methods for the large angle $\theta=0.9702*\pi/2$.[]{data-label="F:CalGdAng"}](Gi4m-80p-4y.eps "fig:") ![Comparison of the behavior of the methods for the large angle $\theta=0.9702*\pi/2$.[]{data-label="F:CalGdAng"}](Gi4m-80p-4yF.eps "fig:") ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- ![Comparison of the behavior of the methods for the small angle $\theta=0.0226*\pi/2$.[]{data-label="F:CalPtAng"}](Gi64m-16p-64y.eps "fig:") ![Comparison of the behavior of the methods for the small angle $\theta=0.0226*\pi/2$.[]{data-label="F:CalPtAng"}](Gi64m-16p-64yF.eps "fig:") -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- We are interested in the smallest eigenvalues of the problem $Sw = \lambda Mw$. Both methods [Arp]{} and [Mel]{} provide an option to choose in which end of the spectrum the wanted eigenvalues are to be searched. Let us mention that these algorithms perform better at computing the largest eigenvalues in general (this is due to the fact that they are ultimately based on the power method). In our case, this is typical and computation nearly always fails if the algorithms are asked to compute the smallest eigenvalues of this problem. For example, with ARPACK, we can observe the following behavior: - in the “large angle" configuration, for $\varepsilon=10^{-4}$ and $N_{\sf sub}=17$ or $45$, the computation fails ; - in the “small angle" configuration, for $\varepsilon=10^{-4}$ and $N_{\sf sub}=27$ or $43$, the computation fails ; if we change the mesh to a coarser one (M16S) with 384 triangles, the computation fails for $N_{\sf sub}=27$ and succeeds for $N_{\sf sub}=43$ with a CPU time of 236 sec. Thus, the correct strategy is to compute the largest eigenvalues of the eigenvalue problem $Mw = \nu Sw$ and retrieve the wanted eigenvalues $\lambda$ by inversion ($\lambda = 1/\nu$ and the eigenvectors are the same). All the computations presented here have been carried on using this strategy. To conclude this remark, let us mention that the computation that took 236 sec. with the wrong strategy, takes 0.47 sec. with the right one. Influence of meshes and polynomial degrees in the finite element method ----------------------------------------------------------------------- We now choose the small angle $\theta=0.0226*\pi/2$ and fix the parameters governing the computation of the eigenvalues: $\varepsilon=10^{-6}$, $N_{\sf sub}=25$, $N_{\sf val}=10$, since there are $10$ eigenvalues $<1$ as shown on Figure \[F:11\]. We have built five nested meshes of the same computational domain $\Omega$, called M4S, M8S (see Figure \[F:mesh4S\_8S\]), M16S, M32S (see Figure \[F:mesh16S\_32S\]) and M64S (see Figure \[F:mesh64S\]). The number $n$ in the name (M$n$S) is the number of segments of the subdivision of the horizontal (and vertical) boundary of $\Omega$. It indicates the characteristic diameter $h$ of the triangles, which is halved from a mesh to the next one in the list. Thus, the number of triangles is multiplied by 4 from a mesh to the next one. We have computed the 10 eigenvalues for $\varepsilon=10^{-8}$ using the finest mesh M64S and considered the first and last eigenvalues obtained as reference values. We denote them by $\lambda_1^{\sf ref}$ and $\lambda_{10}^{\sf ref}$. For each mesh, we let the interpolation degree $k$ vary from 1 to 6, while recording the differences $|\lambda_1 - \lambda_1^{\sf ref}|$ and $|\lambda_{10} - \lambda_{10}^{\sf ref}|$ obtained for each degree. The results are gathered on Figure \[F:converg\]. Each point on the graphs correspond to the result of a computation. Points corresponding to the same mesh have been linked together by a line to make the graphs more readable ; the corresponding degree is written below. The graphs show the convergence of the first and last computed eigenvalues: - with respect to the interpolation degree for a given mesh ; - with respect to the mesh, for a given degree. The number $N$ of degrees of freedom (d.o.f.) of the problem, which is the dimension of the matrices, is roughly proportional to $(kn)^2$, which explains the choice of the abscissa $\log_{10}(N)/2$. The precision attained is about one order of magnitude better for the first eigenvalue $\lambda_1$ than for the last one $\lambda_{10}$. For the first eigenvalue and the coarser meshes, we observe a kind of super convergence for the small degrees, then a linear convergence. The convergence tends to be linear as the mesh becomes finer. For the last eigenvalue $\lambda_{10}$, the convergence is mainly linear. --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- ![Meshes M4S (24 triangles) and M8S (96 triangles).[]{data-label="F:mesh4S_8S"}](M4-1-4.eps "fig:") ![Meshes M4S (24 triangles) and M8S (96 triangles).[]{data-label="F:mesh4S_8S"}](M8-2-8.eps "fig:") --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- ![Meshes M16S (384 triangles) and M32S (1536 triangles).[]{data-label="F:mesh16S_32S"}](M16-4-16.eps "fig:") ![Meshes M16S (384 triangles) and M32S (1536 triangles).[]{data-label="F:mesh16S_32S"}](M32-8-32.eps "fig:") --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ -- ![Convergence towards $\lambda_1\simeq 0.32783$ and $\lambda_{10}\simeq 0.98195$.[]{data-label="F:converg"}](convergL1.eps "fig:") ![Convergence towards $\lambda_1\simeq 0.32783$ and $\lambda_{10}\simeq 0.98195$.[]{data-label="F:converg"}](convergL10.eps "fig:") ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ -- This phenomenon of super convergence is due to a sort of locking which takes place in coarser meshes: such meshes with low degree $k$ are not able to capture the fine scale structure of the eigenvectors (see in Figure \[F:10\] representations of the first eigenvector in the computational domain for small angles). The average rate of convergence with respect to $k^{-1}$ is twice the rate of convergence with respect to $h$, in accordance with a well-known convergence result in finite elements [@Schwab98]. Typically the rate is $1$ in $h$ and $2$ in $k^{-1}$. These somewhat low rates are due to the singularity at the reentrant corner of $\Omega$. This singularity comes from the Laplace singularity , and for $\theta = 0.0226*\pi/2$, the singularity exponent $\frac\pi\omega$ is $\simeq0.5057$. Nevertheless, all our computations are accurate enough to display clearly the asymptotic behavior of the eigenpairs in the small angle and large angle limits, see Figure \[F:EVtg\]. [10]{} . , volume 29 of [*Mathematical Notes*]{}. Princeton University Press, Princeton, NJ 1982. . Quantum bound states in open geometries. (15) (Oct 1991) 8028–8034. . Discrete spectrum of a model schrödinger operator on the half-plane with neumann conditions. (2011). . . (2010) 893–912. . 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The Clarendon Press Oxford University Press, New York 1998. [^1]: We recall that an operator is said to be *Fredholm* if its kernel is finite dimensional, its range is closed and with finite codimension. [^2]: The space ${\mathrm{H}}^{1/2}_{00}(\Gamma)$ is the subspace of ${\mathrm{H}}^{1/2}(\Gamma)$ spanned by the functions $v$ such that $$\int_0^\pi \frac{v^2(\tilde y)} {\tilde y(\pi-\tilde y)} \,\xdif\tilde y <\infty\,.$$

--- abstract: 'Traces – and their extension called combined traces (comtraces) – are two formal models used in the analysis and verification of concurrent systems. Both models are based on concepts originating in the theory of formal languages, and they are able to capture the notions of causality and simultaneity of atomic actions which take place during the process of a system’s operation. The aim of this paper is a transfer to the domain of comtraces and developing of some fundamental notions, which proved to be successful in the theory of traces. In particular, we introduce and then apply the notion of indivisible steps, the lexicographical canonical form of comtraces, as well as the representation of a comtrace utilising its linear projections to binary action subalphabets. We also provide two algorithms related to the new notions. Using them, one can solve, in an efficient way, the problem of step sequence equivalence in the context of comtraces. One may view our results as a first step towards the development of infinite combined traces, as well as recognisable languages of combined traces.' address: - | Faculty of Mathematics and Computer Science\ Nicolaus Copernicus University\ Toru[ń]{}, Chopina 12/18, Poland, and - | School of Computing Science\ Newcastle University\ Newcastle upon Tyne, NE1 7RU, U.K. author: - '[Ł]{}ukasz Mikulski' bibliography: - 'etykaInic.bib' title: Algebraic Structure of Combined Traces --- Introduction ============ The dynamic behaviours of concurrent systems are usually described as sequences of atomic actions of such systems, which leads to its formal language semantics. Using this simple approach we cannot express some phenomena, e.g, concurrency and causality, that are crucial in the process of understanding and analysing concurrent behaviours of a system. In the case of a particular operational model, one can consider extending the sequential description by adding some information about the relevant properties of behaviours. One can do it by considering sequences of steps of actions and by adding some causal dependencies between actions. A well known approach that helps to capture concurrency and causality of a system are traces [@CarFoa69; @Maz77]. Consider, for example, the elementary net system with inhibitor arcs in Example \[e:ENwithInh\]($a$). We have four actions, $a$, $b$, $c$ and $d$, which may be executed in the initial marking, and two actions, $e$ and $f$, which need a previous history of computation to be enabled. Let us focus on action $e$. To enable this action we need to execute actions $a$ and $c$. We can execute them together or in any order. To capture the concurrent behaviour of this computation we need to identify two sequences of executions – $ace$ and $cae$. Using step semantics, which is not necessary in this case, we add also step sequence $(ac)(e)$ as another possible execution. Traces are sufficient to deal with such behaviours. The situation is more complex in the case of action $f$. Now we need three tokens in the pre-set of the considered action, hence actions $b$, $c$ and $d$ should be executed before the action $f$. Because of the presence of inhibitors, there is only one way to execute them sequentially, they should be executed in the order $\mathit{dcbf}$. Note that $\mathit{bdcf}$ or $\mathit{bcdf}$ are not correct sequences of execution. There are, however, other possibilities to execute the four actions in the step semantics. For instance all three actions may be executed simultaneously as a step containing $b$, $c$ and $d$. This gives $(bcd)(f)$ as our allowed sequence of steps. Other step sequences are $(d)(bc)(f)$ and $(cd)(b)(f)$. It is important that action $d$ has to be executed not later than action $c$, and action $c$ has to be executed not later than action $b$. In this case traces are still applicable, but they lose some important behavioural information. Another case is depicted in Example \[e:ENwithInh\]($b$). The upper part of the net is identical to the first case. Here, however, there is a single action $g$ that waits for tokens in all four middle places. In other words, whole tuple $(a,b,c,d)$ has to be executed before action $g$. It is easy to see that because of inhibitors there is no valid sequential execution of the four actions. After executing one of these actions, one of the remaining becomes disallowed. The only possible execution is the step sequence $(abcd)(f)$. Those two situations cannot be precisely described by traces, we need a more complex notion that capture “not later than” relationship between actions. To address this issue one can use a natural generalisation of traces called combined traces (see [@JanKou95]). \[e:ENwithInh\] Two elementary net systems with inhibitor arcs.\ =\[font=\] =\[draw,circle,thick,minimum size=6mm\] =\[draw,regular polygon, regular polygon sides=4,minimum size=10mm\] (p1) \[place,tokens=1\]; (p2) \[place,tokens=1, right of=p1\] ; (p3) \[place,tokens=1, right of=p2\] ; (p4) \[place,tokens=1, right of=p3\] ; (a7) \[transition, below of=p1, yshift=-0.30cm, xshift=-0.05cm, draw=none\] ; (a1) \[transition, below of=p1\] [$a$]{}; (a2) \[transition, below of=p2\] [$b$]{}; (a3) \[transition, below of=p3\] [$c$]{}; (a4) \[transition, below of=p4\] [$d$]{}; (p5) \[place, below of=a1\] ; (p6) \[place, below of=a2\] ; (p7) \[place, below of=a3\] ; (p8) \[place, below of=a4\] ; (a5) \[transition, below of=p6\] [$e$]{}; (a6) \[transition, below of=p7\] [$f$]{}; (p9) \[place, below of=a5\] ; (p10) \[place, below of=a6\] ; (p1) edge \[-&gt;\] (a1); (p2) edge \[-&gt;\] (a2); (p3) edge \[-&gt;\] (a3); (p4) edge \[-&gt;\] (a4); (a1) edge \[-&gt;\] (p5); (a2) edge \[-&gt;\] (p6); (a3) edge \[-&gt;\] (p7); (a4) edge \[-&gt;\] (p8); (p5) edge \[-o\] (a2); (p6) edge \[-o\] (a3); (p7) edge \[-o\] (a4); (p8) edge \[-o\] (a7); (p5) edge \[-&gt;\] (a5); (p7) edge \[-&gt;\] (a5); (p6) edge \[-&gt;\] (a6); (p7) edge \[-&gt;\] (a6); (p8) edge \[-&gt;\] (a6); (a5) edge \[-&gt;\] (p9); (a6) edge \[-&gt;\] (p10); ($a$) =\[font=\] =\[draw,circle,thick,minimum size=6mm\] =\[draw,regular polygon, regular polygon sides=4,minimum size=10mm\] (p1) \[place,tokens=1\]; (p2) \[place,tokens=1, right of=p1\] ; (p3) \[place,tokens=1, right of=p2\] ; (p4) \[place,tokens=1, right of=p3\] ; (a7) \[transition, below of=p1, yshift=-0.30cm, xshift=-0.05cm, draw=none\] ; (a1) \[transition, below of=p1\] [$a$]{}; (a2) \[transition, below of=p2\] [$b$]{}; (a3) \[transition, below of=p3\] [$c$]{}; (a4) \[transition, below of=p4\] [$d$]{}; (p5) \[place, below of=a1\] ; (p6) \[place, below of=a2\] ; (p7) \[place, below of=a3\] ; (p8) \[place, below of=a4\] ; (a5) \[transition, below of=p6, xshift=0.5cm\] [$g$]{}; (p9) \[place, below of=a5\] ; (p1) edge \[-&gt;\] (a1); (p2) edge \[-&gt;\] (a2); (p3) edge \[-&gt;\] (a3); (p4) edge \[-&gt;\] (a4); (a1) edge \[-&gt;\] (p5); (a2) edge \[-&gt;\] (p6); (a3) edge \[-&gt;\] (p7); (a4) edge \[-&gt;\] (p8); (p5) edge \[-o\] (a2); (p6) edge \[-o\] (a3); (p7) edge \[-o\] (a4); (p8) edge \[-o\] (a7); (p5) edge \[-&gt;\] (a5); (p6) edge \[-&gt;\] (a5); (p7) edge \[-&gt;\] (a5); (p8) edge \[-&gt;\] (a5); (a5) edge \[-&gt;\] (p9); ($b$) In this paper, we are concerned with the understanding of the algebraic inner structure of the combined traces (comtraces in short). We start by recalling some standard notions about formal languages, traces and comtraces. In particular, we give the definition of a lexicographical order on step sequences. We then recall the Foata canonical form of a comtrace that turns out to be maximal with respect to their order, and propose another canonical representative - the lexicographical canonical form. Then, we discuss the phenomenon of indivisibility in the case of comtraces and its connections with lexicographical canonical form. In the following sections, we propose an algebraic representation of a comtrace based on projections onto sequential subalphabets, and give a nondeterministic procedure that allows to reconstruct step sequences of the original comtrace. We also give two strategies of determining such reconstruction, each leading to a proper canonical form of a comtrace. In the final section, we describe some natural applications of the algebraic properties developed in this paper, and sketch the directions for further research. The preliminary version of this paper was presented on the CONCUR 2012 conference (Newcastle, UK) and published in local proceedings. The present paper is significantly extended and improved version. Preliminaries {#sect-1} ============= Throughout the paper we use the standard notions of the formal language theory. In particular, by an *alphabet* we mean a nonempty finite set $\Sigma$, the elements of which are called *(atomic) actions*. Finite sequences over $\Sigma$ are called *words*. The set of all finite words, including the empty word $\epsilon$, is denoted by $\Sigma^*$. Let $w=a_1\ldots a_n$ and $v=b_1\ldots b_m$ be two words. Then $$w\circ v=wv=a_1\ldots a_nb_1\ldots b_m$$ is the concatenation of $w$ and $v$. The alphabet $\mathit{alph}(w)$ of $w$ is the set of all the actions occurring within $w$, and $\#_a(w)$ is the number of occurrences of an action $a$ within $w$. By $|w|$ we denote the length of word $w$. More generally, for an object $X$, whenever the notion of size is clear from the contexts, we denote its size by $|X|$. Let $w=a_1\ldots a_n$ be a word. We use the notions of prefix and suffix of the word $w$. For any $k\leq n$, the *k-suffix* of $w$, denoted by ${\mathit{suff_k}}(w)$, is a word $a_k\ldots a_n$. Similarly, the *k-prefix* of $w$, denoted by ${\mathit{pref_k}}(w)$, is the word $a_1\ldots a_k$. We assume that the alphabet $\Sigma$ is given together with a total order $\leq$, called lexicographical order and extend it to the level of words. Such an order is inherited from the first actions on which two words being compared differ. In the case that one word is a prefix of another - the former is the smaller one. The projection onto a binary subalphabet $\{a,b\}$ is the function $\Pi_{a,b}:\Sigma^*\rightarrow\Sigma^*$ defined as follows: $$\Pi_{a,b}(cw)=\left\{ \begin{array}{lcl} c\Pi_{a,b}(w) & \text{ for } & c\in\{a,b\}\\ \Pi_{a,b}(w) & \text{ for } & c\notin\{a,b\}\\ \end{array} \right.$$ and $\Pi_{a,b}(\epsilon)=\epsilon$. In the same way we define a projection onto a unary subalphabet $\{a\}$, denoted by $\Pi_{a,a}:\Sigma^*\rightarrow\Sigma^*$. The algebra of binary relations over set $X$ (i.e., subsets of $X\times X$) is equipped with a concatenation operation $\circ$, where $R_1\circ R_2=\{(x,y){\:|\:}\exists_{z\in X}\;xR_1y\wedge yR_2z\}$. The neutral element for $\circ$ is the identity relation $I_X=\{(x,x){\:|\:}x\in X\}$, the index $X$ is omitted if it is clear from context. The $n$-th power of a relation $R$ is defined as $R^n=R^{n-1}\circ R$ for all $n\geq 1$, where $R^0=I$. The transitive closure of $R$ is $R^+=R^1\cup R^2\cup\ldots$, while its reflexive transitive closure is $R^*=R^0 \cup R^+$. Moreover, for a relation $R\subset X\times X$ we define the reverse of $R$ by $R^{-1}=\{(x,y){\:|\:}(y,x)\in R\}$, and its symmetric closure by $R^{sym} = R \cup R^{-1}$. We also define the largest equivalence relation contained in the reflexive and transitive closure of relation $R$ as $$R^\circledast=\{(x,y){\:|\:}xR^*y \wedge yR^*x\}.$$ The relation $R\subseteq X\times X$ is called symmetric if $R=R^{-1}$, reflexive if $I\subseteq R$, irreflexive if $I\cap R={\varnothing}$, transitive if $R^2\subseteq R$, and acyclic if $R^+$ is irreflexive. Moreover, for every $Y\subseteq X$ we define the restriction of the relation $R\subseteq X\times X$ to the set $Y$ by $$R|_Y=\{(x,y)\in R{\:|\:}x,y\in Y\}.$$ A *directed acyclic graph* is a pair $\mathit{dag}=(X,R)$, where $X$ is a finite set and $R$ is an acyclic irreflexive binary relation on $X$. In a diagrammatical representation, $X$ is the set of vertices while $R$ the set of arcs. A directed acyclic graph $\mathit{po}=(X,\prec)$ is a *poset* if the relation $\prec$ is transitive. An *upper set* is a nonempty subset $U$ of poset $\mathit{po}=(X,\prec)$ such that for every $x\in U$ if $x\prec y$ then $y\in U$. Elementary Net Systems with Inhibitor Arcs ------------------------------------------ In this paper we introduce some algebraic properties of combined traces which are the abstract model that describes causal relationships between executed actions of a concurrent system. The underlying structure, which was a motivation to define combined traces, are elementary net systems with inhibitor arcs. Formally, the *elementary net system with inhibitor arcs* (or $ENI-system$) is a tuple $N=(P,T,F,I,M_0)$, where $P$ and $T$ are two disjoint and finite sets of *places* and *transitions* (or *actions*) respectively. Two other components, $F\subseteq(P\times T)\cup(T\times P)$ and $I\subseteq P\times T$ are relations, called *flow relation* and *inhibition relation*. These relations describe possible dynamic behaviours of a net, which are manifested by executing sets of enabled transitions called *steps*. Such an execution leads from one set of places (called *marking*) to another. The initial marking $M_0\subseteq P$, from which the action of a system begins, is the last element of the tuple $N$. Given an ENI-system $N=(P,T,F,I,M_0)$ and $x\in P\cup T$, the *pre-set* (set of inputs) of $x$, denoted by $^\bullet x$, is defined as $^\bullet x=\{y|(y,x)\in F\}$, while the *post-set* (set of outputs) of $x$, denoted by $x^\bullet$, is defined as $x^\bullet=\{y|(x,y)\in F\}$. We also use the notion $^\bullet x^\bullet$ for the union of the post-set and pre-set of $x$, calling it the *set of neighbouring places/transitions* (or simply the *neighbourhood*). Moreover, if $x\in T$, the *inh-set* (set of inhibitors) of $x$, denoted by $^\circ x$, is defined by $^\circ x=\{y|(y,x)\in I\}$. The set of neighbouring places together with the inh-set forms an *extended neighbourhood* of an action. The dot notations are lifted in the usual way to sets of elements. Hence, by $^\bullet X$ we denote the set $\{y{\:|\:}(y,x)\in F\wedge x\in X\}$, $X^\bullet=\{y{\:|\:}(x,y)\in F\wedge x\in X\}$, and $^\circ X=\{y{\:|\:}(y,x)\in I\wedge x\in X\}$. Graphically, the places are drawn as circles, transitions as rectangles, elements of flow relation as arcs, and elements of inhibition relation as arcs with small circles as arrowheads. Marked places are depicted by drawing small dot called *token* inside. We say that a step $S=\{t_1,t_2,\ldots,t_n\}$ is *enabled* in marking $M$ if $^\bullet S\subseteq M$, $S^\bullet\cap M={\varnothing}$, $^\circ S\cap M={\varnothing}$ and $^\bullet t_i\cap\, ^\bullet t_j={\varnothing}$ for any $i\neq j$. The *execution* of such a step $S$ leads from the marking $M$ to the new marking $M'=(M\setminus\; ^\bullet S) \;\cup\;S^\bullet$. An ENI-system with empty inhibition relation, often considered under the sequential rather than step semantics, is called an *elementary net system* (or $EN-system$). \[e:ENI\] Consider a system $N=(P,T,F,I,M_0)$ depicted below.\ =\[font=\] =\[draw,circle,thick,minimum size=6mm\] =\[draw,regular polygon, regular polygon sides=4,minimum size=10mm\] (p1) \[place,tokens=1, label=left:$p_1$\]; (p2) \[place, below of=p1, label=left:$p_2$\] ; (p8) \[place, below of=p2, tokens=1, label=left:$p_8$\]; (a1) \[transition, right of=p1\] [$a$]{}; (p3) \[place, below of=a1, label=left:$p_3$\] ; (a3) \[transition, below of=p3\] [$c$]{}; (p4) \[place, right of=p3, label=left:$p_4$\] ; (p5) \[place, right of=p4, label=left:$p_5$\] ; (p6) \[place,tokens=1, right of=p5, label=left:$p_6$\] ; (p7) \[place, right of=p6, label=left:$p_7$\] ; (a4) \[transition, above of=p5\] [$d$]{}; (a2) \[transition, below of=p7\] [$b$]{}; (p1) edge \[-&gt;\] (a1); (p8) edge \[-&gt;\] (a3); (p2) edge \[-o\] (a1); (a1) edge \[-&gt;\] (p3); (a3) edge \[-&gt;\] (p2); (p3) edge \[-o\] (a3); (p3) edge \[-o\] (a4); (a4) edge \[-&gt;\] (p4); (p4) edge \[-o\] (a3); (a3) edge \[-&gt;\] (p5); (p5) edge \[-o\] (a2); (a2) edge \[-&gt;\] (p6); (p6) edge \[-&gt;\] (a4); (a4) edge \[-&gt;\] (p7); (p7) edge \[-&gt;\] (a2); The set of places has eight elements (from $p_1$ to $p_8$), the set of transitions has four elements ($T=\{a,b,c,d\}$). In the initial marking, three places are marked – $(p_1,p_6,p_8)$. Therefore, seven steps – including $(a)$, $(d)$ and $(ad)$ – are enabled. Note that after executing transition $d$, transition $a$ remains enabled, however, this does not hold in the opposite direction, i.e. after executing transition $a$ there is a token in place $p_3$ and transition $d$ is no more enabled. Traces {#sect-2} ------ In this section we recall well-known notion of traces (see [@DieRoz95; @Maz77; @Mik08]). Traces are an abstract model describing causal relationships between executed actions in, for example EN-systems. They capture independence, hence the possibility to be executed in any order (and also together) for some actions. Structurally, pairs of actions with disjoint sets of neighbouring places are in the independence relation. A *concurrent alphabet* is a pair $\Psi=(\Sigma,{\mathit{ind}})$, where $\Sigma$ is an alphabet and ${\mathit{ind}}\subseteq \Sigma\times\Sigma$ is an irreflexive and symmetric *independence* relation. The corresponding *dependence* relation is given by ${\mathit{dep}}=(\Sigma\times\Sigma)\setminus {\mathit{ind}}$. A concurrent alphabet $\Psi$ defines an equivalence relation $\equiv^{\Sigma}_\Psi$ identifying words which differ only by the ordering of independent actions. Two words, $w,v\in\Sigma^*$, satisfy $w\equiv^{\Sigma}_\Psi v$ if there exists a finite sequence of commutations of adjacent independent actions transforming $w$ into $v$. More precisely, $\equiv^{\Sigma}_\Psi$ is a binary relation over $\Sigma^*$ which is the reflexive and transitive closure of the relation $\sim^{\Sigma}_\Psi$ such that $w\sim^{\Sigma}_\Psi v$ if there are $u,z\in\Sigma^*$ and $(a,b)\in {\mathit{ind}}$ satisfying $w=uabz$ and $v=ubaz$. Equivalence classes of $\equiv^{\Sigma}_\Psi$ are called *(Mazurkiewicz) traces* and the trace containing a given word $w$ is denoted by $[w]$. The set of all traces over $\Psi$ is denoted by $\Sigma^*/_{\equiv_\Psi^\Sigma}$, and the pair $(\Sigma^*/_{\equiv_\Psi^\Sigma},\circ)$ is a (trace) monoid, where $\tau\circ\tau'=[w\circ w']$, for any words $w\in\tau$ and $w'\in\tau'$, is the concatenation operation for traces. Note that trace concatenation is well-defined as $[w\circ w']=[v\circ v']$, for all $w,v\in\tau$ and $w',v'\in\tau'$. Similarly, for every trace $\tau=[w]$ and every action $a\in \Sigma$, we can define $$\begin{array}{lclclclclcl} \mathit{alph}(\tau) &=& \mathit{alph}(w) & ~~~~~~ & \#_a(\tau) &=& \#_a(w). \end{array}$$ Projections onto unary and binary dependent subalphabets (i.e. $\{a,b\}\subseteq\Sigma$ such that $(a,b)\in{\mathit{dep}}$) are invariants for traces (see [@Mik08]). It is possible to formulate the trace equivalence in terms of projections. Two words $u,w\in\Sigma^*$ are in relation $\equiv_\Psi^\Sigma$ if and only if $$\forall_{(a,b)\in{\mathit{dep}}}\;\Pi_{a,b}(u)=\Pi_{a,b}(w).$$ Following [@Mik08], we define the *projection representation* of $\tau$ as a function $\Pi_\tau:{\mathit{dep}}\rightarrow\Sigma^*$, where $\Pi_\tau(a,b) = \Pi_{a,b}(\tau)$. \[e:traces\] Consider a concurrent alphabet $\Psi$ with four actions $\Sigma = \{a, b, c, d\}$ together with a dependence relation ${\mathit{dep}}$ given by: (n1) [$a$]{}; ; (n2) \[right of=n1\] [$b$]{}; (n3) \[below of=n2\] [$c$]{}; (n4) \[left of=n3\] [$d$]{}; (n1) \[xshift=0.15cm,yshift=0.1cm\] arc (-45:225:0.2cm); (n2) \[xshift=0.15cm,yshift=0.1cm\] arc (-45:225:0.2cm); (n3) \[xshift=0.15cm,yshift=-0.1cm\] arc (45:-225:0.2cm); (n4) \[xshift=0.15cm,yshift=-0.1cm\] arc (45:-225:0.2cm); (n1) – (n2) – (n3) – (n4) – (n1); (n0) ; (n1) \[below of=n0\] [or, equivalently,]{}; (n2) \[below of=n1, yshift=5mm\] [an independence relation]{}; (n1) [$a$]{}; ; (n2) \[right of=n1\] [$b$]{}; (n3) \[below of=n2\] [$c$]{}; (n4) \[left of=n3\] [$d$]{}; (n1) – (n3); (n2) – (n4); Then $w=abbaacd\equiv^{\Sigma}_\Psi abb caad$. The projection representation of a trace $\tau=[w]$ is $$\begin{array}{llllr} ~~~~~~\Pi_{a,a}(\tau)=aaa \; & \;~~ \Pi_{b,b}(\tau)=bb \; & \;~~\Pi_{c,c}(\tau)=c \; & \;~~ \Pi_{d,d}(\tau)=d & \\ ~~~~~~\Pi_{a,b}(\tau)=abbaa\; & \;~~ \Pi_{a,d}(\tau)=aaad\; & \;~~\Pi_{b,c}(\tau)=bbc\; & \;~~ \Pi_{c,d}(\tau)=cd & ~~~~~~ \end{array}$$ A word $w\in\Sigma^*$ is in *Foata canonical form* (see [@DieMet97]) w.r.t. the dependence relation ${\mathit{dep}}$ and a lexicographical order $\leq$ on $\Sigma$, if $w=w_1\ldots w_n$ ($n\geq 0$), where each $w_i$ is a nonempty word such that: [$\bullet$]{} $alph(w_i)$ is pairwise independent and $w_i$ minimal w.r.t. lexicographical order $\leq$ among $[w_i]$ for each $i>1$ and action $a$ occurring in $w_i$, there exists action $b$ occurring in $w_{i-1}$ such that $(a,b)\in {\mathit{dep}}$. Another canonical (normal) form of a trace that one may consider is the *lexicographical canonical form* (see also [@DieMet97]). It is based only on the lexicographical order and is defined as the least representative of a trace with respect to the lexicographical ordering. The intuition behind the Foata canonical form is that it groups actions into maximally concurrent steps, while the lexicographical canonical form is very useful in some combinatorial approaches (see [@MikPiaSmy11]). Each trace contains exactly one sequence in the Foata canonical form, and exactly one sequence in the lexicographical canonical form. It may happen that the two versions of canonical form coincide. Step Traces {#s:stepTraces} ----------- Let us lift the notion of traces from the sequential semantics discussed above to the step semantics. Instead of identifying sequences of actions over alphabet $\Sigma$, we will identify sequences of sets of actions, called steps. We demand that a step should consist of mutually independent actions only. For a given concurrent alphabet $\Psi=(\Sigma,{\mathit{ind}})$ we define a set ${\mathbb{S}}_\Psi$ of all nonempty subsets $A\subseteq\Sigma$ such that for all $a,b\in A$ we have $a\neq b\Rightarrow(a,b)\in{\mathit{ind}}$. If the concurrent alphabet $\Psi$ is clear from the context, we would write ${\mathbb{S}}$ instead of ${\mathbb{S}}_\Psi$. To avoid confusion with the well-established operation of concatenating sets in formal languages theory, we follow Diekert ([@DieMet97]) and denote a step containing actions $a$ and $b$ by $(ab)$ rather then $\{a,b\}$, etc. Finite sequences in ${\mathbb{S}}^*$, including the empty one $\lambda=(\epsilon)$, are called *step sequences*. We now lift a number of notions and notations introduced for words to the level of step sequences. In what follows, $\Psi=(\Sigma,{\mathit{ind}})$ is a *fixed* concurrent alphabet. Let $w=A_1\ldots A_n$ and $v=B_1\ldots B_m$ be two step sequences. Then $w\circ v=wv=A_1\ldots A_nB_1\ldots B_m$ is the concatenation of $w$ and $v$. The alphabet $\mathit{alph}(w)$ of $w$ comprises all actions occurring within $w$, and $\#_a(w)$ is the number of occurrences of an action $a$ within $w$. Moreover, we define the *step alphabet* $Alph(w)\subseteq{\mathbb{S}}$ of a step sequence $w$ as the set of all steps occurring in $w$. Both independence and dependence relations may be extended to the case of steps. Two steps $A,B\in{\mathbb{S}}$ are independent if and only if $A\times B\subseteq{\mathit{ind}}$, otherwise they are dependent. We not only allow to commute, but also to join/split pairs of independent steps. In fact, the commutation of two independent steps may be composed as two join/split operations. More precisely, $\equiv^{\mathbb{S}}_\Psi$ is a binary relation over ${\mathbb{S}}^*$ which is the reflexive, symmetric and transitive closure of the relation $\sim^{\mathbb{S}}_\Psi$ such that $w\sim^{\mathbb{S}}_\Psi v$ if there are $u,z\in{\mathbb{S}}^*$ and $A,B\in{\mathbb{S}}$ satisfying $w=uABz$, $v=u(A\cup B)z$, and $A\times B\subseteq{\mathit{ind}}$. Note that $A\cap B={\varnothing}$, since ${\mathit{ind}}$ is irreflexive. Equivalence classes of $\equiv^{\mathbb{S}}_\Psi$ are called *step traces* (see [@Vog91]). The trace containing a step sequence $w$ is denoted by $[w]$, while set of all step traces – by ${\mathbb{S}}^*/_{\equiv^{\mathbb{S}}_\Psi}$. Step traces are a conservative extension of sequential traces. To justify this statement we prove \[p:stepTraceSwaping\] Let $\Psi=(\Sigma,{\mathit{ind}})$ be a concurrent alphabet, and $w,v$ two step sequences over $\Psi$. If $w=uABz$ and $v=uBAz$, where $u,z\in{\mathbb{S}}^*$ and $A\times B\subseteq{\mathit{ind}}$ then $w\equiv^{\mathbb{S}}_\Psi v$. Directly from the definition, both $w$ and $v$ are in the relation $\sim^{\mathbb{S}}_\Psi$ with $y=u(A\cup B)z$. Since $\equiv^{\mathbb{S}}_\Psi$ is the reflexive, symmetric and transitive closure of $\sim^{\mathbb{S}}_\Psi$, we have $w\equiv^{\mathbb{S}}_\Psi y$ and $v\equiv^{\mathbb{S}}_\Psi y$, so also $w\equiv^{\mathbb{S}}_\Psi v$. We define two operations which help to move from step semantics into sequential semantics and vice versa. Let $A\in{\mathbb{S}}$ be a step and $\leq$ be a total order on $\Sigma$. Using the relation $\leq$ we define $min(A)$, the minimal representative of a step ${\varnothing}\neq A\in{\mathbb{S}}$ as the minimal action in $A$ with respect to $\leq$. Note that $min({\varnothing})$ is not defined. We define the *lexicographical linearization* of step $A$ as $$lex(A)=\left\{ \begin{array}{lcl} \epsilon & \text{ for } & A={\varnothing}\\ min(A)lex(A\setminus min(A)) & \text{ for } & A\neq{\varnothing}.\\ \end{array} \right.$$ We extend the operation $lex$ to step sequences and sets of step sequences in the usual way: $$\begin{aligned} & lex(A_1A_2\ldots A_n)=lex(A_1)lex(A_2)\ldots lex(A_n),\\ & lex(X)=\{lex(w){\:|\:}w\in X\}.\end{aligned}$$ As a reverse operation, we define a *singletonization* of an action $a$ by $sstep(a)=\{a\}$ and extend it to the case of sequences by $sstep(a_1\ldots a_n)=sstep(a_1)\ldots sstep(a_n)$. Since no two dependent actions may occur in the same step, we can easily lift the notion of projections onto unary and binary dependent subalphabets to the case of step sequences being representatives of step traces: $$\Pi_{a,b}(Aw)=\left\{ \begin{array}{lcl} a\Pi_{a,b}(w) & \text{ for } & a\in A\\ b\Pi_{a,b}(w) & \text{ for } & b\in A\\ \Pi_{a,b}(w) & \text{ for } & \{a,b\}\cap A={\varnothing}\\ \end{array} \right.$$ and $\Pi_{a,b}(\lambda)=\epsilon.$ Note that both actions $a$ and $b$ can not simultaneously be in $A$, since they are dependent. \[p:stepTraceLex\] Let $\Psi=(\Sigma,{\mathit{ind}})$ be a concurrent alphabet, and $w,u\in{\mathbb{S}}^*$ two step sequences over $\Psi$. Then $w\equiv^{\mathbb{S}}_{\Psi} u$ if and only if $lex(w)\equiv^\Sigma_{\Psi} lex(u)$. Since every action in sequential semantics can be treated as a singleton step in step semantics, the implication $lex(w)\equiv^\Sigma_{\Psi} lex(u) \Rightarrow w\equiv^{\mathbb{S}}_{\Psi} u$ follows directly from Proposition \[p:stepTraceSwaping\]. Therefore we need to prove that $w\equiv^{\mathbb{S}}_{\Psi} u\Rightarrow lex(w)\equiv^\Sigma_{\Psi} lex(u)$. Recalling the definition of step traces, it is sufficient to show that for $A,B\in{\mathbb{S}}$ such that $A\times B\subseteq {\mathit{ind}}$ we have $lex(A)lex(B)\equiv^\Sigma_{\Psi}lex(A\cup B)$. We make use of the projection formulation for sequential trace equivalence. Since $A\times B\subseteq{\mathit{ind}}$ we have $A\cap B={\varnothing}$. Moreover $A\cup B$ is a step, hence for every dependent pair $(a,b)$ we have $\Pi_{a,b}(A\cup B)$ empty or equal to a single action (also in the degenerated case $a=b$). Therefore for every $(a,b)\in{\mathit{dep}}$ we have either $$\Pi_{a,b}(A\cup B)=\Pi_{a,b}(A) \text{ when } \{a,b\}\cap A\neq{\varnothing}\wedge\{a,b\}\cap B={\varnothing},$$ or $$\Pi_{a,b}(A\cup B)=\Pi_{a,b}(B) \text{ when } \{a,b\}\cap B\neq{\varnothing}\wedge\{a,b\}\cap A={\varnothing}.$$ Hence $\Pi_{a,b}(A\cup B)=\Pi_{a,b}(A)\Pi_{a,b}(B)$ and $lex(A)lex(B)\equiv^\Sigma_{\Psi}lex(A\cup B)$. \[p:stepTraceFixPoints1\] Let $w$ be a step sequence over a concurrent alphabet $\Psi$. Then the sequences of singletons are fixpoints of the function $sstep\circ lex$, i.e. $$sstep\circ lex(w)=w ~~\Longleftrightarrow~~ \forall_{A\in Alph(w)}\; |A|=1.$$ $\Longrightarrow:$\ Let $w\in{\mathbb{S}}^*$ and $A\in Alph(w)$ be such that $|A|>1$. Without loss of generality we may assume that $A$ is the first step in $w$. Then $$lex(w)=lex(Aw')=lex(A)lex(w')=min(A)lex(A\setminus min(A))lex(w'),$$ hence $$sstep\circ lex(w)=sstep(min(A))sstep\circ lex((A\setminus min(A))w').$$ As a result we get that the first step in $sstep\circ lex(w)$ is a singleton, which is in contradiction with the assumption $|A|>1$. Hence $\forall_{A\in Alph(w)}\;|A|=1$.   $\Longleftarrow:$\ The second implication is straightforward, since $|A|=1$ implies $A=\{a\}$ and $a=min(A)$, so $lex(A)=a$ and $sstep\circ lex(A)=sstep(a)=\{a\}=A$. Let $w=A_1\ldots A_n$. Directly from the definitions $$sstep\circ lex(w)=sstep\circ lex(A_1)\ldots sstep\circ lex(A_n)=A_1\ldots A_n=w.\eqno{\qEd}$$ \[p:stepTraceFixPoints2\] Let $u$ be a sequence over a concurrent alphabet $\Psi$. Then $lex\circ sstep(u)=u$. Let $u=a_1\ldots a_n$ and $A_i=\{a_i\}$. Analogously to the proof of Proposition \[p:stepTraceFixPoints1\], we have $a_i=lex(A_i)$ and $sstep(a_i)=A_i$, so $$\begin{gathered} lex\circ sstep(u)=lex\circ sstep(a_1\ldots a_n) =lex(sstep(a_1)\ldots sstep(a_n))\\ =lex(A_1\ldots A_n) =lex(A_1)\ldots lex(A_n)=a_1\ldots a_n=u.\end{gathered}$$ The pair $({\mathbb{S}}^*/_{\equiv^{\mathbb{S}}_\Psi},\circ)$ is a (step trace) monoid, where $\tau\circ\tau'=[w\circ w']$, for any step sequences $w\in\tau$ and $w'\in\tau'$. Step trace concatenation is well-defined as $[w\circ w']=[v\circ v']$, for all $w,v\in\tau$ and $w',v'\in\tau'$. A step trace $\tau$ is a prefix of a step trace $\tau'$ if there is a step trace $\tau''$ such that $\tau\circ\tau''=\tau'$. As in the case of sequential traces, for every step trace $\tau$ and every $a\in \Sigma$, we can define $\mathit{alph}(\tau)=\mathit{alph}(w)$ and $\#_a(\tau)=\#_a(w)$, where $w$ is any step sequence belonging to $\tau$. The situation with the step alphabet, as it is not an invariant of a step trace, is a bit more complex: we define $Alph(\tau)=\bigcup_{w\in\tau}Alph(w)$. \[t:prReprStTr\] Let $\Psi$ be a concurrent alphabet, and $w,u\in {\mathbb{S}}^*$. Then $w\equiv^{{\mathbb{S}}}_\Psi u$ if and only if $\forall_{(a,b)\in{\mathit{dep}}}\;\Pi_{a,b}(w)=\Pi_{a,b}(u)$. The proof follows directly from Proposition \[p:stepTraceLex\] and the projection based definition of trace equivalence (in sequential semantics). Next, we give the canonical (normal) form of a step trace which essentially captures a greedy, maximally concurrent, execution of the actions occurring in the step trace conforming to the independence relations. A step sequence $w = A_1\ldots A_n\in{\mathbb{S}}^*$ is in *Foata canonical form* if, for each $i\leq n$, whenever $Av\equiv^{\mathbb{S}}_\Psi A_i\ldots A_n$ for some $A\in{\mathbb{S}}$ and $v\in{\mathbb{S}}^*$, then $A\subseteq A_i$. One can see that all suffixes and all prefixes of a step sequence in Foata canonical form are also in Foata canonical form, and that each step trace comprises a unique step sequence in Foata canonical form. Note that the following statement holds: \[p:foata4steptraces\] Let $\Psi$ be a concurrent alphabet. A step sequence $w = A_1\ldots A_n\in{\mathbb{S}}^*$ is in *Foata canonical form* if and only if for every $i<n$, there is no ${\varnothing}\neq A\subseteq A_{i+1}$ such that $A_i\times A\subseteq {\mathit{ind}}$.   $\Longrightarrow:$\ Let $w = A_1\ldots A_n$ be in Foata canonical form. Suppose that there are $i<n$ and ${\varnothing}\neq A\subseteq A_i$ such that $A\times A_i\subseteq{\mathit{ind}}$. Then, since for every $A_k$ we have $A_k\times A_k\subseteq({\mathit{ind}}\cup I)$, $$A_iA_{i+1}\ldots A_n\equiv^{\mathbb{S}}_\Psi A_iA(A_{i+1}\setminus A)A_{i+2}\ldots A_n\equiv^{\mathbb{S}}_\Psi (A_i\cup A)(A_{i+1}\setminus A)A_{i+2}\ldots A_n,$$ and by $A\neq{\varnothing}$ we obtain that $w$ is not in Foata canonical form. Hence there are no such $i<n$ and $A\subseteq A_{i+1}$.   $\Longleftarrow:$\ Let $w = A_1\ldots A_n$ and for every $i<n$, there is no ${\varnothing}\neq A\subseteq A_{i+1}$ such that $A_i\times A\subseteq {\mathit{ind}}$. Assume moreover that $Av\equiv^{\mathbb{S}}_\Psi A_i\ldots A_n$ and let $B=A_i\setminus A$. Suppose that $B$ is not empty and let $j$ be the least index such that $B\cap A_j\neq{\varnothing}$. By Theorem \[t:prReprStTr\] not only such $j$ exist, but also we get that $B\times A_k\subseteq{\mathit{ind}}$ for every $i<k<j$. Hence $B\cap A_j$ is a nonempty step contained in $A_j$ and $(B\cap A_j)\times A_{j-1}\subseteq{\mathit{ind}}$, which gives a contradiction with the assumptions, and so $B$ has to be empty, which ends the proof. We can also distinguish one of the representatives of $\tau$, built from singletons. Note that such step sequences may be considered as sequences over $\Sigma$ and compared using lexicographical order $\leq$. Similarly to the case of sequential traces, we call the least (with respect to the order $\leq$) singleton based representative of a step trace $\tau$ its *lexicographical canonical form*. Canonical forms of sequential and step traces connect those two worlds. More precisely, the following hold: \[t:lex4TrAndSTr\] Let $\Psi=(\Sigma,{\mathit{ind}})$ be a concurrent alphabet. Then a step sequence $w$ is in lexicographical canonical form if and only if all $A\in Alph(w)$ are singletons and the sequence $lex(w)$ is in lexicographical canonical form. According to Proposition \[p:stepTraceFixPoints1\], it is only an equivalent reformulation of the definition. \[t:foataSTr2Tr\] Let $\Psi=(\Sigma,{\mathit{ind}})$ be a concurrent alphabet. If a step sequence $w=A_1A_2\ldots A_k$ is in Foata canonical form then the sequence $u=lex(w)$ is in Foata canonical form. We have to prove that sequences $lex(A_1),\ldots,lex(A_k)$ satisfy the conditions from the definition of Foata canonical form in the case of sequential trace. The elements of every $A_i$ are pairwise independent since $A_i$ is a step. Let us suppose that there exist $1<i\leq k$ and $a\in A_i$ such that $a$ is independent with every action $b$ from $A_{i-1}$. Let $A=\{a\}$. Then $A\neq{\varnothing}$, $A\subseteq A_{i}$, and $A_{i-1}\times A\subseteq{\mathit{ind}}$. From the definition of Foata canonical form in the case of step traces we have that $w$ is not in Foata canonical form. Hence, $u$ is indeed in Foata canonical form. \[t:foataTr2STr\] Let $\Psi=(\Sigma,{\mathit{ind}})$ be a concurrent alphabet. If a sequence $u=a_1a_2\ldots a_n$ is in Foata canonical form then there exists a step sequence $w=A_1A_2\ldots A_k$ in Foata canonical form such that $u=lex(w)$. From the definition of Foata canonical form, we know that there exist sequences $u_1,\dots, u_k$ such that the elements of $alph(u_i)$ are pairwise independent for every $0<i\leq k$. Hence, for every $0<i\leq k$ we have that $alph(w_i)$ is a step over $\Psi$. Let $A_i=alph(w_i)$. Suppose that $w$ is not in Foata canonical form. Then, there exist nonempty $A\in{\mathbb{S}}$ and $0<i<k$ such that $A\subseteq A_{i+1}$ and $A_i\times A\subseteq{\mathit{ind}}$. Let $a\in A$. Since $A_i\times A\subseteq{\mathit{ind}}$ there is no $b\in A$ dependent with $a$. Hence $u$ is not in Foata canonical form. This contradicts the assumptions and proves the theorem. We conclude this subsection by formulating and proving a result that establishes a relationship between two semantics in which we can consider traces: \[t:stepEQsequential\] Let $\sigma\in\Sigma^*/_{\equiv^\Sigma_\Psi}$ be a trace (in sequential semantics). Then there exists a unique step trace $\tau$ such that $lex(\tau)=\sigma$. Theorems \[t:foataSTr2Tr\] and \[t:foataTr2STr\] allow us to associate sequential trace $\sigma$ with a trace $\tau$ using their Foata canonical forms. Let $w$ be a lexicographical canonical form of $\tau$. By Theorems \[t:lex4TrAndSTr\] and \[t:prReprStTr\], and definition of sequential traces based on projections, we get that $lex(w)\in \sigma$ and lexicographical canonical forms of $\sigma$ and $\tau$ also overlaps. By Proposition \[p:stepTraceLex\] we conclude that $lex(\tau)\subseteq \sigma$. Let $X=\{sstep(u){\:|\:}u\in\sigma\}$. By Propositions \[p:stepTraceFixPoints2\] and \[p:stepTraceLex\] we get that $X\subseteq\tau$ and $lex(X)=\sigma$. Hence $\sigma=lex(X)\subseteq lex(\tau)$ which ends the proof. Comtraces --------- Whereas traces are satisfactory to describe the concurrent behaviour of [EN-systems]{}, they are not sufficient to capture the behaviour of systems with inhibitor arcs. To deal with such systems, we recall the notion of combined traces (see [@JanKou95]). A *comtrace alphabet* is a triple $\Theta=(\Sigma,{\mathit{sim}},{\mathit{ser}})$, where $\Sigma$ is an arbitrary alphabet and ${\mathit{ser}}\subseteq{\mathit{sim}}\subseteq\Sigma\times\Sigma$ are two relations, respectively called *serialisability* and *simultaneity*; it is assumed that ${\mathit{sim}}$ is irreflexive and symmetric. Intuitively, if $(a,b)\in{\mathit{sim}}$ then $a$ and $b$ may occur simultaneously, whereas $(a,b)\in {\mathit{ser}}$ means that in such a case $a$ may also occur before $b$ (with both executions being equivalent). The set of all (potential) steps over $\Theta$, or *step alphabet*, is then defined as the set ${\mathbb{S}}_\Theta$ comprising all nonempty sets of actions $A\subseteq\Sigma$ such that $(a,b)\in{\mathit{sim}}$, for all distinct $a,b\in A$. If the comtrace alphabet $\Theta$ is clear from the context, we would write ${\mathbb{S}}$ instead of ${\mathbb{S}}_\Theta$. The *comtrace congruence* over $\Theta$, denoted by $\equiv_\Theta$, is the reflexive, symmetric and transitive closure of the relation $\sim_\Theta\subseteq{\mathbb{S}}^*\times{\mathbb{S}}^*$ such that $w\sim_\Theta v$ if there are $u,z\in{\mathbb{S}}^*$ and $A,B\in{\mathbb{S}}$ satisfying $w=uABz$, $v=u(A\cup B)z$ and $A\times B\subseteq {\mathit{ser}}$. Note that $A\cap B={\varnothing}$ as ${\mathit{ser}}$ is irreflexive. Equivalence classes of the relation $\equiv_\Theta$ are called *comtraces* (see [@JanKleKou11]), and the comtrace containing a given step sequence $w$ is denoted by $[w]$. The set of all comtraces is denoted by ${\mathbb{S}}^*/_{\equiv_\Theta}$, and the pair $({\mathbb{S}}^*/_{\equiv_\Theta},\circ)$ is a (comtrace) monoid, where $\tau\circ\tau'=[w\circ w']$, for any step sequences $w\in\tau$ and $w'\in\tau'$. Comtrace concatenation is well-defined as $[w\circ w']=[v\circ v']$, for all $w,v\in\tau$ and $w',v'\in\tau'$. A comtrace $\tau$ is a prefix of a comtrace $\tau'$ if there is a comtrace $\tau''$ such that $\tau\circ\tau''=\tau'$. As in the case of step traces, for every comtrace $\tau$ and every $a\in \Sigma$, we can define $\mathit{alph}(\tau)=\mathit{alph}(w)$ and $\#_a(\tau)=\#_a(w)$, where $w$ is any step sequence belonging to $\tau$. Moreover, $Alph(\tau)=\bigcup_{w\in\tau}Alph(w)$. Next, we give the canonical form of a comtrace which essentially captures a greedy, maximally concurrent, execution of the actions occurring in the comtrace conforming to the simultaneity and serialisability relations. A step sequence $w = A_1\ldots A_n\in{\mathbb{S}}^*$ is in *Foata canonical form* if, for each $i\leq n$, whenever $Av\equiv_\Theta A_i\ldots A_k$ for some $A\in{\mathbb{S}}$ and $v\in{\mathbb{S}}^*$, then $A\subseteq A_i$. This canonical form of a comtrace is extensively discussed in [@JanLe11]. One can see that all suffixes and all prefixes of step sequence in Foata canonical form are also in Foata canonical form, and that each comtrace comprises a unique step sequence in Foata canonical form. Note that an alternative (equivalent) definition of normal form requires that, for every $i<k$, there is no ${\varnothing}\neq A\subseteq A_{i+1}$ such that $A_i\times A\subseteq {\mathit{ser}}$ and $A\times (A_{i+1}{\setminus}A)\subseteq {\mathit{ser}}$. Moreover, in the cases of sequential and step traces we define two canonical forms. The first is, as in the case of comtraces, Foata canonical form, while the latter is called lexicographical. Both of those canonical forms prove to be very elegant and useful theoretical tool (as an example see prove of Theorem \[t:stepEQsequential\]). In the next section we define the lexicographical canonical form of a comtrace. It is one of the main notions introduced and utilised in this paper. But previously, let us discuss in detail direct relationships between atomic actions. Relations between actions ------------------------- In our discussion, we use a number of relations capturing semantically meaningful relationships between individual actions (see also [@MikKou11]): [$\bullet$]{} Dependence ${\mathit{dep}}=(\Sigma\times\Sigma)\setminus{\mathit{sim}}$, and independence ${\mathit{ind}}={\mathit{ser}}\cap{\mathit{ser}}^{-1}$.\ Both relations have their counterparts in trace theory, and we denote them in the same way. If two actions are dependent then they never occur in a common step. Two actions are independent if they can be executed in any order as well as simultaneously (as ${\mathit{ser}}\subseteq{\mathit{sim}}$). Semi-independence ${\mathit{sin}}= {\mathit{sim}}\setminus{\mathit{ser}}$.\ In contrast to the situation found in traces, dependence and independence do not describe all possible relationships between individual actions in comtraces. The remaining ones are called, due to the possibility of occurring together without being fully independent, semi-independent actions. Semi-independent actions may be further divided into symmetric and antisymmetric parts: [$-$]{} Strong simultaneity ${\mathit{ssm}}={\mathit{sim}}\setminus({\mathit{ser}}\cup{\mathit{ser}}^{-1})={\mathit{sin}}\setminus{\mathit{ser}}^{-1}$.\ If two actions are strongly simultaneous then may occur simultaneously but cannot be serialised at all. This means that two occurrences of strongly simultaneous actions which appear together in a step sequence $w$ would appear together in every step sequence belonging to the comtrace $[w]$. Weak dependence ${\mathit{wdp}}={\mathit{ser}}^{-1}\setminus{\mathit{ser}}={\mathit{sin}}\setminus{\mathit{sin}}^{-1}$.\ Two actions are weakly dependent if they can be serialised only in one way. This means that for any two actions $(a,b)\in{\mathit{wdp}}$, if their occurrences appear in the order ‘$a$ followed by $b$’ then they behave like completely dependent actions, while appearing in the order ‘$b$ followed by $a$’ allows one to equivalently execute (if there are no other obstacles) a step $(ab)$. The main motivation to define all those classes was to capture the essence of the interplay between single atomic elements of concurrent systems modelled using comtraces. As a result we achieve the projection representation defined later. Similarly to the case of simultaneity and serialisability, each of proposed relations can be described semantically by specific relationships between pre-sets, post-sets and inh-sets of pairs of actions. Note that if the set of neighbouring places of two actions overlaps, then those places are automatically considered as dependent (like in the case of traces and EN-systems). The main role in the further partition is played by the extended neighbourhoods. To capture dependence we have to add (to the overlapping of strict neighbourhoods) the situation when one action has an input place that is simultaneously an inhibitor for the other. The intersections of post-sets and inh-sets of two different actions are significant if they are not dependent. Namely, if they are totally disjoint, which means that their extended neighbourhoods are disjoint, those two actions are independent. Remaining situations correspond to the cases when two action have disjoint neighbourhoods as well as disjoint pre-sets and inh-sets, but still overlapping extended neighbourhoods and are captured by the semi-independence relation. Note that in the favourable circumstances both of them might be executable (like in the case of independence), but the execution of one of them may disable the execution of the other. If $(^\circ b \cap\;a^\bullet)\neq{\varnothing}\wedge\;(^\circ a \cap\;b^\bullet)={\varnothing}$ then after executing $b$ we can immediately execute $a$, but not vice versa. While if both $^\circ b \cap\;a^\bullet$ and $^\circ a \cap\;b^\bullet$ are nonempty then we cannot split simultaneous execution of $a$ and $b$. The following table gives a straightforward description of all seven relations for ENI-systems. ----------------------------- -------------------------- ------------------------------------------------------------------------------------------------- \[-3mm\] simultaneity $(a,b)\in{\mathit{sim}}$ $^\bullet a^\bullet \cap\;^\bullet b^\bullet={\varnothing}\wedge\; (^\circ a \cap\;^\bullet b)\cup\;(^\circ b \cap\;^\bullet a)={\varnothing}$ \[1mm\] serialisability $(a,b)\in{\mathit{ser}}$ $(a,b)\in{\mathit{sim}}\wedge a^\bullet\cap\;(^\bullet b\cup \,^\circ b)={\varnothing}$ \[1mm\] \[-3mm\] dependence $(a,b)\in{\mathit{dep}}$ $^\bullet a^\bullet \cap\;^\bullet b^\bullet\neq{\varnothing}\vee\; (^\circ a \cap\;^\bullet b)\cup\;(^\circ b \cap\;^\bullet a)\neq{\varnothing}$ \[1mm\] independence $(a,b)\in{\mathit{ind}}$ $(a,b)\notin{\mathit{dep}}\wedge\; (^\circ a \cap\;b^\bullet)\cup\;(^\circ b \cap\;a^\bullet)={\varnothing}$ \[1mm\] semi-independence $(a,b)\in{\mathit{sin}}$ $(a,b)\notin{\mathit{dep}}\wedge\; (^\circ b \cap\;a^\bullet)\neq{\varnothing}$ \[1mm\] strong simultaneity $(a,b)\in{\mathit{ssm}}$ $(a,b)\notin{\mathit{dep}}\wedge\; (^\circ b \cap\;a^\bullet)\neq{\varnothing}\wedge\;(^\circ a \cap\;b^\bullet)\neq{\varnothing}$ \[1mm\] weak dependence $(a,b)\in{\mathit{wdp}}$ $(a,b)\notin{\mathit{dep}}\wedge\; (^\circ b \cap\;a^\bullet)\neq{\varnothing}\wedge\;(^\circ a \cap\;b^\bullet)={\varnothing}$ \[1mm\] ----------------------------- -------------------------- ------------------------------------------------------------------------------------------------- \[e:comtrace\] Consider a comtrace alphabet $\Theta$ for ENI-system $N$ from Example \[e:ENI\]. The simultaneity and serialisability relations are given by:\ ${\mathit{sim}}= $ (n1) [$a$]{}; (n2) \[right of=n1\] [$b$]{}; (n3) \[below of=n2\] [$c$]{}; (n4) \[left of=n3\] [$d$]{}; (n1) – (n2) – (n3) – (n1) – (n4) – (n3);      ${\mathit{ser}}= $ (n1) [$a$]{}; (n2) \[right of=n1\] [$b$]{}; (n3) \[below of=n2\] [$c$]{}; (n4) \[left of=n3\] [$d$]{}; (n4) edge \[-&gt;\] (n1); (n3) edge \[-&gt;\] (n4); (n2) edge \[-&gt;\] (n3); (n1) edge \[-&gt;,out=30,in=150\] (n2); (n1) edge \[&lt;-,out=-30,in=-150\] (n2); \ In the net $N$ we have a pair of independent actions $(a,b)$. Note that their extended neighbourhoods are disjoint. The only pair of different and dependent actions is $(b,d)$. The reason for their dependency is the non-emptiness of their neighbourhoods. All the remaining pairs of different actions are semi-independent. Only one of them, namely $(a,c)$, is strongly simultaneous. Note that the post place of one of these actions is an inhibitor place of another, forming in the net graph a special kind of cycle. Similar behaviour (post place which is simultaneously inhibitor place), may be observed in the remaining cases, namely for pairs $(a,d)$, $(d,c)$ and $(c,b)$. However, we have there an asymmetric situation and those pairs of actions are weakly dependent. The five derived relations on actions are as follows:\ (R1) ; (R2) ; (R3) ; (R4) ; (R5) ; (R2) – (R4); (R2) – (R5); The combined trace of one of the possible executions in the net $N$ is $\tau=\{w,v,u,z\}$, where: $$\begin{array}{lcl} w &=& (d)(ab) \\[1mm] v &=& (d)(a)(b) \\[1mm] u &=& (ad)(b) \\[1mm] z &=& (d)(b)(a)\;. \end{array}$$ Moreover, $u$ is a step sequence in Foata canonical form. Lexicographical canonical form ============================== \ We extend the order on actions to the case of steps (sets of actions). Let $A,B\in{\mathbb{S}}$ be two steps. If the size of $A$ is smaller then the size of $B$ then $A\widehat{\leq} B$. If the sizes are equal, $A\widehat{\leq} B$ if $A=B$ or $A\neq B$ and $min(A\setminus B)\leq min(B\setminus A)$. In this way, $({\mathbb{S}},\widehat{\leq})$ becomes a totally ordered set. Using the order $\widehat{\leq}$ we can define *lexicographical order* on step sequences in the usual way. The *lexicographical canonical form* of a comtrace $\tau$, denoted by $minlex(\tau)$, is the least (with respect to the lexicographical order $\widehat{\leq}$) step sequence contained in the comtrace. Note that, in contrast to the Foata canonical form, the lexicographical canonical form captures one of the most sequential executions of a comtrace. Hence the two canonical forms lie on the opposite sides of the concurrent/sequential spectrum of behaviours. Note that the step sequence $v$ from Example \[e:comtrace\] is in lexicographical canonical form (assuming $a<b<c<d$). \[t:extremalforms\] For a given comtrace $\tau$, its Foata canonical form is the $\widehat{\leq}$-greatest, and its lexicographical canonical form is the $\widehat{\leq}$-least, step sequence contained in $\tau$. The lexicographical canonical form is the $\widehat{\leq}$-least step sequence contained in $\tau$ directly from the definition. We need to prove that Foata canonical form is greater than any other step sequence contained in $\tau$. Let $u=A_1\ldots A_n, v=B_1\ldots B_m$, $u\neq v$, $u\equiv_\Theta v$, and $u$ be in Foata canonical form. Moreover, let $i=min\{k {\:|\:}k\leq n \;\wedge\; A_k\neq B_k\}$. Note that such a number $i$ exists, since $u\neq v$ and $u\equiv_\Theta v$ so one sequence cannot be a prefix of another. We have $A_1\ldots A_{k-1}=B_1\ldots B_{k-1}$, so directly form the definition of Foata canonical form $B_k\neq A_k\wedge B_k\subseteq A_k$. Since $B_k\widehat{\leq}A_k$, we have $B_k\ldots B_m\widehat{\leq}A_k\ldots A_n$, and $v\widehat{\leq}u$. Indivisible steps and sequences ------------------------------- The structure and semantics of relations ${\mathit{sim}}$ and ${\mathit{ser}}$ mean that some actions have to appear simultaneously in every step sequence contained in a comtrace (in other word, they cannot be separated according to the comtrace congruence). A very good example of such actions are those in the ${\mathit{ssm}}$ relation. The strong simultaneity, however, does not exhaust all situations when actions are “glued” together in a permanent manner. Such a behaviour was used in [@MikKou11] to form so called folded actions. It is also worth to observe that the notion of indivisible steps was discussed, in the case of step traces with auto-concurrency, in [@Vog91]. In this section, we discuss the phenomenon of the indivisibility (in the case of comtraces) in depth. Let us consider a step $A\in{\mathbb{S}}$ and a relation $\equiv_A\subseteq A\times A$, such that, for all $a,b\in A$, we have $a\equiv_A b$ if $(a,b)\in({\mathit{sin}}|_A)^\circledast$. Intuitively, the relation $\equiv_A$ joins actions that can be executed simultaneously, but cannot be executed in a sequential way (see Example \[e:indiv\]). Note that, for arbitrary step $A$, the relation $\equiv_A$ is an equivalence relation. We say that a step $A\in{\mathbb{S}}$ is *indivisible* if $\forall_{a,b\in A}\;a\equiv_A b$. The set of all indivisible steps is denoted by $\widehat{{\mathbb{S}}}$. By $indiv(\tau)$ we denote the set of all step sequences contained in a comtrace $\tau$ and built with indivisible steps only. \[e:indiv\] Let us recall the comtrace alphabet from Example \[e:comtrace\] and the relations ${\mathit{sim}}$ and ${\mathit{sin}}$, which are crucial in determining indivisible steps. ${\mathit{sim}}= $ (n1) [$a$]{}; (n2) \[right of=n1\] [$b$]{}; (n3) \[below of=n2\] [$c$]{}; (n4) \[left of=n3\] [$d$]{}; (n1) – (n2) – (n3) – (n1) – (n4) – (n3);      ${\mathit{sin}}= $ (n1) [$a$]{}; (n2) \[right of=n1\] [$b$]{}; (n3) \[below of=n2\] [$c$]{}; (n4) \[left of=n3\] [$d$]{}; (n1) edge \[-&gt;\] (n4); (n4) edge \[-&gt;\] (n3); (n3) edge \[-&gt;\] (n2); (n1) edge \[-&gt;,out=-20,in=110\] (n3); (n1) edge \[&lt;-,out=-70,in=160\] (n3); The set of all possible steps is ${\mathbb{S}}=\{(a),(b),(c),(d),(ab),(ac),(ad),(bc),(cd),(abc),(acd)\}$, while the set of all indivisible steps is $\widehat{{\mathbb{S}}}=\{(a),(b),(c),(d),(ac),(acd)\}$. Note that step $A=(abc)$ is divided by the relation $\equiv_A$ into two indivisible steps $B=(b)$ and $C=(ac)$ and step $B$ occurs not later than step $C$, while step $D=(ab)$ is divided by the relation $\equiv_D$ into two, completely independent, indivisible steps $(a)$ and $(b)$. Moreover, there are only two sequences of indivisible steps contained in the comtrace $\tau$ which is defined in Example \[e:comtrace\]. These two sequences are $v=(d)(a)(b)$ and $z=(d)(b)(a)$. Intuitively, we can treat the indivisible step sequences belonging to $indiv(\tau)$ as classical sequences over the alphabet $\widehat{{\mathbb{S}}}$. Hence we define two complementary relations over this alphabet, the independence relation $\widehat{{\mathit{ind}}}$ and the dependence relation $\widehat{{\mathit{dep}}}$. We say that two indivisible steps $A$ and $B$ are *independent* if $A\times B\subseteq{\mathit{ind}}={\mathit{ser}}\cap{\mathit{ser}}^{-1}$; otherwise two indivisible steps are *dependent*. All steps contained in the lexicographical canonical form of a comtrace are indivisible $(minlex(\tau)\in indiv(\tau))$. Suppose, to the contrary, that $minlex(\tau)=uAv$ contains a non-indivisible step $A$. We conclude from Lemma \[l:division\] that for two disjoint steps $B$ and $C$ we have a step sequence $uBCv\in\tau$ which is different from the step sequence $minlex(\tau)$. Since $B\subseteq A$ and $A\neq B$ we have $uBCv\;\widehat{\leq}\;uAv$ so we found a step sequence contained in $\tau$ that is lexicographically smaller than $minlex(\tau)$, which contradicts our assumption. Hence all steps contained in $minlex(\tau)$ are indivisible. Recall the $lex$ operator defined in Section \[s:stepTraces\]. It allows us to translate a step sequence to a sequence of actions, and was very helpful in dealing with step traces. In the case of comtraces, however, it has rather narrower application. Therefore, we define the *split operator* that translates arbitrary step sequences to step sequences of indivisible steps as $\widehat{ }:{\mathbb{S}}^*\rightarrow\widehat{{\mathbb{S}}}^*$ as $$\widehat{(A_1\ldots A_n)}=\widehat{A_1}\ldots\widehat{A_n} =minlex(A_1)\ldots minlex(A_n).$$ The following facts justify an observation that the split operator does not lead beyond the comtrace, see Proposition \[p:leadout\]. \[l:division\] Let $A\in{\mathbb{S}}\setminus\widehat{{\mathbb{S}}}$ be a step that is not indivisible. Then there exist two steps, $B$ and $C$, such that $A\sim_\Theta BC$. Moreover, $A/_{\equiv_A}=B/_{\equiv_B}\cup C/_{\equiv_C}$. Since $A$ is not indivisible, the relation $\equiv_A$ divides $A$ into at least two equivalence classes. In the following proof we choose an indivisible step, to play a role of $B$. However, at first we separate a special subset of $A$, denoted by $D$. One can think about $D$ as a set of elements from $A$, which form a minimal layer in the graph of the relation $({\mathit{sin}}|_A)^*$. Let $D$ be the set of all actions $b\in A$ such that, $\forall_{a\in A}\;(b,a)\in({\mathit{sin}}|_A)^*\Rightarrow a\in[b]_{\equiv_A}$. Suppose that $D$ is empty. Let us take any $b_1\in A$. Then, by $D={\varnothing}$, there exists $b_2\in A$ such that $b_2\notin[b_1]_{\equiv_A}$ and $(b_1,b_2)\in({\mathit{sin}}|_A)^*$. Continuing in this way, we can construct an infinite sequence of actions $b_i\in A$ such that, for all $i$, $b_{i+1}\notin[b_1]_{\equiv_A} \wedge (b_i,b_{i+1})\in({\mathit{sin}}|_A)^*$. Since $A$ is finite, the elements contained in this sequence have to repeat. Let $b_n=b_m$ and $n<m$. Since $({\mathit{sin}}|_A)^*$ is transitive we have $(b_{n+1},b_n)\in({\mathit{sin}}|_A)^*$ and $(b_m,b_{n+1})\in({\mathit{sin}}|_A)^*$, so $b_{n+1}\in[b_n]_{\equiv_A}$ which contradicts the assumption. Hence $D$ is not empty. Let $d$ be an arbitrary element from $D$ and $B=[d]_{\equiv_A}$. $A$ is not indivisible, hence $A\neq B$. Moreover, directly from the construction of the set $D$, $B\subseteq D$. Let $b\in B$ and $a\in A\setminus B$. From the definition of $D$ we have that $(b,a)\notin({\mathit{sin}}|_A)^*$, so $(b,a)\notin{\mathit{wdp}}$ and $(b,a)\notin{\mathit{ssm}}$. We also have $(a,b)\notin{\mathit{dep}}$ since $a$ and $b$ are both contained in $A$. This gives $$\forall_{a\in A\setminus B} \forall_{b\in B}\;(a,b)\in{\mathit{wdp}}\vee (a,b)\in{\mathit{ind}}.$$ Hence $$\forall_{a\in A\setminus B} \forall_{b\in B}\;(b,a)\in{\mathit{ser}}$$ and finally $A\sim_\Theta B(A\setminus B)$.   It remains to be proven that $A/_{\equiv_A}=B/_{\equiv_B}\cup C/_{\equiv_C}$. According to the definition of the comtrace equivalence, $A\sim_\Theta BC$ implies that $B\times C\subseteq {\mathit{ser}}$. It means that for every pair of actions $b\in B$ and $c\in C$ we have $(b,c)\notin {\mathit{sin}}$. Hence for every $a,b\in A$ we have $a\equiv_A b \wedge a\in B \Rightarrow b\in B$ and $a\equiv_A b \wedge a\in C \Rightarrow b\in C$. It means that the graphs of the relation ${\mathit{sin}}$ restricted to steps $B$ and $C$ not only are vertex induced parts of the graph of the relation ${\mathit{sin}}$ restricted to the step $A$, but also are a division of this graph (i.e., the union of strongly connected components of graphs ${\mathit{sin}}|_B$ and ${\mathit{sin}}|_C$ is equal to the set of strongly connected components of the graph ${\mathit{sin}}|_A$), which end the proof. Let $\tau$ be a comtrace over $\Theta$ and $A\in Alph(\tau)$. Then $$A/_{\equiv_A}\subseteq Alph(\tau).$$ Since $A\in Alph(\tau)$, there exists $w,u\in{\mathbb{S}}$ such that $wAu\in\tau$. Applying Lemma \[l:division\] we can construct the step sequence $A_1\ldots A_n$ composed of indivisible steps only and equivalent to step sequence consisting of $A$ only. Moreover, $A/_{\equiv_A}=\bigcup_{i=1\ldots n} A_i$ and $wA_1\ldots A_nu\in\tau$. As a result we get that $A_i\in Alph(\tau)$, hence $A/_{\equiv_A}\subseteq Alph(\tau)$. \[t:trace4comtrace\] Let $\tau$ be a comtrace. The set $indiv(\tau)$ is a trace (with sequential semantic) over the concurrent alphabet $(\widehat{{\mathbb{S}}},\widehat{{\mathit{dep}}})$. To prove the statement of the theorem it is sufficient to show two facts. Firstly, we need to prove that relation $\widehat{{\mathit{ind}}}$ is symmetric and irreflexive. Secondly, we need to argue that by the repeated transposing of two subsequent and independent actions (in fact indivisible steps) we can reach any of other elements of the set $indiv(\tau)$ and cannot go beyond this set. We start from the first statement. By the definition of ${\mathit{ser}}$ the relation ${\mathit{ind}}={\mathit{ser}}\cap{\mathit{ser}}^{-1}$ is symmetric and irreflexive. Since two indivisible steps $A$ and $B$ are in relation $\widehat{{\mathit{ind}}}$ if all pairs of actions $(a,b)\in A\times B$ are independent, we conclude that the relation $\widehat{{\mathit{ind}}}$ is also symmetric and irreflexive. Let $w=uABv$ be a step sequence from $indiv(\tau)$ and $(A,B)\in\widehat{{\mathit{ind}}}$. By the definition of the $\widehat{{\mathit{ind}}}$ relation we have $AB\sim_\Theta C$ and $BA\sim_\Theta C$, where $C=A\cup B$. Therefore $uABv\equiv_\Theta uBAv$ and the set $indiv(\tau)$ is equal to its own trace closure. The last needed statement follows from Lemma \[l:division\] (about indivisibility of indivisible steps). Let us suppose that there are two comtrace equivalent step sequences $u$ and $v$ belonging to $indiv(\tau)$ that are not trace equivalent. Hence they differ in at least one projection to a binary dependent subalphabet, so there are two occurrences of indivisible steps $A$ and $B$ that appear in the two different orders and are dependent ($(A,B)\in\widehat{{\mathit{dep}}}$). Let $A$ precede $B$ in the step sequence $u$, and $B$ precede $A$ in the step sequence $v$. From the definition of comtrace equivalence there exists a sequence of equivalent step sequences $(w_i)_{i=1\ldots n}$ such that $u=w_1$, $w_i\sim_\Theta w_{i+1}$, and $w_n=v$. In this sequence there has to exist an element $w_i$ where the considered occurrences of indivisible steps were for the last time in the same order as in $u$ ($w_i=w'_iX_iY_iw''_i$ and $w_{i+1}=w'_{i+1}Z_iw''_{i+1}$ and $A\subseteq X_i$ and $B\subseteq Y_i$). Hence $A\times B\subseteq{\mathit{ser}}$. Moreover, there exists an element $w_j$ where the considered occurrences occur for the first time after $w_i$ in the same order as in $v$ ($w_j=w'_jZ_jw''_j$ and $w_{j+1}=w'_{j+1}X_jY_jw''_{j+1}$ and $A\subseteq X_j$ and $B\subseteq Y_j$). Hence also $B\times A\subseteq{\mathit{ser}}$. Therefore $(A,B)\in\widehat{{\mathit{ind}}}$, which gives a contradiction and completes the proof. \[p:leadout\] Let $[w]$ be a comtrace over $\Theta$. Then $$\widehat{w}\in[w]$$ and $$\widehat{w}=w ~~\Leftrightarrow~~ w\in indiv(\tau).$$ Let $w=A_1\ldots A_n$. By the definition of the operator $\widehat{ }$, we get $$\widehat{w}=minlex(A_1)\ldots minlex(A_n).$$ Since $minlex(A_i)\equiv_{\Theta}A_i$ we get $\widehat{w}\equiv_\Theta w$, so $\widehat{w}\in [w]$. Since $|minlex(A_i)|\geq 1$ and $|minlex(A_i)|=1$ if and only if $minlex(A_i)=\widehat{(A_i)}=A_i$ we conclude that $$\widehat{w}=w ~~\Leftrightarrow~~ \forall_i\;\widehat{A_i}=A_i.$$ By Lemma \[l:division\], $\widehat{(A_i)}=A_i$ if and only if $A_i$ is indivisible. Hence $\widehat{w}=w$ if and only if all $A_i$ are indivisible and $$\widehat{w}=w ~~\Leftrightarrow~~ w\in indiv(\tau).\eqno{\qEd}$$ As an immediate corollary of Theorem \[t:trace4comtrace\] and Proposition \[p:leadout\], we can observe that \[c:correspondence\] There is a one to one correspondence between the comtraces over comtrace alphabet $\Theta=(\Sigma,{\mathit{sim}},{\mathit{ser}})$ and traces over concurrent alphabet $\Psi=(\widehat{{\mathbb{S}}},\widehat{{\mathit{dep}}})$ given by the construction of the set of indivisible steps and dependence relation on them. $$\begin{array}{ccc} \tau & \widehat{\tau} & \tau '\\ over & \xlongrightarrow{\widehat{ }} \;\;over\; \longleftrightarrow & over\\ \Theta & \widehat{\Theta} & \Psi \end{array}$$ One can consider using the above correspondence to apply the methods of enumerating all traces of a given size [@MikPiaSmy11] to enumerate comtraces of a given size. Projection Representation of Comtraces ====================================== In the trace theory employing projections onto the cliques of the graph of dependence relation (see also [@Shi85]) turned out to be a very useful tool. We now extend this notion in the case of the binary and unary cliques only (see also [@Mik08]), to define the projection representation of comtraces. In the case of traces, we have only two kinds of relationships between actions. As independent actions may be executed in any order (or together in case of step semantics) one can focus on the order implied by the dependence relation. In the case of comtraces, the situation is more complicated. However, once more we can ignore independent actions and store information about the other three types of relations (dependency, weak dependency and strong simultaneity). Once more, it is sufficient to store the information in the form of sequences. In the case of strong simultaneity, however, we need to add a special symbol $\perp$ that separates the situations of sequential and simultaneous execution of pairs of actions being considered. Let $a,b\in\Sigma$ and $(a,b)\notin{\mathit{ind}}$ (possibly $a=b$). For each such pair we define the projection function $\Pi^\perp_{a,b}:{\mathbb{S}}^*\rightarrow (\Sigma\cup\{\perp\})^*$ as follows. First, for a step $A\in{\mathbb{S}}$ we have $$\Pi^\perp_{a,b}(A)=\left\{ \begin{array}{lll} \epsilon & \text{ for } & \{a,b\}\cap A={\varnothing}\\ a & \text{ for } & a\in A \wedge b\notin A\\ ba & \text{ for } & \{a,b\}\subseteq A \wedge (a,b)\in{\mathit{wdp}}\\ ab & \text{ for } & \{a,b\}\subseteq A \wedge (b,a)\in{\mathit{wdp}}\\ \perp & \text{ for } & \{a,b\}\subseteq A \wedge (a,b)\in{\mathit{ssm}}\\ \end{array} \right.$$ Note that there is a straightforward symmetry, namely for all $(a,b)\notin{\mathit{ind}}$ the equation $\Pi^\perp_{a,b}=\Pi^\perp_{b,a}$ holds. Moreover, according to the definition, we have $\Pi^\perp_{a,a}(A)=\epsilon$ if $a\notin A$ and $\Pi^\perp_{a,a}(A)=a$ if $a\in A$. Then, for a step sequence $w=A_1A_2\ldots A_n$ we have $$\Pi^\perp_{a,b}(w)= \Pi^\perp_{a,b}(A_1)\circ\Pi^\perp_{a,b}(A_2)\circ\ldots\circ\Pi^\perp_{a,b}(A_n).$$ \[t:prrepr\] Let $w,u$ be step sequences over a comtrace alphabet $\Theta=({\mathbb{S}},{\mathit{sim}},{\mathit{ser}})$. Then $w\equiv_\Theta u \;\Leftrightarrow\; \forall_{(a,b)\notin{\mathit{ind}}}\;\Pi^\perp_{a,b}(w)=\Pi^\perp_{a,b}(u)$. $\Rightarrow :$\ We first prove that $$w\equiv_\Theta u \;\Rightarrow\; \forall_{(a,b)\notin{\mathit{ind}}}\;\Pi^\perp_{a,b}(w)=\Pi^\perp_{a,b}(u).$$ According to the definition of comtrace equivalence, it is sufficient to prove the statement in the case of equivalent step sequences $w=A$ and $u=BC$. Let $a,b\in A$. We consider all but one of the possible relationships of these actions (the remaining case is that of independence).\ Case 1: $(a,b)\in {\mathit{dep}}$. Since actions $a$ and $b$ occur simultaneously in the step $A$, this is impossible.\ Case 2: $(a,b)\in {\mathit{ssm}}$. Since actions $a$ and $b$ are strongly simultaneous, Lemma \[l:division\] shows that they both have to occur in step $B$ or $C$. It means that $$\Pi^\perp_{a,b}(BC)=\Pi^\perp_{a,b}(B)\Pi^\perp_{a,b}(C)=\perp\epsilon=\Pi^\perp_{a,b}(A)$$ or $$\Pi^\perp_{a,b}(BC)=\Pi^\perp_{a,b}(B)\Pi^\perp_{a,b}(C)=\epsilon\perp=\Pi^\perp_{a,b}(A).$$ Case 3: $(a,b)\in {\mathit{wdp}}$. Since $B\times C\subseteq{\mathit{ser}}$, it is impossible that $b\in C$ and $a\in B$. If they both belong to one step, we have $$\Pi^\perp_{a,b}(BC)=\Pi^\perp_{a,b}(B)\Pi^\perp_{a,b}(C)=(ba) \epsilon=\Pi^\perp_{a,b}(A)$$ or $$\Pi^\perp_{a,b}(BC)=\Pi^\perp_{a,b}(B)\Pi^\perp_{a,b}(C)=\epsilon (ba)=\Pi^\perp_{a,b}(A)$$ while belonging to the different steps (namely $b\in B$ and $a\in C$) gives $$\Pi^\perp_{a,b}(BC)=\Pi^\perp_{a,b}(B)\Pi^\perp_{a,b}(C)=ba=\Pi^\perp_{a,b}(A),$$ which completes the first part of the proof.\ $\Leftarrow :$\ Now, let us assume that we have two step sequences $u,v\in{\mathbb{S}}^*$ and $$\forall_{(a,b)\notin{\mathit{ind}}}\;\Pi^\perp_{a,b}(v)=\Pi^\perp_{a,b}(u).$$ Without loss of generality we can assume that $u=Au'$ is in the lexicographical canonical form and $v$ consists of indivisible steps only. We claim that then there exist $v',v''\in{\mathbb{S}}^*$ such that $v=v'Av''$, no action occurring in $A$ occurs in $v'$ and $A\times alph(v')\subseteq{\mathit{ind}}$. Directly from the definition of the projection representation we see that all projections onto the subalphabets containing actions from the indivisible step $A$ start with the actions contained in $A$. More precisely, if $a,b\in A$ then $\Pi^\perp_{a,b}(u)$ starts with $ab$, $ba$ or $\perp$, depending on the relation between $a$ and $b$. If $a\in A$ and $b\notin A$ however, $\Pi^\perp_{a,b}(u)$ starts with a single action $a$. Let $v'$ be the longest prefix of $v$ such that $alph(v')\cap A={\varnothing}$ and $v=v'Bv''$. Obviously, all projections onto the subalphabets containing actions from the step $A$ are equal for $v$ and $Bv''$. Moreover, from the definition of the indivisible step, between every two actions $a,b$ contained in $A$ there is a sequence of pairwise different actions $a=a_1,\ldots,a_n=b$ contained in $A$ such that for every $i<n$ we have $(a_{i+1},a_i)\in{\mathit{sin}}$. It means that for every such a pair of consecutive actions we have $\Pi^\perp_{a_i,a_{i+1}}(B)=a_ia_{i+1}$ if $(a_{i+1},a_i)\in{\mathit{wdp}}$ or $\Pi^\perp_{a_i,a_{i+1}}(B)=\perp$ if $(a_{i+1},a_i)\in{\mathit{ssm}}$. Nevertheless, if $a_{i+1}$ is in $B$ then also $a_i$ have to be in $B$. Otherwise $\Pi^\perp_{a_i,a_{i+1}}(B)$ would start with $a_{i+1}$. This proves that, since $A\cap B\neq{\varnothing}$, $A\subseteq B$. Using similar arguments, we can see that since $B$ is indivisible, no other action may occur in $B$ and $A=B$. It remains to be shown that $A\times alph(v')\subseteq{\mathit{ind}}$. Let $a\in A$ and $c\in alph(v')$. Clearly, $c\notin A$ from the definition of sequence $v'$. In the step sequence $v$ the action $c$ appears before action $a$ so, if they are not independent, $\Pi^\perp_{a,c}(v)=\Pi^\perp_{a,c}(u)$ starts with $c$. But $a\in A$ and $c\notin A$, and so $\Pi^\perp_{a,c}(u)$ starts with $a$. This contradicts our assumption that $a$ and $c$ are not independent and proves that $v\equiv_\Theta Av'v''$. Repeating the above reasoning, we obtain that $u$ is the lexicographical canonical form of $v$ which ends the second part of the proof. The projection representation of a comtrace $\tau$ is a function $\Pi^\perp_\tau:(\Sigma\times\Sigma)\setminus{\mathit{ind}}\rightarrow (\Sigma\cup\{\perp\})^*$, given by $\Pi^\perp_\tau(a,b)=\Pi^\perp_{a,b}(\tau)$. Moreover, any function $\Pi^\perp:(\Sigma\times\Sigma)\setminus{\mathit{ind}}\rightarrow (\Sigma\cup\{\perp\})^*$ is called a *projection set*. Clearly, not every projection set is a projection representation of a comtrace. In the next section, we give a procedure that decides whether a given projection set is a projection representation of a comtrace. Moreover, if the answer is positive, the procedure computes a representative of such a comtrace. First, however, we provide the algorithm computing projection representation of a comtrace. This algorithm comes directly from the definition. However, to say anything about the time complexity of the algorithm, it is important to discuss the data structures which might be used by this algorithm. At the beginning, let us consider the input. We get a comtrace alphabet $\Theta$ which consists of the alphabet $\Sigma$ of size $k$ and two relations, ${\mathit{sim}}$ and ${\mathit{ser}}$, of size at most $k^2$ each. We also get a step sequence $w$ which steps consist of $n$ occurrences of atomic actions (elements of $\Sigma$) all together. As a result, we obtain the set of at most $k^2$ sequences (projections onto specified subalphabets). We process the step sequence $w$ step by step, which means that the algorithm is online (i.e. during the computation we achieve correct results for each proper prefix of $w$). The processing of a single step is done according to the definition of projections onto the pairs in the specified relation. It is worth carrying out some preprocessing and, for every action, compute the list of all subalphabets in which it may occur. By storing, for every computed projection, the number of the step when it was most recently updated, we avoid problems with the special cases of relations ${\mathit{wdp}}$ and ${\mathit{ssm}}$ (in these cases two rather than one action may be added to one sequence while processing a single step). \[p:al1comp\] The procedure of computing $\Pi^\perp_\tau$ from a step sequence $w\in\tau$ has the time and memory complexity of $O(nk)$. The proof is straightforward. The algorithm is naturally divided into $n$ stages grouped by steps of input step sequence. In each stage we process a single action and add it to at most $k$ sequences updating at most $k$ counters. Hence each stage can be done in the time linearly proportional to the size of the alphabet. Therefore whole procedure has the time complexity of $O(nk)$. Testing comtrace equivalence can be done in the time complexity of $O(nk)$. Notice that the output of procedure discussed in Proposition \[p:al1comp\] has also memory complexity of $O(nk)$. Hence for two step sequences we can compute their projection representations and compare them sequence by sequence. Reconstructing Step Sequence from Projection Set ------------------------------------------------ The idea of constructing a step sequence from a projection set is based on revealing the first possible step whose projection representation would form a set of prefixes of a given projection set. At first, we identify the set of all possible elements of such a step. We do it in two stages. We first identify the set of conditionally possible actions, i.e. those actions whose first occurrences are the first (or in particular situations the second) actions in all projections, where they could appear. Note that we treat the special symbol $\perp$ as a pair of proper actions, so its occurrence means that both actions might be conditionally possible. After this identification, we remove actions that cannot satisfy some of the necessary conditions. These conditions are related to the cases when the considered action appears as the second action in some sequences connected with the weak dependence relation or are verified positively because of the special symbol $\perp$. As a result of the first stage, we obtain the set of all actions that may appear in the first step of the constructed sequence. The second stage consists of dividing this set into indivisible steps and combining those indivisible steps into one of the allowed steps. The result is obtained by taking advantage of the weak dependence relation inside the set of indivisible steps. It is similar to the ideas behind the proof of Lemma \[l:division\]. Let us look into the details of the proposed procedure. Recall that by ${\mathit{pref_k}}(w)=a_1\ldots a_k$ we denote the k-prefix of $w$. Let $\Pi^\perp$ be a projection set. We say that an action $a\in\Sigma$ is *conditionally possible* for projection set $\Pi^\perp$ if and only if for all $b\in\Sigma$ the following implications are satisfied:\ [$\bullet$]{} $(a,b)\in{\mathit{dep}}\Rightarrow {\mathit{pref_1}}(\Pi^\perp(a,b))=a$ $(b,a)\in{\mathit{wdp}}\Rightarrow {\mathit{pref_1}}(\Pi^\perp(a,b))=a$ $(a,b)\in{\mathit{wdp}}\Rightarrow {\mathit{pref_1}}(\Pi^\perp(a,b))=a \vee {\mathit{pref_2}}(\Pi^\perp(a,b))=ba$ $(a,b)\in{\mathit{ssm}}\Rightarrow {\mathit{pref_1}}(\Pi^\perp(a,b))=a \vee {\mathit{pref_1}}(\Pi^\perp(a,b))=\perp$ We denote all conditionally possible actions as $cpa$ and define the relation $cnd\subseteq\Sigma\times\Sigma$, which describes the conditions that must be satisfied. Only in situations where $$(a,b)\in{\mathit{wdp}}\wedge {\mathit{pref_2}}(\Pi^\perp(a,b))=ba$$ or $$(a,b)\in{\mathit{ssm}}\wedge {\mathit{pref_1}}(\Pi^\perp(a,b))=\perp$$ we say that the existence of action $b$ in the constructed step is a necessary condition for the presence of action $a$ in this step, which is denoted by $(a,b)\in cnd$. We exclude conditionally possible actions with conditions impossible to satisfy to form the set of possible actions. Any action $a\in\Sigma$ that is not conditionally possible in $\Pi^\perp$ is *impossible* in $\Pi^\perp$. Moreover, any action $a$ conditionally possible under impossible condition (i.e. $(a,b)\in cnd$ and $b$ is impossible) is also impossible. Formally, the set of impossible actions for the projection function $\Pi^\perp$ is the smallest set $imp$ that satisfies the following conditions: [$\bullet$]{} $\Sigma\setminus cpa\subseteq imp$ $b\in imp\wedge (a,b)\in cnd\Rightarrow a\in imp$ Let $M(\Pi^\perp)$ be the set of actions which are not impossible (which means that they are possible) for projection set $\Pi^\perp$. The next operation is to choose a subset of $M(\Pi^\perp)$ which could be a first step of the reconstructed step sequence. To do so we take a sequential trace over $\widehat{{\mathbb{S}}}$, given by the step sequence $\widehat{M(\Pi^\perp)}$ (see Corollary \[c:correspondence\]). Note that for any $a\in M(\Pi^\perp)$ we have $\#_a(\widehat{M(\Pi^\perp)})\leq 1$. We take any nonempty trace prefix $B_1\ldots B_n$ of step sequence $\widehat{M(\Pi^\perp)}$ and set $B=\bigcup_{i}\;B_i$ as a requested step. The procedure just described is justified by the following facts: Let $\Pi^\perp$ be a projection set over a comtrace alphabet $\Theta$ and $B\subseteq M(\Pi^\perp)$ a set of actions constructed according to the procedure described above. If $b\in B$ and $a\in M(\Pi^\perp)$ then $$(b,a)\in{\mathit{sin}}^*~~\Longrightarrow~~[a]_{\equiv_{M(\Pi^\perp)}}\subseteq B.$$ Let $\widehat{M(\Pi^\perp)}=B_1\ldots B_n$, where all the $B_m$’s are indivisible. Since $a,b\in M(\Pi^\perp)$ there exist $1\leq p,q\leq n$ such that $a\in B_p$ and $b\in B_q$. By Lemma \[l:division\] $[a]_{\equiv_{M(\Pi^\perp)}}=B_p$. By Corollary \[c:correspondence\], $\widehat{M(\Pi^\perp)}$ forms a sequential trace over $\widehat{{\mathbb{S}}}$. In the above procedure we use one of trace prefixes of $\widehat{M(\Pi^\perp)}$, taking $B$ as the union of all indivisible steps (actions of $\widehat{{\mathbb{S}}}$) contained in this prefix. Hence $B_q\subseteq B$. If $p=q$ we have that $B_p=B_q$ and $B_p=[a]_{\equiv_{M(\Pi^\perp)}}\subseteq B$. Let us consider the case $B_p\neq B_q$. It is sufficient to prove that $B_p$ occurs before $B_q$ in all trace prefixes of $\widehat{M(\Pi^\perp)}$. Since $(b,a)\in\sin^*$, there exists a sequence of actions $b=c_1\ldots c_k=a$ such that $(c_i,c_{i+1})\in\sin$ for every $0<i<k$. Hence there exists a sequence of steps $u=C_1\ldots C_k$ such that $c_i\in C_i$. Clearly, $C_i$ might be equal to $C_{i+1}$, for some $0<i<k$, but surely $C_1\neq C_k$. However, for distinct $i,j$ we have $(C_i,C_j)\in\widehat{{\mathit{dep}}}$. Moreover, each $C_i$ is contained in $Alph(M(\Pi^\perp))$ and if $C_i$ occurs before $C_j$ in $u$, then it also has to occur before $C_j$ in $\widehat{M(\Pi^\perp)}$. If $C_i$ and $C_{i+1}$ are different, then during the division of the step $\widehat{M(\Pi^\perp)}$ (see Lemma \[l:division\]) they have to get to different parts (like steps $B$ and $C$ in Lemma \[l:division\]). Since $(c_i,c_{i+1})\in{\mathit{sin}}$, it is impossible to have $C_i\times C_{i+1}\subseteq{\mathit{ser}}$. This shows that their orders of occurring in $u$ and $\widehat{M(\Pi^\perp)}$ are reversed. Moreover, this remains true for every sequence over $\widehat{{\mathbb{S}}}$ equivalent to $\widehat{M(\Pi^\perp)}$. Finally, we conclude that what we have shown applies not only to consecutive and distinct steps of $u$ but also to all its distinct elements, including $C_1=B_q$ and $C_k=B_p$, which end the proof. Let $w=A_1\ldots A_n$ be a step sequence, and $\Pi^\perp$ be the projection representation of $[w]$. Then $$A_1\subseteq M(\Pi^\perp).$$ Since $\Pi^\perp$ is the projection representation of $[w]$, for all $(a,b)\notin{\mathit{ind}}$ we have $$\Pi^\perp(a,b)=\Pi^\perp_{a,b}(w)=\Pi^\perp_{a,b}(A_1\ldots A_n).$$ Hence all actions contained in $A_1$ are conditionally possible. Moreover, $(a,b)\in cnd$ means that $(a,b)\in{\mathit{wdp}}$ or $(a,b)\in{\mathit{ssm}}$. In the first case, ${\mathit{pref_2}}(\Pi^\perp(a,b))=ba$, so $b\in A_1$. Similarly, if $(a,b)\in{\mathit{ssm}}$ then ${\mathit{pref_2}}(\Pi^\perp(a,b))=\perp$, so $b\in A_1$. Since $(a,b)\in cnd$ and $a\in A_1$ implies $b\in A_1$, and $A_1\subseteq cpa$, we conclude that $A_1\cap imp={\varnothing}$. This proves that $A_1\subseteq M(\Pi^\perp)$. As a result, we can extract step $B$ from $\Pi^\perp$. The extraction function $$extr:((\Sigma\times\Sigma\setminus{\mathit{ind}})^*\rightarrow (\Sigma\cup\perp)^*)\times {\mathbb{S}}\rightarrow((\Sigma\times\Sigma\setminus{\mathit{ind}})^*\rightarrow (\Sigma\cup\perp)^*)$$ for projection set $\Pi^\perp$ and set $B\subseteq M(\Pi^\perp)$ constructed using the procedure described above is defined as: $$extr(\Pi^\perp,B)(a,b)=\left\{ \begin{array}{lcl} \Pi^\perp(a,b) & \text{ for } & |\{a,b\}\cap B|=0\\ {\mathit{suff_2}}(\Pi^\perp(a,b)) & \text{ for } & |\{a,b\}\cap B|=1\\ {\mathit{suff_2}}(\Pi^\perp(a,b)) & \text{ for } & |\{a,b\}\cap B|=2 \wedge (a,b)\in{\mathit{ssm}}\\ {\mathit{suff_3}}(\Pi^\perp(a,b)) & \text{ for } & |\{a,b\}\cap B|=2 \wedge (a,b)\in{\mathit{wdp}}\cup{\mathit{wdp}}^{-1}\\ \end{array} \right.$$ Let us consider the comtrace $\tau$ from Example \[e:comtrace\]. The projection representation of $\tau$ (omitting projections to the unary subalphabets), grouped by the types of relation between the elements of subalphabets on which we project are: $$\begin{array}{lll} {\mathit{dep}}: \;&\; \Pi^\perp_\tau(b,d)=db \;&\; \Pi^\perp_\tau(c,d)=d\\ {\mathit{ssm}}: \;&\; \Pi^\perp_\tau(a,c)=a &\\ {\mathit{wdp}}: \;&\; \Pi^\perp_\tau(c,b)=b \;&\; \Pi^\perp_\tau(d,a)=da \end{array}$$ The set of conditionally possible actions for $\Pi^\perp_\tau$ is $\{a,d\}$, while $(a,d)\in cnd$. Every conditionally possible action is also possible, and so $M(\Pi^\perp_\tau)=\{a,d\}$. This gives the set of two indivisible steps $(a)$ and $(d)$ and, finally, two steps that may appear as the first step of the constructed sequence: $(d)$ and $(ad)$. \[t:extrcorrectness\] Let $\Pi^\perp_\tau$ be the projection representation of a comtrace $\tau$, and $M(\Pi^\perp)$ be a maximal possible step of $\Pi^\perp_\tau$. For every allowed set $B\in{\mathbb{S}}$, we have $$\tau = B\circ \sigma, \text{ where }\Pi^\perp_\sigma=extr(\Pi^\perp_\tau,B).$$ By the Theorem \[t:prrepr\] it is sufficient to prove that $\Pi^\perp_\tau = \Pi^\perp_{B\circ\sigma}$. In other words, we have to show that for all $(a,b)\notin{\mathit{ind}}$, we have $\Pi^\perp_\tau(a,b)=\Pi^\perp_B(a,b)\circ\Pi^\perp_\sigma(a,b)$. The proof can be split in a natural way into three parts, depending on the type of relation between the actions being considered. Let us examine the projections onto $(a,b)\in{\mathit{dep}}$. We have $\Pi^\perp_B(a,b)$ that is equal to the first action of $\Pi^\perp_\tau(a,b)$ if $|B\cap\{a,b\}|=1$, and to $\epsilon$ otherwise. In both cases $\Pi^\perp_B(a,b)\circ \Pi^\perp_\sigma(a,b)=\Pi^\perp_\tau(a,b)$. Almost the same proof works for the remaining two cases, when $(a,b)\in{\mathit{wdp}}$ or $(a,b)\in{\mathit{ssm}}$. By suitably using the extraction function, we can compute any representative of a comtrace $\tau$. In particular, similarly to the case of canonical forms, we can do this using a maximal or minimal strategy. In the maximal strategy, we always take the whole set $M(\Pi^\perp)$ and, as a result, we obtain Foata canonical form of the original comtrace. In the minimal strategy, we take the first step of the step sequence $\widehat{M(\Pi^\perp)}$ and obtain the lexicographical canonical form. The algorithm reconstructing a step sequence from a projection representation of a comtrace follows the notions defined above. From the technical point of view, some concrete decisions concerning data structures are worth noticing. The whole algorithm can be divided into stages. In each stage we compute a set of allowed steps, choose one, and extract it from the projection set. The procedure is repeated until a projection set $\Pi^\perp_i$ or computed set $M(\Pi^\perp_i)$ become empty. In the first case, it returns a step sequence consisting of $n$ occurrences of actions. In the second case, the algorithm returns that an input is not a projection representation of a comtrace. A single stage starts from computing the set of conditionally possible actions and the relation $cnd$ describing the conditions. A good idea is to preprocess, for every action, a list of pointers which helps to investigate only the projections related to this action. Doing so, we can check conditional possibility in the time linearly dependent on the size of alphabet, denoted by $k$. Simultaneously, we build the directed graph of conditions. In the time linearly dependent on the number of arcs in this graph, we remove from the set of conditionally possible actions all impossible ones (browsing, using DFS, all paths which begin in vertices which are not conditionally possible). In the next phase, we compute a vertex induced subgraph of the ${\mathit{sin}}$ relation that contains all possible actions and, once more using DFS, we compute a graph of its strongly connected components (called *condensation graph* [@Deo74]). The condensation graph is an acyclic directed graph of the partial order of the sequential trace associated with $\widehat{M(\Pi^\perp)}$. We choose an arbitrary upper set of the condensation graph, that corresponds to the trace prefix of $\widehat{M(\Pi^\perp)}$. To obtain Foata canonical form, we take the maximal upper set by choosing the whole condensation graph. If we wish to obtain the lexicographical canonical form, we should choose the $\widehat{\leq}$-smallest allowed step. To compute it, we may consider only the maximal elements of provided condensation graph. They correspond to the elements of $\widehat{{\mathbb{S}}}$ which may be placed at the first positions in the sequential trace $\widehat{M(\Pi^\perp)}$. In the last phase, we need to extract the chosen allowed step. We do it according to the definition of the extraction operation. During this phase, we can once more use the precomputed lists of pointers. Projection set $\Pi^\perp$ is the projection representation of a comtrace if and only if the procedure described above ends with the empty projection set. We give the proof only for the case when the maximal strategy is used. Note that the input data is finite and the procedure stops when the set $M(\Pi^\perp)$ is empty for the remaining set of words. From Theorem \[t:extrcorrectness\] we deduce that if the remaining projection set is empty then the input is the projection representation of the constructed comtrace. Suppose that we have nonempty projection set $\Pi^\perp$ that is a projection representation of comtrace $\tau$ and empty set of allowed actions. Let us consider an arbitrary step sequence $u=A_1\ldots A_n$ that is contained in $\tau$, and an arbitrary action $a$ contained in $A_1$. Then, by the definition of projection representation, the action $a$ has to be possibly allowed. This proves that $A_1\subseteq cpa$. Moreover, since in any projection before, or simultaneously with, $a$ may occur only other action from the step $A_1$, if the existence of action $b$ is a necessary condition for the presence of action $a$ (i.e. $(a,b)\in cnd$), then $b$ is also an element of $A_1$. Therefore, none of the actions from step $A_1$ is impossible, which contradicts the emptiness of the set of allowed actions and ends the proof. The procedure of computing canonical forms from a projection representation of a comtrace has the time complexity of $O(nk^2)$. The procedure consist of at most $n$ stages. In each part, we carry out some operations on at most $k^2$ lists and graph of size $k^2$. All graph operations, including computing the compensation graph and choosing minimal or maximal upper set are linear in the size of graph. This gives an overall time complexity of $O(nk^2)$. Traces as a subclass of comtraces --------------------------------- In Section 1 we defined EN-systems as a special case of ENI-systems without inhibitors and with the sequential semantics. We also introduced traces as a model of the causal behaviour of EN-systems. In this section, we show what kind of comtraces are directly related to systems without inhibitors. A comtrace alphabet $\Theta=(\Sigma,{\mathit{sim}},{\mathit{ser}})$ with the empty relation ${\mathit{sin}}$ is called *[radical ]{}comtrace alphabet*. Moreover, comtraces over this alphabet are called *[radical ]{}comtraces*. The radicalism of such comtraces means that the actions may be only dependent or independent, hence they behave exactly like step traces. Later in this section we discuss some properties of this subclass. \[l:singletons\] Let $\tau\in{\mathbb{S}}^*$ be a [radical ]{}comtrace and $w\in indiv(\tau)$. Then each step of $w$ is a singleton. The proof is straightforward. Notice that since the relation ${\mathit{sin}}$ is empty, every action $a$ of every step $A\in{\mathbb{S}}$ forms an indivisible step. Hence all indivisible steps are singletons, which ends the proof. \[c:tracelikealph\] Let $\Theta=(\Sigma,{\mathit{sim}},{\mathit{ser}})$ be a [radical ]{}comtrace alphabet. Then $$lex(\widehat{{\mathbb{S}}})=\Sigma.$$ Note that since the relation ${\mathit{sin}}$ is empty and all steps are singletons, for all steps $A,B\in\widehat{{\mathbb{S}}}$ we have $(A,B)\in\widehat{{\mathit{dep}}}$ if and only if $(lex(A),lex(B))\in{\mathit{dep}}$. Using Theorem \[t:trace4comtrace\] and Lemma \[c:tracelikealph\] we can associate an alphabet of indivisible steps $\widehat{{\mathbb{S}}}$ with $\Sigma$ and [radical ]{}comtrace $\tau$ over a comtrace alphabet $\Theta=(\Sigma,{\mathit{sim}},{\mathit{ser}})$ with a step trace $\sigma$ over the concurrent alphabet $\Psi=(\Sigma,{\mathit{ind}})$. We say that such a step trace $\sigma$ is a *trace representation* of a [radical ]{}comtrace $\tau$. The following facts show this correspondence in details. Let $\Theta=(\Sigma,{\mathit{sim}},{\mathit{ser}})$ be a [radical ]{}comtrace alphabet. Then a set $A\subseteq\Sigma$ is a step in $\Theta$ if and only if $A$ is a step in $\Psi=(\Sigma,\widehat{{\mathit{dep}}})$. It is sufficient to prove that ${\mathit{sim}}={\mathit{ind}}$. Indeed, since ${\mathit{sin}}$ is empty, we have ${\mathit{sim}}\setminus{\mathit{ser}}={\varnothing}$, hence by ${\mathit{ser}}\subseteq{\mathit{sim}}$ we get ${\mathit{sim}}={\mathit{ser}}$. Recall that ${\mathit{sim}}$ is symmetric, and so is ${\mathit{ser}}$. By the definition of relations in comtraces, $${\mathit{ind}}={\mathit{ser}}\cap{\mathit{ser}}^{-1}={\mathit{ser}}={\mathit{sim}}.$$ Let $\tau$ be a [radical ]{}comtrace and $\sigma$ be its step trace representation. Then $\Pi^\perp_\tau=\Pi_\sigma$. Let $\tau=[A_1\ldots A_n]=\sigma$. The relation ${\mathit{sin}}$ is empty, so in the case of comtraces we consider only projections to pair of actions that are dependent. As a result, we conclude that the projections on the same pairs of actions are the same, no matter whether we consider comtraces or step traces, $\Pi^\perp_{a,b}(\tau)=\Pi_{a,b}(\sigma)$ for every $(a,b)\in{\mathit{dep}}$, hence $\Pi^\perp_\tau=\Pi_\sigma$. Let $\tau\in{\mathbb{S}}^*$ be a [radical ]{}comtrace and $\sigma$ be its step trace representation. Their canonical forms (both lexicographical and Foata) are equal. \[c:TLcorrespondence\] The correspondence between comtraces over $\Theta=(\Sigma,{\mathit{sim}},{\mathit{ser}})$ and traces over $\Psi=(\widehat{{\mathbb{S}}},\widehat{{\mathit{dep}}})$ (see Corollary \[c:correspondence\]) collapses in case of [radical ]{}comtraces to $$\begin{array}{ccc} \tau & &\sigma\\ over & \longleftrightarrow & over\\ \Theta & &\Psi \end{array}\;,$$ where $\tau=\sigma$ as sets of step sequences. Summary and future work ======================= In this paper we presented a number of algebraic aspects of combined traces. Similar algebraic tools were successfully used in the study of the Mazurkiewicz traces, a simpler model for capturing and analysing concurrent behaviours. In particular, we defined lexicographical canonical form of a comtrace and its projection representation. We gave two simple algorithms which generate these representations from arbitrary step sequence. Those algorithms seem to have the potential to provide a base for the development of solutions to some natural problems related to the comtrace theory, like model verification [@EspHel08; @RodSchKho13]. In particular, one can use them to design efficient methods for the enumeration of all the representatives of a fixed comtrace, and the enumeration of all comtraces of a given size. Another interesting direction of further studies would be the notion of recognisable and rational languages of combined traces. The projection representation seems to be a good starting point in this area; in particular, if one recalls Zielonka’s asynchronous automata [@Zie87] for traces. Finally, the projection representation may find an application in another important aspect of combined trace theory. A fair strategy of reconstructing step sequences from a projection set might be useful as a starting point in the theory of infinite combined traces. Acknowledgments {#acknowledgments .unnumbered} --------------- I would like to thank Maciej Koutny and anonymous reviewers for their constructive comments, which helped to improve this paper.\ This research was supported by a fellowship funded by the “Enhancing Educational Potential of Nicolaus Copernicus University in the Disciplines of Mathematical and Natural Sciences” Project POKL.04.01.01-00-081/10.

--- address: | National Astronomical Observatories, Chinese Academy of Sciences\ Jia 20 Da-Tun Road, Beijing 100012, China\ author: - 'J.L. Han' title: | The Large-Scale Magnetic Field Structure of Our Galaxy:\ Efficiently Deduced from Pulsar Rotation Measures --- \#1\#2\#3\#4[(\#1) [\#2]{} [**\#3**]{}, \#4]{} Introduction ============ The origin of magnetic fields in the universe is a long-standing problem. It is clear today that magnetic fields play a crucial role in the evolution of molecular clouds and star formation (e.g. Rees 1987). The diffuse magnetic fields are the physical means to confine the cosmic rays (e.g. Strong et al. 2000). The magnetic fields in galactic disks also have a significant contribution to the hydrostatic balance in the interstellar medium (Boulares & Cox 1990). However, it is not clear whether magnetic fields exist in the very early universe, e.g. the recombination phase, and whether the fields affect the structure formation and galaxy formation afterward. To understand the magnetic fields, the first step is to correctly and properly describe their properties based on reliable observations. Magnetic fields of galactic scales ($\sim 10$ kpc) are the most important connection between the magnetic fields at cosmological scales and the fields in currently observable objects. In the last two decades, there have been many observations on magnetic fields in galaxies (see references in reviews by Beck et al. 1996; Han & Wielebinski 2002). Theoretically the magnetic fields in galaxies are believed to be re-generated and maintained by dynamo actions in the interstellar medium (e.g. Ruzmaikin et al. 1988; Kulsrud 1999). The helical turbulence (the $\alpha$-effect) and differential rotation (the $\Omega$-effect) are two key ingredients for dynamos in galaxies (e.g. Krause & Rädler 1980). Our Galaxy is the unique case for detailed studies of magnetic fields, as I will show in this review. It is certainly desirable to know the structure of our Galaxy and to compare it with the magnetic structures. However, as we live near the edge of the disk of the Milky Way, it is impossible for us to get a clear bird-view of the global structure of the whole Galaxy. As has been shown by various tracers, our Galaxy obviously has a few spiral arms with a pitch angle of about 10$^\circ$. But there is no consensus on the number of spiral arms in our Galaxy and whether and how these arms are connected in the opposite side. Apart from the thin disk, there is a thick disk or halo, filled by low-density gas and possibly weak magnetic fields. Pulsars are the best probes for the large-scale magnetic fields in our Galaxy. At present, many new pulsars have been discovered up to distances further than the Galactic center (e.g. Manchester et al. 2001; Morris et al. 2002; Kramer et al. 2003), which can be used to probe the large-scale magnetic fields in about half of the Galactic disk. The magnetic field may further give us some hints to the global structure of our Galaxy. Hundreds of pulsars discovered at high Galactic latitudes (e.g. Edwards et al. 2001) can be used to study the magnetic fields in the Galactic halo. Comments on definitions of useful terms --------------------------------------- Before we start to discuss observational results of magnetic fields, it is worth to clarify the definition of some useful terms. \ How large is the “[*large scale*]{}”? Obviously it should be a scale, relatively much larger than some kind of standard. For example, the large-scale magnetic field of the Sun refers to the global-scale field or the field with a scale-length comparable to the size of Sun, up to $10^9$ m, rather than small-scale magnetic fields in the solar surface. For the magnetic fields of our Galaxy, we should define the [*large scale*]{} as being a [*scale larger than the separation between spiral arms*]{}. That is to say, large scale means a scale larger than 2 or 3 kpc. Note that in the literature, [*large scale*]{} is sometimes used for large [*angular*]{} scale when discussing the structures or prominent features [*in the sky plane*]{}, e.g., the large-scale features in radio continuum radio surveys. These [*large angular-scale*]{} features are often very localized phenomena, and not very large in linear scale. \ Whether a magnetic field is ordered or random depends on the scales concerned. A uniform field at a 1-kpc scale could be part of random fields at a 10-kpc scale, while it is of a very large scale relative to the pc-scale magnetic fields in molecular clouds. [*Uniform fields*]{} are [*ordered fields*]{}. [*Regularly ordered fields*]{} can coherently change their [*directions*]{}, so they may not be [*uniform fields*]{}. Deviations from [*regular fields*]{} or [*orderd fields*]{} are taken as [*random fields*]{}. The fluctuations of fields at scales 10 times smaller than a concerned scale are oftern taken as [*random fields*]{}. Note also that in the literature the measurements of so-called [*polarization vectors*]{} only give the [*orientations*]{} rather than [*directions*]{} of magnetic fields, so they are not real vectors. \ Often the magnetic fields in our Galaxy are expressed in cylindrical coordinates ($\theta,\ r,\ z$). The [*azimuthal component*]{}, $B_\theta$, of magnetic fields dominates in the Galactic disk, where the [*radial and vertical components*]{}, $B_r$ and $B_z$, are generally weak. The [*toroidal component*]{} refers to the structures (without $B_z$ components) confined to a plane parallel to the Galactic plane, while the [*poloidal component*]{} refers to the axisymmetrical field structure around $z$, such as dipole fields (without $B_\theta$ component). Concerning measurements relative to the line of sight, only one field component, either [*perpendicular*]{} or [*parallel*]{} to the line of sight, can be detected by one method (see below).\ The above terms are artificially designed for convenience when studying magnetic fields. Real magnetic fields would be all connected in space, with all components everywhere. Observational tracers of magnetic fields ---------------------------------------- [**Zeeman splitting:**]{}\ It measures the [*parallel component*]{} of magnetic fields in an emission or absorption region by using the the splitting of spectral lines. Up to now, measurements of magnetic fields [*in situ*]{} of masers (e.g. Fish et al. 2003) and molecular clouds (e.g. Bourke et al. 2001) are available. The relationship between the field strength and gas density has been supported by observational data (e.g. Crutcher 1999). Despite strong efforts it failed to relate the magnetic fields in situ and the large-scale fields (see Fish et al. 2003), as suggested by Davies (1974) and later promoted by Reid & Silverstain (1990). \ Dust particles are preferentially orientated due to the ambient magnetic fields. The thermal emission of dust then naturally has linear polarization. Infrared, mm, submm instruments are the best to detect thermal emission. Polarization shows directly magnetic fields projected in the sky plane. Due to the short wavelengths, such a polarized emission does not suffer from any Faraday rotation when passing through the interstellar space. The recent advance in technology has made it possible to make direct polarization mapping at infrared, sub-mm and mm wavebands (e.g. Hildebrand et al. 1998; Novak et al. 2003). At present, measurements can only be made for bright objects, mostly of molecular clouds. But in future it could be more sensitive and powerful to measure even nearby galaxies. A combination of polarization mapping for the perpendicular component of magnetic fields with the parallel components measured from Zeeman splitting will give a 3-D information of magnetic fields in molecular clouds (or galaxies in future), which is certainly crucial to study the role of magnetic fields in the star-formation process. The available measurement, which is really related to large-scale magnetic fields, is the polarization mapping of the central molecular zone by Novak et al. (2003). The results revealed possible toroidal fields parallel to the Galactic disk. This field is probably part of an A0 dynamo field in the Galactic halo, complimented to the poloidal fields traced by vertical filaments in the Galactic center (e.g. Sofue et al. 1987; Yusef-Zadeh & Morris 1987). \ The starlight is scattered by interstellar dust when traveling from a star to the earth. The dust particles are preferentially orientated along the interstellar magnetic fields, which induce the polarization of the scattered star light: more scattering, more polarization. Starlight polarization was the start for the studies of the large-scale magnetic field. Apparently, the starlight can trace prominent magnetic features on large angular scales, over all the sky! Directly from the measurements in the Galactic pole regions, we can easily see the direction of the local magnetic fields in our Galaxy. However, because the measured stars are mostly within 1 or 2 kpc from the Sun, it is not possible to trace magnetic fields further away. Polarization measurements of stars near the Galactic plane give us only the information that the magnetic fields in the Galactic disk are mainly orientated parallel to the Galactic plane. \ Synchrotron radiation from relativistic electrons shows the [*orientation*]{} of magnetic fields in the emission region. Assuming energy equipartition, one may estimate the field strength from the emission flux. Furthermore, from polarized intensity, the energy in the “uniform field” together with that of anisotropic random fields can be estimated. Many nearby galaxies have been observed in polarized radio emission. The emission suffers from Faraday rotatation within the medium of the galaxy and from the Milky Way. After correcting the foreground RMs, one can obtain a map of intrinsic [*orientations*]{} of the [*transverse component*]{} of magnetic fields, though it is often called the “vector” map of the magnetic field. The anisotropic random magnetic fields, such as compressed random fields by large-scale density waves, can also produce such an observed polarization map. So, with a polarization map with a coherent “vector” pattern, one cannot claim the large-scale magnetic field. However, a map of the RM distribution with a regular pattern, especially for the inclined galaxies, provides strong evidence for the large-scale magnetic fields in the halo or the thick disk of a galaxy (see Krause, this volume). As we will show below, the pulsar RM distribution indeed shows large-scale magnetic fields in our Galaxy, with coherent field [*directions*]{} going along the spiral arms. This indirectly proves that the polarization maps of nearby galaxies can be at least partially due to the large-scale magnetic field. The strength of “regular fields” calculated from the polarization percentage may be overestimated, however, as one cannot quantify the contributions from anisotropic random magnetic fields (see Beck et al. 2003). \ Faraday rotation occurs when radio waves travel through the magnetized medium. The RM, which is measured as being the rate of polarization-angle change against the square of wavelength, is an integration of magnetic field strength together with electron density along the line of sight from the source to the observer, i.e., $RM = a \int_{\rm source}^{\rm observer} \; n_e \; B_{||} \; dl. $ Here $a$ is a constant, $n_e$ the electron density, $dl$ is a unit length of the line of sight. Obviously, RMs measure the average diffuse magnetic field if the electron density is known. In the following I will concentrate on how the RMs of pulsars and extragalactic radio sources can be used to probe the large-scale magnetic fields in our Galaxy. Considering various difficulties in other methods, we are really lucky that we can use the RMs of an increasing number of pulsars to probe the large-scale magnetic field in the Galactic disk. It is very hard to conduct observations of Zeeman splitting, and afterwards it is not possible to relate these measurements to the large-scale magnetic fields. Starlight polarization does measure the diffuse magnetic field, but it is not possible to give any other information than the averaged orientation of the field in just 1 or 2 kpc. Diffuse radio emission from our Galaxy can only show the polarized emission from local regions on large [*angular*]{} scales, but not on large linear scales in the Galactic disk. The magnetic field in the Galactic halo: based on the RM sky distribution ========================================================================= In the sky the Milky Way is the largest edge-on Galaxy. This gives us the unique chance to study the magnetic fields in a galactic halo in detail, which is not possible at all for nearby extragalaxies. There is a huge number of extragalactic radio sources as well as hundreds of pulsars, which can be potentially used to probe the magnetic fields in the Galactic halo and in the Galactic disk. The RM of an extragalactic radio source consists of a RM contribution intrinsic to the source, the RM from the intergalactic space from the source to the Galaxy, and the RM within the Galaxy. The first term should be random and hence reasonably small on average, because a source can be randomly orientated in space with any possible field configuration. We observe a random quantity. The second term is ignorable on average. Intergalactic magnetic fields are too weak to be detectable now. A source can be at any possible location in the universe. Even if there is a weak field in the intergalactic space, an integration over the path-length of the intergalactic magnetic fields with random directions together with extremely thin gas should give a quite small combination. Therefore, the common contribution to RMs of extragalactic radio sources is from our Galaxy. So, the [*averaged*]{} sky distribution of RMs of extragalactic sources (see Fig. 1) should be the best presentation for the Galactic magnetic field in the Galactic halo. Though there are many distinguished characteristics related to specific regions in the RM sky, we noticed that the most prominent feature is the antisymmetry in the inner Galactic quadrants (i.e. $|l|<90^{\circ}$). The positive RMs in the regions of ($0^{\circ}<l<90^{\circ},\; b>0^{\circ}$) and ($270^{\circ}<l<360^{\circ}, \; b<0^{\circ}$) indicate that the magnetic fields point towards us, while the negative RMs in the regions of ($0<l<90^{\circ}, \; b<0^{\circ}$) and ($270<l<360^{\circ}, \; b>0^{\circ}$) indicate that the magnetic fields point away from us. Such a high symmetry to the Galactic plane as the Galactic meridian through the Galactic center cannot simply be caused by localized features as previously thought. The antisymmetric pattern is very consistent with the magnetic field configuration of an A0 dynamo, which provides such toroidal fields with reversed directions above and below the Galactic plane (Fig. 1). The toroidal fields possibly extend to the inner Galaxy, even towards the central molecular zone (Novak et al. 2003). This magnetic field model is also supported by the nonthermal radio filaments observed in the Galactic center region for a long time, which have been thought to be indications for the poloidal field in dipole form (Yusef-Zadeh & Morris 1987; Sofue et al. 1987). We noticed that the antisymmetric RM sky is also shown by pulsar RMs at high Galactic latitudes ($|b|>8^{\circ}$). This implies that the magnetic fields responsible for the antisymmetry pattern could be nearer than the pulsars. Indeed, the fields nearer than the pulsars contribute to the RMs, but this is not the only contribution. If it is only a local effect, then there would be no symmetric RM distribution beyond the pulsars. We made computer simulations, which show that large-scale magnetic field structures further away than the pulsars (i.e. the more inner Galaxy) should result in a strong antisymmetry of RM sky within 50$^{\circ}$ from the Galactic center, and that the magnitudes of RMs are systematically increasing towards the Galactic plane and the Galactic center. To judge if antisymmetry is produced by a large-scale magnetic field, it would be necessary to subtract the foreground pulsar RMs induced by local magnetic fields from the RMs of extragalactic radio sources and then check the antisymmetry of the residual RM map. If there is antisymmetry in the residual RM map, then it is large-scale, otherwise it is local. However, both the RM data of pulsars and extragalactic radio sources are so sparse that such a subtraction cannot give a clear result at the moment. We checked the magnitude distribution of RMs of extragalactic radio sources, which is indeed systematically larger than those of pulsars, as expected from the large-scale halo field (Fig.2). More RM data of both pulsars and extragalactic radio sources towards medium Galactic latitudes in the inner Galaxy are desired to check this crucial issue of Galactic dynamo. If such a large-scale magnetic field model is confirmed by more data, then our Galaxy is the first galaxy in which the dynamo signature is clearly identified. In other words, it is the first time to identify a dynamo at a galactic scale. This is very difficult for other galaxies. We studied all possible RM data for M31, and got some evidence for the magnetic field configuration of a possible S0 dynamo (Han et al. 1998). Magnetic fields in the Galactic disk: based on pulsar RMs ========================================================= Pulsars are the best probes for Galactic magnetic fields. First of all, pulsars are highly polarized in general. So their RMs are relatively easy to measure, in contrast to the great difficulties to do Zeeman splitting measurements. Second, pulsars do not have any intrinsic RM. So, what one gets from RMs is just the contribution from the interstellar medium, an integration of diffuse magnetic fields along the path from a pulsar to the observer, rather than [*in situ*]{} measurements in emission regions. Third, the total amount of the electron density between a pulsar and an observer can be measured independently by the pulsar dispersion measure, $DM=\int_{\rm psr}^{\rm obs}\; n_e \; dl$. This leads to a direct measure of the averaged magnetic field along the line of sight by using $\langle B_{||} \rangle = 1.232 RM/DM$. There have been many pulsars discovered, which are widely spread in our Galaxy. After measuring their RMs, it will not be very hard to find a 3-D magnetic field structure in our Galaxy. Historical landmarks of using the pulsar RMs for the large-scale magnetic fields -------------------------------------------------------------------------------- A close look at the historical landmarks of using pulsar RMs to study the Galactic magnetic fields will show the progress in the last two or three decades. Soon after the pulsars were discovered, Lyne & Smith (1969) detected their linear polarization. They noted that this “opens up the possibility of measuring the Faraday rotation in the interstellar medium” which “gives a very direct measure of the interstellar magnetic field”, because $\int_{\rm obs}^{\rm obj} n_e dl $ can be measured by $DM$ so that $\langle B_{||}\rangle$ can be directly obtained from the ratio $RM/DM$. Manchester (1972, 1974) first systematically measured a number of pulsar RMs for Galactic magnetic fields and concluded that the local field (within 2 kpc!) is directed toward about $l\sim90\degr$. Thomson & Nelson (1980) modeled the pulsar RMs mostly within 2 kpc and found the [*first*]{} field reversal near the Carina-Sagittarius arm. The largest pulsar RM dataset was published by Hamilton & Lyne (1987), mostly for pulsars at about 5 kpc and some up to 10 kpc. Then Lyne & Smith (1989) used pulsar RMs to further study the Galactic magnetic field. They confirmed the first field reversal in the inner Galaxy and found evidence for the field reversal in the outer Galaxy by a comparison of pulsar RMs with those of extragalactic radio sources. Rand & Kulkarni (1989) analyzed 185 pulsar RM data and proposed the ring model for the Galactic magnetic field. Rand & Lyne (1994) observed more RMs of distant pulsars and found evidence for the clock-wise field near the Crux-Scutum arm (at about 5 kpc). Han & Qiao (1994) and Indrani & Deshpande (1998) reanalyzed the pulsar RM data and found that the RM data are more consistent with the bisymmetric spiral model than with the ring model. Han et al. (1997) first noticed that the RM distribution of high-latitude pulsars is dominated by the azimuthal field in the halo. Afterwards, any analysis of pulsar RMs for the disk field was limited to pulsars at lower Galactic latitudes ($|b|<8\degr$). Han et al. (1999) then observed 63 pulsar RMs and divided all known pulsar RMs into those lying within higher and lower latitude ranges for studies of the halo and disk field, respectively, and they confirmed the bisymmetric field structure and refined estimates of the vertical field component. The large-scale magnetic field models ------------------------------------- There have been three models to describe the global magnetic field structure of our Galaxy. In the early stage, Simard-Normandin & Kronberg (1980) showed that the RMs of extragalactic radio sources and pulsars are consistent with the bisymmetric spiral model. This was later confirmed by Sofue & Fujimoto (1983). The currently available pulsar RM data are mostly consistent with a bisymmetric spiral model as we will discuss below. While Vallée (1991, 1995) has argued for an axisymmetric spiral model. In this model, the field reversal occurs only in the range of Galactic radii from 5 to 8 kpc (Vallée 1996). No field reversals are allowed beyond 8 kpc or within 5 kpc from the Galactic Center. This is in contrast to the field reversals suggested beyond the solar circle and detected interior to the Crux-Scutum arm (e.g. Han et al. 1999, 2002). We noticed that recent arguments favour no field reversal outside the Perseus arm (e.g. Brown et al. 2003), which need more RM data of pulsars in the Perseus arm to check. Extragalactic radio source data are not enough to make a solid conclusion. The concentric ring model, proposed by Rand & Kulkarni (1989) and Rand & Lyne (1994), has a pitch angle of zero, but the observed pitch angle of fields of $-8^{\circ}$ favours a spiral form of the field structure. Both the ring model and axisymmetric spiral model show that the magnetic field lines go across the spiral arms, which seems not to be physically possible. Current status and future directions ------------------------------------ Up to now, among about $\sim$1450 known pulsars, 535 pulsars have measured values of RM and 373 of them are located at lower latitudes ($|b|<8\degr$). This includes 200 RM data from Parkes observations, which will be published soon (Han et al. in prep.). Significant progress has been made in the last decade on the magnetic fields in the Galactic disk, mainly because many pulsars have been discovered in the nearby half of the whole Galactic disk (e.g. Manchester et al. 1996; Lyne et al. 1998; Manchester et al. 2001) and extensive observations of pulsar RMs (e.g. Hamilton & Lyne 1987; Rand & Lyne 1994; Han et al. 1999) were conducted. Analysis of pulsar RMs needs to consider three important factors for the diagnosis of the large-scale field structure. First, one normally assumes that the azimuthal field component $B_{\phi}$ is greater than the vertical and radial components $B_z$ or $B_r$. This is reasonable and has been justified (Han & Qiao 1994; Han et al. 1999). Second, it is [*the gradient of the average or general tendency of RM variations*]{} versus pulsar DMs that traces the large-scale field. The scatter of the data about this general tendency is probably mostly due to the effect of smaller scale interstellar structure. Finally, the large-scale field structure should produce a coherence in the gradients for many independent lines of sight (see e.g. $l=\pm20\degr$ near the Norma arm in Fig. 4). From the most updated RM distribution (see Fig. 4), we can conclude that magnetic fields between the Perseus arm and Carina-Sagittarius arm have a clock-wise direction when looking from the Northern galactic pole. Apparently this at least holds for about 5 kpc along the spiral arms. Between the Carina-Sagittarius arm and the Crux-Scutum arm, the positive RMs near $l \sim 50^{\circ}$ and negative RMs near $l\sim 315^{\circ}$ show the coherently counter-clockwise magnetic field along the spiral arm over more than 10 kpc! From the RMs of pulsars discovered by the Parkes multibeam survey, the counter-clockwise magnetic field along the Norma arm (i.e. the 3-kpc arm) has been clearly identified (Han et al. 2002). There have been some indications for clockwise magnetic fields between the Crux-Scutum arm and the Norma arm, while more RM data are obviously desired for a definite conclusion. In the outer Galaxy the magnetic fields directions in or outside the Perseus arm have been in controversy recently. The magnetic field reversals suggested by Lyne & Smith (1989) have been confirmed by Han et al. (1999) and Weisberg et al. (2004) using available pulsar RMs mostly near $l\sim70^{\circ}$. While Mitra et al. (2003) and Brown et al. (2003) have argued for no reversal near or outside the Perseus arm from the RM data of pulsars and extragalactic radio sources in the region of $145^{\circ}<l<105^{\circ}$. The average of RM values seems not to be significantly different for the foreground pulsar RMs near the Perseus arm and to the background extragalactic radio sources. This fact probably indicates two field reversals outside the Perseus arm which cancels their RM contributions. It is necesary to compare RM data of pulsars in the Perseus arm and background extragalactic radio sources between $45^{\circ}<l<110^{\circ}$ for that purpose. A solid conclusion about the magnetic field configurations in this region would come out soon after many more pulsars in this region will be discovered in a future Arecibo L-band multibeam pulsar survey. Discussions ----------- Beside the large-scale magnetic field, naturally there are small-scale magnetic fields in our Galaxy. The strength of the large-scale magnetic field has been estimated to be $1.8\pm0.5\;\mu$G (Han & Qiao 1994; Indrani & Deshpande 1998), while the total field strength estimated from cosmic-rays or using the equipartition assumptions is about 6$\mu$G. We have composed the energy spectrum of Galactic magnetic fields at different scales, from 0.5 kpc to 15 kpc (Han et al. 2003). Based on this spectrum, we estimate the fluctuations of magnetic fields have an rms field strength about 6 $\mu$G, which is very consistent with estimates for the total field strength by other methods. This confirms that the magnetic field strengths estimated from pulsar rotation measures are statistically fine for the diffuse interstellar medium. The field strength of regular magnetic fields estimated from the percentage of polarized continuum emission of nearby galaxies then probably has been two or three times overestimated. Recently, Mitra et al. (2003) have shown that two or three pulsar RMs are affected by ${{\rm H\,\scriptstyle II}}$ regions as these pulsars can be easily identified by their large DMs. In fact, the large DM should lead to an overestimated distance for the pulsar in a given electron density model. However, only a very small number of pulsars can be affected by chance, according to simulations made by Cordes & Lazio (2002). The coherent variation of RMs versus DMs in different directions not only provides the information about the large-scale magnetic fields, but also indicates that the data scattering from the general tendency of variation due to small-scale regions does not influence the analysis of RMs for the large-scale magnetic fields. Conclusions =========== Pulsars provide unique probes for the [*large-scale*]{} interstellar magnetic field in the Galactic disk. Other methods seem to have many difficulties for that purpose. The increasing number of RMs, especially of newly discovered distant pulsars, enables us for the first time to explore the magnetic field in nearly one third of the Galactic disk. The fields are found to be [*coherent in directions*]{} over a linear scale of more than $\sim 10$ kpc between the Carina-Sagittarius and Crux-Scutum arms from $l\sim45\degr$ to $l\sim305\degr$ and more than 5 kpc along the Norma arm. The magnetic fields reverse their directions from arm to arm. The coherent spiral structures and field direction reversals, including the newly determined counter-clockwise field near the Norma arm, are consistent with a bisymmetric spiral model for the disk field. At high latitudes, the antisymmetric RM sky is most probably produced by the toroidal field in the Galactic halo. Together with the dipole field in the Galactic center, it strongly suggests that an A0 dynamo is operating in the halo of our Galaxy. Acknowledgments {#acknowledgments .unnumbered} =============== I am very grateful to many colleagues, especially, Prof. R.N. Manchester and Prof. G.J. Qiao for working together with me to improve the knowledge on the magnetic fields of our Galaxy, some of which was presented here. The author is supported by the National Natural Science Foundation of China (10025313) and the National Key Basic Research Science Foundation of China (G19990754) as well as from the partner group of MPIfR at NAOC. I wish to thank the organizers of the conference to invite me to present this review, and many thanks to Dr. Wolfgang Reich and Ms. Gabi Breuer for carefully reading the manuscript. References {#references .unnumbered} ========== Beck R., Brandenburg A., Moss D., Shukurov A., Sokoloff D. . Beck R., Shukurov A., Sokoloff D., Wielebinski R. . Boulares A., Cox D.P. . Bourke T.L., Myers P.C., Robinson G., Hyland A. R. . Brown J. C., Taylor A. R., Wielebinski R., Mueller P. . Cordes J.M., Lazio T.J.W. (2002) , submitted. Crutcher R.M. . Davies R.D. (1974) [*IAU Simp.*]{} [**60**]{}, 275. Edwards R.T., Bailes M., van Straten W., Britton M.C. . Fish V., Reid M.J., Argon A.L., Menten K.M. . Hamilton P.A., Lyne A.G. . Han J.L., Beck R., Berkhuijsen E.M. . Han J.L., Ferriere K., Manchester R.N. (2003) , submitted. Han J.L., Manchester R.N., Lyne A.G., Qiao G.J. . Han J.L., Qiao G.J. . Han J.L., Manchester R.N., Berkhuijsen E.M., Beck R. . Han J.L., Manchester R.N., Qiao G.J. . Han J.L., Wielebinski R. (2002) [*Chinese Journal of A&A*]{} [**2**]{}, 293. Hildebrand R.H., Davidson J.A., Dotson J.L., Dowell C.D., Novak G., Vaillancourt J. . Indrani C., Deshpande A.A. (1998) [*New Astronomy*]{} [**4**]{}, 33. Kramer M., Bell J.F., Manchester R.N. et al. . Krause F., Raedler K.H. 1980, [*Mean-field Magnetohydrodynamics and Dynamo Theory*]{} (Oxford: Pergamon Press). Kulsrud R.M. . Lyne A.G., Smith F.G. . Lyne A.G., Smith F.G. . Lyne A.G., Manchester R.N., Lorimer D.R., Bailes M., D’Amico N., Tauris T.M., Johnston S., Bell J.F., Nicastro L. . Manchester R.N. . Manchester R.N. . Manchester R.N., Lyne A.G., Camilo F. et al. . Manchester R.N., Lyne A.G., D’Amico N., Bailes M., Johnston S., Lorimer D.R., Harrison P.A., Nicastro L., Bell J.F. . Mitra D., Wielebinski R., Kramer M., Jessner A. . Morris D.J., Hobbs G., Lyne A.G. et al. . Novak G., Chuss D.T., Renbarger T., Griffin G.S., Newcomb M.G., Peterson J.B., Loewenstein R.F., Pernic D., Dotson J.L. . Rand R.J., Kulkarni S.R. . Rand R.J., Lyne A.G. . Rees M. . Reid M.J., Silverstein E.M. . Ruzmaikin A.A., Sokolov D.D., Shukurov A.M. (1988) [*Magnetic Fields of Galaxies*]{} (Dordrecht: Kluwer). Simard-Normandin M., Kronberg P.P. . Sofue Y., Fujinmoto M. . Sofue Y., Reich W., Inoue M., Seiradakis J.H. . Strong A.W., Moskalenko I.V., Reimer O. . Thomson R.C., Nelson A.H. . Vallée J.P. . Vallée J.P. . Vallée J.P. . Weisberg J.M., Cordes J.M., Kuan B., Devine K.E., Green J.T., Backer D.C. (2004) , in press. Yusef-Zadeh F., Morris M. .

--- abstract: 'We propose a flexible gradient-based framework for learning linear programs from optimal decisions. Linear programs are often specified by hand, using prior knowledge of relevant costs and constraints. In some applications, linear programs must instead be learned from observations of optimal decisions. Learning from optimal decisions is a particularly challenging bi-level problem, and much of the related [*inverse optimization*]{} literature is dedicated to special cases. We tackle the general problem, learning all parameters jointly while allowing flexible parametrizations of costs, constraints, and loss functions. We also address challenges specific to learning linear programs, such as empty feasible regions and non-unique optimal decisions. Experiments show that our method successfully learns synthetic linear programs and minimum-cost multi-commodity flow instances for which previous methods are not directly applicable. We also provide a fast batch-mode PyTorch implementation of the homogeneous interior point algorithm, which supports gradients by implicit differentiation or backpropagation.' author: - | Yingcong Tan\ Concordia University\ Montreal, Canada\ Daria Terekhov\ Concordia University\ Montreal, Canada\ Andrew Delong\ Concordia University\ Montreal, Canada\ bibliography: - 'File/bibliography.bib' title: Learning Linear Programs from Optimal Decisions ---

--- abstract: 'For simple mechanical systems, bifurcating branches of relative equilibria with trivial symmetry from a given set of relative equilibria with toral symmetry are found. Lyapunov stability conditions along these branches are given.' author: - 'Petre Birtea, Mircea Puta, Tudor S. Ratiu, Răzvan Tudoran' date: 'November 3, 2003' title: Symmetry breaking for toral actions in simple mechanical systems --- Introduction ============ This paper investigates the problem of symmetry breaking in the context of simple mechanical systems with compact symmetry Lie group $G$. Let $\mathbb{T}$ be a maximal torus of $G$ whose Lie algebra is denoted by $\mathfrak{t}$. Denote by $Q $ the configuration space of the mechanical system. Assume that every infinitesimal generator defined by an element of $\mathfrak{t}$ evaluated at a symmetric configuration $q_e \in Q$ whose symmetry subgroup $G_{q_e}$ lies in $\mathbb{T}$ is a relative equilibrium. The goal of this paper is to give sufficient conditions capable to insure the existence of points in this set from which branches of relative equilibria with trivial symmetry will emerge. Sufficient Lyapunov stability conditions along these branches will be given if $G = \mathbb{T}$. The strategy of the method can be roughly described as follows. Denote by $\mathfrak{t} \cdot q_e$ the set of relative equilibria described above. Take a regular element $\mu\in \mathfrak{g}^{\ast }$ which happens to be the momentum value of some relative equilibrium in $\mathfrak{t}\cdot q_{e}$. Choose a one parameter perturbation $\beta(\tau, \mu) \in \mathfrak{g}^\ast$ of $\mu$ that lies in the set of regular points of $\mathfrak{g}^{\ast }$, for small values of the parameter $\tau>0 $. Consider the $G_{q_{e}}$-representation on the tangent space $T_{q_{e}}Q$. Let $v_{q_e}$ be an element in the $\{e\}$-stratum of the representation and also in the normal space to the tangent space at $q_{e}$ to the orbit $G\cdot q_{e}$. Assume that its norm is small enough in order for $v_{q_e}$ to lie in the open ball centered at the origin $0_{q_e} \in T_{q_e}Q $ where the Riemannian exponential is a diffeomorphism. The curve $\tau v_{q_e}$ projects by the exponential map to a curve $q_e( \tau) $ in a neighborhood of $q_e $ in $Q$ whose value at $\tau= 0$ is $q_e$. Note that the isotropy subgroup at every point on this curve, except for $\tau= 0 $, is trivial. We shall search for relative equilibria in $TQ$ starting at points of $\mathfrak{t}\cdot q_e $ such that their base curve in $Q $ equals $q_e(\tau)$ and their momentum values are $\beta(\tau, \mu) $. To do this, we shall choose a curve $\xi(\tau, v_{q_e}, \mu) \in \mathfrak{g} $ uniquely determined by $\beta( \tau, \mu)$; as will be explained in the course of the construction, $\xi(\tau, v_{q_e}, \mu)$ equals the value of the inverse of the locked inertial tensor on $\beta(\tau, \mu)$ for $\tau \neq 0 $. If one can show that the limit of $\xi( \tau, v_{q_e}, \mu) $ exists and belongs to $\mathfrak{t}$ for $\tau \rightarrow 0$, then the infinitesimal generator of this value evaluated at $q_e$ is automatically a relative equilibrium since it belongs to $\mathfrak{t}\cdot q_e$. It will be also shown that the infinitesimal generators of $\xi(\tau, v_{q_e}, \mu)$ evaluated at $q_e(\tau)$ are relative equilibria. This produces a branch of relative equilibria starting at this specific point in $\mathfrak{t}\cdot q_e$ which has trivial isotropy for $\tau>0 $ and which depends smoothly on the additional parameter $\mu \in \mathfrak{g}^\ast$. In this method, there are two key technical problems, namely, the existence of the limit of $\xi( \tau,v_{q_e}, \mu)$ as $\tau \rightarrow 0$ and the extension of the amended potential at points with symmetry. The existence of the limit of $\xi( \tau,v_{q_e}, \mu)$ as $\tau \rightarrow 0$ will be shown using the Lyapunov-Schmidt procedure. To extend the amended potential and its derivative at points with symmetry, two auxiliary functions obtained by blow-up will be introduced. The analysis breaks up in two problems on a space orthogonal to the $G$-orbit. The present paper can be regarded as a sequel to the work of Hernández and Marsden [@hm]. The main difference is that one single hypothesis from [@hm] has been retained, namely that all points of $\mathfrak{t} \cdot q_e $ are relative equilibria. We have also eliminated a strong nondegeneracy assumption in [@hm]. But the general principles of the strategy of the proof having to do with a regularization of the amended potential at points with symmetry, where it is not a priori defined, remains the same. In a future paper we shall further modify this method to deal with bifurcating branches of relative equilibria that have a given isotropy, different from the trivial one, along the branch. The paper is organized as follows. In §\[Lagrangian mechanical systems\] we quickly review the necessary material on symmetric simple mechanical systems and introduce the notations and conventions for the entire paper. Relative equilibria and their characterizations for general symmetric mechanical systems and for simple ones in terms of the augmented and amended potentials are recalled in §\[Relative equilibria\]. Section §\[Some basic results from the theory of group actions\] gives a brief summary of facts from the theory of proper group actions needed in this paper. After these short introductory sections, §\[Regularization of the amended potential criterion\] presents the main bifurcation result of the paper. The existence of branches of relative equilibria starting at certain points in $\mathfrak{t}\cdot q_e$, depending on several parameters and having trivial symmetry off $\mathfrak{t}\cdot q_e$, is proved in Theorem \[principala\], the main result of this paper. In §\[stability section\], using a result of Patrick [@patrick; @thesis], Lyapunov stability conditions for these branches are given if the symmetry group is a torus. Lagrangian mechanical systems {#Lagrangian mechanical systems} ============================= This section summarizes the key facts from the theory of Lagrangian systems with symmetry and sets the notations and conventions to be used throughout this paper. The references for this section are [@f; @of; @m], [@lm], [@marsden; @92], [@ims]. Lagrangian mechanical systems with symmetry -------------------------------------------- Let $Q$ be a smooth manifold, the configuration space of a mechanical system. The [[******]{}fiber derivative]{} or [[******]{}Legendre transform]{} $\mathbb{F}L:TQ\rightarrow T^{\ast }Q$ of $L$ is a vector bundle map covering the identity defined by $$\langle \mathbb{F}L(v_{q}),w_{q}\rangle = \left.\frac{d}{dt}\right |_{t=0}L(v_{q}+tw_{q})$$ for any $v_q, w_q \in TQ$. The [[******]{}energy]{} of $L $ is defined by $E(v_{q})= \langle \mathbb{F}L(v_{q}),v_{q} \rangle -L(v_{q})$, $v_{q}\in T_{q}Q$. The pull back by $\mathbb{F}L $ of the canonical one– and two–forms of $T ^\ast Q $ give the [[******]{}Lagrangian one]{} and [[******]{}two-forms]{} $\Theta _{L}$ and $\Omega_L$ on $TQ$ respectively, that have thus the expressions $$\langle\Theta _{L}(v_{q}),\delta v_{q}\rangle=\langle\mathbb{F}L(v_{q}),T_{v_{q}}\pi _{Q}(\delta v_{q})\rangle,\quad v_{q}\in T_{q}Q,\quad \delta v_{q}\in T_{v_{q}}TQ, \qquad \Omega_{L}=-\mathbf{d}\Theta _{L},$$ where $\pi_Q : TQ \rightarrow Q$ is the tangent bundle projection. The Lagrangian $L $ is called [[******]{}regular]{} if $\mathbb{F}L $ is a local diffeomorphism, which is equivalent to $\Omega_L $ being a symplectic form on $TQ$. The Lagrangian $L $ is called [[******]{}hyperregular]{} if $\mathbb{F}L $ is a diffeomorphism and hence a vector bundle isomorphism. The [[******]{}Lagrangian vector field]{} $X_E $ of $L$ is uniquely determined by the equality $$\Omega _{L}(v_{q})(X_{E}(v_{q}),w_{q})=\langle \mathbf{d}E(v_{q}),w_{q}\rangle, \quad \text{for} \quad v_{q}, \; w_{q}\in T_{q}Q.$$ A [[******]{}Lagrangian dynamical system]{}, or simply a [[******]{}Lagrangian system]{}, for $L$ is the dynamical system defined by $X_{E}$, i.e., $\dot{v} =X_{E}(v)$. In standard coordinates $(q^i, \dot{q}^i) $ the trajectories of $X_E $ are given by the second order equations $$\frac{d}{dt}\frac{\partial L}{\partial \dot{q}^{i}}-\frac{\partial L}{ \partial q^{i}}=0,$$ which are the classical the Euler-Lagrange equations. Let $\Psi: G \times Q \rightarrow Q$ be a smooth left Lie group action on $Q$ and let $L:TQ\rightarrow \mathbb{R}$ be a Lagrangian that is invariant under the lifted action of $G$ to $TQ$. Denote by $\mathfrak{g}$ the Lie algebra of $G$. From the definition of the fiber derivative it immediately follows that $\mathbb{F}L $ is equivariant relative to the lifted $G $–actions to $TQ$ and $T^\ast Q$. The $G$-invariance of $L$ implies that $X_{E}$ is $G$-equivariant, that is, $\Psi_g ^\ast X_E = X_E $ for any $g \in G $. The $G $–action on $TQ $ admits a momentum map given by $$\langle\mathbf{J}_{L}\mathbf{(}v_{q}),\xi \rangle=\langle\mathbb{F}L(v_{q}),\xi _{Q}(q)\rangle, \quad \text{for} \quad v_{q}\in T_{q}Q, \quad \xi \in \mathfrak{g}.$$ where $\xi _{Q}(q) : = d\exp (t\xi) \cdot q/dt|_{t=0}$ is the [[******]{}infinitesimal generator]{} of $\xi \in \mathfrak{g}$, where $\mathfrak{g}$ denotes the Lie algebra of $G $. Recall that the momentum map $\mathbf{J}:T^{\ast}Q\rightarrow \mathfrak{g}^{\ast }$ on $T^{\ast }Q$ is given by $$\langle\mathbf{J(}\alpha _{q}),\xi \rangle=\langle\alpha _{q},\xi _{Q}(q)\rangle, \quad \text{for} \quad \alpha _{q}\in T_{q}^{\ast }Q, \quad \xi \in \mathfrak{g}$$ and hence $\mathbf{J}_{L}= \mathbf{J\circ }\mathbb{F}L$. We shall denote by $g \cdot q: =\Psi(g,q)$ the action of the element $g\in G $ on the point $q \in Q $. Similarly, the lifted actions of $G $ on $TQ$ and $T^\ast Q $ are denoted by $$g \cdot v_q : = T_q \Psi_g(v_q) \quad \text{and} \quad g \cdot \alpha_q : = T^\ast_{g\cdot q} \Psi_{g^{-1}}(\alpha_q)$$ for $g\in G $, $v_q \in T_q Q $, and $\alpha_q \in T_q ^\ast Q $. Simple mechanical systems ------------------------- A [[******]{}simple mechanical system]{} $(Q,\langle \!\langle \cdot ,\cdot \rangle \! \rangle _{Q},V)$ consists of a Riemannian manifold $(Q,\langle \!\langle \cdot ,\cdot \rangle \! \rangle _{Q})$ together with a potential function $V:Q\rightarrow \mathbb{R}$. These elements define a Hamiltonian system on $(T^{\ast }Q,\omega )$ with Hamiltonian given by $ H:T^{\ast }Q\rightarrow \mathbb{R}$, $H(\alpha _{q})= \frac{1}{2} \langle \!\langle \alpha _{q},\alpha _{q}\rangle \! \rangle _{T^{\ast }Q}+V(q)$, where $\alpha_q \in T_q ^\ast Q $ and $\langle \!\langle \cdot ,\cdot \rangle \! \rangle _{T^{\ast }Q}$ is the vector bundle metric on $T^{\ast }Q$ induced by the Riemannian metric of $Q$. The Hamiltonian vector field $X_H $ is uniquely given by the relation $\mathbf{i} _{X_{H}}\omega =\mathbf{d}H$, where $\omega$ is the canonical symplectic form on $ T^{\ast }Q$. The dynamics of a simple mechanical system can also be described in terms of Lagrangian mechanics, whose description takes place on $TQ$. The Lagrangian for a simple mechanical system is given by $L:TQ\rightarrow \mathbb{R}$, $ L(v_{q})=\frac{1}{2}\langle\!\langle v_{q},v_{q}\rangle\!\rangle _{Q}-V(q)$, where $v_q \in T_q Q $. The energy of $L $ is $E(v_q) = \frac{1}{2}\langle\!\langle v_q, v_q \rangle\!\rangle + V(q)$. Since the fiber derivative for a simple mechanical system is given by $\langle\mathbb{F}L(v_{q}),w_{q}\rangle=\langle\!\langle v_{q},w_{q}\rangle\!\rangle _{Q}$, or in local coordinates $\mathbb{F}L\left(\dot{q}^{i}\frac{ \partial }{\partial q^{i}}\right)=g_{ij}\dot{q}^{j}dq^{i}$, where $g_{ij}$ is the local expression for the metric on $Q$, it follows that $L $ is hyperregular. The relationship between the Hamiltonian and the Lagrangian dynamics is the following: the vector bundle isomorphism $\mathbb{F}L$ bijectively maps the trajectories of $X_E$ to the trajectories of $X_H$, $(\mathbb{F}L)^\ast X_H = X_E $, and the base integral curves of $X_E $ and $X_H $ coincide. Simple mechanical systems with symmetry {#Simple mechanical systems with symmetry} --------------------------------------- Let $G$ act on the configuration manifold $Q$ of a simple mechanical system $(Q,\langle\!\langle \cdot ,\cdot \rangle\!\rangle _{Q},V)$ by isometries. The [[******]{}locked inertia tensor]{} $\mathbb{I}:Q\rightarrow \mathcal{L(} \mathfrak{g},\mathfrak{g}^{\ast })$, where $\mathcal{L(} \mathfrak{g},\mathfrak{g}^{\ast })$ denotes the vector space of linear maps from $\mathfrak{g}$ to $\mathfrak{g}^\ast$, is defined by $$\langle\mathbb{I(}q)\xi ,\eta \rangle=\langle\!\langle \xi _{Q}(q),\eta _{Q}(q)\rangle\!\rangle _{Q}$$ for any $q \in Q $ and any $\xi, \eta \in \mathfrak{g} $. If the action is [[******]{}locally free]{} at $q\in Q$, that is, the isotropy subgroup $G_q $ is discrete, then $\mathbb{I(}q)$ is an isomorphism and hence defines an inner product on $\mathfrak{g}$. In general, the defining formula of $\mathbb{I}(q)$ shows that $\ker \mathbb{I}(q) = \mathfrak{g}_q: = \{\xi\in \mathfrak{g} \mid \xi_Q(q) = 0 \}$. Suppose the action is locally free at every point $q \in Q $. Then on can define the [[******]{}mechanical connection]{} $\mathcal{A} \in \Omega^1(Q; \mathfrak{g})$ by $$\mathcal{A}(q)(v_{q})=\mathbb{I(}q)^{-1}\mathbf{J}_{L}(v_{q}), \quad v_{q}\in T_{q}Q.$$ If the $G $–action is free and proper, so $Q \rightarrow Q/G $ is a $G $–principal bundle, then $\mathcal{A}$ is a connection one–form on the principal bundle $Q \rightarrow Q/G$, that is, it satisfies the following properties: - $\mathcal{A}(q) :T_q Q\rightarrow \mathfrak{g}$ is linear and $G$-equivariant for every $q \in Q $, which means that $$\mathcal{A}(g \cdot q)(g\cdot v_q) = \operatorname{Ad}_g[\mathcal{A}(q)(v_q)],$$ for any $v_q \in T_q Q $ and any $g \in G$, where $\operatorname{Ad}$ denotes the adjoint representation of $G $ on $\mathfrak{g}$; - $\mathcal{A}(q)(\xi _{Q}(q))=\xi $, for any $\xi \in \mathfrak{g}$. If $\mu\in \mathfrak{g}^\ast$ is given, we denote by $\mathcal{A}_{\mu } \in \Omega^1(Q)$ the $\mu$–component of $\mathcal{A}$, that is, the one–form on $Q$ defined by $\langle\mathcal{A}_{\mu }(q),v_{q}\rangle =\langle\mu ,\mathcal{A}(q)(v_{q})\rangle$ for any $v_q \in T_qQ$. The $G$-invariance of the metric and the relation $$(\operatorname{Ad}_{g}\xi )_{Q}(q)=g\cdot \xi _{Q}(g^{-1}\cdot q),$$ implies that $$\label{equivariance of I} \mathbb{I(}g\cdot q)=\operatorname{Ad}_{g^{-1}}^{\ast }\circ \mathbb{I(}q)\circ \operatorname{Ad}_{g^{-1}}.$$ We shall also need later the infinitesimal version of the above identity $$\label{infinitesimal equivariance of I} T_q\mathbb{I} \left(\xi_{Q}(q)\right) = -\operatorname{ad}_{\xi }^{\ast} \circ \mathbb{I}(q) - \mathbb{I}(q) \circ \operatorname{ad}_{\xi},$$ which implies $$\label{useful identity for I} \left\langle T_q \mathbb {I} ( \zeta_Q(q)) \xi, \eta \right\rangle = \mathbf{d}\langle\mathbb{I}(\cdot )\xi ,\eta \rangle(q)\left(\zeta _{Q}(q)\right) =\langle\mathbb {I}(q)[\xi,\zeta ],\eta \rangle+\langle\mathbb{I(}q)\xi ,[\eta ,\zeta ]\rangle.$$ for all $q\in Q$ and all $\xi, \eta, \zeta\in \mathfrak{g}$. Relative equilibria {#Relative equilibria} =================== This section recalls the basic facts about relative equilibria that will be needed in this paper. For proofs see [@f; @of; @m], [@lm], [@marsden; @92], [@ims], [@slm]. Basic definitions and concepts ------------------------------ Let $ \Psi: G \times Q \rightarrow Q$ be a left action of the Lie group on the manifold $Q$. A vector field $X:Q\rightarrow TQ$ is said to be $G$-[[******]{}equivariant]{} if $$T_{q}\Psi _{g}(X(q)) = X(\Psi _{g}(q))\quad \text{or,~equivalently,} \quad \Psi_g^\ast X = X$$ for all $q\in Q$ and $g\in G$. If $X$ is $G$-equivariant, then $G$ is said to be a [[******]{}symmetry group]{} of the dynamical system $\dot{q}=X(q)$. A [[******]{}relative equilibrium]{} of a $G$–equivariant vector field $X$ is a point $q_{e}\in Q$ at which the value of $X$ coincides with the infinitesimal generator of some element $\xi \in \mathfrak{g}$, usually called the [[******]{}velocity]{} of $q_e $, i.e., $$X(q_{e})=\xi _{Q}(q_{e}).$$ A relative equilibrium $q_{e}$ is said to be [[******]{}asymmetric]{} if the isotropy subalgebra $\mathfrak{g}_{q_{e}} : = \{ \eta \in \mathfrak{g} \mid \eta_Q(q_e) = 0 \} =\{0\}$, and [[******]{}symmetric]{} otherwise. Note that if $q_{e}$ is a relative equilibrium with velocity $\xi \in \mathfrak{g}$, then for any $g\in G$, $g\cdot q_{e}$ is a relative equilibrium with velocity $\operatorname{Ad}_{g}\xi $. The flow of an equivariant vector field induces a flow on the quotient space. Thus, if the $G $–action is free and proper, a relative equilibrium defines an equilibrium of the induced vector field on the quotient space and conversely, any element in the fiber over an equilibrium in the quotient space is a relative equilibrium of the original system. Relative equilibria in Hamiltonian $G$-systems ---------------------------------------------- Given is a symplectic manifold $(P, \omega)$, a left Lie group action of $G $ on $P $ that admits a momentum map $\mathbf{J}: P \rightarrow \mathfrak{g}^\ast$, that is, $X_{\mathbf{J}^ \xi} = \xi_P $, for any $\xi\in \mathfrak{g}$, where $\mathbf{J}^ \xi(p): = \langle \mathbf{J}(p), \xi \rangle $, $p \in P$, is the $\xi$–component of $\mathbf{J} $. We shall also assume throughout this paper that the momentum map $\mathbf{J}$ is equivariant, that is, $\mathbf{J}(g \cdot p ) = \operatorname{Ad}^\ast_{g^{-1}} \mathbf{J} (p)$, for any $g \in G $ and any $p \in P $. Given is also a $G $–invariant function $H: P \rightarrow \mathbb{R}$. Noether’s theorem states that the $\mathbf{J} $ is conserved along the flow $F_t $ of the Hamiltonian vector field $X_H $. In what follows we shall call the quadruple $(Q,\omega ,H,\mathbf{J},G)$ a [[******]{}Hamiltonian $G$–system]{}. Consistent with the general definition presented above, a point $ p_{e}\in P$ is a [[******]{}relative equilibrium]{} if $$X_{H}(p_{e})\in T_{p_{e}}(G\cdot p_{e}),$$ where $G\cdot p_e : \{ g \cdot p_e \mid g \in G \} $ denotes the $G $–orbit through $p_e $. Relative equilibria are characterized in the following manner. \[characterization of relative equilibria\] (**Characterization of relative equilibria**). Let $p_{e}\in P$ and $ p_{e}(t)$ be the integral curve of $X_{H}$ with initial condition $p_{e}(0)=p_{e}$. Let $\mu:= \mathbf{ J}(p_{e})$. Then the following are equivalent: 1. $p_{e}$ is a relative equilibrium. 2. There exists $\xi \in \mathfrak{g}$ such that $p_{e}(t)=\exp (t\xi )\cdot p_{e}$. 3. There exists $\xi \in \mathfrak{g}$ such that $p_{e}$ is a critical point of the [[******]{}augmented Hamiltonian]{} $$H_{\xi }(p):=H(p)-\langle\mathbf{J(}p)-\mu ,\xi \rangle.$$ Once we have a relative equilibrium, its entire $G$-orbit consists of relative equilibria and the relation between the velocities of the relative equilibria that are on the same $G$-orbit is given by the adjoint action of $G$ on $\mathfrak{g}$. \[orbita\] With the notations of the previous proposition, let $p_{e}$ be a relative equilibrium with velocity $\xi$. Then 1. for any $g\in G$, $g\cdot q_{e}$ is also a relative equilibrium whose velocity is $\operatorname{Ad}_g \xi$; 2. $\xi(q_{e})\in \mathfrak{g}_\mu : = \{\eta \in \mathfrak{g}\mid \operatorname{ad}^\ast_\eta \mu = 0 \}$, the coadjoint isotropy subalgebra at $\mu\in \mathfrak{g}^\ast$, i.e., $\operatorname{Ad}_{\exp t\xi}^{\ast}\mu=\mu$ for any $t \in \mathbb{R}$. Relative equilibria in simple mechanical $G$-systems ---------------------------------------------------- In the case of simple mechanical $G$-systems, the characterization $\mathbf{(iii)}$ in Proposition \[characterization of relative equilibria\] can be simplified in such way that the search of relative equilibria reduces to the search of critical points of a real valued function on $Q$. Depending on whether one  keeps track of the velocity or the momentum of a relative equilibrium, this simplification yields the *augmented* or the *amended* potential criterion, which we introduce in what follows. Let $(Q,\langle\!\langle \cdot ,\cdot \rangle\!\rangle _{Q},V,G)$ be a simple mechanical $G$–system. - For $\xi \in \mathfrak{g}$, the [[******]{}augmented potential]{} $V_{\xi }:Q\rightarrow \mathbb{R}$ is defined by $V_{\xi }(q):=V(q)-\frac{1}{2}\langle \mathbb{I}(q)\xi ,\xi \rangle$. - For $\mu \in \mathfrak{g}^{\ast }$, the [[******]{}amened potential]{} $V_{\mu }:Q\rightarrow \mathbb{R}$ is defined by $V_{\mu }(q):=V(q)+\frac{1}{2}\langle \mathbb{\mu },\mathbb{I}(q)^{-1}\mu \rangle$. Note that the amended potential is defined at $q\in Q$ only if $q$ in an asymmetric point. There is an alternate expression for the amended potential, namely, $V_{\mu }(q)=(H\circ \mathcal{A}_{\mu })(q)$. (**Augmented potential criterion**). \[augmented potential criterion\] A point $(q_{e},p_{e})\in T^{\ast}Q$ is a relative equilibrium if and only if there exists a $\xi \in \mathfrak{g}$ such that: 1. $p_{e}=\mathbb{F}L(\xi _{Q}(q_{e}))$ and 2. $q_{e}$ is a critical point of $V_{\xi }$. (**Amended potential criterion**). \[amended potential criterion\] A point $(q_{e},p_{e})\in T^{\ast }Q$ is a relative equilibrium if and only if there exists a $\mu \in \mathfrak{g} ^{\ast }$ such that: 1. $p_{e}=\mathcal{A}_{\mu }(q_{e})$ and 2. $q_{e}$ is a critical point of $V_{\mu }$. Some basic results from the theory of Lie group actions {#Some basic results from the theory of group actions} ======================================================= We shall need a few fundamental results form the theory of group actions which we now review. For proofs and further information see [@br], [@dk], [@kawakubo], [@or]. Maximal tori ------------ Let $V$ be a representation space of a compact Lie group $G$. A point $v\in V$ is [[******]{}regular]{} if there is no $G$–orbit in $V$ whose dimension is strictly greater than the dimension of the $G$–orbit through $v$. The set of regular points, denoted $V_{reg}$, is open and dense in $V$. In particular, $\mathfrak{g}_{reg}$ and $\mathfrak{g}_{reg}^{\ast}$, denote the set of regular points in $\mathfrak{g}$ and $\mathfrak{g}^{\ast }$ with respect to adjoint and coadjoint representation, respectively. A subgroup of a Lie group is said to be a [[******]{}torus]{} if it is isomorphic to $S^{1}\times \cdot \cdot \cdot \times S^{1}$. Every Abelian subgroup of a compact connected Lie group is isomorphic to a torus. A subgroup of a Lie group is said to be a [[******]{}maximal torus]{} if it is a torus that is not properly contained in some other torus. Every $\xi \in \mathfrak{g}$ belongs to at least one maximal Abelian subalgebra and every $\xi \in \mathfrak{g\cap g}_{reg}$ belongs to exactly one such maximal Abelian subalgebra. Every maximal Abelian subalgebra is the Lie algebra of some maximal torus in $G$. Let $\mathfrak{t}$ be the maximal Abelian subalgebra corresponding to a maximal torus $T$. Then for any $\xi \in \mathfrak{t\cap g}_{reg}$, we have that $G_{\xi }=T$. The space $[\mathfrak{g},\mathfrak{t}]$ is the orthogonal complement to $\mathfrak{t}$ in $\mathfrak{g}$ with respect to any $G$–invariant inner product on $\mathfrak{g}$. Such an inner product exists by compactness of $G$ by simply averaging any inner product on $\mathfrak{g}$. Therefore, we have $\mathfrak{ g} =\mathfrak{t}\oplus [\mathfrak{g},\mathfrak{t}]$. Let $[\mathfrak{g},\mathfrak{t}]^{\circ }$ the annihilator of $[\mathfrak{g},\mathfrak{t}]$. Then $G_{\mu }=T$ for every $\mu \in [\mathfrak{g},\mathfrak{t} ]^{\circ }\cap \mathfrak{g}_{reg}^{\ast }$. Since $[\mathfrak{g},\mathfrak{t}]^{\circ }\cap \mathfrak{g}_{reg}^{\ast }$ is dense in $[\mathfrak{g},\mathfrak{t}]^{\circ }$, it follows that $T\subset G_{\mu }$ for every $\mu \in [\mathfrak{g},\mathfrak{t}]^{\circ }$. Twisted products ---------------- Let $G$ be a Lie group and $H\subset G$ be a Lie subgroup. Suppose that $H$ acts on the left on a manifold $A$. The [[******]{}twisted action]{} of $H$ on the product $G\times A$ is defined by $$h\cdot (g,a)=(gh,h^{-1}\cdot a), \quad h\in H, \quad g\in G, \quad a\in A.$$ Note that this action is free and proper by the freeness and properness of the action on the $G$–factor. The [[******]{}twisted product]{} $G\times _{H}A$ is defined as the orbit space $(G\times A)/H$ of the twisted action. The elements of $G\times _{H}A$ will be denoted by $[g,a],$ $g\in G,$ $a\in A$. The twisted product $G\times _{H}A$ is a $G$–space relative to the left action defined by $g^{\prime }\cdot \lbrack g,a]=[g^{\prime }g,a]$. Also, the action of $H$ on $A$ is proper if and only if the $G$–action on $G\times _{H}A$ is proper. The isotropy subgroups of the $G$–action on the twisted product $G\times _{H}A$ satisfy $$G_{[g,a]}=gH_{a}g^{-1}, \quad g\in G, \quad a\in A.$$ Slices ------ Throughout this paragraph it will be assumed that $\Psi : G \times Q \rightarrow Q $ is a left proper action of the Lie group $G$ on the manifold $Q$. This action will not be assumed to be free, in general. For $q\in Q$ we will denote by $H:=G_{q} := \{ g \in G \mid g \cdot q = q \}$ the isotropy subgroup of the action $\Psi$ at $q $. We shall introduce also the following convenient notation: if $K \subset G $ is a Lie subgroup of $G $ (possibly equal to $G $), $\mathfrak{k}$ is its Lie algebra, and $q\in Q$, then $\mathfrak{k}\cdot q := \{ \eta_Q(q) \mid \eta \in \mathfrak{k} \}$ is the tangent space to the orbit $K\cdot q $ at $q $. A [[******]{}tube]{} around the orbit $G\cdot q$ is a $G$-equivariant diffeomorphism $\varphi :G\times _{H}A\rightarrow U$, where $U$ is a $G$-invariant neighborhood of $G\cdot q$ and $A$ is some manifold on which $H$ acts. Note that the $G$-action on the twisted product $ G\times _{H}A$ is proper since the isotropy subgroup $H$ is compact and, consequently, its action on $A$ is proper. Hence the $G$-action on $G\times _{H}A$ is proper. Let $S$ be a submanifold of $Q$ such that $q\in S$ and $H\cdot S=S$. We say that $S$ is a [[******]{}slice]{} at $q$ if the map $$\varphi :G\times _{H}S\rightarrow U$$ $$\lbrack g,s]\mapsto g\cdot s$$ is a tube about $G\cdot q$, for some $G$–invariant open neighborhood of $G\cdot q$. Notice that if $S$ is a slice at $q$ then $g\cdot S$ is a slice at the point $g\cdot q$. The following statements are equivalent: 1. There is a tube $\varphi :G\times _{H}A\rightarrow U$ about $G\cdot q$ such that $\varphi ([e,A])=S$. 2. $S$ is a slice at $q$. 3. The submanifold $S$ satisfies the following properties: 1. The set $G\cdot S$ is an open neighborhood of the orbit $G\cdot q$ and $S$ is closed in $G\cdot S$. 2. For any $s\in S$ we have $T_{s}Q=\mathfrak{g}\cdot s+T_{s}S$. Moreover, $\mathfrak{g}\cdot s\cap T_{s}S=\mathfrak{h}\cdot s$, where $\mathfrak{h}$ is the Lie algebra of $H $. In particular $T_{q}Q=\mathfrak{g}\cdot q\oplus T_{q}S$. 3. $S$ is $H$-invariant. Moreover, if $s\in S$ and $g\in G$ are such that $g\cdot s\in S$, then $g\in H$. 4. Let $\sigma :U\subset G/H\rightarrow G$ be a local section of the submersion $G\rightarrow G/H$. Then the map $F:U\times S\rightarrow Q$ given by $F(u,s):=\sigma (u)\cdot s$ is a diffeomorphism onto an open set of $Q$. 4. $G\cdot S$ is an open neighborhood of $G\cdot q$ and there is an equivariant smooth retraction $$r:G\cdot S\rightarrow G\cdot q$$ of the injection $G\cdot q\hookrightarrow G\cdot S$ such that $r^{-1}(q)=S$. **(Slice Theorem)** Let $Q$ be a manifold and $G$ be a Lie group acting properly on $Q$ at the point $q\in Q$. Then, there exists a slice for the $G$–action at $q$. \[tube theorem\] **(Tube Theorem)** Let $Q$ be a manifold and $G$ be a Lie group acting properly on $Q$ at the point $q\in Q$, $H:=G_{q}$. There exists a tube $ \varphi :G\times _{H}B\rightarrow U$ about $G\cdot q$ such that $\varphi ([e,0])=q$, $\varphi ([e,B])=:S$ is a slice at $q$; $B$ is an open $H$–invariant neighborhood of $0$ in the vector space $T_{q}Q/T_{q}(G\cdot q) $, on which $H$ acts linearly by $h\cdot (v_{q}+T_{q}(G\cdot q)):=T_{q}\Psi _{h}(v_{q})+T_{q}(G\cdot q)$. If $Q$ is a Riemannian manifold then $B$ can be chosen to be a $G_{q}$–invariant neighborhood of $0$ in $(\mathfrak{g}\cdot q)^{\perp }$, the orthogonal complement to $\mathfrak{g}\cdot q$ in $T_{q}Q$. In this case $ U=G\cdot \operatorname{Exp}_{q}(B)$, where $\operatorname{Exp}_{q}: T_q Q \rightarrow Q $ is the Riemannian exponential map. Type submanifolds and fixed point subspaces ------------------------------------------- Let $G$ be a Lie group acting on a manifold $Q$. Let $H$ be a closed subgroup of $G$. We define the following subsets of $Q$ : $$\begin{aligned} Q_{(H)} &=&\{q\in Q\mid G_{q}=gHg^{-1},g\in G\}, \\ Q^{H} &=&\{q\in Q\mid H\subset G_{q}\}, \\ Q_{H} &=&\{q\in Q\mid H=G_{q}\}.\end{aligned}$$ All these sets are submanifolds of $Q $. The set $Q_{(H)}$ is called the $(H)$–[[******]{}orbit type submanifold]{}, $ Q_{H}$ is the $H$–[[******]{}isotropy type submanifold]{}, and $Q^{H}$ is the $H$–[[******]{}fixed point submanifold]{}. We will collectively call these subsets the [[******]{}type submanifolds]{}. We have: - $Q^{H}$ is closed in $Q$; - $Q_{(H)}=G\cdot Q_{H}$; - $Q_H $ is open in $Q^H $. - the tangent space at $q \in Q_H $ to $Q_H $ equals $$T_{q}Q_{H}=\{v_{q}\in T_{q}Q\mid T_{q}\Psi_{h}(v_{q}) =v_{q},\, \forall h\in H\}=(T_q Q)^H=T_q Q^H;$$ - $T_{q}(G\cdot q)\cap (T_{q}Q)^{H}=T_{q}(N(H)\cdot q)$, where $N(H) $ is the normalizer of $H $ in $G $; - if $H$ is compact then $Q_{H}=Q^{H}\cap Q_{(H)}$ and $Q_H $ is closed in $Q_{(H)}$. If $Q$ is a vector space on which $H$ acts linearly, the set $Q^{H}$ is found in the physics literature under the names of [[******]{}space of singlets]{} or [[******]{}space of invariant vectors]{}. **(The stratification theorem)**. Let $Q$ be a smooth manifold and $G$ be a Lie group acting properly on it. The connected components of the orbit type manifolds $Q_{(H)}$ and their projections onto orbit space $Q_{(H)}/G$ constitute a Whitney stratification of $Q$ and $Q/G$, respectively. This stratification of $Q/G$ is minimal among all Whitney stratifications of $Q/G$. The proof of this result, that can be found in [@dk] or [@pflaum], is based on the Slice Theorem and on a series of extremely important properties of the orbit type manifolds decomposition that we enumerate in what follows. We start by recalling that the set of conjugacy classes of subgroups of a Lie group $G$ admits a partial order by defining $ (K)\preceq (H)$ if and only if $H$ is conjugate to a subgroup of $K$. Also, a point $q\in Q$ in a proper $G$–space $Q$ (or its corresponding $G$–orbit, $ G\cdot q$) is called [[******]{}principal]{} if its corresponding local orbit type manifold is open in $Q$. The orbit $G\cdot q$ is called [[******]{}regular]{} if the dimension of the orbits nearby coincides with the dimension of $ G\cdot q$. The set of principal and regular orbits will be denoted by $ Q_{princ}/G$ and $Q_{reg}/G$, respectively. Using this notation we have: - For any $q\in Q$ there exists an neighborhood $U$ of $q$ that intersects only finitely many connected components of finitely many orbit type manifolds. If $Q$ is compact or a linear space where $G$ acts linearly, then the $G$–action on $Q$ has only finitely many distinct connected components of orbit type manifolds. - For any $q\in Q$ there exists an open neighborhood $U$ of $q$ such that $(G_{q})\preceq (G_{x})$, for all $x\in U$. In particular, this implies that $\dim G\cdot q\leq \dim G\cdot x$, for all $x\in U$. - [[******]{}Principal Orbit Theorem]{}: For every connected component $ Q^{0}$ of $Q$ the subset $Q_{reg}\cap Q^{0}$ is connected, open, and dense in $Q^{0}$. Each connected component $(Q/G)^{0}$ of $Q/G$ contains only one principal orbit type, which is connected open and dense in $(Q/G)^0 $. Regularization of the amended potential criterion {#Regularization of the amended potential criterion} ================================================= In this section we shall follow the strategy in [@hm] to give sufficient criteria for finding relative equilibria emanating from a given one and to find a method that distinguishes between the distinct branches. The criterion will involve a certain regularization of the amended potential. The main difference with [@hm] is that all hypotheses but one have been eliminated and we work with a general torus and not just a circle. The conventions, notations, and method of proof are those in [@hm]. The bifurcation problem ----------------------- Let $(Q,\langle\!\langle \cdot ,\cdot \rangle\!\rangle _{Q},V,G)$ be a simple mechanical $G$-system, with $G$ a compact Lie group with the Lie algebra $\mathfrak{g}$. Recall that the left $G $–action $\Psi:G \times Q \rightarrow Q $ is by isometries and that the potential $V:Q \rightarrow \mathbb{R}$ is $G$–invariant. Let $ q_{e}\in Q$ be a symmetric point whose isotropy group $G_{q_{e}}\subset \mathbb{T}$ is contained in a maximal torus $\mathbb{T}$ of $G$. Denote by $\mathfrak{t} \subset \mathfrak{g}$ the Lie algebra of $\mathbb{T}$; thus $\mathfrak{t}$ is a maximal Abelian Lie subalgebra of $\mathfrak{g}$. Throughout this section we shall make the following hypothesis: **(H)** *every* $v_{q_{e}}\in \mathfrak{t}\cdot q_{e}$ *is a relative equilibrium*. The following result was communicated to us by J. Montaldi. \[montaldi\] In the context above we have that: $$\begin{array}{cc} (i) & \mathbf{d}V(q_{e})=0 \\ (ii) & \mathbb{I(}q_{e}\mathbb{)}\mathfrak{t}\subseteq [\mathfrak{g}, \mathfrak{t}]^{\circ }. \end{array}$$ $(i)$ Because all the elements in $\mathfrak{t}\cdot q_{e}$ are relative equilibria, we have by the augmented potential criterion $\mathbf{d}V_{\xi }(q_{e})=0$, for any $\xi \in \mathfrak{t}$. Consequently for $\xi =0$ we will obtain $0=\mathbf{d}V_{0}(q_{e})=\mathbf{d}V(q_{e})$. $(ii)$ Substituting in the relation , $q$ by $q_{e}$ and setting $\eta =\xi \in \mathfrak{t}$ we obtain: $$\mathbf{d}\langle\mathbb{I(}\cdot \mathbb{)}\xi ,\xi \rangle(q_{e}) (\zeta _{Q}(q_{e})) =\langle \mathbb{I(}q_{e}\mathbb{)[}\xi ,\zeta ],\xi \rangle +\langle\mathbb{I(}q_{e}\mathbb{)}\xi ,[\xi ,\zeta ]\rangle = 2\langle \mathbb{I(}q_{e}\mathbb{)}\xi ,[\xi ,\zeta ]\rangle$$ for any $\xi \in \mathfrak{t}$ and $\zeta \in \mathfrak{g}$. The augmented potential criterion yields $$0 = \mathbf{d} V_{\xi}(q_{e}) =\mathbf{d}V(q_{e})-\frac{1}{2}\mathbf{d}\langle\mathbb{I(}\cdot \mathbb{)}\xi ,\xi \rangle(q_{e}).$$ Since $\mathbf{d}V(q_{e})=0$ by (i), this implies $\mathbf{d}\langle\mathbb{I(}\cdot \mathbb{)}\xi ,\xi \rangle(q_{e})=0$ and consequently $\langle\mathbb{I(}q_{e}\mathbb{)}\xi , [\xi ,\zeta ]\rangle=0$, for any $\xi \in \mathfrak{t}$ and $\zeta \in \mathfrak{g}$. So we have the inclusion $$\mathbb{I(}q_{e}\mathbb{)}\xi \subseteq [\mathfrak{g},\xi ]^{\circ }.$$ Now we will prove that $[\mathfrak{g},\xi ]^{\circ }=[\mathfrak{g},\mathfrak{ t}]^{\circ }$ for regular elements $\xi \in \mathfrak{t}$. For this it is enough to prove that $[\xi ,\mathfrak{g}]=[\mathfrak{t}, \mathfrak{g}]$ for regular elements $\xi \in \mathfrak{t}$. It is obvious that $[\xi ,\mathfrak{g}]\subseteq \lbrack \mathfrak{t}, \mathfrak{g}]$. Equality will follow by showing that both spaces have the same dimension. To do this, let $F_{\xi }:\mathfrak{g}\rightarrow \mathfrak{g}$, $F_{\xi }(\eta):=\operatorname{ad}_{\xi }\eta $, which is obviously a linear map whose image and kernel are $\operatorname{Im} (F_{\xi })=[\xi ,\mathfrak{g}]$ and $\ker (F_{\xi })=\mathfrak{g} _{\xi }$. Because $\xi \in \mathfrak{t}$ is a regular element we have that $\mathfrak{g}_{\xi }=\mathfrak{t}$ and so $\ker (F_{\xi })=\mathfrak{t}$. Thus $\dim (\mathfrak{g})=\dim ( \mathfrak{t})+\dim ([\xi ,\mathfrak{g}])$ and so using the fact that $\dim ( \mathfrak{g})=\dim (\mathfrak{t})+\dim ([\mathfrak{t},\mathfrak{g}])$ (since $\mathfrak{g}=\mathfrak{t}\oplus [\mathfrak{g},\mathfrak{t}]$, $\mathfrak{g}$ being a compact Lie algebra), we obtain the equality $\dim ([\xi ,\mathfrak{g}])=\dim ([\mathfrak{t},\mathfrak{g}])$. Therefore, $[\xi,\mathfrak{g}]=[\mathfrak{t},\mathfrak{g}]$ for any regular element $\xi \in \mathfrak{t}$. Summarizing, we proved $$\mathbb{I(}q_{e}\mathbb{)}\xi \subseteq [\mathfrak{g},\mathfrak{t} ]^{\circ },$$ for any regular element $\xi \in \mathfrak{t}$. The continuity of $\mathbb{I(}q_{e}\mathbb{)}$, the closedness of $[\mathfrak{g},\mathfrak{t}]^{\circ }$, and that fact that the regular elements $\xi \in \mathfrak{t}$ form a dense subset of $\mathfrak{t}$, implies that $$\mathbb{I(}q_{e}\mathbb{)}\xi \subseteq [\mathfrak{g},\mathfrak{t} ]^{\circ },$$ for any $\xi \in \mathfrak{t}$ and hence $\mathbb{I(}q_{e}\mathbb{)} \mathfrak{t}\subseteq [\mathfrak{g},\mathfrak{t}]^{\circ }$. \[same isotropy\] For each $v_{q_{e}}\in \mathfrak{t}\cdot q_{e}$ we have $G_{v_{q_{e}}}=G_{_{q_{e}}}$. The inclusion $G_{v_{q_{e}}}\subseteq G_{_{q_{e}}}$ is obviously true, so it will be enough to prove that $G_{v_{q_{e}}}\supseteq G_{_{q_{e}}}$. To see this, let $g\in G_{_{q_{e}}}$ and $v_{q_{e}}=\xi _{Q}(q_{e})\in \mathfrak{t}\cdot q_{e}$, with $\xi\in \mathfrak{t}$. Then, since $G_{q_e} $ is Abelian, we get $$\begin{aligned} T_{q_{e}}\Psi _{g}\left(v_{q_{e}}\right) &=T_{q_{e}}\Psi _{g}\left(\xi_{Q}(q_{e})\right) =T_{q_{e}}\Psi _{g}\left(\left.\frac{d}{dt}\right|_{t=0}\Psi _{\exp (t \xi )}(q_{e})\right) \\ &=\left.\frac{d}{dt}\right|_{t=0}\left(\Psi _{g}\circ \Psi _{\exp (t\xi )}\right)(q_{e})= \left.\frac{d}{dt}\right| _{t=0}(\Psi _{\exp (t \xi )}\circ \Psi _{g})(q_{e}) \\ &=\left.\frac{d}{dt}\right|_{t=0}\Psi _{\exp (t\xi )}(q_{e})= \xi_Q(q_e) = v_{q_{e}},\end{aligned}$$ that is, $g\cdot v_{q_{e}} = v_{q_{e}} $, as required. The bifurcation problem for relative equilibria on $TQ$ can be regarded as a bifurcation problem on the space $Q\times\mathfrak{g} ^{\ast}$ as the following shows. \[map f\] The map $f:TQ\rightarrow Q\times \mathfrak{g}^{\ast }$ given by $ v_{q}\mapsto (q,\mathbf{J}_{L}(v_{q}))$ restricted to the set of relative equilibria is one to one and onto its image. The only thing to be proved is that the map is injective. To see this, let $(q_{1},(\xi_{1})_{Q}(q_{1}))$ and $(q_{2},(\xi_{2})_{Q}(q_{2}))$ be two relative equilibria such that $f(q_{1},(\xi_{1})_{Q}(q_{1}))= f(q_{2},(\xi_{2})_{Q}(q_{2}))$. Then $q_{1}=q_{2}=:q$ and $\mathbf{J}_{L}(q,(\xi _{1}-\xi _{2})_{Q}(q))= \mathbb{I}(q)(\xi_{1}-\xi_{2})=0$ which shows that $\xi_{1}-\xi_{2}\in \ker \mathbb{I}(q)=\mathfrak{g}_{q}$ and hence $(\xi_{1})_ {Q}(q)=(\xi_{2})_{Q}(q)$. We can thus change the problem: instead of searching for relative equilibria of the simple mechanical system in $TQ$, we shall set up a bifurcation problem on $Q \times \mathfrak{g}^\ast$ such that the image of the relative equilibria by the map $f $ is precisely the bifurcating set. To do this, we begin with some geometric considerations. We construct a $G$-invariant tubular neighborhood of the orbit $G\cdot q_{e}$ such that the isotropy group of every point in this neighborhood is a subgroup of $G_{q_{e}}$. This follows from the Tube Theorem \[tube theorem\]. Indeed, let $B\subset (\mathfrak{g}\cdot q_{e})^{\perp }$ be a $G_{q_{e}}$-invariant open neighborhood of $0_{q_{e}}\in (\mathfrak{g}\cdot q_{e})^{\perp }$ such that on the open $G$-invariant neighborhood $G\cdot \operatorname{Exp}_{q_{e}}(B)$ of $G\cdot q_{e}$, we have $(G_{q_{e}})\preceq (G_{q})$ for every $q\in G\cdot \operatorname{Exp}_{q_{e}}(B)$. Moreover $G$ acts freely on $G\cdot \operatorname{Exp}_ {q_{e}}\left(B\cap (T_{q_{e}}Q)_{\{e\}}\right)$. It is easy to see that $B\times \mathfrak{g}^{\ast }$ can be identified with a slice at $(q_{e},0)$ with respect to the diagonal action of $G$ on $ (G\cdot \operatorname{Exp}_{q_{e}}(B))\times \mathfrak{g}^{\ast }$. The strategy to prove the existence of a bifurcating branch of relative equilibria with no symmetry from the set of relative equilibria $\mathfrak{t} \cdot q_{e}$ is the following. Note that we do not know a priori which relative equilibrium in $\mathfrak{t} \cdot q_{e}$ will bifurcate. We search for a local bifurcating branch of relative equilibria in the following manner. Take a vector $v_{q_{e}}\in B\cap (T_{q_{e}} Q)_{\{e\}}$ and note that $\operatorname{Exp}_{q_e} (v_{q_{e}}) \in Q$ is a point with no symmetry, that is, $G_{\operatorname{Exp}_{q_e} (v_{q_{e}})} = \{e\}$. Then $\tau v_{q_{e}} \in B\cap (T_{q_{e}} Q)_{\{e\}}$, for $\tau\in I $, where $I $ is an open interval containing $[0,1]$, and $\operatorname{Exp}_{q_e} (\tau v_{q_{e}})$ is a smooth path connecting $q_e $, the base point of the relative equilibrium in $\mathfrak{t}\cdot q_e $ containing the branch of bifurcating relative equilibria, to $\operatorname{Exp}_{q_e} (v_{q_{e}}) \in Q$. In addition, we shall impose that the entire path $\operatorname{Exp}_{q_e} (\tau v_{q_{e}})$ be formed by base points of relative equilibria. We still need the vector part of these relative equilibria which we postulate to be of the form $\zeta(\tau)_Q(\operatorname{Exp}_{q_e} (\tau v_{q_{e}}) )$, where $\zeta(\tau) \in \mathfrak{g}$ is a smooth path of Lie algebra elements with $\zeta(0) \in \mathfrak{t}$. Since $\operatorname{Exp}_{q_e} (\tau v_{q_{e}})$ has no symmetry for $\tau> 0 $, the locked inertia tensor is invertible at these points and the path $\zeta(\tau) $ will be of the form $$\zeta(\tau) = \mathbb {I}(\operatorname{Exp}_{q_e} (\tau v_{q_{e}}))^{-1} (\beta(\tau)),$$ where $\beta(\tau) $ is a smooth path in $\mathfrak{g}^\ast$ with $\beta(0) \in \mathbb {I}(q_e) \mathfrak{t}$. Now we shall use the characterization of relative equilibria involving the amended potential to require that the path $\left(\operatorname{Exp}_{q_e}(\tau v_{q_e}), \beta(\tau) \right) \in (G\cdot \operatorname{Exp}_{q_{e}}(B)) \times \mathfrak{g}^\ast $ be such that $f^{-1}(\left(\operatorname{Exp}_{q_e}(\tau v_{q_e}), \beta(\tau) \right)$ are all relative equilibria. The amended potential criterion is applicable along the path $\operatorname{Exp}_{q_e}(\tau v_{q_e})$ for $\tau >0 $, because these points have no symmetry. As we shall see below, we shall look for $\beta(\tau)$ of a certain form and then the characterization of relative equilibria via the amended potential will impose conditions on both $\beta(\tau) $ and $v_{q_e}$. We begin by specifying the form of $\beta(\tau)$. Splittings ---------- We shall need below certain direct sum decompositions of $\mathfrak{g}$ and $\mathfrak{g}^\ast$. The compactness of $G$ implies that $\mathfrak{g}$ has an invariant inner product and that $\mathfrak{g}=\mathfrak{t}\oplus [\mathfrak{g},\mathfrak{t}]$ is an orthogonal direct sum. Let $\mathfrak{k}_{1}\subset\mathfrak{t}$ be the orthogonal complement to $\mathfrak{k}_{0}:=\mathfrak{g}_{q_{e}}$ in $\mathfrak{t}$. Denoting $\mathfrak{k}_{2}:=[\mathfrak{g},\mathfrak{t}]$ we obtain the orthogonal direct sum $\mathfrak{g}=\mathfrak{k}_{0}\oplus \mathfrak{k} _{1}\oplus \mathfrak{k}_{2}$. For the dual of the Lie algebra, let $\mathfrak{m}_{i}:=(\mathfrak{k} _{j}\oplus \mathfrak{k}_{k})^{\circ }$ where $(i,j,k)$ is a cyclic permutation of $(0,1,2)$. Then $\mathfrak{g}^{\ast }=\mathfrak{m} _{0}\oplus \mathfrak{m}_{1}\oplus \mathfrak{m}_{2}$ is also an orthogonal direct sum relative to the inner product on $\mathfrak{g}^\ast$ naturally induced by the invariant inner product on $\mathfrak{g}$. The subspaces defined by the above splittings have the following properties: 1. $\mathfrak{k}_{0}$, $\mathfrak{k}_{1}$, $\mathfrak{k}_{2}$ are $G_{q_{e}}$-invariant and $G_{q_{e}}$ acts trivially on $\mathfrak{k}_{0}$ and $\mathfrak{k}_{1}$; 2. $\mathfrak{m}_{0}$, $\mathfrak{m}_{1}$, $\mathfrak{m}_{2}$ are $G_{q_{e}}$-invariant and $G_{q_{e}}$ acts trivially on $\mathfrak{m}_{0}$ and $\mathfrak{m}_{1}$. **(i)** Because $G_{q_{e}}$ is a subgroup of $ \mathbb{T}$ it is obvious that $G_{q_{e}}$ acts trivially on $\mathfrak{t}=\mathfrak{k}_{0}\oplus \mathfrak{k}_{1}$ and hence on each summand. To prove the $G_{q_{e}}$-invariance of $\mathfrak{k}_{2}=[\mathfrak{g},\mathfrak{t}] $, we use the fact that $\operatorname{Ad}_{g}[\xi _{1},\xi _{2}]=[\operatorname{Ad}_{g}\xi _{1},\operatorname{Ad}_{g}\xi _{2}]$, for any $\xi _{1},\xi _{2}\in \mathfrak{g}$ and $g\in G$. Indeed, if $\xi _{1}\in \mathfrak{g}$, $\xi _{2}\in \mathfrak{t}$, $g\in G_{q_{e}}$ we get $\operatorname{Ad}_{g}[\xi _{1},\xi _{2}]\in \lbrack \mathfrak{g},\mathfrak{t}]=\mathfrak{k}_{2}$. **(ii)** For $g\in G_{q_{e}}$, $\mu \in \mathfrak{m}_{0}$ we have to prove that $ \operatorname{Ad}_{g}^{\ast }\mu \in \mathfrak{m}_{0}$. Indeed, if $\xi =\xi _{1}+\xi _{2}\in \mathfrak{k}_{1}\oplus \mathfrak{k}_{2}$, we have $$\begin{aligned} \langle \operatorname{Ad}_{g}^{\ast }\mu ,\xi \rangle &=\langle \operatorname{Ad}_{g}^{\ast }\mu ,\xi _{1}+\xi _{2}\rangle =\langle \mu, \operatorname{Ad}_{g}(\xi _{1}+\xi _{2})\rangle \\ &=\langle \mu ,\xi _{1}+\operatorname{Ad}_{g}\xi _{2}\rangle =0\end{aligned}$$ since $G_{q_{e}}$ acts trivially on $\mathfrak{k}_{1}$, $\mathfrak{k}_{2}$ is $G_{q_{e}}$–invariant and $\mathfrak{m}_{0}=(\mathfrak{k}_{1}\oplus \mathfrak{k}_{2})^{\circ }$. The same type of proof holds for $\mathfrak{m}_{1}$and $\mathfrak{m}_{2}$. For $g\in G_{q_{e}}$, $\mu \in \mathfrak{m}_{0}$ we have to prove that $ \operatorname{Ad}_{g}^{\ast }\mu =\mu $. Let $\xi =\xi _{0}+\xi _{1}+\xi _{2}\in \mathfrak{g}$, with $\xi_i \in \mathfrak{k}_i $, $i = 0,1,2 $. We have $$\begin{aligned} \langle \operatorname{Ad}_{g}^{\ast }\mu -\mu ,\xi \rangle &=\langle \operatorname{Ad}_{g}^{\ast }\mu ,\xi _{0}+\xi _{1}+\xi _{2}\rangle -\langle \mu ,\xi _{0}+\xi _{1}+\xi _{2}\rangle \\ &=\langle \mu ,\operatorname{Ad}_{g}(\xi _{0}+\xi _{1}+\xi _{2})\rangle - \langle \mu ,\xi _{0}+\xi _{1}+\xi _{2}\rangle \\ &=\langle \mu ,\xi _{0}+\xi _{1}+\operatorname{Ad}_{g}\xi _{2}\rangle - \langle \mu ,\xi _{0}\rangle =\langle \mu ,\xi_{1}+\operatorname{Ad}_{g}\xi _{2} \rangle =0\end{aligned}$$ because $G_{q_{e}}$ acts trivially on $\mathfrak{k}_{0}\oplus \mathfrak{k} _{1}$, $\mathfrak{k}_{2}$ is $G_{q_{e}}$–invariant, and $\mathfrak{m}_{0}=( \mathfrak{k}_{1}\oplus \mathfrak{k}_{2})^{\circ }$. The same type of proof holds for $\mathfrak{m}_{1}$. Recall from §\[Simple mechanical systems with symmetry\] that $\ker \mathbb {I}(q_e) = \mathfrak{g}_{q_e} = \mathfrak{k}_0 $. In particular, $\mathbb {I}(q_e) \mathfrak{k}_0 = \{0\}$. The value of $\mathbb {I}(q_e)$ on the other summands in the decomposition $\mathfrak{g}= \mathfrak{k}_0 \oplus \mathfrak{k}_1 \oplus \mathfrak{k}_2 $ is given by the following lemma. \[moment of inertia isomorphism\] For $i\in \{1,2\}$ we have that $\mathfrak{m}_{i}=\mathbb{I}(q_{e}) \mathfrak{k}_{i}$. Let $\kappa_{i}\in\mathfrak{k}_{i}$ with $i\in \{0,1,2\}$ be arbitrary. Then $$\langle \mathbb{I}(q_{e})\kappa_{1}, \kappa_{0}+\kappa_{2}\rangle =\langle \mathbb{I}(q_{e})\kappa_{1},\kappa_{0}\rangle +\langle\mathbb{I}(q_{e})\kappa_{1},\kappa_{2}\rangle =\langle \mathbb{I}(q_{e})\kappa_{0},\kappa_{1}\rangle +\langle\mathbb{I}(q_{e})\kappa_{1},\kappa_{2}\rangle =0$$ as $\ker \mathbb{I}(q_{e})=\mathfrak{k}_{0}$ and, by Proposition \[montaldi\] (ii), $\mathbb{I}(q_{e})\mathfrak{t}\subset \mathfrak{k}_{2}^{\circ}$. This proves that $\mathbb{I}(q_{e})\mathfrak{k}_{1}\subset \mathfrak{m}_{1}$. Counting dimensions we have that $\dim\mathbb{I}(q_{e})\mathfrak{k}_{1}= \dim \mathfrak{k}_1 - \dim \ker \left(\mathbb {I}(q_e)|_{\mathfrak{k}_1} \right) = \dim \mathfrak{g}-\dim \mathfrak{k}_{0}-\dim \mathfrak{k}_{2} =\dim \mathfrak{m}_{1}$, since $\ker \left(\mathbb {I}(q_e)|_{\mathfrak{k}_1} \right) = \{0\} $. This proves that $\mathfrak{m}_{1}=\mathbb{I}(q_{e}) \mathfrak{k}_{1}$. In an analogous way we prove the equality for $i=2$. In the next paragraph we shall need the direct sum decomposition $\mathfrak{g}^\ast = \mathfrak{m}_1 \oplus \mathfrak{m}$, where $\mathfrak{m}_1 = \mathbb {I}(q_e) \mathfrak{t}$ and $\mathfrak{m} := \mathfrak{m}_0 \oplus \mathfrak{m}_2 $. Let $\Pi_1: \mathfrak{g}^\ast\rightarrow \mathbb {I}(q_e) \mathfrak{t}$ be the projection along $\mathfrak{m} $. Similarly, denote $\mathfrak{k}: = \mathfrak{k}_1 \oplus\mathfrak{k}_2$, and write $\mathfrak{g} = \mathfrak{g}_{q_e} \oplus\mathfrak{k}$. Thus there is another decomposition of $\mathfrak{g}^\ast$, namely, $\mathfrak{g}^\ast = \mathfrak{g}_{q_e} ^\circ \oplus \mathfrak{k}^\circ $. However, for any $\zeta \in \mathfrak{g}_{q_e} $ and any $\xi\in \mathfrak{g}$, we have $\langle \mathbb {I}(q_e) \xi, \zeta \rangle = \langle\!\langle \xi_Q(q_e), \zeta_Q(q_e) \rangle\!\rangle = 0 $ since $\zeta_Q(q_e) = 0 $, which shows that $\mathbb {I}(q_e) \mathfrak{g} \subset \mathfrak{g}_{q_e}^\circ $. Since $ \ker \mathbb {I}(q_e) = \mathfrak{g}_{q_e} $, it follows that $\dim \mathbb {I}(q_e) \mathfrak{g} = \dim \mathfrak{g} - \dim \ker \mathbb {I}(q_e) = \dim \mathfrak{g} - \dim \mathfrak{g}_{q_e} = \dim \mathfrak{g}_{q_e}^\circ $, which shows that $\mathfrak{g}_{q_e}^\circ = \mathbb {I}(q_e)\mathfrak{g}$. Thus we also have the direct sum decomposition $\mathfrak{g}^\ast = \mathbb {I}(q_e)\mathfrak{g} \oplus \mathfrak{k}^\circ $. Note that $\mathbb {I}(q_e) \mathfrak{g} = \mathfrak{m}_1 \oplus \mathfrak{m}_2 $, by Lemma \[moment of inertia isomorphism\] and that $\mathfrak{m}_0 = \mathfrak{k}^\circ $. Summarizing we have: $$\mathfrak{g}^\ast = \mathfrak{m}_0 \oplus \mathfrak{m}_1 \oplus \mathfrak{m}_2 = \mathfrak{k}^\circ \oplus \mathbb {I}(q_e) \mathfrak{g}, \quad \text{where} \quad \mathbb {I}(q_e) \mathfrak{g} = \mathfrak{m}_1 \oplus \mathfrak{m}_2 \quad \text{and} \quad \mathfrak{m}_0 = \mathfrak{k}^\circ.$$ The rescaled equation --------------------- Recall that $B\subset (\mathfrak{g}\cdot q_{e})^{\perp }$ is a $G_{q_{e}}$-invariant open neighborhood of $0_{q_{e}}\in (\mathfrak{g}\cdot q_{e})^{\perp }$ such that on the open $G$-invariant neighborhood $G\cdot \operatorname{Exp}_{q_{e}}(B)$ of $G\cdot q_{e}$, we have $(G_{q_{e}})\preceq (G_{q})$ for every $q\in G\cdot \operatorname{Exp}_{q_{e}}(B)$. Consider the following rescaling: $$v_{q_{e}}\in B\cap (T_{q_{e}}Q)_{\{e\}}\mapsto \tau v_{q_{e}}\in B\cap (T_{q_{e}}Q)_{\{e\}}$$ $$\mu \in \mathfrak{g}^{\ast }\mapsto \beta (\tau ,\mu )\in \mathfrak{g}^{\ast }$$ where, $\tau \in I$, $I$ is an open interval containing $[0,1]$, and $\beta :I\times \mathfrak{g}^{\ast }\rightarrow \mathfrak{g}^{\ast }$ is chosen such that $\beta (0,\mu )=\Pi _{1}\mu $. So, for $(v_{q_{e}},\mu )$ fixed, $(\tau v_{q_{e}},\beta (\tau ,\mu ))$ converges to $(0_{q_{e}},\Pi _{1}\mu )$ as $\tau \rightarrow 0$. Define $$\beta (\tau ,\mu ):=\Pi _{1}\mu +\tau \beta '(\mu )+\tau^{2} \beta ''(\mu )$$ for some arbitrary smooth functions $\beta ',\beta '': \mathfrak{g}^{\ast }\rightarrow \mathfrak{g}^{\ast }$. Since $\mathbb{I}$ is invertible only for points with no symmetry, we want to find conditions on $\beta '$, $\beta ''$ such that the expression $$\label{32} \mathbb{I}(\operatorname{Exp} _{q_{e}}(\tau v_{q_{e}}))^{-1}\beta (\tau ,\mu )$$ extends to a smooth function in a neighborhood of $\tau =0$. Note that $v_{q_{e}}$ is different from $0_{q_{e}}$ since $G_{v_{q_{e}}} = \{e\}$ by construction and $G_{0_{q_{e}}} = G_{q_e} \neq \{e\}$. Define $$\Phi :I\times \left(B\cap (T_{q_{e}}Q)_{\{e\}}\right)\times \mathfrak{g}^{\ast }\times \mathfrak{g}_{q_{e}}\times \mathfrak{k}\rightarrow \mathfrak{g}^{\ast }$$ $$\label{definition of Phi} \Phi (\tau ,v_{q_{e}},\mu ,\xi ,\eta ) :=\mathbb{I}(\operatorname{Exp}_{q_{e}}(\tau v_{q_{e}})) (\xi +\eta )-\beta (\tau ,\mu ).$$ Now we search for the velocity $\xi + \eta $ of relative equilibria among the solutions of $\Phi (\tau ,v_{q_{e}},\mu ,\xi ,\eta )=0$. We shall prove below that $\xi $ and $\eta$ are smooth functions of $\tau $, $v_{q_{e}}$, $\mu$, even at $\tau= 0$. Then shows that $\xi+ \eta $ is a smooth function of $\tau $, $v_{q_{e}}$, $\mu$, for $\tau$ in a small neighborhood of zero. The Lyapunov-Schmidt procedure ------------------------------ To solve $\Phi =0$ we apply the standard Lyapunov-Schmidt method. This equation has a unique solution for $\tau \neq 0$, because $\tau v_{q_{e}}\in B\cap (T_{q_{e}}Q)_{\{e\}}$ so $\mathbb{I}(\operatorname{Exp}_{q_{e}}(\tau v_{q_{e}}))$ is invertible. It remains to prove that the equation has a solution when $\tau =0$. Denote by $D_{\mathfrak{g}_{q_{e}}\times \mathfrak{k}}$ the Fréchet derivative relative to the last two factors $\mathfrak{g}_{q_{e}}\times \mathfrak{k}$ in the definition of $\Phi$. We have $$\ker D_{\mathfrak{g}_{q_{e}}\times \mathfrak{k}}\Phi (0,v_{q_{e}},\mu ,\xi ,\eta )=\ker \mathbb{I}(q_{e})=\mathfrak{g}_{q_{e}}.$$ We will solve the equation $\Phi =0$ in two steps. For this, let $$\Pi :\mathfrak{g}^{\ast }\rightarrow \mathbb{I}(q_{e})\mathfrak{g}$$ be the projection induced by the splitting $\mathfrak{g}^{\ast }=\mathbb{I} (q_{e})\mathfrak{g}\oplus \mathfrak{k}^{\circ }$. **Step1**. Solve $\Pi \circ \Phi =0$ for $\eta $ in terms of $\tau $, $v_{q_{e}}$, $\mu $, $\xi $. For this, let $$\begin{aligned} \widehat{\mathbb{I}}(\operatorname{Exp} _{q_{e}}(\tau v_{q_{e}})) &:= (\Pi\circ \mathbb{I})(\operatorname{Exp}_{q_{e}} (\tau v_{q_{e}}))|_{\mathfrak{k}}: \mathfrak{k}\rightarrow \mathbb {I}(q_e) \mathfrak{g}\\ \overset{\thicksim}{\mathbb{I}}(\operatorname{Exp} _{q_{e}} (\tau v_{q_{e}})) &:=(\Pi \circ \mathbb{I})(\operatorname{Exp}_{q_{e}}(\tau v_{q_{e}}))|_{\mathfrak{g}_{q_{e}}}: \mathfrak{g}_{q_e} \rightarrow \mathbb {I}(q_e) \mathfrak{g}\end{aligned}$$ where $\widehat{\mathbb{I}}(\operatorname{Exp} _{q_{e}}(\tau v_{q_{e}})) $ is an isomorphism even when $\tau =0$. Then we obtain $$\label{pi composed with Phi} (\Pi \circ \Phi )(0,v_{q_{e}},\mu ,\xi ,\eta ) = \Pi [\mathbb{I} (q_{e})(\xi +\eta )-\beta (0,\mu)] =\widehat{\mathbb{I}}(q_{e}) \eta -\Pi _{1}\mu .$$ Denoting $\eta _{\mu }:=\widehat{\mathbb{I}} (q_{e})^{-1}(\Pi _{1}\mu )$, we have $(\Pi \circ \Phi) (0,v_{q_{e}},\mu ,\xi ,\eta _{\mu })\equiv 0$. Denoting by $D_\eta$ the partial Fréchet derivative relative to the variable $\eta \in \mathfrak{k}$ we get at any given point $(0, v_{q_e}^0, \mu^0, \xi^0, \eta^0 )$ $$\label{isomorphism condition for ift} D_\eta (\Pi \circ \Phi )(0, v_{q_e}^0, \mu^0, \xi^0, \eta^0) =\widehat{\mathbb{I}}(q_{e})$$ which is invertible. Thus the implicit function theorem gives a unique smooth function $\eta (\tau ,v_{q_{e}},\mu ,\xi )$ such that $\eta (0, v_{q_e}^0, \mu^0, \xi^0)=\eta^0$ and $$\label{definition of eta} (\Pi \circ \Phi) (\tau ,v_{q_{e}},\mu ,\xi ,\eta (\tau ,v_{q_{e}},\mu ,\xi ))\equiv 0.$$ The function $\eta $ is defined in some open set in $I\times \left(B\cap (T_{q_{e}}Q)_{\{e\}}\right)\times \mathfrak{g}^{\ast}\times \mathfrak{g}_{q_{e}}$ containing $(0, v_{q_e}^0, \mu^0, \xi^0) \in \{0\}\times \left(B\cap (T_{q_{e}}Q)_{\{e\}}\right)\times \mathfrak{g}^{\ast }\times \mathfrak{g}_{q_{e}}$. If we now choose $\eta^0 = \eta_{\mu^0} = \widehat{\mathbb {I}}(q_e) ^{-1}( \Pi_1 \mu ^0)$, then uniqueness of the solution of the implicit function theorem implies that $\eta(0, v_{q_e}, \mu, \xi) = \eta_\mu$ in the neighborhood of $(0, v_{q_e}^0, \mu^0, \xi^0)$. Later we will need the following result. \[belongs\] We have $\eta _{\mu }:=\widehat{\mathbb{I}} (q_{e})^{-1}(\Pi _{1}\mu ) \in \mathfrak{k}_1 \subset \mathfrak{t}$. Since we can write $\mathfrak{t}=\ker \mathbb{I}(q_{e})\oplus \mathfrak{k}_1$ we obtain $$\widehat{\mathbb{I}}(q_{e})\mathfrak{k}_1 =(\Pi \circ \mathbb{I}(q_{e})) \mathfrak{k}_1 =\mathbb{I} (q_{e}) \mathfrak{k}_1=\mathbb{I}(q_{e})(\mathfrak{t}) =\operatorname{Im}\Pi _{1}.$$ Now, because $\widehat {\mathbb{I}}(q_{e})$ is an isomorphism, it follows that $\widehat{\mathbb{I}} (q_{e})^{-1}(\Pi _{1}\mu ) \in \mathfrak{k}_1$. **Step2**. Now we solve the equation $(Id-\Pi)\circ\Phi=0$. For this, let $$\varphi :I\times \left(B\cap (T_{q_{e}}Q)_{\{e\}}\right)\times \mathfrak{g}^{\ast }\times \mathfrak{g}_{q_{e}}\rightarrow \mathfrak{k}^{\circ }$$ $$\label{definition of phi} \varphi (\tau ,v_{q_{e}},\mu ,\xi ) :=(Id-\Pi ) \Phi (\tau ,v_{q_{e}},\mu ,\xi ,\eta (\tau ,v_{q_{e}},\mu ,\xi )).$$ In particular, $\varphi (0,v_{q_{e}},\mu ,\xi )=(Id-\Pi) (\mathbb{I} (q_{e})(\xi +\eta _{\mu })-\Pi _{1}\mu )$. Since $\operatorname{Im}\mathbb{I} (q_{e}) = \operatorname{Im}\Pi$ and $\operatorname{Im}\Pi _{1} = \mathbb {I}(q_e) \mathfrak{t} \subset \mathbb {I}(q_e) \mathfrak{g}$, it follows that $\varphi (0,v_{q_{e}},\mu ,\xi )\equiv 0$. We shall solve for $\xi \in\mathfrak{g}_{q_e}$, in the neighborhood of $(0, v_{q_e}^0, \mu^0, \xi^0)$ found in Step 1, the equation $\varphi (\tau ,v_{q_{e}},\mu ,\xi ) = 0 $. To do this, we shall need information about the higher derivatives of $\varphi $ with respect to $\tau $, evaluated at $\tau =0$. Let $\xi $, $\eta \in \mathfrak{g}$ and $q\in Q$. Suppose that $\mathbf{d}V_{\eta }(q)=0$, where $V_{\eta}$ is the augmented potential and suppose that both $\xi $ and $[\xi ,\eta ]$ belong to $\mathfrak{g}_{q}$. Then $\mathbf{d}\langle \mathbb{I} (\cdot )\xi ,\eta \rangle (q)=0$. Since $\mathbf{d}V_{\eta}(q)=0$, $\eta _{Q}(q)$ is a relative equilibrium by Proposition \[augmented potential criterion\], that is, $X_{H}(\alpha _{q})=\eta_{T^{\ast }Q}(\alpha _{q})$, where $\alpha _{q}=\mathbb{F}L(\eta _{Q}(q))$. Now suppose that both $\xi $, $[\xi ,\eta ]\in \mathfrak{g}_{q}$. Then $$\xi _{T^{\ast }Q}(\alpha _{q})=\left. \frac{d}{dt}\right| _{t=0}\mathbb{F}L(\exp(t\xi )\cdot \eta _{Q}(q))=\mathbb{F}L([\xi ,\eta ]_{Q}(q))=0,$$ where we have used that $g\cdot \eta _{Q}(q)=(\operatorname{Ad}_{g}\eta )_{Q}(g\cdot q)$. It follows that $(\eta +\xi )_{T^{\ast }Q}(\alpha _{q})=X_{H}(\alpha _{q})$ and hence, again by Proposition \[augmented potential criterion\], that $0=\mathbf{d}V_{\eta +\xi }(q)=\mathbf{d}V_{\eta}(q) -\mathbf{d}\langle\mathbb{I}(\cdot )\eta ,\xi \rangle(q)-\frac{1}{2}\mathbf{d}\| \xi _{Q}(\cdot )\| ^{2}(q)$. However, $\mathbf{d}\| \xi _{Q}(\cdot )\| ^{2}(q) = 0 $ since $\xi \in \mathfrak{g}_q $, as an easy coordinate computation shows. Since $\mathbf{d}V_\eta(q) = 0 $ by hypothesis, we have $\mathbf{d}\langle\mathbb{I}(\cdot )\eta ,\xi \rangle(q) = 0 $. Symmetry of $\mathbb {I}(q)$ proves the result. Let now $\xi \in \mathfrak{g}_{q_{e}}$ and $\eta \in \mathfrak{t}$. Since $\mathfrak{g}_{q_e} \subset \mathfrak{t}$, we have $[ \xi, \eta ] = 0 \in \mathfrak{g}_{q_e}$. In addition, hypothesis **(H)** and Proposition \[augmented potential criterion\], guarantee that $\mathbf{d}V_\xi(q_e) = 0 $ which shows that all hypotheses of the previous lemma are satisfied. Therefore, $$\label{differential of i} \mathbf{d}\langle\mathbb{I}(\cdot )\xi,\eta \rangle(q_{e})=0 \quad \text{for} \quad \xi \in \mathfrak{g}_{q_{e}}, \; \eta \in \mathfrak{t}.$$ The bifurcation equation ------------------------ Now we can proceed with the study of equation $ \varphi= (Id-\Pi)\circ\Phi=0$. We have $$\begin{aligned} \label{first tau derivative of phi} \frac{\partial \varphi}{\partial \tau}(\tau, v_{q_e}, \mu, \xi) = (Id - \Pi) &\left[ T_{\tau v_{q_e}}(\mathbb {I} \circ \operatorname{Exp}_{q_e})(v_{q_e})(\xi + \eta(\tau, v_{q_e}, \mu, \xi)) + \mathbb {I}(\operatorname{Exp}_{q_e}(\tau v_{q_e})) \frac{\partial \eta}{\partial \tau}(\tau, v_{q_e}, \mu, \xi) \right. \nonumber \\ &\qquad \left. - \frac{\partial \beta}{\partial \tau}(\tau, \mu) \right].\end{aligned}$$ $\frac{\partial }{\partial \tau }\varphi (0,v_{q_{e}},\mu ,\xi )\equiv -(Id-\Pi )\beta '(\mu )$. Formula gives for $\tau= 0 $ $$\frac{\partial \varphi }{\partial \tau }(0,v_{q_{e}},\mu ,\xi ) =(Id-\Pi ) \left[\big(T_{q_e} \mathbb{I}(v_{q_{e}})\big)(\xi +\eta _{\mu }) +\mathbb{I}(q_{e})\frac{\partial \eta }{\partial \tau }(0,v_{q_{e}},\mu ,\xi )-\frac{\partial \beta }{\partial \tau }(0,\mu )\right].$$ Now, because $\operatorname{Im}\mathbb{I}(q_{e})=\operatorname{Im}\Pi $ we obtain $(Id-\Pi ) \circ \mathbb{I}(q_{e})=0$ and hence the second summand vanishes. From we have that $(T_{q_e}\mathbb{I} (v_{q_{e}}))(\mathfrak{t})\subset \mathfrak{g}_{q_{e}}^{\circ }=\operatorname{Im}\Pi $. Using Proposition \[belongs\] and since $\xi \in \mathfrak{g}_{q_{e}}\subset \mathfrak{t}$, we obtain that $\xi +\eta_{\mu }\in \mathfrak{t}$. Therefore $(Id-\Pi )[(T_{q_e}\mathbb{I}(v_{q_{e}}))(\xi +\eta _{\mu })]=0$. Since $\frac{\partial \beta }{\partial \tau }(0,\mu )=\beta ^{\prime }(\mu )$, we obtain the desired equality. Let us impose the additional condition $\beta '(\mu) \subset\operatorname{Im}\Pi $. Then it follows that $$\varphi (\tau ,v_{q_{e}},\mu ,\xi )=\tau ^{2}\psi (\tau ,v_{q_{e}},\mu ,\xi).$$ for some smooth function $\psi $ where $$\psi (0,v_{q_{e}},\mu ,\xi )=\frac{1}{2}\frac{\partial ^{2}\varphi }{ \partial \tau ^{2}}(0,v_{q_{e}},\mu ,\xi )$$ We begin by solving the equation $$\psi (0,v_{q_{e}},\mu ,\xi )=0$$ for $\xi $ as a function of $v_{q_{e}}$ and $\mu $. Equivalently, we have to solve $$\frac{1}{2}\frac{\partial ^{2}\varphi }{\partial \tau ^{2}}(0,v_{q_{e}},\mu ,\xi )=0.$$ To compute this second derivative of $\varphi$ we shall use . We begin by noting that $\tau\in I \mapsto T_{\tau v_{q_e}}(\mathbb {I} \circ \operatorname{Exp}_{q_e})(v_{q_e})$ is a smooth path in $L(\mathfrak{g}, \mathfrak{g}^\ast)$ and so we can define the linear operator from $\mathfrak{g}$ to $\mathfrak{g}^\ast$ by $$A_{v_{q_e}}: = \left.\frac{\partial}{\partial \tau}\right|_{\tau= 0}T_{\tau v_{q_e}}(\mathbb {I} \circ \operatorname{Exp}_{q_e})(v_{q_e}) \in L(\mathfrak{g}, \mathfrak{g}^\ast).$$ With this notation, formulas , , , and Proposition \[belongs\] yield $$\begin{aligned} \label{second tau derivative of phi at zero} \frac{\partial ^{2}\varphi }{\partial \tau^{2}} (0,v_{q_{e}},\mu, \xi ) &=(Id-\Pi)\left[ A_{v_{q_e}}(\xi +\eta _{\mu}) +2T_{q_e}\mathbb{I}(v_{q_{e}})\frac{\partial \eta}{\partial \tau }(0,v_{q_{e}},\mu ,\xi ) \right.\\ & \qquad \qquad \qquad \left. + \mathbb{I}(q_{e})\frac{\partial ^{2}\eta }{\partial \tau ^{2}} (0,v_{q_{e}},\mu ,\xi )-2\beta ''(\mu )\right] \nonumber \\ &= (Id-\Pi)\left[ A_{v_{q_e}}(\xi +\eta _{\mu}) +2T_{q_e}\mathbb{I}(v_{q_{e}})\frac{\partial \eta}{\partial \tau }(0,v_{q_{e}},\mu ,\xi ) - 2\beta ''(\mu) \right] \nonumber\end{aligned}$$ since $(Id - \Pi)\mathbb{I}(q_{e})\frac{\partial ^{2}\eta }{\partial \tau ^{2}} (0,v_{q_{e}},\mu ,\xi) = 0$. Let $\{\xi _{1},...,\xi _{p}\}$ be a basis of $\mathfrak{g}_{q_{e}}$. Since $\partial^2 \varphi ( \tau, v_{q_e}, \mu, \xi)/\partial \tau ^2 \in \mathfrak{k}^\circ $ and $\mathfrak{g} = \mathfrak{g}_{q_e} \oplus \mathfrak{k}$ the equation $\partial ^2 \varphi(0, v_{q_e}, \mu, \xi)/\partial \tau^2 = 0 $ is equivalent to the following system of $p $ equations $$\left\langle\frac{\partial ^{2}\varphi }{\partial \tau ^{2}}(0,v_{q_{e}},\mu ,\xi ) , \xi_b \right \rangle = 0, \quad \text{for~all} \quad b = 1, \dots, p,$$ which, by , is $$\left\langle (Id-\Pi)\left[ A_{v_{q_e}}(\xi +\eta _{\mu}) +2T_{q_e}\mathbb{I}(v_{q_{e}})\frac{\partial \eta}{\partial \tau }(0,v_{q_{e}},\mu ,\xi ) - 2\beta ''(\mu )\right] ,\xi_b \right\rangle = 0, \quad \text{for~all} \quad b = 1, \dots, p.$$ We shall show that in this expression we can drop the projector $ Id - \Pi $. Indeed, let $\alpha = \alpha_0 + \alpha_1 + \alpha_2 \in \mathfrak{g}^\ast = \mathfrak{m}_0 \oplus \mathfrak{m}_1 \oplus \mathfrak{m}_2 $, where $\alpha_i \in \mathfrak{m}_i $, for $i = 0,1,2 $. Since $\Pi: \mathfrak{g}^\ast \rightarrow \mathbb {I}(q_e) \mathfrak{g} = \mathfrak{m}_1 \oplus \mathfrak{m}_2 $, we have $$\langle (Id - \Pi) \alpha, \xi_b \rangle = \langle \alpha, \xi_b \rangle - \langle \alpha_1, \xi_b \rangle - \langle \alpha_2, \xi_b \rangle = \langle \alpha, \xi_b \rangle$$ because $\langle \alpha_1, \xi_b \rangle = 0$, since $\alpha_1 \in \mathfrak{m}_1 = (\mathfrak{k}_0 \oplus \mathfrak{k}_2)^\circ $, $\xi_b \in \mathfrak{g}_{q_e} = \mathfrak{k}_0 $, and $\langle \alpha_2, \xi_b \rangle = 0 $, since $\alpha_2 \in \mathfrak{m}_2 = (\mathfrak{k}_0 \oplus \mathfrak{k}_1)^\circ $, $\xi_b \in \mathfrak{g}_{q_e} = \mathfrak{k}_0 $. The system to be solved is hence $$\label{linear system to be solved} \left\langle A_{v_{q_e}}(\xi +\eta _{\mu}) +2T_{q_e}\mathbb{I}(v_{q_{e}})\frac{\partial \eta}{\partial \tau }(0,v_{q_{e}},\mu ,\xi ) - 2\beta ''(\mu ) ,\xi_b \right\rangle = 0, \quad \text{for~all} \quad b = 1, \dots, p.$$ In what follows we need the expression for $\frac{\partial \eta} {\partial \tau }(0,v_{q_{e}},\mu ,\xi)$. Differentiating relative to $\tau$ at zero and taking into account and , we get $$\begin{aligned} \label{first tau derivative of eta at zero} &\frac{\partial \eta }{\partial \tau }(0,v_{q_{e}},\mu ,\xi) = - \widehat{\mathbb {I}}(q_e)^{-1}\frac{\partial}{\partial \tau}(\Pi\circ \Phi)(0, v_{q_e}, \mu, \xi, \eta_\mu)\\ &\qquad = - \widehat{\mathbb {I}}(q_e)^{-1} \Pi\left[ T_{q_e}\mathbb {I}(v_{q_e})(\xi+ \eta_\mu) - \beta'(\mu) \right] \nonumber \\ &\qquad = - \left( \widehat{\mathbb{I}}(q_e)^{-1} \circ T_{q_e}\overset{\thicksim} {\mathbb {I}}(v_{q_e})\right) \xi - \left(\widehat{\mathbb {I}}(q_e)^{-1} \circ T_{q_e}\widehat{\mathbb {I}}(v_{q_e}) \circ \widehat{\mathbb{I}} (q_{e})^{-1}\right)(\Pi _{1}\mu ) + \widehat{\mathbb{I}} (q_{e})^{-1}(\beta'(\mu)) \nonumber\end{aligned}$$ since $T_{q_e}\overset{\thicksim}{\mathbb {I}} = \Pi \circ T_{q_e}\mathbb {I}|_{\mathfrak{g}_{q_e}} $ and $T_{q_e}\widehat{\mathbb {I}} = \Pi \circ T_{q_e}\mathbb {I}|_{\mathfrak{k}}$. Expanding $\xi$ in the basis $\{\xi_1, \dots, \xi_p\}$ as $\xi= \alpha^i \xi_i $ and taking into account the above expression, the system is equivalent to the following system of linear equations in the unknowns $\alpha^1, \dots, \alpha^p$ $$A_{a b} \alpha^a + B_b = 0, \quad a,b = 1, \dots , p,$$ where $$\begin{aligned} A_{a b}&: = \left\langle A_{v_{q_e}} \xi_a, \xi_b \right\rangle - 2 \left\langle \left(T_{q_e}\mathbb {I}(v_{q_e}) \circ \widehat{\mathbb {I}}(q_e)^{-1} \circ T_{q_e} \overset{\thicksim} {\mathbb {I}}(v_{q_e})\right) \xi_a, \xi_b \right\rangle \label{A}\\ B_b&:=\left\langle \left(A_{v_{q_e}} \circ \widehat{\mathbb{I}} (q_{e})^{-1}\circ \Pi_1\right) \mu, \xi_b \right\rangle - 2 \left\langle \left(T_{q_e}\mathbb {I}(v_{q_e}) \circ \widehat{\mathbb {I}}(q_e)^{-1} \circ T_{q_e}\widehat{\mathbb {I}}(v_{q_e}) \circ \widehat{\mathbb{I}} (q_{e})^{-1} \circ \Pi_1 \right)\mu, \xi_b \right\rangle \label{B} \\ &\qquad +2 \left\langle \left(T_{q_e}\mathbb {I}(v_{q_e}) \circ \widehat{\mathbb{I}}(q_e)^{-1}\right)\beta'(\mu), \xi_b \right\rangle - \left\langle \beta''(\mu), \xi_b \right\rangle. \nonumber\end{aligned}$$ Denote by $A: = [A_{a b}] $ the $p\times p $ matrix with entries $A_{a b} $. Thus, if $v_{q_{e}}\notin \mathcal{Z}_{\mu}=:\{v_{q_{e}}\in B\cap (T_{q_{e}}Q)_{\{e\}}\mid \det A=0\}$ this linear system has a unique solution for $\alpha^1, \dots , \alpha^p $, that is for $\xi$, as function of $ v_{q_{e}}$, $\mu $. we shall denote this solution by $\xi_0(v_{q_e}, \mu)$. *Summarizing, if $v_{q_e} \notin \mathcal{Z}_{\mu}$, then $\xi_0(v_{q_e}, \mu) $ is the unique solution of the equation* $$\label{second step in LS} \frac{\partial ^{2}\varphi }{\partial \tau ^{2}}(0,v_{q_{e}},\mu ,\xi )=0.$$ \[properties of Z\] The set $\mathcal{Z}_{\mu}$ is closed and $G_{q_{e}}$–invariant in $B\cap (T_{q_e}Q)_{\{e\}}$. The set $\mathcal{Z}_{\mu}$ is obviously closed. Since $\mathfrak{k}$ is $G_{q_{e}}$–invariant it follows that $\mathfrak{k}^\circ$ is $G_{q_e}$–invariant. Formula shows that $\mathbb{I}(q_{e})\mathfrak{g}$ is also $G_{q_e}$–invariant. Thus the direct sum $\mathbb{I}(q_{e}) \mathfrak{g}\oplus \mathfrak{k}^{\circ }$ is a $G_{q_{e}}$–invariant decomposition of $\mathfrak{g}^{\ast }$ and therefore $\Pi : \mathfrak{g}^\ast \rightarrow \mathbb {I}(q_e) \mathfrak{g}$ is $G_{q_{e}}$–equivariant. From the $G_{q_{e}}$–equivariance of $\operatorname{Exp}_{q_{e}}$ and , it follows that $\mathbb{I} (\operatorname{Exp} _{q_{e}}(h\cdot v_{q_{e}}))=\operatorname{Ad}^\ast_{h^{-1}}\circ \mathbb{I}(\operatorname{Exp} _{q_{e}}(v_{q_{e}})) \circ \operatorname{Ad}_{h^{-1}} = \operatorname{Ad}^\ast_{h^{-1}}\circ \mathbb{I}(\operatorname{Exp} _{q_{e}}(v_{q_{e}}))$ for any $h \in G_{q_e}$ since $G_{q_e} \subset \mathbb{T}$ and is therefore Abelian. Thus $$\begin{aligned} \overset{\thicksim}{\mathbb{I}}(\operatorname{Exp} _{q_{e}}(h\cdot v_{q_{e}})) &= \Pi \circ \mathbb{I}(\operatorname{Exp} _{q_{e}}(T_{q_{e}}\Psi _{h}\cdot v_{q_{e}}))|_{\mathfrak{g}_{q_e}} = \Pi\circ \operatorname{Ad}^\ast_{h^{-1}}\circ \mathbb{I}(\operatorname{Exp} _{q_{e}}(v_{q_{e}}))|_{\mathfrak{g}_{q_e}}\\ &= \operatorname{Ad}^\ast_{h^{-1}} \circ \Pi\circ \mathbb{I}(\operatorname{Exp} _{q_{e}}(v_{q_{e}}))|_{\mathfrak{g}_{q_e}} = \operatorname{Ad}^\ast_{h^{-1}} \circ \overset{\thicksim}{ \mathbb{I}}(\operatorname{Exp} _{q_{e}}(v_{q_{e}}))\end{aligned}$$ for all $h\in G_{q_{e}}$ and $v_{q_{e}}\in B$. Replacing here $v_{q_e} $ by $sv_{q_e} $ and taking the $s $–derivative at zero, shows that $T_{q_e}\overset{\thicksim}{\mathbb {I}}(h\cdot v_{q_e}) \xi = \operatorname{Ad}^\ast_{h^{-1}}\left( T_{q_e}\overset{\thicksim}{\mathbb {I}}(v_{q_e}) \xi \right)$ for any $h\in G_{q_{e}}$ and $\xi \in \mathfrak{g}_{q_{e}}$, that is, $T_{q_e}\overset{\thicksim}{\mathbb {I}}(v_{q_e}) \xi $ is $G_{q_{e}}$–equivariant as a function of $v_{q_{e}}$, for all $\xi \in \mathfrak{g}_{q_{e}}$. Similarly $ T_{q_e} \mathbb {I}(h\cdot v_{q_e}) = \operatorname{Ad}^\ast_{h^{-1}} \circ T_{q_e} \mathbb{I}( v_{q_e}) \circ \operatorname{Ad}_{h^{-1}}$. From and the definition of $\widehat{\mathbb {I}}(q_e)^{-1}$, it follows that $\widehat{\mathbb {I}}(q_e)^{-1} = \operatorname{Ad}_h \circ \widehat{\mathbb{I}}(q_e)^{-1} \circ \operatorname{Ad}^\ast_h$ for any $h \in G_{q_e}$. Thus, for $h \in G_{q_e}$, the second summand in $A_{ab}$ becomes $$\begin{aligned} &\left\langle \left(T_{q_e}\mathbb {I}(h\cdot v_{q_e}) \circ \widehat{\mathbb {I}}(q_e)^{-1} \circ T_{q_e} \overset{\thicksim} {\mathbb {I}}(h\cdot v_{q_e})\right) \xi_a, \xi_b \right\rangle \\ &= \left\langle \left(\operatorname{Ad}^\ast_{h^{-1}} \circ T_{q_e} \mathbb{I}( v_{q_e}) \circ \operatorname{Ad}_{h^{-1}} \circ \widehat{\mathbb {I}}(q_e)^{-1} \circ \operatorname{Ad}^\ast_{h^{-1}}\circ T_{q_e}\overset{\thicksim}{\mathbb {I}}(v_{q_e}) \right) \xi_a, \xi_b \right\rangle\\ &= \left\langle \left(\operatorname{Ad}^\ast_{h^{-1}} \circ T_{q_e} \mathbb{I}( v_{q_e}) \circ \widehat{\mathbb {I}}(q_e)^{-1} \circ T_{q_e}\overset{\thicksim}{\mathbb {I}}(v_{q_e}) \right) \xi_a, \xi_b \right\rangle\\ &= \left\langle \left( T_{q_e} \mathbb{I}( v_{q_e}) \circ \widehat{\mathbb {I}}(q_e)^{-1} \circ T_{q_e}\overset{\thicksim}{\mathbb {I}}(v_{q_e}) \right) \xi_a, \operatorname{Ad}_{h^{-1}}\xi_b \right\rangle\\ &= \left\langle \left( T_{q_e} \mathbb{I}( v_{q_e}) \circ \widehat{\mathbb {I}}(q_e)^{-1} \circ T_{q_e}\overset{\thicksim}{\mathbb {I}}(v_{q_e}) \right) \xi_a, \xi_b \right\rangle\end{aligned}$$ since $\operatorname{Ad}_{h^{-1}}\xi_b = 0 $ because $h \in G_{q_e}$ and $\xi_b \in \mathfrak{g}_{q_e}$. This shows that the second summand in $A_{ab}$ is $G_{q_e}$– invariant. Next, we show that the first summand in $A_{ab}$ is $G_{q_e}$– invariant. To see this note that $$\left\langle A_{v_{q_e}}\xi_a, \xi_b \right\rangle = \left.\frac{\partial}{\partial \tau}\right|_{\tau= 0}\left\langle T_{\tau v_{q_e}}(\mathbb {I} \circ \operatorname{Exp}_{q_e})(v_{q_e})\xi_a, \xi_b \right\rangle = \left.\frac{\partial^2}{\partial \tau^2}\right|_{\tau= 0} \left\langle \mathbb{I}(\operatorname{Exp}_{q_e}(\tau v_{q_e}))\xi_a, \xi_b \right\rangle.$$ Therefore, for any $h \in G_{q_e} $ we get from $$\begin{aligned} \left\langle A_{h\cdot v_{q_e}}\xi_a, \xi_b \right\rangle &= \left.\frac{\partial^2}{\partial \tau^2}\right|_{\tau= 0} \left\langle \mathbb{I}(\operatorname{Exp}_{q_e}(\tau h \cdot v_{q_e}))\xi_a, \xi_b \right\rangle = \left.\frac{\partial^2}{\partial \tau^2}\right|_{\tau= 0} \left\langle \mathbb{I}(h \cdot \operatorname{Exp}_{q_e}(\tau v_{q_e}))\xi_a, \xi_b \right\rangle \\ &= \left.\frac{\partial^2}{\partial \tau^2}\right|_{\tau= 0} \left\langle \operatorname{Ad}_{h^{-1}}^\ast \mathbb{I}(\operatorname{Exp}_{q_e}(\tau v_{q_e})) \operatorname{Ad}_{h^{-1}}\xi_a, \xi_b \right\rangle\\ &= \left.\frac{\partial^2}{\partial \tau^2}\right|_{\tau= 0} \left\langle \mathbb{I}(\operatorname{Exp}_{q_e}(\tau v_{q_e})) \operatorname{Ad}_{h^{-1}}\xi_a, \operatorname{Ad}_{h^{-1}}\xi_b \right\rangle\\ &= \left.\frac{\partial^2}{\partial \tau^2}\right|_{\tau= 0} \left\langle \mathbb{I}(\operatorname{Exp}_{q_e}(\tau v_{q_e}))\xi_a, \xi_b \right\rangle = \left\langle A_{v_{q_e}}\xi_a, \xi_b \right\rangle,\end{aligned}$$ as required. \[determination of xi\] The equation $\varphi (\tau ,v_{q_{e}},\mu ,\xi)=0 $ for $(\tau ,v_{q_{e}},\mu ,\xi) \in I\times \left(B\cap (T_{q_{e}}Q)_{\{e\}}\setminus \mathcal{Z}_{\mu}\right) \times \mathfrak{g}^{\ast }\times \mathfrak{g}_{q_{e}}$ has a unique smooth solution $\xi (\tau ,v_{q_{e}},\mu )\in \mathfrak{g}_{q_{e}}$ for $(\tau, v_{q_e}, \mu) \in I\times (B\cap (T_{q_{e}}Q)_{\{e\}}\setminus \mathcal{Z}_{\mu})\times \mathfrak{g}^{\ast }$. Denote by $D_\xi$ the Fréchet derivative relative to the variable $\xi \in \mathfrak{g}_{q_e}$. Recall that $\xi_0(v_{q_e}, \mu) \in \mathfrak{g}_{q_e}$ is the unique solution of the equation $\frac{\partial ^{2} \varphi}{\partial \tau ^{2}}(0,v_{q_{e}},\mu ,\xi )=0$. Formulas and yield $$\begin{aligned} \label{expanded second tau derivative of phi at zero} \frac{\partial ^{2}\varphi }{\partial \tau^{2}} (0,v_{q_{e}},\mu, \xi ) &= (Id-\Pi)\left[ A_{v_{q_e}}(\xi +\eta _{\mu}) - 2\left(T_{q_e}\mathbb{I}(v_{q_{e}}) \circ \widehat{\mathbb{I}}(q_e)^{-1} \circ T_{q_e}\overset{\thicksim} {\mathbb {I}}(v_{q_e})\right) \xi \right. \\ & \phantom{T_{q_e}\overset{\thicksim} {\mathbb{I}}(v_{q_e})} - 2\left(T_{q_e}\mathbb{I}(v_{q_{e}}) \circ \widehat{\mathbb {I}}(q_e)^{-1} \circ T_{q_e}\widehat{\mathbb{I}}(v_{q_e}) \circ \widehat{\mathbb{I}} (q_{e})^{-1}\right)(\Pi _{1}\mu ) \nonumber \\ &\left. \phantom{T_{q_e}\overset{\thicksim} {\mathbb{I}}(v_{q_e})} + 2\left(T_{q_e}\mathbb{I}(v_{q_{e}}) \circ\widehat{\mathbb{I}} (q_{e})^{-1}\right)(\beta'(\mu)) - 2\beta ''(\mu) \right] \nonumber\end{aligned}$$ and hence $$D_\xi\frac{\partial ^{2}\varphi }{\partial \tau^{2}} (0,v_{q_{e}}, \mu, \xi_0(v_{q_e}, \mu)) = (Id-\Pi)\left[ A_{v_{q_e}}|_{\mathfrak{g}_{q_e}} - 2T_{q_e}\mathbb{I}(v_{q_{e}}) \circ \widehat{\mathbb{I}}(q_e)^{-1} \circ T_{q_e}\overset{\thicksim} {\mathbb {I}}(v_{q_e}) \right] : \mathfrak{g}_{q_e} \rightarrow \mathfrak{k}^\circ.$$ We shall prove that this linear map is injective. To see this, note that relative to the basis $\{\xi_1, \dots , \xi_p\}$ of $\mathfrak{g}_{q_e}$ this linear operator has matrix $A $ by . Thus, if $v_{q_e} \notin \mathcal{Z}_{\mu}$, this matrix is invertible. In particular, this linear operator is injective. Since $\mathfrak{g} = \mathfrak{g}_{q_e} \oplus \mathfrak{k}$, it follows that $\dim \mathfrak{g}_{q_e} = \dim \mathfrak{g} - \dim \mathfrak{k} = \dim \mathfrak{k}^\circ $, so the injectivity of the map $D_\xi\frac{\partial ^{2}\varphi }{\partial \tau^{2}} (0,v_{q_{e}}^0, \mu^0, \xi_0(v_{q_e}^0, \mu^0))$ implies that it is an isomorphism. Therefore, if $v_{q_{e}}\in B\cap (T_{q_{e}}Q)_{\{e\}}\setminus \mathcal{Z}_{\mu}$ is near $v_{q_e}^0 $, the implicit function theorem, guarantees the existence of an open neighborhood $V_{0}\subset I\times (B\cap (T_{q_{e}}Q)_{\{e\}}\setminus \mathcal{Z}_{\mu})\times \mathfrak{g}^{\ast }$ containing $(0, v_{q_e}^0, \mu^0) \in \{0\}\times (B\cap (T_{q_{e}}Q)_{\{e\}}\setminus \mathcal{Z}_{\mu}) \times \mathfrak{g}^{\ast }$ and of a unique smooth function $\xi :V_{0}\rightarrow \mathfrak{g}_{q_{e}}$ satisfying $\varphi (\tau ,v_{q_{e}},\mu , \xi(\tau ,v_{q_{e}},\mu) )=0$ such that $\xi(0,v_{q_{e}}^0,\mu^0) = \xi_0(v_{q_e}^0, \mu^0)$. On the other hand, for $\tau \neq 0$, the equation $\varphi (\tau ,v_{q_{e}},\mu ,\cdot )=0$ has a unique solution for $\xi $, namely the $\mathfrak{g}_{q_{e}}$-component of $\mathbb{I}(\operatorname{Exp} _{q_{q}}(\tau v_{q_{e}}))^{-1}\beta (\tau ,\mu )$, which is a smooth function of $\tau ,v_{q_{e}},\mu $. This is true since $\xi + \eta = \mathbb{I}(\operatorname{Exp} _{q_{q}}(\tau v_{q_{e}}))^{-1}\beta (\tau ,\mu )$ by construction and we determined the two components $\xi\in \mathfrak{g}_{q_e} $ and $\eta \in \mathfrak{k}$ in $\mathfrak{g} = \mathfrak{g}_{q_e} \oplus \mathfrak{k}$ via the Lyapunov-Schmidt method, precisely in order that this equality be satisfied. Therefore, the solution $\xi(\tau, v_{q_e}, \mu)$ obtained above by the implicit function theorem must coincide with the $\mathfrak{g}_{q_{e}}$-component of $\mathbb{I}(\operatorname{Exp} _{q_{q}}(\tau v_{q_{e}}))^{-1}\beta (\tau ,\mu )$ for $\tau>0 $. Since this entire argument involving the Lyapunov-Schmidt procedure was carried out for any $(v_{q_e}^0, \mu^0)$, it follows that the equation $\varphi (\tau,v_{q_{e}},\mu ,\xi)=0$ has a unique smooth solution $\xi (\tau ,v_{q_{e}},\mu )\in \mathfrak{g}_{q_{e}}$ for $(\tau, v_{q_e}, \mu) \in I\times (B\cap (T_{q_{e}}Q)_{\{e\}}\setminus \mathcal{Z}_{\mu})\times \mathfrak{g}^{\ast }$. \[remark on smoothness\] The previous proposition says that if we define $$\zeta(\tau ,v_{q_{e}},\mu )=\mathbb{I}(\operatorname{Exp} _{q_{e}}(\tau v_{q_{e}}))^{-1}\beta (\tau ,\mu )$$ on $(I\setminus \{0\})\times (B\cap (T_{q_{e}}Q)_{\{e\}}\setminus \mathcal{Z}_{\mu})\times \mathfrak{g}^{\ast }$, then $\zeta(\tau ,v_{q_{e}},\mu )$ can be smoothly extended for $\tau=0$. We have, in fact, $\zeta(\tau ,v_{q_{e}},\mu )= \xi(\tau ,v_{q_{e}},\mu ) + \eta (\tau ,v_{q_{e}},\mu ,\xi(\tau ,v_{q_{e}},\mu ) )$, where $\eta(\tau ,v_{q_{e}},\mu, \xi )$ was found in the first step of the Lyapunov-Schmidt procedure and $\xi(\tau ,v_{q_{e}},\mu )$ in the second step, as given in Proposition \[determination of xi\]. Note also that $\zeta(0, v_{q_e}, \mu) = \xi_0(v_{q_e}, \mu) + \widehat{\mathbb {I}}(q_e)^{-1} \Pi_1 \mu \in \mathfrak{t}$. A simplified version of the amended potential criterion ------------------------------------------------------- At this point we have a candidate for a bifurcating branch from the set of relative equilibria $\mathfrak{t} \cdot q_e $. This branch will start at $\zeta(0, v_{q_e}, \mu)_Q(q_e) \in \mathfrak{t}\cdot q_e \subset T_{q_e}Q$. By Lemma \[same isotropy\], the isotropy subgroup of $\zeta(0, v_{q_e}, \mu)_Q(q_e) $ equals $G_{q_e} $, for any $v_{q_e} \in B\cap (T_{q_{e}}Q)_{\{e\}}\setminus \mathcal{Z}_{\mu}$ and $\mu\in \mathfrak{g}^\ast$. The isotropy groups of the points on the curve $\zeta(\tau, v_{q_e}, \mu)_Q(\operatorname{Exp}_{q_{e}} (\tau v_{q_e}))$, for $\tau\neq 0 $, are all trivial, by construction. Hence $\zeta(\tau, v_{q_e}, \mu)_Q(\operatorname{Exp}_{q_{e}}(\tau v_{q_e}))$ is a curve that has the properties of the bifurcating branch of relative equilibria with broken symmetry that we are looking for. We do not know yet that all points on this curve are in fact relative equilibria. Thus, we shall search for conditions on $v_{q_e}$ and $\mu$ that guarantee that each point on the curve $\tau\mapsto\zeta(\tau, v_{q_e}, \mu)_Q(\operatorname{Exp}_{q_{e}}(\tau v_{q_e}))$ is a relative equilibrium. This will be done by using the amended potential criterion (see Proposition \[amended potential criterion\]) which is applicable because all base points of this curve, namely $\operatorname{Exp}_{q_e}(\tau v_{q_e})$, have trivial isotropy for $\tau\neq 0$. To carry this out, we need some additional geometric information. From standard theory of proper Lie group actions (see e.g. [@dk], §2.3, or [@kawakubo]) it follows that the map $$\label{corespondenta} [v_{q_{e}},\mu ]_{G_{q_{e}}} \in (B \times \mathfrak{g}^{\ast })/G_{q_{e}} \longmapsto [\operatorname{Exp}_{q_{e}}(v_{q_{e}}),\mu ]_{G} \in ((G \cdot \operatorname{Exp}_{q_e} B) \times \mathfrak{g}^{\ast })/G$$ is a homeomorphism of $(B\times \mathfrak{g}^{\ast })/G_{q_{e}}$ with $((G \cdot \operatorname{Exp}_{q_e} B) \times \mathfrak{g}^{\ast })/G$ and that its restriction to $ ((B\cap (T_{q_{e}}Q)_{\{e\}} \setminus \mathcal{Z}_{\mu})\times \mathfrak{g}^{\ast })/G_{q_{e}}$ is a diffeomorphism onto its image. We think of a pair $(\operatorname{Exp}_{q_e}(v_{q_e}), \mu)$ as the base point of a relative equilibrium and its momentum value. All these relative equilibria come in $G$-orbits. The homeomorphism allows the identification of $G$-orbits of relative equilibria with $G_{q_e}$-orbits of certain pairs $(v_{q_e},\mu)$. We shall work in what follows on both sides of this identification, based on convenience. We will need the following lemma, which is a special case of stability of the transversality of smooth maps (see e.g. [@gp]). Let $G$ be a Lie group acting on a Riemannian manifold $Q$, $q\in Q$, and let $\mathfrak{k}\subset \mathfrak{g}$ be a subspace satisfying $\mathfrak{k}\cap \mathfrak{g}_{q}=\{0\}$. Let $V\subset T_{q}Q$ be a subspace such that $\mathfrak{k}\cdot q\oplus V=T_{q}Q$. Then there is an $\epsilon >0$ such that if $\|v_{q}\| <\epsilon $, $$T_{\operatorname{Exp} _{q}(v_{q})}Q=\mathfrak{k}\cdot \operatorname{Exp} _{q}(v_{q})\oplus (T_{v_{q}}\operatorname{Exp} _{q})V.$$ To deal with $G $-orbits of relative equilibria, we need a different splitting of the same nature. The following result is modeled on a proposition in [@hm]. \[decomposition of the tangent space adapted to sigma bar\] Let $v_{q_e} \in B\cap (T_{q_{e}}Q)_{\{e\}} \setminus \mathcal{Z}_{\mu}$ be given. Consider the principal $G_{q_e}$-bundle $B\cap (T_{q_{e}}Q)_{\{e\}} \setminus \mathcal{Z}_{\mu} \rightarrow [B\cap (T_{q_{e}}Q)_{\{e\}} \setminus \mathcal{Z}_{\mu}]/G_{q_{e}}$ (this is implied by Lemma \[properties of Z\]). Let $\widetilde{U}$ be a neighborhood of $[0_{q_e}] \in (T_{q_e}Q)/G_{q_e}$ and define the open set $U: = \widetilde{U} \cap [B\cap (T_{q_{e}}Q)_{\{e\}} \setminus \mathcal{Z}_{\mu}]/G_{q_{e}} $ in $[B\cap (T_{q_{e}}Q)_{\{e\}} \setminus \mathcal{Z}_{\mu}]/G_{q_{e}} $. Let $\sigma :U\subset [B\cap (T_{q_{e}}Q)_{\{e\}})\setminus \mathcal{Z}_{\mu}]/G_{q_{e}}\rightarrow B\cap (T_{q_{e}}Q)_{\{e\}}\setminus \mathcal{Z}_{\mu}$ be a smooth section, $[v_{_{q_{e}}}]\in U$, and $\overline{\sigma }:=\operatorname{Exp} _{q_{e}}\circ \sigma : U \rightarrow Q$. Then there exists $\epsilon >0$ such that for $ 0<\tau <\epsilon $ sufficiently small, we have $$T_{\overline{\sigma }([\tau v_{q_{e}}])}Q =\mathfrak{t}\cdot \overline{\sigma }([\tau v_{q_{e}}]) \oplus T_{[\tau v_{q_{e}}]} \overline{\sigma}(T_{[\tau v_{q_{e}}]}U) \oplus (T_{\sigma ([\tau v_{q_{e}}])}\operatorname{Exp}_{q_{e}}) (\mathfrak{k} _{2}\cdot q_{e}).$$ Since $\mathfrak{g}= \mathfrak{k}_0 \oplus \mathfrak{k}_1 \oplus \mathfrak{k}_2 $ and $\mathfrak{k}_0 = \mathfrak{g}_{q_e}$ we have $T_{q_{e}}Q=\mathfrak{k}_{1}\cdot q_{e}\oplus \mathfrak{k}_{2}\cdot q_{e}\oplus (\mathfrak{g}\cdot q_{e})^{\perp }$. Apply the above lemma with $\mathfrak{k}=\mathfrak{k}_{1}$ and $V=\mathfrak{k}_{2}\cdot q_{e}\oplus (\mathfrak{g}\cdot q_{e})^{\perp }$. For the $\epsilon >0 $ in the statement choose $\tau $ such that $0< \tau< \epsilon $ and $\| \sigma([ \tau v_{q_e}]) \| < \epsilon$. Then $$\begin{aligned} \label{decomposition of q for the section} T_{\overline{\sigma }([\tau v_{q_{e}}])}Q &=\mathfrak{k}_{1}\cdot \overline{ \sigma }([\tau v_{q_{e}}]) \oplus (T_{\sigma ([\tau v_{q_{e}}])}\operatorname{Exp} _{q_{e}})(\mathfrak{k}_{2}\cdot q_{e} \oplus (\mathfrak{g}\cdot q_e) ^\perp)\\ &= \mathfrak{k}_{1}\cdot \overline{\sigma }([\tau v_{q_{e}}]) \oplus (T_{\sigma ([\tau v_{q_{e}}])}\operatorname{Exp} _{q_{e}})((\mathfrak{g}\cdot q_e) ^\perp ) \oplus (T_{\sigma ([\tau v_{q_{e}}])}\operatorname{Exp} _{q_{e}})(\mathfrak{k}_2 \cdot q_e). \nonumber\end{aligned}$$ since $\operatorname{Exp}_{q_e}$ is a diffeomorphism on $B \subset (\mathfrak{g}\cdot q_e)^\perp$. Since $(\sigma ,U)$ is a smooth local section, $\mathcal{Z}_{\mu}$ is closed and $G_{q_e}$-invariant in $B\cap (T_{q_e}Q)_{\{e\}}$, and $(T_{q_e}Q)_{\{e\}}$ is open in $T_{q_e} Q $, it follows that $B\cap (T_{q_e}Q)_{\{e\}}$ is open in $(\mathfrak{g}\cdot q_e)^\perp$ and thus we get $$(\mathfrak{g}\cdot q_e)^\perp = T_{\sigma ([\tau v_{q_{e}}])}(B\cap (T_{q_{e}}Q)_{\{e\}}\setminus \mathcal{Z}_{\mu}) = T_{[\tau v_{q_{e}}]}\sigma (T_{[\tau v_{q_{e}}]}U) \oplus \mathfrak{k}_{0}\cdot \sigma ([\tau v_{q_{e}}]),$$ where $\mathfrak{k}_{0}\cdot \sigma ([\tau v_{q_{e}}]) = \{\zeta_{T_{q_e}Q} (\sigma([ \tau v_{q_e}]) \mid \zeta\in \mathfrak{k}_0 \}$. The $G_{q_e}$-equivariance of $\operatorname{Exp}_{q_e}$ implies that $$T_{u_{q_{e}}}\operatorname{Exp} _{q_{e}}(\xi _{T_{q_e}Q}(u_{q_{e}}))=\xi _{Q}(\operatorname{Exp} _{q_{e}}(u_{q_{e}})) \quad \text{for~all} \quad \xi \in \mathfrak{k}_{0}, \quad u_{q_{e}}\in T_{q_{e}}Q$$ and hence $$\begin{aligned} \label{decomposition of one summand} &(T_{\sigma ([\tau v_{q_{e}}])}\operatorname{Exp} _{q_{e}})((\mathfrak{g}\cdot q_e) ^\perp ) \\ &\qquad \qquad =(T_{\sigma([\tau v_{q_{e}}])}\operatorname{Exp} _{q_{e}} \circ T_{[\tau v_{q_e}]}\sigma )(T_{[\tau v_{q_{e}}]}U)\oplus (T_{\sigma ([\tau v_{q_{e}}])}\operatorname{Exp} _{q_{e}})(\mathfrak{k}_{0}\cdot \sigma ([\tau v_{q_{e}}])) \nonumber \\ &\qquad \qquad =T_{[\tau v_{q_{e}}]}\overline{\sigma }(T_{[\tau v_{q_{e}}]}U)\oplus \mathfrak{k}_{0}\cdot \overline{\sigma }([\tau v_{q_{e}}]). \nonumber\end{aligned}$$ Introducing in and taking into account that $\mathfrak{t}=\mathfrak{k}_{0}\oplus \mathfrak{k}_{1}$ we get the statement of the proposition. We want to find pairs $(v_{q_e}, \mu) $ such that $\mathbf{d}V_{\beta(\tau, \mu)} (\operatorname{Exp} _{q_{e}}(\tau v_{q_{e}}))=0$ for $\tau> 0 $. Since $V_{\beta(\tau, \mu)}$ is $G_{\beta(\tau, \mu)}$-invariant, this condition will hold if we only verify it on a subspace of $T_{\operatorname{Exp}_{q_e}(\tau v_{q_e})} Q$ complementary to $\mathfrak{g}_{\beta(\tau,\mu)} \cdot \operatorname{Exp} _{q_{e}}(\tau v_{q_{e}}) = \mathfrak{t}\cdot \operatorname{Exp} _{q_{e}}(\tau v_{q_{e}})$. The previous decomposition of the tangent space immediately yields the following result. \[criteriu\] Suppose that $\mu\in \mathfrak{g}^\ast$ is such that $\mathfrak{g}_{\beta( \tau, \mu)} = \mathfrak{t}$ for all $\tau$ in a neighborhood of zero. Let $U$ and $\sigma $ be as in Proposition \[decomposition of the tangent space adapted to sigma bar\], $[v_{q_{e}}]\in U$, and $\overline{\sigma }:=\operatorname{Exp} _{q_{e}}\circ \sigma $. Then there is an $\epsilon >0$ such that $\mathbf{d}V_{\beta (\tau, \mu)}(\overline{\sigma }([\tau v_{q_{e}}])=0$ if and only if $\mathbf{d}(V_{\beta (\tau, \mu )}\circ \overline{\sigma })([\tau v_{q_{e}}])=0$ and $\mathbf{d}(V_{\beta (\tau, \mu )}\circ \operatorname{Exp} _{q_{e}})( \sigma ([\tau v_{q_{e}}]))| _{\mathfrak{k}_{2}\cdot q_{e}}=0$ for $ 0<\tau <\epsilon $. The study of two auxiliary functions ------------------------------------ Let $I $ be an open interval containing zero. Recall that $p = \dim \mathfrak{g}_{q_e} = \dim \mathfrak{m}_0 $. Let $\vartheta_{1}$ be an element of a basis $\{\vartheta _{1},\vartheta _{2},..., \vartheta _{p}\}$ for $\mathfrak{m}_{0}$ and define $\beta :(I\setminus \{0\})\times (\mathfrak{m}_{1}\oplus \mathfrak{m}_{2})\rightarrow \mathfrak{g}^{\ast }$ by $$\beta (\tau ,\mu)=\Pi _{1}\mu+\tau \Pi_{2}\mu +\tau ^{2}\vartheta _{1},$$ where $\Pi_1 : \mathfrak{g}^\ast\rightarrow \mathfrak{m}_1 = \mathbb {I}(q_e) \mathfrak{t}$ and $\Pi_2: \mathfrak{g}^\ast \rightarrow \mathfrak{m}_2 = \mathfrak{t}^\circ $. Notice that this function is a particular case of $$\beta (\tau ,\mu )=\Pi _{1}\mu +\tau \beta ^{\prime }(\mu )+\tau ^{2}\beta ^{\prime \prime }(\mu ),$$ by choosing $\beta' (\mu) = \Pi_2 \mu$ and $\beta '' (\mu) = \vartheta_1 $. Recall that $\mathbb {I}(q_e) = \mathfrak{m}_1 \oplus\mathfrak{m}_2 $ by Lemma \[moment of inertia isomorphism\] and that $\mathbf{J}_L(\mathfrak{g}\cdot q_e) = \mathbb {I}(q_e) \mathfrak{g}$ from the definition of $\mathbf{J}_L $. \[theorem about F\] The smooth function $F_{1}:(I\setminus \{0\})\times U\times \mathbf{J}_{L}(\mathfrak{g}\cdot q_{e})\rightarrow \mathbb{R}$ defined by $$F_{1}(\tau ,[v_{q_{e}}],\mu ):=(V_{\beta (\tau, \mu )}\circ \overline{\sigma })(\tau [v_{q_{e}}]).$$ can be extended to a smooth function on $I\times U\times \mathbf{J}_{L}(\mathfrak{g}\cdot q_{e})$, also denoted by $F_1 $. In addition $$F_{1}(\tau ,[v_{q_{e}}],\mu )=F_{0}(\mu )+\tau ^{2}F(\tau ,[v_{q_{e}}],\mu ).$$ where $F_{0}$, $F$ are defined on $\mathbf{J}_{L}( \mathfrak{g}\cdot q_{e})$ and on $I\times U\times \mathbf{J}_{L}( \mathfrak{g}\cdot q_{e})$ respectively. Denote $v_{q_{e}}:=\sigma ([v_{q_{e}}])\in B\cap (T_{q_{e}}Q)_{\{e\}} \setminus \mathcal{Z}_{\mu}$. One can easily see that $$(V_{\beta (\tau, \mu)}\circ \overline{\sigma })(\tau \lbrack v_{q_{e}}])=V(\operatorname{Exp} _{q_{e}}(\tau v_{q_{e}}))+\frac{1}{2}\left\langle\beta (\tau ,\mu ),\mathbb{I}(\operatorname{Exp}_{q_{e}}(\tau v_{q_{e}}))^{-1} \beta(\tau ,\mu )\right\rangle.$$ By Remark \[remark on smoothness\], the second term is smooth even in a a neighborhood of $\tau= 0 $. Since the first term is obviously smooth, it follows that $V_{\beta (\tau, \mu)}\circ \overline{\sigma }$ is smooth also in a neighborhood of $\tau= 0 $. This is the smooth extension of $F_1 $ in the statement. Let $\{\xi _{1},...,\xi _{p}\}$ be a basis for $\mathfrak{g}_{q_{e}} \subset \mathfrak{t}$. Then, again by Remark \[remark on smoothness\], we have $$\mathbb{I}(\operatorname{Exp}_{q_{e}}(\tau v_{q_{e}}))^{-1}\beta (\tau ,\mu )=\overset{p}{\underset{a=1}{ \sum }}\alpha _{a}(\tau ,v_{q_{e}},\mu )\xi _{a}+\eta \left(\tau ,v_{q_{e}},\mu , \overset{p}{\underset{a=1}{\sum }}\alpha _{a}(\tau ,v_{q_{e}},\mu )\xi _{a}\right)$$ where $\alpha _{1},...,\alpha _{p},\eta $ are smooth real functions of all their arguments. In what follows we will denote $$\eta \left(\tau ,v_{q_{e}},\mu ,\overset{p}{\underset{a=1}{\sum }}\alpha _{a}(\tau ,v_{q_{e}},\mu )\xi _{a}\right) =\eta (\tau,v_{q_{e}},\mu ,\alpha _{1}(\tau ,v_{q_{e}},\mu ),...,\alpha _{p}(\tau ,v_{q_{e}},\mu )).$$ Let $\mu \in \mathbf{J}_{L}(\mathfrak{g}\cdot q_{e})= \mathfrak{m}_1 \oplus\mathfrak{m}_2$ and $v_{q_{e}}\in B\cap (T_{q_{e}}Q)_{\{e\}}\backslash \mathcal{Z}_{\mu}$. Since in the computations that follow, the arguments $v_{q_e}$ and $\mu$ play the role of parameters, we shall denote temporarily $\alpha _{a}(\tau )=\alpha _{a}(\tau, v_{q_{e}},\mu)$, $a\in \{1,...,p\}$, and $\eta (\tau ,\alpha _{1},...,\alpha _{p})=\eta (\tau ,v_{q_{e}},\mu ,\alpha _{1}(\tau ,v_{q_{e}},\mu ),...,\alpha _{p}(\tau ,v_{q_{e}},\mu ))$. Then by we get $$\begin{aligned} \frac{\partial \eta }{\partial \tau }(0,\alpha _{1},..., \alpha _{p}) = &-\overset{p}{\underset{a=1}{\sum }}\alpha_{a} \left(\widehat{\mathbb{I}}(q_{e})^{-1}\circ T_{q_e}\overset{\thicksim}{\mathbb{I}} (v_{q_{e}})\right)\xi _{a} \\ & -\left(\widehat{\mathbb{I}}(q_{e})^{-1} \circ T_{q_e}\widehat{\mathbb{I}}(v_{q_{e}}) \circ \widehat{\mathbb{I}}(q_{e})^{-1} \right)\Pi_{1}\mu +\widehat{\mathbb{I}}(q_{e})^{-1}\Pi_{2}\mu.\end{aligned}$$ Formula shows that $$\frac{\partial \eta }{\partial \alpha_a }(0,\alpha_1, \dots , \alpha_p) = 0$$ Note that $$\left.V_{\beta (\tau ,\mu )}(\operatorname{Exp}_{q_{e}}(\tau v_{q_{e}}))\right|_{\tau =0} = V(q_{e})+\frac{1}{2} \left\langle \Pi_{1}\mu,\widehat {\mathbb{I}}(q_{e})^{-1}\Pi_{1}\mu\right\rangle$$ is independent of $v_{q_{e}}$. This shows that $F_1(0, [v_{q_e}], \mu) = F_0(\mu)$ for some smooth function on $\mathfrak{m}_1 \oplus \mathfrak{m}_2 $. Using Remark \[remark on smoothness\], we get $$\begin{aligned} \left.\frac{d}{d\tau }\right|_{\tau =0}&V_{\beta (\tau ,\mu )}(\operatorname{Exp}_{q_{e}}(\tau v_{q_{e}})) =\mathbf{d}V(q_{e})(v_{q_{e}}) + \frac{1}{2}\left\langle \Pi_{2}\mu ,\overset{p}{ \underset{a=1}{\sum }}\alpha _{a}(0)\xi _{a}+\eta (0,\alpha _{1},...,\alpha _{p})\right\rangle\\ &+\frac{1}{2}\left\langle \Pi_1\mu, \sum_{a=1}^p \frac{\partial \alpha_a}{\partial \tau}(0)\left(\xi_a + \frac{\partial \eta}{\partial \alpha_a}(0, \alpha_1, \dots , \alpha_p) \right) + \frac{\partial \eta}{\partial \tau}(0, \alpha _1, \dots , \alpha_p) \right\rangle.\end{aligned}$$ The first term $\mathbf{d}V(q_{e})=0$ by Proposition \[montaldi\] (i). Since $\eta(0, v_{q_e}, \mu, \xi) = \eta_\mu = \widehat{\mathbb {I}}(q_e)^{-1}\Pi_1 \mu \in \mathfrak{t}$ by Proposition \[belongs\], we get $$\overset{p}{\underset{ a=1}{\sum }}\alpha _{a}(0)\xi _{a}+\eta (0,\alpha _{1},...,\alpha _{p})= \overset{p}{\underset{a=1}{\sum }}\alpha _{a}(0)\xi _{a} +\widehat{\mathbb{I}}(q_{e})^{-1}\Pi_{1}\mu\in \mathfrak{t}.$$ Thus the second term vanishes because $\mathfrak{m}_{2}=\mathfrak{t}^{\circ }$. As $\frac{\partial \eta }{\partial \alpha _{a}}(0, \alpha_1, \dots , \alpha_p) =0$ and $\mathfrak{m}_{1}$ annihilates $\mathfrak{g} _{q_{e}}$, the third term becomes $$\begin{aligned} \left\langle \Pi_{1}\mu,\frac{\partial \eta }{\partial \tau } (0,\alpha _{1},...,\alpha_{p})\right\rangle =&-\overset{p}{\underset{a=1}{\sum}}\alpha _{a} \left\langle \Pi_1\mu, \left(\widehat{\mathbb{I}}(q_{e})^{-1}\circ T_{q_e}\overset{\thicksim}{\mathbb{I}} (v_{q_{e}})\right)\xi _{a}\right\rangle \\ &-\left\langle\Pi_{1}\mu,\left(\widehat{\mathbb{I}}(q_{e})^{-1} \circ T_{q_e}\widehat{\mathbb{I}}(v_{q_{e}}) \circ \widehat{\mathbb{I}}(q_{e})^{-1} \right)\Pi_{1}\mu\right\rangle\\ &+\left\langle\Pi_{1}\mu,\widehat{\mathbb{I}}(q_{e})^{-1}\Pi_{2}\mu \right\rangle.\end{aligned}$$ We will prove that each summand in this expression vanishes. $\bullet$ Since $\langle \mathfrak{m}_0 , \mathfrak{k}_1 \rangle = 0$, we get $$\begin{aligned} &\left\langle \Pi_1\mu, \left(\widehat{\mathbb{I}}(q_{e})^{-1}\circ T_{q_e}\overset{\thicksim}{\mathbb{I}} (v_{q_{e}})\right)\xi _{a}\right\rangle = \left\langle T_{q_e}\overset{\thicksim}{\mathbb{I}} (v_{q_{e}})\xi_{a},\; \widehat{\mathbb{I}}(q_{e})^{-1}\Pi_1\mu \right\rangle \\ &\qquad = \left\langle T_{q_e}{\mathbb{I}} (v_{q_{e}})\xi_{a},\; \widehat{\mathbb{I}}(q_{e})^{-1}\Pi_1\mu \right\rangle = \mathbf{d}\left\langle \mathbb {I}(\cdot) \xi_a, \eta_\mu \right\rangle(q_e)(v_{q_e}) =0\end{aligned}$$ by because $\xi_a \in \mathfrak{g}_{q_e}$ and $\eta_\mu \in \mathfrak{t}$. Thus the first summand vanishes. $\bullet$ The second summand equals $$\left\langle\Pi_{1}\mu,\left(\widehat{\mathbb{I}}(q_{e})^{-1} \circ T_{q_e}\widehat{\mathbb{I}}(v_{q_{e}}) \circ \widehat{\mathbb{I}}(q_{e})^{-1} \right)\Pi_{1}\mu\right\rangle = \left\langle T_{q_e}\widehat{\mathbb{I}}(v_{q_{e}}) \eta_\mu, \eta_\mu \right\rangle = \left\langle T_{q_e}\mathbb{I}(v_{q_{e}}) \eta_\mu, \eta_\mu \right\rangle$$ because $\langle \mathfrak{m}_0, \mathfrak{k}_1 \rangle = 0 $. We shall prove that this term vanishes in the following way. Recall that $\eta_\mu \in \mathfrak{k}_1 \subset \mathfrak{t}$. For any $\zeta \in \mathfrak{t}$, hypothesis **(H)** states that $\zeta_Q(q_e) $ is a relative equilibrium and thus, by the augmented potential criterion (see Proposition \[augmented potential criterion\]), $\mathbf{d}V_\zeta(q_e) = 0 $. Since $$\mathbf{d}V_\zeta(q_e) (u_{q_e}) = \mathbf{d}V(q_e)(u_{q_e}) - \frac{1}{2} \left\langle T_{q_e} \mathbb{I}(u_{q_e}) \zeta, \zeta \right\rangle$$ for any $u_{q_e} \in T_{q_e} Q $ and $\mathbf{d}V(q_e) = 0 $ by Proposition \[montaldi\] (i), it follows that $\left\langle T_{q_e} \mathbb{I}(u_{q_e}) \zeta,\zeta\right\rangle = 0 $. Thus the second summand vanishes. $\bullet$ The third summand is $$\left\langle\Pi_{1}\mu,\widehat{\mathbb{I}}(q_{e})^{-1}\Pi_{2}\mu \right\rangle =\langle \Pi_{2}\mu,\eta_\mu \rangle =0$$ because $\mathfrak{m}_{2}=\mathfrak{t}^{\circ }$. So, we finally conclude that $$\left.\frac{d}{d\tau }\right|_{\tau =0}V_{\beta (\tau ,\mu )}(\operatorname{Exp}_{q_{e}}(\tau v_{q_{e}})) =0$$ and hence, by Taylor’s theorem, we have $$F_{1}(\tau ,[v_{q_{e}}],\mu )=F_{0}(\mu )+\tau ^{2}F(\tau, [v_{q_{e}}],\mu )$$ for some smooth function $F $. \[theorem about G\] The smooth function $G_{1}:(I \setminus \{0\}) \times U\times \mathbf{J}_{L}(\mathfrak{g}\cdot q_{e})\rightarrow \mathfrak{k}_{2}^{\ast }$ defined by $$\left\langle G_{1}(\tau ,[v_{q_{e}}],\mu ),\varsigma \right\rangle =\mathbf{d}(V_{\beta (\tau ,\mu )}\circ \operatorname{Exp}_{q_{e}})(\sigma (\tau \lbrack v_{q_{e}}]))\big( \varsigma _{Q}(q_{e})\big), \quad \varsigma \in \mathfrak{k}_2,$$ can be smoothly extended to a function on $I\times U\times \mathbf{J}_{L}(\mathfrak{g}\cdot q_{e})$, also denoted by $G_1$. In addition, $$G_{1}(\tau ,[v_{q_{e}}],\mu )=\tau G(\tau ,[v_{q_{e}}],\mu )$$ where $G:I\times U\times \mathbf{J}_{L}(\mathfrak{g}\cdot q_{e})\rightarrow \mathfrak{k}_{2}^{\ast }$ is a smooth function. We will show that $G_{1}$ is a smooth function at $\tau =0$ and that $G_{1}(0,[v_{q_{e}}],\mu )=0$. Let $v_{q_{e}}=\sigma ([v_{q_{e}}])$. Then $$\begin{aligned} &\left\langle G_{1}(\tau ,[v_{q_{e}}],\mu ),\varsigma \right\rangle = \mathbf{d}V_{\beta(\tau,\mu)} \left(\operatorname{Exp}_{q_e}(\tau v_{q_e})\right) \left(T_{\tau v_{q_e}} \operatorname{Exp}_{q_e} \big(\varsigma_Q(q_e) \big) \right) \\ &\qquad =\mathbf{d}V(\operatorname{Exp}_{q_{e}}(\tau v_{q_{e}})) \left(T_{\tau v_{q_e}} \operatorname{Exp}_{q_e} \big(\varsigma_Q(q_e) \big) \right) \\ & \qquad \qquad \qquad +\frac{1}{2}\left\langle \beta (\tau ,\mu),\; T_{\operatorname{Exp}_{q_e}( \tau v_{q_e})} (\mathbb{I}(\cdot)^{-1}) \left(T_{\tau v_{q_e}} \operatorname{Exp}_{q_e} \big(\varsigma_Q(q_e) \big) \right) \beta (\tau ,\mu ) \right\rangle \\ &\qquad = \mathbf{d}V \left(\operatorname{Exp}_{q_e}(\tau v_{q_e})\right) \left(T_{\tau v_{q_e}} \operatorname{Exp}_{q_e} \big(\varsigma_Q(q_e) \big) \right) -\frac{1}{2}\left\langle \beta(\tau,\mu),\; \phantom{T_{\operatorname{Exp}_{q_e}( \tau v_{q_e})}\mathbb{I}} \right.\\ &\qquad \qquad \qquad \left. \left[ \mathbb {I}(\operatorname{Exp}_{q_e}(\tau v_{q_e}))^{-1} \circ T_{\operatorname{Exp}_{q_e}( \tau v_{q_e})}\mathbb{I} \left(T_{\tau v_{q_e}} \operatorname{Exp}_{q_e} \big(\varsigma_Q(q_e) \big) \right) \circ \mathbb {I}(\operatorname{Exp}_{q_e}(\tau v_{q_e}))^{-1} \right] \beta( \tau, \mu) \right\rangle \\ &\qquad =\mathbf{d}V \left(\operatorname{Exp}_{q_e}(\tau v_{q_e})\right) \left(T_{\tau v_{q_e}} \operatorname{Exp}_{q_e} \big(\varsigma_Q(q_e) \big) \right) \\ & \qquad \qquad \qquad -\frac{1}{2}\left\langle \zeta (\tau, v_{q_e}, \mu),\; T_{\operatorname{Exp}_{q_e}( \tau v_{q_e})} \mathbb{I} \left(T_{\tau v_{q_e}} \operatorname{Exp}_{q_e} \big(\varsigma_Q(q_e) \big) \right) \zeta (\tau , v_{q_e}, \mu ) \right\rangle ,\end{aligned}$$ where $\zeta (\tau ,v_{q_{e}},\mu) :=\mathbb{I}^{-1}((\operatorname{Exp}_{q_{e}}(\tau v_{q_{e}}))\beta (\tau ,\mu )$. Since $\zeta (\tau ,v_{q_{e}},\mu )$ is smooth in all variables also at $\tau =0$ by Remark \[remark on smoothness\], it follows that $\langle G_{1}(\tau ,[v_{q_{e}}],\mu ),\varsigma \rangle$ is a smooth function of all its variables. This expression at $\tau= 0 $ equals $$\begin{aligned} &\langle G_{1}(0,[v_{q_{e}}],\mu ),\varsigma \rangle =\mathbf{d}V(q_{e}) (\varsigma _{Q}(q_{e})) -\frac{1}{2}\left\langle \zeta (0, v_{q_e}, \mu),\; T_{q_e} \mathbb{I} \left( \varsigma_Q(q_e) \right) \zeta (0 , v_{q_e}, \mu ) \right\rangle \\ &\qquad =\mathbf{d}V(q_{e}) (\varsigma _{Q}(q_{e})) -\frac{1}{2}\left\langle(\mathbb{I}(q_{e})[\zeta (0,v_{q_{e}},\mu ),\varsigma ], \zeta (0,v_{q_{e}},\mu) \right\rangle - \frac{1}{2} \left\langle \mathbb {I}(q_e) \zeta(0,v_{q_{e}},\mu), [ \zeta(0,v_{q_{e}},\mu), \varsigma] \right\rangle\\ &\qquad =\mathbf{d}V(q_{e}) (\varsigma _{Q}(q_{e})) - \left\langle \mathbb {I}(q_e) \zeta(0,v_{q_{e}},\mu), [ \zeta(0,v_{q_{e}},\mu), \varsigma] \right\rangle\end{aligned}$$ by . Since $V$ is $G$-invariant it follows that $\mathbf{d}V(q_{e})(\varsigma _{Q}(q_{e}))=0$. Since $\zeta (0,v_{q_{e}},\mu ) = \xi(0,v_{q_{e}},\mu ) + \eta_\mu \in \mathfrak{g}_{q_e} \oplus \mathfrak{k}_1 = \mathfrak{t}$ (see Remark \[remark on smoothness\]) it follows that $[\zeta (0,v_{q_{e}},\mu ),\varsigma ]\in [ \mathfrak{t}, \mathfrak{g}]$. By Proposition \[montaldi\] (ii), we have $\mathbb{I}(q_{e})\mathfrak{t} \subset [ \mathfrak{g}, \mathfrak{t}]^{\circ }$ and hence the second term above also vanishes. Thus we get $\langle G_{1}(0,[v_{q_{e}}],\mu ),\varsigma \rangle =0$ for any $\varsigma \in \mathfrak{k}_2 $, that is, $G_{1}(0,[v_{q_{e}}],\mu ) = 0 $ which proves the theorem. Bifurcating branches of relative equilibria ------------------------------------------- Let $(Q,\langle\!\langle \cdot ,\cdot \rangle\!\rangle _{Q},V,G)$ be a simple mechanical $G$-system, with $G$ a compact Lie group with the Lie algebra $\mathfrak{g}$. Let $ q_{e}\in Q$ be a symmetric point whose isotropy group $G_{q_{e}}$ is contained in a maximal torus $\mathbb{T}$ of $G$. Denote by $\mathfrak{t} \subset \mathfrak{g}$ the Lie algebra of $\mathbb{T}$. Let $B\subset (\mathfrak{g}\cdot q_{e})^{\perp }$ be a $G_{q_{e}}$–invariant open neighborhood of $0_{q_{e}}\in (\mathfrak{g}\cdot q_{e})^{\perp }$ such that the exponential map is injective on $B $ and for any $ q \in G\cdot \operatorname{Exp}_{q_{e}}(B)$ the isotropy subgroup $G_{q} $ is conjugate to a (not necessarily proper) subgroup of $G_{q_{e}}$. Define the closed $G_{q_e}$–invariant subset $\mathcal{Z}_{\mu^{0}}=:\{v_{q_{e}}\in B\cap (T_{q_{e}}Q)_{\{e\}}\mid \det A=0\}$, where ${\mu^{0}}\in \mathfrak{m}_{1}\oplus \mathfrak{m}_{2}$ is arbitrarily chosen and the entries of the matrix $A $ are given in . Let $U \subset [B\cap (T_{q_{e}}Q)_{\{e\}} \setminus \mathcal{Z}_{\mu^{0}}]/G_{q_{e}} $ be open and consider the functions $F$ and $G $ given in Theorems \[theorem about F\] and \[theorem about G\]. Define $G^{i}:I\times U\times (\mathfrak{m}_{1}\oplus \mathfrak{m}_{2})\rightarrow \mathbb{R}$ by $$G^{i}(\tau,[v_{q_{e}}],\mu_{1}+\mu _{2}):=\langle G(\tau,[v_{q_{e}}],\mu_{1}+\mu _{2}),\varsigma_{i}\rangle,$$ where $\{\varsigma_{i} \mid i=1,... ,\operatorname{dim} \mathfrak{k_{2}}\}$ is a basis for ${\mathfrak k}_{2}$. Choose $([v_{q_{e}}],\mu _{1}+\mu _{2})\in U\times (\mathfrak{m}_{1}\oplus \mathfrak{m}_{2})$ such that $$\frac{\partial F}{\partial u}(0,[v_{q_{e}}],\mu _{1}+\mu _{2})=0,$$ where the partial derivative is taken relative to the variable $u \in U $. Define the matrix $$\Delta_{([v_{q_{e}}],\mu _{1},\mu _{2})} := \left[ \begin{array}{cc} \frac{\partial ^{2}F}{\partial u^{2}}(0,[v_{q_{e}}],\mu _{1}+\mu _{2}) & \frac{\partial^{2}F}{\partial \mu_{2}\partial u}(0,[v_{q_{e}}],\mu _{1}+\mu _{2})\\ \frac{\partial G^{i}}{\partial u}(0,[v_{q_{e}}],\mu _{1}+\mu _{2}) & \frac{\partial G^{i}}{\partial \mu_{2}}(0,[v_{q_{e}}],\mu _{1}+\mu _{2}) \end{array} \right],$$ where the partial derivatives are evaluated at $\tau =0, [v_{q_{e}}],\mu = \mu_1 + \mu_2$. Here $ \frac{\partial} {\partial \mu _{2}}$ denotes the partial derivative with respect to the $\mathfrak{m}_{2}$-component $\mu_2 $ of $\mu $. In the framework and the notations introduced above we will state and prove the main result of this paper. Let $\pi :TQ\rightarrow (TQ)/G$ be the canonical projection and $\mathcal{R} _{e}:=\pi (\mathfrak{t}\cdot q_{e})$. \[principala\] Assume the following: $${\bf (H)}\; \text{every } v_{q_{e}}\in \mathfrak{t}\cdot q_{e} \text{ is a relative equilibrium.}$$ If there is a point $([v_{q_{e}}^{0}],\mu _{1}^{0}+\mu _{2}^{0})\in U\times (\mathfrak{m}_{1}\oplus \mathfrak{m}_{2})$ such that $$\begin{aligned} && \begin{array}{cc} 1) & \frac{\partial F}{\partial u}(0,[v_{q_{e}}^{0}],\mu _{1}^{0}+\mu _{2}^{0})=0, \end{array} \\ && \begin{array}{cc} 2) & G^{i}(0,[v_{q_{e}}^{0}],\mu _{1}^{0}+\mu _{2}^{0})=0 \end{array} \\ && \begin{array}{cccc} 3) & \Delta _{([v_{q_{e}}^{0}],\mu _{1}^{0},\mu _{2}^{0})} & is & non degenerate, \end{array}\end{aligned}$$ then there exists a family of continuous curves $\gamma_{_{({[v_{q_{e}}^{0}]},\mu _{1}^{0},\mu _{2}^{0})}}^{\mu_{1}}:[0,1]\rightarrow (TQ)/G$ parameterized by $\mu_{1}$ in a small neighborhood $\mathcal{V}_{0}$ of $\mu_{1}^{0}$ consisting of classes of relative equilibria with trivial isotropy on $\gamma _{_{({[v_{q_{e}}^{0}]},\mu _{1}^{0},\mu _{2}^{0})}}^{\mu_{1}}(0,1)$ satisfying $$\operatorname{Im}\gamma _{_{({[v_{q_{e}}^{0}]},\mu _{1}^{0}, \mu _{2}^{0})}}^{\mu_{1}}\bigcap \mathcal{R} _{e}=\left\{\gamma _{_{({[v_{q_{e}}^{0}]}, \mu _{1}^{0}, \mu _{2}^{0})}}^{\mu_{1}}(0)\right\}$$ and $\gamma _{_{({[v_{q_{e}}^{0}]}, \mu _{1}^{0}, \mu _{2}^{0})}}^{\mu_{1}}(0) = [ \zeta_Q(q_e)]$, where $\zeta = \widehat{\mathbb {I}}(q_e)^{-1} \mu_1 \in \mathfrak{t}$. For $\mu_{1},\mu_{1}^{\prime}\in \mathcal{V}_{0}$ with $\mu_{1}\neq\mu_{1}^{\prime}$, where $\mathcal{V}_{0}$ is as above, the above branches do not intersect, that is, $$\left\{ \gamma _{_{({[v_{q_{e}}^{0}]}, \mu _{1}^{0}, \mu _{2}^{0})}}^{\mu_{1}}(\tau) \,\Big|\, {\tau\in [0,1]} \right\} \bigcap \left\{\gamma _{_{({[v_{q_{e}}^{0}]}, \mu _{1}^{0}, \mu _{2}^{0})}}^{\mu_{1}^{\prime}}(\tau) \,\Big|\, {\tau\in [0,1]}\right\} = \varnothing.$$ Suppose that $ ([v_{q_{e}}^{0}],\mu _{1}^{0},\mu _{2}^{0}) \neq ([v_{q_{e}}^{1}],\mu _{1}^{1},\mu _{2}^{1})$. (i) If $\mu_{1}^{0}\neq \mu_{1}^{1}$ then the families of relative equilibria do not intersect, that is, $$\left\{\gamma_{_{({[v_{q_{e}}^{0}]}, \mu _{1}^{0}, \mu_{2}^{0})}} ^{\mu_{1}}(\tau) \,\Big|\, {(\tau,\mu_{1})\in [0,1]\times \mathcal{V}_{0}} \right\} \bigcap \left\{\gamma _{_{({[v_{q_{e}}^{1}]}, \mu_{1}^{1}, \mu_{2}^{1})}}^{\mu_{1}^{\prime}}(\tau) \,\Big|\, {(\tau,\mu_{1}^{\prime})\in [0,1]\times \mathcal{V}_{1}}\right\} =\varnothing,$$ where $\mathcal{V}_{0}$ and $\mathcal{V}_{1}$ are two small neighborhoods of $\mu_{1}^{0}$ and $\mu_{1}^{1}$ respectively such that $\mathcal{V}_{0}\cap \mathcal{V}_{1}=\varnothing$. (ii) If $\mu_{1}^{0}=\mu_{1}^{1}=\overline \mu$ and $[v_{q_{e}}^{0}]\neq[v_{q_{e}}^{1}]$ then $\gamma _{_{({[v_{q_{e}}^{0}]},\overline\mu , \mu _{2}^{0})}}^{\overline\mu}(0)=\gamma _{_{({[v_{q_{e}}^{1}]},\overline\mu , \mu _{2}^{1})}}^{\overline\mu}(0)$ and for $\tau>0$ we have $$\left\{\gamma _{_{({[v_{q_{e}}^{0}]},\overline\mu , \mu _{2}^{0})}}^{\overline\mu}(\tau) \,\Big|\, {\tau\in (0,1]}\right\} \bigcap \left\{\gamma _{_{({[v_{q_{e}}^{1}]},\overline\mu , \mu _{2}^{1})}}^{\overline\mu}(\tau) \,\Big|\, {\tau\in (0,1]}\right\} =\varnothing.$$ Let $({[v_{q_{e}}^{0}]},\mu _{1}^{0}+\mu _{2}^{0})\in U\times (\mathfrak{m}_{1}\oplus \mathfrak{m} _{2}) $ be such that the conditions 1-3 hold. Because $\Delta _{({[v_{q_{e}}^{0}]},\mu _{1}^{0}+\mu _{2}^{0})}$ is nondegenerate, we can apply the implicit function theorem for the system $(\frac {\partial F}{\partial u},G^{i})(\tau,[v_{q_{e}}],\mu _{1}+\mu _{2})=0$ around the point $(0,{[v_{q_{e}}^{0}]},\mu _{1}^{0}+\mu _{2}^{0})$ and so we can find an open neighborhood $J\times \mathcal{V}_{0}$ of the point $(0,\mu_{1}^{0})$ in $I\times \mathfrak {m}_{1}$ and two functions $u:J\times \mathcal{V}_{0}\rightarrow U$ and $\mu _{2}:J\times \mathcal{V}_{0}\rightarrow \mathfrak{m}_{2}$ such that $u(0,\mu _{1}^{0})={[v_{q_{e}}^{0}]}$, $\mu _{2}(0,\mu _{1}^{0})=\mu _{2}^{0}$ and $$\begin{aligned} && \begin{array}{cc} i) & \frac{\partial F}{\partial u}(\tau ,u(\tau,\mu _{1} ),\mu _{1}+\mu _{2}(\tau,\mu _{1}))=0 \end{array} \\ && \begin{array}{cc} ii) & G^{i}(\tau ,u(\tau,\mu _{1} ),\mu _{1}+\mu _{2}(\tau,\mu _{1} ))=0. \end{array}\end{aligned}$$ Therefore, from Theorems \[theorem about F\] and \[theorem about G\] it follows that the relative equilibrium conditions of Corollary \[criteriu\] are both satisfied. Thus we obtain the following family of branches of relative equilibria $[(\overline{\sigma}(\tau\cdot u(\tau,\mu_{1})),\beta(\tau,\mu_{1}+\mu_{2}(\tau,\mu_{1})))]_{G}$ parameterized by $\mu_{1} \in \mathcal{V}_{0}$. For $\tau>0$ the isotropy subgroup is trivial and for $\tau=0$ the corresponding points on the branches are $[(\overline{\sigma}([0_{q_{e}}]),\mu_{1}]_{G}=[q_{e},\mu_{1}]_{G}$ which have the isotropy subgroup equal to $G_{q_{e}}$. This shows that there are points in $\mathcal{R}_e $ from which there are emerging branches of relative equilibria with broken trivial symmetry. Using now the correspondence given by Proposition \[map f\] and a rescaling of $\tau$ we obtain the desired family of continuous curves $\gamma _{_{({[v_{q_{e}}^{0}]},\mu _{1}^{0},\mu _{2}^{0})}}^{\mu_{1}}:[0,1]\rightarrow (TQ)/G$ parameterized by $\mu_{1}$ in a small neighborhood $\mathcal{V}_{0}$ of $\mu_{1}^{0}$ consisting of classes of relative equilibria with trivial isotropy on $\gamma _{_{({[v_{q_{e}}^{0}]},\mu _{1}^{0},\mu _{2}^{0})}}^{\mu_{1}}(0,1)$ and such that $$\operatorname{Im}\gamma _{_{({[v_{q_{e}}^{0}]},\mu _{1}^{0},\mu _{2}^{0})}}^{\mu_{1}}\bigcap \mathcal{R} _{e}=\{\gamma_{ _{({[v_{q_{e}}^{0}]},\mu _{1}^{0},\mu _{2}^{0})}}^ {\mu_1}(0)\}$$ and $\gamma_{_{({[v_{q_{e}}^{0}]},\mu _{1}^{0},\mu _{2}^{0})}} ^{\mu_1}(0) = [ \zeta_Q(q_e)]$, where $\zeta = \widehat{\mathbb {I}}(q_e)^{-1} \mu_1$. Equivalently, using the identification given by (\[corespondenta\]) and by Proposition \[map f\] we obtain that the branches of relative equilibria $\gamma _{_{({[v_{q_{e}}^{0}]},\mu _{1}^{0},\mu _{2}^{0})}}^{\mu_{1}}(\tau)\in (TQ)/G$ are identified with $[\sigma(\tau\cdot u(\tau,\mu_{1})),\beta(\tau,\mu_{1}+\mu_{2}(\tau,\mu_{1}))] _{G_{q_{e}}}$. It is easy to see that for $\mu_{1}\neq\mu_{1}^{\prime}$ we have that $\beta(\tau,\mu_{1}+\mu_{2}(\tau,\mu_{1}))\neq\beta(\tau^{\prime}, \mu_{1}^{\prime}+\mu_{2}(\tau,\mu_{1}^{\prime}))$ for every $\tau,\tau^{\prime}\in [0,1]$. Using now the fact that $G_{q_{e}}$ acts trivially on $\mathfrak{m}_{1}$ we obtain $$\left\{ \gamma _{_{({[v_{q_{e}}^{0}]},\mu _{1}^{0},\mu _{2}^{0})}}^{\mu_{1}}(\tau) \,\Big| \, {\tau\in [0,1]} \right\} \bigcap \left\{\gamma _{_{({[v_{q_{e}}^{0}]},\mu _{1}^{0},\mu _{2}^{0})}}^{\mu_{1}^{\prime}}(\tau) \,\Big| \, {\tau\in [0,1]}\right\} = \varnothing.$$ In an analogous way, using the same argument we can prove $(i)$. For $(ii)$ we start with two branches of relative equilibria, $b_{1}(\tau,\overline\mu):=[\sigma(\tau\cdot u(\tau,\overline\mu)),\beta(\tau,\overline\mu+\mu_{2} (\tau,\overline\mu))]_{G_{q_{e}}}$ and $b_{2}(\tau^{\prime},\overline\mu):=[\sigma(\tau^{\prime}\cdot u^{\prime}(\tau^{\prime},\overline\mu)),\beta(\tau^{\prime}, \overline\mu+\mu_{2}(\tau,\overline\mu))]_{G_{q_{e}}}$. For $\tau=\tau^{\prime}=0$ we have $b_{1}(0,\overline\mu)=[0,\overline\mu]_{G_{q_{e}}} =b_{2}(0,\overline\mu)$. We also have $u(0,\overline\mu)=[v_{q_{e}}^{0}]\neq [v_{q_{e}}^{1}]=u^{\prime}(0,\overline\mu)$ and so, from the implicit function theorem, we obtain $u(\tau,\overline\mu)\neq u^{\prime}(\tau^{\prime}, \overline\mu)$ for $\tau,\tau^{\prime}>0$ small enough. Suppose that there exist $\tau,\tau^{\prime}>0$ such that $b_{1}(\tau,\overline\mu)=b_{2}(\tau^{\prime},\overline\mu)$. Then using the triviality of the ${G_{q_{e}}}$-action on $\mathfrak{m}_{0}$ we obtain that $\tau^{2}\nu_{0}={\tau^{\prime}}^{2}\nu_{0}$ and consequently $\tau=\tau^{\prime}$. The conclusion of $(ii)$ follows now by rescaling. We can have two particular forms for the rescaling $\beta $ according to special choices of the groups $G$ and $G_{q_{e}}$, respectively. (a) If $G$ is a torus, then from the splitting $\mathfrak{g}= \mathfrak{k}_{0}\oplus \mathfrak{k}_{1}\oplus \mathfrak{k}_{2}$, where $ \mathfrak{k}_{0}=\mathfrak{g}_{q_{e}}$, $\mathfrak{k}_{0}\oplus \mathfrak{k} _{1}=\mathfrak{t}$, and $\mathfrak{k}_{2}=[\mathfrak{g},\mathfrak{t}]$, we conclude that $\mathfrak{k}_{2}=\{0\}$ (since $\mathfrak{g}= \mathfrak{t}$) and consequently $\mathfrak{m} _{2}=\{0\}$. In this case we will obtain the special form for the rescaling $ \beta :I\times \mathfrak{m}_{1}\rightarrow \mathfrak{g}^{\ast }$, $\beta (\tau ,\mu )=\mu +\tau ^{2}\nu _{0}$. (b) If is $G_{q_{e}}$ a maximal torus in $G$, so $\mathfrak{g}_{q_e} = \mathfrak{t}$, then the same splitting implies that $\mathfrak{k}_{1}=\{0\}$ and consequently $\mathfrak{m}_{1}=\{0\}$. In this case we will obtain the special form for the rescaling $\beta :I\times \mathfrak{m} _{2}\rightarrow \mathfrak{g}^{\ast }$, $\beta (\tau ,\mu )=\tau \mu +\tau ^{2}\nu _{0}$. Stability of the bifurcating branches of relative equilibria {#stability section} ============================================================ In this section we shall study the stability of the branches of relative equilibria found in the previous section. We will do this by applying a result of Patrick [@patrick; @thesis] on $G_{\mu}$-stability to our situation. First we shortly review this result. Let $z_{e}$ be a relative equilibrium with velocity $\xi_{e}$ and $J(z_{e})=\mu_{e}$. We say that $z_{e}$ is [[******]{}formally stable]{} if $\mathbf{d}^{2}(H-J^{\xi_{e}})(z_{e})|_ {T_{z_{e}}J^{-1}(\mu_{e})}$ is a positive or negative definite quadratic form on some (and hence any) complement to ${\mathfrak g}_{\mu_{e}}\cdot z_{e}$ in $T_{z_{e}}J^{-1}(\mu_{e})$. We have the following criteria for formal stability. \[patrick\] Let $z_{e}\in T^{\ast }Q$ be a relative equilibrium with momentum value $\mu _{e} \in \mathfrak{g}^\ast$ and base point $q_e \in Q$. Assume that $\mathfrak{g}_{q_{e}}=\{0\}$. Then $z_{e}$ is formally stable if and only if $\mathbf{d}^{2}V_{\mu }(q_{e})$ is positive definite on one (and hence any) complement $\mathfrak{g}_{\mu }\cdot q_{e}$ in $T_{q_{e}}Q$. To apply this theorem to our case in order to obtain the formal stability of the relative equilibria on a bifurcating branch we proceed as follows. First notice that if we fix $\mu \in \mathfrak{m}_{1}\oplus \mathfrak{ m}_{2}$ and $[v_{q_{e}}]\in U$ as in Theorem \[principala\], we obtain locally a branch of relative equilibria with trivial isotropy bifurcating from our initial set. More precisely, this branch starts at the point $$\left(\widehat{ \mathbb{I}}(q_{e})^{-1}\Pi_{1}\mu \right)_Q(q_e).$$ The momentum values along this branch are $\beta (\tau ,\mu )$, and for $\tau \neq 0$ the velocities have the expression $\mathbb{I} (\operatorname{Exp} _{q_{e}}(\sigma(\tau u(\tau,\mu_{1}))^{-1}\beta (\tau ,\mu )$. The base points of this branch are $\operatorname{Exp}_{q_{e}}(\sigma(\tau u(\tau,\mu_{1}))$. Recall from Corollary \[criteriu\] that we introduced the notation $\overline{\sigma} : = \operatorname{Exp}{q_e} \circ \sigma $ that will be used below. By the definition of $\beta (\tau ,\mu )$ we have $\mathfrak{g}_{\beta (\tau ,\mu )}=\mathfrak{t}$ for all $\tau $, even for $\tau =0$. The base points for the entire branch have no symmetry for $\tau>0 $ so we can characterize the formal stability (in our case the $\mathbb{T}$ -stability) of the whole branch (locally) in terms of Theorem \[patrick\]. We begin by giving sufficient conditions that guarantee the $\mathbb{T}$-stability of the branch, since $G_{\beta(\tau, \mu) } = \mathbb{T} $. To do this, one needs to find conditions that insure that for $\tau \neq 0 $ (where the amended potential exists) $$\mathbf{d}^2 V_{\beta(\tau, \mu)}( \overline{\sigma}(\tau u(\tau,\mu_{1}))|_{T_{[\tau u(\tau,\mu_{1})]} \overline{\sigma}(T_{[\tau u(\tau,\mu_{1})]}U) \oplus (T_{\sigma ([\tau u(\tau,\mu_{1})])}\operatorname{Exp}_{q_{e}}) (\mathfrak{k} _{2}\cdot q_{e})}$$ is positive definite. We do not know how to control the cross terms of this quadratic form. This is why we shall work only with Abelian groups $G $ since in that case the subspace $\mathfrak{k}_2 = \{0\}$ and the second summand thus vanishes. So, let $G $ be a torus $\mathbb{T}$. By Proposition \[decomposition of the tangent space adapted to sigma bar\] and Theorem \[theorem about F\], the second variation $$\mathbf{d}^2 V_{\beta(\tau, \mu)}( \overline{\sigma}(\tau u(\tau,\mu_{1}))|_{T_{[\tau u(\tau,\mu_{1})]} \overline{\sigma}(T_{[\tau u(\tau,\mu_{1})]}U)}$$ coincides for $\tau\neq 0 $, with the second variation $$\label{hessian of f one} \mathbf{d}_{U}^2 F_1(\tau, u(\tau,\mu_{1}), \mu_1 + \mu_2( \tau, \mu_1))|_{T_{[\tau u(\tau,\mu_{1})]}U}$$ of the auxiliary function $F_1 $, where $\mathbf{d}^2_{U}$ denotes the second variation relative to the second variable in $F_1$. But, unlike $V_{\beta(\tau,\mu)}$, the function $F_1$ is is defined even at $\tau = 0$. Recall from Theorem \[theorem about F\] that on the bifurcating branch the amended potential has the expression $$F_{1}(\tau ,u(\tau,\mu_{1}),\mu_1 + \mu_2( \tau, \mu_1) )=F_{0}(\mu_1 + \mu_2( \tau, \mu_1))+\tau ^{2}F(\tau, u(\tau,\mu_{1}),\mu_1 + \mu_2(\tau, \mu_1) ),$$ where $F_0$ is smooth on $\mathbf{J}_L (\mathfrak{g}\cdot q_e) = \mathbb {I}(q_e)\mathfrak{g}$ and $F, F_1 $ are both smooth functions on $I\times U\times \mathbf{J} _{L}(\mathfrak{g}\cdot q_{e})$, even around $\tau =0$. So, if the second variation of $F $ at $(0, [v_{q_e}^0], \mu_1 ^0 + \mu_2 ^0)$ is positive definite, then the quadratic form will remain positive definite along the branch for $\tau > 0 $ small. So we get the following result. Let $\mu_1 ^0 + \mu_2 ^0 \in \mathfrak{m}_{1}\oplus \mathfrak{m}_{2}$ and $[v_{q_{e}}^0]\in U$ be as in the Theorem \[principala\] and assume that $\mathbf{d}^2_U F(0,[v_{q_{e}}^0],\mu_1^0 + \mu_2 ^0 )$ is positive definite. Then the branch of relative equilibria with no symmetry which bifurcate form $\left(\widehat{ \mathbb{I}}(q_{e})^{-1}\mu_1^0 \right)_Q(q_e) $ will be $\mathbb{T}$-stable for $\tau>0$ small. A direct application of this criterion to the double spherical pendulum recovers the stability result on the bifurcating branches proved directly in [@ms]. **Acknowledgments.** We would like to thank J. Montaldi for telling us the result of Proposition \[montaldi\] and for many discussions that have influenced our presentation and clarified various points during the writing of this paper. Conversations with A. Hernández and J. Marsden are also gratefully acknowledged. The third and fourth authors were partially supported by the European Commission and the Swiss Federal Government through funding for the Research Training Network *Mechanics and Symmetry in Europe* (MASIE). The first and third author thank the Swiss National Science Foundation for partial support. [99]{} [P. Birtea and M. Puta]{},\ Departamentul de Matematică, Universitatea de Vest, RO–1900 Timişoara, Romania.\ Email: [[email protected], [email protected]]{}\ [T.S. Ratiu and Răzvan Tudoran]{},\ Section de mathématiques, [É]{}cole Polytechnique F[é]{}d[é]{}rale de Lausanne. CH–1015 Lausanne. Switzerland.\ Email: [[email protected], [email protected]]{}

--- abstract: | Jets and outflows are ubiquitous in the process of formation of stars since outflow is intimately associated with accretion. Free-free (thermal) radio continuum emission in the centimeter domain is associated with these jets. The emission is relatively weak and compact, and sensitive radio interferometers of high angular resolution are required to detect and study it. One of the key problems in the study of outflows is to determine how they are accelerated and collimated. Observations in the cm range are most useful to trace the base of the ionized jets, close to the young central object and the inner parts of its accretion disk, where optical or near-IR imaging is made difficult by the high extinction present. Radio recombination lines in jets (in combination with proper motions) should provide their 3D kinematics at very small scale (near their origin). Future instruments such as the Square Kilometre Array (SKA) and the Next Generation Very Large Array (ngVLA) will be crucial to perform this kind of sensitive observations. Thermal jets are associated with both high and low mass protostars and possibly even with objects in the substellar domain. The ionizing mechanism of these radio jets appears to be related to shocks in the associated outflows, as suggested by the observed correlation between the centimeter luminosity and the outflow momentum rate. From this correlation and that of the centimeter luminosity with the bolometric luminosity of the system it will be possible to discriminate between unresolved HII regions and jets, and to infer additional physical properties of the embedded objects. Some jets associated with young stellar objects (YSOs) show indications of non-thermal emission (negative spectral indices) in part of their lobes. Linearly polarized synchrotron emission has been found in the jet of HH 80-81, allowing one to measure the direction and intensity of the jet magnetic field, a key ingredient to determine the collimation and ejection mechanisms. As only a fraction of the emission is polarized, very sensitive observations such as those that will be feasible with the interferometers previously mentioned are required to perform studies in a large sample of sources. Jets are present in many kinds of astrophysical scenarios. Characterizing radio jets in YSOs, where thermal emission allows one to determine their physical conditions in a reliable way, would also be useful in understanding acceleration and collimation mechanisms in all kinds of astrophysical jets, such as those associated with stellar and supermassive black holes and planetary nebulae. author: - Guillem Anglada - 'Luis F. Rodríguez' - 'Carlos Carrasco-González' date: | Received: 19 September 2017 / Accepted: 15 March 2018\ \ DOI: 10.1007/s00159-018-0107-z title: Radio Jets from Young Stellar Objects --- Introduction {#intro} ============ Until around 1980, the process of star formation was believed to be dominated by infall motions from an ambient cloud that made the forming star at its center grow in mass. Several papers published at that time indicated that powerful bipolar ejections of high-velocity molecular (Snell et al. 1980; Rodríguez et al. 1980) and ionized (Herbig & Jones 1981) gas were also present. The star formation paradigm changed from one of pure infall to one in which infall and outflow coexisted. As a matter of fact, both processes have a symbiotic relation: the rotating disk by which the star accretes provides the energy for the outflow, while this latter process removes the excess of angular momentum that otherwise will impede further accretion. Early studies of centimeter radio continuum from visible T Tau stars made mostly with the Very Large Array (VLA) showed that emission was sometimes detected in them (e.g., Cohen et al. 1982). Later studies made evident that in these more evolved stars the emission could have a thermal (free-free) origin but that most frequently the emission was dominated by a nonthermal (gyrosynchrotron) process (Feigelson & Montmerle 1985) produced in the active magnetospheres of the stars. These non-thermal radio stars are frequently time-variable (e.g., Rivilla et al. 2015) and its compact radio size makes them ideal for the determination of accurate parallaxes using Very Long Baseline Interferometry (VLBI) observations (e.g., Kounkel et al. 2017). In contrast, the youngest low-mass stars, the so-called Class 0 and I objects (Lada 1991; André et al. 1993) frequently exhibit free-free emission at weak but detectable levels (Anglada et al. 1992). In the best studied cases, the sources are resolved angularly at the sub-arcsec scale and found to be elongated in the direction of the large-scale tracers of the outflow (e.g., Rodríguez et al. 1990; Anglada 1996), indicating that they trace the region, very close to the exciting star, where the outflow phenomenon originates. The spectral index at centimeter wavelengths $\alpha$ (defined as $S_\nu \propto \nu^\alpha$), usually rises slowly with frequency and can be understood with the free-free jet models of Reynolds (1986). Given its morphology and spectrum, these radio sources are sometimes referred to as “thermal radio jets”. It should be noted that there are a few Class I objects where the emission seems to be dominantly gyrosynchrotron (Feigelson et al. 1998; Dzib et al. 2013). These cases might be due to a favorable geometry (that is, the protostar is seen nearly pole-on or nearly edge-on, where the free-free opacity might be reduced; Forbrich et al. 2007), or to clearing of circumstellar material by tidal forces in a tight binary system (Dzib et al. 2010) In the last years it has become clear that the radio jets are present in young stars across the stellar spectrum, from O-type protostars (Garay et al. 2003) and possibly to brown dwarfs (Palau et al. 2014), suggesting that the disk-jet scenario that explains the formation of solar-type stars extends to all stars and even into the sub-stellar regime. The observed centimeter radio luminosities (taken to be $S_\nu d^2$, with $d$ being the distance) go from $\sim$100 mJy kpc$^2$ for massive young stars to $\sim$$3 \times 10^{-3}$ mJy kpc$^2$ for young brown dwarfs. High sensitivity studies of Herbig-Haro systems revealed that a large fraction of them showed the presence of central centimeter continuum sources (Rodríguez & Reipurth 1998). With the extraordinary sensitivity of the Jansky VLA (JVLA) and planed future instrumentation it is expected that all nearby (a few kpc) young stellar objects (YSOs) known to be associated with outflows, both molecular and/or optical/infrared, will be detectable as centimeter sources (Anglada et al. 2015). The topic of radio jets from young stars has been reviewed by Anglada (1996), Anglada et al. (2015) and Rodríguez (1997, 2011). The more general theme of multifrequency jets and outflows from young stars has been extensively reviewed in the literature, with the most recent contributions being Frank et al. (2014) and Bally (2016). In this review we concentrate on radio results obtained over the last two decades, emphasizing possible lines of study feasible with the improved capabilities of current and future radio interferometers. Information from radio jets =========================== The study of radio jets associated with young stars has several astronomical uses. Given the large obscuration present towards the very young stars, the detection of the radio jet provides so far the best way to obtain their accurate positions. These observations also provide information on the direction and collimation of the gas ejected by the young system in the last few years and allow a comparison with the gas in the molecular outflows and optical/infrared HH jets, that trace the ejection over timescales one to two orders of magnitude larger. This comparison allows to make evident changes in the ejection direction possibly resulting from precession or orbital motions in binary systems (Rodríguez et al. 2010; Masqué et al. 2012). In some sources it has been possible to establish the proper motions of the core of the jet and to confirm its association with the region studied (Loinard et al. 2002; Rodríguez et al. 2003a; Lim & Takakuwa 2006; Carrasco-González et al. 2008a; Loinard et al. 2010; Rodríguez et al. 2012a, b). In the case of the binary jet systems L1551 IRS5 and YLW 15, the determination of its orbital motions (Lim & Takakuwa 2006; Curiel et al. 2004) allows one to confirm that they are solar-mass systems and that they are very overluminous with respect to the corresponding main sequence luminosity, as expected for objects that are accreting strongly. The free-free radio emission at cm wavelengths has been used for a long time to estimate important physical properties of YSO jets (see below). Recently, the observation of the radio emission from jets has reached much lower frequencies, as in the recent Low Frequency ARray (LOFAR) observations at 149 GHz (2 m) of T Tau (Coughlan et al. 2017). These observations have revealed the low-frequency turnover of the free-free spectrum, a result that has allowed the degeneracy between emission measure and electron density to be broken. Observed properties of radio jets ================================= Properties of currently known angularly resolved radio jets ----------------------------------------------------------- [l@c@c@c@c@c@c@c@c@c@ c@c@c@l]{} & [$L_{\rm bol}$]{} & [$M_\star$]{} & [$d$]{} & [$S_\nu$]{} & & [$\theta_0$]{} & [Size]{} & [$v_j$]{} & [$t_{\rm dyn}$]{} & & [$\dot M_{\rm ion}$]{} & [$r_0$]{} &\ & [($L_\odot$)]{} & [($M_\odot$)]{} & [(kpc)]{} & [(mJy)]{} & [$\alpha$]{} & [(deg)]{} & [(au)]{} & [(kms$^{-1}$)]{} & [(yr)]{} & [$\epsilon$]{} & [($M_\odot$yr$^{-1}$)]{} & [(au)]{} & [Refs.]{}\ HH 1-2 VLA1 & 20 & $\sim$1 & 0.4 & 1 & 0.3 & 19& 200& 270 & 2 & 0.7 & 1$\times$$10^{-8}$ & $\leq$11 & 1, 2, 3, 4\ NGC 2071-IRS3 & $\sim$500 & 4 & 0.4 & 3 & 0.6 &40& 200& 400$^a$ & 1 & 1.0 & 2$\times$$10^{-7}$ & $\leq$18 & 5, 6, 2, 7\ Cep A HW2 & 1$\times$$10^4$ & 15 & 0.7 & 10 & 0.7 &14& 400& 460 & 2 & 0.9 & 5$\times$$10^{-7}$ & $\leq$60 & 8, 9, 10, 11, 12\ HH 80-81 & 2$\times$$10^4$ & 15 & 1.7 & 5 & 0.2 &34& 1500& 1000 & 4 & 0.6 & 1$\times$$10^{-6}$ & $\leq$25 & 13,14,15,16,17,18\ IRAS16547-4247 & 6$\times$$10^4$ & 20 & 2.9 & 11 & 0.5 & 25 & 3000 & 900$^a$ & 8 & 0.9 & 8$\times$$10^{-6}$ & $\leq$310 & 19, 20, 21, 22\ Serpens & 300 & 3 & 0.42 & 2.8 & 0.2 & $<$34 & 280 & 300 & 2 & 0.6 & 3$\times$$10^{-8}$ & $\leq$9 & 23, 24, 25, 26,27\ AB Aur & 38 & 2.4 & 0.14 & 0.14 & 1.1 & $<$39 & 24 & 300$^a$ & 0.2 & 3.5 & 2$\times$$10^{-8}$ & $\leq$3 & 28,29,30\ L1551 IRS5$^b$ & 20 & 0.6 & 0.14 & 0.8 & 0.1 & 36 & 39 & 150$^a$ & 0.6 & 0.6 & 1$\times$$10^{-9}$ & $\leq$1 & 31,32,33\ HH 111$^c$ & 25 & 1.3 & 0.4 & 0.8 & $\sim$1 & $<$79 & 110 & 400 & 0.7 & 2.3 & 2$\times$$10^{-7}$ & $\leq$12 & 34,35,36,37,38\ HL Tau & 7 & 1.3 & 0.14 & 0.3 & $\sim$0.3 & 69 & 27 & 230$^a$ & 0.3 & 0.7 & 2$\times$$10^{-9}$ & $\sim$1.5 & 39,40,41\ IC 348-SMM2E & 0.1 & 0.03 & 0.24 & 0.02 & $\sim$0.4 & 45$^d$ & $<$100 & $\sim$50$^a$ & $<$2 & 0.8 & 2$\times$$10^{-10}$ & $\leq$1 & 42,43,44\ W75N VLA3 & 750 & 6$^d$ & 2.0 & 4.0 & 0.6 & 37 & 420 & 220 & 4.6 & 1.0 & 6$\times$$10^{-7}$ & $\leq$70 & 45,46\ OMC2 VLA11 & 360 & 4$^d$ & 0.42 & 2.2 & 0.3 & 10 & 200 & 100 & 4.6 & 1.0 & 6$\times$$10^{-7}$ & $\leq$70 & 2,47,48\ Re50N & 250 & 4$^d$ & 0.42 & 1.1 & 0.7 & 33 & 450 & 400 & 2.7 & 1.2 & 8$\times$$10^{-8}$ & $\leq$13 & 2,49,50\ \ $^a$Calculated using eq. (\[eq:vj\]).\ $^{b}$[Binary twin jet system. The values listed are the mean value of the two jets.]{}\ $^{c}$[Binary jet system. The values listed are for the dominant east-west jet.]{}\ $^{d}$[Assumed.]{}\ References: [(1) Fischer et al. 2010; (2) Menten et al. 2007; (3) Rodríguez et al. 1990; (4) Rodríguez et al. 2000; (5) Butner et al. 1990; (6) Carrasco González et al. 2012a; (7) Torrelles et al. 1998; (8) Hughes et al. 1995; (9) Patel et al. 2005; (10) Dzib et al. 2011; (11) Curiel et al. 2006; (12) Rodríguez et al. 1994b; (13) Aspin & Geballe 1992; (14) Fernández-López et al. 2011; (15) Rodríguez et al. 1980b; (16) Martí et al. 1995; (17) Martí et al. 1993; (18) Carrasco-González et al. 2012b; (19) Zapata et al. 2015b; (20) Garay et al. 2003; (21) Rodríguez et al. 2008; (22) Rodríguez et al. 2005; (23) Dzib et al. 2010; (24) Rodríguez et al. 1989a; (25) Harvey et al. 1984; (26) Curiel et al. 1993; (27) Rodríguez-Kamenetsky et al. 2016; (28) DeWarf et al. 2003; (29) van den Ancker et al. 1997; (30) Rodríguez et al. 2014a; (31) Liseau et al. 2005; (32) Rodríguez et al. 2003b; (33) Rodrí guez et al. 1998; (34) Reipurth 1989; (35) Lee 2010; (36) Gómez et al. 2013; (37) Rodríguez & Reipurth 1994; (38) Cernicharo & Reipurth 1996; (39) Cohen 1983; (40) ALMA Partnership et al. 2015; (41) A. M. Lumbreras et al., in prep; (42) Palau et al. 2014; (43) Forbrich et al. 2015; (44) Rodríguez et al. 2014b; (45) Persi et al. 2006; (46) Carrasco-González et al. 2010a; (47) Adams et al. 2012; (48) Osorio et al. 2017; (49) Chiang et al. 2015; (50) L. F. Rodríguez et al., in prep.]{} ![Images of selected radio jets. From left to right, and top to bottom: HH 1-2 (Rodríguez et al. 2000); L1551-IRS5 (Rodríguez et al. 2003b); HH 111 (Gómez et al. 2013); DG Tau (Rodríguez et al. 2012b); NGC2071-IRS3 (Carrasco-González et al. 2012a); W75N (Carrasco-González et al. 2010a); IRAS 16547$-$4247 (Rodríguez et al. 2005); IRAS 20126+4104 (Hofner et al. 2007); Cep A HW2 (Rodrí guez et al. 1994b). Jets from low- and intermediate-mass protostars are shown in the top and middle panels, while those from high-mass protostars are shown in the bottom panels. []{data-label="fig:1"}](fig1.pdf){width="100.00000%"} In Table \[tab:1\] we present the parameters of selected radio jets that have been angularly resolved. Several trends can be outlined. The spectral index, $\alpha$, is moderately positive, going from values of 0.1 to $\sim$1, with a median of 0.45. The opening angle of the radio jet near its origin, $\theta_0$, is typically in the few tens of degrees. In contrast, if we consider the HH objects or knots located relatively far from the star, an opening angle an order of magnitude smaller is derived. This result has been taken to suggest that there is recollimation at scales of tens to hundreds of au. The jet velocity is typically in the 100 to 1,000 km s$^{-1}$ range. The ionized mass loss rate, $\dot M_{\rm ion}$, is found to be typically an order of magnitude smaller than that determined from the large scale molecular outflow, and as derived from atomic emission lines, a result that has been taken to imply that the radio jet is only partially ionized ($\sim$ 1-10%; Rodríguez et al. 1990; Hartigan et al. 1994; Bacciotti et al. 1995) and that the mass loss rates determined from them are only lower limits. In Figure \[fig:1\] we show images of several selected radio jets. Proper motions in the jet ------------------------- As noted above, comparing observations taken with years of separation it has been possible in a few cases to determine the proper motions of the core of the jet, whose centroid is believed to coincide within a few au’s with the young star (e.g., Curiel et al. 2003; Chandler et al. 2005; Rodr[í]{}guez et al. 2012a, b). These proper motions are consistent with those of other stars in the region. It is also possible to follow the birth and proper motions of new radio knots ejected by the star (Martí et al. 1995; Curiel et al. 2006; Pech et al. 2010; Carrasco-González et al. 2010a, 2012a; Gómez et al. 2013; Rodríguez-Kamenetzky et al. 2016; Osorio et al. 2017). When radio knots are detected very near the protostar, these are most probably due to internal shocks in the jet resulting from changes in the velocity of the material with time. These shocks are weak, and the emission mechanism seems to be free-free emission from (partially) ionized gas. Their proper motions are directly related to the velocity of the jet material as it arises from the protostar. The observed proper motions of these internal shocks suggest velocities of the jet that go from $\sim$100 km s$^{-1}$ in the low mass stars up to $\sim$1000 km s$^{-1}$ in the most massive objects. So far, in some sources there is also evidence of deceleration far from the protostar. The radio knots observed by Martí et al. (1995) close to the star in the HH 80-81 system move at velocities of $\sim$1000 km s$^{-1}$ in the plane of the sky, while the more distant optical HH objects show velocities of order 350 km s$^{-1}$ (Heathcote et al. 1998; Masqué et al. 2015). A similar case has been observed in the triple source in Serpens, where knots near the protostar appear to be ejected at very high velocities ($\sim$500 km s$^{-1}$), while radio knots far from the protostar move at slower velocities ($\sim$200 km s$^{-1}$; Rodríguez-Kamenetzky et al. 2016). These results suggest that radio knots far from the star are then most probably tracing the shocks of the jet against the ambient medium. In some cases, when the velocity of the jet is high, the emission of these outer knots seems to be of synchrotron nature (negative spectral indices), implying that a mechanism of particle acceleration can take place at these termination shocks. This topic is discussed in more detail below (Section 6). Variability ----------- Since thermal radio jets are typically detected over scales of $\sim$100 au and have velocities in the order of 300 km s$^{-1}$, one expects that if variations are present they will be detectable on timescales of the order of the travel time ($\sim$ a few years) or longer. The first attempts to detect time variability suggested that modest variations, of order 10-20%, could be present in some sources (Martí et al. 1998; Rodríguez et al. 1999; Rodríguez et al. 2000). However, over time a few examples of more extreme variability were detected. The 3.5 cm flux density of the radio source powering the HH 119 flow in B335 was 0.21$\pm$0.03 mJy in 1990 (Anglada et al. 1992), dropping to $\leq$0.08$\pm$0.02 mJy in 1994 (Avila et al. 2001), and finally increasing to 0.39$\pm$0.02 mJy in 2001 (Reipurth et al. 2002). C. Carrasco-González et al. (in prep.) report a factor of two increase in the 1.0 cm flux density of the southern component of XZ Tau over a few months. This increase in radio flux density has been related by these authors to an optical/infrared outburst (Krist et al. 2008) attributed to the periastron passage in a close binary system. The radio source associated with DG Tau A presented an important increase in its 3.5 cm flux density, that was 0.41$\pm$0.04 mJy in 1994 to 0.84$\pm$0.05 mJy in 1996 (Rodríguez et al. 2012b). The radio source associated with DG Tau B decreased from a total 3.5 cm flux density of 0.56$\pm$0.07 mJy in 1994 to 0.32$\pm$0.05 mJy in 1996 (Rodríguez et al. 2012a). In the source IRAS 16293$-$2422A2, Pech et al. (2010) observed an increase in the 3.5 cm flux density from 1.35$\pm$0.08 mJy in 2003 to $>$2.2 mJy en 2009. In these last three sources the observed variations were clearly associated with the appearance or disappearance of bright radio knots in the systems. Despite these remarkable cases, most of the radio jets that have been monitored show no evidence of variability above the 10-20% level (e.g., Rodríguez et al. 2008; Loinard et al. 2010; Carrasco-González et al. 2012a, 2015; Rodríguez et al. 2014a; see Fig. \[fig:n2071var\]). If ejections are present in these systems, they are not as bright as the cases discussed in the previous paragraph. ![Monitoring of the deconvolved size of the major axis (panel a), position angle (panel b), peak intensity (panel c), and flux density (panel d) of NGC2071 IRS 3 at 3.6 cm. Image reproduced with permission from Carrasco-González et al. (2012a), copyright by AAS.[]{data-label="fig:n2071var"}](fig2.pdf){width="60.00000%"} Given that accretion and outflow are believed to be correlated, the variations in the radio continuum emission (tracing the ionized outflow) and in the compact infrared and millimeter continuum emissions (tracing the accretion to the star) are expected to present some degree of temporal correlation. The correlation of the \[OI\] jet brightness with the mid-infrared excess from the inner disk and with the optical excess from the hot accretion layer has been considered as evidence that jets are powered by accretion (Cabrit 2007 and references therein; see also the recent work by Nisini et al. 2018). Also the correlation between the radio and bolometric luminosities (see Section 8.1) in very young objects supports this hypothesis. In these objects this correlation is expected since most of the luminosity comes from accretion, while the radio continuum traces the outflow. However, temporal correlations between the variations in the bolometric luminosity and the radio continuum have not been clearly observed. For example, the infrared source HOPS 383 in Orion was reported to have a large bolometric luminosity increase between 2004 and 2008 (Safron et al. 2015), while the radio continuum monitoring at several epochs before and after the outburst (from 1998 to 2014) shows no significant variation in the flux density (Galván-Madrid et al. 2015). Ellerbroek et al. (2014) could not establish a relation between outflow and accretion variability in the Herbig Ae/Be star HD 163296, the former being measured from proper motions and radial velocities of the jet knots, whereas the latter was measured from near-infrared photometric and Br$\gamma$ variability. Similarly, Connelley & Greene (2014) monitored a sample of 19 embedded (class I) YSOs with near-IR spectroscopy and found that, on average, accretion tracers such as Br$\gamma$ are not correlated in time to wind tracers such as the H$_2$ and \[Fe II\] lines. The infrared source LRLL 54361 shows a periodic variation in its infrared luminosity, increasing by a factor of 10 over roughly one week every 25.34 days. However, the sensitive JVLA observations of Forbrich et al. (2015) show no correlation with the infrared variations. Outflow rotation ---------------- At the scale of the molecular outflows, several cases have been found that show a suggestion of rotation, with a small velocity gradient perpendicular to the major axis of the outflow (e.g., Chrysostomou et a. 2008; Launhardt et al. 2009; Lee et al. 2009; Zapata et al. 2010; Choi et al. 2011; Pech et al. 2012; Hara et al. 2013; Chen et al. 2016; Bjerkeli et al. 2016). Evidence of outflow rotation has also been found in optical/IR microjets from T Tauri stars (e.g., Bacciotti et al. 2002; Anderson et al. 2003; Pesenti et al. 2004; Coffey et al. 2004, 2007). The observed velocity difference across the minor axis of the molecular outflow is typically a few km s$^{-1}$, while in the case of optical/IR microjets (that trace the inner, more collimated component) it can reach a few tens of km s$^{-1}$. These signatures of jet rotation about its symmetry axis are very important because they represent the best way to test the hypothesis that jets extract angular momentum from the star-disk systems. However, there is considerable debate on the actual nature and origin of these gradients (Frank et al. 2014; de Colle et al. 2016). The presence of precession, asymmetric shocks or multiple sources could also produce apparent jet rotation. Also, it is possible that most of the angular momentum could be stored in magnetic form, rather than in rotation of matter (Coffey et al. 2015). To ensure that the true jet rotation is being probed it should be checked that rotation signatures are consistent at different positions along the jet, and that the jet gradient goes in the same direction as that of the disk. This kind of tests have been carried out only in very few objects (see Coffey et al. 2015) with disparate results. In the case of the rotating outflow associated with HH 797 (Pech et al. 2012) a double radio source with angular separation of $\sim3''$ is found at its core (Forbrich et al. 2015), suggesting an explanation in terms of a binary jet system. It is important that the jet is observed as close as possible to the star, where any evidence of angular momentum transfer is still preserved, since far from the star the interaction with the environment can hide and confuse rotation signatures. Therefore, high angular resolution observations of radio recombination lines from radio jets, reaching the region closer to the star, could help to solve these problems. In principle, it could be possible to observe the jet near its launch region and compare the velocity gradients with those observed al larger scales. However, these observations are very difficult and can be feasible only with new instruments such as the Atacama Large Millimeter/submillimeter Array (ALMA) and in the future the Next Generation Very Large Array (ngVLA) and the Square Kilometre Array (SKA). Free-free continuum emission from radio jets ============================================ Frequency dependences --------------------- Reynolds (1986) modeled the free-free emission from an ionized jet. He assumed that the ionized flow begins at an injection radius $r_0$ with a circular cross section with initial half-width $w_0$. He adopted power-law dependences with radius $r$, so that the half-width of the jet is given by $$w = w_0 \Bigl({{r} \over {r_0}}\Bigr)^\epsilon.$$ The case of $\epsilon$ =1 corresponds to a conical (constant opening angle) jet. The electron temperature, velocity, density, and ionized fraction were taken to vary as $r/r_0$ to the powers $q_T$, $q_v$, $q_n$, and $q_x$, respectively. In general. the jets are optically thick close to the origin and optically thin at large $r$. The jet axis is assumed to have an inclination angle $i$ with respect to the line of sight. In consequence, the optical depth along a line of sight through the jet follows a power law with index: $$q_\tau = \epsilon + 2 q_x + 2 q_n - 1.35 q_T.$$ Assuming flux continuity $$q_\tau = -3 \epsilon + 2 q_x - 2 q_v - 1.35 q_T.$$ Assuming the case of an isothermal jet with constant velocity and ionization fraction ($q_T = q_v = q_x$ = 0), the flux density increases with frequency as $$S_\nu \propto \nu^\alpha,$$ where the spectral index $\alpha$ is given by $$\alpha = 1.3 - {{0.7} \over {\epsilon}}.$$ The angular size of the major axis of the jet decreases with frequency as $$\theta_\nu \propto \nu^{-0.7/\epsilon} = \nu^{\alpha - 1.3}.$$ The previous discussion is valid for frequencies below the turnover frequency $\nu_m$, which is related to the injection radius $r_0$. At high enough frequencies, $\nu > \nu_m$, the whole jet becomes optically thin and the spectral index becomes $-0.1$. Physical parameters from radio continuum emission ------------------------------------------------- Following Reynolds (1986) the injection radius and ionized mass loss rate are given by: $$\begin{aligned} \nonumber \left(\frac{r_0}{\rm au}\right) &=& 26 \left[\frac{(2-\alpha) (0.1+\alpha)}{1.3-\alpha}\right]^{0.5} \left[{\left(\frac{S_\nu}{\rm mJy}\right) \left(\frac{\nu}{\rm 10~GHz}\right)^{-\alpha}}\right]^{0.5} \left(\frac{\nu_m}{\rm 10~GHz}\right)^{0.5 \alpha - 1} \\ && \times \left(\frac{\theta_0 \sin i}{\rm rad}\right)^{-0.5} \left(\frac{d}{\rm kpc}\right) \left(\frac{T}{\rm 10^4~K}\right)^{-0.5}, \label{eq:r0}\end{aligned}$$ $$\begin{aligned} \nonumber \left(\frac{\dot M_{\rm ion}}{10^{-6}~M_\odot~\rm yr^{-1}}\right)&=& 0.108 \left[\frac{(2-\alpha) (0.1+\alpha)}{1.3-\alpha}\right]^{0.75} \left[{\left(\frac{S_\nu}{\rm mJy}\right) \left(\frac{\nu}{\rm 10~GHz}\right)^{-\alpha}}\right]^{0.75} \\ \nonumber && \times \left(\frac{v_{j}}{200~\rm km~s^{-1}}\right) \left(\frac{\nu_m}{\rm 10~GHz}\right)^{0.75 \alpha - 0.45} \left(\frac{\theta_0}{\rm rad}\right)^{0.75} \left({\sin i}\right)^{-0.25} \\ && \times \left(\frac{d}{\rm kpc}\right)^{1.5} \left(\frac{T}{\rm 10^4~K}\right)^{-0.075}, \label{eq:mi}\end{aligned}$$ where $\theta_0 = 2 w_0/r_0$ is the injection opening angle of the jet, that usually is roughly estimated using $$\theta_0 = 2~\arctan(\theta_{\rm min}/\theta_{\rm maj}),$$ where $\theta_{\rm min}$ and $\theta_{\rm maj}$ are the deconvolved minor and major axes of the jet. We note that the dimensions of the jet are very compact, comparable or smaller than the beam and as a consequence the value of $\theta_0$ is uncertain. The opening angle determined on larger scales (for example using Herbig-Haro knots along the flow away from the jet core) is usually smaller and at present it is not known if this is the result of recollimation or of an overestimate in the measurement of $\theta_0$. In the case of a conical ($\alpha$ = 0.6) jet, equations (\[eq:r0\]) and (\[eq:mi\]) simplify to: $$\begin{aligned} \nonumber \left(\frac{r_0}{\rm au}\right) &=& 31 \left[{\left(\frac{S_\nu}{\rm mJy}\right) \left(\frac{\nu}{\rm 10~GHz}\right)^{-0.6}}\right]^{0.5} \left(\frac{\nu_m}{\rm 10~GHz}\right)^{-0.7} \\ && \times \left(\frac{\theta_0 \sin i}{\rm rad}\right)^{-0.5} \left(\frac{d}{\rm kpc}\right) \left(\frac{T}{\rm 10^4~K}\right)^{-0.5},\end{aligned}$$ $$\begin{aligned} \nonumber \left(\frac{\dot M_{\rm ion}}{10^{-6}~M_\odot~\rm yr^{-1}}\right)&=& 0.139 \left[{\left(\frac{S_\nu}{\rm mJy}\right) \left(\frac{\nu}{\rm 10~GHz}\right)^{-0.6}}\right]^{0.75} \\ \nonumber && \times \left(\frac{v_{j}}{200~\rm km~s^{-1}}\right) \left(\frac{\theta_0}{\rm rad}\right)^{0.75} \left({\sin i}\right)^{-0.25} \\ && \times \left(\frac{d}{\rm kpc}\right)^{1.5} \left(\frac{T}{\rm 10^4~K}\right)^{-0.075},\end{aligned}$$ Usually a value of $T = 10^4$ K is adopted. In a few cases there is information on the jet velocity, but in general one has to assume a velocity typically in the range from 100 km s$^{-1}$ to 1,000 km s$^{-1}$, depending on the mass of the young star. Following Panoglou et al. (2012) and assuming a launch radius of about 0.5 au (Estalella et al. 2012), the jet velocity can be crudely estimated from $$\label{eq:vj} \biggl({{v_j} \over {\rm km~s^{-1}}}\biggr) \simeq 140 \biggl({{M_*} \over {0.5~M_\odot}}\biggr)^{1/2}.$$ Until recently, the turnover frequency has not been determined directly from observations and is estimated that it will appear above 40 GHz. Only in the case of HL Tau, the detailed multifrequency observations and modeling suggest a value of $\sim$1.5 au (A.M. Lumbreras et al., in prep.). Determining $\nu_m$ is difficult because at high frequencies dust emission from the associated disk starts to become dominant. Future observations at high frequencies made with high angular resolution should allow to separate the compact free-free emission from the base of the jet to that of the more extended dust emission in the disk and give more determinations of $\nu_m$ and of the morphology and size of the gap between the star and the injection radius of the jet. Anyhow, the dependence of $r_0$ on $\nu_m$ is not critical, that is, in order to change $r_0$ one order of magnitude, the turnover frequency should typically change a factor of $\sim$30. Assuming a lower limit of $\sim$10 GHz for the turnover frequency, upper limits of $\sim$10 au have been obtained for $r_0$ (Anglada et al. 1998; Beltrán et al. 2001). In the case of $\dot M_{\rm ion}$ the dependence on $\nu_m$ is almost negligible and disappears for the case of a conical jet ($\alpha$ = 0.6). Using this technique, ionized mass loss rates in the range of $10^{-10}~M_\odot$ yr$^{-1}$ (low-mass objects) to $10^{-5}~M_\odot$ yr$^{-1}$ (high mass objects) have been determined (Rodríguez et al. 1994b; Beltrán et al. 2001; Guzmán et al. 2010, 2012; see Table \[tab:1\]). Radio recombination lines from radio jets ========================================= LTE formulation --------------- Following Reynolds (1986), the flux density at frequency $\nu$, $S_\nu$, from a jet is given by $$S_\nu = \int_0^\infty B_\nu(T) (1 - \exp[-\tau_\nu]) d \Omega ,$$ where $B_\nu(T)$ is the source function (taken to be Planck’s function since we are assuming LTE), $\tau_\nu$ is the optical depth along a line of sight through the jet, and $d \Omega$ is the differential of solid angle. Since $$d \Omega = {{2 w(r)} \over {d^2}} dy,$$ where $d$ is the distance to the source and $y = r \sin i$ is the projected distance in the plane of the sky. Assuming an isothermal jet, the continuum emission is given by $$S_C = B_\nu(T) \int_0^\infty {{2 w(r)} \over {d^2}} (1 - \exp[-\tau_C]) dy,$$ while the line plus continuum emission will be given by $$S_{L+C} = B_\nu(T) \int_0^\infty {{2 w(r)} \over {d^2}} (1 - \exp[-\tau_{L+C}]) dy.$$ The line to continuum ratio will be given by $${{S_L} \over {S_C}} = {{S_{L+C}} \over {S_C}} -1.$$ Using the power law dependences of the variables and noting that the line and the continuum opacities have the same radial dependence, we obtain $${{S_L} \over {S_C}} = {{\int_0^\infty y^{\epsilon} (1 - \exp[-\tau_{L+C}(r_0) y^{\epsilon + 2 q_x + 2 q_n}/ \sin i]) dy} \over {\int_0^\infty y^{\epsilon} (1 - \exp[-\tau_C(r_0) y^{\epsilon + 2 q_x + 2 q_n}/ \sin i]) dy}} - 1.$$ Using the definite integral (Gradshteyn & Ryzhik 1994) $$\int_0^\infty [1-\exp(-\mu x^p)]~x^{t-1}~ dx = -{1\over{|p|}}~ \mu^{-{t/p}}~ \Gamma\left({t\over{p}}\right),$$ valid for $0 < t < -p$ for $p < 0$ and with $\Gamma$ being the Gamma function. We then obtain $${{S_L} \over {S_C}} = \left[{{\tau_{L+C}(r_0)} \over {\tau_C(r_0)}} \right]^{-(\epsilon+1)/(\epsilon + 2 q_x + 2 q_n)} - 1.$$ Finally, $${{S_L} \over {S_C}} = \left[{{\kappa_L} \over {\kappa_C}} + 1 \right]^{-(\epsilon+1)/(\epsilon + 2 q_x + 2 q_n)} - 1.$$ where $\kappa_L$ and $\kappa_C$ are the line and continuum absorption coefficients at the frequency of observation, respectively. Substituting the LTE ratio of these coefficients (Mezger & Höglund 1967; Gordon 1969; Quireza et al. 2006), we obtain: $$\begin{aligned} \nonumber &&{{S_L} \over {S_C}} = \left[0.28 {\left({\nu_L\over{\rm GHz}}\right)^{1.1}} {\left({{T}\over{\rm 10^4\,K}}\right)^{-1.1}} {\left(\frac{\Delta v}{\rm km~s^{-1}}\right)^{-1}}{(1 + Y^+)^{-1}} +1\right]^{-(\epsilon+1)/(\epsilon + 2 q_x + 2 q_n)} -1, \\ &&\end{aligned}$$ where $\nu_L$ is the frequency of the line, $\Delta v$ the full width at half maximum of the line, and $Y^+$ is the ionized helium to ionized hydrogen ratio. For a standard biconical jet with constant velocity and ionized fraction, the equation becomes $${{S_L} \over {S_C}} = \left[0.28 \biggl({{\nu_L} \over {\rm GHz}} \biggr)^{1.1} \biggl({{T} \over {\rm 10^4~K}} \biggr)^{-1.1} \biggl({{\Delta v} \over {\rm km~s^{-1}}}\biggr)^{-1} (1 + Y^+)^{-1} + 1 \right]^{2/3} - 1.$$ In the centimeter regime, the first term inside the brackets is smaller than 1 and using a Taylor expansion the equation can be approximated by $$\label{eq:rrl} {{S_L} \over {S_C}} = 0.19 \biggl({{\nu_L} \over {\rm GHz}} \biggr)^{1.1} \biggl({{T} \over {\rm 10^4~K}} \biggr)^{-1.1} \biggl({{\Delta v} \over {\rm km~s^{-1}}}\biggr)^{-1} (1 + Y^+)^{-1}.$$ Are these relatively weak RRLs detectable with the next generation of radio interferometers? Let us assume that we attempt to observe the H86$\alpha$ line at 10.2 GHz and adopt an electron temperature of $10^4$ K and an ionized helium to ionized hydrogen ratio of 0.1. The line width expected for these lines is poorly known. If the jet is highly collimated, the line width could be a few tens of km s$^{-1}$, similar to those observed in HII regions and dominated by the microscopic velocity dispersion of the gas. Under these circumstances, we expect to see two relatively narrow lines separated by the projected difference in radial velocity of the two sides of the jet. However, if the jet is highly turbulent and/or has a large opening angle we expect a single line with a width of a few hundreds of km s$^{-1}$. Since for a constant area under the line it is easier to detect narrow lines (the signal-to-noise ratio improves as $\Delta v^{-1/2}$), we will conservatively (perhaps pessimistically) assume a line width of 200 km s$^{-1}$. Using equation (\[eq:rrl\]) we obtain that the line-to-continuum ratio will be $S_L/S_C$ = 0.011. The brightest thermal jets (see Table \[tab:1\]) have continuum flux densities of about 5 mJy, so we expect a peak line flux density of 55 $\mu$Jy. These are indeed weak lines. However, the modern wide-band receivers allow the simultaneous detection of many recombination lines. For example, in the X band (8-12 GHz) there are 11 $\alpha$ lines (from the H92$\alpha$ at 8.3 GHz to the H82$\alpha$ at 11.7 GHz). Stacking $N$ different lines will improve the signal-to-noise ratio by $N^{1/2}$, so we expect a gain of a factor of 3.3 from this averaging. At present, the Jansky VLA will give, for a frequency of 10.2 GHz, a channel width of 100 km s$^{-1}$ and, with an on-source integration time of $\sim$45 hours, an rms noise of $\sim$11 $\mu$Jy beam$^{-1}$, sufficient to detect the H86$\alpha$ emission at the 5-$\sigma$ level from a handful of bright jets. Line stacking will improve this signal-to-noise ratio by 3.3. In contrast, the ngVLA is expected to be 10 times as sensitive as the Jansky VLA, which means that a similar sensitivity will be achieved with 1/100 of the time, that is about 0.5 hours. For an on-source integration time of about 14 hours the ngVLA will have an rms noise of $\sim$2 $\mu$Jy beam$^{-1}$ in the line mode previously described and will be able to detect the RRLs from the more typical 1 mJy continuum jet at the 5-$\sigma$ level. Line stacking will increase this detection to a very robust $\sim$16-$\sigma$. This will allow to start resolving the jets into at least a few pixels and start studying the detailed kinematics of the jet. Then, a program of a few hundred hours with the ngVLA will characterize the RRL emission from a wide sample of thermal jets. Similar on-source integration times would be required with the expected sensitivity of the SKA (see Anglada et al. 2015). Stark Broadening ---------------- The collisions between the electrons and the atoms produce a line broadening, known as Stark or pressure broadening, that depends strongly on the electron density of the medium and in particular on the level of the transition. For an RRL with principal quantum number $n$, quantum number decrement $\Delta n$ in a medium with electron density $N_e$ and electron temperature $T_e$, the Stark broadening is given by (Walmsley 1990): $$\left(\delta_S \over {\rm km ~s}^{-1} \right) = 2.72 \left({n \over 42} \right)^{7.5} \Delta n^{-1} \left({N_e \over 10^7 ~{\rm cm}^{-3}} \right) \left({T_e \over 10^4 ~{\rm K}} \right)^{-0.1}.$$ Stark broadening has been detected in the case of HII regions (e.g., Smirnov et al. 1984; Alexander & Gulyaev 2016) and slow ionized winds (e.g., Guzmán et al. 2014). Would it be an important effect in the case of RRLs from ionized jets? In contrast with HII regions, where a uniform electron density can approximately describe the object, thermal jets are characterized by steep gradients in the electron density, that is expected to decrease typically as the distance to the star squared (see discussion above). On the other hand, the inner parts of a thermal jet are optically thick in the continuum and do not contribute to the recombination line emission. All the line emission comes from radii external to the line of sight where the free-free continuum opacity, $\tau_C$, is $\sim$1. The length of the jets along its major axis is of the order of twice the radius at which $\tau_C \simeq 1$. From Table \[tab:1\] we estimate that at cm wavelengths the distance from the star where $\tau_C \simeq 1$ is in the range of 100 to 500 au. Assuming a typical opening angle of $30^\circ$ we find that the physical depth of the jet, $2 w$, at this position ranges from 50 to 250 au. The free-free opacity is given approximately by (Mezger & Henderson 1967): $$\tau_C \simeq 1.59 \times 10^{-10} \left({T_e \over 10^4 ~{\rm K}} \right)^{-1.35} \left({\nu \over {\rm GHz}}\right)^{-2.1} \left({N_e \over {\rm cm}^{-3}}\right)^{2} \left({2 w \over 100 ~{\rm au}}\right).$$ Adopting $T_e$ = $10^4$ K, $\nu$ = 10.2 GHz (the H86$\alpha$ line) and $\tau_C$ = 1, we find that the electron density at this position will be in the range of $5.7 \times 10^5$ cm$^{-3}$ to $1.3 \times 10^6$ cm$^{-3}$. The recombination line emission will be coming from regions with electron densities equal or smaller than these values. Finally, using the equation for the Stark broadening given above, we find that the broadening will be in the range of 33 to 167 km s$^{-1}$. We then conclude that Stark broadening will be important in the case that the line widths are of the order of the microscopic velocity dispersion (tens of km s$^{-1}$) but it will not be dominant in the case that the line widths have a value similar to that of the jet speed. Non-LTE Radio Recombination Lines --------------------------------- The treatment presented here assumes that the recombination line emission is in LTE, which seems to be a reasonable approach. There are, however, cases where the LTE treatment is clearly insufficient and a detailed modeling of the line transfer is required. The more clear case is that of MWC 349A, a photoevaporating disk located some 1.2 kpc away (Gordon 1994). Even when this source is not a jet properly but a photoevaporated wind, we discuss it briefly since it is the best case known of non-LTE radio recombination lines. At centimeter wavelengths, the line emission from MWC 349A is consistent with LTE conditions (Rodríguez & Bastian 1994). However, at the millimeter wavelengths, below 3 mm, this source presents strong, double-peaked hydrogen recombination lines (Martín-Pintado et al. 1989). These spectra arise from a dense Keplerian-rotating disk, that is observed nearly edge-on (Weintroub et al. 2008; Báez-Rubio et al. 2014). The flux density of these features is an order or magnitude larger than expected from the LTE assumption and the inference that maser emission is at work seems well justified. However, the nature of MWC 349A remains controversial, with some groups considering it a young object, while others attribute to it an evolved nature (Strelnitski et al. 2013). Extremely broad millimeter recombination lines of a possible maser nature have been reported from the high-velocity ionized jet Cep A HW2 (Jiménez-Serra et al. 2011). In this case, the difference between the LTE and non-LTE models is less than a factor of two and in some of the transitions it is necessary to multiply the non-LTE model intensities by a factor of several to agree with the observations. The source Mon R2-IRS2 could be a case of weakly amplified radio recombination lines in a young massive star. Jiménez-Serra et al. (2013) report toward this source a double-peaked spectrum. with the peaks separated by about 40 km s$^{-1}$. However, the line intensity is only $\sim$50% larger than expected from LTE. Jiménez-Serra et al. (2013) propose that the radio recombination lines arise from a dense and collimated jet embedded in a cylindrical ionized wind, oriented nearly along the line of sight. In the case of LkH$\alpha$101, Thum et al. (2013) find the millimeter RRLs to be close to LTE and to show non-Gaussian wings that can be used to infer the velocity of the wind, in this case 55 km s$^{-1}$. Finally, Guzmán et al. (2014) detect millimeter recombination lines from the high-mass young stellar object G345.4938+01.4677. The hydrogen recombination lines exhibit Voigt profiles, which is a strong signature of Stark broadening. The continuum and line emissions can be successfully reproduced with a simple model of a slow ionized wind in LTE. Non-thermal emission from radio jets ==================================== As commented above, jets from YSOs have been long studied at radio wavelengths through their thermal free-free emission which traces the base of the ionized jet, and shows a characteristic positive spectral index (the intensity of the emission increases with the frequency, e.g., Rodríguez 1995; Rodríguez 1996; Anglada 1996). However, in the early 1990s, sensitive observations started to reveal that non-thermal emission at centimeter wavelengths might be also present in several YSO jets (e.g., Rodríguez et al. 1989a; Rodríguez et al. 2005; Martí et al. 1993; Garay et al 1996; Wilner et al. 1999). This non-thermal emission is usually found in relatively strong radio knots, away from the core, showing negative spectral indices at centimeter wavelengths (see Fig. \[fig:serpens\]). Oftentimes, these non-thermal radio knots appear in pairs, moving away from the central protostar at velocities of several hundred kilometers per second (e.g., the triple source in Serpens; Curiel et al. 1993). Because of these characteristics, it was proposed that these knots could be tracing strong shocks of the jet against dense material in the surrounding molecular cloud where the protostar is forming. Their non-thermal nature was interpreted as synchrotron emission from a small population of relativistic particles that would be accelerated in the ensuing strong shocks. This scenario is supported by several theoretical studies, showing the feasibility of strong shocks in protostellar jets as efficient particle accelerators (e.g., Araudo et al. 2007, Bosch-Ramon et al. 2010, Padovani et al. 2015, 2016, Cécere et al. 2016). ![Image of the non-thermal radio jet in Serpens. The radio continuum image is shown in contours and was made by combining the data from the S, C, and X bands. This image is superposed on the spectral index image (color scale). Image reproduced with permission from Rodríguez-Kamenetzky et al. (2016), copyright by AAS. []{data-label="fig:serpens"}](fig3.pdf){width="90.00000%"} The possibility of protostellar jets as efficient particle accelerators has been confirmed only recently, with the detection and mapping of linearly polarized emission from the YSO jet in HH 80-81 (Carrasco-González et al. 2010b; see Fig. \[fig:hh80\]). This result provided for the first time conclusive evidence for the presence of synchrotron emission in a jet from a YSO, and then, the presence of relativistic particles. At centimeter wavelengths, where most of the radio jet studies have been carried out, the emission of the non-thermal lobes is usually weaker than that of the thermal core of the jet. Consequently, most of the early detections of non-thermal radio knots were obtained through very sensitive observations resulting from unusually long projects, mainly carried out at the VLA. After the confirmation in 2010 of synchrotron emission in the HH 80-81 radio jet, the improvement in sensitivity of radio interferometers has facilitated higher sensitivity observations at multiple wavelengths of star forming regions, and new non-thermal protostellar jet candidates are emerging (e.g., Purser et al. 2016, Osorio et al. 2017, Hunter et al. 2018; Tychoniec et al. 2018; see Fig. \[fig:hops108\]). It is then expected that the next generation of ultra-sensitive radio interferometers (LOFAR, SKA, ngVLA) will produce very detailed studies on the nature of non-thermal emission in protostellar jets. It is worth noting that, even when non-thermal emission is brighter at lower frequencies, most of the proposed detections have been performed at relatively high frequencies, typically in the 4-10 GHz range. One reason is that studies of radio jets have been focused preferentially on the detection of thermal emission, which is dominant at higher frequencies. Another reason is the lack of sensitivity at low frequencies, specially below 1 GHz, in most of the available interferometers. However, despite the small number of sensitive observations performed at low frequencies, they have produced very interesting results. For example, the only non-thermal radio jet candidate associated with a low-mass protostar known so far has been identified through Giant Metrewave Radio Telescope (GMRT) observations of DG Tau in the 300-700 MHz range (Ainsworth et al. 2014; see Fig. \[fig:dgtau\]). Very recent GMRT observations of HH 80-81 (Vig et al. 2018) at 325 and 610 MHz found negative spectral indices steeper than previous studies at higher frequencies. This has been interpreted as indicating that an important free-free contribution is present at high frequencies even in the non-thermal radio knots. We should then expect more interesting results from low frequency observations using the next generation of low frequency interferometers, such as the Low Wavelength Array (LWA), and the aforementioned LOFAR and SKA. At present, the best studied cases of particle acceleration in protostellar jets are the triple source in Serpens and HH 80-81 (Rodríguez-Kamenetzky et al. 2016, 2017). These sources were observed very recently with the VLA combining high angular resolution and very high sensitivity in the 1-10 GHz frequency range. These observations resolve both jets at all observed frequencies, allowing to study the different emission mechanisms present in the jet in a spatially resolved way. In both cases, strong non-thermal emission is detected at the termination points of the jets, which is consistent with particle acceleration in strong shocks against a very dense ambient medium. Additionally, in the case of HH 80-81, non-thermal emission is observed at several positions along the very well collimated radio jet, suggesting that this object is able to accelerate particles also in internal shocks (Rodríguez-Kamenetzky et al. 2017). Overall, these studies have found that the necessary conditions to accelerate particles in a protostellar jet are high velocities ($\gtrsim$ 500 km s$^{-1}$) and an ambient medium denser than the jet. These conditions are probably well satisfied in the case of very young protostellar jets, which are still deeply embedded in their parental cloud. The discovery of linearly polarized emission in the HH 80-81 radio jet by Carrasco-González et al. (2010b) opened the interesting possibility of studying magnetic fields in protostellar jets. Detection of linearly polarized emission at several wavelengths would allow one to infer the properties of the magnetic field in these jets in a way similar to that commonly employed in AGN jets. The magnetic field strength can be estimated from the spectral energy distribution at cm wavelengths (e.g., Pacholczyk 1970; Beck & Krause 2005), while the magnetic field morphology can be obtained from the properties of the linear polarization (polarization angle, polarization degree and Faraday rotation). For non-relativistic jets, the apparent magnetic field (magnetic field averaged along the line-of-sight) is perpendicular to the direction of the linear polarization. Moreover, theoretical models of helical magnetic fields predict gradients of the polarization degree and Faraday rotation measurements along and across the jet (Lyutikov et al. 2005). Thus, by comparing the observational results with theoretical models, the 3D morphology of the magnetic field can be inferred. Mapping the polarization in a set of YSO jets in combination with detailed theoretical modeling may lead to a deeper understanding of the overall jet phenomenon. However, radio synchrotron emission in YSO jets seems to be intrinsically much weaker and difficult to study than in relativistic jets (e.g., AGN and microquasar jets). So far, only in the case of HH 80-81, one of the brightest and most powerful YSO jets known, it has been possible to obtain enough sensitivity to detect and study its linear polarization, which is only a fraction of the total continuum emission. Subsequent higher sensitivity studies (Rodríguez-Kamenetzky et al. 2016, 2017) have not been able to detect polarization in YSO jets. One of the reasons of this is that, due to the presence of thermal electrons mixed with the relativistic particles, we expect strong Faraday rotation. At the moment, high sensitivities in radio interferometers are obtained by averaging data in large bandwidths. Within these bandwidths, we expect large rotations of the polarization angle, resulting in strong depolarization when the emission is averaged. Then, it is still necessary to observe using long integration times in order to obtain enough sensitivity to detect polarization in smaller frequency ranges. Note that using the Faraday RM-Synthesis tool (Brentjens & de Bruyn 2005) to determine the rotation measure to account for this effect is also hampered by the poor signal-to-noise ratio over narrow frequency ranges in these weak sources. At the moment, probably HH 80-81 is the only object bright enough to perform a study of polarization at several wavelengths; and even in this object, polarization would be detected only after observations of several tens of hours long. Therefore, we will probably have to wait for new ultra-sensitive interferometers in order to perform a polarization study in a large sample of protostellar jets. Masers as tracers of jets ========================= Molecular maser emission at cm wavelengths (e.g., H$_2$O, CH$_3$OH, OH) is often found in the early evolutionary stages of massive protostars. Such maser emission is usually very compact and strong, with brightness temperatures exceeding in some cases 10$^{10}$ K, allowing the observation of outflows at milliarcsecond (mas) scales (1 mas = 1 au at a distance of 1 kpc) using Very Long Baseline Interferometry (VLBI). Sensitive VLBI observations show, in some sources, thousands of maser spots forming microstructures that reveal the 3D kinematics of outflows and disks at small scale (e.g., Sanna et al. 2015). This kind of observations have given a number of interesting results: the discovery of short-lived, episodic non-collimated outflow events (e.g., Torrelles et al. 2001, 2003; Surcis et al. 2014); detection of infall motions in accretion disks around massive protostars (Sanna et al. 2017); the imaging of young ($< 100$ yr) small scale (a few 100 au) bipolar jets of masers (Sanna et al. 2012, Torrelles et al. 2014; Fig. \[fig:masers\]a), and even allowed to analyze the small-scale (1-20 au) structure of the micro bow-shocks (Uscanga et al. 2005; Trinidad et al. 2013; Fig. \[fig:masers\]b); the simultaneous presence of a wide-angle outflow and a collimated jet in a massive protostar (Torrelles et al. 2011; see Fig. \[fig:masers\]c); and polarization studies have determined the distribution and strength of the magnetic field very close to protostars allowing to better understand its role in the star formation processes (Surcis et al. 2009; Vlemmings et al. 2010; Sanna et al. 2015; Fig. \[fig:masers\]d). ![(a) Distribution and proper motions of H$_2$O masers showing a very compact ($\sim$180 au), short-lived ($\sim$40 yr), bipolar jet from a very embedded protostar of unknown nature in the LkH$\alpha$ 234 star-forming region (Torrelles et al. 2014). (b) Micro-bow shock traced by water masers in AFGL2591 VLA 3-N. Arrows indicate the measured proper motions (Trinidad et al. 2013). (c) Maser microstructures tracing a low-collimation outflow around the radio jet associated with the massive protostar Cep A HW2 (Torrelles et al. 2011). (d) Magnetic field structure around the protostar-disk-jet system of Cep A HW2. Spheres indicate the CH$_3$OH masers and black vectors the magnetic field direction. Image adapted from Vlemmings et al. (2010). \[fig:masers\] ](fig7.png){width="\textwidth"} In particular, the combination of continuum and maser studies in Cep A HW2 has been useful to provide one of the best examples of the two-wind model for outflows from massive protostars (Torrelles et al. 2011; Fig. \[fig:masers\]c). In this source the water masers trace the presence of a relatively slow ($\sim$10-70 km s$^{-1}$) wide-angle outflow (opening angle of $\sim102^\circ$), while the thermal jet traces a fast ($\sim$500 km s$^{-1}$) highly collimated radio jet (opening angle of $\sim18^\circ$). This two-wind phenomenon had been previously imaged only in low mass protostars such as L1551-IRS5 (Itoh et al. 2000, Pyo et al. 2005), HH 46/47 (Velusamy et al. 2007), HH 211 (Gueth & Guilloteau 1999, Hirano et al. 2006) and IRAS 04166+2706 (Santiago-García et al. 2009). An extensive study of the connection between high velocity collimated jets and slow un-collimated winds in a large sample of low-mass class II objects has been carried out recently by Nisini et al. (2018); this work has been performed from the analysis of \[OI\]6300 Å line profiles under the working hypothesis that the low velocity component traces a wide wind and the high velocity component a collimated jet. Nature of the Centimeter Continuum Emission in Thermal Radio Jets ================================================================= Observational Properties: Radio to Bolometric Luminosity Correlation\[sect:lbol\] --------------------------------------------------------------------------------- Photoionization does not appear to be the ionizing mechanism of radio jets since, in the sources associated with low luminosity objects, the rate of ionizing UV photons ($\lambda<912$ Å) from the star is clearly insufficient to produce the ionization required to account for the observed radio continuum emission (e.g., Rodríguez et al. 1989b; Anglada 1995). For low-luminosity objects ($1 {\lesssim}L_{\rm bol} {\lesssim}1000~L_\odot$), the observed flux densities at cm wavelengths are several orders of magnitude higher than those expected by photoionization (see Fig. \[fig:cor1\] and left panel in Fig. 5 of Anglada 1995). Ionization by shocks in a strong stellar wind or in the jet itself has been proposed as the most likely possibility (Torrelles et al. 1985, Curiel et al. 1987, 1989, Anglada et al. 1992; see below). Detailed simulations of the two-shock internal working surfaces traveling down the jet flow, and the expected emission of the ionized material at shorter wavelengths have been performed (e.g., the H$\alpha$ and \[OI\]6300 Å line emission of a radiative jet model with a variable ejection velocity by Raga et al. 2007). ![Empirical correlation between the bolometric luminosity and the radio continuum luminosity at cm wavelengths. Data are taken from Table \[tab:cm\]. Triangles correspond to high luminosity objects ($L_{\rm bol} > 1000~L_\odot$), dots correspond to low luminosity objects ($1 \le L_{\rm bol} \le 1000~L_\odot$), and squares to very low luminosity objects ($L_{\rm bol} < 1~L_\odot$). The dashed line is a least-squares fit to all the data points and the grey area indicates the residual standard deviation of the fit. The solid line is a fit to the low-luminosity objects alone. The dot-dashed line corresponds to the expected radio luminosity of an optically thin region photoionized by the Lyman continuum of the star. []{data-label="fig:cor1"}](fig8.pdf){width="95.00000%"} [lccccl]{} & [$d$]{} & [$L_{\rm bol}$]{} & [$S_\nu$]{} & [$\dot P$]{} & [Refs.]{}\ & [(kpc)]{} & [($L_\odot$)]{} & [(mJy)]{} & [($M_\odot\,{\rm yr^{-1}\,km~s^{-1}}$)]{}\ J041757 & 0.140 & 0.003 & 0.14 & $\cdots$ & 79, 83, 84\ J041836 & 0.140 & 0.0033 & 0.13 & $\cdots$ & 84\ J041847 & 0.140 & 0.0041 & 0.10 & $\cdots$ & 84\ J041938 & 0.140 & 0.0062 & 0.10 & $\cdots$ & 84\ IRAM04191 & 0.140 & 0.05 & 0.14 & 1.5$\times10^{-5}$ & 70, 71, 72\ L1014IRS & 0.260 & 0.15 & 0.11 & 1.3$\times10^{-7}$ & 70, 73, 74, 75\ L1148-IRS & 0.140 & 0.13 & 0.080 & 1.0$\times10^{-7}$ & 70, 76, 77\ L1521F-IRS & 0.140 & 0.04 & 0.07 & 1.1$\times10^{-6}$ & 77, 78, 79\ IC348-SMM2E & 0.240 & 0.06 & 0.027 & 1.1$\times10^{-7}$ & 79, 80, 81, 82\ HH30 & 0.140 & 0.42 & 0.042 & 7.5$\times10^{-7}$ & 85, 86, 87\ L1262 & 0.2 & 1.0 & 0.33 & 4.0$\times10^{-5}$ & 47, 50, 51, 142, 46\ VLA1623 & 0.16 & 1.5 & 0.6 & 4.5$\times10^{-4}$ & 16, 35\ L723 & 0.3 & 2.4 & 0.4 & 3.2$\times10^{-4}$ & 10, 11, 33, 28, 132, 155\ L1489 & 0.14 & 4.4 & 0.5 & 2.0$\times10^{-7}$ & 3, 4, 24, 137\ B335 & 0.25 & 4 & 0.2 & 2.0$\times10^{-5}$ & 10, 12, 24, 22, 58, 140, 142\ NGC2264G & 0.8 & 5 & 0.3 & 3.0$\times10^{-3}$ & 17, 18, 36\ AS353A & 0.3 & 8.4 & 0.1 & 4.5$\times10^{-4}$ & 1, 23, 52, 12\ L1448C & 0.35 & 9 & 0.1 & 4.5$\times10^{-4}$ & 14, 15, 34\ L1448N(A) & 0.35 & 10 & 0.9 & 1.2$\times10^{-4}$ & 14, 15, 35, 129, 130, 140\ RNO43 & 0.4 & 12 & 0.5 & 2.1$\times10^{-4}$ & 12, 22, 39\ L483 & 0.2 & 14 & 0.31 & 1.5$\times10^{-5}$ & 47, 45, 24\ L1251B & 0.2 & 14 & 1.2 & 4.9$\times10^{-5}$ & 48, 142, 49\ TTAU & 0.14 & 17 & 5.8 & 4.1$\times10^{-5}$ & 5, 6, 25, 26, 19, 1, 27\ HH111 & 0.46 & 24 & 0.9 & 5.0$\times10^{-5}$ & 20, 21, 23\ IRAS16293 & 0.16 & 27 & 2.9 & 1.6$\times10^{-3}$ & 8, 9, 30, 31, 32\ L1251A & 0.3 & 27 & 0.47 & 8.6$\times10^{-5}$ & 48, 45, 49\ HARO4-255FIR & 0.48 & 28 & 0.2 & 1.3$\times10^{-4}$ & 12, 19, 37, 38\ L1551-IRS5 & 0.14 & 20 & 0.8 & 1.0$\times10^{-4}$ & 1, 6, 28, 101, 102, 103\ HLTAU & 0.16 & 44 & 0.3 & 1.4$\times10^{-6}$ & 5, 19, 1, 29, 23, 7, 119\ L1228 & 0.20 & 7 & 0.15 & 5.4$\times10^{-6}$ & 53, 94, 198\ PVCEP & 0.5 & 80 & 0.2 & 1.0$\times10^{-4}$ & 12, 13, 19, 27\ L1641N & 0.42 & 170 & 0.6 & 9.0$\times10^{-4}$ & 43, 37, 44, 142\ FIRSSE101 & 0.45 & 123 & 1.1 & 2.0$\times10^{-3}$ & 37, 40, 41\ HH7-11 VLA3 & 0.35 & 150 & 0.8 & 1.1$\times10^{-3}$ & 1, 2, 23, 69\ RE50 & 0.46 & 295 & 0.84 & 3.0$\times10^{-4}$ & 42, 37, 40, 23, 41, 116\ SERPENS & 0.415 & 98 & 2.0 & 5.3$\times10^{-4}$ & 110, 111, 112, 156, 92, 93\ IRAS22198 & 1.3 & 1240 & 0.57 & 2.3$\times10^{-4}$ & 88, 89\ NGC2071-IRS3 & 0.39 & 520 & 2.9 & 1.5$\times10^{-2}$ & 54, 90, 91, 136\ HH34 & 0.42 & 15 & 0.160 & 3.0$\times10^{-6}$ & 59, 94, 95\ AF5142 CM1 & 2.14 & 10000 & 1.3 & 2.4$\times10^{-3}$ & 96, 108\ AF5142 CM2 & 2.14 & 1000 & 0.35 & 2.4$\times10^{-3}$ & 96, 108\ CepAHW2 & 0.725 & 10000 & 6.9 & 5.4$\times10^{-3}$ & 104, 105, 106, 107\ IRAS20126 & 1.7 & 13000 & 0.2 & 6.0$\times10^{-3}$ & 152, 153, 199\ HH80-81 & 1.7 & 20000 & 5.0 & 1.0$\times10^{-3}$ & 98, 113, 114, 115\ V645Cyg & 3.5 & 40000 & 0.6 & 7.0$\times10^{-4}$ & 122, 129, 137\ IRAS16547 & 2.9 & 60000 & 8.7 & 4.0$\times10^{-1}$ & 149, 150, 151\ IRAS18566 B & 6.7 & 80000 & 0.077 & 7.2$\times10^{-3}$ & 99, 100, 167\ IRAS04579 & 2.5 & 3910 & 0.15 & 9.0$\times10^{-4}$ & 88, 148, 194\ GGD14-VLA7 & 0.9 & 1000 & 0.18 & 1.6$\times10^{-3}$ & 55, 65, 66, 67, 68, 97\ [lccccl]{} & [$d$]{} & [$L_{\rm bol}$]{} & [$S_\nu$]{} & [$\dot P$]{} & [Refs.]{}\ & [(kpc)]{} & [($L_\odot$)]{} & [(mJy)]{} & [($M_\odot\,{\rm yr^{-1}\,km~s^{-1}}$)]{}\ IRAS23139 & 4.8 & 20000 & 0.53 & 8.0$\times10^{-4}$ & 123, 124, 125\ N7538-IRS9 & 2.8 & 40000 & 3.8 & 2.1$\times10^{-2}$ & 88, 126, 127\ N7538-IRS9 A1 & 2.8 & 40000 & 1.4 & 3.0$\times10^{-2}$ & 88, 126, 127\ IRAS16562 & 1.6 & 70000 & 21.1 & 3.0$\times10^{-2}$ & 131, 133\ I18264-1152 F & 3.5 & 10000 & 0.28 & 1.4$\times10^{-2}$ & 99, 100, 123\ G31.41 & 7.9 & 200000 & 0.32 & 6.0$\times10^{-2}$ & 134, 135, 143, 147\ AF2591-VLA3 & 3.3 & 230000 & 1.52 & 7.7$\times10^{-3}$ & 121, 172, 195\ I18182-1433b & 4.5 & 20000 & 0.6 & 2.8$\times10^{-3}$ & 99, 196\ I18089-1732(1)a & 3.6 & 32000 & 1.1 & $\cdots$ & 196\ G28S-JVLA1 & 4.8 & 100 & 0.02 & $\cdots$ & 197\ G28N-JVLA2N & 4.8 & 1000 & 0.06 & $\cdots$ & 197\ G28N-JVLA2S & 4.8 & 1000 & 0.03 & $\cdots$ & 197\ L1287 & 0.85 & 1000 & 0.5 & 3.3$\times10^{-4}$ & 192, 193, 116, 118\ DG Tau B & 0.15 & 0.9 & 0.31 & 6.0$\times10^{-6}$ & 60, 61, 157, 158\ Z CMa & 1.15 & 3000 & 1.74 & 1.0$\times10^{-4}$ & 122, 159, 160\ W3IRS5(d) & 1.83 & 200000 & 1.5 & 3.0$\times10^{-2}$ & 128, 161, 162\ YLW 16A & 0.16 & 13 & 0.78 & 3.4$\times10^{-6}$ & 137, 163\ YLW 15 VLA1 & 0.12 & 1 & 1.5 & 1.1$\times10^{-5}$ & 138, 164, 165, 166\ L778 & 0.25 & 0.93 & 0.69 & 1.9$\times10^{-6}$ & 4, 137, 168\ W75N(B)VLA1 & 1.3 & 19000 & 4.0 & 1.2$\times10^{-2}$ & 139, 169, 170, 171, 172, 154\ NGC1333 VLA2 & 0.235 & 1.5 & 2.5 & 4.9$\times10^{-4}$ & 140, 141, 173, 174, 175\ NGC1333 IRAS4A1 & 0.235 & 8 & 0.32 & 1.5$\times10^{-4}$ & 140, 173, 176, 177, 178\ NGC1333 IRAS4B & 0.235 & 1.1 & 0.33 & 4.1$\times10^{-5}$ & 140, 173, 177, 178\ L1551 NE-A & 0.14 & 4.0 & 0.39 & 1.5$\times10^{-5}$ & 140, 179, 180, 181, 61, 62\ Haro 6-10 VLA1 & 0.14 & 0.5 & 1.1 & 9.4$\times10^{-6}$ & 146, 12, 182, 183, 184\ L1527 VLA1 & 0.14 & 1.9 & 1.1 & 1.6$\times10^{-4}$ & 144, 145, 146, 179, 184\ OMC 2/3 VLA4 & 0.414 & 40 & 0.83 & 5.2$\times10^{-4}$ & 146, 185\ HH1-2 VLA1 & 0.414 & 23 & 1.2 & 4.6$\times10^{-5}$ & 186, 187, 188, 109\ HH1-2 VLA3 & 0.414 & 84 & 0.40 & 5.7$\times10^{-4}$ & 186, 187, 188\ OMC2 VLA11 & 0.414 & 360 & 2.16 & 3.2$\times10^{-3}$ & 189, 190, 191\ HH46/47 & 0.45 & 12 & 1.8 & 2.0$\times10^{-4}$ & 23, 56, 57, 117, 120\ IRAS20050 & 0.7 & 260 & 1.4 & 5.0$\times10^{-3}$ & 63, 64\ \ 0.2cm References: [ (1) Edwards & Snell 1984; (2) Bachiller & Cernicharo 1990; (3) Rodríguez et al. 1989b; (4) Myers et al. 1988; (5) Calvet et al. 1983; (6) Cohen et al. 1982; (7) Brown et al. 1985; (8) Wootten & Loren 1987; (9) Estalella et al. 1991; (10) Goldsmith et al. 1984; (11) Anglada et al. 1991; (12) Anglada et al. 1992 (13) Levreault 1984; (14) Bachiller et al. 1990; (15) Curiel et al. 1990; (16) André et al. 1990; (17) Margulis et al. 1988; (18) Gómez et al. 1994; (19) Levreault 1988; (20) Reipurth & Olberg 1991; (21) Rodríguez & Reipurth 1994; (22) Cabrit, Goldsmith, & Snell 1988; (23) Reipurth et al. 1993; (24) Ladd et al. 1991; (25) Schwartz et al. 1986; (26) Edwards & Snell 1982; (27) Evans et al. 1986; (28) Mozurkewich et al. 1986; (29) Torrelles et al. 1987; (30) Mundy et al. 1986; (31) Mizuno et al. 1990; (32) Walker et al. 1988; (33) Avery et al. 1990; (34) Bachiller et al. 1991; (35) André et al. 1993; (36) Ward-Thompson et al. 1995; (37) Morgan et al. 1991; (38) Morgan & Bally 1991; (39) Cohen & Schwartz 1987; (40) Fukui 1989; (41) Morgan et al. 1990; (42) Reipurth & Bally 1986; (43) Fukui et al. 1988; (44) Chen, H. et al. 1995; (45) Beltrán et al. 2001; (46) Terebey et al. 1989; (47) Parker et al. 1988; (48) Sato et al. 1994; (49) Kun & Prusti 1993; (50) Yun & Clemens 1994; (51) Parker 1991; (52) Cohen & Bieging 1986; (53) Haikala & Laurenjis 1989; (54) Snell et al. 1984; (55) Rodríguez et al. 1982; (56) Chernin & Masson 1991; (57) S. Curiel, private com.; (58) Moriarty-Schieven & Snell 1989; (59) Antoniucci et al. 2008; (60) Mitchell et al. 1994; (61) Rodríguez et al. 1995; (62) Moriarty-Schieven et al. 1995; (63) Bachiller et al. 1995; (64) Wilking et al. 1989; (65) Little et al. 1990; (66) Gómez et al. 2000; (67) Gómez et al. 2002; (68) Harvey et al. 1985; (69) Rodríguez et al. 1997; (70) Dunham et al. 2008; (71) Choi et al. 2014; (72) André et al. 1999; ]{} -- -- -- -- -- -- -- -- -- -- -- -- : (Continued) \[tab:cmnew3\] \(73) Bourke et al. 2005; (74) Shirley et al. 2007; (75) Maheswar et al. 2011; (76) Kauffmann et al. 2011; (77) AMI Consortium et al. 2011a; (78) Takahashi et al. 2013; (79) Palau et al. 2014; (80) Rodríguez et al. 2014b; (81) Hirota et al. 2008; (82) Hirota et al. 2011; (83) Palau et al. 2012; (84) Morata et al. 2015; (85) Cotera et al. 2001; (86) Pety et al. 2006; (87) G. Anglada et al., in prep.; (88) Sánchez-Monge et al. 2008; (89) Zhang et al. 2005; (90) Carrasco-González et al. 2012a; (91) Stojimirović et al. 2008; (92) Kristensen et al. 2012; (93) van Kempen et al. 2016; (94) Rodríguez & Reipurth 1996; (95) Chernin & Masson 1995,; (96) Zhang et al. 2007; (97) Dzib et al. 2016; (98) Benedettini et al. 2004; (99) Beuther et al. 2002; (100) Rosero et al. 2016; (101) Rodríguez et al. 2003b; (102) Bieging & Cohen 1985; (103) Rodríguez et al. 1986; (104) Hughes 1988; (105) Rodríguez et al. 1994b; (106) Gómez et al. 1999; (107) Curiel et al. 2006; (108) Hunter et al. 1995; (109) Rodríguez et al. 1990; (110) Rodríguez et al. 1989a; (111) Curiel et al. 1993; (112) Curiel 1995; (113) Rodríguez & Reipurth 1989; (114) Martí et al. 1993; (115) Martí et al. 1995; (116) Anglada 1995; (117) Arce et al. 2013; (118) Anglada et al. 1994; (119) Rodríguez et al. 1994a; (120) Zhang et al. 2016; (121) Hasegawa & Mitchell 1995; (122) Skinner et al. 1993; (123) Sridharan et al. 2002; (124) Wouterloot et al. 1989; (125) Trinidad et al. 2006; (126) Tamura et al. 1991; (127) Sandell et al. 2005; (128) Claussen et al. 1994; (129) Girart et al. 1996a; (130) Girart et al. 1996b; (131) Guzmán et al. 2010; (132) Anglada et al. 1996; (133) Guzmán et al. 2011; (134) Osorio et al. 2009; (135) Cesaroni et al. 2010; (136) Torrelles et al. 1998; (137) Girart et al. 2002; (138) Girart et al. 2000; (139) Torrelles et al. 1997; (140) Reipurth et al. 2002; (141) Rodríguez et al. 1999; (142) Anglada et al. 1998; (143) Cesaroni et al. 2011; (144) Rodríguez & Reipurth 1998; (145) Loinard et al. 2002; (146) Reipurth et al. 2004; (147) Mayen-Gijon 2015; (148) Molinari et al. 1996; (149) Garay et al.  2003; (150) Brooks et al. 2003; (151) Rodríguez et al. 2005; (152) Hofner et al. 1999; (153) Trinidad et al. 2005; (154) Torrelles et al. 2003; (155) Carrasco-González et al. 2008b; (156) Rodríguez-Kamenetzky et al. 2016; (157) Rodr[í]{}guez et al. 2012b; (158) Zapata et al. 2015; (159) Evans et al. 1994; (160) Vel[á]{}zquez & Rodr[í]{}guez 2001; (161) Imai et al. 2000; (162) Hasegawa et al. 1994; (163) Sekimoto et al. 1997; (164) Girart et al. 2004; (165) van Kempen et al. 2009; (166) Bontemps et al. 1996; (167) Hofner et al. 2017; (168) Rodr[í]{}guez et al. 1989b; (169) Hunter et al. 1994; (170) Shepherd et al. 2003; (171) Carrasco-Gonz[á]{}lez et al. 2010a; (172) Rygl et al. 2012; (173) Plunkett et al. 2013; (174) Sadavoy et al. 2014; (175) Downes & Cabrit 2007; (176) Dunham et al. 2014; (177) Ching et al. 2016; (178) Y[i]{}ld[i]{}z et al. 2012; (179) Froebrich 2005; (180) Lim et al. 2016; (181) Takakuwa et al. 2017; (182) Doppmann et al. 2008; (183) Roccatagliata et al. 2011; (184) Hogerheijde et al. 1998; (185) Yu et al. 2000; (186) Fischer et al. 2010; (187) Rodr[í]{}guez et al. 2000; (188) Correia et al. 1997; (189) Furlan et al. 2016; (190) Osorio et al. 2017; (191) Takahashi et al. 2008; (192) Yang et al. 1991; (193) Wu et al. 2010; (194) Xu et al. 2012; (195) Johnston et al. 2013; (196) Zapata et al. 2006; (197) C. Carrasco-González et al., in prep; (198) Skinner et al. 2014; (199) Shepherd et al. 2000. In Figure \[fig:cor1\] we plot the observed cm luminosity ($S_\nu d^2$) as a function of the bolometric luminosity for the sources listed in Table \[tab:cm\] (squares, dots, and triangles correspond respectively to very low, low, and high luminosity objects). For most of the sources in this plot, $S_\nu$ is the flux density at 3.6 cm, but some data at 6, 2, and 1.3 cm are included to construct a larger sample; these data points are used to estimate the 3.6 cm luminosity using the spectral index, when known, or assuming a value of $\sim$0.5. As can be seen in Figure \[fig:cor1\], the observed cm luminosity (data points) is uncorrelated with the cm luminosity expected from photoionization (dot-dashed line), further indicating that this is not the ionizing mechanism. However, as the figure shows, the observed cm luminosity is indeed correlated with the bolometric luminosity ($L_{\rm bol}$). A fit to the 48 data points with $1~L_\odot {\lesssim}L_{\rm bol} {\lesssim}1000~L_\odot$ (dots) gives: $${\biggl(\frac{S_\nu d^2}{\rm mJy~kpc^2}\biggr)} = 10^{-1.93\pm0.14} \biggl(\frac{L_{\rm bol}}{L_\odot}\biggr)^{0.54\pm0.08}.$$ As can be seen in the figure, the correlation also holds for both the most luminous (triangles) and the very low luminosity (squares) objects, suggesting that the mechanism that relates the bolometric and cm luminosities is shared by all the YSOs. A fit to all the 81 data points ($10^{-2}~L_\odot {\lesssim}L_{\rm bol} {\lesssim}10^6~L_\odot$) gives a better fit, with a similar result, $$\label{eq:lbol} {\biggl(\frac{S_\nu d^2}{\rm mJy~kpc^2}\biggr)} = 10^{-1.90\pm0.07} \biggl(\frac{L_{\rm bol}}{L_\odot}\biggr)^{0.59\pm0.03}.$$ A correlation between bolometric and cm luminosities was noted by Cabrit & Bertout (1992) from a set of $\sim$25 outflow sources, quoting a slope of $\sim$0.8 in a log-log plot. Skinner et al. (1993) suggested a correlation between the 3.6 cm luminosity and the bolometric luminosity with a slope of 0.9 by fitting a sample of Herbig Ae/Be stars and candidates with 11 detections and 7 upper limits. The fit to the 29 outflow sources presented in Anglada (1995) gives a slope of 0.6 (after updating some distances and flux densities; a slope of 0.7 was obtained with the values originally listed in Table 1 of that paper), similar to the value given in equation (\[eq:lbol\]). Shirley et al. (2007) obtained separate fits for the 3.6 cm and for the 6 cm data, obtaining slopes of 0.7 and 0.9, respectively (although these fits are probably affected by a few outliers with anomalously high values of the flux density, taken from the compilation of Furuya et al. 2003). Because radio jets have positive spectral indices with typical values around $\sim0.4$, it is expected that the 3.6 cm flux densities are $\sim$20% higher than the 6 cm flux densities, but the slopes of the luminosity correlations are expected to be similar, provided the scattering in the values of the spectral index is small. L. Tychoniec et al. (in prep.) analyze a large homogeneous sample of low-mass protostars in Perseus, obtaining similar slopes of $\sim$0.7 for data at 4.1 and 6.4 cm. These authors note, however, that the correlations are weak for these sources that cover a relatively small range of luminosities. These results suggest that there is a general trend over a wide range of luminosities, but with an intrinsic dispersion. Recently, Moscadelli et al. (2016) derived a slope of 0.5 from a small sample (8 sources) of high luminosity objects with outflows traced by masers. Also, recent surveys targeted towards high-mass protostellar candidates (but without a confirmed association with an outflow) also show radio continuum to bolometric luminosity correlations with similar slopes of $\sim$0.7 (Purser et al. 2016). Recently, Tanaka et al. (2016) modeled the evolution of a massive protostar and its associated jet as it is being photoionized by the protostar, making predictions for the free-free continuum and RRL emissions. These authors find global properties of the continuum emission similar to those of radio jets. The radio continuum luminosities of the photoionized outflows predicted by the models are somewhat higher than those obtained from the empirical correlations for jets but much lower than those expected for optically thin HII regions. When including an estimate of the ionization of the ambient clump by photons that escape along the outflow cavity the predicted properties get closer to those of the observed UC/HC HII regions. Photoevaporation has not been included. This kind of models, including additional effects such as the photoevaporation, are a promising tool to investigate the transition from jets to HII regions in massive protostars. Purser et al. (2016) found a number of objects with radio luminosities intermediate between optically thin HII regions and radio jets that these authors interpret as optically thick HII regions. We note that this kind of objects could be in an evolutionary stage intermediate between the jet and the HII region phases, and their properties could be predicted by models similar to those of Tanaka et al. (2016). From a sample of outflow sources of low and very low bolometric luminosity, but using low angular resolution data ($\sim30''$) at 1.8 cm, a correlation between the cm luminosity and the internal luminosity[^1] with a slope of either 0.5 or 0.6 was found (AMI Consortium et al. 2011a, b), depending, respectively, on the use of either the bolometric luminosity or the IR luminosity as an estimate of the internal luminosity. Morata et al. (2015) report cm emission from four proposed proto-brown dwarf (proto-BD) candidates, that appear to follow the general trend of the luminosity correlation but showing some excess of radio emission. Further observations are required to confirm the nature of these objects as proto-BDs, to better determine their properties such as distance and intrinsic luminosity, as well as their radio jet morphology. If confirmed as bona fide proto-BDs that follow the correlation, this would suggest that the same mechanisms are at work for YSOs and proto-BDs, supporting the idea that the intrinsic properties of proto-BDs are a continuation to smaller masses of the properties of low-mass YSOs. It is interesting that the only two *bona fide  young brown dwarfs detected in the radio continuum fall well in this correlation (Rodrí guez et al. 2017).* Recently, it has been realized that young stars in more advanced stages, such as those surrounded by transitional disks[^2] are also associated with radio jets that have become detectable with the improved sensitivity of the JVLA (Rodríguez et al. 2014a, Macías et al. 2016). In these objects, accretion is very low but high enough to produce outflow activity detectable through the associated radio emission at a level of $\lesssim$0.1 mJy. These radio jets follow the general trend of the luminosity correlation but appear to be radio underluminous with respect to the correlation that was established from data corresponding to younger objects. This fact has been interpreted as indicating that it is the accretion component of the luminosity that is correlated with the outflow (and, thus, with the radio flux of the jet). Accretion luminosity is dominant in the youngest objects, from which the correlation was derived, while in more evolved objects the stellar contribution (which is not expected to be correlated with the radio emission) to the total luminosity becomes more important. Thus, it is expected that in more evolved objects the observed radio luminosity is correlated with only a fraction of the bolometric luminosity. In summary, the radio luminosity ($S_\nu d^2$) of thermal jets associated with very young stellar objects is correlated with their bolometric luminosity ($L_{\rm bol}$). This correlation is valid for objects of a wide range of luminosities, from high to very low luminosity objects, and likely even for proto-BDs. This suggests that accretion and outflow processes work in a similar way for objects of a wide range of masses and luminosities. Accretion appears to be correlated with outflow (which is traced by the radio luminosity). As the young star evolves, accretion decreases and so does the radio luminosity. However, for these more evolved objects the accretion luminosity represents a smaller fraction of the bolometric luminosity (the luminosity of the star becomes more important) and they become radio underluminous with respect to the empirical correlation, which was derived for younger objects. Observational Properties: Radio Luminosity to Outflow Momentum Rate Correlation \[sect:pp\] ------------------------------------------------------------------------------------------- The radio luminosity of thermal radio jets is also correlated with the properties of the associated outflows. A correlation between the momentum rate (force) in the outflow, $\dot P$, derived from observations, and the observed radio continuum luminosity at centimeter wavelengths, $S_{\nu}d^2$, was first noted by Anglada et al. (1992) and by Cabrit & Bertout (1992). Anglada et al. (1992) considered a sample of 16 sources of low bolometric luminosity (to avoid a contribution from photoionization) and found a correlation $(S_\nu d^2/{\rm mJy~kpc^2}) = 10^{2.4\pm1.0}\,(\dot P/M_{{\odot}}~{\rm yr^{-1}})^{0.9\pm0.3}$. Since the outflow momentum rate estimates have considerable uncertainties (more than one order of magnitude, typically), $\dot P$ was fitted (in the log-log space) taking $S_{\nu}d^2$ as the independent variable, as it was considered to be less affected by observational uncertainties. The correlation was confirmed with fits to larger samples (Anglada 1995, 1996; Anglada et al. 1998; Shirley et al. 2007; AMI Consortium et al.  2011a, b, 2012). The best fit to the low luminosity sources ($1 {\lesssim}L_{\rm bol} {\lesssim}1000~L_\odot$) presented in Table \[tab:cm\] gives (see also Fig. \[fig:cor2\]): $${\biggl(\frac{S_\nu d^2}{\rm mJy~kpc^2}\biggr)} = 10^{2.22\pm0.46} \biggl(\frac{\dot P}{M_\odot~{\rm yr^{-1}~km~s^{-1}}}\biggr)^{0.89\pm0.16}. \label{eq:dotp1}$$ ![Empirical correlation between the outflow momentum rate and the radio continuum luminosity at cm wavelengths. Data are taken from Table \[tab:cm\]. Triangles correspond to high luminosity objects ($L_{\rm bol} > 1000~L_\odot$), dots correspond to low luminosity objects ($1 \le L_{\rm bol} \le 1000~L_\odot$), and squares to very low luminosity objects ($L_{\rm bol} < 1~L_\odot$). The dashed line is a least-squares fit to all the data points and the grey area indicates the residual standard deviation of the fit. The solid line is a fit to the low-luminosity objects alone. The dot-dashed line corresponds to the radio luminosity predicted by the models of Curiel et al. (1987, 1989). []{data-label="fig:cor2"}](fig9.pdf){width="95.00000%"} This correlation between the outflow momentum rate and the radio luminosity has been interpreted as evidence that shocks are the ionizing mechanism of jets. Curiel et al. (1987, 1989) modeled the scenario in which a neutral stellar wind is ionized as a result of a shock against the surrounding high-density material, assuming a plane-parallel shock. From the results obtained in this model, ignoring further details about the radiative transfer and geometry of the emitting region, and assuming that the free-free emission is optically thin, a relationship between the momentum rate in the outflow and the centimeter luminosity can be obtained (see Anglada 1996, Anglada et al. 1998): $$\left(\frac{S_{\nu}d^2}{{\rm mJy~kpc^2}}\right) = 10^{3.5}\,\eta \left(\frac{\dot{P}}{M_{{\odot}}~{\rm yr^{-1}}~\mbox{{km~s$^{-1}$}}} \right), \label{eq:curiel}$$ where $\eta = \Omega/{4\pi}$ is an efficiency factor that can be taken to equal the fraction of the stellar wind that is shocked and produces the observed radio continuum emission. Despite the limitations of the model, and the simplicity of the assumptions used to derive equation (\[eq:curiel\]), its predictions agree quite well with the results obtained from a large number of observations (equation \[eq:dotp1\]), for an efficiency $\eta \simeq 0.1$. González & Cantó (2002) present a model in which the ionization is produced by internal shocks in a wind, resulting of periodic variations of the velocity of the wind at injection. Cabrit & Bertout (1992) and Rodríguez et al. (2008) noted that high-mass protostars driving molecular outflows appear to follow the same radio luminosity to outflow momentum rate correlation as the sources of low luminosity. As can be seen in Figure \[fig:cor2\], high luminosity objects fall close to the fit determined by the low luminosity objects. Actually, a fit including all the sources in the sample of Table \[tab:cm\], including both low and high luminosity objects, gives a quite similar result, $${\biggl(\frac{S_\nu d^2}{\rm mJy~kpc^2}\biggr)} = 10^{2.97\pm0.27} \biggl(\frac{\dot P}{M_\odot~{\rm yr^{-1}~km~s^{-1}}}\biggr)^{1.02\pm0.08}, \label{eq:dotp}$$ as it was also the case for the bolometric luminosity correlation (Fig \[fig:cor1\]). Thus, radio jets in massive protostars appear to be ionized by a mechanism similar to that acting in low luminosity objects. Radio jets would represent a stage in massive star formation previous to the onset of photoionization and the development of an HII region. Actually, the correlations can be used as a diagnostic tool to discriminate between photoionized (HII regions) versus shock-ionized (jets) sources (see Tanaka et al. 2016 and the previous discussion in Sect. \[sect:lbol\]). As in the radio to bolometric luminosity correlation, the very low luminosity objects also fall in the outflow momentum rate correlation. These results are interpreted as indicating that the mechanisms for accretion, ejection and ionization of outflows are very similar for all kind of YSOs, from very low to high luminosity protostars. A direct measure of the outflow momentum rate is difficult for objects in the last stages of the star formation process, when accretion has decreased to very small values and outflows are hard to detect. For these objects, the weak radio continuum emission can be used as a tracer of the outflow. Recent results obtained for sources associated with transitional disks (Rodríguez et al. 2014a, Macías et al. 2016) indicate that the observed radio luminosities are consistent with the outflow momentum rate to radio luminosity correlation being valid and the ratio between accretion, and outflow being similar in these low accretion objects than in younger protostars. On the Origin of the Correlations \[sect:origin\] ------------------------------------------------- As has been shown above, the observable properties of the cm continuum outflow sources indicate that these sources trace thermal free-free emission from ionized collimated outflows (jets). Both theoretical and observational results suggest that the ionization in thermal jets is only partial ($\sim$1-10%; Rodríguez et al. 1990, Hartigan et al.  1994, Bacciotti et al. 1995). The mechanism that is able to produce the required ionization, even at these relatively low levels, is still not fully understood. As photoionization cannot account for the observed radio continuum emission of low luminosity objects (see above), shock ionization has been proposed as a viable alternative mechanism (Curiel et al. 1987; González & Cantó 2002). The correlations described in Sections \[sect:lbol\] and \[sect:pp\] are related to the well-known correlation between the momentum rate observed in molecular outflows and the bolometric luminosity of the driving sources, first noted by Rodríguez et al. (1982). More recent determinations of this correlation give $(\dot P/M_{{\odot}}~{\rm yr^{-1}~km~s^{-1}})$ = $10^{-4.36\pm0.12}$ $(L_{\rm bol}/L_\odot)^{0.69\pm0.05}$ (Cabrit & Bertout 1992) or $(\dot P/M_{{\odot}}~{\rm yr^{-1}~km~s^{-1}})$ = $10^{-4.24\pm0.32}$ $(L_{\rm bol}/L_\odot)^{0.67\pm0.13}$ (Maud et al. 2015, for a sample of massive protostars). The empirical correlations with cm emission described by equations \[eq:lbol\] and \[eq:dotp\] result in an expected correlation $(\dot P/M_{{\odot}}~{\rm yr^{-1}~km~s^{-1}})$ = $10^{-4.77\pm0.14}$ $(L_{\rm bol}/L_\odot)^{0.58\pm0.08}$, which is in agreement with the results obtained directly from the observations. We interpreted the correlation of the outflow momentum rate with the radio luminosity (eq. \[eq:dotp\]) as a consequence of the shock ionization mechanism working in radio jets, and the correlation of the bolometric and radio luminosities (eq. \[eq:lbol\]) as a consequence of the accretion and outflow relationship. In this context, the well-known correlation between the momentum rate of molecular outflows and the bolometric luminosity of their driving sources can be interpreted as a natural consequence of the other two correlations. Additional topics ================= Fossil outflows --------------- In the case of regions of massive star formation there are also clear examples of jets that show the morphology and spectral index characteristic of this type of sources. Furthermore, these massive jets fall in the correlations previously discussed. However, there is a significant number of molecular outflows in regions of massive star formation where it has not been possible to detect the jet. Instead, ultracompact HII regions are found near the center of these outflows (e.g., G5.89-0.39, Zijlstra et al. 1990; G25.65+1.05 and G240.31+0.07, Shepherd & Churchwell 1996; G45.12+0.13 and G45.07+0.13, Hunter et al. 1997; G192.16$-$3.82, Devine et al. 1999; G213.880$-$11.837, Qin et al. 2008; G10.6-0.4, Liu et al. 2010; G24.78+0.08, Codella et al. 2013; G35.58$-$0.03, Zhang et al. 2014). We can think of two explanations for this result. One is that the central source has evolved and the jet has been replaced by an ultracompact HII region. From momentum conservation the outflow will continue coasting for a large period of time, becoming a fossil outflow in the sense that it now lacks an exciting source of energy. The alternative explanation is that a centimeter jet is present in the region but that the much brighter HII region makes it difficult to detect it. This is a problem that requires further research. It should also be noted that in two of the best studied cases, G25.65+1.05 and G240.31+0.07, high angular resolution radio observations (Kurtz et al. 1994; Chen et al. 2007; Trinidad et al. 2011) have revealed fainter sources in the region that could be the true energizing sources of the molecular outflows. If this is the case, the outflows cannot be considered as fossil since they would have an associated active jet. Jets or ionized disks? ---------------------- The presence in star forming regions of an elongated centimeter source is usually interpreted as indicating the presence of a thermal jet. This interpretation is typically confirmed by showing that the outflow traced at larger scales by molecular outflows and/or optical/IR HH objects aligns with the small scale radio jet. In the sources where the true dust disk is detected and resolved, it is found to align perpendicular to the outflow axis. However, in a few massive objects there is evidence that the elongated centimeter source actually traces a photoionized disk (S106IR: Hoare et al. 1994; S140-IRS: Hoare 2006; Orion Source I: Reid et al. 2007; see Fig. \[fig:iondisk\]). These objects show a similar centimeter spectral index to that of jets and one cannot discriminate using this criterion. There is also the case of NGC 7538 IRS1, a source that has been interpreted as an ionized jet (Sandell et al. 2009) or modeled as a photoionized accretion disk (Lugo et al. 2004), although it is usually referred to as an ultracompact HII region (Zhu et al. 2013). ![The radio sources associated with S140 IRS1 (left) and with source I in Orion (right). These sources of radio continuum emission are believed to trace photoionized disks and not jets. Image reproduced with permission from \[left\] Hoare (2006), copyright by AAS; and \[right\] based on Reid et al. (2007). \[fig:iondisk\] ](fig10a.pdf "fig:"){width="48.00000%"} ![The radio sources associated with S140 IRS1 (left) and with source I in Orion (right). These sources of radio continuum emission are believed to trace photoionized disks and not jets. Image reproduced with permission from \[left\] Hoare (2006), copyright by AAS; and \[right\] based on Reid et al. (2007). \[fig:iondisk\] ](fig10b.pdf "fig:"){width="49.00000%"} A possible way to favor one of the interpretations is to locate the object in a radio luminosity versus bolometric luminosity diagram as has been discussed above. It is expected that a photoionized disk will fall in between the regions of thermal jets and UC HII regions in such a diagram. Another test to discriminate between thermal jets and photoionized disks would be to eventually detect radio recombination lines from the source. In the case of photoionized disks lines with widths of tens of km s$^{-1}$ are expected, while in the case of thermal jets the lines could exhibit widths an order of magnitude larger. It should be noted, however, that in the case of very collimated jets the velocity dispersion and, thus, the observed line widths, will be narrow. The possibility of an ionized disk is also present in the case of low mass protostars. The high resolution images of GM Aur presented by Macías et al. (2016) show that, after subtracting the expected dust emission from the disk, the centimeter emission from this source is composed of an ionized radio jet and a photoevaporative wind arising from the disk perpendicular to the jet (see Fig. \[fig:gmaur\]). It is believed that extreme-UV (EUV) radiation from the star is the main ionizing mechanism of the disk surface. Dust emission at cm wavelengths is supposed to arise mainly from grains that have grown up to reach pebble sizes (e.g., Guidi et al. 2016). ![Decomposition of the emission of GM Aur at cm wavelengths. The free-free emission at 3 cm of the radio jet is shown in white contours and that of the photoevaporative wind from the disk is shown in black contours. The dust emission from the disk at 7 mm is shown in color scale. The free-free emission of the two ionized components was separated by fitting two Gaussians to the 3 cm image after subtraction of the contribution at 3 cm of the dust of the disk, estimated by scaling the 7 mm image with the dust spectral index obtained from a fit to the spectral energy distribution. Image reproduced with permission from Macías et al. (2016), copyright by AAS. \[fig:gmaur\] ](fig11.pdf){width="90.00000%"} Conclusions =========== The study of jets associated with young stars has contributed in an important manner to our understanding of the process of star formation. We list below the main conclusions that arise from these studies. 1\. Free-free radio jets are typically found in association with the forming stars that can also power optical or molecular large-scale outflows. While the jets trace the outflow over the last few years, the optical and molecular outflows integrate in time over centuries or even millenia. 2\. The radio jets provide a means to determine accurately the position and proper motions of the stellar system in regions of extremely high obscuration. 3\. The core of these jets emits as a partially optically thick free-free source. However, in knots along the jet (notably in the more massive protostars) optically thin synchrotron emission could be present. Studies of this non-thermal emission will provide important information on the role of magnetic fields in these jets. 4\. At present there are only tentative detections of radio recombination lines from the jets. Future instruments such as SKA and the ngVLA will allow a new avenue of research using this observational tool. 5\. The radio luminosity of the jets is well correlated both with the bolometric luminosity and the outflow momentum rate of the optical or molecular outflow. This is a result that can be understood theoretically for sources that derive most of its luminosity from accretion and where the ionization of the jet is due to shocks with the ambient medium. These correlations extend from massive young stars to the sub-stellar domain, suggesting a common formation mechanism for all stars. GA acknowledges support from MINECO (Spain) AYA2014-57369-C3-3-P and AYA2017-84390-C2-1-R grants (co-funded by FEDER). LFR acknowledges support from CONACyT, Mexico and DGAPA, UNAM. CC-G acknowledges support from UNAM-DGAPA-PAPIIT grant numbers IA102816 and IN108218. We thank an anonymous referee for his/her useful comments and suggestions. [^1]: The internal luminosity is the luminosity in excess of that supplied by the interstellar radiation field. [^2]: Transitional disks are accretion disks with central cavities or gaps in the dust distribution that are attributed to disk clearing by still forming planets.

--- abstract: 'Given a type $A$ in homotopy type theory (HoTT), we can define the *free $\infty$-group on $A$* as the loop space of the suspension of $A + 1$. Equivalently, this free higher group can be defined as a higher inductive type ${\mathbf{F}(A)}$ with constructors ${\mathsf{unit}}: {\mathbf{F}(A)}$, ${\mathsf{cons}}: A \to {\mathbf{F}(A)} \to {\mathbf{F}(A)}$, and conditions saying that every ${\mathsf{cons}}(a)$ is an auto-equivalence on ${\mathbf{F}(A)}$. Assuming that $A$ is a set (i.e. satisfies the principle of unique identity proofs), we are interested in the question whether ${\mathbf{F}(A)}$ is a set as well, which is very much related to an open problem in the HoTT book [@hott-book Ex. 8.2]. We show an approximation to the question, namely that the fundamental groups of ${\mathbf{F}(A)}$ are trivial, i.e. that ${\mathopen{}\left\Vert {\mathbf{F}(A)}\right\Vert_{1}\mathclose{}}$ is a set.' author: - Nicolai Kraus - Thorsten Altenkirch bibliography: - 'master.bib' title: Free Higher Groups in Homotopy Type Theory --- [^1] Introduction ============ An important feature of *Martin-Löf type theory* (MLTT) is the identity type which makes it possible to express equality inside type theory. More precisely, if $A$ is a type (in any context), and $x,y : A$ are terms, then $\mathsf{Id}_A(x,y)$ is a type whose inhabitants represent proofs that $x$ and $y$ are equal. As it is common nowadays, we write $x=_A y$ or $x=y$ instead of $\mathsf{Id}_A(x,y)$, and call its elements simply *equalities*. *Homotopy type theory* (HoTT) embraces the fact that $x=y$ may come with interesting structure. This means that, in many cases, we do not only care about the question whether $x$ and $y$ are equal, but also *how* or *in which ways* they are equal. For example, due to Voevodsky’s univalence axiom, ${\mathbf 2}=_{\mathcal{U}}{\mathbf 2}$ is equivalent to ${\mathbf 2}$. Here, ${\mathbf 2}$ is the type of booleans, ${\mathcal{U}}$ is a type universe, and the two equalities stem from the two ways in which ${\mathbf 2}$ is equivalent to itself. Another non-trivial example is ${\mathsf{base}}=_{{\mathbb{S}^{1}}} {\mathsf{base}}$ which turns out to be equivalent to the type of integers [@Licata:2013:CFG:2591370.2591407], where ${\mathbb{S}^{1}}$ is the “circle” in HoTT. A type where each such type of equalities can have at most one inhabitant is called a *set*, and further said to satisfy the principle of *unique identity proofs* (UIP). An example for a set is the type of natural numbers: $0 =_{\mathbb N}1$ is equivalent to the empty type ${\ensuremath{\mathbf{0}}}$, while $0 =_{\mathbb N}0$ is equivalent to the unit type ${\ensuremath{\mathbf{1}}}$, and so on. Many algebraic structures can be implemented straightforwardly if we are happy to do everything with sets; for example, [@hott-book Def 6.11.1] defines a set-level group to be a tuple $(G, \circ, u, \cdot^{-1}, \ldots)$, where $G$ is a set, together with a multiplication operation $\circ : G \times G \to G$, a unit element $u$, an inversion operator $G \to G$, and equalities expressing the usual laws. The book then further shows how one can construct the free set-level group over a given set (see [@hott-book Chp 6.11]). More interesting and challenging is it to define higher-level structures, not restricted to sets. Since we use HoTT (rather than, say, set theory) as our foundation, it is natural to attempt this. For example, it has been known for some time that externally, every type carries the structure of an *$\infty$-groupoid* [@lumsdaine_weakOmegaCatsFromITT; @bg:type-wkom]. Internally, we can play a trick and use the following definition, which probably can be considered “HoTT folklore”: \[def:cheating-group-def\] An *$\infty$-group* is a type $G$ which is equivalent to a loop space. More precisely, $G$ is a group means that we have a connected type $X$ and a point $x:X$ together with an equivalence $G \simeq (x=x)$. If $G$ is an $\infty$-group represented by $(X,x)$ and $H$ is a second $\infty$-group represented by $(Y,y)$, then a *homomorphism* $G \to_{\infty\mathsf{grp}} H$ is a pointed function $(X,x) \to_\bullet (Y,y)$. In other words, an *$\infty$-group* is a type that admits a *delooping*. Clearly, the unit element of this group is ${\mathsf{refl}}_x$, and composition is given by composition of equalities. Some theory of higher groups in homotopy type theory has very recently been developed by Buchholtz, van Doorn and Rijke [@buchVDrijke:highergroups]. It is worth noting that Definition \[def:cheating-group-def\] makes use of the fact that a suitable notion of an (untruncated) group already naturally exists in HoTT, which is not the case for many other interesting structures. Defining untruncated algebraic structures in general and *directly* is an important open problem in HoTT. To see why this is hard, let us start from the set-level definition $(G,\circ, u, \cdot^{-1}, \ldots)$ above and remove the condition that $G$ is a set. The equalities which guarantee that the multiplication is associative, the unit is neutral, and inverses cancel, are not sufficient anymore; they would not give a well-behaved definition of a higher group. For example, one may ask oneself how one would prove that $x \circ (y \circ (z \circ w))$ equals $((x \circ y) \circ z) \circ w$: there are two canonical ways, and these should coincide (“MacLane’s pentagon”), but any such rule that we add would have to satisfy its own coherences. It is currently unknown whether it is possible to complete this sort of definition in a satisfactory way. In a nutshell, the problem is that the usual definitions would, if expressed internally in type theory, amount to infinite structures of coherences. In classical homotopy theory, these conditions are often organised in the form of an *operad* [@sta:h-spaces; @ada:infinite-loop-spaces], but a representation of such structures that can be written down in type theory has not been discovered so far. This is certainly not for a lack of trying; cf. the much-discussed open problem of defining *semisimplicial types* [@uf:semisimplicialtypes]. In this paper, we study the *free $\infty$-group* over a type $A$. It is folklore in homotopy type theory that a suitable definition of the free $\infty$-group over $A$ is given by the loop space of a “wedge of $A$-many circles”, or put differently, the suspension ${\mathbf \Sigma}(A + {\ensuremath{\mathbf{1}}})$. For us, it will however be more helpful to give a more explicit construction of this free higher group which we call ${\mathbf{F}(A)}$. We define ${\mathbf{F}(A)}$ to be the *higher inductive type* (HIT) [@hott-book Chp 6] which as constructors has a neutral element ${\mathsf{unit}}: {\mathbf{F}(A)}$ and a multiplication operation ${\mathsf{cons}}: A \to {\mathbf{F}(A)} \to {\mathbf{F}(A)}$, together with conditions ensuring that each ${\mathsf{cons}}(a) : {\mathbf{F}(A)} \to {\mathbf{F}(A)}$ is an equivalence. This definition encodes the a priori infinite tower of coherence condition suitably and it will turn out that it is equivalent to the loop space of ${\mathbf \Sigma}(A + {\ensuremath{\mathbf{1}}})$. The most basic properties that one would expect from a free higher group are easy to prove. More intriguing is the question what the free $\infty$-group has to do with the free set-based group. Clearly, we would want the former to be a generalisation of the latter. The most obvious way of interpreting this is to ask whether, for a *set* $A$, the free higher group ${\mathbf{F}(A)}$ coincides with the construction of the set-based higher group. It turns out that the central question is the following: \[eq:main-question\] If $A$ is a set, is ${\mathbf{F}(A)}$ a set as well? One reason why we believe that Question \[eq:main-question\] is hard is that a slight generalisation of it is a known open problem in HoTT which has been recorded in the book (see [@hott-book Ex. 8.2]). To be precise, the open problem asks whether, for a set $A$, the suspension ${\mathbf \Sigma}(A)$ is a $1$-type; our question is equivalent to asking whether ${\mathbf \Sigma}(A + {\ensuremath{\mathbf{1}}})$ is a $1$-type. A positive answer to the open problem would imply a positive answer to our Question \[eq:main-question\], but we do not expect that our question is fundamentally easier. (Recall from the book [@hott-book] that the suspension ${\mathbf \Sigma}(A)$ is the HIT with constructors ${\mathsf{N}}, {\mathsf{S}}: {\mathbf \Sigma}(A)$ and ${\mathsf{mer}}: A \to {\mathsf{N}}= {\mathsf{S}}$, for “north”, “south”, “meridian”.) The core of our paper consists of a proof of a weakened version, a first approximation, of Question \[eq:main-question\]. Our main result can be phrased as follows: [theorem]{}[mainthm]{} \[thm:mainthm\] If $A$ is a set, then all fundamental groups of ${\mathbf{F}(A)}$ are trivial. In other words, ${\mathopen{}\left\Vert {\mathbf{F}(A)}\right\Vert_{1}\mathclose{}}$ is a set. Our strategy to prove this is to define a simple reduction system together with a *non-recursive approximation*, written ${\mathbf{N}(A)}$, to the free $\infty$-group. These are both based on the usual construction of the free monoid on $A$, that is, ${\mathsf{List}}(A)$. The proof of Theorem \[thm:mainthm\], with all the tools and strategies that we need to develop, constitutes the main part of the paper. The reason why our strategies are not sufficient to provide a full answer to Question \[eq:main-question\] is that ${\mathbf{N}(A)}$ is really only an approximation. Defining ${\mathbf{F}(A)}$ in a non-recursive way (i.e. without using some sort of induction that quantifies over elements of ${\mathbf{F}(A)}$ itself) seems to, as least as far as we can see, correspond exactly to expressing the infinite coherence tower “directly”. We would not be surprised if it turned out that Question \[eq:main-question\] was in fact independent of “standard HoTT” (the type theory developed in the book [@hott-book]), and if the status of Question \[eq:main-question\] was related to the status of semisimplicial types. We will come back to this in the conclusions of the paper. #### Setting {#setting .unnumbered} The type theory that we consider in this paper is the standard homotopy type theory developed in the book [@hott-book]. This means that we have *univalent universes* ${\mathcal{U}}$, *function extensionality*, and *higher inductive types* (HITs). Regarding notation, we strive to stay close to [@hott-book], with the exception that we write $(a : A) \to B(a)$ instead of $\Pi(a:A).B(a)$. All performed constructions will preserve the universe level, hence there is no risk at omitting it. Uncurrying is done implicitly, allowing us to write $f(a,b)$ instead of $f(a)(b)$. #### Outline {#outline .unnumbered} We give the precise definition of the free $\infty$-group ${\mathbf{F}(A)}$ in Section \[sec:freeHigherGroups\], together with some simple observations. The statements in this section (apart from the definition of ${\mathbf{F}(A)}$ and its universal property) are not important for the main part of the paper, the proof of Theorem \[thm:mainthm\], which is given in Section \[sec:mainsection\]. As a corollary of the constructions, we get that ${\mathopen{}\left\Vert {\mathbf{F}(A)}\right\Vert_{1}\mathclose{}}$ coincides with the set-based construction of the free group, and ${\mathbf{F}(A)}$ does under the assumption that Question \[eq:main-question\] has a positive answer. In Section \[sec:conclusions\], we make some concluding remarks and discuss related open problems in homotopy type theory. Free $\infty$-Groups {#sec:freeHigherGroups} ==================== Definition and First Properties ------------------------------- Let us start with an explicit construction of the free higher group as a higher inductive type ${\mathbf{F}(A)}$, since this is the central concept of the paper. We use a point constructor ${\mathsf{unit}}: {\mathbf{F}(A)}$ for the neutral element, and a constructor ${\mathsf{cons}}: A \to {\mathbf{F}(A)} \to {\mathbf{F}(A)}$ which “multiplies” an element of $A$ with any other group element. We write ${\mathsf{cons}}_a : {\mathbf{F}(A)} \to {\mathbf{F}(A)}$ instead of ${\mathsf{cons}}(a)$ since, most of the time, we regard $a$ as a fixed variable. The trick which completes the definition in an elegant way, due to Paolo Capriotti, is to add the condition that ${\mathsf{cons}}_a$ is an equivalence for every $a$. This cannot be done directly (at least not according to the usual intuitive rules for presentations of higher inductive types), but it can be “unfolded” and expressed via a suitable collection of constructors. Let us show the concrete definition. \[def:FA\] The *free $\infty$-group* over a given type $A$ is the following higher inductive type ${\mathbf{F}(A)}$: $$\begin{aligned} & \textbf{data} \; {\mathbf{F}(A)} \; \textbf{where} \\ & \; {\mathsf{unit}}: {\mathbf{F}(A)} \\ & \; {\mathsf{cons}}: A \to {\mathbf{F}(A)} \to {\mathbf{F}(A)} \\ & \; {\mathsf{icons}}: A \to {\mathbf{F}(A)} \to {\mathbf{F}(A)} \\ & \; \mu_1 : (a : A) \to (x : {\mathbf{F}(A)}) \to {\mathsf{icons}}_a ({\mathsf{cons}}_a (x)) = x \\ & \; \mu_2 : (a : A) \to (x : {\mathbf{F}(A)}) \to {\mathsf{cons}}_a ({\mathsf{icons}}_a (x)) = x \\ & \; \mu : (a : A) \to (x : {\mathbf{F}(A)}) \to \mathsf{ap}_{{\mathsf{cons}}(a)}(\mu_1(a,x)) = \mu_2(a,{\mathsf{icons}}_a(x)) \end{aligned}$$ At first glance, the above HIT appears complicated and rather unappealing. Due to the “unfolding”, the underlying idea that we have discussed above is somewhat hidden. The constructors ${\mathsf{unit}}$ and ${\mathsf{cons}}$ are the standard ones that one would use to define the type of lists over $A$, ${\mathsf{List}}(A)$, or, in other words, the free monoid over $A$. The remaining four constructors $({\mathsf{icons}}, \mu_1, \mu_2, \mu)$ simply say that, for every $a : A$, the function ${\mathsf{cons}}_a : {\mathbf{F}(A)} \to {\mathbf{F}(A)}$ is a *half-adjoint equivalence* as defined in [@hott-book Chp 4]. This is the “unfolding” mentioned before; note that we could equally well have used other definition of equivalences, such as the “bi-invertible” or “contractible fibre” constructions. In any case, this means that we can think of ${\mathbf{F}(A)}$ as being fully described as a triple $({\mathsf{unit}}, {\mathsf{cons}}, {\mathsf{iseq}})$, with ${\mathsf{iseq}}: (a : A) \to {\mathsf{isequiv}}({\mathsf{cons}}_a)$. To make use of the type ${\mathbf{F}(A)}$, we need to know an elimination principle for it. This can be stated as an induction (dependent elimination) principle, which is how it is done in the book [@hott-book]. More concise, and (we would say) conceptually clearer, is the approach of phrasing it using a universal property, in other words, a recursion (non-dependent) elimination principle with a uniqueness property. The equivalence between these approaches for inductive types has been discussed by Awodey, Gambino, and Sojakova [@awodeyGamSoja_indTypesInHTT], for some HITs, by Sojakova [@DBLP:journals/corr/Sojakova14], and a restricted version for set-truncated HITs can be found in [@ACDKNF]. For *concrete* HITs, such as our ${\mathbf{F}(A)}$, it is straightforward to derive the various elimination principles from each other. We state the universal property using the presentation as a *homotopy-initial algebras* [@awodeyGamSoja_indTypesInHTT]: \[principle:hinitial\] We say that an *${\mathbf{F}(A)}$-algebra structure* on a type $X$ consists of a point $u : X$, a map $f : A \to X \to X$, and a proof $p : (a: A) \to {\mathsf{isequiv}}(f(a))$. We say that the type of *${\mathbf{F}(A)}$-algebra morphisms* between $(X,u,f,p)$ and $(Y,v,g,q)$ consists of triples $(h,r,s)$, where $h : X \to Y$, $r : f(u) = v$, and $s : h \circ f = g \circ h$. Then, the induction principle of ${\mathbf{F}(A)}$ is equivalent to saying that $({\mathbf{F}(A)}, {\mathsf{unit}}, {\mathsf{cons}}, {\mathsf{iseq}})$ is *homotopy initial*, i.e. that for any $(X,u,f,p)$, the type of morphisms from $({\mathbf{F}(A)}, {\mathsf{unit}}, {\mathsf{cons}}, {\mathsf{iseq}})$ to $(X,u, f,p)$ is contractible. We will come back to ${\mathbf{F}(A)}$-algebras later. An obvious question is whether ${\mathbf{F}(A)}$ really deserves to be called the *free $\infty$-group* on $A$. There are two points: first, we need to check that it is a higher group in the sense of Definition \[def:cheating-group-def\], and second, we have to justify the attribute *free*. For the first point, note that the suspension ${\mathbf \Sigma}(A + {\ensuremath{\mathbf{1}}})$ has an equivalent description which can be obtained by essentially collapsing the point ${\mathsf{S}}$ with the path given by the unit type: it is the HIT ${\mathsf{W}}(A)$ with a single point constructor ${\mathsf{N}}: {\mathsf{W}}(A)$ and a family of loops indexed over $A$, as in ${\mathsf{loops}}: A \to {\mathsf{N}}= {\mathsf{N}}$, a *wedge of $A$-many circles*. Note that ${\mathsf{W}}(A)$ is automatically connected. A further side remark is that ${\mathsf{W}}({\ensuremath{\mathbf{1}}})$ is canonically equivalent to the circle ${\mathbb{S}^{1}}$. We then observe: \[lem:FA-is-group\] The free $\infty$-group ${\mathbf{F}(A)}$ is an $\infty$-group, with ${\mathsf{W}}(A)$ as its delooping. The canonical equivalence $e : {\mathbf{F}(A)} \to ({\mathsf{N}}=_{{\mathsf{W}}(A)} {\mathsf{N}})$ maps the structure as one would expect, i.e.  we have $e({\mathsf{unit}}) = {\mathsf{refl}}$ and $e({\mathsf{cons}}_a(x)) = {\mathsf{loops}}(a) { \mathchoice{\mathbin{\raisebox{0.5ex}{$\displaystyle\centerdot$}}} {\mathbin{\raisebox{0.5ex}{$\centerdot$}}} {\mathbin{\raisebox{0.25ex}{$\scriptstyle\,\centerdot\,$}}} {\mathbin{\raisebox{0.1ex}{$\scriptscriptstyle\,\centerdot\,$}}} }e(x)$. Note that this statement is completely independent from the rest of the paper. This is a relatively straightforward generalisation of the proof that the loop space of ${\mathbb{S}^{1}}$ is equivalent to the type of integers. The proof is an application of [@hott-book Lem 8.9.1] and does not provide much insight, which is why we choose to omit it. For a detailed argument, one can easily adapt the proof given by Brunerie (for an only slightly different statement) in [@DBLP:journals/corr/abs-1710-10307 Sec 6]. From the above lemma, we can in particular observe that ${\mathbf{F}({\ensuremath{\mathbf{1}}})}$ is a presentation of the type of integers. Next, we need to justify why we call ${\mathbf{F}(A)}$ the *free* higher group. The following presentation of the argument was suggested by Paolo Capriotti. Let us consider the following diagram: $$\label{eq:adj-diag} \begin{tikzpicture}[x=1cm,y=1cm,baseline=(current bounding box.center)] \node (A) at (0,0) {${\mathcal{U}}$}; \node (B) at (2,0) {${{\mathcal{U}}_\bullet}$}; \node (C) at (4,0) {${{\mathcal{U}}_\bullet}$}; \node (AB) at (1,0) {$\bot$}; \node (BC) at (3,0) {$\bot$}; \draw[->, bend left=30] (A) to node [above] {$+ {\ensuremath{\mathbf{1}}}$} (B); \draw[->, bend left=30] (B) to node [above] {${\mathbf \Sigma}$} (C); \draw[->, bend left=30] (B) to node [below] {${\mathsf{p}_1}$} (A); \draw[->, bend left=30] (C) to node [below] {$\Omega$} (B); \end{tikzpicture}$$ Here, ${{\mathcal{U}}_\bullet}$ is the universe of pointed types. The function $(+ {\ensuremath{\mathbf{1}}}) : {\mathcal{U}}\to {{\mathcal{U}}_\bullet}$ maps a type $X$ to $(X+{\ensuremath{\mathbf{1}}}, {\mathsf{inr}}(\star))$, while projection ${\mathsf{p}_1}$ simply forgets the point. As in [@hott-book Chp 6.5], we regard the suspension as a function ${\mathbf \Sigma}: {{\mathcal{U}}_\bullet}\to {{\mathcal{U}}_\bullet}$, mapping $(X,x)$ to $({\mathbf \Sigma}(X),{\mathsf{N}})$, and $\Omega$ is the loop space. For $X : {\mathcal{U}}$ and $Y,Z : {{\mathcal{U}}_\bullet}$, it is easy to see that there is a canonical equivalence $$\label{eq:fst-adj} \left(X \to {\mathsf{p}_1}Y\right) \simeq \left((X + {\ensuremath{\mathbf{1}}}) \to_\bullet Y\right),$$ and by [@hott-book Lem 6.5.4], we have $$\label{eq:snd-adj} \left(Y \to_\bullet \Omega(Z)\right) \simeq \left({\mathbf \Sigma}(Y) \to_\bullet Z\right).$$ The above diagram should for our purpose only be regarded as an illustration of these two equivalences. Talking about the adjunctions more precisely is difficult since the correct notions would be $\infty$-categorical. This leads into a territory that is vastly unexplored in homotopy type theory [@capKra_semisegal], although higher adjunctions can be represented using only a finite amount of data [@Riehl2016802]; here, we do not go further into this. Let $G$ be a given $\infty$-group, represented by $(Z,z)$. This means that we have $G \simeq (z=z)$. We can then calculate: $$\begin{alignedat}{3} &&&& \quad & {\mathbf{F}(A)} \to_{\infty\mathsf{grp}} G \\ && \mbox{by Def \ref{def:cheating-group-def} and Lem~\ref{lem:FA-is-group}} \quad &\simeq && ({\mathsf{W}}(A), {\mathsf{N}}) \to_\bullet (Z,z) \\ && \mbox{} &\simeq && {\mathbf \Sigma}(A + {\ensuremath{\mathbf{1}}}) \to_\bullet (Z,z) \\ && \mbox{by \eqref{eq:snd-adj}} \quad &\simeq && (A + {\ensuremath{\mathbf{1}}}) \to_\bullet \Omega(Z,z) \\ && \mbox{by \eqref{eq:fst-adj}} \quad &\simeq && A \to (z=z) \\ && \mbox{} &\simeq && A \to G. \end{alignedat}$$ Thus, ${\mathbf{F}}: {\mathcal{U}}\to \infty\mathsf{GRP}$ is “morally” left adjoint to the forgetful functor which returns the underlying type of a higher group. On Alternative Constructions {#subsec:alternatives} ---------------------------- As a preparation for the development in Section \[sec:mainsection\], and to better understand the difficulties with ${\mathbf{F}(A)}$, let us attempt to construct ${\mathbf{F}(A)}$ in a different way. Let us write ${{A}^{\pm}}$ for $A + A$; we call ${{A}^{\pm}}$ the type of *elements of $A$ with a sign*, and we think of ${\mathsf{inl}}(a)$ as $a$ and ${\mathsf{inr}}(a)$ as $a^{-1}$. For $a : {{A}^{\pm}}$, we write ${{ \accentset{\mbox{\large\bfseries .}}{a}}}$ for the element we get by changing the sign. Of course, this means that ${{ \accentset{\mbox{\large\bfseries .}}{{{ \accentset{\mbox{\large\bfseries .}}{a}}}}}} = a$. Elements of the free group ${\mathbf{F}(A)}$ are, at least intuitively, lists over ${{A}^{\pm}}$. The difficulty is that different lists may represent the same group element. This happens, for example, for $[{\mathsf{inl}}(a), {\mathsf{inr}}(a)]$ (i.e. $a \cdot a^{-1}$) and the empty list, both of which represent the unit of the group. We can avoid this problem by quotienting to identify the list $[x_0, \ldots, x_k, a, {{ \accentset{\mbox{\large\bfseries .}}{a}}}, x_{k+1}, \ldots, x_n]$ with the list $[x_0, \ldots, x_n]$. This quotient will be a set by definition of the quotient (set quotient) operation. If we are happy to work only with sets, and to set-truncate everything, then this is entirely possible, and in fact, it is a construction of the set-based free group given in [@hott-book Thm 6.11.7]. If, like in this paper, we do not want to restrict ourselves to sets, we might think of taking a HIT which has path constructors for each such pair of lists, without set-truncating. The problem is that we need coherences: if we use a path constructor to reduce one redex and then a second, we should get the same equality as if we reduce the second redex and then the first. When looking at *three* redexes, we need to express that these equalities “fit together”, and so on. This is an instance of the problem of infinite coherences which seem to be hard and possibly impossible to express in HoTT. In Section \[sec:mainsection\], we will perform a finite approximation of this construction in order to show Theorem \[thm:mainthm\], although we will see that a couple of additional arguments are required to complete the proof. Alternatively, we could think to only consider lists over ${{A}^{\pm}}$ in *normal form*, i.e. lists which come together with a proof that they do not contain a redex. The type of lists over $A$ in normal form is a set (assuming that $A$ is a set), and the presentation is indeed fully coherent. The trouble is that we are in general unable to define a suitable binary operation on this set, i.e. we are lacking a group operation. If we have two lists in normal form, their concatenation might not be in normal form, and for arbitrary types, we have no way of calculating a normal form or even checking whether we already have a normal form. Unsurprisingly, the approach with normal forms works if $A$ has decidable equality: If $A$ has decidable equality, in the sense that $$(a_1, a_2 : A) \to (a_1 = a_2) + ((a_1 = a_2) \to {\ensuremath{\mathbf{0}}}),$$ then ${\mathbf{F}(A)}$ has decidable equality as well. Moreover, ${\mathbf{F}(A)}$ is in this case canonically equivalent to the set-truncated construction of the free group as given in [@hott-book Chp 6.11]. Thanks to [@hott-book Thm 6.11.7], we can take set-quotiented lists (as described above) as the definition of the set-truncated free group. Using decidable equality of $A$, it is easy to see that this quotient is equivalent to the type of lists in normal form; let us write ${\mathsf{LNF}(A)}$ for the latter type. An element of ${\mathsf{LNF}(A)}$ is a list together with a propositional property, and we have an embedding ${\mathsf{LNF}(A)} \to {{\mathsf{List}}({{A}^{\pm}})}$. What is left to do is to compare ${\mathsf{LNF}(A)}$ with ${\mathbf{F}(A)}$. Note that ${\mathsf{LNF}(A)}$ is a set without being explicitly set-truncated. There is a canonical ${\mathbf{F}(A)}$-algebra structure on ${\mathsf{LNF}(A)}$, giving rise to a map ${\mathbf{F}(A)} \to {\mathsf{LNF}(A)}$. Further, one can construct a function ${{\mathsf{List}}({{A}^{\pm}})} \to {\mathbf{F}(A)}$, by induction on the list. The empty list is mapped to ${\mathsf{unit}}$, ${\mathsf{inl}}(a)$ translates into an application of ${\mathsf{cons}}_a$, and ${\mathsf{inr}}(a)$ becomes ${\mathsf{icons}}_a$. These functions give rise to an equivalence ${\mathsf{LNF}(A)} \simeq {\mathbf{F}(A)}$, and since ${\mathsf{LNF}(A)}$ has decidable equality, ${\mathbf{F}(A)}$ enjoys the same property. The fundamental group of the free $\infty$-group {#sec:mainsection} ================================================ In this section, the core of the paper, we develop a couple of techniques that, when combined, allow us to prove Theorem \[thm:mainthm\]. For the whole section, let us assume that $A$ is a given set. Given lists $x,y : {{\mathsf{List}}({{A}^{\pm}})}$, we write $xy$ for their *concatenation*, i.e. the list we get by simply joining the two lists as in $[a_1,a_2] [a_3, a_4] = [a_1,a_2,a_3,a_4]$. Since this operation is associative (up to a canonical and fully coherent equality), we omit brackets and write $xyz$ for both $(xy)z$ and $x(yz)$. Given $a : {{A}^{\pm}}$, we regard $a$ as a one-element list and allow ourselves to write e.g. $xayz$ or $a{{ \accentset{\mbox{\large\bfseries .}}{a}}} y$. A simple reduction system in type theory ---------------------------------------- As discussed in Section \[subsec:alternatives\] above, we can think of elements of ${\mathbf{F}(A)}$ as lists over ${{A}^{\pm}}$, and the main problem is that different lists represent the same group element. This motivates the development of a system of reductions. \[def:Red\] The type family ${\mathsf{Red}}: {{\mathsf{List}}({{A}^{\pm}})} \to {\mathcal{U}}$, which expresses that a list represents the same group element as the empty list (i.e. the neutral element of the group ${\mathbf{F}(A)}$), is defined as follows. We first define an auxiliary family ${\mathsf{R}}: {\mathbb N}\to {{\mathsf{List}}({{A}^{\pm}})} \to {\mathcal{U}}$ by induction on the natural numbers: $$\begin{aligned} {3} & {\mathsf{R}}_0 (x) &&{:\equiv}\; \; && {\mathsf{length}}(x) = 0 \\ & {\mathsf{R}}_{2+n}(x) && {:\equiv}&& \Sigma (a : {{A}^{\pm}}), (y,z : {{\mathsf{List}}({{A}^{\pm}})}), \\ & &&&& \phantom{\Sigma} \left({\mathsf{length}}(y) + {\mathsf{length}}(z) = n\right), \\ & &&&& \phantom{\Sigma} \left(x = y {}a {}{{ \accentset{\mbox{\large\bfseries .}}{a}}} {}z\right), \\ & &&&& \phantom{\Sigma} {\mathsf{R}}_n (y {}z)\end{aligned}$$ Using this, we set ${\mathsf{Red}}(x) {:\equiv}{\mathsf{R}}_{{\mathsf{length}}(x)}(x)$. If we have indexed inductive families in the theory, we can alternatively define ${\mathsf{Red}}$ directly as such a family generated by $$\begin{aligned} {2} & \mathsf{zero} : &\;& {\mathsf{Red}}({\mathsf{nil}}) \\ & \mathsf{step} : &\;& (y,z : {{\mathsf{List}}({{A}^{\pm}})}) \to (a : {{A}^{\pm}}) \to {\mathsf{Red}}(y {}z) \to {\mathsf{Red}}(y {}a {{ \accentset{\mbox{\large\bfseries .}}{a}}} {}z).\end{aligned}$$ The two definitions are essentially the same, only represented in different ways. In both cases, given $r : {\mathsf{Red}}(x)$, we say that $r$ witnesses that $x$ can be *reduced* to the empty list and we call $r$ a *reduction sequence*. We view it as a sequence consisting of *steps*, each of which removes a single *redex* $a {{ \accentset{\mbox{\large\bfseries .}}{a}}}$. An example of a reduction sequence $r: {\mathsf{Red}}(a {{ \accentset{\mbox{\large\bfseries .}}{a}}} b c {{ \accentset{\mbox{\large\bfseries .}}{c}}} {{ \accentset{\mbox{\large\bfseries .}}{b}}})$ could be pictured as follows, where each step is represented by an arrow $\leadsto$ annotated with the redex it reduces: $$\label{eq:sample-red-sequence} a {{ \accentset{\mbox{\large\bfseries .}}{a}}} b c {{ \accentset{\mbox{\large\bfseries .}}{c}}} {{ \accentset{\mbox{\large\bfseries .}}{b}}} \; \stackrel{c {{ \accentset{\mbox{\large\bfseries .}}{c}}}}\leadsto \; a {{ \accentset{\mbox{\large\bfseries .}}{a}}} b {{ \accentset{\mbox{\large\bfseries .}}{b}}} \; \stackrel{a {{ \accentset{\mbox{\large\bfseries .}}{a}}}}\leadsto \; b {{ \accentset{\mbox{\large\bfseries .}}{b}}} \; \stackrel{b {{ \accentset{\mbox{\large\bfseries .}}{b}}}}\leadsto \; {\mathsf{nil}}.$$ There are a couple of points that we want to point out explicitly. 1. In the above example and in the discussions to come, $a : {{A}^{\pm}}$ is already positive or negative, which means that every redex is of the form $a {{ \accentset{\mbox{\large\bfseries .}}{a}}}$; the possibility ${{ \accentset{\mbox{\large\bfseries .}}{a}}} a$ is already covered. 2. The number of steps of $r : {\mathsf{Red}}(x)$ is simply half of the length of the list $x$, which means that all elements of ${\mathsf{Red}}(x)$ have the same number of steps. In particular, it is easy to prove that ${\mathsf{Red}}(x)$ is empty if ${\mathsf{length}}(x)$ is odd. 3. For a given list $x$, there is no way to *compute* a reduction sequence, since we do not know whether an occurring pair $bc$ forms a redex. A reduction $r: {\mathsf{Red}}(x)$ encodes equalities which guarantee that all redexes that it reduces are really redexes. Deciding whether $bc$ is a redex would require decidable equality on $A$ (but of course, we can always check whether an element of ${{A}^{\pm}}$ is positive or negative, and this analysis might give us that $bc$ is definitely not a redex). 4. For a given $x$, equality on ${\mathsf{Red}}(x)$ *is* decidable. This is because a sequence encodes the positions of the redexes that it reduces, and positions are decidable, while the (in general undecidable) equalities on ${{A}^{\pm}}$ are propositions. Similarly, if we have $r : {\mathsf{Red}}(xby)$, we can say in which step $b$ is reduced, since this is encoded by a position. Let us remind ourselves that the goal of the paper is to show that ${\mathbf{F}(A)}$ has trivial fundamental groups. This is a statement about equalities between equalities. If we think of a reduction sequence as a proof that a list represents the neutral group element, i.e. as something giving rise to an equality proof, it is hopefully intuitive that we now want to discuss the relationship between different reduction sequences. In a nutshell, we want to give a criterion which guarantees that two reduction sequences give rise to equal equalities. To do so, we consider *transformations*: \[def:transformations\] Let $w : {{\mathsf{List}}({{A}^{\pm}})}$ be a list and $r : {\mathsf{Red}}(w)$ a reduction sequence. We consider the following two operations, each of which allows us to create a new reduction sequence in ${\mathsf{Red}}(w)$ from $r$: 1. Swap two consecutive independent steps in $r$. More precisely, if $r$ is a sequence of the form $$\label{eq:seq-1-to-transform} \ldots \; \leadsto xa {{ \accentset{\mbox{\large\bfseries .}}{a}}} y b {{ \accentset{\mbox{\large\bfseries .}}{b}}} z \; \stackrel{b {{ \accentset{\mbox{\large\bfseries .}}{b}}}}\leadsto \; xa {{ \accentset{\mbox{\large\bfseries .}}{a}}} yz \; \stackrel{a {{ \accentset{\mbox{\large\bfseries .}}{a}}}}\leadsto \; xyz \leadsto \ldots,$$ we can change it to $$\label{eq:seq-2-to-transform} \ldots \; \leadsto xa {{ \accentset{\mbox{\large\bfseries .}}{a}}} y b {{ \accentset{\mbox{\large\bfseries .}}{b}}} z \; \stackrel{a {{ \accentset{\mbox{\large\bfseries .}}{a}}}}\leadsto \; xyb {{ \accentset{\mbox{\large\bfseries .}}{b}}}z \; \stackrel{b {{ \accentset{\mbox{\large\bfseries .}}{b}}}}\leadsto \; xyz \leadsto \ldots.$$ Analogously, we can change into . 2. If a step reduces a redex $a {{ \accentset{\mbox{\large\bfseries .}}{a}}}$ in a list of the form $xa {{ \accentset{\mbox{\large\bfseries .}}{a}}} ay$, we can change this step to remove the redex ${{ \accentset{\mbox{\large\bfseries .}}{a}}} a$ instead, or vice versa. This means that $$\ldots \; \leadsto xa {{ \accentset{\mbox{\large\bfseries .}}{a}}} ay \; \stackrel{a {{ \accentset{\mbox{\large\bfseries .}}{a}}}}\leadsto \; xay \; \leadsto \ldots$$ can be changed to $$\ldots \; \leadsto xa {{ \accentset{\mbox{\large\bfseries .}}{a}}} ay \; \stackrel{{{ \accentset{\mbox{\large\bfseries .}}{a}}}a}\leadsto \; xay \; \leadsto \ldots,$$ or vice versa. We say that $r : {\mathsf{Red}}(w)$ can be *transformed* into $s : {\mathsf{Red}}(w)$ if there is a finite chain of these operations that changes $r$ into $s$. After what we said in the paragraph before Definition \[def:transformations\], the best we could hope for is that *any reduction sequence can be transformed into any other reduction sequence* (of the same list $w$). Indeed, this is what we will show. We start with a technical lemma which will not only help us to prove what we just said (Corollary \[cor:transform-any\]), but also another useful consequence (Corollary \[cor:not-stuck\]). \[lem:normalising-reductions\] Assume we are given a list of the form $x a {{ \accentset{\mbox{\large\bfseries .}}{a}}} y$, i.e. a list in ${{A}^{\pm}}$ with an explicitly given redex $a {{ \accentset{\mbox{\large\bfseries .}}{a}}}$. Assume further that we have a reduction sequence $s : {\mathsf{Red}}(x a {{ \accentset{\mbox{\large\bfseries .}}{a}}} y)$. It is possible to transform $s$ into a reduction sequence which reduces the redex $a {{ \accentset{\mbox{\large\bfseries .}}{a}}}$ in the first step, i.e. starts with $xa {{ \accentset{\mbox{\large\bfseries .}}{a}}}y \; \stackrel{a {{ \accentset{\mbox{\large\bfseries .}}{a}}}}\leadsto \; xy \; \leadsto \; \ldots$. Let $x$, $a$, $y$, and $r : {\mathsf{Red}}(xa {{ \accentset{\mbox{\large\bfseries .}}{a}}}y)$ be given. Let us write $m$ for the number of the step in which $a$ is reduced, and $n$ for the number of the step in which ${{ \accentset{\mbox{\large\bfseries .}}{a}}}$ is reduced. There are three cases: - If $m = n$, then the redex $a {{ \accentset{\mbox{\large\bfseries .}}{a}}}$ is reduced in step $n$. If $n = 0$, there is nothing to do. Otherwise, we can swap this step with step $(n-1)$, since the two steps will be independent of each other. Swapping a further $(n-1)$ times, we can move the step reducing $a {{ \accentset{\mbox{\large\bfseries .}}{a}}}$ to the beginning of the sequence. - If $m > n$, then $a {{ \accentset{\mbox{\large\bfseries .}}{a}}}$ are not reduced together, but ${{ \accentset{\mbox{\large\bfseries .}}{a}}}$ is reduced with some ${{ \accentset{\mbox{\large\bfseries .}}{{{ \accentset{\mbox{\large\bfseries .}}{a}}}}}}$ to its right instead. Note that ${{ \accentset{\mbox{\large\bfseries .}}{{{ \accentset{\mbox{\large\bfseries .}}{a}}}}}} = a$. Before step $n$, the list thus has to be of the form $u a {{ \accentset{\mbox{\large\bfseries .}}{a}}} a v$, and step $n$ consists of reducing ${{ \accentset{\mbox{\large\bfseries .}}{a}}} a$. We define $r_1$ to be the reduction sequence which is identical to $r$ in every step expect in step $n$ where it reduces $a {{ \accentset{\mbox{\large\bfseries .}}{a}}}$; this is the second of the two possible operations in Definition \[def:transformations\]. We are now in case one ($m=n$). - The case $m < n$ is analogous to the case $m > n$. \[cor:transform-any\] Any reduction sequence can be transformed into any other reduction sequence. More precisely, for $w : {{\mathsf{List}}({{A}^{\pm}})}$ and $r,s : {\mathsf{Red}}(w)$, we can transform $r$ into $s$. A reduction sequence is given by a chain of reduction steps, and the number of steps in $r$ and $s$ are equal (both are $\mathsf{length}(w)/2$). Thus, it is sufficient to transform $r$ into a sequence which consists of the same steps as $s$. By the above lemma, we can transform $r$ into a sequence $r'$ which in the first step reduces whichever redex $s$ reduces in the first step. Applying the same argument to the “tail” of the sequences (note that $r'$ and $s$, each with the first step removed, still reduce the same list), we get a transformation into a sequence which in every step mirrors the reduction of $s$ and is thus equal to $s$. A second easy consequence is that, if a list is reducible, then we cannot “get stuck” while reducing: we can start reducing at an arbitrary position without risking of ending up with an unreducible list. Note that we write $B \leftrightarrow C$ for $(B \to C) \times (C \to B)$. \[cor:not-stuck\] For any lists $y, z$ and $a : {{A}^{\pm}}$, we have $${\mathsf{Red}}(y {}a {}{{ \accentset{\mbox{\large\bfseries .}}{a}}} {}z) \leftrightarrow {\mathsf{Red}}(y {}z).$$ The direction $\leftarrow$ is immediate, by adding a single reduction step reducing $a {{ \accentset{\mbox{\large\bfseries .}}{a}}}$. The direction $\rightarrow$ is an application of Lemma \[lem:normalising-reductions\]. Note that Corollary \[cor:transform-any\] subtly but crucially depends on the assumption that $A$ is a set, while Lemma \[lem:normalising-reductions\], as formulated, would work for arbitrary types $A$. It is true independently of $A$ that a reduction sequence is given by a chain of reduction steps. A reduction step encodes the *position* at which the reduction is taking place (say, the length of the list $y$ in Definition \[def:Red\]), together with a proof that the reduction is possible (i.e. a proof that the pair at the position is actually a redex). The second part amounts to an equality in ${{A}^{\pm}}$ (since “$ab$ being a redex” means $a = {{ \accentset{\mbox{\large\bfseries .}}{b}}}$); thus, it is a proposition if $A$ is a set. In this case, a reduction step is determined by the position, and a reduction sequence is determined by the chain of positions which it encodes. The proof of Corollary \[cor:transform-any\] relies on this. Lemma \[lem:normalising-reductions\] holds even without the requirement of $A$ being a set. However, note that the proof of Lemma \[lem:normalising-reductions\], when it uses the second operation in Definition \[def:transformations\], has to construct a new equality (this is hidden in the sentence “Before step $n$, the list thus has to be of the form $u a {{ \accentset{\mbox{\large\bfseries .}}{a}}} a v$”). Therefore, the new sequence constructed in Lemma \[lem:normalising-reductions\] *will* reduce $a {{ \accentset{\mbox{\large\bfseries .}}{a}}}$ in the first step, but the proof that $a {{ \accentset{\mbox{\large\bfseries .}}{a}}}$ is indeed a redex could be a nontrivial one. A non-recursive approximation to the free $\infty$-group -------------------------------------------------------- We are ready to define a *non-recursive approximation* to the free group ${\mathbf{F}(A)}$, a HIT that we call ${\mathbf{N}(A)}$. By *non-recursive*, we mean that constructors of ${\mathbf{N}(A)}$ do not use points or paths of ${\mathbf{N}(A)}$ in their arguments. \[def:NA\] We define ${\mathbf{N}(A)}$ to be the HIT with the following constructors: $$\begin{alignedat}{2} &\eta : &\; & {{\mathsf{List}}({{A}^{\pm}})} \to {\mathbf{N}(A)} \\ &\tau : && (x : {{\mathsf{List}}({{A}^{\pm}})}) \to (a : {{A}^{\pm}}) \to (y : {{\mathsf{List}}({{A}^{\pm}})}) \\ &&& \; \to \eta(x {}a {}{{ \accentset{\mbox{\large\bfseries .}}{a}}} {}y) = \eta(x {}y) \\ &{\mathsf{sw}}: && (x : {{\mathsf{List}}({{A}^{\pm}})}) \to (a : {{A}^{\pm}}) \to (y : {{\mathsf{List}}({{A}^{\pm}})}) \\ &&& \; \to (b : {{A}^{\pm}}) \to (z : {{\mathsf{List}}({{A}^{\pm}})}) \\ &&& \; \to \tau(x,a,yb {{ \accentset{\mbox{\large\bfseries .}}{b}}} z) { \mathchoice{\mathbin{\raisebox{0.5ex}{$\displaystyle\centerdot$}}} {\mathbin{\raisebox{0.5ex}{$\centerdot$}}} {\mathbin{\raisebox{0.25ex}{$\scriptstyle\,\centerdot\,$}}} {\mathbin{\raisebox{0.1ex}{$\scriptscriptstyle\,\centerdot\,$}}} }\tau(xy,b,z) = \tau(xa {{ \accentset{\mbox{\large\bfseries .}}{a}}} y, b, z) { \mathchoice{\mathbin{\raisebox{0.5ex}{$\displaystyle\centerdot$}}} {\mathbin{\raisebox{0.5ex}{$\centerdot$}}} {\mathbin{\raisebox{0.25ex}{$\scriptstyle\,\centerdot\,$}}} {\mathbin{\raisebox{0.1ex}{$\scriptscriptstyle\,\centerdot\,$}}} }\tau(x,a,yz) \\ &{\mathsf{ov}}: && (x : {{\mathsf{List}}({{A}^{\pm}})}) \to (a : {{A}^{\pm}}) \to (y : {{\mathsf{List}}({{A}^{\pm}})}) \\ &&& \; \to \tau(x,a,ay) = \tau(xa,{{ \accentset{\mbox{\large\bfseries .}}{a}}},y) \\ &{\mathsf{tr}}: &&{\mathsf{is}\mbox{-}{1}\mbox{-}\mathsf{type}}({\mathbf{N}(A)}) \end{alignedat}$$ We can think of ${\mathbf{N}(A)}$ (without the last constructor) as a “wild” quotient of ${{\mathsf{List}}({{A}^{\pm}})}$. Recall that we said that lists over ${{A}^{\pm}}$ correspond to very intentional representations of group elements. The HIT with constructors $\eta$ and $\tau$ can be thought of as a “level $0$ approximation” to a fully coherent non-recursive quotient of ${{\mathsf{List}}({{A}^{\pm}})}$: we identify some lists which represent the same group element, but the equalities are incoherent. This is partially remedied by the constructors ${\mathsf{sw}}$ (“swap”) and ${\mathsf{ov}}$ (“overlap”), ensuring that the equalities generated by $\tau$ satisfy basic coherence. They can be pictured as follows: $$\begin{tikzpicture}[x=4cm,y=-1.5cm,baseline=(current bounding box.center)] \node (XAYBZ) at (0,0) {$\eta(xa {{ \accentset{\mbox{\large\bfseries .}}{a}}} y b {{ \accentset{\mbox{\large\bfseries .}}{b}}} z)$}; \node (XYBZ) at (1,0) {$\eta(xy b {{ \accentset{\mbox{\large\bfseries .}}{b}}} z)$}; \node (XAYZ) at (0,1) {$\eta(xa {{ \accentset{\mbox{\large\bfseries .}}{a}}} yz)$}; \node (XYZ) at (1,1) {$\eta(xyz)$}; \node (SW) at (0.5,0.5) {${\mathsf{sw}}(x,a,y,b,z)$}; \draw[->] (XAYBZ) to node [above] {$\tau(x,a,yb{{ \accentset{\mbox{\large\bfseries .}}{b}}} z)$} (XYBZ); \draw[->] (XAYBZ) to node [left] {$\tau(xa {{ \accentset{\mbox{\large\bfseries .}}{a}}}y, b, z)$} (XAYZ); \draw[->] (XYBZ) to node [right] {$\tau(x,a,yb{{ \accentset{\mbox{\large\bfseries .}}{b}}} z)$} (XYZ); \draw[->] (XAYZ) to node [below] {$\tau(xa {{ \accentset{\mbox{\large\bfseries .}}{a}}}y, b, z)$} (XYZ); \end{tikzpicture}$$ $$\begin{tikzpicture}[x=4cm,y=-2cm,baseline=(current bounding box.center)] \node (XAAY) at (0,0) {$\eta(xa {{ \accentset{\mbox{\large\bfseries .}}{a}}} a y)$}; \node (XAY) at (1,0) {$\eta(x a y)$}; \node (OV) at (0.5,0) {${\mathsf{ov}}(x,a,y)$}; \draw[->, bend left=20] (XAAY) to node [above] {$\tau(x,a,ay)$} (XAY); \draw[->, bend right=20] (XAAY) to node [below] {$\tau(xa, {{ \accentset{\mbox{\large\bfseries .}}{a}}}, y)$} (XAY); \end{tikzpicture}$$ ${\mathsf{sw}}$ and ${\mathsf{ov}}$ themselves are not directly guaranteed to be coherent; if we omit the constructor ${\mathsf{tr}}$, we can think of ${\mathbf{N}(A)}$ as a “level $1$ approximation”. ${\mathsf{tr}}$ ensures that all higher equalities hold, by forcing ${\mathbf{N}(A)}$ to be $1$-truncated. The statement that ${\mathbf{N}(A)}$ is an approximation to the free higher group can then be made by drawing a connection to ${\mathopen{}\left\Vert {\mathbf{F}(A)}\right\Vert_{1}\mathclose{}}$, which we will do later. If a list can be reduced, then in ${\mathbf{N}(A)}$, it is indistinguishable from the empty list: \[lem:redIsNeutral\] We have a function $${\mathsf{red}\mbox{-}\mathsf{is}\mbox{-}\mathsf{neutral}}: (z : {{\mathsf{List}}({{A}^{\pm}})}) \to {\mathsf{Red}}(z) \to \eta(z) = \eta({\mathsf{nil}}).$$ We need to analyse the element $r : {\mathsf{Red}}(z)$. It encodes a finite number of reduction steps. The first reduction step shows that $z$ is of the form $z = x a {{ \accentset{\mbox{\large\bfseries .}}{a}}} y$, thus the constructor $\tau(x,a,y)$ (transported along the equality $z=xa {{ \accentset{\mbox{\large\bfseries .}}{a}}} y$) provides us with the equality $\eta(z) = \eta(xy)$. Similarly, each of the remaining reduction steps encoded in $r$ shows how $\tau$ can be applied, and the concatenation of all these equalities yields $\eta(z) = \eta({\mathsf{nil}})$. If ${\mathsf{Red}}$ is defined as an indexed inductive family, ${\mathsf{red}\mbox{-}\mathsf{is}\mbox{-}\mathsf{neutral}}(x)(r)$ can be constructed by induction on $r$, and the induction step is given by the constructor $\tau$. Not only can we show that reducible lists are equal to ${\mathsf{nil}}$ in ${\mathbf{N}(A)}$, it is also the case that the concrete witness of reducibility does not matter: \[lem:redIsNeutral-wconst\] For any given $x$, the function $${\mathsf{red}\mbox{-}\mathsf{is}\mbox{-}\mathsf{neutral}}(x) : {\mathsf{Red}}(x) \to \eta(x) = \eta({\mathsf{nil}})$$ is weakly constant, in the following sense: $$(r, s : {\mathsf{Red}}(x)) \to {\mathsf{red}\mbox{-}\mathsf{is}\mbox{-}\mathsf{neutral}}(x)(r) = {\mathsf{red}\mbox{-}\mathsf{is}\mbox{-}\mathsf{neutral}}(x)(s).$$ The constructors ${\mathsf{sw}}$ and ${\mathsf{ov}}$ ensure that, if two reduction sequences can be transformed into each other, then they lead to equal proofs of $\eta(x) = \eta({\mathsf{nil}})$. More precisely, the first operation in Definition \[def:transformations\] is exactly covered by the constructor ${\mathsf{sw}}$, while the second operation is covered by ${\mathsf{ov}}$. The statement thus follows from Corollary \[cor:transform-any\]. The point of ${\mathbf{N}(A)}$ is that it is easier to reason about ${\mathbf{N}(A)}$ than about ${\mathbf{F}(A)}$, thanks to the absence of recursive constructors; one can say that ${\mathbf{N}(A)}$ attempts to bridge the gap between ${{\mathsf{List}}({{A}^{\pm}})}$ and ${\mathbf{F}(A)}$. We first define a property stating that an element of ${\mathbf{N}(A)}$ can be reduced. We write ${\mathsf{hProp}}$ for ${\Sigma \left(X:{\mathcal{U}}\right).\,} {\mathsf{is}\mbox{-}\mathsf{prop}}(X)$ as usual. The family ${{\mathopen{}\left\Vert -\right\Vert_{}\mathclose{}}} \circ {\mathsf{Red}}: {{\mathsf{List}}({{A}^{\pm}})} \to {\mathsf{hProp}}$ extends to a family ${\mathsf{red}}: {\mathbf{N}(A)} \to {\mathsf{hProp}}$ as in the following commuting triangle: $$\begin{tikzpicture}[x=2.5cm,y=-1.2cm,baseline=(current bounding box.center)] \node (LSA) at (0,0) {${{\mathsf{List}}({{A}^{\pm}})}$}; \node (TYPE) at (1,0) {${\mathcal{U}}$}; \node (PROP) at (2,0) {${\mathsf{hProp}}$}; \node (NA) at (0,1) {${\mathbf{N}(A)}$}; \draw[->] (LSA) to node [above] {${\mathsf{Red}}$} (TYPE); \draw[->] (TYPE) to node [above] {${{\mathopen{}\left\Vert -\right\Vert_{}\mathclose{}}}$} (PROP); \draw[->] (LSA) to node [left] {$\eta$} (NA); \draw[->, dotted] (NA) to node [below] {${\mathsf{red}}$} (PROP); \end{tikzpicture}$$ We do induction on ${\mathbf{N}(A)}$. Clearly, we have to set ${\mathsf{red}}(\eta(x)) {:\equiv}{{\mathopen{}\left\Vert {\mathsf{Red}}(x)\right\Vert_{}\mathclose{}}}$. The proof obligation of the constructor $\tau$ is met by Corollary \[cor:not-stuck\]. The remaining two constructors are trivial, since they ask for equalities between elements of propositions. To avoid confusion with elements of ${{\mathsf{List}}({{A}^{\pm}})}$, which we call $x,y,z,\ldots$, we use Greek letters for elements of ${\mathbf{N}(A)}$. If $\gamma : {\mathbf{N}(A)}$ is reducible, it is equal to the neutral element: \[lem:NA-red-equal\] There is a function of type $$(\gamma : {\mathbf{N}(A)}) \to {\mathsf{red}}(\gamma) \to \gamma = \eta({\mathsf{nil}}).$$ We do induction on $\gamma$. First, we consider the case $\gamma \equiv \eta(x)$, and we want to find $f_x : {\mathsf{red}}(\eta(x)) \to \eta(x) = \eta({\mathsf{nil}})$. Recall that a weakly constant function into a set (which the codomain here is) factors through the propositional truncation [@lmcs:3217], hence since ${\mathsf{red}}(\eta(x)) \equiv {{\mathopen{}\left\Vert {\mathsf{Red}}(x)\right\Vert_{}\mathclose{}}}$ by definition, Lemma \[lem:redIsNeutral-wconst\] gives us a function $$f_x : {\mathsf{red}}(\eta(x)) \to \eta(x) = \eta({\mathsf{nil}})$$ such that $f_x({{\mathopen{}\left|r\right|_{}^{}\mathclose{}}}) = {\mathsf{red}\mbox{-}\mathsf{is}\mbox{-}\mathsf{neutral}}(x)(r)$. We want to extend this function to ${\mathbf{N}(A)}$. Induction on $\gamma$ requires us to provide constructions corresponding to $\tau$, ${\mathsf{sw}}$, and ${\mathsf{ov}}$. The latter two are contractible, and we do not need to worry about them. The proof obligation for $\tau$ says that, for any $y,a,z$, and witnesses $s : {\mathsf{red}}(\eta(yz))$, $s' : {\mathsf{red}}(\eta(ya {{ \accentset{\mbox{\large\bfseries .}}{a}}} z))$, the triangle $$\begin{tikzpicture}[x=4.5cm,y=-1.3cm,baseline=(current bounding box.center)] \node (T00) at (0,0) {$\eta(y a {{ \accentset{\mbox{\large\bfseries .}}{a}}} z)$}; \node (T10) at (1,0) {$\eta(yz)$}; \node (T11) at (1,1) {$\eta({\mathsf{nil}})$}; \draw[->] (T00) to node [below left] {$f_{y a {{ \accentset{\mbox{\large\bfseries .}}{a}}} z} (s')$} (T11); \draw[->] (T10) to node [right] {$f_{yz}(s)$} (T11); \draw[->] (T00) to node [above] {$\tau(y,a,z)$} (T10); \end{tikzpicture}$$ commutes. This is a proposition, thus we can assume that $s$, $s'$ come from actual reduction sequences, i.e. we have $r : {\mathsf{Red}}(yz)$ with ${{\mathopen{}\left|r\right|_{}^{}\mathclose{}}} = s$ and $r' : {\mathsf{Red}}(ya {{ \accentset{\mbox{\large\bfseries .}}{a}}} z)$ with ${{\mathopen{}\left|r'\right|_{}^{}\mathclose{}}} = s'$. This simplifies the triangle to: $$\begin{tikzpicture}[x=4.5cm,y=-1.3cm,baseline=(current bounding box.center)] \node (T00) at (0,0) {$\eta(y a {{ \accentset{\mbox{\large\bfseries .}}{a}}} z)$}; \node (T10) at (1,0) {$\eta(yz)$}; \node (T11) at (1,1) {$\eta({\mathsf{nil}})$}; \draw[->] (T00) to node [below left] {${\mathsf{red}\mbox{-}\mathsf{is}\mbox{-}\mathsf{neutral}}(y a {{ \accentset{\mbox{\large\bfseries .}}{a}}} z)(r')$} (T11); \draw[->] (T10) to node [right] {${\mathsf{red}\mbox{-}\mathsf{is}\mbox{-}\mathsf{neutral}}(yz)(r)$} (T11); \draw[->] (T00) to node [above] {$\tau(y,a,z)$} (T10); \end{tikzpicture}$$ If we write $r'' : {\mathsf{Red}}(ya {{ \accentset{\mbox{\large\bfseries .}}{a}}} z)$ for the sequence $r$, extended by the single step reducing $a {{ \accentset{\mbox{\large\bfseries .}}{a}}}$ in the beginning, we see that composition of the horizontal and vertical arrow give ${\mathsf{red}\mbox{-}\mathsf{is}\mbox{-}\mathsf{neutral}}(ya {{ \accentset{\mbox{\large\bfseries .}}{a}}} z)(r'')$. Thus, Lemma \[lem:redIsNeutral-wconst\] yields the required commutativity of the triangle. This allows us to conclude: \[lem:NA-locally-set\] ${\mathbf{N}(A)}$ is a set locally at $\eta({\mathsf{nil}})$, in the sense that ${\eta({\mathsf{nil}}) =_{{\mathbf{N}(A)}} \eta({\mathsf{nil}})}$ is contractible. If equality is implied by a “reflexive mere relation”, then the type is a set ([@hott-book Thm 7.2.2.], sometimes called “Rijke’s theorem”). Here, we need the local formulation of this statement as given in [@nicolai:thesis], together with Lemma \[lem:NA-red-equal\]. Next, we want to extend this observation and show that ${\mathbf{N}(A)}$ is a set. In general, if a type $X$ is a set locally at $x_0 : X$ and we have an equivalence $e : X \to X$, then $X$ is also a set locally at $e(x_0)$, using that $\mathsf{ap}_e$ will be an equivalence. Therefore, if for every $y: X$ there is an equivalence mapping $x_0$ to $y$ (we can say that such an $X$ is “homogeneous”), then $X$ is a set; and in fact, it is enough if for a given $y$ the equivalence merely exists (i.e. hidden with a truncation). This is one motivation for the following technical lemma, where we construct equivalences ${\mathbf{N}(A)} \to {\mathbf{N}(A)}$. Another motivation is that these equivalences are the main part of an ${\mathbf{F}(A)}$-algebra structure on ${\mathbf{N}(A)}$, but we will come back to this later. \[lem:NA-has-equiv\] There is a function $f : {{A}^{\pm}} \to {\mathbf{N}(A)} \to {\mathbf{N}(A)}$ such that, for every $c : {{A}^{\pm}}$, the map $f_c : {\mathbf{N}(A)} \to {\mathbf{N}(A)}$ is an equivalence with $f_{{{ \accentset{\mbox{\large\bfseries .}}{c}}}}$ as its inverse. Further, the construction can be done such that, for every $x : {{\mathsf{List}}({{A}^{\pm}})}$, we have $f_c(\eta(x)) \equiv \eta(cx)$. Let $c:A$ be given. We need to define $f_c : {\mathbf{N}(A)} \to {\mathbf{N}(A)}$, i.e. for a given $\alpha : {\mathbf{N}(A)}$, we need $f_c(\alpha) : {\mathbf{N}(A)}$. This can be done by recursion on $\alpha$ in the obvious way: - We set $f_c(\eta(x)) {:\equiv}\eta(cx)$. - Next, we need a witness of $f_c(\eta(xa {{ \accentset{\mbox{\large\bfseries .}}{a}}} y)) = f_c(\eta(xy))$. Slightly abusing notation, we write $f_c(\tau(x,a,y))$ for this[^2], and we set $f_c(\tau(x,a,y) {:\equiv}\tau(cx,a,y)$. - Similarly, we set $f_c({\mathsf{sw}}(x,a,y,z)) {:\equiv}{\mathsf{sw}}(cx,a,y,z)$; - and $f_c({\mathsf{ov}}(x,a,y)) {:\equiv}{\mathsf{ov}}(cx,a,y)$; - and finally, we have $f_c({\mathsf{tr}}) {:\equiv}{\mathsf{tr}}$. We need to show that $f_c$ is an inverse of $f_{{{ \accentset{\mbox{\large\bfseries .}}{c}}}}$. It is sufficient to show that, for $\alpha : {\mathbf{N}(A)}$, we have $f_c(f_{{{ \accentset{\mbox{\large\bfseries .}}{c}}}}(\alpha)) = \alpha$ and $f_{{{ \accentset{\mbox{\large\bfseries .}}{c}}}}(f_c(\alpha)) = \alpha$. Let us concentrate on the first of these, as the second is no more than a copy which switches the sign of $c$. Note that the goal is an equality in the $1$-type ${\mathbf{N}(A)}$ and thus a set. Thus, when we do induction on $\alpha$, in order to construct a function $h : (\alpha : {\mathbf{N}(A)}) \to f_c(f_{{{ \accentset{\mbox{\large\bfseries .}}{c}}}}(\alpha)) = \alpha$, the proof obligations for ${\mathsf{sw}}$, ${\mathsf{ov}}$, and ${\mathsf{tr}}$ are trivial. For $\eta$ and $\tau$, the constructions work as follows: - For $\eta$, we need $h(\eta(x))$ of type $f_c(f_{{{ \accentset{\mbox{\large\bfseries .}}{c}}}}(\eta(x))) = \eta(x)$, which reduces to $\eta(c {{ \accentset{\mbox{\large\bfseries .}}{c}}} x) = \eta(x)$. Therefore, we can set $h(\eta(x)) {:\equiv}\tau({\mathsf{nil}}, c,x)$. - For $\tau$, we need to construct $h(\tau(x,a,y))$ which shows that $h(\eta(xa {{ \accentset{\mbox{\large\bfseries .}}{a}}} y))$ and $h(\eta(xy))$ are equal as paths over $\tau(x,a,y)$. After unfolding what this means, we see that the type of $h(\tau(x,a,y))$ is: $$\mathsf{ap}_{f_{c {{ \accentset{\mbox{\large\bfseries .}}{c}}}}}(\tau(x,a,y)) { \mathchoice{\mathbin{\raisebox{0.5ex}{$\displaystyle\centerdot$}}} {\mathbin{\raisebox{0.5ex}{$\centerdot$}}} {\mathbin{\raisebox{0.25ex}{$\scriptstyle\,\centerdot\,$}}} {\mathbin{\raisebox{0.1ex}{$\scriptscriptstyle\,\centerdot\,$}}} }\tau({\mathsf{nil}},c,xy) = \tau({\mathsf{nil}},c,xa{{ \accentset{\mbox{\large\bfseries .}}{a}}} y) { \mathchoice{\mathbin{\raisebox{0.5ex}{$\displaystyle\centerdot$}}} {\mathbin{\raisebox{0.5ex}{$\centerdot$}}} {\mathbin{\raisebox{0.25ex}{$\scriptstyle\,\centerdot\,$}}} {\mathbin{\raisebox{0.1ex}{$\scriptscriptstyle\,\centerdot\,$}}} }\tau(x,a,y).$$ By the given construction of $f_c$ above, this simplifies to $$\tau(c {{ \accentset{\mbox{\large\bfseries .}}{c}}} x,a,y) { \mathchoice{\mathbin{\raisebox{0.5ex}{$\displaystyle\centerdot$}}} {\mathbin{\raisebox{0.5ex}{$\centerdot$}}} {\mathbin{\raisebox{0.25ex}{$\scriptstyle\,\centerdot\,$}}} {\mathbin{\raisebox{0.1ex}{$\scriptscriptstyle\,\centerdot\,$}}} }\tau({\mathsf{nil}},c,xy) = \tau({\mathsf{nil}},c,xa {{ \accentset{\mbox{\large\bfseries .}}{a}}} y) { \mathchoice{\mathbin{\raisebox{0.5ex}{$\displaystyle\centerdot$}}} {\mathbin{\raisebox{0.5ex}{$\centerdot$}}} {\mathbin{\raisebox{0.25ex}{$\scriptstyle\,\centerdot\,$}}} {\mathbin{\raisebox{0.1ex}{$\scriptscriptstyle\,\centerdot\,$}}} }\tau(x,a,y),$$ which is given by ${\mathsf{sw}}({\mathsf{nil}},c,x,a,y)$. From Lemma \[lem:NA-has-equiv\], it is very easy to derive an ${\mathbf{F}(A)}$-algebra structure. We will record this later in Corollary \[cor:NA-has-FA-structure\]. Before going there, we draw another immediate conclusion: \[lem:NA-is-set\] The type ${\mathbf{N}(A)}$ is a set. It suffices to show the that, for any given $\alpha : {\mathbf{N}(A)}$, the type ${\alpha =_{{\mathbf{N}(A)}} \alpha}$ is contractible. We do induction on $\alpha$. Since the goal is a proposition which becomes trivial for all higher constructors, we only need to show the statement for the point constructor $\eta$. Thus, assuming $x : {{\mathsf{List}}({{A}^{\pm}})}$, we need to show that $\eta(x) =_{{\mathbf{N}(A)}} \eta(x)$ is contractible. We do induction again, this time on the list $x$. If $x$ is the empty list ${\mathsf{nil}}$, then the statement is given by Lemma \[lem:NA-locally-set\]. Otherwise, $x$ is $ay$ with $a : {{A}^{\pm}}$. Consider the equivalence $f(a) : {\mathbf{N}(A)} \to {\mathbf{N}(A)}$ from Lemma \[lem:NA-has-equiv\]. It gives us an equivalence $\mathsf{ap}_{f(a)} : \eta(y) = \eta(y) \to \eta(ay) = \eta(ay)$, the domain of which is contractible by the induction hypothesis. Connection between approximations of the free group --------------------------------------------------- In order to make use of ${\mathbf{N}(A)}$ and the results we have found so far, we show in this section that ${\mathopen{}\left\Vert {\mathbf{F}(A)}\right\Vert_{1}\mathclose{}}$ is equivalent to ${\mathbf{N}(A)}$. A direct proof via “maps in both directions which are inverse to each other” would in principle be possible. Our calculations however led to a very messy argument, which did not provide much insight. In this paper, we therefore proceed a bit differently: after constructing ${\mathbf{F}(A)}$- and ${\mathbf{N}(A)}$-algebra structures on both ${\mathbf{N}(A)}$ and ${\mathopen{}\left\Vert {\mathbf{F}(A)}\right\Vert_{1}\mathclose{}}$ (which corresponds to constructing the two functions), we show that the structures are “compatible”, i.e. that a certain ${\mathbf{N}(A)}$-algebra map is also an ${\mathbf{F}(A)}$-algebra morphism. We will later explain in detail what this means. Recall from the statement of Principle \[principle:hinitial\] that an *${\mathbf{F}(A)}$-algebra structure* on a type $X$ consists of a point $u : X$ and a family $f : A \to X \to X$ such that each $f_a$ is an equivalence on $X$, witnessed by some $p: (a : A) \to \mathsf{isequiv}(f_a)$. An ${\mathbf{F}(A)}$-algebra is a type $X$ with such a structure, i.e. a tuple $(X,u,f,p)$. Also recall that an ${\mathbf{F}(A)}$-algebra morphism between $(X,u,f,p)$ and $(Y, v, g, q)$ is a triple $(h, r , s)$, where $h : X \to Y$, $r : f (u) = v$, and $s : h \circ f = g \circ h$. Similarly, we say that a type $Y$ carries an ${\mathbf{N}(A)}$-algebra structure if we have a tuple $(e,t,s,o,h)$ mirroring the constructors of ${\mathbf{N}(A)}$, with ${\mathbf{N}(A)}$-algebra morphisms defined in the obvious way. Then, $({\mathbf{N}(A)}, \eta, \tau, {\mathsf{sw}}, {\mathsf{ov}}, {\mathsf{tr}})$ is homotopy initial among all ${\mathbf{N}(A)}$-algebras. From Lemma \[lem:NA-has-equiv\], we immediately get a canonical ${\mathbf{F}(A)}$-algebra. Note that here and later we write $\_$ (blank) for a “nameless” component which should be clear from the context. \[cor:NA-has-FA-structure\] We have an ${\mathbf{F}(A)}$-algebra $({\mathbf{N}(A)}, \eta({\mathsf{nil}}), \overline{f}, \_)$, where $\overline f$ is given by the function Lemma \[lem:NA-has-equiv\] composed with the embedding ${\mathsf{inl}}: A \to {{A}^{\pm}}$ of $a$ into “positively signed $a$”. Since ${\mathbf{F}(A)}$ carries the initial such structure, we get a canonical map ${\mathbf{F}(A)} \to {\mathbf{N}(A)}$. It does not seem to be the case in general that truncations preserve algebra structure, since this seems to require a choice principle; see e.g. the infinitary branching trees in [@alt-kap:tt-in-tt; @alt-dan-kra:partiality]. Fortunately, it is very simple in our case: \[lem:trFA-has-FA-str\] The type ${\mathopen{}\left\Vert {\mathbf{F}(A)}\right\Vert_{1}\mathclose{}}$ carries an ${\mathbf{F}(A)}$-algebra structure, and ${{\mathopen{}\left|-\right|_{}^{}\mathclose{}}} : {\mathbf{F}(A)} \to {\mathopen{}\left\Vert {\mathbf{F}(A)}\right\Vert_{1}\mathclose{}}$ is an ${\mathbf{F}(A)}$-algebra morphism. This follows easily from the fact that ${{\mathopen{}\left|-\right|_{}^{}\mathclose{}}}$ preserves equivalences. Corollary \[cor:NA-has-FA-structure\] can be reversed if we add a truncation: \[lem:FG-has-NR-structure\] The type ${\mathopen{}\left\Vert {\mathbf{F}(A)}\right\Vert_{1}\mathclose{}}$ carries an ${\mathbf{N}(A)}$-algebra structure. Doing this in detail is tedious, but there is no hidden difficulty. The components corresponding to the constructors $(\eta, \tau, {\mathsf{sw}}, {\mathsf{ov}})$ could all be constructed using ${\mathbf{F}(A)}$ directly, we simply need to throw in ${{\mathopen{}\left|-\right|_{}^{}\mathclose{}}} : {\mathbf{F}(A)} \to {\mathopen{}\left\Vert {\mathbf{F}(A)}\right\Vert_{1}\mathclose{}}$ at the right places. The component corresponding to $\eta$, which has type $$e : {{\mathsf{List}}({{A}^{\pm}})} \to {\mathopen{}\left\Vert {\mathbf{F}(A)}\right\Vert_{1}\mathclose{}},$$ is simply given by composing instances of ${{\mathopen{}\left|{\mathsf{cons}}\right|_{}^{}\mathclose{}}}$ or ${{\mathopen{}\left|{\mathsf{icons}}\right|_{}^{}\mathclose{}}}$ with each other, one for each element of the list $x$ (we use ${{\mathopen{}\left|{\mathsf{cons}}\right|_{}^{}\mathclose{}}}$ for positive list elements and ${{\mathopen{}\left|{\mathsf{icons}}\right|_{}^{}\mathclose{}}}$ for negative ones), and applying them on the unit element ${{\mathopen{}\left|{\mathsf{unit}}\right|_{}^{}\mathclose{}}}$. We write $e(x) {:\equiv}{{\mathopen{}\left|\vec {{\mathsf{cons}}_x}\right|_{}^{}\mathclose{}}}$ for this. For example, if $x$ is the list $a{{ \accentset{\mbox{\large\bfseries .}}{b}}}c$ (where $a$, $b$, $c$ are now all assumed to be positive), then $e(x) \equiv {{\mathopen{}\left|\vec{{\mathsf{cons}}_x}\right|_{}^{}\mathclose{}}} \equiv {{\mathopen{}\left|{\mathsf{cons}}_a\right|_{}^{}\mathclose{}}}({{\mathopen{}\left|{\mathsf{icons}}_b\right|_{}^{}\mathclose{}}}({{\mathopen{}\left|{\mathsf{cons}}_c\right|_{}^{}\mathclose{}}}({{\mathopen{}\left|{\mathsf{unit}}\right|_{}^{}\mathclose{}}})))$. The component corresponding to $\tau$, which has type $$t : (x : {{\mathsf{List}}({{A}^{\pm}})}) \to (a : {{A}^{\pm}}) \to (y : {{\mathsf{List}}({{A}^{\pm}})}) \to e(xa {{ \accentset{\mbox{\large\bfseries .}}{a}}} y) = e(xy),$$ is then given by “whiskering” as in (let us for simplicity assume that $a$ is positive): $$t(x,a,y) {:\equiv}\mathsf{ap}_{{{\mathopen{}\left|\vec{{\mathsf{cons}}(x)}\right|_{}^{}\mathclose{}}}}({{\mathopen{}\left|\mu_2\right|_{}^{}\mathclose{}}}(a,y))$$ The components for ${\mathsf{sw}}$ and ${\mathsf{ov}}$ are essentially naturality of whiskering and $\mu$, respectively, while the fact that we have $1$-truncated ${\mathbf{F}(A)}$ gives us the component for the constructor ${\mathsf{tr}}$. Using that ${\mathbf{F}(A)}$ carries the (homotopy) initial ${\mathbf{F}(A)}$-algebra structure, and ${\mathbf{N}(A)}$ the (homotopy) initial ${\mathbf{N}(A)}$-algebra structure, the statements of Corollary \[cor:NA-has-FA-structure\] and Lemma \[lem:FG-has-NR-structure\] give us maps $h$ and $k$ as follows: $$\label{eq:two-maps} \begin{tikzpicture}[x= 1.15cm , y = -.5cm,baseline=(current bounding box.center)] \node[] (A) at (0,0) {${\mathbf{F}(A)}$}; \node[] (B) at (3,0) {${\mathbf{N}(A)}$}; \node[] (C) at (6,0) {${\mathopen{}\left\Vert {\mathbf{F}(A)}\right\Vert_{1}\mathclose{}}$}; \draw[->] (A) to node[above] {map of ${\mathbf{F}(A)}$-algs} (B); \draw[->] (B) to node[above] {map of ${\mathbf{N}(A)}$-algs} (C); \node[] (L1) at (1.5,0.5) {$h$}; \node[] (L2) at (4.5,0.5) {$k$}; \end{tikzpicture}$$ In the next lemma, we show that *both* these functions are maps of ${\mathbf{F}(A)}$-algebras. This will be sufficient to show that ${\mathbf{N}(A)}$ is a retract of ${\mathopen{}\left\Vert {\mathbf{F}(A)}\right\Vert_{1}\mathclose{}}$. It was a suggestion by Paolo Capriotti that this lemma might lead to a cleaner proof of the property we ultimately want, which, we think, is indeed the case. \[lem:map-is-double-alg\] The map $k$ in is a map of ${\mathbf{F}(A)}$-algebras, with respect to the ${\mathbf{F}(A)}$-algebra structures constructed in Corollary \[cor:NA-has-FA-structure\] and Lemma \[lem:trFA-has-FA-str\]. We need to show that the points and the equivalences are preserved, independently of each other. The point in ${\mathbf{N}(A)}$ is $\eta({\mathsf{nil}})$, which is mapped to the ${{\mathopen{}\left|{\mathsf{unit}}\right|_{}^{}\mathclose{}}}$ as required. For the equivalence, we only need to check that the underlying functions match accordingly. This corresponds to showing commutativity of the following square, for any given $c : A$: $$\label{eq:f-maps} \begin{tikzpicture}[x= 1cm , y = -0.4cm,baseline=(current bounding box.center)] \node[] (A) at (0,0) {${\mathbf{N}(A)}$}; \node[] (B) at (4,0) {${\mathbf{N}(A)}$}; \node[] (C) at (0,4) {${\mathopen{}\left\Vert {\mathbf{F}(A)}\right\Vert_{1}\mathclose{}}$}; \node[] (D) at (4,4) {${\mathopen{}\left\Vert {\mathbf{F}(A)}\right\Vert_{1}\mathclose{}}$}; \draw[->] (A) to node[above] {$f_c$ (Lem~\ref{lem:NA-has-equiv})} (B); \draw[->] (C) to node[below] {${{\mathopen{}\left|{\mathsf{cons}}_c\right|_{}^{}\mathclose{}}}$} (D); \draw[->] (A) to node[left] {$k$} (C); \draw[->] (B) to node[right] {$k$} (D); \end{tikzpicture}$$ We do induction on $\alpha : {\mathbf{N}(A)}$. The goal is an equality in a $1$-type, i.e. a set, which means that we only have to check the constructors $\eta$ and $\tau$. Tracing the explicit construction in Lemma \[lem:FG-has-NR-structure\] through the square, we can check directly that the square commutes in both cases (strictly speaking, in the case for $\tau$, it is a cube): $$\label{eq:f-maps1} \begin{tikzpicture}[x= 1cm , y = -0.4cm,baseline=(current bounding box.center)] \node[] (A) at (0,0) {$\eta(x)$}; \node[] (B) at (4,0) {$\eta(cx)$}; \node[] (C) at (0,4) {${{\mathopen{}\left|\vec{{\mathsf{cons}}_x}\right|_{}^{}\mathclose{}}}({{\mathopen{}\left|{\mathsf{unit}}\right|_{}^{}\mathclose{}}})$}; \node[] (D) at (4,4) {${{\mathopen{}\left|{\mathsf{cons}}_c\right|_{}^{}\mathclose{}}}({{\mathopen{}\left|\vec{{\mathsf{cons}}_x}\right|_{}^{}\mathclose{}}}({{\mathopen{}\left|{\mathsf{unit}}\right|_{}^{}\mathclose{}}}))$}; \draw[|->] (A) to node[above] {} (B); \draw[|->] (C) to node[below] {} (D); \draw[|->] (A) to node[left] {} (C); \draw[|->] (B) to node[right] {} (D); \end{tikzpicture}$$ and: $$\label{eq:f-maps2} \begin{tikzpicture}[x= 1cm , y = -0.4cm,baseline=(current bounding box.center)] \node[] (A) at (0,0) {$\tau(x,a,y)$}; \node[] (B) at (4,0) {$\tau(cx,a,y)$}; \node[] (C) at (0,4) {$\mathsf{ap}_{|\vec{{\mathsf{cons}}(x)}|}(|\mu_2|(a,y))$}; \node[] (D) at (4,4) {$\mathsf{ap}_{|{\mathsf{cons}}_c|(|\vec{{\mathsf{cons}}(x)}|)}(|\mu_2|(a,y))$}; \draw[|->] (A) to node[above] {} (B); \draw[|->] (C) to node[below] {} (D); \draw[|->] (A) to node[left] {} (C); \draw[|->] (B) to node[right] {} (D); \end{tikzpicture}$$ The commutativity is judgmental in the first square, and the second square only uses the usual equality $\mathsf{ap}_g \circ \mathsf{ap}_f = \mathsf{ap}_{g \circ f}$. This finally allows us to show: By the previous lemma, the composition of the maps in is an ${\mathbf{F}(A)}$-algebra map. But so is the map ${{\mathopen{}\left|-\right|_{}^{}\mathclose{}}} : {\mathbf{F}(A)} \to {\mathopen{}\left\Vert {\mathbf{F}(A)}\right\Vert_{1}\mathclose{}}$ by Lemma \[lem:trFA-has-FA-str\]. Since ${\mathbf{F}(A)}$ is the *initial* such algebra, these two functions must coincide, which means that ${{\mathopen{}\left|-\right|_{}^{}\mathclose{}}} : {\mathbf{F}(A)} \to {\mathopen{}\left\Vert {\mathbf{F}(A)}\right\Vert_{1}\mathclose{}}$ factors through ${\mathbf{N}(A)}$. We know from Lemma \[lem:NA-is-set\] that ${\mathbf{N}(A)}$ is a set. This implies that ${\mathopen{}\left\Vert {\mathbf{F}(A)}\right\Vert_{1}\mathclose{}}$ is a set, which is the second part of the theorem. To see the first part, take $q : {\mathbf{F}(A)}$. ${\mathbf{F}(A)}$ having trivial fundamental groups means that ${\mathopen{}\left\Vert q =_{{\mathbf{F}(A)}} q\right\Vert_{0}\mathclose{}}$ is contractible. By [@hott-book], we have $${\mathopen{}\left\Vert q =_{{\mathbf{F}(A)}} q\right\Vert_{0}\mathclose{}} \; \simeq \; \left( {{\mathopen{}\left|q\right|_{}^{}\mathclose{}}} =_{{\mathopen{}\left\Vert {\mathbf{F}(A)}\right\Vert_{1}\mathclose{}}} {{\mathopen{}\left|q\right|_{}^{}\mathclose{}}}\right).$$ The second type is contractible since ${\mathopen{}\left\Vert {\mathbf{F}(A)}\right\Vert_{1}\mathclose{}}$ is a set. Having proved the main result, we add two results that now have become very easy: \[lem:FANA\] For a set $A$, the two approximations of the free higher group which we have considered are equivalent, i.e. ${\mathopen{}\left\Vert {\mathbf{F}(A)}\right\Vert_{1}\mathclose{}} \simeq {\mathbf{N}(A)}$. From the argument in the proof of the previous theorem, we can follow that ${\mathopen{}\left\Vert {\mathbf{F}(A)}\right\Vert_{1}\mathclose{}}$ is a retract of ${\mathbf{N}(A)}$. Thus, we still need to show that the composition ${\mathbf{N}(A)} \to {\mathopen{}\left\Vert {\mathbf{F}(A)}\right\Vert_{1}\mathclose{}} \to {\mathbf{N}(A)}$ is the identity. But now that we know that everything is a set, it is easy to do this by induction on ${\mathbf{N}(A)}$. The type ${\mathopen{}\left\Vert {\mathbf{F}(A)}\right\Vert_{1}\mathclose{}}$ is equivalent to the purely set-based free group over $A$ as constructed in [@hott-book Chp 6.11]. If Question \[eq:main-question\] can be answered positively, then our free group does indeed generalise the free group construction of [@hott-book] from only sets to arbitrary types. Since ${\mathbf{N}(A)}$ is a set by Lemma \[lem:NA-is-set\], it is easy to see that it is equivalent to the set-quotient of ${{\mathsf{List}}({{A}^{\pm}})}$ by the relation that identifies a list with the list then one gets after reducing; this is essentially because, when we know that ${\mathbf{N}(A)}$ is a set, the constructors ${\mathsf{sw}}$ and ${\mathsf{ov}}$ become obsolete, and what remains is just this set-quotient. But this set-quotient is exactly the purely set-based free group of [@hott-book] by [@hott-book Thm 6.11.7]. If Question \[eq:main-question\] turns out to have a positive answer, then ${\mathbf{F}(A)}$ and ${\mathopen{}\left\Vert {\mathbf{F}(A)}\right\Vert_{1}\mathclose{}}$ are equivalent, and everything that holds for the latter is also true for the former. Conclusions {#sec:conclusions} =========== The central and guiding question of this paper was the problem of showing that the free $\infty$-group ${\mathbf{F}(A)}$ over a set is a set as well. We have proved a first approximation of this, namely that ${\mathbf{F}(A)}$ has trivial fundamental groups. This is done entirely in “book HoTT”, the type theory developed in [@hott-book]. It would be very interesting to formalise the complete argument in a proof assistant, and we expect that this would be challenging. For example, the use of list concatenation in the constructors of the higher inductive type ${\mathbf{N}(A)}$ would lead to many application of *transport* (*substitution*). It is likely that a different representation of ${\mathbf{N}(A)}$ and the reduction relation would enable a more elegant formalisation. For a human reader, the presentation in terms of lists is the most intuitive and understandable one that we could think of. Brunerie has discussed the James construction in homotopy type theory [@DBLP:journals/corr/abs-1710-10307]. In this context, a type $A$ with a point $\star_A : A$ is given, and the higher inductive type $JA$ is defined to be the *free monoid* over $A$ where $\star_A$ plays the role of the neutral element. Brunerie then constructs a non-recursive version of $JA$. Of special interest for him is the case that $A$ is connected (i.e. ${\mathopen{}\left\Vert A\right\Vert_{0}\mathclose{}}$ is contractible), and in this case, $JA$ becomes very similar to our free group. However, connectedness would be a very unnatural assumption in the present paper; in fact, since we are interested in the case that $A$ is a set, our case of interest is orthogonal to Brunerie’s. If $A$ is not known to be connected, then, compared to our ${\mathbf{F}(A)}$, $JA$ is lacking the condition that every ${\mathsf{cons}}_a$ is invertible, which is the main source of difficulty in our work. Related to the current paper is also previous work by Capriotti, Vezzosi, and the current first author [@capKraVez_elimTruncs]. That work gives a necessary and sufficient condition for a function $X \to Y$ to factor through ${\mathopen{}\left\Vert X\right\Vert_{n}\mathclose{}}$, assuming that $Y$ is $(n+1)$-truncated. In the current paper, we have been particularly interested in the situation that $n$ is $0$, $X$ is the “level $0$ approximation” of the free group (see the description after Definition \[def:NA\]), and $Y$ is ${\mathopen{}\left\Vert {\mathbf{F}(A)}\right\Vert_{1}\mathclose{}}$. The reason why we have not directly applied the result of [@capKraVez_elimTruncs] is that, in our case of interest, showing the mentioned condition is tricky. This difficulty corresponds to what in our presentation has made the more refined approximation with the constructors ${\mathsf{sw}}$ and ${\mathsf{ov}}$ necessary. We do not know whether there is an alternative proof of our main result which uses [@capKraVez_elimTruncs] directly. Let us further analyse the methods we have used in the paper. In principle, the strategy which we have developed should be applicable to more general results than the one we have proved; for example, with some more effort, we expect that it should be possible to show that ${\mathopen{}\left\Vert {\mathbf{F}(A)}\right\Vert_{2}\mathclose{}}$ and ${\mathopen{}\left\Vert {\mathbf{F}(A)}\right\Vert_{3}\mathclose{}}$ (which are better approximations to ${\mathbf{F}(A)}$) are sets. The obvious attempt to do this is to work with a “better” non-recursive approximation, i.e. a refined version of ${\mathbf{N}(A)}$ which would use higher path constructors to guarantee the coherence of ${\mathsf{sw}}$ and ${\mathsf{ov}}$. One would then include a $2$- or $3$-truncation constructor instead of the $1$-truncation constructor ${\mathsf{tr}}$. It seems plausible that this could work; for example, instead of constructing a weakly constant function $$\label{eq:Red-to-eq} {\mathsf{Red}}(x) \to \eta(x) = \eta({\mathsf{nil}})$$ as in Lemma \[lem:redIsNeutral-wconst\], we would have to construct a constant function satisfying one or more coherence conditions [@kraus_generaluniversalproperty], and the new constructors of ${\mathbf{N}(A)}$ would be chosen in such a way that this would be possible. The additional value that such a generalisation would give us is unclear. Of course, what we want is to show that ${\mathbf{F}(A)}$ is a set, not just a finite truncation of it. If we try to use our approach, it seems we would need to find a way to encode the *whole infinite* tower of coherences in a non-recursive type, and it looks suspiciously similar to the long-standing open problem of defining semisimplicial types in HoTT [@uf:semisimplicialtypes]. (To clarify, we would not need a single HIT with infinitely many constructors, since we could take a sequential colimit; and the absence of a general version of Whitehead’s principle does not seem to be a problem as long as we can show that satisfies the coherence conditions given in [@kraus_generaluniversalproperty], which does not rely on hypercompleteness either.) Our problem of showing that ${\mathbf{F}(A)}$ is a set is not much different from the open problem of HoTT which asks whether the suspension of a set is a $1$-type; as we have already discussed, what we are asking is essentially whether a the suspension of a set with a distinguished isolated point is a $1$-type. Thus, our question is slightly weaker and, as far as we can see, an answer to the weaker question would not be sufficient to answer the more general question.[^3] However, it seems plausible that an approach similar to ours is applicable to the more general question as well. In this case, being able to encode infinite towers of coherences could potentially be key to both open problems, although of course there would still be a lot of work to do (which might or might not even be impossible). Theories such as Voevodsky’s *homotopy type system* (HTS) [@voe:hts], *two-level type theory* (2LTT) [@alt-cap-kra:two-level; @ann-cap-kra:two-level] or *computational higher type theory* [@2017arXiv171201800A] are variations of standard HoTT in which such infinite structures can be constructed. We believe it would be worth investigating whether the “suspension of a set” problem can be resolved in such systems. Our preliminary investigations hint that it is at least be possible to define a “completely non-recursive” version of ${\mathbf{F}(A)}$, which would be a starting point. However, actually *using* this construction to mimic the proof that we have given in this paper is, of course, a completely different story. If (we are now in the realm of complete speculation) it turns out that HTS can prove that the suspension of a set is a $1$-type, it would be even more interesting whether “standard HoTT” can do it as well. This is because one would need to come up with a completely different argument in standard HoTT, and if it turns out that the open problem is independent of standard HoTT, hope for a conservativity result would be lost for all theories that are powerful enough to encode semisimplicial types. Recall that we have a conservativity result for 2LTT, due to Capriotti [@paolo:thesis], which says that a fibrant type in 2LTT can only be inhabited if the corresponding type in HoTT (assuming it exists) is inhabited as well. This however only works for a version of 2LTT where we do *not* have semisimplicial types in the usual formulation (we only have semisimplicial types indexed over the pretype of strict natural numbers, but not over the type fibrant natural numbers). Thus, in this version of 2LTT, the sketched approach to solve the open problem regarding the suspension of a set would not work. This might be more than a coincidence. ### Acknowledgements {#acknowledgements .unnumbered} We are very grateful to Paolo Capriotti for many discussions on the topic, and for several remarks which have influenced this paper. Most importantly, the construction of a free group using the constructors of the free monoid and conditions ensuring that ${\mathsf{cons}}_a$ is an equivalence is due to Paolo, as well as the decomposition shown in . It was also him who suggested using the statement of Lemma \[lem:map-is-double-alg\] to complete the main proof of the paper, which has led to a cleaner proof than if we had done it with a more direct argument. We further thank Rafaël Bocquet for discussions on free groups, and the anonymous reviewers whose comments have helped us to improve the presentation and readability of the paper. [^1]: This work was supported by the Engineering and Physical Sciences Research Council (EPSRC), grant reference EP/M016994/1, and by USAF, Airforce office for scientific research, award FA9550-16-1-0029.\ *Disclaimer as requested by publisher:* This paper has been published in the proceedings of LiCS’18, <https://doi.org/10.1145/3209108.3209183>. [^2]: The more accurate notation might be $\mathsf{ap}_{f_c}(\tau(x,a,y))$. [^3]: Related is the discussion *Does “adding a path” preserve truncation levels?* at <https://groups.google.com/forum/#!topic/homotopytypetheory/gVmcvaOeD5c>.

CERN-TH/2001-267\ ROMA-1325/01\ SISSA-77/2001/EP\ [**Anomalies in orbifold field theories**]{} [C.A. Scrucca$^{a}$, M. Serone$^{b}$, L. Silvestrini$^{c}$ and F. Zwirner$^{c}$]{}\ ${}^a$ [*CERN, CH-1211 Geneva 23, Switzerland*]{} ${}^b$ [*ISAS-SISSA, Via Beirut 2-4, I-34013 Trieste, Italy*]{}\ [*INFN, sez. di Trieste, Italy*]{} ${}^c$ [*Dip. di Fisica, Univ. di Roma “La Sapienza” and INFN, sez. di Roma*]{}\ [*P.le Aldo Moro 2, I-00185, Rome, Italy*]{} **Abstract** > We study the constraints on models with extra dimensions arising from local anomaly cancellation. We consider a five-dimensional field theory with a $U(1)$ gauge field and a charged fermion, compactified on the orbifold $S^1/(\Z_2 \times \Z_2^\prime)$. We show that, even if the orbifold projections remove both fermionic zero modes, there are gauge anomalies localized at the fixed points. Anomalies naively cancel after integration over the fifth dimension, but gauge invariance is broken, spoiling the consistency of the theory. We discuss their implications for realistic supersymmetric models with a single Higgs hypermultiplet in the bulk, and possible cancellation mechanisms in non-minimal models. Introduction ============ Theories formulated in $D>4$ space-time dimensions may lead to a geometrical understanding of the problems of mass generation and symmetry breaking. Orbifold compactifications [@Dixon] of higher-dimensional theories are simple and efficient mechanisms to reduce their symmetries and to generate four-dimensional (4-D) chirality. Phenomenologically interesting orbifold models can be formulated, either as explicit string constructions or as effective higher-dimensional field theories. The field-theoretical approach to orbifolds is currently fashionable because of its apparent simplicity and flexibility. However, it is well known that the rules for the construction of consistent string-theory orbifolds are quite stringent, and automatically implement a number of consistency conditions in the corresponding effective field theories: in particular, the cancellation of gauge, gravitational and mixed anomalies. Since anomalies are infrared phenomena, if we start from a consistent string model (‘top–down’ approach), anomaly cancellation must find an appropriate description in the effective field theory. Such a description, however, may be non-trivial, as for the Green–Schwarz [@gs] or the inflow [@ch] mechanisms. If, instead, we decide to work directly at the field-theory level (‘bottom–up’ approach), great care is needed, since orbifold projections do not necessarily preserve the quantum consistency of a field theory (as discussed, for example, in [@hw]). In particular, the question of anomaly cancellation must be explicitly addressed. A first step in this direction was taken in ref. [@Cohen], which discussed the chiral anomaly in a five-dimensional (5-D) theory compactified on the orbifold $S^1/\Z_2$. It was found that, in such a simple context, naive 4-D anomaly cancellation is sufficient to ensure 5-D anomaly cancellation. For a 5-D fermion of unit charge, and a chiral action of the $\Z_2$ projection, the 5-D anomaly is localized at the orbifold fixed points, and is proportional to the 4-D anomaly: \[prevres\] \_M J\^M (x,y) = [1 4]{} [Q]{}(x,y) , where[^1] $J^M$ is the 5-D current and \[qcal\] [Q]{}(x,y) = F\_(x,y) F\^(x,y) is proportional to the 4-D chiral anomaly from a charged Dirac spinor in the external gauge potential $A_{\mu}(x,y)$. In our notation: $M=[(\mu=0,1,2,3),4]$; $x\equiv (x^{0,1,2,3})$ are the first four coordinates, $y \equiv x^4$ is the fifth coordinate, compactified on a circle of radius $R$; $y=0,\pi R$ are the two fixed points with respect to the $Z_2$ symmetry $y \rightarrow - y$; $g_5$ is the 5-D gauge coupling constant. In this letter we show that the phenomenon discussed in [@Cohen] does not persist in more general cases. To be definite, we consider a 5-D field theory with a $U(1)$ gauge field $A_M$ and a massless fermion $\psi$ of unit charge, compactified on the orbifold $S^1/(\Z_2 \times \Z_2^\prime)$. The action of the two parities are $y \rightarrow -y$ and $y' \rightarrow - y'$, respectively, where $y' = y - \pi R /2$. Both the gauge and the fermion fields are taken to be periodic on the circle. We decompose the Dirac spinor $\psi$ into left and right spinors with parities $(+,-)$ and $(-,+)$, respectively: $\psi \equiv \psi^{+-} + \psi^{-+}$. Notice that a standard fermion mass term is forbidden by the $\Z_2 \times \Z_2^\prime$ symmetry. As for the gauge field, we assign $(+,+)$ parities to $A_\mu$, $(-,-)$ to $A_4$. Although the theory has no massless 4-D chiral fermion, a non-vanishing anomaly is induced, given by eq. (\[ano\]) below. The theory can be trivially supersymmetrized, by embedding its field content in a $U(1)$ vector multiplet and a charged hypermultiplet. From the point of view of anomalies, our simple example reproduces the essential features of a recently proposed phenomenological model [@Barber], whose light spectrum contains just the states of the Standard Model (SM), with an anomaly-free fermion content. The underlying 5-D theory is supersymmetric, with vector multiplets containing the SM gauge bosons, and hypermultiplets containing the SM quarks and leptons. In addition, the model of ref. [@Barber] has just one charged hypermultiplet, which contains the SM Higgs boson. Such a model has received some attention because it may give a prediction for the Higgs mass, even though it was recently shown [@Nilles] that the Higgs self-energy receives a quadratically divergent one-loop contribution. The latter corresponds to the appearance of a Fayet–Iliopoulos (FI) term, with divergences localized at the orbifold fixed points, which immediately hints at a possible connection with anomalies. The content of the present letter is organized as follows. We begin by showing that, even if there are no 4-D massless fermions in the spectrum, our simple 5-D theory is actually anomalous. Localized anomalies, with opposite signs, appear at the fixed points of the two orbifold projections. The integrated anomaly vanishes, reflecting the absence of any one-loop anomaly among 4-D massless states, but there are anomalous triangle diagrams when at least one of the external states is a massive Kaluza–Klein (KK) mode. We focus our attention on the $U(1)^3$ gauge anomaly, which we explicitly compute along the lines of [@Cohen]. In realistic extensions, such as [@Barber], similar results would hold for the $U(1)_Y^3$, $U(1)_Y$–$SU(2)^2_L$ and $U(1)_Y$–gravitational anomalies. We then argue that this anomaly leads to a breakdown of 4-D gauge invariance. Hence, in its minimal form, the model is inconsistent, even as an effective low-energy theory. Next, we consider the supersymmetric extension of our simple theory, and compute the precise expression for the one-loop FI term. Finally, we discuss the possible modifications that could restore the consistency of the theory. $U(1)$ anomalies {#sec:anomalies} ================ In this section, we take the theory defined in the Introduction and we compute the $U(1)^3$ anomaly, following closely the method and the notation of ref. [@Cohen] (for an early computation of this type, see also ref. [@Binetruy]). The KK wavefunctions $\xi^{ab}$ for fields $\varphi^{ab}$ of definite $\Z_2 \times \Z_2^\prime$ parities $(a,b=\pm)$ are defined as: $$\begin{aligned} \xi^{++}_{n \ge 0}(y) \a\equiv\a \frac{\eta_n}{\sqrt{\pi R}} \cos \frac {2 n y}R \, , \;\;\;\;\; \xi^{+-}_{n>0}(y) \equiv \frac{1}{\sqrt{\pi R}} \cos \frac {(2 n -1)y}R \, , \nn \\ \xi^{--}_{n>0}(y) \a\equiv\a \frac{1}{\sqrt{\pi R}} \sin \frac {2 n y}R \, , \;\;\;\;\; \xi^{-+}_{n>0}(y) \equiv \frac{1}{\sqrt{\pi R}} \sin \frac {(2 n -1)y}R \,, \label{eq:icchese} \end{aligned}$$ where $\eta_n$ is $1/\sqrt{2}$ for $n=0$ and $1$ for $n>0$. They form a complete orthonormal basis of periodic functions on $S^1$, with given $\Z_2 \times \Z_2^\prime$ parities. The Fourier modes of a field $\varphi^{ab}$ are defined as: $$\begin{aligned} \varphi^{ab}_{n}(x) \a \equiv \a\int_0^{2 \pi R} \!\!\! dy\, \xi^{ab}_n(y) \, \varphi^{ab}(x,y) \,, \label{eq:fthiggs} \end{aligned}$$ and have a mass given by $m_{2n+(ab-1)/2}$, where $m_n = n/R$. In the gauge $A_4 = 0$, the 4-D Lagrangian for the Fourier modes $\psi_n \equiv \psi_n^{+-} + \psi_n^{-+}$ can be written as \[eq:lagrangian\] [L]{}= \_[m,n]{} |\_m \_n , where $\Aslash_{mn} \equiv \Aslash_{mn}^{+-} P_+ + \Aslash_{mn}^{-+} P_-$, with $P_\pm = (1 \pm \gamma_5)/2$, and [^2] A\^\_[mn]{}(x) \_0\^[2 R]{} dy \^\_m(y) \^\_n(y) A\_(x,y) in terms of the $U(1)$ connection $A_\mu$. Interpreting $\psi_n$ as a single fermion with a flavour index and chiral couplings to the gauge field $A^{\mu}_{mn}$ through the currents ${J^{\mu}_{\pm}}_{mn} = \bar \psi_m \gamma^\mu P_\pm \psi_n$, it is straightforward to adapt the standard computation of anomalies to obtain: \_\_[mn]{} = (m\_[2m-1]{} J\^4\_[mn]{} + m\_[2n-1]{} J\^4\_[mn]{}) \_[k&gt;0]{} F\_[mk]{}\^ F\^\_[kn]{} , \[mm\] where ${J^4_{\pm}}_{mn} = \bar \psi_m i\gamma_5P_\pm \psi_n$. Equation (\[mm\]) can be easily Fourier-transformed back to configuration space by convolution with $\xi^{\pm \mp}_m(y)\, \xi^{\pm \mp}_n(y)$. Using completeness, this yields \_M J\^[M]{}\_(x,y) = 12 \_[k&gt;0]{} \[\_k\^(y)\]\^2 [Q]{}(x,y) , where the quantity ${\cal Q}$ was defined in eq. (\[qcal\]). The anomaly in the vector current $J^{M}(x,y) = J^M_+(x,y) + J^M_-(x,y)$ is then proportional to \_[k &gt; 0]{} = 14 e\^[-2iy/R]{} \_[l=-]{}\^(y - l R/2) , hence \_M J\^[M]{} (x,y) = 18 (x,y) . \[ano\] Therefore, although the integrated anomaly vanishes, there are anomalies, localized at the fixed points, that are equal in magnitude to $1/4$ (or $1/2$ if we sum the contribution from identified fixed points) of the anomaly from a 4-D Weyl fermion. The full 5-D theory is thus inconsistent (at least in its minimal form). Let us now rewrite eq. (\[ano\]) in terms of standard Fourier modes of the current and gauge fields. Recalling that both have $(+,+)$ parities, the Fourier transform of (\[ano\]) takes the form: q\_M J\^M\_n(q) = 1[g\_5]{} \_[i,j=0]{}\^ q\_M T\^[M ]{}\_[n i j]{}(p,q) A\_[i]{}(p) A\_[j]{}(q-p) , where q\_M T\^[M ]{}\_[n i j]{}(p,q) = \_n \_i \_j \_[n+i+j,odd]{} \^ p\_ q\_, \[qT\] with $g_4 = g_5/\sqrt{2 \pi R}$. This quantity encodes the triangular anomaly between three external KK modes of the photon with indices $(n,i,j)$, as illustrated in fig. 1. (280,130)(0,-10) (35,47)[$T_{nij}^{M,N,P}(p,q) =$]{} (113,60)[$A^M_i(p)$]{} (215,10)[$A^N_j(q)$]{} (215,90)[$A^P_n(-p-q)$]{} (150,50)(200,80) (200,80)(200,20) (200,20)(150,50) (120,50)(150,50)[2]{}[6]{} (200,80)(215,105)[2]{}[6]{} (200,20)(215,-5)[2]{}[6]{} -10pt This anomaly vanishes for $n+i+j=$ even, and in particular for $n = i = j = 0$, reflecting the fact that there is no 4-D anomaly for the massless modes: all non-vanishing anomalous diagrams involve at least one massive mode. These diagrams make the full theory inconsistent. However, it may be asked whether the low-energy effective theory obtained by integrating out all massive modes could be consistent. This is not the case, because gluing such diagrams through heavy lines produces 4-D gauge symmetry breaking effective interactions among zero-modes. Consider for instance a 3-loop diagram obtained by gluing two anomalous triangles through two massive photons, as depicted in fig. 2. This represents a contribution to the two-point function $\Pi^{\mu \nu}$ of the zero-mode photon that violates gauge invariance. The non-vanishing longitudinal component of $\Pi^{\mu \nu}$ is encoded in $q_\mu q_\nu \Pi^{\mu \nu}(q)$, which feels only the anomalous part of the triangular subdiagram [@agwgj]. Another example is the four-point function involving two longitudinal and two transverse zero-mode photons, which receives a non-vanishing finite two-loop contribution controlled by the anomaly. (270,90)(0,13) (0,47)[$\Pi^{\mu \nu}(q) =$]{} (60,60)[$A^\mu_0(q)$]{} (240,60)[$A^\nu_0(-q)$]{} (100,50)(150,80) (150,80)(150,20) (150,20)(100,50) (70,50)(100,50)[2]{}[6]{} (150,80)(180,80)[2]{}[6]{} (150,20)(180,20)[2]{}[6]{} (230,50)(260,50)[2]{}[6]{} (180,80)(230,50) (230,50)(180,20) (180,20)(180,80) -10pt These gauge anomalies could be computed in an independent way by using their well-known relation with chiral anomalies and index theorems, which is particularly clear in Fujikawa’s approach. In this formalism, the integrated chiral anomaly of a 4-D Dirac fermion is encoded in the quantity ${\rm Tr}_{D=4}\,[\gamma_5] = {\rm index}(\Dslash)$, where $\Dslash$ is the Dirac operator. In the case of a 5-D theory compactified on $S^1$, the anomaly vanishes, since the Hilbert space splits into two identical components of opposite chirality and ${\rm Tr}_{D=5}\,[\gamma_5] = 0$. This can be easily extended to an $S^1/(\Z_2 \times \Z_2^\prime)$ orbifold compactification. The trace must now be restricted to invariant states only; this can be achieved by inserting into the unconstrained trace a $\Z_2 \times \Z_2^\prime$ projector $P$. Denoting by $g$ and $g^\prime$ the generators of $\Z_2$ and $\Z_2^\prime$ respectively, the explicit expression of this projector is $P=\frac 14(1+g+g^\prime + g g^\prime)$. Each element in $P$, when inserted in the trace, leads to a so-called equivariant index of the Dirac operator. This has a non-vanishing support only at the fixed points of the element. The identity in $P$ gives a vanishing result as in the $S^1$ compactification. Similarly, the $g g^\prime$ element also gives a vanishing contribution, because it generates a translation along the compact direction that does not affect chirality. On the other hand, the elements $g$ and $g^\prime$ act chirally on the Dirac fermion $\psi$ ($g \psi g^{-1} = \gamma_5 \psi$, $g^\prime \psi {g^\prime}^{-1} = -\gamma_5 \psi$) and give a non-vanishing contribution. Both have two fixed points, and the integrated anomaly is thus : \_[D=5]{}\[P\_5\] = 14 \_[i=1]{}\^4 [index]{}()|\_[y\_i]{}\ =14 d\^4x , where the relative sign between the contributions associated with $g$ and $g^\prime$ is due to their opposite action on fermions. This leads to (\[ano\]). Supersymmetric models ===================== The result found for the anomalies in section \[sec:anomalies\] can be trivially extended to supersymmetric configurations, where the $U(1)$ gauge field belongs to a 5-D $N=1$ vector multiplet and the Dirac fermion $\psi$ to a charged hypermultiplet. As such, all the above considerations apply also to the model of ref. [@Barber]. In particular, focusing on the Higgs hypermultiplet, doublet under $SU(2)_L$ with hypercharge $Y=+1$, we get a localized $U(1)_Y^3$ anomaly that is twice the one in (\[ano\]). The reader may wonder whether such localized anomaly has any relation with the one-loop FI term recently found in [@Nilles]. The method of the previous section can be easily extended to the computation of the full one-loop FI term. The relevant part of the Lagrangian is \[lagfi\] [L]{}=-\_[m,n]{} (\_m\^[++]{})\^\_n\^[++]{}\ - \_[m,n]{} (\_m\^[–]{})\^\_n\^[–]{} , where $\phi_m^{\pm\pm}$ are the modes, defined according to eqs. (\[eq:icchese\]) and (\[eq:fthiggs\]), of the two scalars in the Higgs hypermultiplet, and D\^\_[mn]{}(x) \_0\^[2 R]{} dy \^\_m(y) \^\_n(y) D(x,y) , where $D(x,y)$ is the third component of the triplet of $N=2$ auxiliary fields. Considering again the mode indices as flavour indices, we find for the FI term: \[eq:FI-inizio\] [F]{}(x) = \_[n0]{} T\_n (D\_[nn]{}\^[++]{} - D\_[nn]{}\^[–]{})(x) , where T\_n = ig\_5 1[p\^2 - m\_[2n]{}\^2]{} = (\^2 - m\_[2n]{}\^2 ) , \[Tn\] where $\Lambda$ is an ultraviolet cutoff. By Fourier-transforming back to configuration space, we can write (x) = \_0\^[2 R]{} dy (y) D(x,y) , where the exact profile of $\xi(y)$ can be explicitly evaluated. By first summing over the KK states, the 4-D momentum integral is convergent for generic $y$, yielding: (y) = \_[n0]{}T\_n= , \[fiexp\] where $\tilde y = y - \pi R /2 \sum_{l>0} \theta(y - l\pi R/2)$ is the restriction of $y$ to the interval $[0,\pi R/2[$. Equation (\[fiexp\]) diverges as $\tilde y^{-3}$ when $\tilde y$ tends to $0$, which corresponds to $y$ approaching one of the four fixed points $y_i = (i-1)\pi R/2$ $(i=1,2,3,4)$. Away from the fixed points, there is only a finite bulk contribution. The divergent part of $\xi(y)$ is easily evaluated by going back to (\[Tn\]) and using: \_[k 0]{} = 14 \_[i=1]{}\^4 (y - y\_i) ,\ \_[k 0]{} m\_[2k]{}\^2 = - 1[16]{} \_[i=1]{}\^4 \^(y-y\_i) . Hence, the structure of the FI term is: (x) = \_[i=1]{}\^4 + \_0\^[2 R]{} dy K(y) D(x,y) , with $K(y)$ being a finite function. Therefore, the divergent part of the induced FI term is localized at the orbifold fixed points, as the anomaly. This is a remnant of the relation between FI terms and mixed $U(1)$–gravitational anomalies in supersymmetric theories. Outlook ======= We have seen that orbifold field theories can be anomalous even in the absence of an anomalous spectrum of zero modes. It is then important to understand whether there exist anomaly cancellation mechanisms, and whether they can be consistently implemented: for definiteness, we discuss this issue by making reference once more to the case of $S^1/(\Z_2 \times \Z_2^\prime)$. One possibility would be to add localized fermions at the fixed points, analogous to the twisted sectors of string compactifications. However, as we have seen in section 2, a bulk fermion produces only half of the anomaly of a Weyl fermion at each fixed point. Therefore, this possibility may be generically cumbersome to realize without the guidance of an underlying string theory. Another possibility would be to implement an anomaly cancellation mechanism of the Green–Schwarz [@gs] or inflow [@ch] type. The former would lead to a spontaneous breaking of the $U(1)$ gauge symmetry [@gs4]. For the latter, we must cope with the fact that a 5-D Chern–Simons term $\epsilon^{MNOPQ} A_M F_{NO} F_{PQ}$ (see [@Arkani-Hamed:2001tb]) cannot be added to the bulk Lagrangian, because it is not invariant under the two orbifold projections. However, we can imagine more general possibilities. For example [^3], we could introduce a bosonic field $\chi$ with $(-,-)$ periodicities and try to introduce the Chern–Simons term in combination with $\chi$. The field $\chi$ should then dynamically get a vacuum expectation value with a non-trivial $y$-profile $$\label{eq:epsi} \langle \chi(y) \rangle \propto \left\{ \begin{array}{cc} +1\,, & 0<y<\frac{\pi R}{2} \qquad (\mathrm{mod.}\; \pi R) \\ -1\,, & \frac{\pi R}{2}<y< \pi R \qquad (\mathrm{mod.}\; \pi R) \end{array} \right.$$ (breaking spontaneously the $\Z_2\times \Z_2^\prime$ discrete symmetry), thereby generating a sort of magnetic charge for the fixed points and leaving only very massive fluctuations. The resulting Chern–Simons term could then cancel, for an appropriate value of the coefficient, the one-loop anomaly. Notice that this mechanism can work only when the integrated anomaly vanishes. It is interesting to observe that the presence and the structure of the anomalies that we have found in the $S^1/(\Z_2 \times \Z_2^\prime)$ orbifold could have been anticipated [^4] by analysing the intermediate models on $S^1/\Z_2$ or $S^1/\Z_2^\prime$. Indeed, the $S^1/\Z_2$ and $S^1/\Z_2^\prime$ models have anomalies given by eq. (\[prevres\]) and localized at the two $\Z_2$ and $\Z_2^\prime$ fixed points respectively, but with opposite sign, reflecting the difference between the $\Z_2$ and $\Z_2^\prime$ actions on the fermions. For supersymmetric models, all the above considerations apply, but supersymmetry poses further constraints. It seems then quite difficult to get a consistent SUSY field theory on the $S^1/(\Z_2 \times \Z_2^\prime)$ orbifold with a single bulk Higgs hypermultiplet. On the contrary, the addition of a second Higgs hypermultiplet in the bulk, as in [@Pomarol], would cancel at the same time the anomaly and the one-loop-induced FI term. It therefore seems that the necessity of having two Higgs doublets in 4-D supersymmetric extensions of the SM persists also in these higher-dimensional constructions. Acknowledgements {#acknowledgements .unnumbered} ================ We would like to thank J.-P. Derendinger, F. Feruglio, P. Gambino, G. Isidori, A. Pomarol, A. Uranga and especially R. Rattazzi for interesting discussions. This work was partially supported by the RTN European Programs “Across the Energy Frontier”, contract HPRN-CT-2000-00148, and “The Quantum Structure of Space-Time”, contract HPRN-CT-2000-00131. M.S. and L.S. acknowledge partial support by CERN. C.A.S. and M.S. thank INFN and the Phys. Dept. of Rome Univ. “La Sapienza”, where part of this work was done. [99]{} L. J. Dixon, J. A. Harvey, C. Vafa and E. Witten, Nucl. Phys. B [**261**]{} (1985) 678. M. B. Green and J. H. Schwarz, Phys. Lett. B [**149**]{} (1984) 117. C. G. Callan and J. A. Harvey, Nucl. Phys. B [**250**]{} (1985) 427. P. Horava and E. Witten, Nucl. Phys. B [**460**]{} (1996) 506, hep-th/9510209;\ Nucl. Phys. B [**475**]{} (1996) 94, hep-th/9603142. N. Arkani-Hamed, A. G. Cohen and H. Georgi, Phys. Lett. B [**516**]{} (2001) 395, hep-th/0103135. R. Barbieri, L. J. Hall and Y. Nomura, Phys. Rev. D [**63**]{} (2001) 105007, hep-ph/0011311. D.M. Ghilencea, S. Groot Nibbelink and H.P. Nilles, hep-th/0108184. P. Binetruy, C. Deffayet, E. Dudas and P. Ramond, Phys. Lett. B [**441**]{} (1998) 163, hep-th/9807079. D. J. Gross and R. Jackiw, Phys. Rev. D [**6**]{} (1972) 477;\ L. Alvarez-Gaumé and E. Witten, Nucl. Phys. B [**234**]{} (1984) 269. M. Dine, N. Seiberg and E. Witten, Nucl. Phys. B [**289**]{} (1987) 589. M. Gunaydin, G. Sierra and P. K. Townsend, Nucl. Phys. B [**242**]{} (1984) 244. N. Arkani-Hamed, T. Gregoire and J. Wacker, hep-th/0101233. A. Pomarol and M. Quiros, Phys. Lett. B [**438**]{} (1998) 255, hep-ph/9806263;\ A. Delgado, A. Pomarol and M. Quiros, Phys. Rev. D [**60**]{} (1999) 095008, hep-ph/9812489;\ R. Barbieri, L. J. Hall and Y. Nomura, hep-ph/0106190;\ N. Arkani-Hamed, L. J. Hall, Y. Nomura, D. R. Smith and N. Weiner, Nucl. Phys. B [**605**]{} (2001) 81, hep-ph/0102090;\ V. Di Clemente, S. F. King and D. A. Rayner, hep-ph/0107290. [^1]: We work on the orbifold covering space $S^1$, and we normalize the $\delta$-functions so that, for $y_0 \in [0,2 \pi R )$ and $0< \epsilon < 2 \pi R - y_0$, $\int_{- \epsilon}^{2 \pi R - \epsilon} d y \delta(y - y_0) f(y) = f(y_0)$. [^2]: Notice that the $A_{\mu\,mn}^{\pm\mp}$ are not Fourier modes of the type (\[eq:fthiggs\]), but can be easily related to them. One finds: $A^{\mu \pm \mp}_{mn} = (\eta_{|m-n|}^{-1} A^\mu_{|m-n|} \pm \eta_{|m+n-1|}^{-1} A^\mu_{|m+n-1|})/\sqrt{\pi R}$. [^3]: We thank R. Rattazzi for having suggested this possibility to us. [^4]: We are grateful to A. Uranga for discussions on this argument.

--- author: - Okyu Kwon - 'Jae-Suk Yang[^1]' title: Information flow between stock indices --- Introduction {#intro} ============ Economic systems have recently become an active field of research for physicists striving to transfer concepts and methodologies from statistical physics such as phase transition, fractal theory, spin models, complex networks, and information theory to the analysis of economic problems [@eguiluz; @krawiecki; @chowdhury; @takaishi; @kaizoji02; @kaizoji04; @palagyi; @kaizoji01; @wsjung1; @wsjung2; @jsyang3; @arthur; @mategna; @bouchaud; @mandel; @kullmann; @giada; @jbpark; @jwlee; @jsyang2; @kaizoji03; @matal; @yang; @silva]. Among the numerous methodologies put forward, time series analysis has proven to be one of the most efficient methods and is widely applied to the examination of characteristics of stock and foreign exchange markets. In order to analyze financial time series, a range of statistical measures have been introduced, including probability distribution [@jsyang2; @kaizoji03; @matal; @yang; @silva; @stanley1; @mccauley], autocorrelation [@yang], multi-fractal [@kkim], complexity [@jbpark; @jwlee; @jsyang2], entropy density [@jwlee; @jsyang2], and transfer entropy [@jsyang3]. Information is a keyword in analyzing financial market data or in estimating the stock price of a given company. It is quantified through a variety of methods such as cross-correlation, autocorrelation, and complexity. However, while they may be appropriate measures for the observation of the internal structure of information flow, they fail to illuminate the directionality of information flow. Schreiber [@schreiber] introduced transfer entropy, which measures the dependency in time between two variables and notes the directionality of information flow. This concept of transfer entropy has already been applied to the analysis of financial time series. Marschinski and Kantz [@marschinski] calculated information flow between the Dow Jones and DAX stock indexes to better observe interactions between the two huge markets. Kwon and Yang [@jsyang3] measured the direction of information flow between the composite stock index and individual stock prices, while the information flow among individual stocks in a stock market has been estimated by Baek and coauthors [@baek] in order to measure the internal structure of a stock market. -------------- ---- -------- ------------- Americas 1 MERV Argentina 2 BVSP Brazil 3 GSPTSE Canada 4 MXX Mexico 5 GSPC U.S. 6 DJA U.S. 7 DJI U.S. Asia/Pacific 8 AORD Australia 9 SSEC China 10 HSI China 11 BSESN India 12 JKSE Indonesia 13 KLSE Malaysia 14 N225 Japan 15 STI Singapore 16 KS11 Korea 17 TWII Taiwan Europe 18 ATX Austria 19 BFX Belgium 20 FCE.NX France 21 GDAXI Germany 22 AEX Holland 23 MIBTEL Italy 24 SSMI Switzerland 25 FTSE U.K. -------------- ---- -------- ------------- : List of 25 markets. We obtain data from the website http://finance.yahoo.com.[]{data-label="table:market"} Through transfer entropy, this paper focuses quantitatively on the direction of information flow between 25 stock markets to determine which market serves as a source of information for global stock indices. Drawn from the economic system, a plethora of empirical data reflecting economic conditions can be obtained. The time series of a composite stock price index provides prime data accurately reflecting economic conditions. Therefore, we analyzed the daily time series of the 25 stock indices listed in Table \[table:market\] for the period of 2000 – 2007 using transfer entropy in order to examine the information flow between stock markets and identify the hub. Transfer entropy ================ Transfer entropy, which measures the directionality of a variable with respect to time was recently introduced by Schreiber [@schreiber] based on the probability density function (PDF). Let us consider two discrete and stationary process, $I$ and $J$. Transfer entropy relates $k$ previous samples of process $I$ and $l$ previous samples of process $J$ and is defined as follows: $$T_{J \rightarrow I} = \sum p(i_{t+1}, i_t^{(k)}, j_t^{(l)}) \log \frac{p( i_{t+1} \mid i_t^{(k)}, j_t^{(l)})}{p( i_{t+1} \mid i_t^{(k)})},$$ where $i_t$ and $j_t$ represent the discrete states at time $t$ of $I$ and $J$, respectively. $i_t^{(k)}$ and $j_t^{(l)}$ denote $k$ and $l$ dimensional delay vectors of two time consequences $I$ and $J$, respectively. The joint PDF $p(i_{t+1}, i_t^{(k)}, j_t^{(l)})$ is the probability that the combination of $i_{t+1}$, $i_t^{(k)}$ and $j_t^{(l)}$ have particular values. The conditional PDF $p(i_{t+1} \mid i_t^{(k)}, j_t^{(l)})$ and $p( i_{t+1} \mid i_t^{(k)})$ are the probability that $i_{t+1}$ has a particular value when the value of previous samples $i_t^{(k)}$ and $j_t^{(l)}$ are known and $i_t^{(k)}$ are known, respectively. The transfer entropy with index $J \rightarrow I$ measures the extent to which the dynamics of process $J$ influences the transition probabilities of another process $I$. Reverse dependency is calculated by exchanging $i$ and $j$ of the joint and conditional PDFs. The transfer entropy is explicitly asymmetric under the exchange of $i_t$ and $j_t$. It can thus provide information regarding the direction of interaction between two time series. The transfer entropy is quantified by information flow from $J$ to $I$. The transfer entropy can be calculated by subtracting the information obtained from the last observation of $I$ exclusively from the information about the latest observation $I$ obtained from the final joint observation of $I$ and $J$. This is the nexus of transfer entropy. Therefore, transfer entropy can be rephrased as $$T_{J \rightarrow I} = h_I(k) - h_{IJ}(k,l),\label{eq:TE}$$ where $$\begin{aligned} %\begin{array}{cc} h_I(k)=-\sum p(i_{t+1}, i_t^{(k)}) \log p( i_{t+1} \mid i_t^{(k)}) \\ h_{IJ}(k,l)=-\sum p(i_{t+1}, i_t^{(k)}, j_t^{(l)}) \log p( i_{t+1} \mid i_t^{(k)}, j_t^{(l)}). %\end{array}\end{aligned}$$ Empirical data analysis {#data} ======================= We analyzed the daily data records of 25 stock exchange indices for the period January 2000 – December 2007. We adopted the log returns to describe a financial time series $x(n)\equiv \ln (S(n)) - \ln (S(n-1))$. Where $S_n$ means the index of $n$-th trading day. We partitioned the real value $x(n)$ into discretized value $A(n)$; $A(n) = 0$ for $x(n) \leq -d$ (decrease), $A(n) = 1$ for $ -d < x(n) < d$ (intermediate), $A(n) = 2$ for $x(n) \geq d/2$ (increase). We set $d=0.04$ in our analysis. At that value, the probabilities of three states are approximately same for all indices. Additionally, we set $k=l=1$. Fig. \[fig:te\] is the plots of the outgoing and the incoming transfer entropy. Numbers in $x$-axis indicate markets listed in Table \[table:market\] while the dotted lines in the graphs serve to distinguish the continents. Each stock market interacts with 24 other stock markets, and the transfer entropy for all possible pairs can be calculated. Fig. \[fig:te\](a) represents the transfer entropy from the market indicating $x$-axis to the other 24 markets, that is, the outgoing transfer entropy. Fig. \[fig:te\](b) represents the transfer entropy from the other 24 markets to the market in $x$-axis, that is, the incoming transfer entropy. The markets in the Americas have high values of outgoing transfer entropy, while the value of the transfer entropy for the markets in Asia remains low. Moreover, the incoming transfer entropy for Asia is higher than that of the Americas and Europe. From Fig. \[fig:te\], we can determine that the directionality of influence for a market index is generally from the Americas to Asia. To clarify the tendencies of information flow, we shuffled the data and verified the transfer entropy. In Fig. \[fig:te\](c) and Fig. \[fig:te\](d), the transfer entropy for the shuffled data is generally approaching zero, information flow between markets has nearly disappeared. [ ]{} Fig. \[fig:matrix\] is the gray-scale map of the transfer entropy between stock markets using the same data as in Fig. \[fig:te\]. The direction of information flow is from the $x$-axis to $y$-axis. The darker the lattice, the lower the transfer entropy. The markets in the Americas influence the markets in both Asia and Europe, but influence on the Asian markets is particularly strong. The European markets also influence the Asian markets, but to a lesser degree than do the American markets. [ ]{} Fig. \[fig:corrmat\] is the gray-scale map of the cross-correlations between stock markets from log-return time series $x(n)$. The cross-correlation does not represent directionality between markets. Therefore, the graph is symmetrical, centering around the diagonal. In this graph, a light lattice means that the cross-correlation is high. The American and European markets are highly correlated between themselves, while the Asian markets are not when compared with the American and European markets. Also, the American and Asian markets are rarely correlated with each other. From these results, we can induce (or that the comparatively mature American and European markets are well clustered between themselves, while the relatively emerging Asian markets are not. Furthermore, between the American and Asian markets there exists low cross-correlation and high transfer entropy, demonstrating an imbalance of information flow. To observe schematically the relation of information flow between the stock markets, the minimum spanning tree is depicted in Fig. \[fig:mst\]. The colors of nodes indicate the continent housing the markets. The minimum spanning tree is constructed by connecting the highest outgoing (Fig. \[fig:mst\](a)) and incoming (Fig. \[fig:mst\](b)) transfer entropy. In Fig. \[fig:mst\](a), the GSPC is located at the focus of the network, connected with the Asian and European markets. The GSPC plays the role of information source among stock markets. On the other hand, in Fig. \[fig:mst\](b) the Australian market, AORD, is located in the center of the minimum spanning tree for incoming transfer entropy. Conclusions {#conclusion} =========== We observed the transfer entropy and the cross-correlation of the daily data of 25 stock markets to investigate the intensity and the direction of information flow between stock markets. The value of the outgoing transfer entropy from the American markets is high, as it is from the European markets. The information flow mainly streams to the Asia/Pacific region. However, the transfer entropy between markets on the same continent is not high. We also reveal that markets on the same continent have higher values of cross-correlation for log-return. This indicates that intra-continental market indices fluctuate simultaneously with common deriving mechanisms. The cross-correlation between America and Europe is relatively high. Therefore, we can see that American and European stock markets fluctuate in tune with common deriving mechanisms. However, the Asia/Pacific markets fluctuate simultaneously with other common deriving mechanisms. Consequently, we can predict that the deriving mechanism for America and Europe does not depend on the Asia/Pacific market situation but rather that Asia is affected by the American and European market situation. From Fig. \[fig:mst\], we can confirm the above description. The GSPC is located in the center of the star network, influencing all the other markets. Therefore, the Asian markets represent similar patterns of time series, although they do not actively interact with each other. Transfer entropy is a proper measure for the investigation of the information flow between global stock markets. Through this measure we can quantify information transportation between underlying mechanisms that govern stock markets even though we cannot concretely identify the mechanism. This work is supported in part by the Creative Research Initiatives of the Korea Ministry of Science and Technology and the Second Brain Korea 21 project. [0]{} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [^1]: Corresponding author.

--- abstract: 'Principles of a new approach (binary geometrophysics) are presented to construct the unified theory of spacetime and the familiar kinds of physical interactions. Physically, the approach is a modified S-matrix theory involving ideas of the multidimensional geometric models of physical interactions of Kaluza-Klein’s type as well as Fokker-Feynman’s action-at-a-distance theory. Mathematically, this is a peculiar binary geometry being described in algebraic terms. In the present approach the binary geometry volume is a prototype of three related notions: the S-matrix, the physical action (Lagrangian) of both strong and electroweak interactions, and the multidimensional metric. A transition from microworld geometrophysics to the conventional physical theory in classical spacetime are characterized.' author: - | Yu.S. Vladimirov\ Physics Faculty, Moscow State University title: PRINCIPLES OF A UNIFIED THEORY OF SPACETIME AND PHYSICAL INTERACTIONS --- =-15mm =-20mm 10000 Introduction ============ In attempts to construct microworld physics for the last half of a century hopes have been pinned successively on a few key ideas and principles. So, in the fifties those were the principles of quantum field theory, in the sixties attempts were made to construct the S-matrix approach and quantum theory axiomatics, in the seventies and early eighties a primary emphasis was placed upon group methods and gauge theory of physical interactions, at the very end of the 20th century hopes were pinned first on supesymmetric theories and then on superstrings and p-branes. At the same time, other (side) concepts: Fokker-Feynman’s action-at-a-distance theory, Penrose’s twistor theory, multidimensional geometrical models of Kaluza-Klein’s type etc. were investigated. In all these investigations a relation to the coordinate spacetime was formulated in any event. The latter was a priori assumed four-dimensional, either was generalized to superspaces or multidimensional manifolds, or it was considered that for constructing microworld physics needless to proceed from the coordinate spacetime, and the momentum space should be used as the basis for constructing a theory (as was declared by Chew in the S-matrix approach \[1\]), either it was proposed to start from complex notions of another kind, e.g. from twistors in Penrose’s program \[2, 3\], then arriving at both quantum theory and the classical spacetime. In the present paper a new approach is proposed to construct microworld physics, involving a number of ideas and methods of the previous research and allowing one to consider the essence of spacetime and physical interactions from another viewpoint. The theory being put forward is called [*binary geometrophysics*]{}. Its premises have somewhat in common with Penrose’s twistor program, however our starting points are more abstract and richer in their consequences. Binary geometrophysics is closest to the S-matrix theory. The word “binary” reflects two sets of states of the system, i.e. the initial (i) and the final (out) ones, forming the basis of the theory. The theory itself is a peculiar binary geometry describing microworld physics. As known, the idea of S-matrix approach was put forward by J. Wheeler \[4\] and W. Heisenberg \[5\] and was developed into the S-matrix in the sixties in the works by a number of authors (e.g., see \[1, 6\]). Its basis constituted principles of Lorentz invariance, analyticity, causality etc. Particles and their characteristics were related to poles in the complex plane. Binary geometrophysics and the S-matrix theory share, apart from the two kinds of states, a possibility of constructing a theory “independently of whether microscopic spatial-temporal continuum does exist or not” \[1, p.17\]. In binary geometrophysics there is no a priori spacetime for microworld. It arises only in the relations between macro-objects (according to the idea of a macroscopic nature of the classical spascetime \[7\]). On the very elementary level there becomes invalid the differential and integral calculus alongside spatial-temporal continuum. There remain only algebraic methods. To construct binary geometrophysics, the mathematical (algebraic) formalism of Yu.I. Kulakov’s binary physical structures \[8, 9, 10\] has been used. (The latter was previously developed for other purposes.) As in the S-matrix theory, of importance here is a symmetry, however, in our theory there emerges a more general than the Lorentzian symmetry from which one may turn to Lie groups, in particular, to the Lorentz group and internal symmetries being used in gauge theories. In the present approach the function continuity condition is used only at the first stage to find algebraic laws of structures (relation systems), and then from merely algebraic considerations one may arrive at a definition of particles and a relationship between constants and charges in strong and electroweak intractions. But above all, in the framework of binary geometrophysics, from the unified expressions prototypes of three related notions: the S-matrix, the action (Lagrangian) for strong and electroweak interactions of elementary particles and the multidimensional metric are derived. The ideas and methods used in the multidimensional geometrical models of physical interactions of Kaluza-Klein’s type theories (e.g., see \[13, 14\]) and Fokker-Feynman’s action-at-a-distance theory \[15–17\] prove necessary in constructing binary geometrophysics \[11, 12\]. It should be emphasized that binary geometrophysics has a relational pattern. Relations between elements, i.e. primary notions of the theory, play the key part in it. Specifically, the microanalogue of reference frames in Relativity \[18\], called a binary system of complex relations (BSCR), is put in the forefront. If micro-oobjects are denoted by the symbol $\mu$, and macro-objects by the symbol $m$, then the starting points of the [*relational*]{} theory should be defined by the symbol $R_{\mu}(\mu)$, where $\mu$ in parentheses means that micro-objects are considered in it, and the subscript $\mu$ means that this is done with respect to micro-objects as well. In such terms classical mechanics, describing macro-objects with respect to macroinstruments, is denoted as $R_m(m)$, and quantum mechanics – as $R_m(\mu)$. The first two sections present principal ideas and principles of binary geometrophysics at the level $R_{\mu}(\mu)$ in the absence of notions of the classical spacetime, and in the fourth section a transition to the conventional theory $R_m(m)$ in the presence of these notions is discussed. Binary Geometry of the Microworld ================================= Briefly outline the principal notions of BSCR forming the basis of the theory $R_{\mu}(\mu)$ and give their physical interpretation. [**1. Two Sets of Elements**]{} It is postulated that there are two sets of elements. Denote the first set by a symbol ${\cal M}$, and the second one by ${\cal N}$. Elements of the first set are denoted by Latin letters ($i,j,k,\ldots$), and the elements of the second one – by Greek letters ($\alpha, \beta, \gamma,\ldots$). Between any pair of the elements of different sets a pair relation – some complex number $u_{i\alpha}$ is given (see Fig. 1). The elements of the two sets have the following physical meaning. The elements of the first set ${\cal M}$ characterize [*initial states*]{} of particles, and the elements of the second one ${\cal N}$ – [*final states*]{}. Thus, an idea of the S-matrix approach (more exactly, an elementary chain of any evolution), i.e an initial-to-final transition, proves to be laid in the very fundamental notions of BSCR. The complex relations between elements in the two states are a prototype of both the amplitudes of a transition between states and the notions of momenta, and finally particle coordinates. [**2. Law and Fundamental Symmetry**]{} It is postulated that there exists some algebraic [*law*]{}, connecting all possible relations between any $r$ elements of the set ${\cal M}$ and $s$ elements of the set ${\cal N}$: $$\Phi_{(r,s)}(u_{i\alpha}, u_{i\beta},\ldots, u_{k\gamma})=0.$$ The integers $r$ and $s$ characterizes [*rank*]{} ($r,s$) of BSCR. The essential point of the theory is a requirement of [*fundamental symmetry*]{} lying in the law (1) to be valid, while substituting the given set of elements by any others in the corresponding sets. The fundamental symmetry and the continuity condition allow functional-differential equations to be derived from them and the form of both pair relations $u_{i\alpha}$ as well as the function $\Phi_{(r,s)}$ itself to be found (see \[8, 9, 10\]). (92.33,55.00) (55.17,15.50)[(69.67,19.00)\[\]]{} (55.00,45.00)[(70.00,20.00)\[\]]{} (69.33,30.67)[(20.67,33.33)\[\]]{} (29.33,17.33) (40.00,17.67) (51.33,17.67) (64.00,17.67) (75.33,18.00) (85.00,17.67) (29.33,44.00) (39.67,43.67) (51.67,43.67) (64.00,43.67) (75.33,43.33) (85.33,43.67) (24.67,12.67)[(30.67,9.33)\[cc\]]{} (24.67,39.00)[(30.67,8.67)\[cc\]]{} (14.00,15.33)[(0,0)\[cc\][${\cal M}$]{}]{} (14.00,44.33)[(0,0)\[cc\][${\cal N}$]{}]{} (39.67,51.00)[(0,0)\[cc\][s elements]{}]{} (39.67,9.33)[(0,0)\[cc\][r elements]{}]{} (69.33,9.67)[(0,0)\[cc\][BSCR basis]{}]{} (32.00,14.67)[(0,0)\[cc\][$i$]{}]{} (42.67,15.00)[(0,0)\[cc\][$k$]{}]{} (53.67,15.00)[(0,0)\[cc\][$j$]{}]{} (66.33,15.33)[(0,0)\[cc\][$m$]{}]{} (72.33,15.67)[(0,0)\[cc\][$n$]{}]{} (82.33,15.67)[(0,0)\[cc\][$l$]{}]{} (32.33,45.67)[(0,0)\[cc\][$\alpha$]{}]{} (42.67,45.33)[(0,0)\[cc\][$\beta$]{}]{} (54.00,46.00)[(0,0)\[cc\][$\gamma$]{}]{} (66.33,45.00)[(0,0)\[cc\][$\mu$]{}]{} (72.67,45.00)[(0,0)\[cc\][$\nu$]{}]{} (82.33,45.33)[(0,0)\[cc\][$\lambda$]{}]{} (26.33,30.67)[(0,0)\[cc\][$u_{i\alpha}$]{}]{} (66.00,31.00)[(0,0)\[cc\][$u_{m\mu}$]{}]{} In binary geometrophysics BSCR of symmetric ranks $(r,r)$ are used, with nondegenerate and degenerate systems of relations being distinguished. For the nongeranerate BSCR the law is written via a determinant of pair relations: $$\Phi_{(r,r)}(u_{i\alpha},u_{i\beta},\ldots)= \left|\begin{array}{cccc} u_{i\alpha} & u_{i\beta} & \cdots & u_{i\gamma} \\ u_{k\alpha} & u_{k\beta} & \cdots & u_{k\gamma} \\ \cdots & \cdots & \cdots & \cdots \\ u_{j\alpha} & u_{j\beta} & \cdots & u_{j\gamma} \end{array}\right|=0,$$ where the pair relations are represented in the form $$u_{i\alpha}=\sum_{l=1}^{r-1}i^l\alpha^l.$$ Here $i^1, \ldots, i^{r-1}$ – $(r-1)$ parameters of the element $i$, and $\alpha^1,\ldots$, $\alpha^{r-1}$ – $(r-1)$ parameters of the element $\alpha$. [**3. Elementary Basis**]{} The origin of the element parameters should be especially dwelled on. They are analogues of the notions of coordinates in the conventional geomentry. To arrive at them, in the law (1) one should separate $r-1$ elements of the set ${\cal M}$ and $s-1$ elements of the set ${\cal N}$ and consider them to be [*standard*]{}. Fig. 1 denotes these elements by letters $m$, $n$, $\mu$, $\nu$. Then this law may be interpreted as a relationship determining a pair relation between two nonstandard elements (say, the elements $i$ and $\alpha$) in terms of their relations to standard elements (see Fig. 1). Relations between the standard elements themselves may be considered given forever. Then the pair relation $u_{i\alpha}$ proves to be characterized by $s-1$ parameters of the element $i$ (its relations to $s-1$ standard elements of the set ${\cal N}$) and similar $r-1$ parameters of the element $\alpha$. The system of the standard elements comprises [*elementary basis of BSCR*]{}. [**4. Fundamental Relations**]{} In the BSCR of rank $(r,r)$ an important part is played by [*fundamental*]{} $(r-1)\times (r-1)$-[*relations*]{}, being nonzero minors of the maximal order in the determinant of the law (2). For BSCR of any rank $(r,r)$ the fundamenta relations are written in terms of a product of two determinants composed of parameters of one sort: $$\left[{\alpha\atop i}{\beta\atop k}{\cdots\atop\cdots}\right] \equiv \left|\begin{array}{ccc} u_{i\alpha} & u_{i\beta} & \cdots \\ u_{k\alpha} & u_{k\beta} & \cdots \\ \cdots & \cdots & \cdots \end{array}\right|= \left|\begin{array}{ccc} i^1 & k^1 & \cdots \\ i^2 & k^2 & \cdots \\ \cdots & \cdots & \cdots \end{array}\right|\times \left|\begin{array}{ccc} \alpha^1 & \beta^1 & \cdots \\ \alpha^2 & \beta^2 & \cdots \\ \cdots & \cdots & \cdots \end{array}\right|.$$ [**5. Linear Transformations**]{} Transitions from one elementary basis to another may be shown (see \[11\]) to be described by linear transformations of element parameters of two sets: $$i^{\prime s}= C^s_r i^r; \ \ \ \alpha^{\prime s}=C^{\star s}_r \alpha^r,$$ where $C^s_r$ и $C^{\star s}_r$ are the coefficients defining a class of binary systems (standard elements) being used. Restrict ourselves to the case of complex conjugate coefficients. [**6. Two-Component and Finsler Spinors**]{} Low-rank BSCR notions are, in fact, used in modern theoretical physics. In In particular, the two-component spinor theory \[19\] naturally arises in the framework of the minimamal nondegenerate rank (3,3) BSCR. Really, according to this theory, the elements are characterized by pairs of complex parameters $ i\rightarrow (i^1, i^2)$, $ \alpha\rightarrow (\alpha^1, \alpha^2)$, i.e. are vectors of the two-dimensional complex space where the linear transformations (5) are defined. Restrict ourselves to the liner transformations leaving invariant each of the $2\times 2$ determinants on the left in (4). Since each of such determinants is an antisymmetric bilinear form: $ i^1 k^2-i^2 k^1=Inv; \alpha^1\beta^2-\alpha^2\beta^1=Inv$, the definition of two-component spinors becomes evident. For a selected class of transformations (5) the coefficients satisfy the condition $C^1_1C^2_2-C^1_2C^2_1=1,$ i.e. these transformations belong to a six-parameter group $SL(2,C)$. Such transformations single out the privileged class of standard elements corresponding to inertial frames of reference in General Relativity. In the framework of rank (3,3) BSCR, the transformations keeping invariant both the antisymmetric forms on the right in (4) and the pair relations $u_{i\alpha}$ form the three-parameter group $SU(2)$ corresponding to rotations in three-dimensional space. As follows from the foregoing, one may state that [*four-dimensionality of the classical spacetime and the signature*]{} $(+ - - -)$ [*are due to the rank 3,3) of the first nondegenerate BSCR*]{}. At this level of theory development in the framework of rank (3,3) BSCR we, in fact, arrive at the notions comprising Penrose’s twistor theory. However in binary geometrophysics, relation systems of a higher rank $(r,r)$ are considered. In them the elements are described by $r-1$-dimensional vectors. Requiring the corresponding antisymmetric forms in (4) under the transformations (5), we arrive at a nonstandard generalization of the two-component spinors which are naturally called [*Finsler $(r-1)$-component spinors*]{} \[21\]. These transformations form a group $SL(r-1,C)$. [**7. Description of Elementary Particles**]{} In binary geometrophysics the elementary particles are described by a few elements in each of the two sets. In the framework of a simplified model based on the rank (3,3) BSCR, massive leptons (electrons) are described by pairs of elements, and neutrinos – by one element in each of the sets. Let the electron $(e)$ be described by two pairs of elements: $i$, $k$ and $\alpha$, $\beta$, then it may be characterized in terms close to the conventional ones, i.e. by the four-component column and line: $$e=\left(\begin{array}{c} i^1 \\ i^2 \\ \beta_1 \\ \beta_2 \end{array}\right)= \left(\begin{array}{c} i^1 \\ i^2 \\ \beta^2 \\ -\beta^1 \end{array}\right); \ \ \ e^{\dag}=(\alpha^1, \alpha^2, k_1, k_2),$$ where the quantities with subscripts denote covariant components of spinors. On the elements describing the leptons imposed are constraints along the horizontal (in one set of elements) and along the vertical (in two sets). Thus, for [*free*]{} leptons the vertical constraints mean that the parameters of two pairs of elements in the two sets are complex conjugate to one another, following the rules of quantum mechanics. For the electron in (6) this means that $i^s=\stackrel{\ast}{\alpha^s}$, $k^s= \stackrel{\ast}{\beta^s}$. The horizontal constraints connect the elements $i$ and $k$. They correspond to Dirac’s equations for free particles in a momentum space \[11\]. [**8. Momentum Space (Velocity Space)**]{} In the rank (3,3) BSCR, of element parameters – two-component spinors – the four-vectors [*physically interpreted as velocity (or momentum) components*]{} of particles are conventionally constructed. Introducing Dirac’s four-row matrices in the corresponding representation, the electron four-velocity components may be presented as follows: $$\begin{aligned} u^0 & = & \frac{1}{2}(\overline{e}\gamma^0 e)=\frac{1}{2}(i^1\alpha^1+ i^2\alpha^2 + k^1\beta^1+ k^2\beta^2); \nonumber \\ u^1 & = & \frac{1}{2}(\overline{e}\gamma^1 e)= \frac{1}{2}(i^1\alpha^2+i^2\alpha^1+ k^1\beta^2+ k^2\beta^1); \\ u^2 & = & \frac{1}{2}(\overline{e}\gamma^3 e)= \frac{i}{2}(i^1\alpha^2- i^2\alpha^1+ k^1\beta^2 - k^2\beta^1); \nonumber \\ u^3 & = & \frac{1}{2}(\overline{e}\gamma^3 e)= \frac{1}{2}(i^1\alpha^1-i^2\alpha^2+k^1\beta^1- k^2\beta^2), \nonumber \end{aligned}$$ where $\overline{e}=e^{\dag}\gamma^0$. Defining the matrix $\gamma^5$, the left and right components of the electron may be conventionally introduced, then the left component of the electron is described by the pair of elements $i$, $\alpha$, and the right component is described by the pair $k$, $\beta$. It is evident that there is only one component for the neutrino. In the general case, when the particles are described by a triple of elements, a generalization of the above rules is necessary. The above-mentioned transition from parameters to velocities may be intrepreted as a transition from a binary geometry to a unary one by a specific glueing of two pairs of elements of the two BSCR sets into new elements of one set being described by Lobachevsky’s geometry. It is easy to verify that for one particle being characterized by the expression (6) we have $$\left|\begin{array}{cc} u_{i\alpha} & u_{i\beta} \\ u_{k\alpha} & u_{k\beta} \end{array}\right|\equiv \left[{\alpha\atop i}{\beta\atop k}\right]=g_{\mu\nu}u^{\mu}u^{\nu}.$$ For a free particle the four-dimensional velocities defined in such a way possess the well-known property $ u^{\mu}u_{\mu}=Const=1.$ Introducing the second particle $e_2$ being described by the parameters: $j$, $s$, $\gamma$, $\delta$, the scalar product of velocities (momenta) of two particles may be defined as $$u^{\mu}_1u_{2\mu} = \frac{1}{4} (\overline{e}_1\gamma^{\mu}e_1) (\overline{e}_2 \gamma_{\mu}e_2)= \frac{1}{2}\left(\left[{\alpha\gamma\atop ij}\right]+ \left[{\alpha\delta\atop is}\right]+ \left[{\beta\gamma\atop kj}\right]+ \left[{\beta\delta\atop ks}\right]\right),$$ where the expression in square brackets denote the fundamental $2\times 2$- rank (3,3) BSCR relations defined in (4). For interacting particles the conditions of complex conjugation of elements of the two sets are not satisfied any more. Introducing for particles in initial and final states the complex conjugate quantities and constructing from them, using the well-known formulas, four-velocities (momenta), we arrive at the characteristics of particles before and after interaction typical for the S-matrix. [**9. Choice of BSCR Rank**]{} In our works (see \[11, 12\]) properties of physical theories based on the BSCR of ranks (2,2), (3,3), (4,4), (5,5), (6,6) and higher have been analyzed step by step. The analysis has shown that the prototype of the known kinds of physical interactions is constructed in the framework of [*rank (6,6) BSCR.*]{} Therewith individual elementary particles (fermions) should be described by triples of elements in each of two sets ${\cal M}$ and ${\cal N}$. This corresponds to the familiar notions of the structure of baryons consisting of three quarks. In this approach these notions are extended to leptons as well. [**10. Base $6\times 6$-Relations**]{} As a prototype of such related notions as the S-matrix or the action (Lagrangian) for two interacting particles there should be some expression containing two triples, i.e. six elements of one set ${\cal M}$ and six elements of the set ${\cal N}$. Such is the [*base $6\times 6$-relations*]{} being written in the framework of the rank (6,6) BSCR as follows $$\left\{{\alpha\atop i}{\beta\atop k}{\gamma\atop j}{\delta\atop s}{\lambda\atop l}{\rho\atop r}\right\}\equiv \left|\begin{array}{ccccccc} 0 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & u_{i\alpha} & u_{i\beta} & u_{i\gamma} & u_{i\delta} & u_{i\lambda} & u_{i\rho} \\ 1 & u_{k\alpha} & u_{k\beta} & u_{k\gamma} & u_{k\delta} & u_{k\lambda} & u_{k\rho} \\ 1 & u_{j\alpha} & u_{j\beta} & u_{j\gamma} & u_{j\delta} & u_{j\lambda} & u_{j\rho} \\ 1 & u_{s\alpha} & u_{s\beta} & u_{s\gamma} & u_{s\delta} & u_{s\lambda} & u_{s\rho} \\ 1 & u_{l\alpha} & u_{l\beta} & u_{l\gamma} & u_{l\delta} & u_{l\lambda} & u_{l\rho} \\ 1 & u_{r\alpha} & u_{r\beta} & u{r\gamma} & u_{r\delta} & u_{r\lambda} & u_{r\rho} \end{array}\right|=$$ $$=\left|\begin{array}{ccc|ccc} i^1 & k^1 & j^1 & s^1 & l^1 & r^1 \\ i^2 & k^2 & j^2 & s^2 & l^2 & r^2 \\ \hline i^3 & k^3 & j^3 & s^3 & l^3 & r^3 \\ i^4 & k^4 & j^4 & s^4 & l^4 & r^4 \\ i^5 & k^5 & j^5 & s^5 & l^5 & r^5 \\ 1 & 1 & 1 & 1 & 1 & 1 \end{array}\right|\times \left|\begin{array}{ccc|ccc} \alpha^1 & \beta^1 & \gamma^1 & \delta^1 & \lambda^1 & \rho^1 \\ \alpha^2 & \beta^2 & \gamma^2 & \delta^2 & \lambda^2 & \rho^2 \\ \hline \alpha^3 & \beta^3 & \gamma^3 & \delta^3 & \lambda^3 & \rho^3 \\ \alpha^4 & \beta^4 & \gamma^4 & \delta^4 & \lambda^4 & \rho^4 \\ \alpha^5 & \beta^5 & \gamma^5 & \delta^5 & \lambda^5 & \rho^5 \\ 1 & 1 & 1 & 1 & 1 & 1 \end{array}\right|,$$ where the vertical lines underline the fact that the first particle is described by the elements $i$, $k$, $j$, $\alpha$, $\beta$, $\gamma$, and the second one is described by the elements $s$, $l$, $r$, $\delta$, $\lambda$, $\rho$. This expression is invariant under the transformations of parameters of the group $SL(5,C)$ and is a specific volume in the binary geometry of rank (6,6). The physical interpretation of the base $6\times 6$-relation is illustrated by the digrams of Fig. 2, where on the left the 12-tail structure of binary geometrophysics is depicted. Two triples of the lower lines describe initial states of two particles, and two upper triples – their final states. A generalization of Feynman’s type diagram is presented in the middle, and a standard diagram of particle scattering is given on the right. (111.00,50.67) (89.00,16.00)[(1,3)[4.67]{}]{} (94.00,31.00)[(-1,2)[6.00]{}]{} (109.33,16.00)[(-1,3)[4.67]{}]{} (104.33,31.00)[(1,2)[6.67]{}]{} (39.67,30.67)[(1,0)[15.33]{}]{} (81.00,30.67)[(1,0)[9.67]{}]{} (61.50,31.00) (73.50,31.00) (94.00,31.00) (104.00,31.00) (66.33,31.00)[(0,0)\[cc\][$\sim$]{}]{} (69.00,31.00)[(0,0)\[cc\][$\sim$]{}]{} (96.67,30.33)[(0,0)\[cc\][$\sim$]{}]{} (99.33,30.33)[(0,0)\[cc\][$\sim$]{}]{} (102.00,30.33)[(0,0)\[cc\][$\sim$]{}]{} (13.67,41.33)[(0,0)\[cc\][$\beta$]{}]{} (19.67,41.67)[(0,0)\[cc\][$\gamma$]{}]{} (25.33,41.67)[(0,0)\[cc\][$\delta$]{}]{} (31.00,41.33)[(0,0)\[cc\][$\lambda$]{}]{} (35.00,38.33)[(0,0)\[cc\][$\rho$]{}]{} (11.00,23.00)[(0,0)\[cc\][$i$]{}]{} (25.33,18.33)[(0,0)\[cc\][$s$]{}]{} (31.33,19.00)[(0,0)\[cc\][$l$]{}]{} (35.00,22.00)[(0,0)\[cc\][$r$]{}]{} (19.67,18.67)[(0,0)\[cc\][$j$]{}]{} (53.67,41.00)[(0,0)\[cc\][$\alpha$]{}]{} (61.00,41.33)[(0,0)\[cc\][$\gamma$]{}]{} (74.00,41.67)[(0,0)\[cc\][$\delta$]{}]{} (79.00,42.00)[(0,0)\[cc\][$\lambda$]{}]{} (82.67,41.67)[(0,0)\[cc\][$\rho$]{}]{} (53.67,20.67)[(0,0)\[cc\][$i$]{}]{} (61.33,20.67)[(0,0)\[cc\][$j$]{}]{} (73.67,20.67)[(0,0)\[cc\][$s$]{}]{} (78.33,21.33)[(0,0)\[cc\][$l$]{}]{} (82.33,21.67)[(0,0)\[cc\][$r$]{}]{} (95.00,19.67)[(0,0)\[cc\][$P^{\mu}_{(1)}$]{}]{} (93.33,43.00)[(0,0)\[cc\][$P^{\prime\mu}_{(1)}$]{}]{} (104.33,19.67)[(0,0)\[cc\][$P^{\mu}_{(2)}$]{}]{} (105.67,43.00)[(0,0)\[cc\][$P^{\prime\mu}_{(2)}$]{}]{} (9.00,39.33)[(0,0)\[cc\][$\alpha$]{}]{} (15.00,19.67)[(0,0)\[cc\][$k$]{}]{} (57.67,20.67)[(0,0)\[cc\][$k$]{}]{} (57.33,41.67)[(0,0)\[cc\][$\beta$]{}]{} Transition from the Bionary Volume to the S-Matrix Prototype or the Action (Lagrangian) ======================================================================================= To pass from the base $6\times 6$-relation to prototypes of the S-matrix or the action (lagrangian) of two particles, the following set of approaches, procedures and principles should be used. [**1. Splitting Procedures**]{} First of all, it is necessary to perform a procedure of splitting (or reduction) corresponding to the $4+s$-splitting procedure into four-dimensional spacetime and additional dimensions in multidimensional geometrical models of Kaluza-Klein’s type theory. It lies in separating the parameters with indices 1 and 2 called [*external ones*]{}, from three rest parameters with indices 3, 4, 5 called [*internal ones*]{}. From the external parameters four-dimensional momenta (velocities) are constructed both in the framework of the rank (3,3) BSCR according (7), and from the internal ones charges of elementary particles are constructed. In fact, this procedure means splitting the initial rank (6,6) BSCR into two subsystems: a rank (3,3) BSCR (with two parameters) and a rank (4,4) BSCR (with three parameters). As a result of splitting, the initial transformation group $SL(5,C)$ is narrowed down up to two subgroups: $SL(2,C)$ – for external parameters and $SL(3,C)$ (or a narrower one $SU(3)$) – for internal parameters. To construct a prototype of the action, the base $6\times 6$-relation should be represented in the form reduced to four-dimensional spacetime when parameters with indices 1 and 2 are singled out, and the final expression has the Lorentz-invariant ($SL(2,C)$-invariant) form. This may be performed by expansion of the determinants on the right in (10) with repect first two lines. Multiplying them, we arrive at a set of 225 expressions of the form $$\left\{{\alpha\atop i}{\beta\atop k}{\gamma\atop j}{\delta\atop s}{\lambda\atop l}{\rho\atop r}\right\} =\sum^{225}\left[{\alpha\atop i}{\delta\atop s}\right] \left({\beta\atop k}{\gamma\atop j}{\lambda\atop l} {\rho\atop r}\right),$$ where the square brackets denote the fundamental $2\times 2$-relations constructed from parameters with indices 1 and 2, and the paretheses denote combinations of internal parameters of the form $$\left({\beta\atop k}{\gamma\atop j} {\lambda\atop l}{\rho\atop r}\right)=\left|\begin{array}{cc|cc} k^3 & j^3 & l^3 & r^3 \\ k^4 & j^4 & l^4 & r^4 \\ k^5 & j^5 & l^5 & r^5 \\ 1 & 1 & 1 & 1 \end{array}\right|\times \left|\begin{array}{cc|cc} \beta^3 & \gamma^3 & \lambda^3 & \rho^3 \\ \beta^4 & \gamma^4 & \lambda^4 & \rho^4 \\ \beta^5 & \gamma^5 & \lambda^5 & \rho^5 \\ 1 & 1 & 1 & 1 \end{array}\right|.$$ [**2. Classification and Physical Interpretation of Terms of the Base $6\times 6$-Relation**]{} The set of 225 terms of the form (11) should be divided into 9 subsets depicted as subblocks of a $15\times 15$-matrix separated by horizontal and vertical lines: $$\left\{{\alpha\atop i}{\beta\atop k}{\gamma\atop j}{\delta\atop s}{\lambda\atop l}{\rho\atop r}\right\}= \left(\begin{array}{ccc|ccccccccc|ccc} \star & \star & \star & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \star & \star & \star \\ \star & \star & \star & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \star & \star & \star \\ \star & \star & \star & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \star & \star & \star \\ \hline \cdot & \cdot & \cdot & \star & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \star & \cdot & \star & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \star & \cdot & \cdot & \cdot & \star & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \star & \cdot & \star & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \star & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \star & \cdot & \star & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \star & \cdot & \cdot & \cdot & \star & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \star & \cdot & \star & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \star & \cdot & \cdot & \cdot \\ \hline \star & \star & \star & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \star & \star & \star \\ \star & \star & \star & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \star & \star & \star \\ \star & \star & \star & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \star & \star & \star \end{array}\right)\equiv$$ $$\equiv \left(\begin{array}{c|c|c} M(4,0) & +M(3,1) & +M[(2)(2)]+ \\ \\ \hline +M(3,1) & +M(2,2) & +M(1,3)+ \\ \\ \hline +M[(2),(2)] & +M(1,3) & +M(0,4) \end{array}\right),$$ where individual terms are denoted by points or astersks. The latter mark terms of utmost importance in this theory (with a special order of terms). The subblocks differ in the character of the fundamental $2\times 2$-relations. The middle $9\times 9$-subblock contains terms of the form $\left[{\cdot\atop\cdot} {|\atop |}{\cdot\atop\cdot}\right]$, where the vertical line divided the parameters characterizing two particles. Such terms describe vector-vector interactions of two pareticles. In the gauge theory they correspond to the interactions via intermediate vector bosons (gluons, photons, Z- or W-bosons). The left lower or right upper $3\times 3$-subblocks $M[(2),(2)]$ contain terms of the form $\left[{\cdot \ \cdot\atop\overline{\cdot \ \cdot}} \right]$, where the horizontal line divides the parameters characterizing two particles. Such terms describe scalar interactions of two particles. They correspond to Higgs’ scalar bosons. These subblocks will be called mass ones. The diagonal left upper $M(4,0)$ and right lower $M(0,4)$ subblocks contain external parameters of only one of the particles. They correspond to the action (Lagrangian) of “free” particles. Such submartices will be called free for the first and second particles. The rest four extreme $3\times 9$- и $9\times 3$-subblocks contain three parameters of one particle and one parameter of another particle. There are criteria allowing contributions of such terms to be excluded. [**3. Unified Principle of Describing Baryons and Leptons**]{} The theory being presented here permits interactions (processes involving) baryons, massive leptons and neutrinos to be described in the same way. All particles mentioned are described by triples of elements in each of the two sets (states). The baryons are generally characterized by all three two-component columns of the external parameters containing nonzero components. Thus, e.g., in the set ${\cal M}$ the baryon (b) is characterized by a rectangular $3\times 5$-matrix of the parameters of its consistuents: $$(b)\Rightarrow\left(\begin{array}{ccc} i^1 & k^1 & j^1 \\ i^2 & k^2 & j^2 \\ \hline i^3 & k^3 & j^3 \\ i^4 & k^4 & j^4 \\ i^5 & k^5 & j^5 \end{array}\right)\equiv \left(\begin{array}{ccc} i^1 & k^1 & j^1 \\ i^2 & k^2 & j^2 \\ \hline c^3_{(1)} & c^3_{(2)} & c^3_{(3)} \\ c^4_{(1)} & c^4_{(2)} & c^4_{(3)} \\ c^5_{(1)} & c^5_{(2)} & c^5_{(3)} \end{array}\right),$$ where two upper lines correspond to external two-component spinors, and the three rest – to additional parameters (three-component Finsler spinors) describing internal degrees of freedom (charges) of an interacting particle. For massive leptons (electrons (e)) the matrix has the same form (14), however, one of the upper two-component columns consists of zero external parameters. Let it be the third column: $j^1=j^2=0$. For neutrinos $(\nu)$, being also described by $\times 5$-матрицей (14), two two-component columns of the external parameters are zero. Let it be the first two columns: $i^1=i^2=0$, $k^1=k^2=0$. For the leptons defined in such a way the number nonzero terms in the base $6\times 6$-relation (13) reduces considerably. It should be noted that the number zero columns of the external parameters is an invariant property of particles with respect to distinguished parameter transfomation groups. The particle definitions by external parameters given here are in agreement with those presented in paragraph 7 of the previous section. [**4. Exchange Character of Physical Interactions**]{} If one write the base $6\times 6$-relation for two particles with the same internal parameters, then it vanishes due to antisymmetry of determinant columns. A nonzero result arises in using the exchange mechanism of physical interactions. It is based on the postulate that, according to values of the external parameters, the particles may be in two kinds of states: in the $U$-state (“normal”) or in one of the $X$-states (“excited”). The interaction process consists in an exchange of states between particles. The “normal” or $U$-state is characterized by a nonzero $3\times 3$-determinant of the internal parameters. The analysis of possible simplest kinds of $X$-states (taking account of the principle of correspondence to the conventional theory) shows that the $X$-states may be of two kinds. One type ($X_C$) is also characterized by a nonzero determinant of three columns of the external parameters, and for another type ($X_N$) such a deteminant is zero. [**4a) $X_C$-states (charged)**]{} The simplest variant of determining $X_C$ states leading to nonzero base $6\times 6$-relations lies in changing the sign of of the three columns of the internal parameters of the $U$-state. In all there are three such possibilities (channels) which, according to the column number with changed sign, are called $X_X$-, $X_Y$- and $X_Z$-states. It can be shown that for these $X_C$-states only three pairs of terms in the $M(2,2)$-matrix prove to be nonzero. They are located in its $3\times 3$-subblocks similarly to nonzero elements in six nondiagonal Gell-Mann’s matrices $\lambda_n$. Such terms correspond to interactions via charged vector bosons in the conventional theory. [**4b) $X_N$-states (neutral)**]{} In $X_N$-states a pair or all three columns (three-dimensional vectors) of the internal parameters are collinear. In the simplest case on may assume that all three vectors (of the column) $\vec{c}^{\prime}_{(s)}$, where $s=1,2,3$, are collinear, i.e. representable as $ \vec{c}^{\prime}_{(s)}=C^{\prime}_s\vec{c}^{\prime}$, where $\vec{c}^{\prime}$ is a three vector, and $C^{\prime}_s$ are three coefficients. The coefficients $C^{\prime}_s$ in diagonal summands of the submatrix $M(2,2)$ may be shown to enter only as differences, i.e. only two combinations of them are independent. One may choose the combinations as follows: $$C^{\prime}=\frac{1}{2}(C^{\prime}_1+ C^{\prime}_2-2C^{\prime}_3); \ \ \ \tilde{C}^{\prime}= - \frac{1}{2}(C^{\prime}_1-C^{\prime}_2).$$ Two particular cases: А) $C^{\prime}\neq 0; \ \ \tilde{C}^{\prime}=0$; и В) $C^{\prime}= 0; \ \ \tilde{C}^{\prime} \neq 0$ define two channels: an А-channel with the corresponding $X_A$-state and В) a channel with the corresponding $X_B$-state, which should be compared to two channels of interactions via neutral vector bosons in the conventional theory. The two combinations of the coefficients in (15) correspond to two Gell-Mann’s diagonal matrices: $\lambda_3$ and $\lambda_8$ in the conventional representation. The above considerations characterize, so far only qualitatively, an essence of five channels (three charged and two neutral) in the known types of physical interactions. Details will be presented in the next article. [**5. Intermediate Vector Boson Treatment**]{} In the given approach, corresponding to (Fokker-Planck’s) action-at-a-distance concept, there are not intermediate carriers of interactions (vector bosons). To them correspond the above channels of interactions (types of $X$-states of elementary particles). For strong interactions eight gluons correspond to these channels: three pairs of charged gluons corresponding to the $X_X$-, $X_Y$- and $X_Z$-states, and two neutral А- и В-gluons corresponding to the states $X_A$ and $X_B$. Similarly “intermediate” vector bosons “carrying” electroweak interactions. [**6. “Matreshka” Principle**]{} In the conventional gauge field theory three types of spaces are, in fact, used to describe interactions: 1) an external one corresponding to the classical four-dimensional spacetime with the Lorentz transformation group ($SL(2,C)$ group), 2) an internal space of electroweak interactions in which there takes place the group $SU(2)\times U(1)$ and 3) an inernal (chromatic) space of stong interactions with the group $SU(3)$. The theory is being constructed as a composition of these spaces, which is called [*the “bricks” principle*]{} Binary geometrophysics assumes a more economical approach of electroweak interactions being considered as truncated strong interactions, which is achieved by fixing one of the lines of the internal parameters in (14) (let it will be a line of the parameters with index 3), when the group $SL(3,C)$ (or $SU(3)$) of admissible transformations of the internal parameters is narrowed down up to the subgroup $SL(2,C)$ (or $SU(2)$) of the internal parameters with indices 4 and 5. It turns out that in such a truncated theory there remain valid considerations presented above of two types of interactions – analogues of those via charged and neutral vector bosons, with the difference that for one particle generation only one channel (of three $X_C$) survives, which corresponds to interactions via one pair of charged vector bosons. In such a theory, electromagnetic interactions may be shown to correspond to strong ones via neutral A-gluons, weak interactions via neutral vector Z-bosons to strong ones via B-gluons, and electroweak interactions via charged vector $W^{\pm}$-bosons to stong ones via one of the pairs of charged vector gluons (say, $X^{\pm}$-gluons). Such a principle of actual embedding of one type of physical interation to another may be called the [*“matreshka” princile*]{}, an alternative to the conventional “bricks” principle. [**7. Physical Interaction Channel Symmetry Principle**]{} The above procedures and principles allow prototypes of the action (Lagrangian) of strong and electroweak interaction of baryons in terms of quarks (their consistuents) as well as electroweak interaction of leptons (massive ones and neutrinos) between one another and with baryons (quarks) to be written down in the same way. These prototypes are constructed as combinations of products of the four-velocities (of the external parameters) of the particle (quark) components with the corresponding coefficients of the internal parameters having a physical meaning of charges in strong and electroweak interactions. However, therewith there remain uncertain quantities and values of the independent constants and charges in the corresponding interactions. This gap is eliminated by using the physical interaction channel symmetry. It turns out that for obtaining the familiar relations between charges it suffices to know, first, the fact of an existence of the above channels, second, the character of the presence of charges in the corresponding interactions (multiplicativity in charges in the interactions via charged vector bosons) and, third, the conditions of total symmetry of the above channels for all quarks in strong interactions or for left components of particles (quarks or leptons) in electroweak interactions. Therewith strong interactions are unambigously shown to be characterized by only one constant corresponding to the constant $g_0$ in chromodynamics, and electroweak ones – by two constants corresponding to the familiar charges $g_1$ and $g_2$ in Weinberg-Salam’s model. In so doing it is natural that one relationship for two interaction constants is written down, which permits Weiberg’s angle to be calculated theoretically. [**8. Description of Elementary Particle Generations**]{} The approach proposed (in the framework of binary geometrophysics) allows one to resolve (or opens a new avenue of attack on) a number of problems of the modern theory of physical interactions. For example, there opens a possibility of theoretical justification of the existence of just three generations of elementary particles. Simpifying the situation, one may argue that the availabitity of three generations of elementary particles (quarks and leptons) is closely related to the existence of three channels of strong interactions of quarks via charged vector bosons. As mentioned here, in electroweak interactions of the particles of the same generation an analogue of only one pair of charged gluons in the form of W-bosons; in interactions of the particles of two other generations there appear analogues of two other pairs of charged gluons. For leptons, in particular, the definitions of three generations are also related to arrangement of zero columns of the external parameters. It should be particularly emphasized that the approach proposed corresponds basically to the conventional gauge models of physical interactions (e.g., see \[22, 23\]). In its framework notions of antiquarks and antileptons are easily defined, besides introduced are mesons as well as particles being described by pairs of elements in each of two sets of the rank (6,6) BSCR, with, to fit the conventional approach, on of these elements corresponding to the definition of a quark, and another – of an antiquark. A number of other problems will be elucidated in a succession of articles. Architectonics of Binary Geometrophysics ======================================== The aforesaid relates only to the fundamentals of binary geometrophysics. The simplest act of evolution – a transition of two interacting particles from one state to another has been discussed, with elementary particles being as elementary bases for the latter. This always meant three specific subsets of elements comprising the first and second interacting particles and elementary basis. An importamt problem of binary geometrophysics program lies in constructing a transition from an algebraic theory of the level $R_{\mu}(\mu)$ to the conventional spatial-temporal and other attendant notions of modern physics. The most essential points here are as follows. First of all, to the three specific susets of elements mentioned the fourth subset, formed by all other elements that did not enter in these three subsets, should be added. The first three of these subsets may be simple (individual particles) or rather complex up to macro-objects, whereas the added fourth subset is extremely complex by definition. To turn to the conventional physical theories, it is necessary to take the steps describable as the following chain: $$R_{\mu}(\mu)\stackrel{1}{\rightarrow} R^{\prime}_{\mu}(\mu)\stackrel{2}{\rightarrow} R^M_{\mu}(\mu)\stackrel{3}{\rightarrow} R^M_m(\mu)\stackrel{4}{\rightarrow} R^M_m(m).$$ This route has been described in our book \[12\] on the basis of a simplified model in the framework of the rank (4,4) BSCR. In such an approach the base $6\times 6$-relations (or $4\times 4$-relations in the simplified model) generate an interval squared of Kaluza-Klein’s multidimensional metric: $$(\mbox{Base} \ 6\times 6-\mbox{relation})\rightarrow d\Sigma^2=G_{AB}dx^Adx^B,$$ where $G_{AB}$ are the multidimensional metric tensor components, and indices $A$ and $B$ run the values: 0, 1, 2, 3, 4, 5, $\cdots$. Therewith part of terms with the indices $A=\mu$ and $B=\nu$, taking the values: 0, 1, 2, 3, is due to an angular diagonal subblock of the base $6\times 6$-relation $M(4,0)$, the terms with mixed indices (one index is four-dimensional, and another index with the values 5, 6, $\cdots$) are due to the middle submatrix $M(2,2)$. An especial role plays diagonal components of the multidimensional metric with indices $A,B>3$. The prototypes of coordinate shifts are formally introduced in terms of four-dimensional momenta $p^{\mu}=mdx^{\mu}/d\Sigma $, where $d\Sigma=dx^4$ is the prototype of the multidimensional interval. The square-law characteristic of the multidimensional metric prototype is due to the square-law of particle momenta in a number of terms of the base $6\times 6$-relations. The additional momentum components and shift prototypes are obtained from the internal parameters of elements. [**1. The first step**]{}, in fact, has been discussed in the previous section of the article. It consisted in passing from the general notions of the theory of relations between abstract elements to prototypes of the action (Lagrangian) of interacting particles and other conventional physical notions. This step may be denoted as a transition $R_{\mu}(\mu)\stackrel{1}{\rightarrow} R^{\prime}_{\mu}(\mu)$, where $R^{\prime}_{\mu}(\mu)$ means the theory obtained from rank (6,6) BSCR, as a result of applying the procedures and principles of the previous section. At this stage being described in algebraic terms there do not exist most of the habitual notions of modern physics. First of all, there is no a notion of the classical spacetime as well as many attendant macronotions: there are no metric, causality, world lines; the notions of wave functions, particles and fields – interaction carriers are absent; there are no field propagators and singular functions troubling physicists very much in the twentieth century. There are no many other notions, which makes the conventional field theory unthinkable. Finally, gravitational interactions are absent at this level. [**2. The second step (chain)**]{} consists in taking account of all particle of the surrounding world. In modern literature this is called Mach’s principle \[24\]. This step may denoted as a transition $ R^{\prime}_{\mu}(\mu)\stackrel{2}{\rightarrow} R^M_{\mu}(\mu),$ where to the two above factors we added one more, i.e. the external world, designated by the subscript $M$. The basic idea of this step lies in a transition from one base $r\times r$-relation to the sum of such base relations which necessarily contain a fixed number each of elements of an isolated (first) particle in each of two BSCR sets ${\cal M}$ и ${\cal N}$. Symbolically, we represent this transition by the formula: $$\left\{{\alpha\atop i}{\beta\atop k}{\gamma\atop j}{|\atop |}{\delta\atop s}{\lambda\atop l}{\rho\atop r}\right\}^{\prime}\rightarrow \Sigma(1,1^{\prime})=\sum^{World}_{s,l,r; \delta\lambda\rho}\left\{{\alpha\atop i}{\beta\atop k}{\gamma\atop j}{|\atop |}{\delta\atop s}{\lambda\atop l}{\rho\atop r}\right\}^{\prime},$$ where the prime means the base $6\times 6$-relations interpreted according to the previous section. Here the summing is performed over all elements of two BSCR sets ${\cal M}$ and ${\cal N}$, except for elements determining the first particle, i.e. over all particles of the surrounding world. In this case we are dealing with a set of second particles (but not one particle) each of which contributes to the value of the prototype of multidimensional interval of the first particle. It may be singled out by considering separately contributions of the base $6\times 6$-relations containing elements of the second particle. In the framework of this step (stage of theory development) one needs to perform the procedure of $4+1+\cdots$-splitting corresponding to reducibility (splitting) of the rank (6,6) BSCR into two subsystems of ranks (3,3) and (4.4). In Kaluza-Klein’s multidimensional theory context this procedure corresponds to employment of the monadic, dyadic, $\cdots$, s-аdic methods of splitting \[14, 18\] of the n-dimensional manifold into the four-dimensional spacetime and additional dimensions orthogonal to it. It should be especially emphasized that [**only after application of the splitting procedure one may speak about appearance of gravitational interaction.**]{} In the expression (16) the contributions of other particles are due to electroweak interactions. After the splitting procedure there arise prototypes of the components of the four-dimensional curved metric being defined by a sum of Minkowski space and quadratic contributions of mixed components of the metric $G_{\mu a}$, where $a=4, 5, \cdots$, which corresponds to the form of Kaluza-Klein’s metric \[14\]. Therewith the resulting components of the metric for additional dimensions, physically interpretable as prototypes of the electromagnetic and other “intermediate vector fields” field strengths, contain sums of linear contributions of surrounding particles. Thus, [*both electromagnetic and gravitational interactions are determined by the same contributions, but they are summing up linearly for an electromagnetic interaction, and quadratically for a gravitational one*]{}. Remind that the square of a sum of terms is inequal to a sum of squares of the same terms. This means that a linear sum of contributions of electromagnetic interaction may be zero, whereas the sum of squares of the same contributions will be always nonzero, i.e. gravitation interaction will exist even in the absence of an observable electromagnetic field. This conclusion determines a new glance to the nature of gravity as well as the essence of Einstein’s General Relativity. In particular, this determines another approach to many fundamental problem of the modern theory of gravity. After the first stage and performing the splitting procedure in the theory there arise large sums of the terms corresponding to the surrounding world. In Wheeler-Feynman’s absorber theory \[16\], in the framework of Fokker-Planck’s at-a-distance theory, such sums are, in fact, approximated by integrating over matter uniformly distributed in the whole space. [**3. The third step**]{} consists in a transition from the elementary bases $\mu$ to rather complex systems of base elements and in the limit to a macroinstrument $m$: $R^M_{\mu}(\mu)\stackrel{3}{\rightarrow} R^M_m(\mu),$ which, in fact, corresponds to quantum mechanics. Symbollically, this step may be described by the formula: $$\Sigma(1,1^{\prime})\rightarrow \sum_{basis}\left(\sum^{World}_{s,l,r; \delta\lambda\rho}\left\{{\alpha\atop i}{\beta\atop k}{\gamma\atop j}{|\atop |}{\delta\atop s}{\lambda\atop l}{\rho\atop r}\right\}^{\prime}\right)\rightarrow d\Sigma^2(1,1^{\prime}).$$ Here there arise one more summing over the elementary bases comprising a macroinstrument. The sums are complicated considerably, however, at this stage the new sums are approximated by integrating over the momenta of transfer from some objects to others. These sums correspond to Fourier integrals in the conventional representations of vector field potentials. Only after performing the second stage there arise a possibility of speaking about the wave functions of spinor particles and about boson fields – interaction carriers. At this stage there arise notions of retarded and advanced interactions (potentials). The latter are eliminated according to Feynman-Wheeler’s procedure of allowing for world’s absolute absorber. At this stage there arise quantum mechanics and quantum field theory. [**4. The fourth step**]{} consists in a transition from consideration of microparticles to description of micro-objects: $R^M_m(\mu)\stackrel{4}{\rightarrow} R^M_m(m),$ which means a transition to classical physics. The latter is realized by summing (averaginig) over all particles comprising macro-objects (a isolated particle). Strictly speaking, only after this stage the classical spacetime is said to be constructed from primary relations. It is here that there arise notions of distances and time intervals meant in the previous paragraph, when speaking of the wave functions and boson interaction field-carriers. Conclusion ========== In the present article the main principles forming the basis for a new approach to constructing the unified theory of spacetime and physical interactions have been indicated, only a brief characteristic of the most essential results obtained in the framework of binary geometrophysics are given. A thorough presentation will be given in a run of articles devoted to separate above-mentioned problems. The author thanks the participants of the Russian Gravitational Society Seminar at Physics Faculty, Moscow State University, as well as the participants of the Theoretical Physics Laboratory Seminar at Joint Institute for Nuclear Research in Dubna for detailed discussions and valuable comments. [99]{} Chew G.F. The Analytic S-Matrix. W.A.Benjamin, INC., New-York – Amsterdam, 1966. Penrose R. Structure of Spacetime. W.A.Benjamin, INC., New-York – Amsterdam, 1968. Penrose R., MacCallum M.A.H., Phys. Repts., [**6C**]{}, 241, (1972). Wheeler J.A., Phys. Rev., [**52**]{}, 1107, (1937). Heisenberg W., Zs. Phys., [**120**]{}, 513, 673 (1943). Eden R.S., Landshoff P.V., Olive D.I., Polkinghorne J.C. The Analytic S-Matrix. Cambridge, 1966. Zimmerman E.J., Amer.J.Philos., [**30**]{}, 97-105 (1962). Kulakov Yu.I., Doklady AN SSSR, [**201**]{}. No.3, 570-572, (1971) (in Russian). Mikhailichenko G.G., Doklady AN SSSR, [**206**]{}. No.5, 1056-1058, (1972) (in Russian). Mikhailichenko G.G. 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--- abstract: 'We calculate the magnetic susceptibility and $g$-factor of the isolated C$_{60}^{-}$ ion at zero temperature, with a proper treatment of the dynamical Jahn-Teller effect, and of the associated orbital angular momentum, Ham-reduced gyromagnetic ratio, and molecular spin-orbit coupling. A number of surprises emerge. First, the predicted molecular spin-orbit splitting is two orders of magnitude smaller than in the bare carbon atom, due to the large radius of curvature of the molecule. Second, this reduced spin-orbit splitting is comparable to Zeeman energies, for instance, in X-band EPR at 3.39KGauss, and a field dependence of the $g$-factor is predicted. Third, the orbital gyromagnetic factor is strongly reduced by vibron coupling, and so therefore are the effective weak-field $g$-factors of all low-lying states. In particular, the ground state doublet of C$_{60}^{-}$ is predicted to show a negative $g$-factor of $\sim -0.1$.' address: - '$^1$ S.I.S.S.A., Via Beirut 4, I-34013 Trieste, Italia' - '$^2$ Istituto Nazionale di Fisica della Materia (INFM)' - '$^3$ I.C.T.P., P.O. Box 586, I-34014 Trieste, Italia' - '$^4$ E.S.R.F., B.P. 220, F 38043 Grenoble Cedex, France' - '$^5$ M.P.I. für Festkörperforschung, D-70569 Stuttgart, F.R.G.' author: - 'Erio Tosatti$^{1,2,3}$[^1], Nicola Manini$^{1,2,4}$[^2],and Olle Gunnarsson$^{5}$[^3]' title: 'SURPRISES IN THE ORBITAL MAGNETIC MOMENT AND $g$-FACTOR OF THE DYNAMIC JAHN-TELLER ION C$_{60}^-$' --- Introduction ============ A neutral, isolated fullerene molecule is an eminently stable and symmetrical system. The fullerene ions (either negative or positive) along with the electronically excited neutral molecule, particularly the long-lived triplet exciton, undergo instead Jahn-Teller (JT) distortions. The negative ion and the triplet exciton, respectively with $t_{1u}$ and $t_{1g}$ symmetry, will distort according to a linear combination of the eight $H_g$ molecular modes. The JT distortion of the positive ion, with $h_u$ symmetry, involves also the six $G_g$ modes in addition to the eight $H_g$ ones. Although accurate numerical values of all couplings are not yet available in all cases, the static JT energy gains are believed to be roughly in the order of 0.1eV. This value is comparable with the typical vibrational frequency, and the coupling is generally of intermediate strength. Several descriptions of the static JT effect in fullerene ions can be found in the literature.[@Negri; @vzr; @Schl; @Johnson] As pointed out more recently, however, a static JT description, where the pseudorotational motion of the carbon nuclei is treated classically, and then quantized separately in the Born-Oppenheimer approximation is, at least in the isolated molecule, and at zero temperature fundamentally inadequate. In other words, the fullerene ions are expected to be genuine [*dynamical* ]{}Jahn-Teller (DJT) systems, where different but equivalent distorted configurations, (forming the usual static JT manifold[@Ceulemans]), are not independent of one another, but are in fact connected by nonzero transition amplitudes[@Wang]. This in turn requires giving up Born-Oppenheimer, and fully quantizing electronic and ionic motions together, which is the essence of DJT. The physical understanding of DJT in C$_{60}$ ions is greatly eased by initially assuming the strong-coupling limit. In this limit, as it turns out, a [*modified*]{} Born-Oppenheimer approximation can again be recovered, provided a suitable gauge field, reflecting the electronic [*Berry phase*]{},[@Berry] is added to the nuclear motion. This situation, discussed originally for the triatomic molecule,[@Delacretaz] and subsequently for other JT systems[@o'brien] has been recently the object of a close scrutiny in fullerene, especially in the negative ions[@AMT; @MTA; @Ihm] and, to a lesser degree, in the positive ion.[@delos] It is found in particular that, if treated in the strong-coupling limit, the odd-charged fullerene ions, in particular the singly-charged C$_{60}^{-}$[@AMT; @Reno] and C$_{60}^{+}$, [@delos; @ChoB95], the Hund’s rule [*triplet*]{} ground state of doubly charged C$_{60}^{2-}$, as well as the neutral $t_{1g}$ triplet exciton,[@triplet:note], must possess this kind of Berry phase. By contrast, the Berry phases cancel out in the [*singlet*]{} configuration of even-charged ions, such as C$_{60}^{2-}$, and C$_{60}^{2+}$. Although, as stated above, the true electron-vibron coupling in C$_{60}$ is in reality only of intermediate strength, the presence or absence of a Berry phase in the strong-coupling limit DJT implies a number of physical consequences, which persist at realistic couplings, and whose importance has been discussed in detail elsewhere. Properties affected include basic ground-state features such as symmetry[@ham] and energy,[@MTA; @Reno] spectral features (including characteristic splittings of the lowest vibron excitations with skipping of even angular momenta[@AMT; @MTA; @Reno; @Gunnarsson]), scattering anomalies such as suppression of ordinary s-wave attachment of low-energy electrons,[@attachment] and the prediction of orbitally-related electron pairing phenomena in idealized molecular metal lattices with weak electron hopping between molecules.[@MTD; @SAMTP; @SMPT; @Fabrizio] For these and other reasons it seems important to understand completely and quantitatively the DJT effect of single fullerene ions, experimentally as well as theoretically. To this date, however, and in spite of a large amount of data collected on fullerene and especially on fulleride systems, there is still frustratingly little [*direct*]{} evidence that the JT effect in fullerene ions is, to start with, really dynamical, as theory predicts. In the solid state, for example, Raman data on metallic fullerides such as K$_3$C$_{60}$[@Zhou2] fail to show the characteristic vibron splittings expected for the isolated C$_{60}^{3-}$ ions.[@MTA; @Reno] This probably means that in true metallic fullerides the molecular JT effect may be profoundly affected and modified by the large crystal fields, as well as by the strong, rather than weak, intermolecular electron hopping. Even in nonmetallic fullerides like K$_4$C$_{60}$, where the insulating behavior is almost certainly due to a molecular JT effect,[@MTA; @Kerkoud] it is presently not at all clear whether the quantum dynamical features are present or suppressed. Fullerene ions have also been widely studied in solutions[@Dubois] and in solid ionic salts[@kato] but again no specific DJT signature has been pinpointed, so far. The main target remains therefore [*gas-phase fullerene*]{}: the DJT signatures should be unmistakable in ions of either sign, and in triplet-excited neutral molecules, particularly if in the future Raman excitations could be studied.[@Reno] What is available so far are essentially only gas-phase photoemission spectra of C$_{60}^{-}$,[@Gunnarsson] and of C$_{60}$.[@bruhwiler] Encouragingly, the former are fit very well indeed by a DJT theory.[@Gunnarsson] Nonetheless, this can only be considered as indirect evidence. The positive ions results have not yet been analyzed, although the appropriate DJT theory has been formulated.[@delos] We therefore wish to consider here other properties of the fullerene ions, among those crucially affected by the DJT effect, which could at least in principle be accessed either in gas-phase, or in ideally inert matrices, or in suitable salts with especially small crystal-field effects. One such quantity is precisely the molecular magnetic moment. The magnetic moment of a [*static*]{} JT molecule is strictly the spin moment. The orbital degeneracy is removed, so long as static-JT energies are, as in the case of C$_{60}$, sufficiently large. The magnetic moment in a [*dynamic*]{}-JT molecule, conversely, is a compound of spin and orbital moments, since here the quantum effects fully restore the original orbital symmetry.[@ham] The calculation of the magnetic moment and effective $g$-factor of fullerene ions in their DJT ground state is precisely the subject of this work. Let us consider for a start the orbital magnetic moment of a molecule. Qualitatively speaking, the proportionality factor between magnetic moment and the mechanical angular momentum can be thought as some [*effective*]{} Bohr magneton $e\hbar /2m^{*}$, where $m^{*}$ is the mass of the orbiting electron ($e$ is the electron charge and S.I. units are used throughout). In an atom, $m^{*}=m_e$, the free electron mass. In a DJT molecule, however, orbital electron motion involves nuclear motion as well, since electronic and vibrational modes are entangled. Therefore, we expect $m^{*}>m_e$, with a corresponding reduction of the orbital magnetic moment. This is a classic example of the so-called “Ham reduction factor”,[@ham65] well-known in DJT systems.[@Englman] The consequence of orbital reduction is that, while the quantum of mechanical angular momentum is of course universal and equal to $\hbar$, that of the magnetic moment for a DJT molecule is [*not universal*]{}. Below, we will calculate quantitatively the orbital magnetic moment for C$_{60}^{-}$. Moreover, since the orbital moment is not easily accessible experimentally, while the total magnetic moment is commonly measured, spin-orbit coupling will have to be introduced, to determine the correct composition of the (DJT-reduced) orbital moment, and of the spin moment. The spin-orbit coupling within the $t_{1u}$ orbital of C$_{60}$ is quantitatively unknown. Here we calculate it, and find some surprises. First, the calculated molecular spin-orbit splitting is very small, roughly one hundredth than in the bare carbon atom. This is related to the larger radius of curvature of the molecular orbit. Second, this reduced spin-orbit splitting is now wholly comparable to typical Zeeman energies, for instance,in X-band EPR at 0.339T. Hence, a field dependence of the low-temperature susceptibility and of apparent $g$-factors is predicted, at least in an idealized gas-phase EPR experiment. Third, we find that the orbital gyromagnetic factor is strongly reduced by vibron coupling, and so therefore are the effective weak-field $g$-factors for all the low-lying states. The ground state $g$-factor, in particular, is predicted to be slightly [*negative*]{}, about -0.1. The model {#model:sect} ========= The basic model Hamiltonian we consider has the following standard structure: $$\label{Hschemat:eq}H=H^0+H^{e-v}+H^{so}+H^B\ .$$ The JT part $H^0+H^{e-v}$ has been introduced and discussed in previous papers.[@AMT; @MTA; @Gunnarsson] We report here the basic version for the coupling to a single $H_g$ vibrational (quadrupolar) mode: $$\begin{aligned} H^0 & = & \hbar \omega \sum_{m=-2}^2(b_m^{\dagger }b_m+\frac 12)+(\epsilon -\mu) \sum_{m=-1}^1\sum_{\sigma =\uparrow,\downarrow } c_{m,\sigma}^{\dagger }c_{m,\sigma }\ , \nonumber\\ \label{Hesplic:eq} H^{e-v} & = & g \frac{\sqrt{3}}2\hbar \omega \sum_{m_1,m_2,\sigma} (-1)^{m_2}<1,m_1;1,m_2\mid 2,m_1+m_2> \\ & & \ \ \ \times \left[ b_{m_1+m_2}^{\dagger} +(-1)^{m_1+m_2}b_{-m_1-m_2}\right] c_{m_1,\sigma }^{\dagger}c_{-m_2,\sigma}\ +\ ... \nonumber\end{aligned}$$ $H^0$ describes the free (uncoupled) electrons and the fivefold-degenerate vibration of frequency $\omega$. $H^{e-v}$ introduces a (rotationally invariant) standard linear coupling between the electronic state and the vibrational mode. The dimensionless linear coupling parameter is indicated as $g$ (not to be confused with the magnetic factors $g_L$). Here we neglect higher-order terms in the boson operators, indicated with continuation dots. Orbital currents are associated with the partly filled $t_{1u}$ level, which is known to derive essentially from a superatomic $L=5$ orbital of C$_{60}$ as a whole.[@Savina; @Troullier] These orbital currents give rise to a magnetic moment, which we now wish to calculate. Electron spin also contributes to the total magnetic moment. Although uninfluenced by JT coupling (the Hamiltonian (\[Hesplic:eq\]) conserves spin), spin is coupled to the orbital motion [*via*]{} spin-orbit coupling $H^{so}=\lambda ({\vec L}\cdot {\vec S})$. The general Hamiltonian finally includes Zeeman coupling to an external magnetic field $B$ along the $z$ axis $$\label{HZeeman1:eqn}H^B=-\mu_B B (g_LL_z+g_SS_z)$$ where $\mu_B=e\hbar /2m$. For generic $g_L$ and $g_S$ factors, the appropriate value for the $g_J$ factor of the spin-orbit coupled state ($\left| L-S\right| \leq J\leq \left| L+S\right| $) is[@Abragam] $$\label{CoupledGfac:eqn} g_J=\frac{g_L+g_S}2+\frac{L(L+1)-S(S+1)}{2J(J+1)}\left(g_L-g_S\right) \ .$$ For C$_{60}^{-}$, $L=1$, $S=\frac 12$, $J=\frac 12$, $\frac 32$, and $g_S=2.0023$ as appropriate for a free spin. Thus, in order to obtain the $g_J $ factors of the individual spin-orbit split states $J=\frac 12$ and $J=\frac 32$, we only need to calculate the value of $g_L$. Two main phenomena should affect the value of the orbital $g_L$-factor: the ${\cal D}^{(L=5)}$ parentage of the $t_{1u}$ (${\cal D}^{(L=1)}$) state, and the DJT coupling with the vibrons. An old paper by Cohan[@Cohan] provides tables with the icosahedral decomposition of spherical states up to $L=15$, as expansion coefficients on an unnormalized and real basis $$\label{UnnormSpHar:eqn} \widetilde{Y}_{L,|M|}^{C/S}=\left[ \frac{4\pi }{2L+1}\ \frac{\left(L+|M|\right) !}{\left(L-|M|\right) !}\right]^{\frac12} \frac{Y_{L,M}\pm Y_{L,-M}}{2(-1)^{\frac 34\pm \frac 14}}$$ The tabulated wave functions which transform as $t_{1u}$ are: $$\begin{aligned} \psi_0&\propto& 2160\ \widetilde{Y}_{5,0}^C\ +\ \widetilde{Y}_{5,5}^C \nonumber\\ \psi_{C/S}&\propto& 72\ \widetilde{Y}_{5,1}^{C/S}\ \mp \ \widetilde{Y}_{5,4}^{C/S}\ , \label{PsiC/S:eqn}\end{aligned}$$ which can be normalized to obtain $$\begin{aligned} \psi _0&=& {\cal C}\ \left[ 2160\ Y_{5,0}\ +\frac{\sqrt{10!}}2 \left(Y_{5,5}+Y_{5,-5}\right) \right] = \frac 6{\sqrt{50}}Y_{5,0}+ \sqrt{\frac 7{50}}\left(Y_{5,5}+Y_{5,-5}\right) \nonumber\\ \label{PSIC/S:eqn} \psi _{C/S}&=&{\cal C}\ \left[ 72\sqrt{\frac{6!}{4!}}\ \left(Y_{5,1}\pm Y_{5,-1}\right) \ \mp \sqrt{\frac{9!}{1!}} \left(Y_{5,4}\pm Y_{5,-4}\right) \right] =\\ & & \frac 1{\sqrt{2}~(-1)^{\frac34 \pm \frac14}}\left[ \sqrt{\frac 3{10}}\ \left(Y_{5,1}\pm Y_{5,-1}\right) \ \mp \sqrt{\frac 7{10}}\left( Y_{5,4}\pm Y_{5,-4}\right) \right] \ , \nonumber\end{aligned}$$ yielding $$\label{PSIPM:eqn} \psi _{\mp 1}= \frac{\psi _C\pm i\psi _S}{\sqrt{2}}= \sqrt{\frac 3{10}}\ Y_{5,\mp 1}-\sqrt{\frac 7{10}}Y_{5,\pm 4}\ .$$ On the $\left\{ \psi _M\right\} _{M=-1,0,1}$ basis of $t_{1u}$, the orbital Zeeman coupling with the external magnetic field is diagonal: $$\label{gEffPar:eqn}\left\langle \psi _M\right| H_L^B\left| \psi_{M^{\prime }}\right\rangle =-g_1\mu_B\left[ \frac 3{10}(-1)+\frac 7{10}\cdot (4)\right] B\ M\ \delta_{MM^{\prime }}=-\frac 52g_1\mu_B B\ M\ \delta_{MM^{\prime }}$$ This formula implicitly defines an effective orbital $\widetilde{g}_1$ factor $\frac 52g_1$, ($\alpha = \frac 52$, in the language of Ref.[@Abragam]). It is convenient to define for the $t_{1u}$ orbital an effective angular momentum $\vec {\widetilde{L}}$, whose $z$-component has values -1,0,1 on the $\left\{ \psi _M\right\}_{M=-1,0,1}$ basis, in terms of which the orbital Zeeman interaction (\[HZeeman1:eqn\]) is rewritten $$\label{HZeeman2:eqn} H_L^B=-\widetilde{g}_L \mu_B B\widetilde{L}_z= -\alpha g_L \mu_B B\widetilde{L}_z$$ It is clear that the enhancement is due to the $|M|=4$ component in the the $t_{1u}$ wave function, which is really $L=5$, but is regarded formally as an effective $\widetilde{L}=1$ state. Reference [@Troullier] also provides the explicit spherical parentage of the LUMO orbital, now in a solid-state environment. By computing the spectrum of $L_z$ on the basis provided in that work, we obtain a somewhat smaller value for $\alpha $, namely $1.86$. However, for gas-phase C$_{60}^{-}$ the group-theoretical value for a strictly $L=5$ parentage, $\alpha =2.5$, is probably more accurate. We now concentrate on the second effect, namely that of the coupling of the electronic state with the vibrons. For clarity we start considering a single $H_g$ vibron coupled to the $t_{1u}$ level, as in Eq. (\[Hschemat:eq\]). We represent the $L_z$ operator in second quantization as $\sum_\sigma \left(c_{1\sigma }^{\dagger }c_{1\sigma }-c_{-1\sigma }^{\dagger }c_{-1\sigma }\right)$ (same notation as in Eq. (\[Hesplic:eq\])) and we measure the magnetic energy $\mu_B B$ in units of the energy scale of the vibron, $\hbar \omega $, and indicate it with ${\cal B}\equiv \frac{\mu_B B}{\hbar\omega}$. As in Ref. [@AMT], it is instructive to treat first the weak-JT-coupling limit. We solve the quantum problem in perturbation theory to second order in the e-v coupling parameter $g$, this time including $H^B$ (which is diagonal on the basis $\left| \psi_M\right\rangle$) in the unperturbed Hamiltonian. Thus we consider $H=\left(H^0+H^B\right) +H^{e-v}$, and apply nondegenerate perturbation theory to second order within the threefold space of the $t_{1u} $ level (\[PSIC/S:eqn\]). The second-order energy shift caused by $H^{e-v}$ to level $|\psi _M>$ ($M=-1,0,1$, unperturbed energy $\left(\frac 52-\widetilde{g}_1M\cdot {\cal B}\right) \hbar \omega $) is: $$\label{5.3}\Delta _M^{(2)}=\left\langle \psi _M\right| H^{e-v}\frac 1{\left( \frac 52-\widetilde{g}_1M\cdot {\cal B}\right) \hbar \omega {-}\left(H^0+H^B\right) }H^{e-v}\left| \psi _M\right\rangle$$ while off diagonal terms $\Delta _{MM^{\prime }}^{(2)}$ vanish since $H^{e-v}$ is rotationally invariant. This shift can be rewritten as $$\label{shift}\Delta _M^{(2)}=-\frac 34 g^2\hbar \omega \sum_{m=-1,0,1}\frac{(\langle 1,m;1,-M|2,m-M\rangle)^2} {1+\widetilde{g}_1{\cal B}(M-m)}$$ By substituting the Clebsch-Gordan coefficients, and carrying out the sum over $m$, for each fixed value of $M$, we get $$\begin{aligned} \label{shift0} \Delta _0^{(2)}&=&-\frac 34g^2\hbar \omega \left[ \frac 23+ \frac 12\left( \frac 1{1-\widetilde{g}_1{\cal B}}+\frac 1{1+\widetilde{g}_1 {\cal B}}\right) \right] =-\frac 34g^2\hbar \omega \left[ \frac 53+{\cal O}({\cal B} ^2)\right] \\ \Delta _{\pm 1}^{(2)}&=&-\frac 34g^2\hbar \omega \left[ \frac 16+\frac 12\frac 1{1\pm \widetilde{g}_1{\cal B}}+ \frac 1{1\pm 2\widetilde{g}_1{\cal B}}\right] = -\frac 34g^2\hbar \omega \left[ \frac 53\mp \frac 52 \widetilde{g}_1{\cal B}+{\cal O}({\cal B}^2)\right] \nonumber\end{aligned}$$ The weak-field ${\cal B}$-expansion, is done here under the customary assumption that the magnetic energy is the smallest energy scale in the problem (we will return to this point later, however). The final result for the energy to first order in $\cal B$ of the three $t_{1u}$- derived levels is finally $$\label{EsecOrd:eqn}E_M^{(2)}=\left(\frac 52-\frac 54 g^2 \right) \hbar \omega -M\left(1-\frac{15}8g^2\right) \widetilde{g}_1{\cal B\ }\hbar \omega \ .$$ The result of e-v coupling is a reduction of both zero-point energy ($-\frac 54 g^2\hbar \omega$), and magnetic moment. By identification we obtain $$\label{Geff1} g_1^{{\rm {eff}}}=\left(1-\frac{15}8g^2\right) \widetilde{g}_1= \left(1-\frac{15}8g^2\right) \frac 52g_1= \left(1-\frac{15}8g^2\right) \frac 52\ ,$$ the desired perturbative result for the reduction of the $g_1$-factor due to weak coupling to an $H_g$ mode. The factor $\frac 52$ reflects the $L=5$ parentage and $\left(1-\frac{15}8g^2\right) $ is the (weak-coupling) Ham reduction factor[@Abragam] of this DJT problem, correctly coincident with that obtained for a general vector observable by Bersuker and Polinger.[@Bersuker] As anticipated, the reduction factor reflects the increased “effective mass” of the $t_{1u}$ electron, as it carries along some ionic mass while orbiting. However, the coupling in C$_{60}$ is not really weak, and perturbation theory is essentially only of qualitative value. For quantitative accuracy, we can instead solve the problem by numerical (Lanczos) diagonalization,[@AMT; @Gunnarsson; @Reno] which is feasible up to realistically large coupling strengths. On a basis of states $$\label{Basis}\Psi =\sum \epsilon _{k_1\mu _1,...k_N\mu _N,M,\sigma }b_{k_1\mu _1}^{\dagger }...b_{k_N\mu _N}^{\dagger }c_{M\sigma }^{\dagger }|0>$$ (where $|0>$ is the state with no vibrons and no electrons), truncated to include up to some maximum number $N$ of vibrons, ($N$ must be larger for larger coupling) we diagonalize the Hamiltonian operator (\[Hschemat:eq\]), and take the numerical derivative of the ground-state energy with respect to the magnetic field ${\cal B}$. Again, we consider here only the orbital part, and ignore spin for the time being. In Fig. \[Gfactor:fig\] we plot the resulting reduction of $g_1$-factor as a function of $g^2$ for a single $H_g$ mode. The initial slope at $g=0$ coincides correctly with $-\frac{15}8$, while at larger coupling, the behavior is compatible with the expected Huang-Rhys–type decrease, $\sim \exp (-\chi g^2).$ We now repeat the same diagonalization including all the eight $H_g$ vibrons with their realistic couplings, as extracted by fitting gas-phase photoemission spectra of C$_{60}^{-}$.[@Gunnarsson] Including, as in our previous calculation of ground state and excitation energies,[@Gunnarsson; @Reno] up to $N$=5 vibrons for an accuracy of better than two decimal figures, we obtain[@GfacScaling:nota] for the orbital factor of the $t_{1u}$ LUMO of C$_{60}$ a final value of $0.17$, whence $$\label{Geff2}g_1^{{\rm {eff}}}=0.17\widetilde{g}_1=0.17\dot\frac 52 \simeq 0.43\ .$$ With this orbital $g_L$-factor, we can now move on to compute the overall $g_J$-factor in a realistic situation, where however spin-orbit must be included. Spin-orbit coupling in the $t_{1u}$ LUMO, and results {#so:sect} ===================================================== The magnitude of the spin-orbit coupling $\lambda$ in the $t_{1u}$ state of C$_{60}$ is not known. We estimate it by using straightforward tight-binding, as follows. Starting from $2s$ and $2p_x,2p_y,2p_z$ orbitals for each C atom, and including spin degeneracy, we diagonalize the 480x480 first-neighbor hopping Hamiltonian matrix to obtain all the molecular orbitals. Spin-orbit in this scheme is obtained by adding to the hopping Hamiltonian a local coupling on each individual carbon in the form $$H^{so}=\lambda_{\rm at} \sum_i {\bf L}_i\cdot {\bf S}_i \ .$$ The level splitting introduced by this term defines the precise value of spin-orbit coupling for each molecular orbital. We are dealing with $\pi$-states, which are unaffected by spin-orbit in a planar case, such as in graphite. However, in fullerene, due to curvature, there will be an effect. In particular the splitting between the LUMO states $^2{t_{1u}}_{\frac 12}$ and $^2{t_{1u}}_{\frac 32}$ gives the spin-orbit coupling for the LUMO. Our calculation yields $\lambda =0.9\cdot 10^{-2}\lambda_{\rm at}$ for the $t_{1u}$ state of C$_{60}$ when all bonds are assumed to have equal lengths, slightly increasing to $$\lambda =1.16\cdot 10^{-2}\lambda_{\rm at} \ ,$$ when bond alternation is included. Since $\lambda_{\rm at}=<2p_z\left| {\bf L\cdot S}\right| 2p_x> \simeq 13.5$ cm$^{-1}$,[@Friedrich] we conclude that the effective spin-orbit splitting of a $t_{1u}$ electron in C$_{60}^{-}$ is of the order of $0.16$ cm$^{-1}=19\mu eV$. This value is exceedingly small, due both to the small value of $\lambda$ in carbon, a low-Z element, and (mainly) to the large curvature radius of C$_{60}$. For relatively large radius $R$, as appropriate to fullerenes and nanotubes, one can expect a small spin-orbit effect in $\pi$-states, of order $\lambda \sim 1/R^{2}$, the lowest power of curvature which is independent of its [*sign*]{}. The $\pi$-electron radius of C$_{60}$, $R \sim$ 5Å, is one order of magnitude larger than in the carbon atom, correctly suggesting a reduction of two orders of magnitude from the atom to C$_{60}$. Larger splittings of 30-50cm$^{-1}$ observed in luminescence spectra had earlier been attributed to spin-orbit.[@Gasyna] These values are incompatible with our estimate, and we conclude that these splittings must be of different origin, unless an enhancement of two orders of magnitude over the gas phase could somehow arise due to the host matrix. We can now include this spin-orbit coupling in the calculation of the full $g$-factor. A strong–spin-orbit approach[@ob92] $\lambda\gg E_{\rm JT}$, relevant for some JT transition impurities, is not useful here, since clearly $\lambda \ll E_{\rm JT}\approx 140$meV[@Gunnarsson; @Reno] $ \sim \hbar \omega $. In this case, the purely orbital description of the Berry phase DJT of Refs. [@AMT; @MTA] provides the correct gross features, to which spin-orbit adds small splittings. These splittings are controlled by the $g_J$ factors of Eq. (\[CoupledGfac:eqn\]). In our $t_{1u}\bigotimes H_g$ case, an effective “$\widetilde{L}=1$” ground state is turned, for positive $\lambda$, into a “$J=\frac 12$” ground-state doublet and a ”$J=\frac 32$” excited quartet. If we assume the usual weak-field limit $\mu_B B\ \ll\ \lambda$, we can recast Eq. (\[CoupledGfac:eqn\]) in the form $$\label{GJ1/2} g_{\frac 12}=-\frac 13g_S+\frac 43g_L\simeq -\frac 23+\frac 43 g_L$$ and $$\label{GJ3/2} g_{\frac 32}=\frac 13g_S+\frac 23g_L\simeq \frac 23+\frac 23 g_L\ .$$ According to these linear formulae, the orbital reduction factor $g_1$ (Fig. \[Gfactor:fig\]) is easily deformed on the vertical axis to give $g_J$. Since $g_L$ decreases from 1 to 0 for increasing JT coupling, we see that while $g_{\frac 32}$ is always positive, $g_{\frac 12}$ may instead become [*negative*]{} at large JT coupling. For example, if the $L=5$ parentage enhancement is neglected, then $g_{\frac 12}$ ranges from $\frac 23$ at zero coupling to $-\frac 23$ at strong coupling. Including the $\frac 52$ orbital parentage factor, $g_{\frac 12}$ finally varies from $\frac 83$ to $-\frac 23$. The physical reason for a possible overall negative $g$-factor for $J=\frac 12$ at large e-v coupling, where $g_L \sim e^{-g^2}$, is also clear. The $J=\frac 12$ overall[*mechanical*]{} angular momentum is dominated by $L=1$ orbital component. The [*magnetic*]{} moment is instead dominated by the spin component, due to the strong reduction of the orbital part. But in the $J=\frac 12$ state, spin and orbit are coupled (mainly) upside down, whence the sign inversion. Inserting the orbital $g_L$ factor (\[Geff2\]) in Eqs. (\[GJ1/2\],\[GJ3/2\]), we get the final effective $g$-factor for the ground state manifold of gas-phase C$_{60}^{-}$ $$\label{Geff3} g_{\frac 12}^{{\rm {eff}}}=-0.1\ ,\ \ g_{\frac 32}^{{\rm {eff}}}=0.95\ .$$ Intermediate field {#intfield:sect} ================== We have just obtained in Sect. \[so:sect\] a slightly negative value for the $g$-factor of the “$J=\frac 12$” ground state. This result, valid in the approximation of $\mu_B B\ \ll\ \lambda \ \ll \ E_{\rm JT}$, becomes unapplicable as soon as $\mu_B B\ \sim\ \lambda$. In an ideal EPR-like experiment on gas-phase C$_{60}^-$, the resonant quantum $h\nu$ is easily comparable with, or larger than, the spin-orbit frequency scale $\lambda/h = 4.7$GHz. For example, standard X-band EPR employs a larger frequency of 9.5GHz, which resonates at $B=0.339$T, for a free spin. For ease of comparison in Fig. \[specEPR:fig\] we report the full spectrum of the Zeeman– and spin-orbit–split low-energy states, calculated under the assumption that $\mu_B B\ \ll\ E_{\rm JT}$, but for general spin-orbit strength, and increasing magnetic field. Here, arrows indicate the symmetry-allowed microwave absorption transitions, matching an arbitrary excitation frequency 9.5GHz. EPR-like lines should ideally appear at the corresponding values of the field. Of the seven lines expected, two correspond to (apparent) $g$-factors vastly larger than 2, and five to $g$-factors vastly smaller than 2. These $g$-factors are only apparent, since they depend on the field (since it is not weak), and through it on the frequency chosen. As the figure indicates, the weak-field limit for C$_{60}^{-}$ should only really be achieved with fields in the order of a few hundred Gauss. Discussion and conclusions {#discuss:sect} ========================== Three main predictions result from the present calculation. First, that the molecular spin-orbit splitting is very small, a fraction of a degree Kelvin. Second, the reduced spin-orbit splitting is now wholly comparable to typical Zeeman energies, for fields of a few KGauss. Hence, a strong and uncommon field-dependence of the $g$-factors is predicted. Third, we find that the orbital gyromagnetic factor is strongly reduced by vibron coupling, and so therefore are the effective weak-field $g$-factors for all the low-lying states. In particular, the isolated C$_{60}^{-}$ ion in its “$J=1/2$” ground state should be essentially non magnetic, in fact slightly diamagnetic, if cooled below $T\approx \frac{E_{3/2}-E_{1/2}}{k_B} \sim 0.2$K. For C$_{60}^{-}$ in solid ionic salts at 77K and room temperature, Kato [*et al.*]{}[@kato] have found $g^{{\rm {eff}}}\simeq 1.999$, with slight anisotropies due to the lattice. Similar $g^{{\rm {eff}}}$values are also obtained for C$_{60}^{-}$ in molecular sieves[@Keizer], as well as in various solvents and salts[@Penicaud; @Schell-Sorokin; @Khaled; @Boyd]. These $g$-factors are relatively close to the bare-spin value, implying that the dynamical orbital effects discussed above are apparently quenched by coupling to the matrix. In the photoexcited triplet state of neutral C$_{60}$, some evidence has been found for nonthermal jumps between JT valleys[@mehring], but apparently none for orbital magnetism. In a majority of these systems, all goes as if the extra electron of C$_{60}^{-}$, or the extra electron-hole triplet pair of C$_{60}^{-}$ occupied a nondegenerate level, as expected in the static JT case. A detailed discussion of the quenching of quantum orbital effects is beyond the scope of this work. However, we think that coupling to the host matrix, however weak it may be in some cases, must be responsible for the apparent quenching from DJT to static JT. One possibility, for example, is that the extra electron on C$_{60}^{-}$ acts to strongly polarizes the surrounding, which in turn slows down and damps the quantum mechanical electron tunneling between different JT valleys. Insofar as these couplings seem ubiquitous and fatal to the DJT, we would provisionally tend to conclude that gas-phase studies may represent the only serious possibility for the observation of orbital moments in fullerene ions. Stern-Gerlach–like measurements of magnetic moment in gas-phase C$_{60}^{-}$, or other similar experiments, are therefore called for to provide a definitive confirmation of the striking quantum orbital effects described in this paper. Since to our knowledge this would be new, we feel that such experiments should be considered, even if very difficult to carry out. Gas-phase measurements would presumably be easier in the $^3t_{1g}$ triplet exciton state of [*neutral*]{} C$_{60}$. This state has a number of similarities[@triplet:note] to that of C$_{60}^{-}$ which we have just described. However, here $S=1$ and $\widetilde{L}=1$, leading to a $J=0$ singlet ground state[@singlet:note] for the triplet exciton with DJT and spin-orbit coupling, and the magnetic anomalies will have to be sought in the lowest excited states. We acknowledge support from NATO through CRG 92 08 28, and the EEC, through Contracts ERBCHRXCT 920062, and ERBCHRXCT 940438. [10]{} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [^1]: Email: [email protected] [^2]: Email: [email protected] [^3]: Email: [email protected]

--- abstract: 'We present preliminary results for the heavy-light leptonic decay constants in the presence of three light dynamical flavors. We generate dynamical configurations with improved staggered and gauge actions and analyze them for heavy-light physics with tadpole improved clover valence quarks. When the scale is set by $m_\rho$, we find an increase of $\approx\!23\%$ in $f_B$ with three dynamical flavors over the quenched case. Discretization errors appear to be small ($\ltwid3\%$) in the quenched case but have not yet been measured in the dynamical case.' address: - '[-0.10in[0.07in Department of Physics, Washington University, St. Louis, MO 63130, USA]{}]{}' - 'Department of Physics, University of Arizona, Tucson, AZ 85721, USA' - 'Department of Physics, Indiana University, Bloomington, IN 47405, USA' - 'Physics Department, University of Colorado, Boulder, CO 80309, USA' - 'Physics Department, University of Utah, Salt Lake City, UT 84112, USA' - 'SCRI, Florida State University, Tallahassee, FL 32306-4130, USA' - 'RIKEN BNL Research Center, Upton, New York 11973, USA' - 'Department of Physics, University of California, Santa Barbara, CA 93106, USA' author: - 'C. Bernard,-0.03in -0.03in[^1] T. Burch,-0.03in S. Datta,-0.03in T. DeGrand,-0.03in C. DeTar,-0.03in Steven Gottlieb,$\!\null^{\rm c}$ -2pt Urs M. Heller,-0.03in K. Orginos,-0.03in R. Sugar0.005in and D. Toussaint$\null^{\rm b}$ -2pt' title: 'Heavy-light decay constants with three dynamical flavors' --- The computation of leptonic decay constants for heavy-light mesons plays a key role in the extraction of CKM matrix elements from experiment and has been a focus of lattice QCD calculations for many years. Here, we report on the first attempt to compute such decay constants in the presence of $N_F\!=\!3$ light sea quarks ($u$, $d$, $s$). For the generation of the dynamical lattices, we use the “Asqtad” action [@IMP_KS], which consists of order $\alpha_s a^2, a^4$ improved staggered quarks and order $\alpha_s^2 a^2, a^4$ Symanzik improved glue [@IMP_GAUGE; @ALFORD]. This action has small discretization errors for many quantities [@IMP_SCALING; @MILC_POTENTIAL]. For the valence quarks, we employ a tadpole-improved (Landau link) clover action and the Fermilab formalism [@EKM]. The analysis is done in a “partially quenched” manner: the valence quark masses are extrapolated to their physical values with the sea quark masses held fixed. The chiral extrapolation of the sea quark masses (at fixed lattice spacing) is performed afterwards. Table \[tab:lattices\] shows the parameters of our lattices and the current state of this project. As we decrease the dynamical quark masses from large values in the three-flavor case, we keep the three masses degenerate until the physical strange quark mass is reached. We then keep the strange quark mass fixed as we further decrease $m_{u,d}$. Configurations with varying dynamical quark mass (including quenched and two-flavor configurations) have been matched in lattice spacing using the quantity $r_1$ [@MILC_POTENTIAL], which is defined from the static quark potential at shorter distance than $r_0$ [@SOMMER]. From the value $r_0\approx 0.5$ fm we have found $r_1\approx 0.35$ fm [@MILC_SPECTRUM] in full QCD. Note that the errors in these phenomenological values may be as large as $10\%$ [@SOMMER]; we therefore prefer to use $m_\rho$ in setting the absolute scale. Details of the lattice generation and light quark results are in Ref. [@MILC_SPECTRUM]. Our analysis of heavy-light decay constants follows Refs. [@MILC_PRL; @MILC_LAT00]; we describe only the important differences below. ------------------- --------- ---------- ------ ------- dynamical $\beta$ $a$ (fm) \# \# $am_{u,d}/am_s$ gen. $f_B$ $\infty$/$\infty$ 8.00 0.13 408 290 0.02/$\infty$ 7.20 0.13 536 411 0.40/0.40 7.35 0.13 332 – 0.20/0.20 7.15 0.13 341 341 0.10/0.10 6.96 0.13 339 – 0.05/0.05 6.85 0.13 425 425 0.04/0.05 6.83 0.13 351 – 0.03/0.05 6.81 0.13 564 193 0.02/0.05 6.79 0.13 486 486 0.01/0.05 6.76 0.13 407 399 $\infty$/$\infty$ 8.40 0.09 417 200 0.031/0.031 7.18 0.09 162 40 0.0124/0.031 7.11 0.09 25 – ------------------- --------- ---------- ------ ------- : Status of heavy-light running. We show the nominal value of $a$ determined from $r_1=0.35$ fm. All 0.13 fm lattices are $20^3\times64$; all 0.09 fm lattices are $28^3\times96$. “\# gen.” is the number of configurations that have been generated; “\# $f_B$” is the number on which $f_B$ has been calculated. \[tab:lattices\] -1.2truecm On all lattice sets, we compute quark propagators for 5 light and 5 heavy masses. This gives good control over both chiral and heavy quark interpolations/extrapolations. In Ref. [@MILC_PRL], with 3 light quark masses, the chiral extrapolation was a major source of systematic error. Here, changing from linear to quadratic chiral fits of decay constants changes the results by $\ltwid2\%$. The (tadpole improved) perturbative renormalization of the heavy-light axial current, $Z_A^{\rm tad}$, has not yet been calculated. At present, we therefore proceed as follows at $a\!\approx\!0.13$ fm: \(1) Define boosted coupling $\alpha^P_s$ from the plaquette following Refs. [@ALFORD; @WEISZ], and then define the 1-loop coefficient $\zeta_A$ by $Z_A^{\rm tad} = 1 + \alpha_s^P \zeta_A$. (2) Fix the scale from $m_\rho$ on each set, with valence quarks extrapolated to physical values. \(3) Fix $\zeta_A$ by demanding that $f_B^{\rm quench}=169$ MeV (the MILC continuum-extrapolated result [@EXPLAINfB]). This gives $Z_A^{\rm tad}$ on all the 0.13 fm lattices, since $\zeta_A$ is independent of the number of dynamical flavors. Note that $\zeta_A$ is dependent on the heavy quark mass through the dimensionless quantity $am$. To test scaling we first repeat the above procedure for $f_D$ at 0.13 fm and then interpolate to the correct quark mass for $f_B$ at 0.09 fm. There is a 2% change in $\zeta_A$ for $f_B$ between the two lattice spacings. Because this change results in only a 0.3% effect on $f_B$, we believe that systematic effects in the procedure are negligible. Note also that comparisons of $f_B$ with $f_B^{\rm quench}$ or changes of $f_B$ with $a$ are only very weakly dependent on our assumption about the continuum value of $f_B^{\rm quench}$. Further, the ratio $f_{B_s}/f_B$ is essentially independent of perturbation theory. In the plots below, we show two sets of errors. The smaller ones represent the statistical errors from the fits, computed by jackknife. We use the same fitting ranges in time for all lattices with the same spacing. The larger error bars take into account variations caused by changing the fitting ranges while keeping confidence levels and statistical errors reasonable. Preliminary data for $f_B$ with the scale set by $m_\rho$ is shown in Fig. \[fig:fb\]. There is clear evidence for an increase in $f_B$ as the dynamical quark mass is decreased. There is also an increase in the $N_F\!=\!3$ results over those with $N_F\!=\!2$. Note that the quenched results at the two lattice spacings are consistent; they differ by $\ltwid 3\%$. -.1truein Figure \[fig:fb\_r1\] shows $f_B$ with the scale set by $r_1=0.35$ fm (the phenomenological error in this value of $r_1$ is not included). There is now little if any dependence on the dynamical quark mass: Holding $r_1$ fixed forces the short distance potentials to match quite closely, which in turn keeps $f_B$, the “wave function at the origin,” more or less fixed. In the physical case (three dynamical quarks extrapolated to physical masses) the values of $f_B$ in Figs. \[fig:fb\] and \[fig:fb\_r1\] are consistent. The ratio  is shown in Fig. \[fig:fbsofb\]. No obvious dependence on the number or masses of dynamical quarks is apparent. We therefore fit all the data to a constant. For the moment, we assume that the scaling errors in $f_B$ or  are no larger than the difference in our quenched results at 0.13 fm and 0.09 fm. With this assumption (which will be tested in the coming year) we arrive at the following [*preliminary*]{} results: $f_B/f_B^{\rm quench}=1.23(4)(6)$ and $\efbsofb=1.18(1)({}^{+4}_{-1})$, where the first error is statistical and the second is systematic, including discretization and chiral extrapolation (valence and dynamical) errors, the uncertainties in $\kappa_s$ (for ), and a rough estimate of perturbative errors. A direct determination of $f_B$ itself with three dynamical flavors must await the calculation of the renormalization factors. We thank NPACI, The Alliance, PSC, NERSC, LANL, Indiana University, and Washington University (CSPC) for computing resources. This work was supported in part by the DOE and NSF. [9]{} S. Naik, (1989) 238; K. Orginos, D. Toussaint and R.L. Sugar, Phys. Rev. D [**60**]{} (1999) 054503; Nucl. Phys. B (Proc. Suppl.) [**83-84**]{} (2000) 878; G.P. Lepage, Phys. Rev. D [**59**]{} (1999) 074501. K. Symanzik, in [*Recent Developments in Gauge Theories*]{}, eds.G. ’t Hooft , 313 (Plenum,$\!$ 1980); Nucl. Phys. [**B226**]{} (1983) 187; M. Lüscher and P. Weisz, Comm. Math. Phys. [**97**]{} (1985) 19; Phys. Lett. [**158B**]{} (1985) 250. M. Alford, , Phys. Lett. [**361B**]{} 87 (1995). The MILC collaboration: C. Bernard , Phys. Rev. D [**61**]{}, 111502(R) (2000). The MILC collaboration: C. Bernard , Phys. Rev. D [**62**]{}, 034503 (2000). A. El-Khadra, A. Kronfeld and P. Mackenzie, (1997) 3933. R. Sommer, (1994) 839. The MILC collaboration: C. Bernard , Phys. Rev. D [**64**]{} (2001) 054506. The MILC collaboration: C. Bernard , (1998) 4812. The MILC collaboration: C. Bernard ,  346. P. Weisz and R. Wohlert, (1984) 397. Ref. [@MILC_LAT00] quotes $f_B^{\rm quench}\!=\!173(6)(16)$ MeV, where $f_\pi$ sets the scale for the central value. With an $m_\rho$ scale, this becomes 169 MeV. [^1]: presented by C. Bernard

--- abstract: 'We construct geometric examples of N-differential graded algebras such as the algebra of differential forms of depth $N$ on an affine manifold, and $N$-flat covariant derivatives.' author: - Mauricio Angel and Rafael Díaz title: 'N-flat connections' --- Introduction {#introduction .unnumbered} ============ The applications of the theory of complexes and homological algebra have touched many branches of mathematics such as topology, geometry and mathematical physics. The possibility of developing an homological algebra for the equation $d^N=0$ for $N\geq 3$ has been around at least since 1940 in works by Mayer \[M\]. However, N-homological algebra for $N\geq 3$ only aroused the interest that it deserves in 1991 when Kapranov’s paper \[K\] appeared. Fundamental works by Dubois-Violette appeared in \[DV1\], \[DV2\] soon after. The key notion is that of a q-differential graded algebra (q must be a primitive N-root of unity), which consists of a ${\mathbb{Z}}$-graded vector space $V$ together with an operation $m:V\otimes V\to V$ and an degree one map $d:V\to V$ such that: 1) $m$ is associative, 2) $d$ satisfy the q-Leibniz rule $d(ab)=d(a)b+q^{deg(a)}ad(b)$ and 3) $d^N=0$. Notice that in the definition of a q-differential graded algebra not only the equation $d^2=0$ is deformed to $d^N=0$, but also the Leibniz rule is substituted by the q-Leibniz rule. The theory of q-differential graded algebras has been further developed in many papers such as \[AB\], \[DV3\], \[DVK\], \[KW\], \[KN\], \[S\]. The foundations of a theory of N-differential graded algebras (N-dga) has been written down in \[AD\]. A N-dga consists of a ${\mathbb{Z}}$-graded vector space $V$ together with an associative operation $m:V\otimes V\to V$ and a degree one map $d:V\to V$ such that: 1) $d^N=0$ and 2) $d(ab)=d(a)b+(-1)^{deg(a)}ad(b)$. Notice that in the definition of a N-dga the equation $d^2=0$ is replaced by $d^N=0$, whereas the Leibniz rule is not modified. This fact explain why our definition is better suited for the use of differential geometric techniques. At first sight it looks difficult to came up with examples of N-dga. The purpose of this paper is to introduce geometric examples of N-differential graded algebras, which show that that this sort of algebras appear naturally in a wide variety of contexts. Our examples range from the algebra of differential forms of depth N (see Section 4) to the theory of N-flat covariant derivatives (see Section 1). N-covariant differentials ========================= The purpose of this paper is to introduce geometric examples of N-differential graded algebras (N-dga). Let us first formally introduce the notion of N-dga \[AD\]. Let ${\mathbf{k}}$ be a commutative ring with unit. \[Ndga\] A [**N-differential graded algebra**]{} or N-dga over ${\mathbf{k}}$, is a triple $({A^\bullet},m,d)$ where $m:A^k\otimes A^l\to A^{k+l}$ and $d:A^k\to A^{k+1}$ are ${\mathbf{k}}$-modules homomorphisms satisfying 1. $({A^\bullet},m)$ is a graded associative algebra. 2. $d$ satisfies the graded Leibniz rule $d(ab)=d(a)b+(-1)^{\bar{a}}ad(b)$. 3. $d^{N}=0$, i.e., $({A^\bullet},d)$ is a $N$-complex. A 1-dga is a graded associative algebra. A 2-dga is a differential graded algebra. Definition above may be justified categorically as follows: consider the category ${\mathbf{NComp_{\mathbf{k}}}}$ of nilpotent differential graded ${\mathbf{k}}$-modules. Objects in ${\mathbf{NComp_{\mathbf{k}}}}$ are pairs $(V,d)$ where $V$ is a ${\mathbb{Z}}$-graded ${\mathbf{k}}$-module $V=\displaystyle{\oplus_{i\in{\mathbb{Z}}}V^i}$ together with a degree one map $d:V\to V$ such that $d^N=0$ for some $N\geq 2$. ${\mathbf{NComp_{\mathbf{k}}}}$ is a symmetric monoidal category since if $(V_1,d_1)$ is such that $d_1^{N_1}=0$ and $(V_2,d_2)$ is such that $d_2^{N_2}=0$ then $(V_1\otimes V_2,d_1\otimes Id+Id\otimes d_2)$ satisfy $(d_1\otimes Id+Id\otimes d_2)^{N_1+N_2-1}=0$. N-differential graded algebras (for $N\geq 2$) are the monoids in ${\mathbf{NComp_{\mathbf{k}}}}$. A N-dga $({A^\bullet},m,d)$ is called [**proper**]{} if $d^{N-1}\neq 0$. Let $M$ be a smooth finite dimensional manifold and ${\mathcal{E}}=M\times E$ be a trivial bundle over $M$. The reader may assume that ${\mathcal{E}}$ is a global bundle and that $M$ is actually a neighborhood on which ${\mathcal{E}}$ is trivial. Since our results are covariant they will hold globally as well. The space $\Omega^{\bullet}(M,End({\mathcal{E}}))$ of $End({\mathcal{E}})$-valued forms on $M$ is endowed with a differential graded algebra structure, with the product given by $$\begin{array}{c} \wedge:\Omega^{\bullet}(M,End({\mathcal{E}}))\otimes\Omega^{\bullet}(M,End({\mathcal{E}}))\to\Omega^{\bullet}(M,End({\mathcal{E}}))\\ (\alpha\otimes\psi)(\beta\otimes\phi)\longmapsto (\alpha\wedge\beta)\otimes \psi\phi.\\ \end{array}$$ The space of ${\mathcal{E}}$-valued forms $\Omega^{\bullet}(M,{\mathcal{E}})$ is endowed with a differential graded module structure over $\Omega^{\bullet}(M,End({\mathcal{E}}))$ and also over the differential graded algebra $\Omega^{\bullet}(M)$ of differential forms on $M$, given by $$\begin{array}{c} \Omega^{\bullet}(M)\otimes\Omega^{\bullet}(M,{\mathcal{E}})\to\Omega^{\bullet}(M,{\mathcal{E}})\\ \alpha(\beta\otimes\phi)\longmapsto (\alpha\wedge\beta)\otimes\phi.\end{array}$$ $$\begin{array}{c} \Omega^{\bullet}(M,End({\mathcal{E}}))\otimes\Omega^{\bullet}(M,{\mathcal{E}})\to\Omega^{\bullet}(M,{\mathcal{E}})\\ (\alpha\otimes\Phi)(\beta\otimes\phi)\longmapsto (\alpha\wedge\beta)\otimes\Phi(\phi)\\ \end{array}$$ Recall that a covariant derivative $\nabla$ on ${\mathcal{E}}$ is a linear map $\nabla:\Omega^\bullet(M,E)\to\Omega^\bullet(M,E)$ of degree 1 such that $$\nabla(a\alpha)=(da)\alpha+(-1)^{deg(a)}a\nabla\alpha, \text{for all $a\in\Omega^\bullet(M)$, $\alpha\in\Omega^\bullet(M,E)$}.$$ Since ${\mathcal{E}}$ is trivial, $\nabla$ may be written as $\nabla=d+\omega$ for some $\omega\in\Omega^1(M,End({\mathcal{E}}))$, where $d$ is the de Rham differential and $\omega$ is the connection one-form of $\nabla$. For any $\alpha\in\Omega^k(M,E)$, $\nabla\alpha\in\Omega^{k+1}(M,E)$ is the $E$-valued form given by $$\nabla\alpha=d\alpha+\omega\wedge\alpha.\label{Cov}$$ Taking the covariant derivative of (\[Cov\]) we get $$\nabla^2(\alpha)=d\omega\wedge\alpha+\omega\wedge\omega\wedge\alpha=(d\omega+\omega\wedge\omega)\wedge\alpha=F_\omega\wedge\alpha.$$ The 2-form $F_\omega\in\Omega^2(M,End({\mathcal{E}}))$ is called the curvature of $\nabla$. A connection $\omega$ is said to be [**flat**]{} if $\nabla^2=0$ or equivalently if $F_\omega=0$. We say that a connection $\omega$ is [**N-flat**]{} if $\nabla^{N}=0$, where $\nabla=d+\omega$ and $N\geq 2$. A 2-flat connection is just a flat connection. We have the following \[MC2N\] Let $M$ be a manifold and $\mathcal{E}=M\times E$ be a trivial bundle over $M$. A connection $\omega$ is 1. $2N$-flat if and only if $F_\omega^N=0$. 2. $(2N+1)$-flat if and only if $F_\omega^N\nabla=0$. [**proof**]{} Suppose that $K=2N+n$, $n\in\{0,1\}$, then $$\nabla^{K}=(d+\omega)^{2N+n}=(d(\omega)+\omega\wedge\omega)^N\nabla^n=F_\omega^N\nabla^n.\blacklozenge$$ Consider ${\mathbb{R}}^4$ with coordinates $(x_1,x_2,x_3,x_4)$. For $\omega_1,\omega_2:{\mathbb{R}}^4\to {\mathbb{R}}$ we consider the connection $\omega=\omega_1dx_1+\omega_2dx_2$. A simple calculation shows that $$F_\omega=\Bigl(\frac{\partial\omega_1}{\partial x_2}- \frac{\partial\omega_2}{\partial x_1}\Bigr)dx_1\wedge dx_2\neq 0, \ \text{if}\ \frac{\partial\omega_1}{\partial x_2}-\frac{\partial\omega_2}{\partial x_1}\neq 0.$$ For example one can take $\omega_1=x_2$ and $\omega_2=-x_1$, then $F_\omega=2dx_1\wedge dx_2\neq 0$. But $F_\omega^2=0$ which implies that $\omega$ is a 4-flat connection and $(\Omega^\bullet({\mathbb{R}}^4,End({\mathbb{R}}^4)),\nabla)$ is a proper 4-dga. The previous example is an instance of the following result. Let $M$ be a manifold and assume that $TM=A\oplus B$, where $dim(A)=a$, $dim(B)=b$, and $\omega$ is a connection on $TM$ such that if $F_\omega(\alpha,\beta)\neq 0$ then $\alpha\in A$ and $\beta\in A$. In this case $\nabla=d+\omega$ is $2N$-flat for $N>a$. Let now $M$ be a $n$-dimensional smooth manifold and ${\mathcal{E}}=M\times E$ a trivial bundle on it. Using local coordinates $x_1,\cdots,x_n$ a connection $\omega\in\Omega^1(M,End({\mathcal{E}}))$ may be written as $\omega=\omega_idx^i$. The 2-form $d\omega+\omega\wedge\omega$ can be written as $F_{ij}dx^i\wedge dx^j$ where $F_{ij}=\partial_i \omega_j-\partial_j \omega_i+ [\omega_i,\omega_j]$. Furthermore $$(F_{ij}dx^i\wedge dx^j)^k=\sum_{A\subseteq [n]}\biggl(\sum_{\alpha\in P(A)}\prod_{i=1}^k sign(\alpha)F_{a_i,b_i}\biggr)dx_{s_1}\wedge...\wedge dx_{s_{2k}}$$ where $[n]$ denotes the set $\{1,2,...,n\}$, the cardinality of $A$ is $2k$ and $P(A)$ is the set of ordered pairings of $A=\{s_1<\cdots<s_{2k}\}$. An ordered pairing $\alpha\in P(A)$ is a sequence $\{(a_i,b_i)\}_{i=1}^k$ such that $A=\bigsqcup_{i=1}^k \{a_i,b_i\}$ and $a_i<b_i$. Theorem \[MC2N\] part (1) implies $(\Omega^{\bullet}(M,{\mathcal{E}}),d+\omega)$ is a $2k$-dga if and only if $$\sum_{\alpha\in P(A)}sign(\alpha)\prod_{i=1}^k F_{a_i,b_i}=0,$$ for all $A\subseteq [n]$ with cardinality $2k$. If $dim(M)=2n$, then $(\Omega^{\bullet}(M,{\mathcal{E}}),d+\omega)$ is a $2n$-dga if and only if $$\sum_{\alpha\in P([2n])}sign(\alpha)\prod_{i=1}^n F_{a_i,b_i}=0.$$ Let $(M,g)$ be a Riemannian manifold. The tangent bundle $TM$ has a canonical covariant derivative $\nabla$, called the Levi-Civita connection. Suppose that the Riemannian metric is given in local coordinates $x_1,\cdots,x_n$ by the positive definite symmetric matrix $g_{ij}$: $$ds^2=\sum g_{ij}dx^idx^j.$$ Denote by $\partial_1,\cdots,\partial_n$ the corresponding coordinate vector fields, then the covariant derivatives can be expressed $$\nabla_{\partial_i}\partial_j=\sum_k\Gamma^i_{jk}\partial_k\hspace{1cm}\Gamma^i_{jk}=\sum_l\frac{1}{2}(\partial_kg_{lj}+\partial_jg_{lk}-\partial_{l}g_{jk})g^{il},$$ where $(g^{lk})$ is the inverse matrix to $(g_{kl})$. In this case the curvature 2-form is given by $$R=R_{kl}dx_k\wedge dx_l=R_{jkl}^i\partial_j\otimes dx_i\otimes dx_k\wedge dx_l,$$ where $$R_{jkl}^i=\partial_k\Gamma^i_{jl}-\partial_l\Gamma^i_{jk}+\Gamma^h_{jl}\Gamma^i_{hk}-\Gamma^h_{jk}\Gamma^i_{hl}.$$ $(\Omega^\bullet(M,TM),\nabla)$ is a $2k$-dga if and only if $$\sum_{\alpha\in P(A)}sign(\alpha)\prod_{i=1}^k R_{a_ib_ik}^l=0,$$ for all $A\subseteq [n]$ with cardinality $2k$. Consider the space $S^2\times T^2$ with coordinates $(x_1,x_2,x_3,x_4)$, $0< x_1< 2\pi$, $-\pi/2< x_2< \pi/2$, $0<x_3<1$ and $0<x_4<1$. The metric on $S^2\times T^2$ is given in those coordinates by $$ds^2=sin^2(x_2)(dx_1)^2+(dx_2)^2+(dx^3)^2+(dx^4)^2.$$ The curvature is $$R=\left(\begin{array}{cc} 0& 1 \\ -sin^2(x_2)&0 \\ \end{array}\right)dx_1\wedge dx_2 \neq 0$$ Since $R^2=0$, then $(\Omega(S^2\times{\mathbb{R}}^2,T(S^2\times T^2)),\nabla)$ is a 4-dga. We generalize the previous example as follows: Let $(M,g)$ be a Riemannian manifold and assume that $TM=A\oplus B$ is an orthogonal decomposition, where $dim(A)=a$, $dim(B)=b$, and $g$ is flat on $B$, then $(\Omega(M,TM),\nabla)$ is a $2N$-dga for $N\geq a$. Recall the definition of the generalized cohomology groups of a N-complex $(A,d)$ \[Kap\] $$_pH^{i}(A)=\frac{Ker\{d^{p}:A^{i}\to A^{i+p}\}}{Im\{d^{N-p}:A^{i-N+p}\to A^{i}\}}.$$ The total object $\mbox{{\bf H$^\bullet$}}(A)$ of the cohomology associated to a $N$-complex is $\mbox{{\bf H$^\bullet$}}(A)=\bigoplus_{m=0}^{\infty}\mbox{\bf H$^m$}(A)$, where $$\mbox{\bf H$^m$}(A)=\bigoplus_{2i-p=m}{ _pH^{i}(A)}.$$ Consider the torus $T^4$ with global coordinates $(\theta_1,\theta_2,\theta_3,\theta_4)$ satisfying $(\theta_1,\theta_2,\theta_3,\theta_4)=(\theta_1+1,\theta_2+1,\theta_3+1,\theta_4+1)$. For $$E_{11}=\Bigl(\begin{matrix} 1&0\\0&0\end{matrix}\Bigr)\hspace{1cm}E_{12}=\Bigl(\begin{matrix} 0&1\\0&0\end{matrix}\Bigr)$$ We define $\nabla=d+E_{11}d\theta_1+E_{12}d\theta_2$, then since $[E_{11},E_{12}]=E_{12}$ we have $\nabla^2=E_{12}d\theta_1\wedge d\theta_2\neq 0$, nevertheless it is obvious that $\nabla^4=0$ thus $(\Omega(T^4,{\mathbb{C}}^2),\nabla)$ is a (proper) 4-dga. We have $(d\theta_1\wedge d\theta_2)(d\theta_1)=0$ which implies that $d\theta_1\in Ker(\nabla^2)$, and $d\theta_1$ is not in $Im(\nabla^{2})$, because otherwise $\nabla^2\beta$ is a 2-form for all functions $\beta$. Then the cohomology group $_2H^1(T^4)\neq 0$. In \[AD\] we showed that if $(A,d)$ is a N-dga then ([**H**]{}$(A),d$) is a $(N-1)$-dga, thus we see that $\mbox{{\bf H} }(\Omega(T^4,{\mathbb{C}}^2),\nabla)(\neq 0)$ is a (proper) 3-dga. Let $M$ be a manifold and ${\mathcal{E}}_i$ trivial bundles on $M$, $i=1,2$. For $\nabla^{{\mathcal{E}}_i}$ covariant derivatives on ${\mathcal{E}}_i$, we have a natural covariant derivative $\nabla^{{\mathcal{E}}_1\otimes{\mathcal{E}}_2}$ on the tensor product ${\mathcal{E}}_1\otimes{\mathcal{E}}_2$ given by $$\nabla^{{\mathcal{E}}_1\otimes{\mathcal{E}}_2}(s_1\otimes s_2)=\nabla^{{\mathcal{E}}_1}s_1\otimes s_2+s_1\otimes\nabla^{{\mathcal{E}}_2}s_2,$$ for $s_i\in\Omega(M,{\mathcal{E}}_i)$. The next result is an analog to Theorem 8 \[AD\] and we omit the proof, If $(\Omega(M,{\mathcal{E}}_1),\nabla^{{\mathcal{E}}_1})$ is a N-dga and $(\Omega(M,{\mathcal{E}}_2),\nabla^{{\mathcal{E}}_2})$ is a M-dga, then $(\Omega(M,{\mathcal{E}}_1\otimes{\mathcal{E}}_2),\nabla^{{\mathcal{E}}_1\otimes{\mathcal{E}}_2})$ is a $(M+N-1)$-dga. Equivalently, if $\nabla^{{\mathcal{E}}_1}$ is $M$-flat and if $\nabla^{{\mathcal{E}}_2}$ is $N$-flat then $\nabla^{{\mathcal{E}}_1\otimes{\mathcal{E}}_2}$ is $(M+N-1)$-flat. N-Chern-Simons actions ====================== Let $M$ be a manifold, $dim(M)=2K+1$ and $\mathcal{E}=M\times E$ be a trivial bundle over $M$. For a connection $\omega$ on ${\mathcal{E}}$, consider its K-Curvature $F_{\omega}^{K}=(d\omega+\omega\wedge\omega)^{K}$. On $\Omega^\bullet(M,E)$ exists a linear functional $\int_M Tr\!\!:\Omega^\bullet(M,End(E))\to {\mathbb{R}}$ of degree $2K\!+\!1$, i.e., $\int b=0$ if $\bar{b}\neq 2K+1$, given by $$\omega\longmapsto\int_MTr(\omega), \text{for all $\omega\in\Omega^\bullet(M,End(E))$}.$$ The functional $\int_MTr$ satisfy the conditions of \[AD\], 1. $\int_MTr$ is non degenerate, that is, $\int_MTr(\alpha\wedge\beta)=0$ for all $\alpha$, then $\beta=0$. 2. $\int_M Tr(d(\alpha))=0$ for all $\alpha$, where $d=d_{End({\mathcal{E}})}$. 3. $\int_MTr$ is cyclic, this is $$\int_MTr(\alpha_1\alpha_2...\alpha_n)=(-1)^{\bar{\alpha_1}(\bar{\alpha_2}+\cdots+\bar{\alpha_n})}\int_MTr( \alpha_2...\alpha_n\alpha_1).$$ We define the [**Chern-Simons**]{} functional $cs_{2,2K}:\Omega^\bullet(M,End(E))\to{\mathbb{R}}$ by $$cs_{2,2K}(\omega)=2K\!\int_M Tr(\!\pi (\#^{-1}(\omega(d\omega+\omega^2)^K)))$$ where 1. ${\mathbb{R}}\!\!<\!\!\omega,d\omega\!\!>$ denotes the free ${\mathbb{R}}$-algebra generated by symbols $\omega$ and $d\omega$. 2. $\#:{\mathbb{R}}\!<\!\omega,d\omega\!>\longrightarrow{\mathbb{R}}\!<\!\omega,d\omega\!>$ is the linear map defined by $$\#(\omega^{i_1}d(\omega)^{j_1}...\omega^{i_k}d(\omega)^{j_k})= (i_1+..+i_k+j_1+..+j_k)\omega^{i_1}d(\omega)^{j_1}.. \omega^{i_k}d(\omega)^{j_k}.$$ 3. $\pi:{\mathbb{R}}\!<\!\omega,d(\omega)\!>\longrightarrow \Omega^\bullet(M,End(E))$ is the canonical projection. We have the following result Let $K\geq 1$ be an integer. The Chern-Simons functional $cs_{2,2K}$ is the Lagrangian for the 2K-Maurer-Cartan equation, i.e., $\omega\in \Omega^\bullet(M,End(E))$ is a critical point of $cs_{2,2K}$ if and only if $F_\omega^{K}=0$. For $K=2,\ 3,\ 4$ the Chern-Simons functional $cs_{2,2K}(a)$ is given by $$\begin{aligned} cs_{2,4}(\omega)&=&\int_MTr(\frac{4}{3}\omega(d(\omega))^2+2\omega^3d(\omega)+\frac{4}{5}\omega^5).\\ cs_{2,6}(\omega)&=&\int_MTr(\frac{3}{2}\omega(d(\omega))^3+\frac{18}{5}\omega^3(d(\omega))^2+ 3\omega^5d(\omega)+\frac{6}{7}\omega^7). \\ cs_{2,8}(\omega)&=&\int_MTr(\frac{8}{5}\omega(d(\omega))^4+\frac{16}{3}\omega^3(d(\omega))^3+ \frac{48}{7}\omega^5(d(\omega))^2+4\omega^7d(\omega)+\frac{8}{9}\omega^9).\\\end{aligned}$$ (K,N)-flat connections ====================== Suppose now that we choose a nilpotent derivation $\delta\in\Omega^1(M,End(\mathcal{E}))$ instead of the exterior differential, in this case any covariant derivative on $M\times E$ still may be written $\nabla=\delta+\omega$ for some $\omega\in\Omega^1(M,End(\mathcal{E}))$. Before continue we will review some notations from \[AD\]. For $s=(s_1,...,s_n)\in {\mathbb{N}}^n$ we set $l(s)=n$, the length of the vector $s$, and $|s|=\sum_i{s_i}$. For $1\leq i <n,\ s_{>i}$ denotes the vector given by $s_{>i}=(s_{i+1},...,s_n)$, for $1<i\leq n,\ s_{<i}$ stands for $s_{<i}=(s_1,...,s_{i-1})$, we also set $s_{>n}=s_{<1}=\emptyset$. ${\mathbb{N}}^{(\infty)}$ denotes the set $\bigsqcup_{n=0}^{\infty}{\mathbb{N}}^n$, where by convention ${\mathbb{N}}^{0}=\{\emptyset\}$. For $e\in\Omega^1(M,End(\mathcal{E}))$, we define $e^{(s)}=e^{(s_1)}...e^{(s_n)}$, where $e^{(a)}=d_{End}^{a}(e)$ if $a\geq 1$, $e^{(0)}=e$ and $e^{\emptyset}=1$. In the case that $e_\omega\in\Omega^1(M,End(\mathcal{E}))$ given by $$e_\omega(\alpha)=\omega\wedge\alpha,\hspace{.3cm}\omega\in\Omega^1(M)$$ then $e_\omega^{(a)}=d_{End}^{a}(e_\omega)$ reduce to $e_\omega^{(a)}=e_{d^{a}(\omega)}$, thus $$e_\omega^{(s)}=e_\omega^{(s_1)}\cdots e_\omega^{(s_n)}=e_{d^{s_1}(\omega)}\cdots e_{d^{s_n}(\omega)}.$$ For $N\in{\mathbb{N}}$ we define $E_N=\{s\in{\mathbb{N}}^{(\infty)}:|s|+l(s)\leq N\}$ and for $s\in E_N$ we define $N(s)\in{\mathbb{Z}}$ by $N(s)=N-|s|-l(s)$. We introduce the discrete quantum mechanical system $L$ by 1. $V_{L}={\mathbb{N}}^{(\infty)}$. 2. There is a unique directed edge in $L$ from vertex $s$ to $t$ if and only if $t\in\{(0,s),s,(s+e_i)\}$ where $e_i=(0,..,\underset{\scriptsize{i-th}}{\underbrace{1}},..,0)\in{\mathbb{N}}^{l(s)}$. 3. Edges in $L$ are weighted according to the following table $s(e_i)$ $t(e_i)$ $v(e_i)$ ---------- ----------- ----------------------- $s$ $(0,s)$ $1$ $s$ $s$ $(-1)^{|s|+l(s)}$ $s$ $(s+e_i)$ $(-1)^{|s_{<i}|+i-1}$ The set $P_N(\emptyset,s)$ consist of all paths $\gamma=(e_1,...,e_N)$, such that $s(e_1)=\emptyset$ and $t(e_N)=s$. For $\gamma\in P_N(\emptyset,s)$ we define the weight $v(\gamma)$ of $\gamma$ as $$v(\gamma)=\prod_{i=1}^{N}v(e_i).$$ For example if we have $\delta^K=0$ then the $N$-curvature is given by \[MCMN\] Let $M$ be a manifold and $\mathcal{E}=M\times E$ be a trivial bundle over $M$. Given any connection $\omega$ consider the covariant differential $\nabla=\delta+\omega$, then the N-curvature is given by $$\nabla^N=\sum_{k=0}^{N-1}c_k\delta^k$$ where $$c_k=\sum_{\begin{subarray}{c} s\in E_N\\ N(s)=k \\ s_i<K\\ \end{subarray}}c(s,N)\omega^{(s)} \hspace{.5cm}\text{and}\hspace{.5cm} c(s,N)=\sum_{\gamma\in P_N(\emptyset,s)}v(\gamma).$$ Suppose that we have a derivation $\delta$ such that $\delta^3=0$, and we want to deform it into a covariant 3-differential $\nabla=\delta+\omega$, $\omega\in\Omega^1(M)$. So we required that $\nabla^3=0$. By Theorem \[MCMN\] we must have $\displaystyle{\sum_{k=0}^{2}c_k\delta^k=0}$, let us calculate the coefficients $c_k$. Notice that $$E_3=\{\emptyset, (0), (1), (2),(0,0), (1,0), (0,1), (0,0,0)\}$$ Let us first compute ${\mathbf{c_0}}$. In this case we have four vectors in $E_3$ such that $N(s)=0$, these are $(2), (1,0), (0,1), (0,0,0)$. For $s=(2)$ there is only one path from $\emptyset$ to $(2)$ of length 3,\ $\emptyset\to(0)\to(1)\to(2)$ with weight is 1 and $\omega^{(2)}=\delta^2(\omega)$, thus $c(s,3)=\delta^2(\omega)$. For $s=(1,0)$ there is only one path from $\emptyset$ to $(1,0)$ of length 3,\ $\emptyset\to(0)\to(0,0)\to(1,0)$ with weight 1 and $\omega^{(1,0)}=\delta(\omega)\omega$, thus $c(s,3)=\delta(\omega)\omega$. For $s=(0,1)$ the paths from $\emptyset$ to $(0,1)$ of length 3 are\ $\emptyset\to(0)\to(0,0)\to(0,1)$with weight -1.\ $\emptyset\to(0)\to(1)\to(0,1)$with weight 1.\ $\omega^{(0,1)}=\omega\delta(\omega)$, then $c(s,3)=0$, because the sum of the weight of the 2 paths is 0. For $s=(0,0,0)$ the only path from $\emptyset$ to $(0,0,0)$ of length 3 is\ $\emptyset\to(0)\to(0,0)\to(0,0,0)$the weight is 1 and $\omega^{(0,0,0)}=\omega^3$, thus $c(s,3)=\omega^3$. We proceed to compute ${\mathbf{c_1}}$. In this case we have two vectors in $E_3$ such that $N(s)=1$, $(1)$ and $(0,0)$ For $s=(1)$, the paths from $\emptyset$ to $(1)$ of length 3 are\ $\emptyset\to\emptyset\to(0)\to(1)$with weight 1.\ $\emptyset\to(0)\to(0)\to(1)$with weight -1.\ $\emptyset\to(0)\to(1)\to(1)$with weight 1.\ $\omega^{(1)}=\delta(\omega)$ and thus $c(s,3)=\delta(\omega)$. For $s=(0,0)$ the paths from $\emptyset$ to $(0,0)$ of length 3 are\ $\emptyset\to(0)\to(0)\to(0,0)$with weight -1.\ $\emptyset\to\emptyset\to(0)\to(0,0)$with weight 1.\ $\emptyset\to(0)\to(0,0)\to(0,0)$with weight 1.\ $\omega^{(0,0)}=\omega^2$ and thus $c(s,3)=\omega^2$. Finally we compute ${\mathbf{c_2}}$. In this case we have one vector in $E_3$ such that $N(s)=2$, $(0)$ For the case $s=(0)$ the paths from $\emptyset$ to $(0)$ of length 3 are\ $\emptyset\to\emptyset\to\emptyset\to(0)$with weight 1.\ $\emptyset\to\emptyset\to(0)\to(0)$with weight -1.\ $\emptyset\to(0)\to(0)\to(0)$with weight 1.\ $\omega^{(0)}=\omega$ and thus $c(s,3)=\omega$. Now the 3-curvature is given by $$\nabla^3=(\delta^2(\omega)+\delta(\omega)\omega+\omega^3)+(\delta(\omega)+\omega^2)\delta+\omega \delta^2.$$ Infinitesimal deformations. Suppose that we have a K-differential $\delta$ and for a connection $\omega$ we consider the infinitesimal deformation $\nabla=\delta+t\omega$, where $t^2=0$. $\omega$ is N-flat if we have $\sum_kc_k\delta^k=0$, but in this case $$(t\omega)^{(s)}=(t\omega)^{(s_1)}\cdots(t\omega)^{(s_{l(s)})}=t^{l(s)}\omega^{(s)}=0\hspace{.3cm}\text{unless $l(s)\leq 1$}.$$ Thus $E_N$ is given by $$E_N=\{(1),\cdots,(K-1)\}$$ and $c_k=c((N-(k+1)),N)\omega^{(N-(k+1))}$ for all $k$ such that $2\leq N-k\leq K$. Differential forms of depth N ============================= We consider $U\subseteq{\mathbb{R}}^k$ an open set. We use local coordinates $x_1,\cdots,x_k$ on $U$. For $(N_1,\cdots,N_k)\in{\mathbb{N}}^k_{\geq 2}$ we define the algebra $\Omega_{(N_1,\cdots,N_k)}(U)$ of differential forms of depth $(N_1,\cdots,N_k)$ on ${\mathbb{R}}^k$ as follows: an element $\alpha\in \Omega_{(N_1,\cdots,N_k)}(U)$ is given by $$\alpha=\sum_{I}\alpha_Idx^I$$ where $I:D(I)\subset [k]\to{\mathbb{N}}^+$, and for all $i\in D(I)$ we have $I(i)\in[N_i-1]$, $\alpha_I\in C^\infty(U)$ and $\displaystyle{dx^I=\prod_{i\in D(I)}d^{I(i)}x_i}$. We declare that $deg(x_i)=0$ and $deg(d^kx_i)=k$ then $\Omega_{(N_1,\cdots,N_k)}({\mathbb{R}}^n)$ is a graded algebra with the product given by: for $\alpha,\beta\in\Omega_{(N_1,\cdots,N_k)}({\mathbb{R}}^n)$, $\alpha=\sum_I\alpha_Idx^I$ and $\beta=\sum_J\beta_Jdx^J$ $$\alpha\beta=\sum_{I}(\alpha\beta)_Idx^I,$$ where $$(\alpha\beta)_I=\sum_{K_1\sqcup K_2=D(I)}sgn(K_1,K_2,I)\alpha_{I|K_1}\beta_{I|K_2},$$ and $\displaystyle{sgn(K_1,K_2,I)=\prod_{\begin{subarray}{c}i\in K_1,\ j\in K_2 \\ i>j \end{subarray}}(-1)^{I(i)I(j)}}$. For $\displaystyle{\alpha=\sum_{I}\alpha_{I}dx^I}$ we define the form $d\alpha$ by $$d\alpha=\sum_{1\leq s\leq k}\frac{\partial \alpha_I}{\partial x_s}dx_s\wedge dx^I+\sum_{s\in D(I)}(-1)^{\sum_{t<s}I(t)}\alpha_Idx^{I+\delta_s}$$ where $\delta_s:[k]\to{\mathbb{N}}$ is given by $e_s(j)=\left\{\begin{array}{cc} 1 & j=s \\ 0 & j\neq s .\end{array}\right.$ We have the following $\Omega_{(N_1,\cdots,N_k)}({\mathbb{R}}^n)$ is a $(N_1+\cdots+N_k-k)$-dga. [**proof**]{} Since $\Omega_{(N_1,\cdots,N_k)}(x_1,...,x_k)=\Omega_{N_1}(x_1)\hat{\otimes}\cdots\hat{\otimes}\Omega_{N_k}(x_k)$, we just need to consider the one variable case $\Omega_N(x)={\mathbb{R}}[x,dx,...,d^{N-1}x]/<d^ixd^jx>$, for $1\leq i\leq N-1$. For $\alpha=\sum_{i=0}^{N-1}f_i(x)d^i(x)$ we have $$d^k\alpha=\frac{\partial f_0}{\partial x}d^kx+\sum_{i=1}^{N-k-1}f_i(x)d^{i+k}x.$$ Taking $k=N$ we see that $\Omega_N(x)$ is a $N$-complex, the Leibniz rule and associativity of the product are easy to check, thus $\Omega_N(x)$ is a N-dga. By \[AD, Theorem 8\] $\Omega_{(N_1,\cdots,N_k)}({\mathbb{R}}^n)$ is a $(N_1+\cdots+N_k-k)$-dga.$\blacklozenge$ We shall use the notation $\Omega_N({\mathbb{R}}^k):=\Omega_{N,\cdots,N}({\mathbb{R}}^k)$. Let us recall the definition of affine varieties. $M$ is an affine variety if there is an open covering $\Lambda=\{U_i\}$ of $M$ and diffeomorphisms $\varphi_i:U_i\to{\mathbb{R}}^n$, with $\varphi_i(U_i)$ open, such that $$\varphi_j\circ\varphi_i^{-1}:\varphi_i(U_i\cap U_j)\longrightarrow \varphi_j(U_i\cap U_j)$$ satisfy $\varphi_j\circ\varphi_i^{-1}(x)=A_{ij}x+b_{ij}$, $A_{ij}\in Gl_n({\mathbb{R}})$ and $b_{ij}\in{\mathbb{R}}^n$. $f:{\mathbb{R}}^k\to{\mathbb{R}}^k$ is an affine map, i.e., $f(x)=Ax+b$ for $A\in Gl_k({\mathbb{R}}^k)$, $b\in{\mathbb{R}}^k$. The map given by $$f^*(x_i)=f^i\hspace{.5cm}\text{and}\hspace{.5cm}f^*(d^jx_i)=\frac{\partial f^i}{\partial x_j}d^kx_j,$$ can be extended to a linear map $f:\Omega_N({\mathbb{R}}^k)\to\Omega_N({\mathbb{R}}^k)$ such that 1) $df^*=f^*d$ and 2) $f^*(\alpha\wedge\beta)=f^*(\alpha)\wedge f^*(\beta)$, for all $\alpha,\ \beta\in\Omega_N({\mathbb{R}}^k)$. Let $M$ be an affine manifold. We define the algebra $\Omega_N(M)$ of differential forms of depth N on $M$ as follows, $\Omega_N(M)$ consist of tuples $\alpha=(\alpha_U)_{U\in\Lambda}$, where $\alpha_U\in\Omega_N(\varphi_U(U))$, satisfying the following compatibility condition $$(\varphi_V\circ\varphi_U^{-1})^*(\alpha_V|_{\varphi_V(U\cap V)})=\alpha_U|_{\varphi_U(U\cap V)}.$$ For $U,V\in\Lambda$ such that $U\cap V\neq\emptyset$. Our final result is the following Let $M$ an affine manifold with $dim(M)=m$. $\Omega_N(M)$ is a $(m-1)N$-dga and $\mathbf{H}_N(M)$ is a $((m-1)N-1)$-dga. [00]{} V. Abramov, *On realizations of exterior calculus with dN =0*, Czechoslovak Journal of Physics, 48 (1998) 1265-1272. V. Abramov, N. Bazunova, *Exterior calculus with $d^3=0$ on a free associative algebra and reduced quantum plane*, Proceedings of XIV Max Born Symposium, New Symmetries and Integrable Models, Karpacz, Poland, 21-24.09.1999, World Scientific, Singapore-New Jersey-London-Hong Kong, 2000, pp. 3-7. V. Abramov, R. Kerner, *On certain realizations of the q-deformed exterior differential calculus*, Reports on Math. Phys., 43 (1999) 179-194 M. Angel, R. Díaz, *N-differential graded algebras*, Preprint math.DG/0504398. N. Berline, E. Getzler, M. Vergne, *Heat Kernels and Dirac Operators*, Grundlehren 298, Springer-Verlag, Berlin-Heidelberg-New York, 1992. T. Frankel, *The geometry of physics: an introduction*, Cambridge University Press, 1997. M. Dubois-Violette, *Generalized differential spaces with $d^N=0$ and the q-differential calculus*, Czech J. Phys. 46 (1996) 1227- 1233. M. Dubois–Violette, *Generalized homologies for $d^N=0$ and graded q-differential algebras*, Contemporary Mathematics 219, American Mathematical Society 1998, M. Henneaux, J. Krasil’shchik, A. Vinogradov, Eds., p. 69-79. M. Dubois-Violette, *Lectures on differentials, generalized differentials and some examples related to theoretical physics*, Contemporary Mathematics 294, American Mathematical Society 2002, R. Coquereaux, A. Garcia, R. Trinchero Eds, p. 59-94. M. Dubois-Violette, R. Kerner, *Universal q-differential calculus and q-analog of homological algebra*, Acta Math. Univ. Comenianae Vol. LXV, 2 (1996), 175-188. M.M. Kapranov, *On the q-analog of homological algebra*, Preprint q-alg/9611005. C. Kassel, M. Wambst, *Algèbre homologique des N-complexes et homologie de Hochschild aux racines de l’unité*, Publ. Res. Inst. Math. Sci. Kyoto University 34 (1998), nº 2, 91-114. R. Kerner, B. Niemeyer, *Covariant q-differential calculus and its deformations at $q^N=1$*, Lett. in Math. Phys., 45,161-176, (1998). W. Mayer, *A new homology theory I, II*, Annals of Math. 43 (1942) 370-380 and 594-605. A. Sitarz; *On the tensor product construction for q-differential algebras*, Lett. Math. Phys. 44 1998. $$\begin{array}{c} \mbox{Mauricio Angel. Universidad Central de Venezuela (UCV).} \ \ \mbox{\texttt{[email protected]}} \\ \mbox{Rafael D\'\i az. Universidad Central de Venezuela (UCV).} \ \ \mbox{\texttt{[email protected]}} \\ \end{array}$$

--- address: - 'Mathematics & Computer Science Department, Open University, 1 University Road, P.O. Box 808, Raanana 53537, ISRAEL.' - | Mathematics Department\ Princeton University\ Fine Hall, Washington Road, Princeton, NJ 08544-1000, USA. author: - Manor Mendel - Assaf Naor bibliography: - 'from-from-to-to.bib' title: A relation between finitary Lipschitz extension moduli --- [^1] This note contains a simple and elementary observation in response to [@Bas17]. To explain it, we need to first briefly recall standard notation (introduced in [@Mat90]) related to Lipschitz extension moduli. Suppose that $({\mathcal{M}},d_{\mathcal{M}})$ and $({\mathcal{N}},d_{\mathcal{N}})$ are metric spaces and ${\mathscr{C}}{\subseteq}{\mathcal{M}}$. The Lipschitz constant of a mapping ${\upphi}:{\mathscr{C}}\to {\mathcal{N}}$ is denoted $\|{\upphi}\|_{{\mathrm{Lip}}({\mathscr{C}};{\mathcal{N}})}$. Thus, $\|{\upphi}\|_{{\mathrm{Lip}}({\mathscr{C}};{\mathcal{N}})}\in [0,\infty]$ is the infimum over those $L\in [0,\infty]$ such that $d_{\mathcal{N}}({\upphi}(x),{\upphi}(y)){\leqslant}Ld_{\mathcal{M}}(x,y)$ for all $x,y\in {\mathscr{C}}$. Denote by ${\mathsf{e}}({\mathcal{M}},{\mathscr{C}};{\mathcal{N}})\in [1,\infty]$ the infimum over those $K\in [1,\infty]$ such that for every ${\upphi}:{\mathscr{C}}\to {\mathcal{N}}$ with $\|{\upphi}\|_{{\mathrm{Lip}}({\mathscr{C}};{\mathcal{N}})}<\infty$ there exists $\Phi:{\mathcal{M}}\to {\mathcal{N}}$ that extends ${\upphi}$, i.e., $\Phi(x)={\upphi}(x)$ for all $x\in {\mathscr{C}}$, and satisfies $\|\Phi\|_{{\mathrm{Lip}}({\mathcal{M}};{\mathcal{N}})}{\leqslant}K\|{\upphi}\|_{{\mathrm{Lip}}({\mathscr{C}};{\mathcal{N}})}$. Note that when ${\mathcal{N}}$ is complete, ${\mathcal{N}}$-valued Lipschitz functions on ${\mathscr{C}}$ automatically extend to the closure of ${\mathscr{C}}$ while preserving the Lipschitz constant, so one usually assumes here that ${\mathscr{C}}$ is closed. Given $n\in {\mathbb N}$, the finitary modulus ${\mathsf{e}}_n({\mathcal{M}};{\mathcal{N}})$ is defined to be the supremum of ${\mathsf{e}}({\mathcal{M}},{\mathscr{C}};{\mathcal{N}})$ over all those subsets ${\mathscr{C}}{\subseteq}{\mathcal{M}}$ of cardinality at most $n$. Analogously, denote by ${\mathsf{e}}^n({\mathcal{M}};{\mathcal{N}})$ the supremum of ${\mathsf{e}}({\mathscr{C}}\cup\{x_1,\ldots,x_n\},{\mathscr{C}};{\mathcal{N}})$ over all closed subsets ${\mathscr{C}}{\subseteq}{\mathcal{M}}$ and all $x_1,\ldots,x_n\in {\mathcal{M}}{\smallsetminus}{\mathscr{C}}$. The Lipschitz extension modulus ${\mathsf{e}}_n({\mathcal{M}};{\mathcal{N}})$ has been investigated extensively over the past several decades, though major fundamental questions about it remain open; a thorough description of what is known in this context appears in [@NR17]. A variant of the modulus ${\mathsf{e}}^n({\mathcal{M}};{\mathcal{N}})$ (related to the stronger requirement that Lipschitz retractions exist) was studied in [@Gru60; @Lin64]. However, it seems that the quantity ${\mathsf{e}}^n({\mathcal{M}};{\mathcal{N}})$ did not receive further scrutiny in the literature, prior to the recent preprint [@Bas17]. The purpose of this note is to derive the following simple upper bound on ${\mathsf{e}}^n({\mathcal{M}};{\mathcal{N}})$ in terms of ${\mathsf{e}}_n({\mathcal{M}};{\mathcal{N}})$, thus allowing one to use the available literature on ${\mathsf{e}}_n({\mathcal{M}};{\mathcal{N}})$ to bound ${\mathsf{e}}^n({\mathcal{M}};{\mathcal{N}})$, and in particular to improve some of the estimates in [@Bas17]; see Remark \[rem:quote\] below. Many natural questions related to upper and lower bounds on ${\mathsf{e}}^n({\mathcal{M}};{\mathcal{N}})$ remain open and warrant future investigation. \[claim\] ${\mathsf{e}}^n({\mathcal{M}};{\mathcal{N}}){\leqslant}{\mathsf{e}}_n({\mathcal{M}};{\mathcal{N}})+2$ for every $n\in {\mathbb N}$ and every two metric spaces $({\mathcal{M}},d_{\mathcal{M}}),({\mathcal{N}},d_{\mathcal{N}})$. \[rem:quote\] Fix an integer $n{\geqslant}3$. By combining Claim \[claim\] with [@LN05 Theorem 1.10], it follows that[^2] $${\mathsf{e}}^n({\mathcal{M}};Z)\lesssim \frac{\log n}{ \log\log n}$$ for every metric space ${\mathcal{M}}$ and every Banach space $Z$. This answers (for sufficiently large $n$) a question that was asked in [@Bas17 page 3]. By combining Claim \[claim\] with [@MP84 Theorem 2.12] it follows that $$\forall\, p\in (1,2],\qquad {\mathsf{e}}^n(\ell_p;\ell_2)\lesssim_p (\log n)^{\frac{1}{p}-\frac12}.$$ Also, by combining Claim \[claim\] with [@JL84] it follows that ${\mathsf{e}}^n(\ell_2;Z)\lesssim\sqrt{\log n}$ for every Banach space $Z$. A “dual” version of this estimate follows by combining Claim \[claim\] with [@LN05 Theorem 1.12], which yields that ${\mathsf{e}}^n(Z;\ell_2)\lesssim\sqrt{\log n}$, thus improving (for sufficiently large $n$) over the bound ${\mathsf{e}}^n(\ell_2;Z){\leqslant}\sqrt{n+1}$ of [@Bas17 Theorem 1.2]; more generally, this implies that ${\mathsf{e}}^n(\ell_p;Z)\lesssim_p (\log n)^{1/p}$ for every $p\in (1,2]$. Fix ${{\updelta}}\in (0,1)$, a closed subset ${\mathscr{C}}{\subseteq}{\mathcal{M}}$, and $x_1,\ldots,x_n\in {\mathcal{M}}{\smallsetminus}{\mathscr{C}}$. Since ${\mathscr{C}}$ is closed, we have $d_{\mathcal{M}}(x_j,{\mathscr{C}})>0$ for all $j\in {\mathbb N}$. Hence, there exist $y_1,\ldots,y_n\in {\mathscr{C}}$ such that $$\label{eq:all j} \forall\, j\in {\{1,\ldots,n\}},\qquad d_{\mathcal{M}}(y_j,x_j){\leqslant}(1+{{\updelta}})d_{\mathcal{M}}(x_j,{\mathscr{C}}).$$ Suppose that ${\upphi}:{\mathscr{C}}\to {\mathcal{N}}$ is a Lipschitz mapping. Denote $$\label{eqLdef KL} K{\stackrel{\mathrm{def}}{=}}{\mathsf{e}}_n({\mathcal{M}};{\mathcal{N}})\qquad\mathrm{and}\qquad L{\stackrel{\mathrm{def}}{=}}\|{\upphi}\|_{{\mathrm{Lip}}({\mathscr{C}};{\mathcal{N}})}.$$ There is $\Psi:\{y_1,\ldots,y_n\}\cup\{x_1,\ldots,x_n\}\to {\mathcal{N}}$ such that $\Psi(y_j)={\upphi}(y_j)$ for all $j\in {\{1,\ldots,n\}}$ and $$\label{eq:psi lip} \|\Psi\|_{{\mathrm{Lip}}(\{y_1,\ldots,y_n\}\cup\{x_1,\ldots,x_n\};{\mathcal{N}})}{\leqslant}(1+{{\updelta}})K\|{\upphi}\|_{{\mathrm{Lip}}(\{y_1,\ldots,y_n\};{\mathcal{N}})}{\leqslant}(1+{{\updelta}})KL.$$ Define $\Phi:{\mathscr{C}}\cup\{x_1,\ldots,x_n\}\to {\mathcal{N}}$ by setting $$\Phi(z){\stackrel{\mathrm{def}}{=}}\left\{\begin{array}{ll}\Psi(z)&\mathrm{if}\ z\in \{x_1,\ldots,x_n\},\\ {\upphi}(z)&\mathrm{if}\ z\in {\mathscr{C}}.\end{array}\right.$$ By design, $\Phi$ extends both ${\upphi}$ and $\Psi$. For every $z\in {\mathscr{C}}{\smallsetminus}\{y_1,\ldots,y_n\}$ and $j\in {\{1,\ldots,n\}}$ we have $$\begin{aligned} \nonumber d_{\mathcal{N}}\big(\Phi(z),\Phi(x_j)\big)&{\leqslant}d_{\mathcal{N}}\big(\Phi(z),\Phi(y_j)\big)+d_{\mathcal{N}}\big(\Phi(y_j), \Phi(x_k)\big)\\ \label{use 23} &{\leqslant}Ld_{\mathcal{M}}(z,y_j)+(1+{{\updelta}})KL d_{\mathcal{M}}(x_j,y_j)\\ \label{use nearest} &{\leqslant}L\big(d_{\mathcal{M}}(z,x_j)+d_{\mathcal{M}}(x_j,y_j)\big)+(1+{{\updelta}})^2KLd_{\mathcal{M}}(x_j,{\mathscr{C}})\\\label{use nearest 2}&{\leqslant}L\big(d_{\mathcal{M}}(z,x_j)+(1+{{\updelta}})d_{\mathcal{M}}(x_j,{\mathscr{C}})\big)+(1+{{\updelta}})^2KLd_{\mathcal{M}}(x_j,z)\\ \nonumber &{\leqslant}L\big(d_{\mathcal{M}}(z,x_j)+(1+{{\updelta}})d_{\mathcal{M}}(x_j,z)\big)+(1+{{\updelta}})^2KLd_{\mathcal{M}}(x_j,z) \\ &=\big(2+{{\updelta}}+(1+{{\updelta}})^2K\big)Ld_{\mathcal{M}}(z,x_j),\label{that's all}\end{aligned}$$ where  uses the definition of $L$ and , and both  and  use . Since $\Phi$ extends both ${\upphi}$ and $\Psi$, it is $L$-Lipschitz on ${\mathscr{C}}$ and $(1+{{\updelta}})KL$-Lipschitz on $\{x_1,\ldots,x_n\}$. Therefore, due to  we have $\|\Phi\|_{{\mathrm{Lip}}({\mathscr{C}}\cup\{x_1,\ldots,x_n\};{\mathcal{N}})}{\leqslant}(2+{{\updelta}}+(1+{{\updelta}})^2K)L$. Hence ${\mathsf{e}}^n({\mathcal{M}};{\mathcal{N}}){\leqslant}2+{{\updelta}}+(1+{{\updelta}})^2{\mathsf{e}}_n({\mathcal{M}};{\mathcal{N}})$ and the desired estimate follows by letting ${{\updelta}}\to 0$. By [@Bas17 Theorem 1.1] we have ${\mathsf{e}}^n({\mathcal{M}},{\mathcal{N}}){\leqslant}n+1$ for every $n\in {\mathbb N}$ and all pairs of metric spaces $({\mathcal{M}},d_{\mathcal{M}}), ({\mathcal{N}},d_{\mathcal{N}})$. At the same time, it could be the case that ${\mathsf{e}}_n({\mathcal{M}},{\mathcal{N}})=\infty$; this is so for example when $n=2$, ${\mathcal{M}}={\mathbb R}$ and ${\mathcal{N}}=\{0,1\}$, because there is no nonconstant continuous function from ${\mathbb R}$ to $\{0,1\}$. Hence, in general one cannot reverse the assertion of Claim \[claim\] so as to obtain an estimate of the form ${\mathsf{e}}_n({\mathcal{M}},{\mathcal{N}}){\leqslant}f({\mathsf{e}}^{g(n)}({\mathcal{M}},{\mathcal{N}}))$ for some $f:[1,\infty)\to [1,\infty)$ and $g:{\mathbb N}\to {\mathbb N}$. However, it would be worthwhile to obtain good asymptotic bounds on such $f$ and $g$ for meaningful subclasses of the possible metric spaces $({\mathcal{M}},d_{\mathcal{M}}), ({\mathcal{N}},d_{\mathcal{N}})$, e.g. when ${\mathcal{N}}$ is a Banach space. [^1]: M. M. was supported by the BSF. A. N. was supported by the BSF, the Packard Foundation and the Simons Foundation. The research that is presented here was conducted under the auspices of the Simons Algorithms and Geometry (A&G) Think Tank. [^2]: We use throughout the following (standard) asymptotic notation. Given two quantities $Q,Q'>0$, the notations $Q\lesssim Q'$ and $Q'\gtrsim Q$ mean that $Q{\leqslant}\mathsf{K}Q'$ for some universal constant $\mathsf{K}>0$. The notation $Q\asymp Q'$ stands for $(Q\lesssim Q') \wedge (Q'\lesssim Q)$. If we need to allow for dependence on certain parameters, we indicate this by subscripts. For example, in the presence of an auxiliary parameter ${\uppsi}$, the notation $Q\lesssim_{\uppsi}Q'$ means that $Q{\leqslant}c({\uppsi})Q' $, where $c({\uppsi}) >0$ is allowed to depend only on ${\uppsi}$, and similarly for the notations $Q\gtrsim_{\uppsi}Q'$ and $Q\asymp_{\uppsi}Q'$.

--- abstract: 'We present a class of topological plasma configurations characterized by their toroidal and poloidal winding numbers, $n_t$ and $n_p$ respectively. The special case of $n_t=1$ and $n_p=1$ corresponds to the Kamchatnov-Hopf soliton, a magnetic field configuration everywhere tangent to the fibers of a Hopf fibration so that the field lines are circular, linked exactly once, and form the surfaces of nested tori. We show that for $n_t \in \mathbb{Z}^+$ and $n_p=1$ these configurations represent stable, localized solutions to the magnetohydrodynamic equations for an ideal incompressible fluid with infinite conductivity. Furthermore, we extend our stability analysis by considering a plasma with finite conductivity and estimate the soliton lifetime in such a medium as a function of the toroidal winding number.' author: - Amy Thompson - Joe Swearngin - Alexander Wickes - Dirk Bouwmeester title: Constructing a class of topological solitons in magnetohydrodynamics --- Introduction ============ A *hopfion* is a field configuration whose topology is derived from the Hopf fibration. The Hopf fibration is a map from the 3-sphere ($\mathbb{S}^3$) to the 2-sphere ($\mathbb{S}^2$) such that great circles on $\mathbb{S}^3$ map to single points on $\mathbb{S}^2$. The circles on $\mathbb{S}^3$ are called the *fibers* of the map, and when projected stereographically onto $\mathbb{R}^3$ the fibers correspond to linked circles that lie on nested, toroidal surfaces and fill all of space. The fibers can be physically interpreted as the field lines of the configuration, giving the hopfion fields their distinctive toroidal structure [@Irvine2008]. Hopfions have been shown to represent localized topological solitons in many areas of physics - as a model for particles in classical field theory [@Skyrme1962], fermionic solitons in superconductors [@Ran], particle-like solitons in superfluid-He [@Volovik1977], knot-like solitons in spinor Bose-Einstein (BE) condensates [@Kawaguchi2008] and ferromagnetic materials [@Dzyloshinskii1979], and topological solitons in magnetohydrodynamics (MHD) [@Kamchatnov1982]. The Hopf fibration can also be used in the construction of finite-energy radiative solutions to Maxwell’s equations and linearized Einstein’s equations [@Swearngin2013]. Some examples are Ra[ñ]{}ada’s null EM hopfion [@Ranada1989; @Ranada2002] and its generalization to torus knots [@Irvine2008; @Irvine2013; @Kobayashi2013]. Topological solitons are metastable states. They are not in an equilibrium, or lowest energy, state, but are shielded from decay by a conserved topological quantity. The energy $E$ is a function of a scale factor, typically the size $R$ of the soliton, so that the field could decrease its energy by changing this parameter. However, the topological invariant fixes the length scale and thus the energy. In condensed states (superconductors, superfluids, BE condensates, and ferromagnets) the topological structure is physically manifested in the order parameter, which is associated to a topological invariant. For example, the hopfion solutions in ferromagnets are such that the Hopf fibers correspond to the integral curves of the magnetization vector $\vec m$. The associated Hopf invariant is equal to the linking number of the integral curves of $\vec m$. For many systems the solution can still decay by a continuous deformation while conserving the topological invariant. Another physical stabilization mechanism is needed to inhibit collapse [@Kalinkin2007]. For example, this can be achieved for superconductors with localized modes of a fermionic field [@Pismen2002], for superfluids by linear momentum conservation [@Volovik1977], for BE condensates with a phase separation from a second condensate [@Battye2002], and for ferromagnets with conservation of the spin projection $S_z$ [@Zhmudskii1999]. In MHD, the topological structure is present in the magnetic field. The topological soliton of Kamchatnov has a magnetic field everywhere tangent to a Hopf fibration, so that the integral curves of the magnetic field lie on nested tori and form closed circles that are linked exactly once. The Hopf invariant is equal to the linking number of the integral curves of the magnetic field, which is proportional to the magnetic helicity. In addition to the topological invariant, another conserved quantity is required. MHD solitons can be stabilized if the magnetic field has a specific angular momentum configuration which will be discussed below. Because of the importance of topology in plasma dynamics, there has previously been interest in generalizing the Kamchatnov-Hopf soliton [@Semenov2002]. The topology of field lines has been shown to be related to stability of flux tube configurations, with the helicity placing constraints on the relaxation of magnetic fields in plasma [@Candelaresi2012; @Candelaresi2011influence]. Magnetic helicity gives a measure of the structure of a magnetic field, including properties such as twisting, kinking, knotting, and linking [@Berger1984; @Berger1999]. Simulations have shown that magnetic flux tubes with linking possess a longer decay time than similar configurations with zero linking number [@DelSordo2010; @Candelaresi2011helical; @Candelaresi2011trefoil]. Recently, higher order topological invariants have been shown to place additional constraints on the evolution of the system [@Candelaresi2012; @Yeates2010; @Yeates2011]. The work presented in this paper distinguishes itself from these topological studies of discrete flux tubes in the sense that we are considering the topology and stability of continuous, space-filling magnetic field distributions. Furthermore, our results are analytic, rather than based on numerical simulations.  There are many applications where magnetic field topology has a significant effect on the stability and dynamics of plasma systems. For example, toroidal magnetic fields increase confinement in fusion reactors [@White2001book; @Haeseleer1991book], and solving for the behavior of some magnetic confinement systems is only tractable in a coordinate system based on a known parameterization of the nested magnetic surface topology [@Boozer1982; @Hamada1962; @Haeseleer1991book]. In astrophysics, the ratio of the toroidal and poloidal winding of the internal magnetic fields impacts many properties of stars, including the shape [@Chandrasekhar2013; @Wentzel1961] and momentum of inertia [@Mestel1981], as well as the gravity wave signatures [@Lasky2013] and disk accretion [@Ghosh1978] of neutron stars. The new class of stable, analytic MHD solutions presented in this paper may be of use in the study of fusion reactions, stellar magnetic fields, and plasma dynamics in general. The MHD topological soliton is intimately related to the radiative EM hopfion solution. The EM hopfion constructed by Ra[ñ]{}ada is a null EM solution with the property that the electric, magnetic, and Poynting vector fields are tangent to three orthogonal Hopf fibrations at $t=0$. The electric and magnetic fields deform under time evolution, but their field lines remain closed and linked with linking number one. The Hopf structure of the Poynting vector propagates at the speed of light *without deformation*. The EM hopfion has been generalized to a set of null radiative fields based on torus knots with an identical Poynting vector structure [@Irvine2013]. The electric and magnetic fields of these toroidal solutions have integral curves that are not single rings, but rather each field line fills out the surface of a torus. The time-independent magnetic field of the topological soliton is the magnetic field of the radiative EM hopfion at $t=0$ $$\mathbf{B}_{\rm soliton}(\mathbf{{x}}) = \mathbf{B}_{\rm hopfion}(t=0,\mathbf{{x}}). \label{Bsoliton}$$ The soliton field is then sourced by a stationary current $$\mathbf{j}(\mathbf{{x}}) = \frac{1}{\mu_0} \mathbf{\nabla} \times \mathbf{B}_{ \rm soliton}(\mathbf{{x}}). \label{jsoliton}$$ We will use this relationship, along with the generalization of the EM hopfion to toroidal fields of higher linking number, in order to generalize the Kamchatnov-Hopf topological soliton to a class of stable topological solitons in MHD. We will also discuss how the helicity and angular momentum relate to the stability of these topological solitons. Generalization of the Kamchatnov-Hopf Soliton =============================================  We construct the generalized topological soliton fields using Eqns. (\[Bsoliton\]) and (\[jsoliton\]) applied to the null radiative torus knots. The time-independent magnetic field of the soliton is identical to the magnetic field of the radiative torus knots at $t=0$. The magnetic field is sourced by a current, resulting in a stationary solution. The torus knots are constructed from the Euler potentials: $$\begin{aligned} \label{alpha} \alpha =& \frac{(r^2 -t^2- 1) + 2 \imath z}{r^2-(t-\imath)^2}\\ \beta =& \frac{2 (x - \imath y)}{r^2-(t-\imath)^2}. \label{beta}\end{aligned}$$ where $r^2=x^2+y^2+z^2$. As Ref. [@Irvine2013] points out, at $t=0$ these are the stereographic projection coordinates on $\mathbb{S}^3$. The magnetic field of the torus knots is obtained from the Euler potentials for the Riemann-Silberstein vector ${\mathbf{F} = \mathbf{E} + \imath \mathbf{B}}$.[^1] The solitons are found by taking the magnetic field of the torus knots at $t=0$ $$\begin{aligned} \label{Bfield} \mathbf{B} = Im[\nabla \alpha^{n_{t}} \times \nabla \beta^{n_{p}}]\mid_{t=0}.\end{aligned}$$ Each $(n_{t},n_{p})$ with $n_t,n_p=1,2,3...$ represents a solution to Maxwell’s equations. A single magnetic field line fills the entire surface of a torus. These tori are nested and each degenerates down to a closed core field line that winds $n_t$ times around the toroidal direction and $n_p$ times around the poloidal direction, as illustrated in Fig. \[fig:surfaces\]. A complete solution for a given $(n_t,n_p)$ is composed of pairs of these nested surfaces that are linked and fill all of space as shown in Fig. \[fig:toroidal\]. For $n_g=gcd(n_{t}, n_{p})$, the solution is a magnetic field with $2n_g$ linked core field lines (knotted if $n_t>1$ and $n_p>1$). If $n_{t}=1$ and $n_{p}=1$, the solution is the Kamchatnov-Hopf soliton. We will analyze these fields and how the linking of field lines affects the stability of magnetic fields in plasma. In particular, for $n_{p}=1$ and $n_t \in \mathbb{Z}^+$, we will show that these fields can be used to construct a new class of stable topological solitons in ideal MHD. The solutions with $n_p \neq 1$ are not solitons in plasma, and their instability will be discussed in section \[sec:instability\]. Stability Analysis ================== In this section we assume the plasma is an ideal, perfectly conducting, incompressible fluid. In a fluid with finite conductivity, the magnetic field energy diffuses. Under this condition, one can estimate the lifetime of the soliton as will be shown in section \[sec:lifetime\]. First we consider the case where the poloidal winding number $n_{p}=1$ and the toroidal winding number $n_{t}$ is any positive integer. These will be shown to represent stable topological solitons in ideal MHD. In the next section, we will consider the solutions with $n_{p} \neq 1$. Using the method in this paper, these do not represent stable solitons, and we will discuss how this instability relates to the angular momentum. To analyze the stability of these solutions, following the stability analysis in Ref.[@Kamchatnov1982][^2], we study the two scaled quantities of the system - the length scale $R$ which corresponds to the size of the soliton and $B_0$ which is the magnetic field strength at the origin. (The length scale R is also the radius of the sphere $\mathbb{S}^3$ before stereographic projection.) First we change to dimensionful coordinates by taking $$\begin{aligned} \{ x,y,z \} \rightarrow & \{ \tfrac{x}{R},\tfrac{y}{R},\tfrac{z}{R} \} \\ | \mathbf{B}\left( 0,0,0 \right)| = & B_0.\end{aligned}$$ The stability depends on three quantities - energy, magnetic helicity, and angular momentum - which are functions of $R$ and $B_0$. For a perfectly conducting plasma, the magnetic helicity $h_m$ is an integral of motion and is thus conserved. The magnetic helicity is also a topological invariant proportional to the linking number of the magnetic field lines. If the field can evolve into a lower energy state by a continuous deformation (therefore preserving the topological invariant) then it will be unstable. However, we will show that such a deformation does not exist because the angular momentum $M$ is also conserved and serves to inhibit the spreading of the soliton.  The magnetic helicity is defined as $$\begin{aligned} h_m =& \int{\mathbf{A} \cdot \mathbf{B} d^3x} \end{aligned}$$ where $\mathbf{A} =Im[\alpha^{n_t} \nabla \beta^{n_p}]$ is the vector potential. From Eqns. -, it follows that $$\begin{aligned} h_m =& \frac{2n_{t}}{(n_{t}+1)} \pi^2 B_0^2 R^4. \label{soliton_helicity}\end{aligned}$$ The MHD equations for stationary flow are satisfied for a fluid with velocity $$\begin{aligned} \mathbf{v} = \pm \frac{\mathbf{B}}{(\mu_0 \rho)^{\frac{1}{2}}}.\end{aligned}$$ The energy of the soliton is given by $$\begin{aligned} E =& \int{\left( \frac{\rho v^2}{2}+ \frac{B^2 }{2 \mu_0} \right) d^3x} \\ =& \int{ \frac{B^2}{\mu_0} d^3x} \notag\end{aligned}$$ so that $$\begin{aligned} E =& \frac{2 n_{t} \pi^2}{\mu_0} B_0^2 R^3 \\ \propto & \frac{h_m}{R}. \notag\end{aligned}$$ The angular momentum is $$\begin{aligned} \mathbf{M} =& \rho \int{[\mathbf{x} \times \mathbf{v}] d^3x} \\ =& \left(\frac{\rho}{\mu_0}\right)^{1/2} 4 n_{t} \pi^2 B_0 R^4 \hat{y} \notag\end{aligned}$$ where we took the positive velocity solution. We find that the conserved quantities $h_m$ and $\mathbf{M}$ fix the values of $R$ and $B_0$, $$\begin{aligned} R =& \left(\frac{1}{8\pi^2 n_{t}(n_{t}+1)} \left(\frac{\mu_0}{\rho}\right)\frac{|M|^2}{h_m}\right)^{\frac{1}{4}},\\ B_0 =& 2 n_{t}(n_{t}+1)\left(\frac{\rho}{\mu_0}\right)^{1/2} \frac{h_m}{|M|}, \notag\end{aligned}$$ thus inhibiting energy dissipation. This shows that the solution given in Eqns. - (and shown in Fig. \[fig:toroidal\]) represents a class of topological solitons characterized by the parameter $n_{t} \in \mathbb{Z}^+$ for $n_{p}=1$. Angular Momentum and Instability for $n_p \neq 1$ \[sec:instability\] --------------------------------------------------------------------- For $n_{p} \neq 1$, the angular momentum for all $n_{t}$ is zero. Some examples of fields with $n_{t}=1$ and different $n_{p}$ values are shown in Fig. \[fig:poloidal\]. The field lines fill two sets of linked surfaces. For a given pair of linked surfaces, the field in each lobe wraps around the surface in opposite directions. In Fig. \[fig:poloidal\] the red and blue surfaces wind in opposite directions. This means that the contribution to the angular momentum of the two field lines cancels. In this case the length scale is not fixed by the conserved quantities. The energy can therefore decrease by increasing the radius and the fields are not solitons. Finite Conductivity and Soliton Lifetime \[sec:lifetime\] ========================================================= To include losses due to diffusion, we need to consider a plasma with finite conductivity. We can estimate the soliton lifetime by dividing the energy by $dE/dt$, calculated before any energy dissipation [@Kamchatnov1982]. Since this is the maximum rate of energy dissipation, we can obtain a lower bound on the time it takes for the total energy to dissipate. Thus, $$\begin{aligned} \frac{dE}{dt} =& \frac{1}{\sigma} \int j^2 d^3x \\ =& (3 n_{t} + 7 n_{t}^2 + 5 n_{t}^3) \frac{\pi^2 B_0^2 R}{\mu_0^2 \sigma}.\end{aligned}$$ The resulting lifetime is $$t_{n_{t}} \geq \frac{3 n_{t}}{3 n_{t} + 7 n_{t}^2 + 5 n_{t}^3} \mu_0 \sigma R^2.$$ For higher $n_t$, the lifetime decreases although the helicity in Eqn. increases. This result is interesting as we would have expected from the results regarding flux tubes mentioned previously that the lifetime would increase with increasing helicity. Conclusion ========== We have shown how to construct a new class of topological solitons in plasma. The solitons consist of two linked core field lines surrounded by nested tori that fill all of space. The solutions are characterized by the toroidal winding number of the core field lines and have poloidal winding number one in order to have non-zero angular momentum. We have shown that the conservation of linking number and angular momentum give stability to the solitons in the ideal case. For a plasma with finite conductivity, we have estimated the lifetime of the solitons and found that the lifetime decreases with increasing helicity. Finally, we note that there may be related generalizations of the hopfion fields in other physical systems, such as superfluids, Bose-Einstein condensates, and ferromagnetic materials. The authors would like to acknowledge discussions with J.W. Dalhuisen and C.B. Smiet. This work is supported by NWO VICI 680-47-604 and NSF Award PHY-1206118. [36]{}ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\ 12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty @noop [****, ()]{} [****,  ()](\doibase 10.1016/0029-5582(62)90775-7) @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [“,” ]{} (),  @noop [****, ()]{} [****,  ()](\doibase 10.1002/0471231495.ch2) @noop [****, ()]{} @noop [****, ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****, ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [**]{} (, ) @noop [**]{} (, ) @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****, ()]{} @noop [****,  ()]{} @noop [****,  ()]{} [^1]: Note that the Riemann-Silberstein construction is a non-standard use of Euler potentials. We are following the method in Ref. [@Irvine2013]. [^2]: Note that Ref. uses CGS units and we use SI units in our analysis. The reference also has a typo - Eqn. (45) should have a factor of $R^2$ instead of $R$.

--- abstract: 'We introduce the notion of multipass automata as a generalization of pushdown automata and study the classes of languages accepted by such machines. The class of languages accepted by deterministic multipass automata is exactly the Boolean closure of the class of deterministic context-free languages while the class of languages accepted by nondeterministic multipass automata is exactly the class of poly-context-free languages, that is, languages which are the intersection of finitely many context-free languages. We illustrate the use of these automata by studying groups whose word problems are in the above classes.' address: - 'Dipartimento di Ingegneria, Università del Sannio, C.so Garibaldi 107, 82100 Benevento, Italy' - 'Institut de Recherche Mathématique Avancée, UMR 7501, Université de Strasbourg et CNRS, 7 rue René-Descartes,67000 Strasbourg, France' - 'Laboratoire de Recherche en Informatique, Université Paris-Sud 11, 91405 Orsay Cedex, France' - 'Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green Street (MC-382) Urbana, Illinois 61801- 2975, USA' - ' Department of Mathematical Sciences, Stevens Institute of Technology, 1 Castle Point Terrace Hoboken, New Jersey 07030, USA' author: - 'Tullio Ceccherini-Silberstein' - Michel Coornaert - Francesca Fiorenzi - 'Paul E. Schupp' - 'Nicholas W. M. Touikan' title: Multipass automata and group word problems --- Introduction ============ The main purpose of this paper is to introduce a very natural machine model, that of *multipass automata*. These are essentially like pushdown automata except that they are able to read the input tape several times. It turns out that the class ${\mathcal{DM}}$ of languages accepted by deterministic multipass automata is exactly the Boolean closure of the class of deterministic context-free languages, which we denote by $\mathcal{BDC}$. The class $\mathcal{NM}$ of languages accepted by nondeterministic multipass automata is exactly the class ${\mathcal{PCF}}$ of *poly-context-free* languages, that is, languages which are the intersection of finitely many context-free languages. It follows from [@wotschke1] that the following strict inclusions hold $$\label{e:wotschke} {\mathcal{DM}}= {\mathcal{BDC}}\subsetneq {\mathcal{NM}}= {\mathcal{PCF}}\subsetneq {\mathcal{BC}}$$ where ${\mathcal{BC}}$ denotes the Boolean closure of all the context-free languages. Wotschke also proved that the classes of context-free languages and deterministic multipass languages are incomparable in the sense that neither one is contained in the other. We also mention that Wotschke [@wotschke2] introduced a notion of *Boolean acceptance* and presented a characterization of ${\mathcal BC}$ and of ${\mathcal{PCF}}$ in terms of Boolean acceptance. Our basic references about context-free languages are the monographs by Harrison [@Harrison] and by Hopcroft and Ullman [@HU]. As a motivating example for multipass automata, consider the word problem for the free abelian group of rank two with presentation $G = \langle a,b; ab=ba \rangle$. The associated word problem is the language consisting of all words over the alphabet $\Sigma = \{a, a^{-1},b, b^{-1} \}$ which have exponent sum $0$ on both $a$ and $b$. The pumping lemma for context-free languages shows that this word problem is not context-free. However, this word problem is accepted by a multipass automaton $M$ working as follows. Given an input word $w \in \Sigma^*$, on the first pass $M$ checks if the exponent sum on $a$ in $w$ is $0$ and, on the second pass, $M$ checks if the exponent sum on $b$ in $w$ is $0$. Then $M$ accepts $w$ at the end of the second pass if and only if both conditions are met. In this paper we focus on applying multipass automata to study group word problems, although we believe that they will be useful in many other areas. Brough [@Brough] studied the class $\mathcal{PG}$ of finitely generated groups whose word problem is a poly-context-free language. Holt, Rees, Röver and Thomas [@HRRT] studied the class ${\mathcal{CG}}$ of finitely generated groups whose word problem is the complement of a context-free language. This class has several properties in common with the class of poly-context-free groups. But to show that groups are in ${\mathcal{CG}}$ one generally complements nondeterministic context-free languages, so there are groups in ${\mathcal{CG}}$ which are not in ${\mathcal{PG}}$, for example the standard restricted wreath product $\mathbb{Z} \wr \mathbb{Z}$ [@Brough]. The Muller-Schupp theorem [@MS] shows that a finitely generated group $G$ has context-free word problem if and only if it is virtually free, that is, $G$ contains a free subgroup of finite index. Since the word problem for a finitely generated virtually free group is a deterministic context-free language, nondeterminism does not increase the class of finitely generated groups with context-free word problem. This raises the question of the situation for ${\mathcal{PG}}$. Brough [@Brough] conjectured that the class $\mathcal{PG}$ coincides with the class $\mathcal{D}$ of (finitely generated) groups which are virtually a finitely generated subgroup of a direct product of free groups. We define $\mathcal{BDG}$ to be the class of finitely generated groups whose word problem is in ${\mathcal{BDC}}$, the Boolean closure of deterministic context-free languages. All groups in $\mathcal{D}$ are in ${\mathcal{BDG}}$ so the conjecture would again show that nondeterminism does not increase the class of groups considered. We will prove that if $G \in \mathcal{BDG}$ (respectively $\mathcal{PG}$) and $S$ is a subgroup of finite index, and $\varphi$ is an automorphism of $G$ of finite order with $\varphi(S) = S$ then the HNN-extension $$H = \langle G,t; tst^{-1} = \varphi(s), s \in S \rangle$$ is again in $\mathcal{BDG}$ (respectively $\mathcal{PG}$). This theorem seems to us to use the maximum power of multipass automata. While we initially did not believe that all groups covered in the theorem were in $\mathcal{D}$ it turns out that if we start with a group $G$ in $\mathcal{D}$ then the HNN extension $H$ is indeed again in $\mathcal{D}$, that is, the class $\mathcal{D}$ itself is closed under taking the mentioned HNN extensions. The paper concludes with a proof of this rather delicate algebraic fact. So it turns out that starting with finitely generated virtually free groups, none of the closure properties for the class ${\mathcal{PG}}$ which we establish take us outside the class $\mathcal{D}$. It now seems to us that Brough’s Conjecture is probably true. We point out that the complement of the word problem for a direct product of finitely many free groups is a context-free language and the class ${\mathcal{CG}}$ of finitely generated groups whose word problem is the complement of a context-free language is closed under taking finite extensions and finitely generated subgroups [@HRRT]. Thus Brough’s Conjecture also implies that ${\mathcal{PG}}\subsetneq {\mathcal{CG}}$. There is still a good method for showing that languages are not poly-context-free, namely, one can use Parikh’s theorem. We mention that Gorun [@gorun] used properties of the Parikh map to prove that a certain bounded matrix language is not poly-context-free. Brough [@Brough] developed a very detailed analysis of semi-linear sets which allows her to prove a hierarchy theorem for poly-context-free languages. For example, while the word problem for the free abelian group of rank $k$ is an intersection of $k$ context-free languages it is *not* an intersection of $k-1$ context-free languages. Brough [@Brough] also proves that none of the solvable Baumslag-Solitar groups $\langle b,t; tb^m t^{-1} = b^n \rangle$ where $0 < |m| < |n|$ are poly-context free. The paper is organized as follows: we first define deterministic and nondeterministic multipass automata and study the closure properties of the corresponding classes of languages which they define. After proving the characterizations of the language classes, ${\mathcal{DM}}= {\mathcal{BDC}}$ and $\mathcal{NM} = {\mathcal{PCF}}$, we show that both classes are closed under interleaved products and left quotients by finite sets. We then study group word problems. Whether or not a group is in $\mathcal{BDG}$ or in $\mathcal{PG}$ is independent of the group presentation (Theorem \[t:multipass-w-p\]) and both classes are closed under taking direct products (Corollary \[c:product\]), finite extensions (Theorem \[t:finite-extension\]) and finitely generated subgroups (Theorem \[t:multipass-w-p\]). These easy results are to expected for a class of groups with word problems in a reasonable formal language class and were known for $\mathcal{PG}$ and ${\mathcal{CG}}$. We then show (Theorem \[t:HNN\]) that ${\mathcal{BDG}}$ and ${\mathcal{PG}}$ are closed under taking the HNN-extensions mentioned above. For $n \ge 2$ we construct a faithful representation of the Baumslag-Solitar group ${\mbox{\rm{BS}}}(1,n^2)$ into ${\mbox{\rm{SL}}}(2, \mathbb{Z}[\frac{1}{n}])$ showing that those matrix groups do not have [poly-context-free ]{}word problem. In the last section we prove the rather delicate algebraic fact we alluded to above, namely that the class $\mathcal{D}$ is closed under taking the mentioned HNN extensions. Multipass automata and closure properties ========================================= Deterministic multipass automata -------------------------------- We first consider deterministic multipass automata. Let $\Sigma$ be a finite input alphabet and let $k \ge 1$ be a positive integer. A *deterministic* $k$-*pass automaton* is a tuple $$M = ([k],Q, \Sigma, \Gamma, \sharp, \delta, q_0, H_a,H_r)$$ where $[k] = \{1,2,\ldots,k\}$ is the *pass-counter* and, as usual, $Q$ is a finite set of *states* and $q_0 \in Q$ is the *initial state*. The machine $M$ starts its first pass at the beginning of the input tape in state $q_0$ with an empty stack. The finite set $\Gamma \supseteq \Sigma$ is the *stack alphabet*. We introduce the notation $\Sigma_{\varepsilon} := \Sigma \bigcup \{\varepsilon\}$ and $\Gamma_{\varepsilon} := \Gamma \bigcup \{\varepsilon\}$, where $\varepsilon \in \Sigma^*$ denotes the empty word. The *end-marker* $\sharp$ is a letter not in $\Sigma$, and given any input word $w \in \Sigma^*$, the machine $M$ will process $w\sharp$, the word $w$ followed by $\sharp$. The distinct *halting states* $H_a$ and $H_r$ are not in $Q$. We define the *transition function* $\delta$ by cases. When not reading the end-marker the transition function is a map $$\delta \colon [k] \times Q \times {\Sigma_{\varepsilon}}\times {\Gamma_{\varepsilon}}\to Q \times \Gamma^* \cup \emptyset$$ subject to the restrictions given below. As usual, the interpretation of $$\delta(j,q, \sigma, \gamma) = (q', \zeta)$$ where $j \in [k]$, $q, q' \in Q$, $\sigma \in \Sigma$, $\gamma \in {\Gamma_{\varepsilon}}$, and $\zeta \in \Gamma^*$, is that if the machine is on its $j$-th pass, in state $q$, and reading the letter $\sigma$ on the input tape with $\gamma$ on top of the stack, then the automaton changes state to $q'$, replaces $\gamma$ by the word $\zeta$ and advances the input tape. Note that $\gamma$ may be $\varepsilon$ and the machine may continue working when the stack is empty. Indeed, our machines start each pass with empty stack. The interpretation of $$\delta(j,q, \varepsilon, \gamma) = (q', \zeta)$$ is, similarly, that if the machine is running its $j$-th pass, in state $q \in Q$, and reading a letter $\gamma \in \Gamma$ on top of the stack, then, independent of the input symbol being read, the automaton changes state to $q'$, and replaces $\gamma$ by $\zeta \in \Gamma^*$. Such transitions are called $\varepsilon$-*transitions* and, in this case, the reading head does not advance on the tape. Since we are considering *deterministic* machines, we require that if there is a transition $\delta(j,q, \varepsilon, \gamma)$ then $\delta(j,q,\sigma,\gamma)$ is empty for all $\sigma \in \Sigma$. Thus the machine has no choice between reading a letter and advancing the tape or making an $\varepsilon$ transition. Also, for given $i \in \{1,2, \ldots, k\}$, $q \in Q$ and $\gamma$ on top of the stack, the machine always has either a transition reading the input letter and advancing the tape or an $\varepsilon$-transition. Note also that the machine cannot make an $\varepsilon$-transition when the stack is empty. When the automaton reads the end-marker $\sharp$ on a nonfinal pass $j < k$ the transition function is a map $$\delta \colon \{1,2,\ldots, k-1\} \times Q \times \{\sharp\} \times {\Gamma_{\varepsilon}}\to Q.$$ The interpretation of $\delta(j,q, \sharp, \gamma) = q'$ is that on reading the end-marker $\sharp$ on finishing the $j$-th pass, in state $q \in Q$ with $\gamma \in {\Gamma_{\varepsilon}}$ on top of the stack, the automaton changes state to $q'$ to begin the next pass. As part of the definition of the way the machine works, the reading head is automatically reset to the beginning of the input, the stack is emptied, and the pass-counter is advanced to $j+1$. Note that although the pass-counter is read by the machine, it functions automatically. When reading the end-marker on the last pass the transition function is a map $$\label{e:sharp} \delta \colon \{k\} \times Q \times \{\sharp\} \times {\Gamma_{\varepsilon}}\to \{H_a,H_r\}.$$ On reading the end-marker on the last pass the machine must either halt in state $H_a$ and accept or halt in state $H_r$ and reject. Note that it is only on reading the end-marker on the last pass that the machine can go to either $H_a$ or $H_r$ and acceptance is thus completely determined by the last transition. A deterministic $k$-pass automaton $M$ *accepts* a word $w \in \Sigma^*$ if and only if when $M$ is started in its initial state with an empty stack and with $w\sharp$ written on the input tape, then $M$ halts in the accepting state $H_a$ at the end of its $k$-th pass. We write $M \vdash w$ if $M$ accepts $w$ and denote by $$L(M) := \{w \in \Sigma^*: M \vdash w\}$$ the *language accepted* by $M$. A language which is accepted by a deterministic $k$-pass automaton is called a *deterministic $k$-pass language*. A *deterministic multipass language* is a language accepted by a deterministic $k$-pass automaton for some $k \geq 1$. We denote by $\mathcal{DM}$ the class of all deterministic multipass languages. Nondeterministic multipass automata ----------------------------------- A *nondeterministic* $k$-pass automaton is a tuple $$M = ([k], Q, \Sigma, \Gamma, \sharp, \delta, q_0, H_a, H_n)$$ where the notation is as before but the special state $H_n$ is a *no decision* state. When not reading the end-marker the transition function is a map $$\delta \colon [k] \times Q \times {\Sigma_{\varepsilon}}\times {\Gamma_{\varepsilon}}\to \mathcal{P}_f(Q \times \Gamma^*)$$ where $\mathcal{P}_f(Q \times \Gamma^*)$ denotes the collection of all finite subsets of $Q \times \Gamma^*$. The interpretation of $$\delta(j,q, \sigma, \gamma) = \{(q_1,\zeta_1),(q_2,\zeta_2),\ldots,(q_r,\zeta_r)\}$$ is that if the machine $M$ is on its $j$-th pass in state $q$ and reads the letter $\sigma$ on the input tape with the letter $\gamma \in {\Gamma_{\varepsilon}}$ on top of the stack, then the automaton can choose any of the pairs $(q_i,\zeta_i)$ and go to $q_i$ as its next state, replace $\gamma$ by the word $\zeta_i$ in the top of the stack, and advance the input tape. The interpretation of $$\delta(j,q, \varepsilon, \gamma) = \{(q_1,\zeta_1),(q_2,\zeta_2),\ldots,(q_r,\zeta_r)\}$$ is, similarly, that if the machine is on its $j$-th pass in state $q$ and with $\gamma$ on top of the stack, then, independent of the input symbol being read, the automaton can choose any of the pairs $(q_i,\zeta_i)$ and go to $q_i$ as its next state, replace $\gamma$ by $\zeta_i$, but the reading head does not advance on the tape. There may now be both transitions $\delta(j,q,\sigma,\gamma)$ and $\delta(j,q,\varepsilon,\gamma)$. If there is no transition from a given configuration the machine halts. We again require that the machine cannot make an $\varepsilon$-transition when the stack is empty. On reading the end-marker on a nonfinal pass $j < k$ the transition function is a map $$\delta \colon \{1,2,\ldots,k-1\} \times Q \times \{\sharp\} \times {\Gamma_{\varepsilon}}\to \mathcal{P}(Q).$$ The interpretation of $\delta(j,q,\sharp,\gamma) = Q_j \subseteq Q$ is that when the automaton reads the end-marker on a nonfinal pass $j < k$, in state $q$ with $\gamma$ on top of the stack, the automaton can change state to any $q' \in Q_j$ to begin the next pass. As before, the reading head is automatically reset to the beginning of the input, the stack is emptied, and the pass-counter is advanced to $j+1$. On reading the end-marker on the final pass, the transition is a map $$\delta \colon \{k\} \times Q \times \{\sharp\} \times {\Gamma_{\varepsilon}}\to \{H_a, H_n\}.$$ When the automaton reads the end-marker on the last pass in state $q \in Q$, with $\gamma \in {\Gamma_{\varepsilon}}$ on the top of the stack, the machine must either halt in state $H_a$ and accept or halt in the no-decision state $H_n$. It is only on reading the end-marker on the last pass that the machine can go to either $H_a$ or $H_n$. A nondeterministic $k$-pass automaton $M$ *accepts* a word $w \in \Sigma^*$ if and only if when $M$ is started in its initial state with an empty stack and with $w\sharp$ written on the input tape, some possible computation of $M$ on $w\sharp$ halts in the accepting state $H_a$. As in the deterministic case, we then write $M \vdash w$ if $M$ accepts $w$, and denote by $L(M) := \{w \in \Sigma^*: M \vdash w\}$ the *language accepted* by $M$. A language which is accepted by a nondeterministic $k$-pass automaton is called a *nondeterministic $k$-pass language*. A *nondeterministic multipass language* is a language accepted by a nondeterministic $k$-pass automaton for some $k \geq 1$. We denote by $\mathcal{NM}$ the class of all nondeterministic multipass languages. Of course, $\mathcal{DM} \subseteq \mathcal{NM}$. Properties of deterministic and nondeterministic multipass languages -------------------------------------------------------------------- A basic fact about the class of languages accepted by deterministic pushdown automata is that the class is closed under complementation. The only obstacle to proving this is that at some point the automaton might go into an unbounded sequence of ${\varepsilon}$-transitions. Call a deterministic [pushdown automaton ]{}*complete* if it always reads its entire input. Similarly, we call a multipass automaton *complete* if it always reads the end-marker on every pass. A basic lemma [@HU Lemma 10.3] shows that for any deterministic [pushdown automaton ]{}there is a complete [pushdown automaton ]{}accepting the same language. The proof for deterministic multipass automata is essentially the same but even easier since we do not have to worry about final states. \[l:complete\] For every deterministic multipass automaton there is a complete multipass automaton accepting the same language. Let $M = ([k],Q, \Sigma, \Gamma, \sharp, \delta, q_0, H_a,H_r)$ be a deterministic $k$-pass automaton. We construct a complete $k$-pass automaton $M' = ([k],Q', \Sigma, \Gamma, \sharp, \delta', q_0, H_a,H_r)$ as follows. $Q' := Q \sqcup \{r\}$, where $r$ is a new rejecting state. The machine $M'$ basically works as $M$ so that for the new transition map $\delta'$ we have $$\delta'(j,q,\sigma,\gamma) = \delta(j,q,\sigma,\gamma)$$ and $$\delta'(j,q,\sharp,\gamma) = \delta(j,q,\sharp,\gamma)$$ for all $j \in [k]$, $q \in Q$, $\sigma \in \Sigma$, and $\gamma \in \Gamma$. However, if on some pass $j \in [k]$ and in some state $q \in Q$ with a letter $\gamma$ on top of the stack, $M$ would start an unbounded sequence of ${\varepsilon}$-transitions without erasing that occurrence of $\gamma$ then $M'$ will instead enter the rejecting state $r$ where it remains and then simply reads all input letters for all remaining passes until it reads the end-marker on the last pass and then rejects. For example, if we have the sequence of transitions $$\delta(j,q_i,\varepsilon, \gamma_i) = (q_{i+1}, \zeta_{i+1}),$$ where $q_i \in Q$ ($q_0 = q$), $\gamma_i \in \Gamma$ ($\gamma_0 = \gamma$), $\zeta_{i+1} \in \Gamma^*$, $i \in {\mathbb{N}}$, and there eixts $i_0 > 0$ such that $q_{i_0} = q$ and $\zeta_{i_0} = \zeta'\gamma$ for some $\zeta' \in \Gamma^*$, we would then set $$\begin{split} \delta'(j',q,\varepsilon,\gamma) & := (r,\gamma)\\ \delta'(j',r,\sigma,\gamma) & := (r,\gamma)\\ \delta'(j',r,\sharp,\gamma) & := \begin{cases} r & \mbox{ if } j' < k\\ H_r & \mbox{ if } j' = k \end{cases} \end{split}$$ for all $j'=j,j+1, \ldots, k$, $q \in Q$, $\sigma \in \Sigma$, and $\gamma \in \Gamma \bigcup \{\varepsilon\}$. It is clear that $L(M') = L(M)$. This construction can be made effective but we only need the stated result. \[p:dmp-complement\] The class $\mathcal{DM}$ of deterministic multipass languages is closed under complementation. Let $L \subseteq \Sigma^*$ be a deterministic multipass language and let $M$ be a complete deterministic multipass automaton accepting $L$. The complement $\neg L:=\Sigma^* \setminus L$ of $L$ is accepted by the multipass automaton $M^\neg$ which is the same as $M$ except that it does the opposite of what $M$ does on reading the end-marker on the final pass, that is, it exchanges the accepting and rejecting states. \[p:det-multi\] The class of deterministic context-free languages coincides with the class of deterministic $1$-pass languages and the class of context-free languages coincides with the class of nondeterministic $1$-pass languages. This result is intuitively clear but we need to check some details. Let $L \subseteq \Sigma^*$ be a deterministic context-free language and let $M = (Q,\Sigma,\Gamma, \delta, q_0, F, Z_0)$ be a complete deterministic pushdown automaton accepting $L$ by entering a final state. Note that $M$ has start symbol $Z_0$ on the stack and scans the whole input. Then $L$ is accepted by the deterministic $1$-pass automaton $$M' = ([1], Q', \Sigma, \Gamma, \sharp, \delta', q_0', H_a, H_r)$$ defined as follows. The set of states of $M'$ is $Q':=Q \sqcup \{q_0'\}$. Then, on reading the first letter of the input, $M'$ adds $Z_0$ to the stack followed by whatever $M$ would add to the stack, so $$\delta'(1, q_0', \sigma, \varepsilon) := (q, Z_0\zeta) \mbox{ \ \ if \ \ } \delta(q_0,\sigma, Z_0) = (q,\zeta)$$ for all $\sigma \in \Sigma$, where $q \in Q$ and $\zeta \in \Gamma^*$. Then, $M'$ simulates $M$: $$\delta'(1,q,\sigma,\gamma) = \delta(q,\sigma,\gamma)$$ and, finally, $$\delta'(1,q,\sharp,\gamma) = \begin{cases} H_a & \mbox{ \ \ if \ \ } q \in F\\ H_r & \mbox{ otherwise,} \end{cases}$$ for all $q \in Q$, $\sigma \in \Sigma$, and $\gamma \in \Gamma$. Thus $L = L(M')$ is a deterministic $1$-pass language. The proof is essentially the same for the nondeterministic case. However we have to replace $Q'$ by $Q'':= Q \sqcup \overline{Q} \sqcup \{q_0',r\}$, where $\overline{Q}$ is a disjoint copy of $Q$. Then, on a sequence of ${\varepsilon}$-transitions of $M$, the $1$-pass automaton $M'$ remembers if $M$ ever entered a final state in $F$ during the sequence (this is achieved by movind from state $q$ to its “accepting” copy $\overline{q} \in \overline{Q}$). Then, if $M'$ reads $\sharp$ as its next letter it accepts if $M$ did enter a final state or rejects if not. Also, if $M$ emptied its stack without entering a final state on its current input, then $M'$ goes to a rejecting state $r \in Q'$ in which it remains and simply reads the remaining input until it encounters $\sharp$ and then rejects. Conversely, let $L$ be a deterministic $1$-pass language and let $$M = ([1], Q, \Sigma, \Gamma, \sharp, \delta, q_0, H_a, H_r)$$ be a deterministic $1$-pass automaton accepting $L$ which, by Lemma \[l:complete\], we may suppose being complete. We now construct a (complete) deterministic [pushdown automaton ]{} $M' = (Q', \Sigma, \Gamma', \delta', q_0,F,Z_0)$ which accepts $L$ as follows. $Q' := \{q_0'\} \sqcup (Q \times \Gamma_{\varepsilon})$ and $\Gamma' := \Gamma \sqcup \{Z_0\}$. $M'$ starts with a new beginning symbol $Z_0 \notin \Gamma$, which it will never erase and which it treats it the same way as $M$ treats an empty stack, and then simulates $M$: thus $$\delta'(q_0, \sigma, Z_0) := ((q',\gamma'), \zeta \gamma') \mbox{ \ \ if \ \ } \delta(1,q_0, \sigma, \varepsilon) = (q',\zeta \gamma')$$ and $$\delta'((q, \gamma''), \sigma, \gamma) := ((q', \gamma'''), \zeta \gamma''') \mbox{ \ \ if \ \ } \delta(1,q, \sigma, \gamma) = (q', \zeta \gamma'''),$$ where $q'\in Q$, $\gamma', \gamma''' \in \Gamma_{\varepsilon}$, and $\zeta \in \Gamma^*$, for all $q \in Q$, $\sigma \in \Sigma \cup \{\varepsilon\}$ and $\gamma, \gamma'' \in \Gamma$. The set $F$ of final states of $M'$ is defined by $$F := \{(q,\gamma) \in Q': \delta(1,q,\sharp,\gamma) = H_a\}.$$ Then, it is then clear that $M'$ accepts $L$ by final state, so that $L = L(M')$. Again, the proof is essentially the same for the nondeterministic case. $M'$ will now have two copies of the state set of $M$, one accepting and one rejecting and simulate $M$. However, on ${\varepsilon}$-transitions of $M$, the pushdown automaton $M'$ always goes to a non-accepting copy of the corresponding state. On a transition advancing the tape, if $M$ would accept when reading $\sharp$ as the next letter, $M'$ goes to the accepting copy of the state of $M'$ and otherwise to the non-accepting copy. \[p:union-int\] Both the classes of deterministic and nondeterministic multipass languages are closed under union and intersection. Let $L_i \subseteq \Sigma^*$ be accepted by a deterministic $k_i$-pass automaton $M_i$ for $i = 1,2$. We suppose that $M_1$ and $M_2$ have disjoint state sets $Q_1$ and $Q_2$, respectively, and set $k := k_1 + k_2$. We construct a $k$-pass automaton $M$ accepting the union $L_1 \cup L_2$ as follows. $M$ will have state set $Q_1 \cup Q_2 \cup \{a\}$, where $a$ is a special accepting state. On the first $k_1$ passes $M$ simulates $M_1$. On reading the end-marker $\sharp$ at the end of pass $k_1$, if $M_1$ would accept then $M$ goes to the accepting state $a$ in which it remains while reading the input for the remaining $k_2$ passes, and then accepts on the $k$-th pass. If $M_1$ would reject then $M$ goes to the first initial state of $M_2$ and then simulates $M_2$ on the next $k_2$ passes. On reading the end-marker $\sharp$ at the end of the $k$th pass, if $M_2$ would accept then $M$ accepts, otherwise $M$ rejects. This shows that the language accepted by $M$ is $L_1 \cup L_2$. We can similarly construct a $k$-pass automaton $M$ accepting the intersection $L_1 \cap L_2$ as follows. The argument is essentially the same as before, but now $M$ has state set $Q_1 \cup Q_2 \cup \{r\}$ where $r$ is a rejecting state. At the end of pass $k_1$, if $M_1$ would reject, then $M$ goes to the rejecting state $r$ in which it remains while reading the input for the remaining $k_2$ passes, and then rejects on the final pass. On the other hand, if $M_1$ accepts then, for the remaining $k_2$ passes, $M$ just simulates $M_2$ and, in particular, does whatever $M_2$ at the end of the final pass. It is then clear that the language accepted by $M$ is $L_1 \cap L_2$. We now consider the nondeterministic case. So, let $L_i \subseteq \Sigma^*$ be accepted by nondeterministic $k_i$-pass automaton $M_i$ for $i = 1,2$. We set $k:=k_1+k_2$. Then the $k$-pass nondeterministic automaton $M$ accepting $L_1 \cup L_2$ works as follows. $M$ simulates $M_1$ on the first $k_1$ passes and if the computation of $M_1$ would accept, then, as before, $M$ goes to an accepting state in which it remains while reading the input for the remaining $k_2$ passes, and then accepts on the $k$-th pass. If not, then $M$ goes to the first initial state of $M_2$ and then simulates $M_2$ on the next $k_2$ passes. Similarly, for the intersection, $M$ accepts exactly if computations of $M_1$ and $M_2$ both accept. From the previous results we immediately deduce the following. \[c:boolean\] The class $\mathcal{DM}$ of languages accepted by deterministic multipass automata contains the Boolean closure of the class of deterministic context-free languages. The class $\mathcal{NM}$ of languages accepted by nondeterministic multipass automata contains the closure of context-free languages under union and intersection. We now turn to proving the converse of Corollary \[c:boolean\]. \[t:char-multip\] The class $\mathcal {DM}$ of deterministic multipass languages is the Boolean closure of the class of deterministic context-free languages and the class $\mathcal{NM}$ of nondeterministic multipass languages is the class $\mathcal{PCF}$ of poly-context-free languages. Let $M = ([k],Q, \Sigma, \Gamma, \sharp, \delta, q_0, H_a,H_r)$ be a $k$-pass automaton, which we assume to be complete if $M$ is deterministic. If $w \in \Sigma^*$ and $M \vdash w$, define an $M$-*accepting profile* of $w$ as an ordered sequence of $k$ triples: $$p(w) = \left((q_{1,0},\gamma_{1},q_{1,1}), (q_{2,0},\gamma_{2},q_{2,1}), \ldots,(q_{k,0},\gamma_k,q_{k,1})\right)$$ where $q_{i,0} \in Q$ is the initial state beginning the $i$-th pass and $q_{i,1} \in Q$ (respectively $\gamma_i \in {\Gamma_{\varepsilon}}$) is the control state (respectively the top letter of the stack) on reading the end-marker on the $i$-th pass in an accepting computation of $M$ on $w$ (so that $\delta(k,q_{k,1},\sharp,\gamma_k) = H_a$). Note that if $M$ is deterministic there is of course only one such profile. Let $P(M) = \{p(w): w \in L(M)\}$ denote the set of all such $M$-accepting profiles, and note that $P(M)$ is finite since $P(M) \subseteq (Q \times {\Gamma_{\varepsilon}}\times Q)^k$. Let $P(M) = \{p_1,p_2, \ldots, p_t\}$ and suppose that $$p_i = \left((q_{1,0}^i, \gamma_{1}^i, q_{1,1}^i), (q_{2,0}^i, \gamma_{2}^i, q_{2,1}^i), \ldots, (q_{k,0}^i, \gamma_{k}^i, q_{k,1}^i)\right)$$ for $i = 1,2,\ldots,t$. Let $L_{i,j}$ be the language consisting of all words $w \in \Sigma^*$ accepted by the $1$-pass automaton $M_{i,j}$ which, starting in state $q_{j,0}^i$, simulates $M$ on reading $w\sharp$ on the $j$-th pass, and then accepts exactly when reading the end-marker in state $q_{j,1}^i$ with $\gamma_{j}^i$ on top of the stack. Note that for $w \in \Sigma^*$ one has $p(w) = p_i$ if and only if $w \in \bigcap_{j=1}^k L_{i,j}$. As a consequence, $$\label{eq:profiles} L(M) = \bigcup_{i = 1}^{t} \bigcap_{j=1}^k L_{i,j}.$$ Since De Morgan’s laws allow us to move negations inside unions and intersections in a Boolean expression, the Boolean closure of the class of deterministic context-free languages is the same as its closure under union and intersection. Since the class of arbitrary context-free languages is closed under unions, we can distribute unions past intersections and the closure of the class of context-free languages under unions and intersections is the same as its closure under intersections. This completes the proof of the theorem. Recall [@Harrison] that a gsm, a *generalized sequential machine*, $$S = (Q, \Sigma, \Delta, \lambda, q_0)$$ is a deterministic finite automaton with output. On reading a letter $\sigma \in \Sigma$ in state $q$, the machine outputs a word $\lambda(q, \sigma) \in \Delta^*$. Since $S$ is deterministic it defines a map $g \colon \Sigma^* \to \Delta^*$. The convention is that $g(\varepsilon) = \varepsilon$. For example, a homomorphism $\phi: \Sigma^* \to \Delta^*$ can be defined by a one-state gsm: on reading a letter $\sigma$ the gsm outputs $\phi(\sigma)$. It is well-known that both deterministic and nondeterministic context-free languages are closed under inverse gsm mappings. (If $L$ is context-free, incorporate $S$ into a pushdown automaton $M$ accepting $L$, and when $S$ would output $u$ simulate $M$ on reading $u$.) Since inverse functions commute with Boolean operations we have: \[p:inverse\] The classes ${\mathcal{DM}}$ and ${\mathcal{PCF}}$ are both closed under inverse gsm mappings. In particular, ${\mathcal{DM}}$ and ${\mathcal{PCF}}$ are both closed under inverse monoid homomorphisms. Let $\Sigma_i, i =1,2, \ldots,r$ be finite alphabets and let $L_i \subseteq \Sigma_i^*$, $i=1,2, \ldots,r$. Let $\Sigma = \bigcup_{i = 1}^{r}\Sigma_i$ and denote by $\pi_i \colon \Sigma^* \to \Sigma_i^*$ the monoid homomorphism defined by setting $$\pi_i(a) = \begin{cases} a & \mbox{ if } a \in \Sigma_i\\ \varepsilon & \mbox{ otherwise.} \end{cases}$$ We call the language $$L = \{w \in \Sigma^*: \pi_i(w) \in L_i, i=1,2, \ldots,r\}$$ the *interleaved product* of the languages $L_i$. Note that in the definition above there is no hypothesis on how the $\Sigma_i$ overlap. If the alphabets are all disjoint then $L$ is the *shuffle product* of the $L_i$. On the other hand, if the alphabets are all the same then $L$ is the intersection of the $L_i$. There does not seem to be a standard name if the overlap of the alphabets is arbitrary. \[p:interleaved\] The classes ${\mathcal{BDC}}$ and ${\mathcal{PCF}}$ are closed under interleaved product. With the notation above, the interleaved product of the languages $L_i$ is $$\bigcap_{i =1}^{r} {\pi_i}^{-1}(L_i)$$ The statement then follows from Proposition \[p:inverse\] and Proposition \[p:union-int\]. Recall [@Harrison] that if $K$ and $L$ are subsets of $\Sigma^*$ then the *left quotient* of $L$ by $K$ is the language $$K^{-1} L = \{w \in \Sigma^*: \exists u \in K \mbox{ such that } uw \in L\}.$$ \[p:leftquotient\] The classes ${\mathcal{BDC}}$ and ${\mathcal{PCF}}$ are closed under left quotients by finite sets. Let $L \subseteq \Sigma^*$ be in ${\mathcal{BDC}}$ (respectively ${\mathcal{PCF}}$) and let $K$ be a finite subset of $\Sigma^*$. Since $K^{-1} L = \bigcup_{u \in K}\{u\}^{-1}L$ and the classes ${\mathcal{BDC}}$ and ${\mathcal{PCF}}$ are closed under finite unions (cf. Proposition \[p:union-int\]), we may suppose that $K = \{u\}$. We then define a two-state generalized sequential machine $S$ defining a mapping $g \colon \Sigma^* \to \Sigma^*$ as follows. On reading a letter $\sigma$ in its initial state, $S$ outputs $u\sigma$ and then goes to its second state, in which it will stay. In its second state $S$ simply outputs $\sigma$ on reading $\sigma$. Since $u \in L \Leftrightarrow \varepsilon \in \{u\}^{-1}L$ and $\varepsilon \in L \Leftrightarrow \varepsilon \in g^{-1}(L)$ (recall that $g(\varepsilon) = \varepsilon$), it is then clear that $$\{u\}^{-1}L = \begin{cases} g^{-1}(L) \cup \{\varepsilon\} & \mbox{ if } u \in L \mbox{ and } \varepsilon \notin L\\ g^{-1}(L) \setminus \{\varepsilon\} & \mbox{ if } u \notin L \mbox{ and } \varepsilon \in L\\ g^{-1}(L) & \mbox{ otherwise.} \end{cases}$$ Now, since $\{\varepsilon\}$ and its complement are regular languages in $\Sigma^*$ and the classes ${\mathcal{BDC}}$ and ${\mathcal{PCF}}$ are closed under finite unions/intersections (cf. Proposition \[p:union-int\]), the result then follows from Proposition \[p:inverse\]. Group word problems =================== Let $G = \langle X;R\rangle$ be a finitely generated group presentation. We denote by $\Sigma:= X \bigcup X^{-1}$ the associated *group alphabet*. Then the *Word Problem* of $G$ (cf. [@Anisimov; @CCFS]), relative to the given presentation, is the language $${\mbox{\rm{WP}}}(G:X;R):= \{w \in \Sigma^*: w = 1 \mbox{ in } G\} \subseteq \Sigma^*$$ where for $w,w'\in \Sigma^*$ we write “$w = w'$ in $G$” provided $\pi(w) = \pi(w')$, where $\pi \colon \Sigma^* \to G$ denotes the canonical monoid epimorphism. \[t:multipass-w-p\] Let $G = \langle X;R \rangle$ be a finitely generated group. Then whether or not the associated word problem ${\mbox{\rm{WP}}}(G:X;R)$ is in ${\mathcal{BDC}}$ (respectively ${\mathcal{PCF}}$ is independent of the given presentation. Moreover, if the word problem of $G$ is in ${\mathcal{BDC}}$ (respectively ${\mathcal{PCF}}$), then every finitely generated subgroup of $G$ also has word problem in ${\mathcal{BDC}}$ (respectively ${\mathcal{PCF}}$). Suppose that the word problem ${\mbox{\rm{WP}}}(G:X;R)$ is a deterministic (respectively nondeterministic) multipass language. Let $H = \langle Y;S \rangle$ be a finitely generated group and suppose that there is an injective group homomorphism $\phi \colon H \to G$. Let $\Sigma = X \bigcup X^{-1}$ and $Z= Y \bigcup Y^{-1}$ denote the corresponding group alphabets. For each $y \in Y$ let $w(y) \in \Sigma^*$ be a word representing the group element $\phi(y) \in G$ and consider the unique monoid homomorphism $\psi \colon Z^* \to \Sigma^*$ satisfying $\psi(y) = w(y)$ and $\psi(y^{-1}) = w(y)^{-1}$ for all $y \in Y$. Let $w \in Z^*$. Then $w \in {\mbox{\rm{WP}}}(H:Y;S)$ if and only if $\psi(w) \in {\mbox{\rm{WP}}}(G:X;R)$, that is, ${\mbox{\rm{WP}}}(H:Y;S) = \psi^{-1}({\mbox{\rm{WP}}}(G:X;R))$. From Proposition \[p:inverse\] we then deduce that ${\mbox{\rm{WP}}}(H:Y;S)$ is deterministic (respectively nondeterministic) multipass. This proves the second part of the statement. Taking $H$ isomorphic to $G$ gives the first part of the statement. We denote by ${\mathcal{BDG}}$ the class of all finitely generated groups having a deterministic multipass word problem and by ${\mathcal{PG}}$ the class of all finitely generated groups having a nondeterministic multipass, equivalently poly-context-free, word problem. Since the word problem of the direct product of two groups given by presentations on disjoint sets of generators is the shuffle product of the word problems of the two groups, from Proposition \[p:interleaved\] we immediately deduce: \[c:product\] The classes ${\mathcal{BDG}}$ and ${\mathcal{PG}}$ are closed under finite direct products. That is, if two groups $G_1$ and $G_2$ are in ${\mathcal{BDG}}$ (respectively ${\mathcal{PG}}$) then $G_1 \times G_2$ is in ${\mathcal{BDG}}$ (respectively ${\mathcal{PG}}$). Stallings’ example [@Stallings] of a finitely generated subgroup of $F_2 \times F_2$ which is not finitely presented is the kernel of the homomorphism $$F_2 \times F_2 \to {\mathbb{Z}}$$ defined by mapping every free generator of each factor of $F_2 \times F_2$ to a fixed generator of the infinite cyclic group ${\mathbb{Z}}$. Thus ${\mathcal{BDG}}$ and ${\mathcal{PG}}$ contain groups which are *not* finitely presentable. \[c:abelian\] All finitely generated abelian groups are in ${\mathcal{BDG}}$. \[t:finite-quotients\] The classes ${\mathcal{BDG}}$ and ${\mathcal{PG}}$ are closed under quotients by *finite* normal subgroups. Let $G = \langle X; R \rangle$ be in ${\mathcal{BDG}}$ (respectively ${\mathcal{PG}}$) and let $N = \{1 = n_1, n_2, \ldots, n_r\}$ be a finite normal subgroup of $G$. Then the quotient group $H = G/N$ admits the presentation $$\label{e:pres-H} H = \langle X; R \cup N \rangle.$$ Let $\Sigma = X \bigcup X^{-1}$ denote the group alphabet for both $G$ and $H$ and let $W$ be the word problem of $G$. If $w \in \Sigma^*$ then $w = 1$ in $H$ if and only if $w \in N$ in $G$. Let $K$ be the finite set of *inverses* of elements of $N$. Then $w \in N$ if and only if $w \in K^{-1}W \cup W$, and the result follows from Proposition \[p:leftquotient\] and Proposition \[p:union-int\]. It is known that both the class of finitely generated groups with context-free co-word problem [@HRRT] and the class of groups with [poly-context-free ]{}word problem [@Brough] are closed under finite extensions. \[t:finite-extension\] The classes ${\mathcal{BDG}}$ and ${\mathcal{PG}}$ are closed under finite extensions. Let $G = \langle X;R \rangle$ be a finitely generated group with a subgroup $H = \langle Y; S \rangle$ of finite index which is in ${\mathcal{BDG}}$ (respectively ${\mathcal{PG}}$). Note that $H$ must be finitely generated. Let $\{1 = c_1,c_2,\ldots,c_k \}$ be a set of representatives of the left cosets of $H$ in $G$. For each coset representative $c_i$ and each $x \in X^{\pm 1}$ there exist a unique representative $c_j$, $j = j(i,x)$, and a word $u_{i,x}$ in the generators $Y$ of $H$ such that $c_i x = c_j u_{i,x}$. Let $M$ be a deterministic (respectively nondeterministic) multipass automaton accepting the word problem ${\mbox{\rm{WP}}}(H:Y;S)$ of $H$. Define a multipass automaton $\widehat{M}$ accepting the word problem ${\mbox{\rm{WP}}}(G:X;R)$ of $G$ as follows. The state set of ${\widehat{M}\ }$ will keep track of the coset representative of the word read so far. If the coset is $c_i$ and ${\widehat{M}\ }$ reads a letter $x \in X^{\pm 1}$, then ${\widehat{M}\ }$ changes the current coset to $c_j$, $j = j(i,x)$, and simulates $M$ on reading $u_{i,x}$. At the end of its final pass, ${\widehat{M}\ }$ accepts exactly if the current coset is $c_1$ and $M$ would accept. We can immediately use the above theorem to prove the following: \[t:smidirect\] Let $G_1$ and $G_2$ be two finitely generated groups in ${\mathcal{BDG}}$ (respectively ${\mathcal{PG}}$). Suppose that $G_2$ acts on $G_1$ by a *finite* group of automorphisms. Then the corresponding semi-direct product $G_1 \rtimes G_2$ is also in ${\mathcal{BDG}}$ (respectively ${\mathcal{PG}}$). Since $G_2$ acts on $G_1$ by a finite group of automorphisms, the subgroup $H \leq G_2$ which fixes all the elements of $G_1$ has finite index in $G_2$ and therefore is in ${\mathcal{BDG}}$ (respectively ${\mathcal{PG}}$) by Theorem \[t:multipass-w-p\]. As a consequence, the subgroup $G_1 \rtimes H$ is just $G_1 \times H$ and has finite index in $G_1 \rtimes G_2$ and the statement follows from Corollary \[c:product\] and Theorem \[t:finite-extension\]. If $G = \langle X;R\rangle$ is a finitely generated group and $\varphi$ is an automorphism of $G$ then the *mapping torus* of $\varphi$ is the HNN-extension $$\langle G,t; t x t^{-1} = \varphi(x), x \in X \rangle.$$ \[c:mapping-torus\] Let $G$ be a group in ${\mathcal{BDG}}$ (respectively ${\mathcal{PG}}$) and suppose that $\phi$ is an automorphism of $G$ of *finite order*. Then the *mapping torus* of $\phi$ is also in ${\mathcal{BDG}}$ (respectively ${\mathcal{PG}}$). HNN-extensions and doubles ========================== We have seen that the classes ${\mathcal{BDG}}$ and ${\mathcal{PG}}$ are closed under taking mapping tori of automorphisms of finite order. In this section we prove a much more general result. \[t:HNN\] Let $G = \langle X; R \rangle$ in ${\mathcal{BDG}}$ (respectively ${\mathcal{PG}}$) and $S$ a subgroup of $G$ of *finite index*. Suppose also that $\varphi \colon G \to G$ is an automorphism of finite order such that $\varphi(S) = S$. Then the HNN-extension $$H = \langle G,t; tst^{-1} = \varphi(s), s \in S\rangle$$ is again in ${\mathcal{BDG}}$ (respectively ${\mathcal{PG}}$). First recall that a subgroup $S$ of finite index contains a characteristic subgroup $C$ of finite index. Let $G/C =: K = \{1_K = k_1, k_2,\ldots,k_r\}$ be the corresponding finite quotient group and let $\psi \colon G \to K$ be the associated epimorphism. Since $C$ is characteristic, $\varphi$ induces an automorphism $\overline{\varphi} \colon K \to K$. Finally, let $J:=\psi(S) \leq K$ denote the image of $S$ in $K$. By Britton’s Lemma, if a word $w = 1$ in $H$, then all occurrences of the stable letter $t$ must cancel by successive $t$-reductions. Now for $u \in G$ we have $t^{\epsilon} u t^{-\epsilon} = \varphi^{\epsilon}(u)$ (where $\epsilon = \pm 1$) if and only if $u \in S$, or equivalently, if and only if $\psi(u) \in J$. Let $\Sigma(X):= X \bigcup X^{-1}$ be the group alphabet for the base group $G$ and let $M$ be a multipass automaton accepting ${\mbox{\rm{WP}}}(G:X;R) \subseteq \Sigma(X)^*$, say with stack alphabet $\Delta$. We now describe a new multipass automaton $\widehat M$ with input alphabet $\Sigma:= \Sigma(X) \bigcup \{t,t^{-1}\}$ and stack alphabet $\Delta \bigcup (\{t, t^{-1}\} \times K)$, which will accept the word problem of $H$. An input word has the form $w = u t^{\epsilon_1} u_1t^{\epsilon_2} \cdots u_{n-1} t^{\epsilon_{n}}u_{n}$, where $u,u_i \in \Sigma(X)^*$ and $\epsilon_i = \pm 1$ for $i=1,2,\ldots,n$. The first pass of $\widehat{M}$ is to decide whether or not all the $t$’s cancel. Let $p \geq 1$ denote the order of the automorphism $\varphi$. As $\widehat{M}$ proceeds from left to right, it will count the algebraic sums $m_i:= \epsilon_1 + \epsilon_2 + \cdots + \epsilon_i$ mod $p$ for $i=1,2,\ldots,n$. To begin, $\widehat{M}$ ignores $u$, that is, any letters from the base group until the first $t^{\epsilon_1}$, and then adds $(t^{\epsilon_1}, 1_K)$ to the stack and counts $\epsilon_1$ mod $p$. Next, $\widehat{M}$ keeps track of the element $k:= \psi(\varphi^{\epsilon_1}(u_1)) = \overline{\varphi}^{\epsilon_1}(\psi(u_1)) \in K$. There are now two cases. If $k \notin J$ or if $\epsilon_2 = \epsilon_1$, then $\widehat{M}$ replaces $(t^{\epsilon_1}, 1_K)$ with $(t^{\epsilon_1}, k) (t^{\epsilon_2}, 1_K)$ on the stack. If $k \in J$ and $\epsilon_2 = - \epsilon_1$ then $\widehat{M}$ erases $(t^{\epsilon_1}, 1_K)$ from the stack which then remains empty. In either case, $\widehat{M}$ now counts $m_2=\epsilon_1 + \epsilon_2$ mod $p$. Suppose that the machine has just finished processing the $i$-th $t$-symbol, so that the count is $m_i = (\epsilon_1 + \epsilon_2 + \cdots + \epsilon_i)$ mod $p$, and the stack is either empty or the top symbol on the stack is $(t^{\epsilon}, k)$. If the stack is empty, then as before, $\widehat M$ reads the base letters until the next $t^{\eta}$-symbol is encountered and then puts $(t^{\eta}, 1_K)$ on the stack. If there is $(t^{\epsilon}, k)$ on top of the stack, the machine keeps track of the the image $k':= \psi(\varphi^{m_i}(u))= \overline{\varphi}^{m_i}(\psi(u)) \in K$ of the product $u$ of generators in the base from that point until the next occurrence, say $t^\eta$, of a $t$-symbol. If $\eta = -\epsilon$ and $kk' \in J$, then $\widehat M$ does the following. It removes $(t^{\epsilon}, k)$ on top of the stack and, if the stack is empty then $\widehat M$ continues processing the input; if there is now a $(t^\beta, k'')$ on the top of the stack, then $\widehat M$ replaces that symbol by $(t^\beta, k''kk')$. On the other hand, if $\eta = \epsilon$ or $kk' \notin J$ then $\widehat M$ erases $(t^{\epsilon}, k)$ on top of the stack and replaces it by $(t^{\epsilon}, kk')(t^{\eta}, 1_K)$. On reading the end-marker, all the $t$’s have cancelled if and only if the stack is empty. If the stack is nonempty $\widehat{M}$ goes to a rejecting state that will reject on reading the end-marker on the final pass. If the stack is empty, since $t^\epsilon s t^{-\epsilon} = \varphi^\epsilon(s)$ in $G$ for all $s \in S$ and $\epsilon = \pm 1$, then after cancelling all $t$-symbols, we have $w = u \varphi^{m_1}(u_1)\varphi^{m_2}(u_2)\cdots \varphi^{m_n}(u_n)$ in $G$. Then $\widehat{M}$ ignores $t$-symbols and just simulates $M$ to check that the word $u\varphi^{m_1}(u_1)\varphi^{m_2}(u_2)\cdots \varphi^{m_n}(u_n)$ is equal to the identity in $G$. \[d:double-twisted\] Let $G$ (respectively $\overline{G}$) be a group and let $S \leq G$ be a subgroup. Let also $\varphi \colon G \to G$ be an automorphism of finite order such that $\varphi(S) = S$. Suppose there exists an isomorphism $g \mapsto \overline{g}$ of $G$ onto $\overline{G}$ and denote by $\overline{S} \leq \overline{G}$ the image of the subgroup $S$. Consider the free product with amalgamation $$D := \langle G * \overline{G}; \varphi(s) = \overline{s}, s \in S\rangle.$$ If $\varphi$ is the identity of $G$, then $D$ is called a *double* of $G$ over the subgroup $S$. In general, if $\varphi$ is not the identity, then $D$ is called a *double* of $G$ over the subgroup $S$ *twisted* by the automorphism $\varphi$. Note that for a double we require that the amalgamation is induced by the global isomorphism between $G$ and $\overline{G}$. \[t:double\] Let $G$ be a group in ${\mathcal{BDG}}$ (respectively ${\mathcal{PG}}$) and suppose that $S$ is a subgroup of finite index in $G$. Let also $\varphi \colon G \to G$ be an automorphism of finite order such that $\varphi(S) = S$. Then $D := \langle G * \overline{G}; \varphi(s) = \overline{s}, s \in S\rangle$ is again in ${\mathcal{BDG}}$ (respectively ${\mathcal{PG}}$). Note that $D$ embeds in the HNN-extension $$H = \langle G,t; t s t^{-1} = \varphi(s), s \in S\rangle$$ via the map defined by $g \mapsto g$ and $\overline{g} \mapsto tg^{-1}t^{-1}$ for all $g \in G$ and $\overline{g} \in \overline{G}$. This is a general fact that does not require any particular hypothesis on $G$ or $S$. But if $G$ is in ${\mathcal{BDG}}$ (respectively ${\mathcal{PG}}$) and $S$ is a subgroup of finite index, then $H$ is in ${\mathcal{BDG}}$ (respectively ${\mathcal{PG}}$) by Theorem \[t:HNN\]. Since $D$ is isomorphic to a finitely generated subgroup of $H$, we deduce from Theorem \[t:multipass-w-p\] that $D$ is also in ${\mathcal{BDG}}$ (respectively ${\mathcal{PG}}$). Parikh’s theorem and some examples of groups not in ${\mathcal{PG}}$ ==================================================================== We start by recalling some known preliminary facts. Let $\Sigma = \{a_1,a_2,\ldots,a_r\}$. Then $\Sigma^*$ is the *free monoid of rank $r$*. The free *commutative* monoid of rank $r$ is $\mathbb{N}^r$ with vector addition. The natural *abelianization map* $\pi \colon \Sigma^* \to \mathbb{N}^r$ is defined by $\pi(a_i) = \boldsymbol{e}_i$, where the vector $\boldsymbol{e}_i \in \mathbb{N}^r$ has $1$ in the $i$-th coordinate and $0$ elsewhere, for all $i=1,2,\ldots,r$. Note that if $w \in \Sigma^*$ then the $i$-th coordinate of $\pi(w)$ equals the number of occurrences of the letter $a_i$ in $w$. In formal language theory the abelianization map is called the *Parikh mapping* because of a remarkable theorem of Parikh [@Parikh; @P2; @Harrison] which we now review. \[d:linear\] A *linear* subset of $\mathbb{N}^r$ is a set of the form $$\label{e:linear} S = S(\boldsymbol{v}_0;\boldsymbol{v}_1, \boldsymbol{v}_2, \ldots, \boldsymbol{v}_m) := \{\boldsymbol{v}_0 + n_1 \boldsymbol{v}_1 + \dots + n_m \boldsymbol{v}_m: n_1,n_2 \ldots, n_m \in \mathbb{N}\}$$ where $m \geq 0$ and $\boldsymbol{v}_i \in \mathbb{N}^r$ for all $i=0,1,\ldots,m$. A *semi-linear* subset of $\mathbb{N}^r$ is a finite union of linear subsets of $\mathbb{N}^r$. Let $L \subseteq \Sigma^*$ be a context-free language. Then $\pi(L) \subseteq \mathbb{N}^r$ is semi-linear and there is a regular language $R \subseteq \Sigma^*$ such that $\pi(R) = \pi(L)$. Thus abelianizations cannot distinguish between regular languages and context-free languages! It is this property that one can use to show that certain groups are not in ${\mathcal{PG}}$. A clear proof of Parikh’s theorem is found in [@P2]. It is easy to show that any semi-linear subset of $\mathbb{N}^r$ is the image of a regular language under the abelization map $\pi$. Indeed, in the notation of Definition \[d:linear\], let $\boldsymbol{v}_i = (v_{i,1} v_{i,2},\ldots,v_{i,r})$ for $i=0,1,\ldots,m$. Consider the regular language $$V:= (a_{1}^{v_{0,1}} a_{2}^{v_{0,2}} \cdots a_{r}^{v_{0,r}}) (a_{1}^{v_{1,1}}a_{2}^{v_{1,2}} \cdots a_{r}^{v_{1,r}})^* \cdots (a_{1}^{v_{m,1}} a_{2}^{v_{m,2}} \cdots a_{r}^{v_{m,r}})^*.$$ Looking back to , it is clear that $\pi(V) = S$. Since the class of regular languages is closed under union, any semi-linear set is the image of a regular language under $\pi$. It is well-known [@Harrison] that a Boolean combination of semi-linear sets is again semi-linear. Indeed, the semi-linear sets are exactly the sets definable in Presburger arithmetic [@GS]. Let $L = \bigcap_{j=1}^q L_j$ be a poly-context-free language. Now *suppose* that the restriction to $\bigcup_{j=1}^q L_j$ of the Parikh map $\pi$ is injective. Then $$\pi(L) = \pi(\bigcap_{j=1}^q L_j) = \bigcap_{j=1}^q \pi(L_j)$$ and thus $\pi(L)$ is semi-linear. This gives a general strategy for showing that some word problems cannot be poly-context-free. Brough [@Brough] proved the general result that none of the Baumslag-Solitar groups $$BS(1,n) = \langle b,t; t b^m t^{-1} = b^n \rangle$$ where $ 1 \le |m| < |n|$ are in ${\mathcal{PG}}$. We illustrate the techinique outlined above by giving a simple proof of the following: None of the Baumslag-Solitar groups $${\mbox{\rm{BS}}}(1,n) = \langle b,t; t b t^{-1} = b^n \rangle , n \ge 2$$ have poly-context-free word problem. Fix $n \ge 2$. Let $\Sigma = \{b,b^{-1},t, t^{-1}\} $ be the associated group alphabet and let $W \subseteq \Sigma^*$ be the word problem for the Baumslag-Solitar group ${\mbox{\rm{BS}}}(1,n)$. We write $t^+$ (respectively $t^-$, respectively $b^-$) to denote the set of all words of the form $t^s$ (respectively $t^{-s}$, respectively $b^{-s}$), where $s \in {\mathbb{N}}$. We then consider the regular language $$R:= t^+bt^-b^- \subseteq \Sigma^*.$$ Observe that $W':=W \cap R$ is exactly $\{t^s b t^{-s} b^{-n^s} : s \ge 1 \}$. Suppose by contradiction that the word problem $W$ for ${\mbox{\rm{BS}}}(1,n)$ is poly-context-free. Then $W = \bigcap_{j=1}^q L_j$, where $L_j \subseteq \Sigma^*$ is context-free for $j=1,2,\ldots,q$. Set $L_j' := L_j \cap R$ and observe that each $L_j'$ is context-free. We thus have $$W'= \bigcap_{j=1}^q L_j' \subseteq R$$ Now consider the Parikh map $\pi \colon \Sigma^* \to {\mathbb{N}}^4$ and observe that $\pi\vert_R$ is injective. Since $W'$ is poly-context-free, it follows that $\pi(W') \subseteq {\mathbb{N}}^4$ is semi-linear and there is a regular language $U \subseteq \Sigma^*$ such that $$\label{e:U-parikh} \pi(U) = \pi(W') = \{(1,n^s,s,s): s \geq 1\} \subseteq {\mathbb{N}}^4.$$ One has to be careful since letters in words in $U$ may occur in a very different order than in words of $W'$. (For example, the words $t^s b t^{-s} b^{-n^s} \in W'$ and $b^{-n^s} (t t^{-1})^s b \in W \setminus W'$ have the same image under the Parikh map.) Let $C$ be the pumping lemma constant for $U$. Chose $s$ so that $n^s > n^{s-1} + C$. Then we can find a word $u \in U$ such that $\pi(u) = \pi(t^s b t^{-s} b^{-n^s}) = (1,n^s,s,s) \in {\mathbb{N}}^4$. Then, by the pumping lemma, $u$ has a decomposition $u = xyz$ where $|xy| \le C, |y| \ge 1$ and (by pumping down) $u' := xz \in U$. We claim that no word obtained by deleting at least one and no more than $C$ letters from $u$ can have the same Parikh vector as any word in $W'$. First, we cannot delete $b$ since all words in $W'$ contain exactly one occurrence of $b$. Suppose we delete $0 < j \leq s$ occurrences of $t$ or $t^{-1}$. Then we must delete the same number of each to have a word with the same $\pi$-image as a word in $W'$. Let us denote by $0 \leq \ell \leq C$ the number of occurrences of $b^{-1}$ that have also been deleted, so that $\pi(u') = (1,n^s - \ell, s-j,s-j)$. Since $\pi(u') \in \pi(U)$, gives us $n^s - \ell = n^{s-j} \leq n^{s-1} < n^s - C$, so that $\ell > C$, a contradiction. Finally, if we only cancel some occurrences of $b^{-1}$, then again yields a contradiction. This shows that $W$ is not [poly-context-free ]{}. There is a faithful representation of the Baumslag-Solitar group ${\mbox{\rm{BS}}}(1,n)$, $n \geq 2$, into ${\mbox{\rm{GL}}}(2,{\mathbb{Z}}[\frac{1}{n}])$ which has been studied in connection with diffeomorphisms of the circle [@BW; @GL]. For each $n \ge 2$ we construct a faithful representation of the group ${\mbox{\rm{BS}}}(1,n^{2})$ into ${\mbox{\rm{SL}}}(2,{\mathbb{Z}}[\frac{1}{n}])$ by setting $$\label{e:rep-s} b \mapsto \left( \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix} \right) \ \ \mbox{ and } \ \ t \mapsto \left(\begin{matrix} n & 0 \\ 0 & \frac{1}{n} \end{matrix} \right).$$ It is easy to verify that the proposed mapping into ${\mbox{\rm{SL}}}(2,{\mathbb{Z}}[\frac{1}{n}])$ preserves the defining relation by multiplying out the matrices for the product $tbt^{-1}$. So the representation is well-defined. The representation of ${\mbox{\rm{BS}}}(1,n^{2})$ into ${\mbox{\rm{SL}}}(2,{\mathbb{Z}}[\frac{1}{n}])$ defined in is faithful. Let $w \in {\mbox{\rm{BS}}}(1,n^2)$ be an arbitrary reduced word. First note that from one immediately deduces that $$b^m \mapsto \left( \begin{matrix} 1 & m \\ 0 & 1 \end{matrix} \right) \ \ \mbox{ and } \ \ t^{-\ell} \mapsto \left(\begin{matrix} \frac{1}{n^{\ell}} & 0 \\ 0 & n^{\ell} \end{matrix} \right),$$ for all $m, \ell \in {\mathbb{Z}}$, so no nonzero power of $b$ or of $t$ goes to the identity. Thus, we can suppose that $w$ is not equal to a power of $b$ or $t$. Up to replacing $w$ with $w^{-1}$ if necessary, we may suppose that $w$ has nonpositive exponent sum on $t$. Since the defining relation for ${\mbox{\rm{BS}}}(1,n^2)$ can be written $tb = b^{n^2} t$, we can move all occurrences of $t^{+1}$ in $w$ to the right, yielding a word of the form $w_1t^\alpha$, where $w_1$ does not contain any positive powers of $t$ and $\alpha \geq 0$. Thus $w$ is conjugate to $w':=t^\alpha w_1$. Moving $t^\alpha$ to the right, all positive occurrences of $t$ will cancel in $w'$, since $w$ has nonpositive exponent sum on $t$. Since $w$ is not a power of $b$, we have that $w$ or $w^{-1}$ is conjugate to a word of the form $b^{m_1}t^{-\ell_1}b^{m_2}t^{-\ell_2}\cdots b^{m_r}t^{-\ell_r}$ where $m_i \in {\mathbb{Z}}$ and $\ell_i > 0$ for all $i=1,2,\ldots, r$. The image of such an element is a matrix of the form $$\left(\begin{matrix} \frac{1}{n^{\ell}} & * \\ 0 & n^{\ell} \end{matrix} \right)$$ where $\ell = \ell_1 + \ell_2 + \cdots + \ell_r > 0$, and therefore is not the identity matrix. This shows that is faithful. The groups $SL(2,\mathbb{Z}[\frac{1}{n}])$ are interesting here because Serre [@Serre] showed that the groups ${\mbox{\rm{SL}}}(2,{\mathbb{Z}}[\frac{1}{p}])$, $p$ a prime, are free products of two copies of ${\mbox{\rm{SL}}}(2,\mathbb{Z})$ amalgamating subgroups of finite index. *However*, the amalgamation is *not* induced by an isomorphism between the two copies and so these groups are *not* doubles. Since a double of ${\mbox{\rm{SL}}}(2,\mathbb{Z})$ over a subgroup of finite index is in ${\mathcal{PG}}$ (cf. Theorem \[t:double\]), the groups ${\mbox{\rm{SL}}}(2,{\mathbb{Z}}[\frac{1}{p}])$ are on the borderline of not being in ${\mathcal{PG}}$. Some decision problems ====================== Many classical decision problems are already undecidable for direct products of free groups. Mikhailova’s theorem [@LS] shows that $F_2 \times F_2$ has specific finitely generated subgroups with unsolvable *membership problem*. C.F. Miller III [@Miller] showed that $F_3 \times F_3$ contains subgroups with unsolvable *conjugacy problem*. Miller [@Miller] also showed that if $n \ge 5$ then the *generating problem* for $F_n \times F_n$ is undecidable. Thus the conjugacy, membership and generating problems are generally unsolvable for groups in $\mathcal{D}$. The *order problem* for a finitely generated group is the problem of deciding whether or not arbitrary elements have infinite order. Holt, Rees, Röver and Thomas [@HRRT] prove the interesting result that groups with [context-free ]{}co-word problem have solvable order problem. We use the main ideas of their proof to establish the same result for groups in ${\mathcal{BDG}}$. \[t:order-p\] A group in ${\mathcal{BDG}}$ has solvable order problem. Let $G =\langle X;R\rangle \in {\mathcal{BDG}}$ and denote by $\Sigma = X \bigcup X^{-1}$ the associated group alphabet. Since the class of deterministic multipass languages is closed under complementation (cf. Proposition \[p:dmp-complement\]), the complement $C:= \Sigma^* \setminus {\mbox{\rm{WP}}}(G:X;R)$ of the word problem of $G$ is also deterministic multipass. By virtue of Theorem \[t:char-multip\] we can find context-free languages $L_1, L_2, \ldots, L_n \subseteq \Sigma^*$ such that $C = \bigcap_{i=1}^n L_i$. For any $w \in \Sigma^*$ let $w^+:= \{w^n: n \geq 1\} \subseteq \Sigma^*$ denote the regular language consisting of all positive powers of $w$. Then $w$ represents a group element of infinite order if and only if the multipass language $L_{w}:= (w^+ \cap C)$ equals $w^+$. But $L_w = w^+$ if and only if $(w^+ \cap L_i) = w^+$ for all $i=1,2,\ldots,n$. Since the class of context-free languages is closed under intersection with regular languages, each language $L_i':= w^+ \cap L_i$ is context-free (in fact, a grammar ${\mathcal{G}}_i'$ for $L_i'$ can be effectively constructed from a grammar ${\mathcal{G}}_i$ for $L_i$). Recall that a language $L \subseteq \Sigma^*$ is said to be *bounded* provided there are words $z_1,z_2,\ldots,z_m \in \Sigma^*$ such that $L \subseteq z_1^* z_2^* \cdots z_m^*$. Thus $w^+$ and therefore $L_1', L_2',\ldots,L_n'$ are bounded context-free languages. Although equality is generally undecidable for [context-free ]{}languages, results of Ginsburg [@Ginsburg] show that there is an algorithm which, given a bounded [context-free ]{}language and another [context-free ]{}language, decides if they are equal. So we can decide if $w^+ = L_i'$ for each $i=1,2,\ldots,n$ and thus whether or not $w$ represents a group element of infinite order. Note that taking the complement of the word problem is essential for the above proof. It is well known that the membership problem for deterministic context-free languages is decidable in linear time [@Harrison Section 5.6]. Essentially the same argument gives the following: \[t:membership\] If $M$ is a complete, deterministic multipass automaton, the membership problem for $L(M)$ is solvable in linear time. Since $M$ is complete there is a bound $B$ on the number of $\epsilon$-transitions which $M$ can make before either advancing the tape or erasing the symbol which was on the top of the stack when $M$ began the sequence of $\epsilon$-transitions. Let $C$ be the maximum number of symbols which $M$ can add to the stack on a single transition. Then on a single pass on an input of length $n$, at most $CBn$ symbols can be added to the stack. So in the complete run of $M$ on the input at most $kCBn$ symbols can be added to the stack and if it takes $B$ moves to erase each symbol then $M$ makes at most $kCB^2n$ transitions. The Cocke-Kasami-Younger algorithm [@Harrison; @HU] shows that the membership problem for an arbitrary context-free language is decidable in time $\mathcal{O}(n^3)$. Since we need only check subsequent terms in an intersection of context-free languages, it follows that the membership problem for any poly-context-free language is decidable in time $\mathcal{O}(n^3)$. In formal language theory it is well-known that deciding whether or not the intersection of two deterministic context-free languages is empty is undecidable. That is, there does not exist an algorithm which, when given two deterministic push-down automata $M_1$ and $M_2$ over the same input alphabet, decides whether or not $L(M_1) \cap L(M_2) =\emptyset$. The standard construction shows that one can represent valid computations of Turing machines as the intersection of two deterministic context-free languages [@HU]. Since the intersection of two deterministic context-free languages is a deterministic multipass language, we have: \[p:emptyness\] The emptiness problem for deterministic multipass languages is undecidable. That is, there does not exist an algorithm which, when given a deterministic multipass automaton $M$ decides whether or not $L(M) =\emptyset$. The class $\mathcal{D}$ and HNN extensions ========================================== We now show that the class $\mathcal{D}$ of groups that are virtually finitely generated subgroups of direct product of free groups is closed under the class of HNN extensions given in Theorem \[t:HNN\]. In fact we will prove a slightly more general statement: taking such HNN extensions virtually amounts to taking a direct product with a free group of finite rank. \[t:HNN-closed\] Let $G$ be finitely generated group in $\mathcal{D}$ with $S$ a subgroup $G$ of finite index. Let $\varphi \colon G \to G$ be an automorphism of finite order of $G$ such that $\varphi(S) = S$. Let $H$ be the HNN-extension $$H = \langle G,t; tst^{-1} = \varphi(s), s \in S\rangle.$$ There is a subgroup $E \leq S \leq G$ that is a finite index characteristic subgroup of $G$. Furthermore for any such subgroup $E$, $H$ has a finite index subgroup $H'$ isomorphic to $E \times F$, where $F$ is a free group of finite rank. Let $E \leq S \leq G \leq H$ be as in the statement of the theorem. The proof is divided into five steps.  \ *Step 1. The existence of $E$ and its normality in $H$.* Suppose that $S \leq G$ has index $f$. Then taking $E$ to be the intersection of all index $f$ subgroups of $G$ gives the desired property. Since $E$ is characteristic and thus normal in $G$ and $\varphi$ is an automorphism of $G$, $t$ normalizes $E$. $E$ is therefore normalized by a generating set of $H$ and is thus normal in $H$.  \ *Step 2. The quotient $H/E$ is virtually free.* This quotient splits as the HNN extension $$H/E = \langle \bar{G}, t ; t\bar{s}t^{-1} = \bar{\varphi}(\bar{s}), \bar{s} \in \bar{S}\rangle,$$ where $\bar{G}$ denotes the quotient $G/E$ and $\bar{s},\bar{S}$ denote the image of $s\in S$, the image of the subgroup $S$ in $\bar{G}$ respectively. $H/E$ is therefore the fundamental group of a finite graph of groups with finite vertex groups and therefore (c.f. [@Serre II.2.6, Proposition 11 and Corollary]) contains a finite index free subgroup $F_k$ of rank $k$.  \ *Step 3. The structure of the preimage $\hat{F_k}$ as a semidirect product.* Denote by $\hat{F_k}$ the preimage of $F_k$ in $H$ via the canonical epimorphism $H \to H/E$. Then $\hat{F_k}$ has finite index in $H$ and furthermore, since $F_k$ is free, the short exact sequence $$1 \to E \to \hat{F_k} \to F_k \to 1$$ splits. Specifically there is a set of elements $\{f_1,\ldots,f_k\} \subseteq \hat F$ such that 1. $\langle f_1,\ldots,f_k \rangle \cong F_k$, 2. $\hat{F_k} = \langle E, f_1,\ldots,f_k \rangle$, and 3. $\hat{F_k} \cong E \rtimes_\rho \langle f_1,\ldots,f_k \rangle$, where $\rho: \langle f_1,\ldots,f_k \rangle \to \mathrm{Aut}(E)$ is the representation induced by conjugation in $H$. Note that $\rho$ is well-defined since $E$ is normal in $\hat F$. We will now study the representation $\rho$ more closely.  \ *Step 4. The action of $H$ on $E$ and outer automorphisms.* Since $E$ is normal in $H$ there is a natural conjugation representation $\rho_H: H \to \mathrm{Aut}(E)$. We now study the map $\bar\rho_H: H \to \mathrm{Out}(E)$, where $\mathrm{Out}(E) = \mathrm{Aut}(E)/\mathrm{Inn}(E)$. Our goal is to show that its image is finite. Denote by $\rho_G$ the restriction of $\rho_H$ to $\rho_G$. We first note that, since $\varphi$ is an automorphism of the entire base group $G$, the following calculation holds for all $k\in {\mathbb{Z}}, g \in G$ and $e \in E$: $$\label{e:count-outer} \left(\varphi^k\circ\rho_G(g)\right)(e) = \varphi^k\left(g e g^{-1} \right) = \varphi^k(g) \varphi^k(e) \varphi^k(g^{-1}) = \left(\rho_G(\varphi(g))\circ\varphi^k\right)(e).$$ Since every element of $H$ can be written as a product of elements of the form $t^kg$, with $k \in {\mathbb{Z}}$ and $g\in G$, and since $$\left(t^k g\right)e\left(g^{-1} t^{-k}\right) = \left(\varphi^k\circ\rho_G(g)\right)(e),$$ the image $\rho_H(H)$ of $H$ in $\mathrm{Aut}(E)$ is generated by elements of the form $\varphi^k\circ\rho_G(g)$. The identity (\[e:count-outer\]) enables us to collect $\varphi^k$s to the left, so we can express every element of $\rho_H(H)$ in a normal form: $$\varphi^n \circ \rho_G(g),$$ with $1\leq n \leq \mathrm{order}(\varphi)$ and $g \in G$. If $g' = ge$, for some $g\in G, e \in E$ then $\rho_G(g')\mathrm{Inn}(E) = \rho_G(g)\mathrm{Inn}(E)$. It follows that if $g_1,\ldots,g_{[G:E]}$ is a finite set of left coset representatives of $E$ in $G$, then the finite set of automorphisms $$\left\{ \varphi^i \circ \rho_G(g_j) \mid 1\leq i \leq \mathrm{order}(\varphi), 1\leq j \leq [G:E] \right\}$$ gives a complete set of representatives of the image of $H$ in $\mathrm{Out}(E)$. The image of $\bar{\rho_H}:H \to \mathrm{Out}(E)$ is therefore finite.  \ *Step 5. Untwisting the semidirect product.* The twisting representation $\rho$ in $\hat{F_k} \cong E \rtimes_\rho \langle f_1,\ldots,f_k \rangle$ is the restriction of $\rho_H$ to the subgroup $\langle f_1,\ldots,f_k \rangle$. It follows from the previous part that the image of $\bar\rho: \langle f_1,\ldots,f_k \rangle \to \mathrm{Out}(E)$ is finite; thus the kernel $K = \ker(\bar\rho)$ is of finite index in $\langle f_1,\ldots,f_k \rangle$. Let $X= \{k_1,\ldots,k_m\}$ be a free generating set for $K$. On one hand the subgroup $H' = \langle E,k_1,\ldots,k_m \rangle \cong E \rtimes_{\rho|_K} K$ is of finite index in $\hat{F_k}$, and therefore in $H$. On the other hand, since $K = \ker(\bar\rho)$, for each $k_i\in X$ there exists some $e_i \in E$ such that for all $e \in E$, $$k_i e k_i^{-1} = e_i e e_i^{-1}.$$ It follows that $$H' = \langle E, k_1e_1^{-1},\ldots, k_me_m^{-1}\rangle \cong E\times F,$$ where $F \cong K$ is a free group of rank $m$. For the proof to work, it is essential that the isomorphism $\varphi:E\to E$ in the HNN extension extends to an isomorphism $\varphi:G \to G$. If this is not the case, then the identity (\[e:count-outer\]) need not hold, which in turn prevents rewriting into normal form. In particular the image $\bar{\rho_H}(H)$ could be infinite, which makes it impossible to virtually untwist the semidirect product in Step 5. We point out that the class $\mathcal{D}$ is closed under taking quotients by finite normal subgroups. If $G$ has a subgroup $S$ of finite index which is a subgroup of a direct product of free groups then $S$ is torsion-free. So $N \cap S = \{1\}$ and $S$ is embedded in the quotient $G/S$ and again has finite index. In conclusion, if we start with a virtually free group, none of the known closure properties of the class ${\mathcal{PG}}$ which we have discussed take us outside the class $\mathcal{D}$. This seems fairly strong evidence in favor of Brough’s Conjecture.\ [**Acknowledgments.**]{} We wish to thank Ralph Strebel for very helpful discussions and his interest in our work as well as for some simplifications of an earlier version of the proof of Theorem \[t:HNN-closed\]. We also express our gratitude to the two referees for their careful reading of our manuscript and their valuable comments. [20]{} A.V. Anisimov, *Über Gruppen-Sprachen*, Kibernetika **4** (1971), 18–24. 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M.A. Harrison, Introduction to Formal Language Theory, Addison-Wesley, Reading, MA, 1978. D.F. Holt, S. Rees, C. Rover and R.M. Thomas, *Groups with context-free co-word problem* J. London Math. Soc. (2) **71** (2005), 643-657. J.E. Hopcroft and J.D. Ullman, Introduction to Automata Theory, Languages and Computation, Addison-Wesley, Reading, MA, 1979. R.C. Lyndon and P.E. Schupp, Combinatorial Group Theory, Springer-Verlag Classics in Mathematics, New York, 2000. C.F. Miller III, On Group-theoretic Decision Problems and Their Classification, Annals of Mathematics Studies **68**, Princeton University Press, Princeton, 1971. D.E. Muller and P.E. Schupp, *Groups, the theory of ends and context-free languages*, J. Comput. System Sci. **26** (1983), 295–310. R.J. Parikh, *Language-generating devices*, MIT Res. Lab. Electon. Quart. Prog. Report, **60** (1961), 199-212. R.J. Parikh, *On context-free languages*, J. Assoc. Computing Machinery **13** (1966), 570-581. 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--- abstract: 'In our work we define a new algebra of operators as a substitute for fuzzy logic. Its primary purpose is for construction of binary discriminators for phonemes based on spectral content. It is optimized for design of non-parametric computational circuits, and makes uses of 4 operations: $\min$, $\max$, the difference and generalized additively homogenuous means.' author: - 'Ondrej Šuch[^1][^2]' bibliography: - 'alg\_bib.bib' title: 'Phoneme discrimination using $KS$-algebra I.' --- Introduction ============ Probability, statistics and in particular Bayesian statistics are disciplined foundation for understanding uncertainty phenomena. Fuzzy logic is a generalization of Boolean logic that purports to provide another way to describe and reason with uncertainties [@zad95], [@zad08]. A very useful feature of fuzzy logic is that it can be implemented in analog circuits. It serves as a basis for fuzzy set theory and since its inception has found thousands of applications ranging from modelling and prediction to industrial controllers. There is really not a single fuzzy logic, but rather many variations. Most of them however use $\min$ and $\max$ operators [@ZavalaNieto]. Recently, a new way to compute voltage-mode $\min$ and $\max$ operations using memristors [@chua71] has been proposed in [@arxiv]. Unlike previous implementations using bipolar junction transistors [@Yamakawa93], [@Yamakawa1], and CMOS transistors [@Fatt94], [@Baturone97], the proposed circuit is passive. This has prompted research inquiries about computational power of circuits whose elements are mostly $\min$ and $\max$ operators, extending classical research on sorting networks [@bat68] that use $\min$ and $\max$ exclusively. First steps were taken in works [@Foltan], [@FoltanSmiesko], [@FoltanPhd], [@Badura1], [@BaduraPhd] where authors find pattern recognition circuits using fuzzy logic operators. Their application domain is speech recognition. In this note we lay out our arguments why we believe that fuzzy logic may not be the best choice to use in this context. We propose an alternative that we call $KS$-algebra ${{\cal MM}}$. The key principles guiding our proposal are: - simplicity, keeping the family of operations small to allow for more efficient optimization using evolutionary algorithms, which are commonly used to search for optima in discrete spaces, [@Sekanina1],[@EvolutionTexas] - richness, that would allow expression of desirable properties, such as being able to capture the concept of formant [@Mariani1 page 12], or provide invariance under volume adjustment, - nonlinearity, by which we mean using nonlinear operators, such as $\min$ and $\max$, as well as the absence of continuous parameters, which are currently hard to implement in analog electrical circuits. This is the first paper in a series in which we will investigate phoneme discrimination circuits using this algebra. Let us note that it is possible to use the algebra also to design circuits for feature detection (e.g. lines, edges) in vision applications, but speed of computation when simulated on digital computers is not competitive. Basic properties of human speech processing =========================================== In order to design an efficient speech recognition framework it is helpful to understand underlying mechanisms that allow humans process audio signal. If the framework takes the mechanisms into acccount, it is more likely to exhibit performance closer to humans. This is demonstrated by improved performance of mel-based cepstral coefficients, PLP [@Hermansky90] or RASTA [@Hermansky92]. From physical point of view sound is simply an oscillation of air pressure. It is perceived by humans via a complex series of transformations. First, air pressure oscillation is conducted (and certain frequencies partially amplified) via ear canal to bones of middle ear. They transfer oscillations into vibrations of the basiliar membrane that in turn stimulates inner hair cells. By release of neurotransmitters mechanical energy is converted to electrical signals that are further processed by central neural system in which neurons communicate primarily by firing pulses. Increase of stimulation in inner hair cells translates to increase of firing rate of a neuron. There is however a limit to a neuron’s firing rate due to its refractor period. The crucial point is that from computational point of view organ of Corti performs *Fourier transform*. That is, each section of the basiliar membrane is tuned to a narrow frequency band and neural response is commensurate to energy in that band. It is therefore natural to construct phonemic classifiers based on spectral data. Nonlinear means =============== Our primary motivation is to discriminate between phonemes. It is well known that location of formants is an important characteristic of vowels. Thus would like to find ways other than LPC to quantify “peakness” of formants in the spectral envelope. One way to do that is to start with arithmetic means $$\begin{aligned} M(x_1, x_2, \ldots, x_n) &= \dfrac{x_1 + x_2 + \cdots + x_n}{n}. \intertext{and look for their generalizations. One obvious one is to use weighted linear combinations} L(x_1, \ldots, x_n) &= a_1 x_1 + \ldots a_n x_n\end{aligned}$$ which leads to approaches like logistic regression or support vector machines. We wish to investigate nonlinear functions. In nonlinear algebras, useful simplifying laws such as commutativity or distributivity of operations do not need to hold. In order to obtain a reasonably small set of functions in nonlinear algebra we will work with functions derived from *symmetric* operators. These are functions $f$ invariant with respect to argument transpositions $$f(x_1,\ldots, x_i, x_{i+1}, \ldots)= f (x_1,\ldots, x_{i+1}, x_{i}, \ldots)$$ and thus also with respect to any permutations of its arguments. In fact, they are uniquely defined, once one specifies them on the whole cone $x_1 \leq x_2 \cdots \leq x_n$. Let us recall the classically known generalizations of the arithmetic mean. Quadratic mean is given by formula $$\begin{aligned} M_2(x_1, x_2, \ldots, x_n) &= \sqrt{ \dfrac{x_1^2 + x_2^2 + \cdots + x_n^2}{n}}, \intertext{the geometric mean by } G(x_1, x_2, \ldots, x_n) &= \sqrt[n]{x_1 \cdot x_2 \cdots x_n}, \intertext{and the harmonic mean by} H(x_1, x_2, \ldots, x_n) &= \Bigl( \dfrac{x_1^{-1} + x_2^{-1} + \cdots + x_n^{-1}}{n}\Bigr)^{-1},\end{aligned}$$ The following ordering between these means holds: $$\begin{aligned} H({{\bf x}}) \leq G({{\bf x}}) \leq M({{\bf x}})\leq M_2({{\bf x}}). \label{eqn:gah}\end{aligned}$$ One can define still more general means of which arithmetic, quadratic and harmonic means are special cases. Namely, for $\alpha \not = 0$ and ${{\bf x}}> 0$ set $$M_\alpha(x_1, \ldots, x_n) = \Bigl( \dfrac{x_1^\alpha + \cdots + x_n^\alpha}{n} \Bigr)^{1/\alpha}$$ One can show that in fact $$\lim_{\alpha \rightarrow 0} M_\alpha(x_1, \ldots, x_n) = G(x_1, \ldots,x_n)$$ so that the geometric mean is also a member of the family. By L’Hôspital rule we have $$\begin{aligned} \lim_{\alpha\rightarrow 0} \log M_\alpha({{\bf x}}) &= \lim_{\alpha\rightarrow 0}\dfrac{\log \sum_i x_i^\alpha}{\alpha} \\ &=\lim_{\alpha\rightarrow 0} \dfrac{\sum_i x_i^\alpha \ln x_i}{\sum_i x_i^\alpha}\\ &= \dfrac{\sum_i \ln x_i}{n} = \ln \sqrt[n]{x_1\cdots x_n}\end{aligned}$$ Inequalities are just special cases of the following theorem: If $\alpha < \beta$ then for nonnegative $x_1, \ldots, x_n$ we have $$M_\alpha(x_1, \ldots, x_n) \leq M_\beta (x_1, \ldots, x_n).$$ For $\alpha \not = 1$ the means $M_\alpha$ are not linear, meaning the equality $$M_\alpha({{\bf x}}+ {{\bf x}}') = M_{\alpha}({{\bf x}}) + M_\alpha({{\bf x}}')$$ does not need to hold. However they are multiplicatively homogeneous, meaning $$M_\alpha ( c{{\bf x}})= c M_\alpha({{\bf x}}).$$ Since sound data is usually provided in logarithmic scale one may desire to possess means that are additively homogeneous so that $$F (c \cdot (1, \ldots,1) + {{\bf x}})= c + F({{\bf x}})$$ A family of such means is given by the formula $$\begin{aligned} A_\alpha(x_1, \ldots, x_n) &= \ln(M_\alpha(\exp(x_1), \ldots, \exp(x_n))) \intertext{where in particular} A_0(x_1, \ldots, x_n ) &= \dfrac{x_1 + \ldots + x_n}n, \label{A0} \\ \lim_{\alpha\rightarrow -\infty}A_\alpha(x_1,\ldots,x_n) &= \min(x_1, \ldots, x_n), \label{Amin} \\ \lim_{\alpha\rightarrow \infty}A_\alpha(x_1,\ldots,x_n) &= \max(x_1, \ldots, x_n). \label{Amax}\end{aligned}$$ Quantiles ========= Minimum and maximum arise naturally as the limiting cases of nonlinear means studied in the previous section, since $$\begin{aligned} \lim_{\alpha\rightarrow -\infty} M_\alpha(x_1, \ldots, x_n) = \min(x_1, \ldots, x_n)\\ \lim_{\alpha\rightarrow\infty} M_\alpha(x_1, \ldots, x_n) = \max(x_1, \ldots, x_n)\end{aligned}$$ A simple, but important observation is that both $\min$ and $\max$ do not generate new values. In particular evaluation of any expression of variables $x_1, \ldots, x_n$ using only composition of $\min$ and $\max$ operators results in one of $x_1, \ldots, x_n$. A natural class of symmetric operators in the algebra generated by $\min$ and $\max$ are *quantiles*. In fact, under continuity requirements they are the only symmetric $n$-ary operators in that algebra. From computer science point of view it is natural to ask how to compute quantiles using a given class of $\min$ and $\max$ gates. In this context one may ask for instance how many gates one needs, or what is the depth of the resulting feed-forward circuit. The answer differ depending on the kinds of $\min$ and $\max$ operators one allows. Suppose for instance that one allows $\min$ and $\max$ gates of arbitrary arity. Obviously $n$-ary $\min$ and $n$-ary $\max$ are both representable by one gate. Any other quantile can be represented by circuits of depth two, for instance the second largest element function can be found by computing either of the following two expressions: $$\begin{aligned} \min\bigl(\max(x_2, \ldots, x_n), \max(x_1, x_3, \ldots,x_n), \max(x_1,x_2, x_4, \ldots), \ldots\bigr)\\ \max\bigl(\min(x_1,x_2), \min(x_1,x_3), \ldots, \min(x_i, x_j), \ldots\bigr)\end{aligned}$$ The restriction to using binary $\min$ and $\max$ gates is more delicate. Results about computations of quantiles can be deduced from classical studies on sorting networks. Well-known is bitonic sorting network of Batcher, which yields explicit circuits for quantiles of $n$ variables of depth $O(\log^2n)$. Theoretical improvement came in work [@AKS] which showed that sorting networks with $O(n \log n)$ comparators and depth $O(\log n)$ exist, which is asymptotically optimal. However the implied constant is quite large and research in this area is ongoing. For values of $n$ up to 8, optimal networks are known. For instance, a sorting network with 8 inputs and optimal depth 6 is shown in Figure \[fig:sorting8\]. ![Sorting network with 8 inputs and depth 6. Inputs are supplied on the left, outputs (sorted values) can be read out on the right. Each vertical wire signifies reordering of inputs on corresponding horizontal wires.[]{data-label="fig:sorting8"}](sorting8.pdf){width="70.00000%"} Fuzzy logic =========== Mathematical functions introduced so far ($M_\alpha$, $A_\alpha$ and quantiles) are essentially monotonous. When spectral data increase they increase as well. In order to gain *contrast* it is useful to have a nonsymmetric operation of say two variables, “increasing” in the first and “decreasing” in the second. A well known framework that uses such functions is fuzzy logic. Fuzzy logic dates back to works of Lukasiewicz and Post. H. Weyl in 1940 proposed a fuzzy logic where propositions are assigned values in the unit interval. He generalized ordinary Boolean logic operators as follows: $$\begin{gathered} a\text{~and~}b = \min(a,b) \\ a\text{~or~}b = \max(a,b) \\ \text{not~}a = 1 -a \\ a\text{~implies~}b = 1 -a + min(a,b) \\ = min(1, 1-a+b),\end{gathered}$$ where $a$ and $b$ take values in the interval $[0,1]$. Many other operators in fuzzy logic have been since proposed. For instance instead one can use $t$-norms or $t$-conorms. Fuzzy logic gained prominence in the foundational paper on fuzzy sets written by L. Zadeh [@zad65]. In this paper he used fuzzy logic as a means to represent vague linguistical and cognitive notions. Eliminating negation ==================== We note that using min and max in fuzzy logic has been shown to be the only choice under natural assumptions [@MR0414314]. Using negation is however problematic for several reasons. The first reason arises from implementation issues one encounters when translating fuzzy logic expressions into electronic circuits. Both negation and implication such as Lukasiewicz implication need to be implemented with active electronic circuits. This issue can be partially addressed. Passive circuits whose inputs are normalized to $[0,1]$ can compute with negation $\bar a = \text{not~}a = 1 -a$, if those are prepared beforehand due to identities $$\begin{aligned} \overline{\min(x,y)} &= \max(\overline x, \overline y)\\ \overline{\max(x,y)} &= \min(\overline x, \overline y)\end{aligned}$$ This idea is a special case of negation conversion for MIN-MAX-AVG circuits described in [@blum] (see Figure \[fig:BlumNegation\]). ![How to convert MIN-MAX-NEGATION-AVG circuit to MIN-MAX-AVG circuit[]{data-label="fig:BlumNegation"}](BlumNegation.png){width="70.00000%"} The second reason lies in conflict with our understanding of auditory perception. Introducing negation implies providing lower *and* upper bounds on spectral data. Establishing a lower bound may be reasonable, because it should correspond to the smallest intensity of audible sound. However, an upper bound should probably reflect the largest sound a human ear can sustain without damage, or possibly the intensity threshold at which a harmonic causes the maximum possible firing rate in the auditory nerve. Both of these quantities are very hard to measure and are probably quite variable. It is therefore unclear why one should let such an upper bound affect information processing in phoneme classification circuits. A final disagreement is of philosophical nature. Fuzzy logic traditionally deals with uncertain notions primarily of linguistic nature, such as “tall”, or “warm”. Spectral content of sound is of course also uncertain, due to uncertainty principle of Fourier transform. However the uncertainty is of a different kind, taking form of a random variable [@Brockwell Chapter 10]. We thus propose to leave the realm of fuzzy logic and instead use the difference operator (this has value $x-y$ for a pair of real valued variables $x$ and $y$). If this operator is applied to two additively homogeneous expressions such as quantiles or generalized means $A_\alpha$ described earlier, it turns volume sensitive expressions into volume invariant expressions. This invariance is likely to be a desirable feature of classificators. We thus arrive at the definition of *KS-algebra* ${{\cal MM}}$. An element of the algebra is any expression of spectral components of sound and the zero value, using operators - the minimum $\min(x_1, \ldots, x_n)$, - the maximum $\max(x_1, \ldots, x_n)$, - the difference $x_1 - x_2$, - the additively homogeneous means $A_\alpha$. In view of from , , , the addition of operators in the last category can be seen as constructing a link between linear circuits and $\min$–$\max$ circuits, allowing for direct comparison of performance of linear and nonlinear circuits. Binary Classifiers arising from ${{\cal MM}}$ ============================================= An interesting experiment is described in work [@Hermansky94], [@Hermansky11]. A sentence was processed by a filter with frequency response approximating inverse to spectral envelope of a vowel. In spite of this, the vowel was recognized by speakers. It is thus not the absolute spectral content that determines phonemes, but rather relative. This is confirmed for instance also in work [@Prendergast12]. In our context, a binary classifier will consist of two ingredients. The first one is a function $f$ of KS-algebra algebra ${{\cal MM}}$. The second ingredient describes what one concludes given an evaluation of $f$ on spectral data ${{\bf s}}$. We can distinguish the following natural variants of classifiers. - A [**${\bf Z}$-classifier**]{} is described by a function $f\in {{\cal MM}}$. It decides that a phoneme is of the first class if $f({{\bf s}}) < 0$, and decides that is of the second class if $f({{\bf s}}) > 0$. - A [**${\bf B}$-classifier**]{} consists of a pair $(f, c)$, where $f \in {{\cal MM}}$ and $c$ is a real number. It decides that a phoneme is of the first class if $f({{\bf s}}) < c$, and concludes that is of the second class if $f({{\bf s}}) > c$. - an [**${\bf A}$-classifier**]{} consists of a pair $(f, c)$, where $f \in {{\cal MM}}$ and $c$ is a real number. It decides that a phoneme is of the first class if $f_1({{\bf s}}) < c$, but makes no conclusion, if this condition is not met. From $A$-classifiers we will distinguish the subclass of $A^+$ classifiers that are charactized by the condition $c\leq 0$. Let us remark that unlike models with continuous parameters, the classificators are really described by their structure and support, the latter being defined as follows. $$\begin{aligned} {\textrm{supp}}(0) &= \{\} \\ {\textrm{supp}}(s_i) &= \{i\} \\ {\textrm{supp}}(\min(f_1, f_2)) &= {\textrm{supp}}(f_1) \cup {\textrm{supp}}(f_2) \\ {\textrm{supp}}(\max(f_1, f_2)) &= {\textrm{supp}}(f_1) \cup {\textrm{supp}}(f_2) \\ {\textrm{supp}}(A_\alpha(f_1, \ldots, f_n)) &= \bigcup_i {\textrm{supp}}(f_i)\end{aligned}$$ [^1]: O.Šuch is with Slovak Academy of Sciences, Banská Bystrica, Slovakia, [[email protected]]{} [^2]: Work on this paper was partially supported by research grant VEGA 2/0112/11

--- abstract: 'This investigation is a part of a research program aiming to characterize the extreme behavior possible in hydrodynamic models by analyzing the maximum growth of certain fundamental quantities. We consider here the rate of growth of the classical and fractional enstrophy in the fractional Burgers equation in the subcritical and supercritical regimes. Since solutions to this equation exhibit, respectively, globally well-posed behavior and finite-time blow-up in these two regimes, this makes it a useful model to study the maximum instantaneous growth of enstrophy possible in these two distinct situations. First, we obtain estimates on the rates of growth and then show that these estimates are sharp up to numerical prefactors. This is done by numerically solving suitably defined constrained maximization problems and then demonstrating that for different values of the fractional dissipation exponent the obtained maximizers saturate the upper bounds in the estimates as the enstrophy increases. We conclude that the power-law dependence of the enstrophy rate of growth on the fractional dissipation exponent has the same global form in the subcritical, critical and parts of the supercritical regime. This indicates that the maximum enstrophy rate of growth changes smoothly as global well-posedness is lost when the fractional dissipation exponent attains supercritical values. In addition, nontrivial behavior is revealed for the maximum rate of growth of the fractional enstrophy obtained for small values of the fractional dissipation exponents. We also characterize the structure of the maximizers in different cases.' author: - 'Dongfang Yun and Bartosz Protas[^1]' title: Maximum Rate of Growth of Enstrophy in Solutions of the Fractional Burgers Equation --- [**[Keywords:]{}**]{} Fractional Burgers equation; extreme behavior; enstrophy growth; numerical optimization; gradient methods [**[AMS subject classifications:]{}**]{} 35B45, 35Q35, 65K10 Introduction {#sec:intro} ============ One of the key questions studied in the mathematical analysis of evolutionary partial differential equations (PDEs) is the existence of solutions, both locally and globally in time. The motivation is that, in order to justify the application of different PDEs as models of natural phenomena, these equations must be guaranteed to possess meaningful solutions for physically relevant data. In addition, characterization of the extreme behavior which can be exhibited by the solutions of different PDEs is also relevant for our understanding of the worst-case scenarios which can be realized in the actual physical systems these PDEs describe. These two types of questions can be investigated by studying the time evolution of suitable Sobolev norms of the solutions. In particular, should a given Sobolev norm of the solution become unbounded at a certain time due to a spontaneous formation of a singularity, this will signal that the solution is no longer defined in that Sobolev space; this loss of regularity is referred to as “blow-up”. An example of an evolutionary PDE model with widespread applications whose global-in-time existence remains an open problem is the three-dimensional (3D) Navier-Stokes system describing the motion of viscous incompressible fluids. Questions of existence of solutions to this system are usually studied for problems defined on unbounded or [periodic domains]{} $\Omega$, i.e., $\Omega = \RR^d$ or $\Omega = \SS^d$, where $d=2,3$. Unlike the two-dimensional (2D) problem where smooth solutions are known to exist globally in time [@kl04], in 3D existence of such solutions has been established for short times only [@d09]. Establishing global existence of smooth solutions in 3D is one of the key open questions in mathematical fluid mechanics and, in fact, its importance for mathematics in general has been recognized by the Clay Mathematics Institute as one of its “millennium problems” [@f00]. Suitable weak solutions were shown to exist in 3D for arbitrarily long times [@l34], however, such solutions may not be regular in addition to being nonunique. Similar questions also remain open for the 3D Euler equation [@gbk08]. While many angles of attack on this problem have been pursued in mathematical analysis, one research direction which has received a lot of attention focuses on the evolution of the [*enstrophy*]{} $\E(\u)$ which for an incompressible velocity field $\u(t,\cdot) \; : \; \Omega \rightarrow \RR^d$ at a given time $t$ is defined as $\E(\u(t)) := (1/2) \int_{\Omega} | \bnabla\times\u(t,\x)|^2 \, d\Omega = (1/2) \| \bnabla\u(t,\cdot)\|_{L^2(\Omega)}^2$, where “$:=$” means “equals to by definition”, i.e., it is proportional to the square of the $L^2$ norm of the vorticity $\bnabla\times\u$. The reason why this quantity is interesting in the context of the 3D Navier-Stokes equation is due to a conditional regularity result proved by Foias and Temam [@ft89] who showed that the solution remains smooth (i.e., stays in a suitable Gevrey regularity class) as long as the enstrophy remains bounded, i.e., for all $t$ such that $\E(\u(t)) < \infty$. In other words, a loss of regularity must be manifested by the enstrophy becoming infinite. While there exist many different conditional regularity results, this one is particularly useful from the computational point of view as it involves an easy to evaluate quadratic quantity. Analogous conditional regularity results, although involving other norms of vorticity, were also derived for the 3D Euler equation (e.g., the celebrated Beale-Kato-Majda (BKM) criterion [@bkm84]). In order to assess whether or not the enstrophy can blow up in finite time one needs to study its instantaneous rate of growth $d\E/dt$ which can be estimated as [@d09] $$\frac{d\E}{dt} < C \, \E^3, \label{eq:dEdt3D}$$ for some $C>0$ (hereafter $C$ will denote a generic positive constant whose actual value may vary between different estimates). It was shown in [@l06; @ld08], see also [@ap16], that this estimate is in fact sharp, in the sense that, for each [given enstrophy $\bar{\E}>0$]{} there exists an incompressible velocity field [${\widetilde{\mathbf{u}}_{\bar{\E}}}$]{} with [$\E({\widetilde{\mathbf{u}}_{\bar{\E}}}) =\bar{\E}$]{}, such that [$d\E({\widetilde{\mathbf{u}}_{\bar{\E}}}) / dt \sim \bar{\E}^3$ as $\bar{\E}\rightarrow +\infty$]{}. The fields [${\widetilde{\mathbf{u}}_{\bar{\E}}}$]{} were found by numerically solving a family of variational maximization problems for different values of [$\bar{\E}$]{} (details of this approach will be discussed further below). However, the corresponding [*finite-time*]{} estimate obtained by integrating in time takes the form $${\E(\u(t)) \leq \frac{\bar{\E}}{\sqrt{1 - \frac{C}{2}\,\bar{\E}^2\, t}}}, \quad t \ge 0, \label{eq:Et3D}$$ and it is clear that based on this estimate alone it is not possible to ensure a prior boundedness of enstrophy in time. Thus, the question of finite-time blow-up may be recast in terms of whether or not it is possible to find initial data $\u_0$ such that the corresponding flow evolution saturates the right-hand side of . We emphasize that for this to happen the rate of growth of enstrophy given in would need to be sustained over a [*finite*]{} window of time, rather than just instantaneously, a behavior which has not been observed so far [@ap16] (in fact, a singularity may arise in finite time even with enstrophy growth occurring at a slower sustained rate of $d\E / dt \sim \E^{\gamma}$ where $\gamma > 2$ [@ap16]). The question of the maximum enstrophy growth has been tackled in computational studies, usually using initial data chosen in an ad-hoc manner, producing no evidence of a finite-time blow-up in the 3D Navier-Stokes system [@bp94; @p01; @oc08; @opc12; @dggkpv13]. However, for the 3D Euler system the situation is different and the latest computations reported in [@lh14a; @lh14b] indicate the possibility of a finite-time blow-up. A new direction in the computational studies of extreme behavior in fluid flow models relies on the use of variational optimization approaches to systematically search for the most singular initial data. This research program, initiated in [@l06; @ld08], aims to probe the sharpness, or realizability, of certain fundamental estimates analogous to and and defined for various hydrodynamic models such as the one-dimensional (1D) viscous Burgers equation and the 2D/3D Navier-Stokes system. Since the 1D viscous Burgers equation and the 2D Navier-Stokes system are both known to be globally well-posed in the classical sense [@kl04], there is no question about the finite-time blow-up in these problems. However, the relevant estimates for the growth of certain Sobolev norms, both instantaneously and in finite time, are obtained using very similar functional-analysis tools as estimates and , hence the question of their sharpness is quite pertinent as it may offer valuable insights about estimates –. An estimate such as (or ) is declared “sharp”, if for increasing values of $\E$ (or [$\bar{\E}$]{}) the quantity on the left-hand side (LHS) exhibits the same power-law dependence on $\E$ (or [$\bar{\E}$]{}) as the upper bound on the right-hand side (RHS). What makes the fractional Burgers equation interesting in this context is that it is a simple model which exhibits either a globally well-posed behavior or finite-time blow-up depending on the value of the fractional dissipation exponent. Therefore, it offers a convenient testbed for studying properties of estimates applicable in these distinct regimes. Assuming the domain $\I := (0,1)$ to be periodic, we write the 1D fractional Burgers equation as \[eq:fburgers\] $$\begin{aligned} {2} & u_t+ uu_x + \nu\,{(-\Delta)}^\alpha u = 0, && t>0,\ x\in \I, \label{eq:fburgers_a}\\ & \text{Periodic Boundary Conditions}, \quad && t>0, \label{eq:fburgers_b}\\ & u(0, x) = u_0(x), && x\in \I \label{eq:fburgers_c}\end{aligned}$$ for some viscosity coefficient $\nu > 0$ and with ${(-\Delta)}^\alpha$ denoting the fractional Laplacian which for sufficiently regular functions $v$ defined on a periodic domain and $\alpha \ge 0$ is defined via the relation $$\mathcal{F}\left[{(-\Delta)}^\alpha v\right](\xi) = \left({2\pi|\xi|}\right)^{2\alpha} \mathcal{F}[v](\xi), \label{eq:F}$$ where $\mathcal{F}[\cdot](\xi)$ represents the Fourier transform. We [remark]{} that in the special cases of $\alpha = 0 $ and $\alpha = 1$ the fractional Laplacian reduces to, respectively, the identity operator and the (negative) classical Laplacian. [Is is interesting to note that in addition to $\int_0^1 u\, dx$ the quantity $\int_0^1 {(-\Delta)}^{(1-\alpha)}u\,dx$ is also conserved during evolution governed by system . Furthermore, in the periodic setting, $\int_0^1 u\, dx = 0$ also implies that $\int_0^1 {(-\Delta)}^{(1-\alpha)}u\,dx = 0$.]{} We now define the associated \[eq:EEa\] $$\begin{aligned} {2} &\text{(classical) enstrophy:}& \qquad \E(u) &:= \frac{1}{2}\int_0^1 {|u_x|}^2\,dx, \quad\qquad \text{and} \label{eq:E} \\ &\text{fractional enstrophy:}& \qquad \E_\alpha(u) &:= \frac{1}{2}\int_0^1 {\left| {(-\Delta)}^\frac{\alpha}{2}u \right|}^2\,dx. \label{eq:Ea}\end{aligned}$$ It should be noted that $\E(u)$ and $\E_{\alpha}(u)$ become equivalent when $\alpha=1$, which is a consequence of the relation $$\int_0^1 {(-\Delta)}^\frac{1}{2}u \cdot {(-\Delta)}^\frac{1}{2}u\,dx= \int_0^1 u \cdot {(-\Delta)}^1u\,dx = \int_0^1 u_x \cdot u_x\,dx \label{eq:E1}$$ following from the properties of the fractional Laplacian . Evidently, when $\alpha = 1$, system reduces to the classical Burgers equation for which a number of relevant results have already been obtained in the seminal studies [@l06; @ld08]. It was shown in these investigations that the rate of growth of the classical enstrophy $\E(u)$ is subject to the following bound $$\frac{d \E}{dt} \leq \frac{3}{2}{\left( \frac{1}{\pi^2\nu} \right)}^\frac{1}{3} \E^\frac{5}{3}. \label{eq:dEdt1D}$$ By considering a family of variational optimization problems $$\begin{aligned} & \max\limits_{{ u\in H^2(\I) }} \frac{d\E(u)}{dt} \\ & \text{subject to}\ \E(u)={\bar{\E}} \end{aligned}, \label{eq:maxdEdt}$$ parameterized by [$\bar{\E}>0$]{}, in which [$H^s(\I)$, $s\in\RR$,]{} is the Sobolev space of functions defined on the periodic interval $\I$ [and possessing square-integrable derivatives of up to (fractional) order $s$]{} [@af05], it was then demonstrated that estimate is in fact sharp. Remarkably, the authors in [@l06; @ld08] were able to solve problem analytically in closed form (although the structure of the maximizers is quite complicated and involves special functions). When using the instantaneously optimal initial states [${\widetilde{u}_{\bar{\E}}}$]{} obtained for different values of [$\bar{\E}$]{} as the initial data $u_0$ for Burgers system with $\alpha = 1$, the maximum enstrophy growth $\max_{t\ge 0} \E(u(t)) -{\bar{\E}}$ achieved during the resulting flow evolution was proportional to ${\bar{\E}}$. The question about the [*maximum*]{} enstrophy growth achievable in finite time was investigated in [@ap11a] where the following estimate was obtained $${\max_{{t>0}} \E(u(t))\leq {\left[ \bar{\E}^\frac{1}{3}+ \frac{1}{16} {\left( \frac{1}{\pi^2\nu} \right)}^\frac{4}{3}\bar{\E} \right]}^3 \ {{\buildrel{\bar{\E} \rightarrow \infty}\over\longrightarrow\;}} \ \frac{1}{4096} {\left( \frac{1}{\pi^2\nu} \right)}^4 \bar{\E}^3.} \label{eq:Et1D}$$ To probe its sharpness, a family of variational optimization problems $$\begin{aligned} & \max\limits_{\phi\in H^1(\I)} \left[{\E(u(T))} - {\bar{\E}}\right] \\ & \text{subject to}\ \E(\phi)={\bar{\E}} \end{aligned}, \label{eq:maxdEt}$$ where $\phi$ is the initial data for the Burgers system, i.e., $u_0 = \phi$ in , was solved numerically for a broad range of initial enstrophy values [$\bar{\E}$]{} and lengths $T$ of the time window. It was found that the maximum finite-time enstrophy growth [$\max_{T>0} \max_{\phi} \E(u(T)) - \bar{\E}$]{} scales as [$\bar{\E}^{3/2}$ and these observations were later]{} rigorously justified by Pelinovsky in [@p12] using the Laplace method combined with the Cole-Hopf transformation. In a related study [@p12b], a dynamical-systems approach was used to reveal a self-similar structure of the maximizing solution in the limit of large enstrophy. This asymptotic solution was shown to have the form of a viscous shock wave superimposed on a linear rarefaction wave. In that study similar maximizing solutions were also constructed on the entire real line. [The observed dependence of the maximum finite-time growth of enstrophy $\max_{T>0} \max_{\phi} \E(u(T)) - \bar{\E}$ on $\bar{\E}$ is thus]{} significantly weaker than the maximum growth stipulated by estimate in the limit ${\bar{\E}} \rightarrow \infty$, demonstrating that this estimate is [*not*]{} sharp and may be improved (which remains an open problem). The question how the maximum finite-time growth of enstrophy in the Burgers system may be affected by additive stochastic noise was addressed in [@pp15]. Using an approach based on Monte-Carlo sampling, it was shown that the stochastic excitation does not decrease the extreme enstrophy growth, defined in a suitable probabilistic setting, as compared to what is observed in the deterministic case. The question of the extreme behavior in 2D Navier-Stokes flows was addressed in [@ap13a; @ap13b]. Since on 2D unbounded and doubly-periodic domains the enstrophy may not increase, the relevant quantity in this setting is the [*palinstrophy*]{} defined as the $L^2$ norm of the vorticity gradient. In [@ap13a] it was shown, again by numerically solving suitably defined variational maximization problems, that the available bounds on the rate of growth of palinstrophy are sharp and that, somewhat surprisingly, the corresponding maximizing vorticity fields give rise to flow evolutions which also saturate the estimates for the palinstrophy growth in finite time. Thus, paradoxically, as far as the sharpness of the finite-time estimates is concerned, the situation in 2D is more satisfactory than in 1D. The goal of the present investigation is to advance the research program outlined above by considering the extreme behavior possible in the fractional Burgers system when $\alpha \in [0,1]$. The reason why this problem is interesting from the point of view of this research program is that, as discussed in [@adv07; @kns08; @ddl09], the fractional Burgers system is globally well-posed when $\alpha \ge 1/2$ and exhibits a finite-time blow-up in the supercritical regime when $\alpha < 1/2$ (it was initially demonstrated in [@kns08] that the blow-up occurs in the Sobolev space $H^s(\I)$, $s>3/2-2\alpha$, and this results was later refined in [@ddl09] where it was shown that under certain conditions on the initial data the blow-up occurs in $W^{1,\infty}(\I)$ for all $\alpha < 1/2$; eventual regularization of solutions after blow-up was discussed in [@ccs10]). Thus, the behavior changes fundamentally when the fractional dissipation exponent $\alpha$ is reduced below $1/2$ (this aspect was also illustrated in [@a15]). Furthermore, there is also a certain similarity with the 3D Navier-Stokes system which is known to be globally well-posed in the classical sense in the presence of fractional dissipation with $\alpha \ge 5/4$ [@kp12]. Our specific objectives are therefore twofold: - first, we will obtain upper bounds on the rate of growth of enstrophy generalizing estimate to the fractional dissipation case with $\alpha \in [0,1]$ in ; this will be done separately for both the classical and fractional enstrophy, cf.  and , and - secondly, we will probe the sharpness, in the sense defined above, of these new estimates by numerically solving corresponding variational maximization problems; in this latter step we will also provide insights about the structure of the optimal states saturating different bounds. Based on this, we will conclude how the maximum instantaneous growth of enstrophy changes between the regimes with globally well-posed behavior and with finite-time blow-up. The structure of the paper is as follows: in the next section we derive upper bounds on the rate of growth of the classical and fractional enstrophy as functions of the fractional dissipation exponent $\alpha$, then in Section \[sec:method\] we provide details of the computational approach designed to probe the sharpness of these bounds, whereas in Section \[sec:results\] we present numerical results obtained for the two cases; discussion and final conclusions are deferred to Section \[sec:final\]. Upper Bounds on the Rate of Growth of the Classical and Fractional Enstrophy {#sec:bounds} ============================================================================ In this section we first use system to obtain expressions for the rate of growth of the classical and fractional enstrophy and in terms of the state variable $u$. Next, we derive estimates on these rates of growth in terms of the instantaneous enstrophy values $\E$ and ${\E_{\alpha}}$. These results are stated in the form of theorems in two subsections below. In order to obtain an expression for the growth rate $d\E/dt$ of the classical enstrophy , we multiply the fractional Burgers equation by $(-u_{xx})$, integrate the resulting relation over the periodic interval $\I$ and then perform integration by parts to obtain $$\begin{aligned} \frac{d\mathcal{E}}{dt}& =\frac{1}{2}\frac{d}{dt}\int_0^1 {|u_x|}^2\,dx \\ & = \int_0^1 u_{xx} u u_x\,dx + \nu\int_0^1 u_{xx} {(-\Delta)}^\alpha u \,dx \\ &= -\frac{1}{2}\int_0^1 u_x^3\,dx- \nu\int_0^1 {\left[ {(-\Delta)}^{\frac{\alpha}{2}} u_x\right]}^2 \,dx \\ & =: \mathcal {R}_{\mathcal {E}}(u)\,. \end{aligned} \label{eq:R}$$ Analogously, in order to obtain an expression for the growth rate $d\E_{\alpha} / dt$ of the fractional enstrophy , we multiply the fractional Burgers equation by ${(-\Delta)}^\alpha u$ and after performing similar steps as above we obtain $$\begin{aligned} \frac{d\mathcal{E}_\alpha}{dt}& = \frac{1}{2}\frac{d}{dt}\int_0^1 {\left| {(-\Delta)}^\frac{\alpha}{2}u \right|}^2\,dx \\ & = -\int_0^1 {(-\Delta)}^\alpha u\cdot u u_x \,dx - \nu\int_0^1 {\left[ {(-\Delta)}^\alpha u \right]}^2 \,dx \\ & =: \mathcal {R}_{\mathcal {E}_\alpha}(u)\,. \end{aligned} \label{eq:Ra}$$ Estimate of $d\E/dt$ {#sec:estimR} -------------------- We begin by estimating the cubic integral in which is addressed by \[R3\] For $\alpha\in (\frac{1}{6}, 1]$ and a sufficiently smooth function $u$ defined on the periodic interval $\I$, we have $${\| u \|}_{L^3(\mathcal{I})}^3 \leq C_1 \, {\| u \|}_{L^2(\mathcal{I})}^{3-\frac{1}{2\alpha}}\ {\| {(-\Delta)}^\frac{\alpha}{2} u \|}_{L^2(\mathcal{I})}^\frac{1}{2\alpha} \label{eq:R3}$$ with some constant $C_1$ depending on $\alpha$. In [@l06; @ld08] the following estimate was established $${\| u \|}_{L^3(\mathcal{I})}^3 \leq \frac{2}{\sqrt{\pi}} {\| u \|}_{L^2(\mathcal{I})}^\frac{5}{2}\,{\| u_x \|}_{L^2(\mathcal{I})}^\frac{1}{2},$$ from which it follows, upon noting that ${\| u_x \|}_{L^2(\mathcal{I})}={\| {(-\Delta)}^\frac{1}{2} u \|}_{L^2(\mathcal{I})}$, that inequality holds when $\alpha = 1$. Since $u$ is defined on the periodic interval, it has a discrete Fourier series representation $$u(x)=\sum_k \widehat{u}_k\,e^{2\pi i kx}\,, \label{eq:uhat}$$ where $k \in \NN$ is the wavenumber and $\widehat{u}_k$ the corresponding Fourier coefficient. [In the case when $\alpha>1/2$, ]{}we split the sum at $k=\kappa$, so that $$\begin{aligned} |u(x)|&=\left|\sum_k \widehat{u}_k e^{2\pi i k x}\right|\\ &\leq \sum_{|k|\leq\kappa} |\widehat{u}_k| + \sum_{|k|>\kappa} |\widehat{u}_k|\\ &=\sum_{|k|\leq\kappa} 1\cdot|\widehat{u}_k|+ \sum_{|k|>\kappa} {(2\pi |k|)}^{-\alpha} {(2\pi |k|)}^{\alpha} |\widehat{u}_k|\\ &\leq \sqrt{\sum_{|k|\leq\kappa}1^2 \sum_{|k|\leq\kappa}{|\widehat{u}_k|}^2} + \sqrt{\sum_{|k|>\kappa} {(2\pi |k|)}^{-2\alpha} \sum_{|k|>\kappa} {(2\pi |k|)}^{2\alpha}{|\widehat{u}_k|}^2}\\ & \leq {(2\kappa)}^\frac{1}{2} {\| u \|}_{L^2} + {\left(\frac{2}{2\alpha-1}\right)}^\frac{1}{2} {(2\pi)}^{-\alpha} {\kappa}^\frac{-2\alpha+1}{2} {\| {(-\Delta)}^\frac{\alpha}{2}u \|}_{L^2}\,, \end{aligned} \label{eq:usplit}$$ where $\kappa$ is a parameter to be determined, and the Plancherel theorem, Cauchy-Schwarz inequality as well as the inequality $$\sum_{|k|>\kappa} {(2\pi|k|)}^{-2\alpha}\leq 2\,{(2\pi)}^{-2\alpha}\int_\kappa^\infty {x}^{-2\alpha}\,dx = \frac{2\,{(2\pi)}^{-2\alpha}}{2\alpha-1} {\kappa}^{-2\alpha+1},\quad \alpha > \frac{1}{2}$$ were used to obtain the last inequality in . The upper bound in is minimized by choosing $\kappa={ {(2\alpha-1)}^\frac{1}{2\alpha} {\| {(-\Delta)}^\frac{\alpha}{2}u \|}_{L^2}^\frac{1}{\alpha} } / \left( 2\pi{\| u \|}_{L^2}^\frac{1}{\alpha} \right)$ which produces $${\| u \|}_{L^\infty(\mathcal{I})}\leq C\,{\| u \|}_{L^2(\mathcal{I})}^{1-\frac{1}{2\alpha}}\,{\| {(-\Delta)}^\frac{\alpha}{2}u \|}_{L^2(\mathcal{I})}^\frac{1}{2\alpha},\quad \text{where} \quad C=\frac{2\alpha}{{(2\alpha-1)}^{1-\frac{1}{4\alpha}}\sqrt{\pi}}.$$ Then, we finally obtain $${\| u \|}_{L^3(\mathcal{I})}^3 \leq {\| u \|}_{L^\infty(\mathcal{I})}\,\int_0^1 u^2\,dx \leq C \, {\| u \|}_{L^2(\mathcal{I})}^{3-\frac{1}{2\alpha}}\ {\| {(-\Delta)}^\frac{\alpha}{2}u \|}_{L^2(\mathcal{I})}^\frac{1}{2\alpha},$$ which proves that inequality holds for $1/2<\alpha<1$. When $\alpha=1/2$, we have for an arbitrary $x_0 \in \I$ $$\begin{aligned} u^2(x_0) & \leq 2\int_0^{x_0}u(x)u'(x) dx \\ & \leq 2 \int_0^1 \sum_{k,\,j} 2\pi|k|\,\widehat{u}_k\,\widehat{u}_j\, e^{2\pi i(k-j) x}\, dx \\ & \leq 4\pi\sum_k |k||\widehat{u}_k|^2 \\ & = 2{\| {(-\Delta)}^\frac{1}{4}{u} \|}_{L^2(\mathcal{I})}^2\,, \end{aligned} \label{eq:u2}$$ where the last equality is obtained by the Plancherel theorem. It then follows that the upper bound on $u$ is given by $${\| u \|}_{L^\infty(\mathcal{I})} \leq \sqrt{2} \, {\| {(-\Delta)}^\frac{1}{4}u \|}_{L^2(\mathcal{I})}$$ and we have $${\| u \|}_{L^3(\mathcal{I})}^3 \leq {\| u \|}_{L^\infty(\mathcal{I})}\,\int_0^1 u^2\,dx \leq \sqrt{2}\,{\| u \|}_{L^2(\mathcal{I})}^2\,{\| {(-\Delta)}^\frac{\alpha}{2}u \|}_{L^2(\mathcal{I})}\, .$$ When $1/6 < \alpha < 1/2$, the fractional Gagliardo-Nirenberg inequality established in [@hyz12] yields $${\| u \|}_{L^3(\mathcal{I})}^3 \leq C\,{\| u\|}_{L^2(\mathcal{I})}^{3-\frac{1}{2\alpha}}\,{\| {(-\Delta)}^\frac{\alpha}{2}u \|}_{L^2(\mathcal{I})}^\frac{1}{2\alpha}, \quad \text{where} \quad C=\frac{1}{\sqrt{2\pi}}{\left[ \frac{\Gamma(\frac{1-\alpha}{2})}{\Gamma(\frac{1+\alpha}{2})} \right]}^\frac{1}{2\alpha}\,.$$ We note that this fractional Gagliardo-Nirenberg inequality was originally obtained in [@hyz12] for an unbounded domain $\RR^d$ with some $0< d \in \NN$ and here we restrict it to the periodic interval $\I$. As a result, the constant $C$ may not be optimal. The lemma is thus proved. We remark that when $\I=\mathbb{R}$ the generalized Gagliardo-Nirenberg inequality from [@mrr13; @hmow11] can be applied to deduce the same inequality as in Lemma \[R3\], but without an explicit expression for the prefactor. We are now in the position to state \[dEdt\] For $\alpha\in (\frac{1}{4}, 1]$ the rate of growth of enstrophy is subject to the bound $$\begin{aligned} \frac{d\mathcal {E}}{dt} & \leq \sigma_1 \,\mathcal {E}^{\gamma_1}, \quad \text{where} \quad \gamma_1 = \frac{6\alpha-1}{4\alpha-1} \quad \text{and} \\ \sigma_1 &= \left\{ \begin{alignedat}{2} &\frac{4\alpha-1}{(2\alpha-1)2^\frac{2\alpha+1}{4\alpha-1}\nu^\frac{1}{4\alpha-1}\pi^\frac{2\alpha}{4\alpha-1}}, && \mathrm{for}\ \frac{1}{2}<\alpha\leq 1\,, \\ &\frac{1}{2\nu}, && \mathrm{for}\ \alpha=\frac{1}{2}\,,\\ &\frac{ 4\alpha-1 }{ 2^\frac{8\alpha+1}{4\alpha-1}\pi^\frac{2\alpha}{4\alpha-1}\alpha^\frac{4\alpha}{4\alpha-1}\nu^\frac{1}{4\alpha-1} } {\left[ \frac{\Gamma(\frac{1-\alpha}{2})}{\Gamma(\frac{1+\alpha}{2})} \right]}^\frac{2}{4\alpha-1}, & \qquad & \mathrm{for}\ \frac{1}{4}<\alpha<\frac{1}{2}\,. \end{alignedat} \right. \end{aligned} \label{eq:dEdt}$$ Applying inequality to estimate the cubic integral , we have $$\frac{d\mathcal{E}}{dt} \leq \frac{C_1}{2} {\| u_x \|}_{L^2(\mathcal{I})}^{3-\frac{1}{2\alpha}}\ {\| {(-\Delta)}^\frac{\alpha}{2} u_x \|}_{L^2(\mathcal{I})}^\frac{1}{2\alpha} - \nu\,{\| {(-\Delta)}^\frac{\alpha}{2} u_x \|}_{L^2(\mathcal{I})}^2\,, \label{eq:dEdt2}$$ where $C_1$ is defined in . Then, Young’s inequality is used to estimate the first term on the RHS of such that the second term is eliminated (we refer the reader to [@l06; @ld08] for details of this step which is analogous to the case with $\alpha = 1$). We note that this last step is valid only when $\alpha > 1/4$ and we finally obtain $$\frac{d\mathcal{E}}{dt} \leq \frac{(4\alpha-1) }{ {(8\alpha)}^\frac{4\alpha}{4\alpha-1} \nu^\frac{1}{4\alpha-1}}\,C_1^\frac{4\alpha}{4\alpha-1} \, {\| u_x \|}_{L^2(\mathcal{I})}^{\frac{2(6\alpha-1)}{4\alpha-1}},\quad \alpha>1/4$$ which is equivalent to . The theorem is thus proved. [As regards the range of applicability of estimate , we note that $\lim_{\alpha \rightarrow (1/4)^+} \gamma_1(\alpha) = \infty$, so $1/4$ represents a natural lower bound on $\alpha$, cf. Figure \[fig:estim\](b).]{} Estimate of $d\E_{\alpha}/dt$ {#sec:estimRa} ----------------------------- As above, we begin by estimating the cubic integral in which is addressed by \[R3a\] For $\alpha\in (\frac{3}{4}, 1]$ and a sufficiently smooth function $u$ defined on the periodic interval $\I$ we have $$\left| \int_0^1 {(-\Delta)}^\alpha u \cdot u u_x \, dx \right| \leq C_{\alpha} \,{\| {(-\Delta)}^\frac{\alpha}{2} u \|}_{L^2(\mathcal{I})}^{\frac{8\alpha-3}{2\alpha}}\ {\| {(-\Delta)}^\alpha u \|}_{L^2(\mathcal{I})}^\frac{3-2\alpha}{2\alpha} \label{eq:R3a}$$ with some constant $C_{\alpha}$ depending on $\alpha$. Based on the discrete Fourier series representation of $u$, we have $$\begin{aligned} |u_x(x)|&=\left|\sum_k (2\pi i k)\widehat{u}_k e^{2\pi ikx}\right|\\ &\leq \sum_{|k|\leq\kappa} 2\pi |k| |\widehat{u}_k| + \sum_{|k|>\kappa} 2\pi |k| |\widehat{u}_k|\\ &=\sum_{|k|\leq\kappa} {(2\pi|k|)}^{1-\alpha} {(2\pi |k|)}^\alpha |\widehat{u}_k| + \sum_{|k|>\kappa} {(2\pi |k|)}^{1-2\alpha} {(2\pi |k|)}^{2\alpha} |\widehat{u}_k|\\ &\leq \sqrt{\sum_{|k|\leq\kappa} {(2\pi|k|)}^{2-2\alpha} \sum_{|k|\leq\kappa} {(2\pi|k|)}^{2\alpha}{|\widehat{u}_k|}^2} + \\ &\ \ \ \ \sqrt{\sum_{|k|>\kappa} {(2\pi |k|)}^{2-4\alpha} \sum_{|k|>\kappa} {(2\pi |k|)}^{4\alpha}{|\widehat{u}_k|}^2}\\ & \leq \sqrt{2}{(2\pi)}^{1-\alpha} \kappa^{\frac{3}{2}-\alpha} {\| {(-\Delta)}^\frac{\alpha}{2} u \|}_{L^2(\mathcal{I})} + \sqrt{\frac{2}{4\alpha-3}} {(2\pi)}^{1-2\alpha} \kappa^{{\frac{3}{2}-2\alpha}} {\| {(-\Delta)}^\alpha u \|}_{L^2(\mathcal{I})}\,, \end{aligned} \label{eq:ux}$$ where $\kappa$ is a splitting parameter to be determined, and the Plancherel theorem, Cauchy-Schwarz inequality and as well as inequalities \[eq:ineq1\] $$\begin{aligned} & \sum_{|k|\leq\kappa} {(2\pi|k|)}^{2-2\alpha} \leq 2 \kappa {(2\pi \kappa)}^{2-2\alpha} = 2{(2\pi)}^{2-2\alpha}\kappa^{3-2\alpha}\,, \label{eq:ineq1a} \\ & \sum_{|k|>\kappa} {(2\pi |k|)}^{2-4\alpha} \leq 2{(2\pi)}^{2-4\alpha}\int_\kappa^\infty x^{2-4\alpha} dx = \frac{2{(2\pi)}^{2-4\alpha}}{4\alpha-3} \kappa^{3-4\alpha}, \quad \alpha>\frac{3}{4} \label{eq:ineq1b}\end{aligned}$$ were applied to obtain the last inequality in . The upper bound in is minimized by choosing $\kappa={ {(4\alpha-3)}^\frac{1}{2\alpha} {\| {(-\Delta)}^\alpha u \|}_{L^2(\mathcal{I})}^\frac{1}{\alpha} } / \allowbreak \left( 2\pi{(3-2\alpha)}^\frac{1}{\alpha}{\| {(-\Delta)}^\frac{\alpha}{2} u \|}_{L^2(\mathcal{I})}^\frac{1}{\alpha} \right)$ which yields $$\begin{aligned} { {\| u_x \|}_{L^\infty(\mathcal{I})} }& \leq C_{\alpha} \, {\| {(-\Delta)}^\frac{\alpha}{2} u \|}_{L^2(\mathcal{I})}^{\frac{4\alpha-3}{2\alpha}}\ {\| {(-\Delta)}^\alpha u \|}_{L^2(\mathcal{I})}^\frac{3-2\alpha}{2\alpha}, \quad \text{where} \\ C_{\alpha} & = \frac{ 2\alpha }{ \sqrt{\pi}{(4\alpha-3)}^\frac{6\alpha-3}{4\alpha}{(3-2\alpha)}^\frac{3-2\alpha}{2\alpha} }. \end{aligned} \label{eq:uLinf}$$ We then finally obtain $$\begin{aligned} \left| \int_0^1 {(-\Delta)}^\alpha u \cdot u u_x \, dx \right| & \leq {\| u_x \|}_{L^\infty(\mathcal{I})}\ \left| \int_0^1 {\left|{(-\Delta)}^\frac{\alpha}{2} u\right|}^2 dx \right| \\ &\leq C_{\alpha} {\| {(-\Delta)}^\frac{\alpha}{2} u \|}_{L^2(\mathcal{I})}^{\frac{8\alpha-3}{2\alpha}}\ {\| {(-\Delta)}^\alpha u \|}_{L^2(\mathcal{I})}^\frac{3-2\alpha}{2\alpha},\end{aligned}$$ which proves the lemma. We are now in the position to state \[dEadt\] For $\alpha\in (\frac{3}{4}, 1]$ the rate of growth of fractional enstrophy is subject to the bound $$\begin{aligned} \frac{d\E_{\alpha}}{dt} & \leq \sigma_{\alpha} \,\E_{\alpha}^{\gamma_{\alpha}}, \quad \text{where} \quad \gamma_{\alpha} = \frac{8\alpha-3}{6\alpha-3} \quad \text{and} \\ \sigma_{\alpha} &= \frac{ 2^\frac{4\alpha-3}{6\alpha-3}(6\alpha-3) }{ \pi^\frac{2\alpha}{6\alpha-3}\nu^\frac{3-2\alpha}{6\alpha-3}{(3-2\alpha)}^\frac{3-2\alpha}{6\alpha-3}(4\alpha-3) }. \end{aligned} \label{eq:dEadt}$$ Applying inequality to estimate the cubic integral in , we have $$\frac{d\mathcal{E}_\alpha}{dt} \leq C_{\alpha} \, {\| {(-\Delta)}^\frac{\alpha}{2} u \|}_{L^2(\mathcal{I})}^{\frac{8\alpha-3}{2\alpha}}\ {\| {(-\Delta)}^\alpha u \|}_{L^2(\mathcal{I})}^\frac{3-2\alpha}{2\alpha} - \nu {\| {(-\Delta)}^\alpha u \|}_{L^2(\mathcal{I})}^2\,, \label{eq:dEadt2}$$ where $C_{\alpha}$ is defined in . Then, Young’s inequality is used to estimate the first term on the RHS of such that the second term is eliminated and we finally obtain . The theorem is thus proved. [As regards the somewhat narrow range of applicability of estimate , we remark that it is a consequence of the limitations of the “spectral splitting” approach used to bound $|u_x(x)|$ in , cf. inequality .]{} Relation Between New Estimates and and Classical Estimate ---------------------------------------------------------- In estimates and , assuming the viscosity coefficient $\nu$ is fixed, the exponents and prefactors are functions of the fractional dissipation order $\alpha$, i.e., $\gamma_1 = \gamma_1(\alpha)$, $\sigma_1 = \sigma_1(\alpha)$, $\gamma_{\alpha} = \gamma_{\alpha}(\alpha)$ and $\sigma_{\alpha} = \sigma_{\alpha}(\alpha)$. Their dependence on $\alpha \in (1/4,1]$ and $\alpha \in (3/4,1]$, which are the respective ranges of the fractional dissipation orders for which the estimates and are valid, is shown in Figure \[fig:estim\]. It is clear that $\gamma_1(\alpha), \gamma_{\alpha}(\alpha) \rightarrow 5/3$ as $\alpha \rightarrow 1$, indicating that our upper bounds and are consistent with the original estimate obtained in [@l06; @ld08]. Analogous property holds for the prefactor $\sigma_1(\alpha)$, but not for $\sigma_{\alpha}(\alpha)$. Sharpness of these estimates will be assessed in Sections \[sec:resultsR\] and \[sec:resultsRa\]. Methodology for Probing Sharpness of Estimates and {#sec:method} =================================================== In this section we discuss the approach which will allow us to verify whether or not the upper bounds and on the rate of growth of the classical and fractional enstrophy are sharp. An estimate of the type or is considered “sharp” when the upper bound on its [right-hand side]{} can be realized by certain fields $u$ with prescribed (classical or fractional) enstrophy. In other words, in regard to estimate , if we can find a family of fields [$\tilde{u}_{\bar{\E}}$]{} such that [$\E(\tilde{u}_{\bar{\E}}) = \bar{\E}$]{} and [ $\R_{\E}(\tilde{u}_{\bar{\E}}) \rightarrow \sigma_1 \, \bar{\E}^{\gamma_1}$ ]{} when [$\bar{\E} \rightarrow \infty$]{}, then this estimate is declared sharp (analogously for estimate ). We note that, given the power-law structure of the upper bounds in and , the question of sharpness may apply independently to both the exponents $\gamma_1$ and $\gamma_{\alpha}$ as well as the prefactors $\sigma_1$ and $\sigma_{\alpha}$ (with the caveat that if the exponent is not sharp, then the question about sharpness of the corresponding prefactor becomes moot). A natural way to systematically search for fields saturating an estimate is by solving suitable variational maximization problems [@l06; @ld08; @ap11a; @ap13a] and such problems corresponding to estimates and are introduced below. Variational Formulation {#sec:variational} ----------------------- For given [$\bar{\E} > 0$]{} and [$\bar{\E}_\alpha > 0$ on the RHS]{} in estimates and , we define the respective maximizers as $$\begin{aligned} & {\widetilde{u}_{\bar{\E}}} = \operatorname*{arg\,max}\limits_{u\in {\mathcal {S}_{\bar{\E}}}} \mathcal {R}_{\mathcal {E}}(u)\,, \\ & {\mathcal {S}_{\bar{\E}}}=\left\{u\in H^{1+\alpha}(\mathcal{I}): \frac{1}{2}\int_0^1 {\left|u_x\right|}^2\,dx={\bar{\E}}\right\}\,, \end{aligned} \label{eq:maxR}$$ and $$\begin{aligned} & { \widetilde{u}_{\bar{\E}_\alpha} } = \operatorname*{arg\,max}\limits_{u\in { \mathcal {S}_{\bar{\E}_\alpha}}} \mathcal {R}_{\mathcal {E}_\alpha}(u)\,, \\ & {\mathcal {S}_{\bar{\E}_\alpha}} = \left\{ \begin{alignedat}{2} & \left\{u\in H^{2\alpha}(\mathcal{I}): \frac{1}{2}\int_0^1 {\left| {(-\Delta)}^\frac{\alpha}{2}u \right|}^2\,dx={\bar{\E}_\alpha} \right\}& \qquad & \text{for} \ \alpha>\frac{1}{2}, \\ & \left\{u\in H^1(\mathcal{I}): \frac{1}{2}\int_0^1 {\left| {(-\Delta)}^\frac{\alpha}{2}u \right|}^2\,dx={\bar{\E}_\alpha} \right\} && \text{for} \ \alpha\leq \frac{1}{2}, \end{alignedat} \right. \end{aligned} \label{eq:maxRa}$$ where “$\operatorname*{arg\,max}$” denotes the state realizing the maximum, whereas ${\mathcal {S}_{\bar{\E}}}$ and ${\mathcal {S}_{\bar{\E}_\alpha}}$ represent the constraint manifolds. The choice of the Sobolev spaces in the definitions of these manifolds is dictated by the minimum regularity of $u$ required in order to make the expressions for $\R_{\E}(u)$ and $\R_{\E_{\alpha}}(u)$, cf.  and , well defined. [While establishing rigorously the solvability of optimization problems and , especially for large values of [$\bar{\E}$]{} and [$\bar{\E}_\alpha$]{}, is a difficult task (which is outside the scope of the present study), the computational results reported in Section \[sec:results\] indicate that the maxima defined by these problems are indeed attained.]{} A numerical approach to solution of maximization problems and is presented below. Gradient-Based Solution of Problems and {#sec:grad} ---------------------------------------- To fix attention, we first focus on the solution of problem . For a fixed [$\bar{\E}$]{}, its maximizers can be computed as ${{\widetilde{u}_{\bar{\E}}}} = \lim_{n\rightarrow \infty} u^{(n)}$, where the consecutive approximations $u^{(n)}$ are defined via the following iterative procedure (in practice, only a finite number of iterations is performed) \[eq:iter\] $$\begin{aligned} u^{(n+1)} & = \mathbb{P}\mathcal {S}\left(u^{(n)} + \tau_n\nabla^{H^{1+\alpha}}\mathcal {R}_{\mathcal {E}}(u^{(n)}) \right), \qquad n=1,2,\dots, \label{eq:itera}\\ u^{(1)} & = u^0\,, \label{eq:iterb}\end{aligned}$$ in which $u^0$ is the initial guess chosen such that $u^0 \in {\mathcal {S}_{\bar{\E}}}$ [and $\int_0^1 u^0 \,dx = 0$]{}, $\nabla^{H^{1+\alpha}}\R_{\E}(u^{(n)})$ is the Sobolev gradient of $\R_{\E}(u)$ evaluated at the $n$th iteration with respect to the suitable Sobolev topology, $\tau_n$ is the corresponding length of the step and ${\mathbb{P}\mathcal {S}}\; : \; H^{1+\alpha}(\I) \rightarrow {\mathcal {S}_{\bar{\E}}}$ is the projection operator ensuring that every iterate $u^{(n)}$ is restricted to the constraint manifold ${\mathcal {S}_{\bar{\E}}}$. For simplicity of presentation, in we used the steepest-ascent approach, however, in practice one can use a more advanced maximization approach characterized by faster convergence, such as for example the conjugate gradients method [@nw00]. For the maximization problem the maximizer [$\widetilde{u}_{\bar{\E}_\alpha}$]{} can be computed with an algorithm similar to , but involving the constraint manifold [$\mathcal {S}_{\bar{\E}_\alpha}$]{}, the gradient $\nabla^{{H^{2\alpha}}}\R_{{\E_{\alpha}}}$ (or, $\nabla^{{H^{1}}}\R_{{\E_{\alpha}}}$) and the projection operator ${\mathbb{P}\mathcal {S}}_{\alpha}$, all of which are defined below. As regards evaluation of the gradients $\nabla^{H^{1+\alpha}}\R_{\E}$ and $\nabla^{{H^{2\alpha}}}\R_{{\E_{\alpha}}}$ (or, $\nabla^{{H^{1}}}\R_{{\E_{\alpha}}}$), the starting point are the Gâteaux differentials of the objective functions $\eqref{eq:R}$ and , defined as $$\mathcal {R}'_{\mathcal {E}}(u; v) := \lim_{\epsilon\rightarrow 0} \frac{\mathcal {R}_{\mathcal {E}}(u+\epsilon v)- \mathcal {R}_{\mathcal {E}}(u)}{\epsilon} \quad \text{and} \quad \mathcal {R}'_{\mathcal {E}_\alpha}(u; v) := \lim_{\epsilon\rightarrow 0} \frac{\mathcal {R}_{\mathcal {E}_\alpha}(u+\epsilon v)-\mathcal {R}_{\mathcal {E}_\alpha}(u)}{\epsilon}$$ which can be evaluated as follows \[eq:dR\] $$\begin{aligned} \mathcal {R}'_{\mathcal {E}}(u; v) & = \int_0^1 \left[3u_xu_{xx}+2\nu {(-\Delta)}^\alpha u_{xx}\right]v \, dx\,, \label{eq:dRa} \\ \mathcal {R}'_{\mathcal {E}_\alpha}(u; v) & = \int_0^1 \left[ {(-\Delta)}^\alpha u_x\cdot u - {(-\Delta)}^\alpha(u\cdot u_x)-2\nu {(-\Delta)}^{2\alpha}u \right]v \, dx\,, \label{eq:dRb}\end{aligned}$$ where integration by parts has been used to factorize the “direction” $v$. Next, recognizing that, for a fixed $u$ and when regarded as functions of the second argument $v$, the Gâteaux differentials – are bounded linear functionals on the given Sobolev spaces, we can invoke the Riesz theorem [@l69] to obtain \[eq:Riesz\] $$\begin{aligned} {3} \mathcal {R}'_{\E}(u; v) &= {\left\langle \nabla^{L^2}\mathcal {R}_{\mathcal {E}}(u), v \right\rangle}_{L^2(\mathcal{I})} & &= {\left\langle \nabla^{H^{1+\alpha}}\mathcal {R}_{\mathcal {E}}(u), v \right\rangle}_{H^{1+\alpha}(\mathcal{I})}\,,& & \label{eq:Riesza} \\ \mathcal {R}'_{{\E_{\alpha}}}(u; v) &= {\left\langle \nabla^{L^2}\mathcal {R}_{{\E_{\alpha}}}(u), v \right\rangle}_{L^2(\mathcal{I})} & &= {\left\langle \nabla^{H^{2\alpha}}\mathcal {R}_{{\E_{\alpha}}}(u), v \right\rangle}_{H^{2\alpha}(\mathcal{I})}, & & \quad \text{for} \ \alpha>\frac{1}{2}\,, \label{eq:Rieszb} \\ && &= {\left\langle \nabla^{H^{1}}\mathcal {R}_{{\E_{\alpha}}}(u), v \right\rangle}_{H^{1}(\mathcal{I})}, & & \quad \text{for} \ \alpha \le \frac{1}{2}\,, \label{eq:Rieszc}\end{aligned}$$ where the Sobolev inner products are defined as \[eq:ip\] $$\begin{aligned} {\left\langle u, v \right\rangle}_{H^{1+\alpha}(\mathcal{I})} & := \int_0^1 u\,v + \ell_1^2 u_x\,v_x + \ell_2^{2+2\alpha} {(-\Delta)}^\frac{\alpha}{2} u_x\cdot {(-\Delta)}^\frac{\alpha}{2} v_x\,dx\,, \label{eq:ipa} \\ {\left\langle u, v \right\rangle}_{H^{2\alpha}(\mathcal{I})} & := \int_0^1 u\,v + \ell_1^2 u_x\,v_x + \ell_2^{4\alpha} {(-\Delta)}^{\alpha-\frac{1}{2}} u_x\cdot {(-\Delta)}^{\alpha-\frac{1}{2}} v_x\,dx\,, \label{eq:ipb} \\ {\left\langle u, v \right\rangle}_{H^1(\mathcal{I})} & := \int_0^1 u\,v + \ell_1^1 u_x v_x \, dx\, \label{eq:ipc}\end{aligned}$$ in which $\ell_1$ and $\ell_2$ are constants with the dimension of a length-scale (the significance of the choice of these constants will be discussed below). In order to characterize the Sobolev gradients defined in the spaces $H^{1+\alpha}(\I)$, $H^{2\alpha}(\I)$ and $H^{1}(\I)$, cf. –, we first derive expressions for the $L^2$ gradients from – \[eq:gradL2\] $$\begin{aligned} \nabla^{L^2}\mathcal {R}_{\mathcal {E}}(u) & = 3u_xu_{xx}+2\nu {(-\Delta)}^\alpha u_{xx}\,, \label{eq:gradL2a} \\ \nabla^{L^2}\mathcal {R}_{\mathcal {E}_\alpha}(u) & = {(-\Delta)}^\alpha u_x\cdot u - {(-\Delta)}^\alpha(u\cdot u_x)-2\nu {(-\Delta)}^{2\alpha}u \label{eq:gradL2b}\end{aligned}$$ and then invoke Riesz relations – which, upon performing integration by parts and noting that $v$ is arbitrary, yields \[eq:gradH\] $$\begin{aligned} {2} \left[ {1} - l_1^2 \Delta - l_2^{2+2\alpha} {(-\Delta)}^\alpha \Delta \right] \nabla^{H^{1+\alpha}}\mathcal {R}_{\mathcal {E}}(u) & = \nabla^{L^2}\mathcal {R}_{\mathcal {E}}(u)\,, & \qquad & \text{on} \ \I, \label{eq:gradHa} \\ \left[ {1} - l_1^2 \Delta - l_2^{4\alpha} {(-\Delta)}^{2\alpha-1} \Delta \right] \nabla^{H^{2\alpha}}\mathcal {R}_{\mathcal {E}_\alpha}(u) & = \nabla^{L^2}\mathcal {R}_{\mathcal {E}_\alpha}(u)\,, & & \text{on} \ \I, \label{eq:gradHb} \\ \left[ {1} - l_1^2 \Delta \right] \nabla^{H^{1}}\mathcal {R}_{\mathcal {E}_\alpha}(u) & = \nabla^{L^2}\mathcal {R}_{\mathcal {E}_\alpha}(u)\,, & & \text{on} \ \I, \label{eq:gradHc} \\ & \hspace*{-3.0cm} \text{Periodic Boundary Conditions.} && \nonumber\end{aligned}$$ These boundary-value problems allow us to determine the required Sobolev gradients in terms of the $L^2$ gradients –. We now return to the question of the choice of the length-scale parameters $\ell_1$ and $\ell_2$. As is evident from the form of these expressions, for different values of $\ell_1$ and $\ell_2$ inner products – define [*equivalent*]{} norms as long as $\ell_1, \ell_2 \in (0,\infty)$. On the other hand, as demonstrated in [@pbh04], computation of Sobolev gradients by solving elliptic boundary-value problems – can be interpreted as application of spectral low-pass filters to the $L^2$ gradients with the parameters $\ell_1$ and $\ell_2$ defining the cut-off length-scales. Thus, while for sufficiently “good” initial guesses $u^0$ iterations of the type with different values of $\ell_1$ and $\ell_2$ lead to the [*same*]{} maximizer [${\widetilde{u}_{\bar{\E}}}$]{}, the actual rate of convergence usually depends very strongly on the choice of these parameters [@ap11a; @ap13a]. Since the constraints defining the manifolds in and are quadratic, the corresponding projection operators are naturally defined in terms of the following rescalings (normalizations) $$\begin{aligned} \mathbb{P}\mathcal {S}(u) & := \sqrt{ \frac{{\bar{\E}}}{ \frac{1}{2}\int_0^1 {|u_x|}^2\,dx } }\ u \,, \label{eq:PN} \\ \mathbb{P}\mathcal {S}_\alpha(u) & := \sqrt{ \frac{{\bar{\E}_\alpha}}{ \frac{1}{2}\int_0^1 {\big| {(-\Delta)}^\frac{\alpha}{2}u \big|}^2\,dx } }\ u\ \label{eq:PNa}\end{aligned}$$ [which in the language of optimization on manifolds can be interpreted as “retractions” from the tangent subspace to the constraint manifold [@ams08]. The form of expressions and is particularly simple, because the constraint manifolds ${\mathcal {S}_{\bar{\E}}}$ and ${\mathcal {S}_{\bar{\E}_\alpha}}$, cf.  and , can be regarded as “spheres” in their respective functional spaces.]{} Finally, the optimal step length $\tau_n$ in is found by solving an arc-minimization problem $$\tau_n := \operatorname*{arg\,max}\limits_{\tau>0} \left\{ \mathcal {R}_{\mathcal {E}} \left[ \mathbb{P}\mathcal {S}\left( u^{(n)}+\tau\nabla^{H^{1+\alpha}} \mathcal {R}_{\mathcal {S}}(u^{(n)}) \right) \right]\right\}\, \label{eq:taun}$$ which is an adaptation of standard line minimization [@nw00] to the case with quadratic constraints. This step is performed with a straightforward generalization of Brent’s method [@numRecipes] and an analogous approach is also used to compute the step size when solving problem . Tracing Solutions Branches via Continuation {#sec:continuation} ------------------------------------------- Families of maximizers [${\widetilde{u}_{\bar{\E}}}$]{} corresponding to a range of enstrophy values ${\bar{\E}} = \E^{(m)}$, $m=0,1,\dots$, are obtained using a continuation approach where the solution $\widetilde{u}_{\E^{(m)}}$ determined by the iteration process at some enstrophy value $\E^{(m)}$ is used as the initial guess $u^0$ for iterations at the next, slightly larger, value $\E^{(m+1)} = \E^{(m)} + \Delta\E$ for some $\Delta\E > 0$. By choosing sufficiently small steps $\Delta\E$, one can ensure that the iterations at a given enstrophy value are rapidly convergent. The same continuation approach is also used to compute the families of the maximizers [$\widetilde{u}_{\bar{\E}_\alpha}$]{}. Rates of Growth of Enstrophy for $\alpha = 0$ {#sec:a0} --------------------------------------------- To close this section, we provide some comments about the rates of growth of the classical and fractional enstrophy in the case when $\alpha = 0$. As regards the first quantity, from we see that $$\R_{\E}(u) \, \le \, \frac{1}{2} \| u_x \|_{L^3(\I)}^3 - \nu \| u_x \|_{L^2(\I)}^2 \label{eq:R0}$$ which, in view of the property $\| v \|_{L^p(\I)} \le \| v \|_{L^q(\I)}$ true for $p \le q$ [@af05], implies that for $u \in {\mathcal {S}_{\E}}$, $\R_{\E}(u)$ may not be bounded and hence the maximization problem does not have solutions when $\alpha = 0$. Concerning the rate of growth of the fractional enstrophy, from we obtain $$\frac{d\K(u)}{dt} = -\K(u), \label{eq:Ra0}$$ where $\K(u) := \E_0(u) = (1/2) \int_0^1 u^2 \, dx$ is the kinetic energy and we used the property that $\int_0^1 uuu_x\, dx = 0$. Then, the maximization problem takes the form $\max_{u, \; \K(u) = {\bar{\K}}} \left[ - \K(u)\right]$ for some ${\bar{\K}} > 0$, and it is clear that any function $u \in H^1(\I)$ such that $\K(u) = {\bar{\K}}$ is a solution. Thus, when $\alpha = 0$, the maximization problem has infinitely many solutions. Computational Results {#sec:results} ===================== In this section we present and analyze the results obtained by solving problems and for a broad range of [$\bar{\E}$]{} and [$\bar{\E}_\alpha$]{} and for different values of the fractional dissipation exponent $\alpha \in [0,1]$. We begin by describing the numerical techniques employed to discretize the approach introduced in Section \[sec:grad\] and then summarize the values of the different numerical parameters used. For a given field $u$, the gradient expressions – are evaluated using a spectrally accurate Fourier-Galerkin approach in which the nonlinear terms are computed in the physical space with dealiasing based on the $2/3$ rule [@b01]. A similar Fourier-Galerkin approach is also used to solve the boundary-value problems – for the Sobolev gradients. As will be discussed in more detail below, the maximizers [${\widetilde{u}_{\bar{\E}}}$]{} and [${\widetilde{u}_{\bar{\E}_{\alpha}}}$]{} corresponding to increasing values of, respectively, [$\bar{\E}$]{} and [$\bar{\E}_\alpha$]{} are characterized by shock-like steep fronts of decreasing thickness. Resolving these regions accurately requires numerical resolutions (given in terms of the numbers $N$ of Fourier modes) increasing with [$\bar{\E}$]{} and [$\bar{\E}_\alpha$]{}. In our computations we used the resolutions $N = 512,\dots,8388608$ which were refined adaptively based on the criterion that the Fourier coefficients corresponding to several largest resolved wavenumbers be at the level of the round-off errors, i.e., $|\hat{u}_k| \sim \O(10^{-14})$ for $k \lessapprox N$. [By carefully performing such grid refinement it was possible to assert that in all cases the computed approximations converge to well-defined maximizers.]{} In all cases considered the viscosity was set to $\nu = 0.01$. The length-scale parameters $\ell_1$ and $\ell_2$ in the definitions of the Sobolev inner products were chosen to maximize the rate of convergence of iterations for given values of $\alpha$, [$\bar{\E}$]{} or [$\bar{\E}_\alpha$]{}. The best results were obtained for $\ell_1 \in [1,10]$ and $\ell_2 \in [10^{-6},1]$ with smaller values corresponding to larger [$\bar{\E}$]{} and [$\bar{\E}_\alpha$]{}. Since the gradient-based method from Section \[sec:grad\] may find [*local*]{} maximizers only, in addition to the continuation approach described in Section \[sec:continuation\], we have also started iterations with several different initial guesses $u^0$ intended to nudge the iterations towards other possible local maximizers (typically, $u^0(x) = A \sin(2\pi k x)$ with $k=1,2,\dots$ and $A$ chosen so that $\E(u^0)$ or ${\E_{\alpha}}(u^0)$ was equal to a prescribed value). However, these attempts did not reveal any additional maximizers other than the ones already found using the continuation approach. Maximum Growth Rate of Classical Enstrophy {#sec:resultsR} ------------------------------------------ We consider solutions to maximization problem for $\alpha \in (1/4,1]$, which is the range of fractional dissipation exponents for which estimate is valid. The maximum rate of growth $\R_{\E}({{\widetilde{u}_{\bar{\E}}}})$ is shown as a function of [$\bar{\E}$]{} for different values of $\alpha$ in Figure \[fig:R\]. In this figure we also indicate the upper bounds from estimate . The corresponding maximizers [${\widetilde{u}_{\bar{\E}}}$]{} are shown both in the physical and spectral space in Figure \[fig:tuE\] for $ {\bar{\E}}=5,50,500$ and $\alpha \in [0.5,1]$. It is evident from this figure that the sharp fronts in these maximizers become steeper with increasing [$\bar{\E}$]{} and this effect is more pronounced for smaller values of $\alpha$. This aspect is further illustrated in Figure \[fig:tuE2\] where the maximizers are shown for $\alpha=0.3,0.4$ and small enstrophy values. Needless to say, accurate determination of maximizers [${\widetilde{u}_{\bar{\E}}}$]{} for such small values of the fractional dissipation exponent $\alpha$ requires a very refined numerical resolution, cf. Figure \[fig:tuE2\](d), making the optimization problem harder and more costly to solve. This also explains why for small values of $\alpha$ the data for $\R_{\E}({{\widetilde{u}_{\bar{\E}}}})$ in Figure \[fig:R\] is available only for small [$\bar{\E}$]{}. The relation $\R_{\E}({{\widetilde{u}_{\bar{\E}}}})$ versus [$\bar{\E}$]{} is characterized by certain generic properties evident for all values of $\alpha$ — while for small [$\bar{\E}$]{} the quantity $\R_{\E}({{\widetilde{u}_{\bar{\E}}}})$ exhibits a steep growth with [$\bar{\E}$]{}, for larger values of [$\bar{\E}$]{} it develops a power-law dependence on [$\bar{\E}$]{}. This behavior can be quantified by fitting the relation $\R_{\E}({{\widetilde{u}_{\bar{\E}}}})$ versus [$\bar{\E}$]{} locally with the formula ${\widetilde{\sigma}}_1 {\bar{\E}}^{{\widetilde{\gamma}}_1}$ and determining the parameters ${\widetilde{\sigma}}_1$ and ${\widetilde{\gamma}}_1$ as functions of [$\bar{\E}$]{} via a least-squares procedure. The dependence of thus determined prefactor ${\widetilde{\sigma}}_1$ and exponent ${\widetilde{\gamma}}_1$ on [$\bar{\E}$]{} is shown for different $\alpha$ in Figures \[fig:fit1\](a,c,e) and \[fig:fit1\](b,d,f), respectively. In these figures we also indicate the values of $\sigma_1$ and $\gamma_1$ obtained in estimate . We observe, that as [$\bar{\E}$]{} increases, both the prefactor and the exponent obtained via the least-squares fit approach well-defined limiting values. These limiting values are then compared against the relations $\sigma_1 = \sigma_1(\alpha)$ and $\gamma_1 = \gamma_1(\alpha)$ from estimate in Figures \[fig:sigmagamma1\](a,b). It is clear from Figure \[fig:sigmagamma1\](b) that there is a good quantitative agreement between the exponent in estimate and the computational results. On the other hand, in Figure \[fig:sigmagamma1\](a) we see that the numerically determined prefactors are smaller than the prefactor derived in estimate , although they do exhibit similar trends with $\alpha$ (except for the discontinuity of the latter at $\alpha = 1/2$). This demonstrates that exponent $\gamma_1$ in estimate is sharp, while prefactor $\sigma_1$ might be improved. We add that we also attempted to solve the maximization problem for $\alpha \in (0,1/4]$, however, we were unable to obtain converged solutions. In agreement with the discussion of the case $\alpha = 0$ in Section \[sec:a0\], this indicates that $d\E/dt$ may be unbounded for $\alpha \le 1/4$. ![Dependence of the maximum enstrophy rate of growth $\R_{\E}({{\widetilde{u}_{\bar{\E}}}})$, obtained by solving optimization problems , on [$\bar{\E}$]{} for different values of $\alpha$ (solid lines). The dashed lines represent the corresponding upper bounds from estimate .[]{data-label="fig:R"}](./Figures/Standard_Powerlaw_curve_01-eps-converted-to.pdf){width="95.00000%"} Maximum Growth Rate of Fractional Enstrophy {#sec:resultsRa} ------------------------------------------- While estimate was established for $\alpha \in (3/4,1]$, in order to obtain insights about the maximum growth rate of the fractional enstrophy for a broad range of fractional dissipation exponents, in this section we solve the maximization problems for $\alpha \in [0,1]$. The obtained maximum growth rate $\R_{{\E_{\alpha}}}({{\widetilde{u}_{\bar{\E}_{\alpha}}}})$ is shown as a function of [$\bar{\E}_\alpha$]{} in Figures \[fig:Ra\](a) and \[fig:Ra\](b) for $\alpha \in (3/4,1]$ and $\alpha \in (1/10,3/4]$, respectively. In the first figure we also indicate the predictions of estimate . For small values of $\alpha$ and [$\bar{\E}_\alpha$]{} we also observed the presence of another branch of maximizers characterized by negative values of $\R_{{\E_{\alpha}}}({{\widetilde{u}_{\bar{\E}_{\alpha}}}})$ — this data is shown in Figure \[fig:Ra2\](a) using a restricted range of [$\bar{\E}_\alpha$]{} for clarity, whereas the positive branches obtained for the same (small) values of $\alpha$ are presented in Figure \[fig:Ra2\](b). We thus see that for small $\alpha$ and [$\bar{\E}_\alpha$]{} the maximization problem admits two distinct families of local maximizers. The maximizers [${\widetilde{u}_{\bar{\E}_{\alpha}}}$]{} corresponding to the data in Figures \[fig:Ra\](a,b) are shown both in the physical and spectral space in Figure \[fig:tuEa\] for ${\bar{\E}_\alpha} = 5,50,500$ and $\alpha = 0.1,0.2,\dots,0.9$. We observe that, interestingly, as $\alpha$ decreases the sharp fronts in the maximizers disappear and are replaced with oscillations (Figures \[fig:tuEa\](a,c,e)). This behavior is also reflected in the spectra of the maximizers which become less developed as $\alpha \rightarrow 0$ (Figures \[fig:tuEa\](b,d,f)). The maximizers [${\widetilde{u}_{\bar{\E}_{\alpha}}}$]{} obtained at the same values of $\alpha$ and [$\bar{\E}_\alpha$,]{} and corresponding to the positive and negative branches of $\R_{{\E_{\alpha}}}({{\widetilde{u}_{\bar{\E}_{\alpha}}}})$, cf. Figures \[fig:Ra2\](a) and \[fig:Ra2\](b), are shown in Figure \[fig:tuEa2\]. We see that the maximizers for which $\R_{{\E_{\alpha}}}({{\widetilde{u}_{\bar{\E}_{\alpha}}}}) < 0$ have a simpler structure and for all considered values of $\alpha $ and [$\bar{\E}_\alpha$]{} are essentially indistinguishable from $A \sin(2\pi x)$ for some $A>0$. The relation $\R_{{\E_{\alpha}}}({{\widetilde{u}_{\bar{\E}_{\alpha}}}})$ versus [$\bar{\E}_\alpha$]{} (the upper branch shown in Figure \[fig:Ra\](a,b)) reveals similar properties as observed in the case of the classical enstrophy discussed in Section \[sec:resultsR\], namely, an initially steep growth followed by saturation with a power-law behavior when ${\bar{\E}_\alpha} \rightarrow \infty$. Performing local least-squares fits to these relations with the formula ${\widetilde{\sigma}}_{\alpha} {\bar{\E}_\alpha}^{{\widetilde{\gamma}}_{\alpha}}$, as described in Section \[sec:resultsR\], we can calculate how the actual prefactors ${\widetilde{\sigma}}_{\alpha}$ and exponents ${\widetilde{\gamma}}_{\alpha}$ depend on [$\bar{\E}_\alpha$]{} and these results are shown in Figure \[fig:fita\](a,b), where we have also indicated, for $\alpha \in (3/4,1]$, the predictions of estimate . We see that, as ${\bar{\E}_\alpha} \rightarrow \infty$, both ${\widetilde{\sigma}}_{\alpha}$ and ${\widetilde{\gamma}}_{\alpha}$ approach well-defined values, which are in turn shown in Figure \[fig:sigmagammaa\](a,b) as functions of $\alpha$ together with the prefactors and the exponents obtained in estimate . We see in Figure \[fig:sigmagammaa\](b) that for $0.9 \lessapprox \alpha \lessapprox 1$ the numerically obtained exponents ${\widetilde{\gamma}}_{\alpha}$ match the exponents $\gamma_{\alpha}$ from estimate and a difference appears for $0.8 \lessapprox \alpha \lessapprox 0.9$ which grows as $\alpha$ decreases. For $0.7 \lessapprox \alpha \lessapprox 0.8$ the numerically determined exponents ${\widetilde{\gamma}}_{\alpha}$ are a decreasing function of $\alpha$ which saturates at a constant value of approximately $3/2$ when $\alpha \lessapprox 0.7$. At the same time, the prefactors ${\widetilde{\sigma}}_{\alpha}$ obtained numerically are by a few orders of magnitude smaller than the prefactors predicted by estimate over the entire range of $\alpha$, although they do exhibit qualitatively similar trends with $\alpha$. For $\alpha \in [0,3/4]$, which is outside the range of validity of estimate , the numerically obtained exponents ${\widetilde{\gamma}}_{\alpha}$ are constant, indicating that, somewhat surprisingly, in this range $\R_{{\E_{\alpha}}}({{\widetilde{u}_{\bar{\E}_{\alpha}}}})$ does not depend on the fractional dissipation exponent $\alpha$. The corresponding numerically obtained prefactors ${\widetilde{\sigma}}_{\alpha}$ reveal a decreasing trend with $\alpha$. We add that these trends are accompanied by the maximizers [${\widetilde{u}_{\bar{\E}_{\alpha}}}$]{} becoming more regular as $\alpha$ decreases (cf. Figure \[fig:tuEa\]). We thus conclude that the exponent $\gamma_{\alpha}$ in estimate is sharp over a part of the range of validity of this estimate and appears to overestimate the actual rate of growth of fractional enstrophy for smaller values of $\alpha$. Over the range of $\alpha$ where the exponent $\gamma_{\alpha}$ is sharp, the prefactor $\sigma_{\alpha}$ may be improved. Summary and Discussion {#sec:final} ====================== While the estimates on the rate of growth of the classical and fractional enstrophy obtained in Theorems \[dEdt\] and \[dEadt\] are not much different from similar results already available in the literature [@kns08; @ddl09], the key finding of the present study is that these estimates are in fact sharp, in the sense that for different $\alpha$ the exponents $\gamma_1$ and $\gamma_{\alpha}$ in and capture the correct power-law dependence of the maximum growth rates $\R_{\E}({{\widetilde{u}_{\bar{\E}}}})$ and $\R_{{\E_{\alpha}}}({{\widetilde{u}_{\bar{\E}_{\alpha}}}})$ on [$\bar{\E}$]{} and [$\bar{\E}_\alpha$]{}, respectively (the second estimate was found to be sharp only over a part of the range of $\alpha$ for which it is defined). This was demonstrated by computationally solving suitably defined constrained optimization problems and then showing that the maximizers obtained under constraints on [$\E$]{} and [$\E_\alpha$]{} saturate the upper bounds in the estimates for different values of $\alpha$. Therefore, the conclusion is that the mathematical analysis on which Theorems \[dEdt\] and \[dEadt\] are based may not be fundamentally improved, other than a refinement of exponent $\gamma_{\alpha}$ in for $3/4 < \alpha \lessapprox 0.9$ and an improvement of the prefactors in and . In regard to the maximum rate of growth of the classical enstrophy, it was found that for $\alpha \rightarrow (1/4)^+$ the exponent $\gamma_1$ in estimate becomes unbounded, cf. Figure \[fig:sigmagamma1\](b), which together with the computational evidence obtained for $\alpha \in [0,1/4]$ suggests that $d\E/dt$ may be unbounded for $\alpha$ in this range. This would indicate that for $\alpha \in [0,1/4]$ system is not even locally well posed in $H^1(\I)$. Concerning the maximum rate of growth of the fractional enstrophy, a surprising result was obtained for $\alpha \in [0,3/4)$, where the exponent in the upper bound on $d{\E_{\alpha}}/dt$ was found to be independent of $\alpha$ (cf. Figure \[fig:sigmagammaa\](b)). This indicates that, unlike in the case of the classical enstrophy, in this range of $\alpha$ the problem does not become more singular with the decrease of $\alpha$ and this is in fact also reflected in the maximizers [${\widetilde{u}_{\bar{\E}_{\alpha}}}$]{} becoming more regular as $\alpha \rightarrow 0$. [In addition, this also suggests that it should be possible to obtain rigorous bounds on $d{\E_{\alpha}}/ dt$ valid for $\alpha \le 3 / 4$, although they would likely need to be derived using techniques other than those employed in the proof of Theorem \[dEadt\].]{} [It should be emphasized that although most of the individual inequalities used in the proofs of Theorems \[dEdt\] and \[dEadt\] are known to be sharp, the fact that the upper bounds in and were found to be sharp as well is not trivial. This is because, in general, these individual inequalities may be saturated by [*different*]{} fields which may belong to different function spaces and hence it is not obvious whether sharpness is preserved when these inequalities are “chained” together to form estimates and . ]{} On the methodological side, it ought to be emphasized that gradient-based iterations may only identify [ *local*]{} maximizers and in general it is not possible to ascertain whether these maximizers are also global. However, our careful search based on the continuation approach (cf. Section \[sec:continuation\]) and, independently, using several different initial guesses $u^0$ did not reveal any additional maximizers (other than the maximizers obtained via a trivial rescaling of the solutions as discussed in detail in [@ap11a]). An exception to this was the solution of the maximization problem for small $\alpha$ and [$\bar{\E}_\alpha$]{} where a branch of maximizers such that $\R_{{\E_{\alpha}}}({{\widetilde{u}_{\bar{\E}_{\alpha}}}}) < 0$ was also found. The presence of this additional branch appears related to the degenerate nature of the maximization problem which for $\alpha = 0$ has an uncountable infinity of trivial solutions (cf. Section \[sec:a0\]). As regards the research program discussed in Introduction, the key finding of the present study is that exponents $\gamma_1$ in the upper bound on $d\E/dt$ have the same dependence on $\alpha$ and remain sharp in the subcritical, critical and parts of the supercritical regime. Thus, the loss of global well-posedness as $\alpha$ is reduced to values below $1/2$ cannot be detected based on the instantaneous rate of growth of enstrophy $d\E/dt$. 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--- author: - Carlo Heissenberg title: 'Asymptotic Symmetries of Yang-Mills, Gravity, Two-Forms and Higher-Spins' --- {width=".3\textwidth"} XXXII Ciclo Introduction {#introduction .unnumbered} ============ How much physical information can be extracted from local symmetries? This question surely bears relevance to our understanding of theoretical physics, due to the prominent examples of well-established physical theories taking the form of gauge field theories: Maxwell’s electrodynamics and general relativity at the classical level, the standard model of elementary particles at the quantum level. By their very nature, local gauge transformations leave all observable quantities invariant, so that they can be defined only by introducing nonobservable fields, namely gauge fields acting as force mediators and charged matter fields, that transform nontrivially under the local gauge group. The Hamiltonian and the equations of motion are invariant under such transformations. Therefore, due to the intrinsic redundancy introduced in the dynamical problem by the gauge symmetry, the time evolution of gauge fields is not well defined, already at the classical level, unless one introduces a gauge fixing condition that breaks local gauge invariance to a residual subset, which may or may not coincide with the set of global transformations depending on the strength of such a condition. This picture may thus superficially leave the impression that local gauge transformations are only a convenient artifact that permits to write down interesting theories, but which does not carry any intrinsic physical relevance. On the contrary, the physical meaning of the global counterparts of local gauge symmetries is rather well-understood. They give rise to very important conserved observable quantities, at the classical level: the electric charge $Q$ for electrodynamics, the color charge $Q^A$ for Yang-Mills theory, energy-momentum $P_\mu$ and angular momentum $M_{\mu\nu}$ for general relativity in the case of asymptotically flat spacetimes. In quantum theories, unbroken global symmetries arising from local gauge symmetries induce a decomposition of the Hilbert space into superselection sectors and allow one to derive selection rules in scattering processes. However, motivated by recently-established connections with observable phenomena, in particular with soft factorization theorems for scattering amplitudes and with memory effects, a larger class of local transformations appears worthy of attention as far as direct physical information is concerned: asymptotic symmetries. Such symmetries were originally introduced in the sixties by Bondi Metzner, Van der Burg and Sachs [@BMS; @Sachs_Waves; @Sachs_Symmetries] in the context of general relativity, within the study of asymptotically flat spacetimes, namely of the solutions of the Einstein equations that describe weakly radiating mass distributions—for instance, a planet orbiting around the sun, a black-hole merger or a pair of coalescing neutron stars. As such, these systems possess a metric tensor that tends to the flat one as one follows the outflowing gravitational waves and goes “very far” from the sources, thus approaching null infinity. More precisely, after specifying properly-defined coordinates, the components of the metric tensor approach those of the Minkowski one as the radial coordinate $r$ tends to infinity for fixed retarded time $u$, up to corrections that become negligibly small in this limit. The delicate point in the definition of asymptotically flat spacetimes, and the important result of the above authors, is precisely the identification of the proper assignment of “falloff conditions” on these corrections, specifying how a nontrivial geometry becomes Minkowski as $r\to\infty$; in four spacetime dimensions, this falloff is $\mathcal O(1/r)$ for the normalized corrections to the angular components. Asymptotic symmetries of asymptotically flat spacetimes arise as those diffeomorphisms that map the set of asymptotically flat spacetimes to itself and transform a given solution of the Einstein equations to a physically inequivalent one. Familiar examples of these symmetries are given by the ordinary translations, rotations and boosts, namely the isometries of flat spacetime, which indeed preserve (asymptotic) flatness, while still altering physically relevant quantities, *e.g.* the angular momentum and the energy-momentum. In fact, in the sixties, it was expected that Poincaré symmetries could be uniquely selected as the asymptotic symmetries of asymptotically flat spacetimes. However, this is not the case. In four spacetime dimensions, one cannot adopt falloff conditions so stringent as to reduce the asymptotic symmetries of gravity to the Poincaré group without giving up on gravitational waves, and the asymptotic symmetries of gravity take the form of an infinite-dimensional enhancement of Poincaré, termed $BMS$ group. It was later realized that the concept of asymptotic symmetry can be in principle extended to any gauge theory, in the following way. One first specifies, after fixing the gauge (the analog of choosing suitable coordinates), a physically relevant set of solutions to the equations of motion by assigning falloff conditions on the gauge fields, *i.e.* by specifying the manner in which they reduce to a trivial (flat) configuration as one goes very far from matter sources. Asymptotic symmetries are then defined as those local symmetries that preserve the set of solutions under consideration and that map a given solution to a different, inequivalent one. A prominent feature of these symmetries is that they often comprise an infinite-dimensional enhancement of standard global symmetries [@Strominger_YM; @Einstein-YM.Barnich; @Strominger_QED; @StromingerKac], again in analogy with the gravitational setup. The concept of asymptotic symmetry appears thus to be both quite flexible, as it may find applicability in many interesting theories, and very rich, at least from a formal point of view. Actually, as we anticipated, the interest in asymptotic symmetries is not merely formal, since these symmetries have been shown to imply observable consequences: memory effects in classical gauge theories and soft theorems in scattering amplitudes. Mainly thanks to the works of Strominger and collaborators, the interplay among these three features of gauge theories, sometimes termed the *infrared triangle*, triggered a significant trend of original research, the main aspects of which are reviewed in [@Strominger_rev]. More specifically, it has been observed that the passage of a wave packet near a test charge can leave a permanent observable imprint on the physical properties of the probe, *i.e.* a memory effect, that can be understood in terms of the underlying action of an asymptotic symmetry. The detection of a memory effect is understood as a concrete manifestation of the fact that the underlying gauge field underwent a vacuum transition: the passage of radiation induced a sharp transformation from a given radiative vacuum to an inequivalent vacuum, connected to the previous one by an asymptotic symmetry. In this respect, one can clearly see that asymptotic symmetries are akin to *spontaneously broken* global symmetries. The first instance of memory effect again dates back to studies in general relativity, in particular [@Zeldovich; @ChristodoulouMem], where memory was identified in the form of a permanent relative displacement induced by the passage of gravitational radiation on a system of detectors. The underlying connection with $BMS$ symmetry was realized only much later in [@memory]. An electromagnetic analog of gravitational wave memory was instead put forward in [@Bieri:2013hqa], where it occurs as a velocity kick, and the corresponding asymptotic-symmetry interpretation followed in [@MagicaSabrina]. Another interesting type of memory arising in the electromagnetic theory is the one encoded in the phases of suitably-placed superconducting nodes [@Susskind], while a Yang-Mills counterpart of this phase memory is given by color memory [@StromingerColor], which occurs as a color rotation in the Hilbert space of two test quarks. Many other types of memories have been discussed, see *e.g.* [@Mao_em; @Mao_note; @Hamada:2017atr; @Afshar:2018sbq; @Chumemory], and occur both in linearized theories, with massive or massless background sources, and in nonlinear theories. More precisely, memory effects induced by radiation emitted by massive sources are identified as *linear* or *ordinary* and can be regarded as a picture of the movements of bulk sources that is stored in the properties of faraway probes, while memory effects associated to massless sources or nonlinear wave-like perturbations give rise to the so-called *nonlinear* or *null* memory. At the level of scattering amplitudes, asymptotic symmetries bear relevance in connection with factorization results that are valid when an external massless particle is evaluated in the low-energy regime: soft theorems. These identities have been remarked to be equivalent to semiclassical Ward identities stemming from asymptotic symmetries, so that soft theorems can be actually seen as an expression of the invariance of the $S$ matrix under these infinite-dimensional asymptotic symmetries. More specifically, Weinberg’s soft photon and soft graviton theorems [@Weinberg_64; @Weinberg_65] have been shown to be equivalent to the invariance of the $S$ matrix under asymptotic $U(1)$ symmetries [@Strominger_QED; @Campiglia] and under $BMS$ symmetries [@Strominger_Invariance; @Strominger_Weinberg], respectively. This link has been extended to many different contexts and has been generalized to encompass subleading corrections, in the soft frequency, to Weinberg’s result [@soft-subleading; @Bern:2014vva; @soft_QED_Strominger; @Sen:2017nim; @LaddhaSen:2017ygw]. Moreover, the presence of these symmetries appears to be at the basis of the universality of certain soft theorems, whose validity does not depend on the specific interactions under consideration but rather only on the gauge symmetry of the theory. Compatibly with the spontaneous breaking of asymptotic symmetries, soft photons and gravitons have been interpreted as the massless bosons whose existence is then ensured by the Goldstone theorem. Since they unveil a new aspect of the infrared physics associated to gauge mediators, asymptotic symmetries also gave rise to a resurgence of interest in the definition of asymptotic states in QED, gravity and QCD [@Sever; @PerryStates; @HiraiSugiDressed; @Tristan] and in the infrared problem, which lies at the heart of a possible nonperturbative definition of the $S$ matrix that does not rely on inclusive processes (see also [@Strocchi-Erice] for a detailed review). Asymptotic symmetries possess another piece of appeal, not unrelated to the previous aspects which we have touched upon, in relation with the so-called black-hole information paradox. It has been observed that a proper description of black-hole solutions may require taking into account additional symmetries, *i.e.* asymptotic symmetries acting at null infinity and near the horizon, and hence additional conserved charges [@Barnich-Brandt] compared to the one at the basis of the proof of the no-hair theorem. However, the presence of additional soft charges associated to asymptotic symmetries in itself may be insufficient to the purpose of fully resolving the information paradox [@Perry; @PorratiMir; @PerrySuperrot; @PorratiWig; @StromingerInfo; @PerryEntropy]. Up to now, we have been considering the case of gauge theories defined in four spacetime dimensions. Indeed, the attention of the literature focused at first on the arena of four-dimensional (asymptotically) Minkowski spacetime, where certain issues concerning the asymptotic behavior of the fields and the calculation of asymptotic charges simplify, and attempts to carry out the analysis in arbitrary dimensions have faced a number of difficulties since their inception. Indeed, considering for the moment the gravitational case, a first aspect is that, in dimensions $D>4$, it is indeed possible to impose falloff conditions, termed radiation falloffs, that select Poincaré as the asymptotic symmetry group of asymptotically flat spacetimes, without spoiling the description of gravitational waves [@gravity_evenD_1; @Hollands_SI; @angular-momentum; @No_ST]. Thus, infinite-dimensional asymptotic symmetries of higher-dimensional gravity appear to be less fundamental than in $D=4$ as they can be safely trivialized without losing interesting solutions. Actually, from the point of view of general relativity itself, it may appear rather natural to do so, also because radiation falloffs are instrumental in ensuring the finiteness of energy fluxes and of other physical quantities as $r\to\infty$ , while being compatible with the finiteness of Poincaré charges. Quantitatively, the radiation falloff for the angular components of the normalized metric fluctuation is $\mathcal O(1/r^{\frac{D-2}{2}})$. These features are shared by electromagnetism, where, for a time, it appeared impossible to retrieve an infinite-dimensional asymptotic symmetry group in any spacetime dimension, without derogating from the falloff conditions that ensure the finiteness of fluxes/charges at infinity. A different approach was then put forward, for the case of even-dimensional spacetimes, in [@alternative-bnd; @StromingerQEDevenD], where such falloff conditions were relaxed, from $\mathcal O(1/r^{\frac{D-2}{2}})$ to $\mathcal O(1/r)$, while at the same time introducing constraints on gauge-invariant quantities that ensure the finiteness of physical observables, in order to allow for the presence of infinite-dimensional asymptotic symmetries. Indeed, the absence of the latter in higher dimensions would be at odds with the validity of soft theorems, which hold independently of the dimension of spacetime. Therefore, from this point of view, it appears more natural to adopt (if possible) falloff conditions that do not select standard global symmetries only, *even if* this can be in principle achieved without renouncing the description of radiation. Recent achievements on the connection between soft theorems and asymptotic symmetries in spacetimes of arbitrary dimensions, either even or odd, have been presented in [@He-Mitra-photond+2; @He-Mitra-gauged+2; @He-Mitra-magneticd+2] for the case of Maxwell and Yang-Mills theories, by analyzing the asymptotic behavior of field strengths rather than the gauge fields themselves, while the canonical realization of the asymptotic symmetries for the Maxwell theory in any dimension was given in [@HenneauxQEDanyD]. Another partially unsatisfactory feature of the early investigations of the asymptotic structure of gauge theories in higher dimensions was the purported absence of memory effects [@AltroWald; @BMS-memory; @Garfinkle:2017fre; @WaldOddD]. The elusive nature of memory in higher dimensions, which was later elucidated in [@Mao_EvenD; @StromingerGravmemevD; @SatishchandranWald] in the case of even-dimensional spacetimes, can be traced back to the fact that, despite being imprints left by the passage of radiation, memory effects do not scale asymptotically in the same manner as radiation itself, $\mathcal O(1/r^{\frac{D-2}{2}})$. Instead, they possess a much faster falloff, $\mathcal O(1/r^{D-3})$, subleading in $1/r$ for $D>4$, which is associated to stationary solutions and, in general, to static or “DC” effects; this is also the asymptotic behavior corresponding to fields giving rise to finite and nontrivial global charges and is therefore termed Coulombic falloff. The interpretation of the gravitational memory effect experienced by geodesic detectors in terms of an underlying symmetry acting at large $r$ was in particular clarified in [@StromingerGravmemevD]. Contrary to the four-dimensional case, the corresponding symmetry responsible for the vacuum transition underwent by the gravitational field in higher, even dimensions is not a true asymptotic symmetry, since in particular it does not give rise to a nonvanishing soft charge; nevertheless, it has been noted that a tight link exists between this effect and the corresponding soft theorem [@Mao_EvenD]. Difficulties associated to the discussion of memory effects in odd dimensions, instead, can be essentially ascribed to the phenomenon of dispersion that characterizes the propagation of waves in those spacetimes, *i.e.* the failure of the Huygens principle, due to which an idealized point-like perturbation gives rise to disturbances that persist after the passage of the first wavefront. This peculiarity causes a blurring of ordinary memory effects in odd dimensions, as also noted in [@SatishchandranWald], but may be resolved, as we shall see, by analyzing null memory, where null sources effectively reach the region near the probe and bypass the problem introduced by dispersion phenomena. An interesting feature of the higher-dimensional context, as already exhibited by the example of memory effects, is thus the interplay between radiation and Coulombic terms, $\mathcal O(1/r^{\frac{D-2}{2}})$ and $\mathcal O(1/r^{D-3})$ respectively, which indeed become separated into two different orders of $1/r$ only for $D>4$ and whose identification would be therefore precluded in a purely four-dimensional approach. Another issue pertaining to this interplay is related to the possibility of defining nonzero and finite asymptotic charges in the presence of radiation. Indeed, asymptotic charges are generically finite as $r\to\infty$ only when sourced by Coulombic fields, while in the presence of radiation fields, which are leading in $1/r$, they exhibit seemingly divergent contributions. This problem is particularly interesting in the case of asymptotic symmetries that do not reduce to the standard global symmetries and in nonlinear theories. In these cases, no general results prevent the charges from diverging in the limit $r\to\infty$ in the presence of radiation contributions. Furthermore, waves in nonlinear theories are actually charged under the global group (for instance they carry energy-momentum in general relativity); therefore, they will induce a leak of global charges, as they reach null infinity, and a dependence of the latter on retarded time $u$. As we have seen, the analysis of asymptotic symmetries, charges and soft theorems focused at first on the case of four spacetime dimensions, but a lot of mileage has been covered since then in the context of higher dimensions. The extension of this program to the case of spins greater than two is instead a much less-explored subject. However, this research direction is worth pursuing essentially for the very same reason that motivates the higher-dimensional investigations: soft theorems are valid not only for an arbitrary dimension but also for an arbitrary spin. Therefore one may wonder whether or not they can be understood in terms of underlying asymptotic symmetries also in the case of higher-spin theories. Another physical motivation for the extension to higher spins is provided by the role that such symmetries play in the understanding of asymptotic states and of the infrared problem in lower-spin gauge theories. In view of both the impossibility of long-range higher-spin forces already implied by Weinberg’s results, and of the established lore against interacting higher-spin theories in flat space [@BBS], one may envision that unraveling the asymptotic structure of higher-spin theories could lead to some clarifications on their infrared structure. The relevance of the role played by the infrared limit in connection with the puzzling issues concerning higher-spin interactions is also clearly indicated by the two main frameworks where the most direct obstructions are seemingly evaded. On the one hand, as originally proposed by Fradkin and Vasiliev in their seminal paper [@Vasi], a nonvanishing, negative cosmological constant can indeed act as an infrared regulator for some of the most immediate singularities met by massless higher spins in interaction with gravity on flat backgrounds. This suggestion paved the way to a complete on-shell description of gauge-invariant, nonlinear higher-spin dynamics, encoded in the Vasiliev equations, up to the more recent developments in the context of AdS/CFT (see *e.g.* [@BCIValgebras; @HOLO; @Did]). On the other hand, another context in which higher-spin interactions can be naturally formulated is string theory, where higher spins are lifted to being massive, the string tension playing the role in this context of the needed infrared regulator (see *e.g.* [@GSW]). A long-standing conjecture concerns the interpretation of massive higher spins as the manifestation of some underlying higher-spin gauge symmetry breaking mechanism [@Gross; @Sagnotti], and one may speculate that a deeper understanding of the infrared physics of higher-spin interactions should be ultimately related to this picture. Let us also mention that an underlying driving force for the study of asymptotic symmetries on asymptotically flat spacetimes in any dimension and for gauge theories of any spin is the possibility to find a suitable generalization of the AdS/CFT duality to the case of Minkowski spacetime, the so-called *flat space holography*, which could be exploited to gain insight into the structure of bulk theories, such as quantum gravity or higher spins, by studying their dual boundary theory at null infinity. At the gravitational level in four dimensions this program is currently being pursued [@FSHolo1; @FSHolo2; @FSHolo3; @FSHolo4] and the infinitesimal symmetries of the boundary theory take the form of a semidirect sum of supertranslations, an infinite-dimensional enhancement of Poincaré translations occurring within the $BMS$ group, and superrotations, Virasoro symmetries that locally generalize ordinary rotations and boosts [@Barnich_Revisited; @Barnich_BMS/CFT; @Barnich_Superrotations; @Strominger-semiVir; @StromingerKac]. Results {#results .unnumbered} ------- In this Ph.D. thesis we aim, first, to review the main aspects of the above-described connection between asymptotic symmetries and observable effects in the context of gravity, electromagnetism and Yang-Mills theory in four dimensions and, second, to present original results concerning the extension of this program to the case of arbitrary-dimensional spacetimes and to higher-spin gauge theories. Motivated by the appeal possessed by memory effects in connection with the asymptotic structure of gauge theories, we present a collection of results concerning memory effects in higher-dimensional scalar, electromagnetic and non-Abelian theories and on their interpretation in terms of residual symmetries acting at large distances in even dimensions [@Memory-io]. In this context, we derive explicit formulas for kick memory effects due to the interaction of test particles with massless scalar fields and electromagnetic fields emitted by specific background sources in any dimensions, either even or odd; we observe that null memory can be employed as a tool for obtaining nontrivial memory kicks in odd spacetime dimensions. We then analyze the significance that these memory effects bear with respect to the residual symmetries of Maxwell and Yang-Mills theories in the Lorenz gauge, including a discussion of (Abelian) phase memory and (non-Abelian) color memory, in arbitrary even dimensions. We obtain an interpretation of spin-one memories in terms of underlying symmetries acting at large $r$, which, as in the case of gravitational memories, act at Coulombic order and do not comprise *bona fide* asymptotic symmetries for even $D>4$ because their charge tends to zero as $r\to\infty$. Concerning the discussion of asymptotic observables at null infinity, we provide an explicit evaluation of the global color charge, energy flux and color flux for Yang-Mills theory in any spacetime dimension [@Cariche-io]. We also propose a strategy for the definition of surface charges associated to infinite-dimensional asymptotic symmetries at null infinity that overcomes the above-mentioned difficulties [@Memory-io]. The main idea of the proposal is to evaluate these charges by taking the limit $r\to\infty$ in a region where radiation is absent, so that no divergence may arise in this step, and then define their evolution with respect to retarded time $u$ by means of the asymptotic expansion of the equations of motion in powers of $1/r$, even in the presence of radiation. Another issue which we address is the following seeming paradox arising in the case of a massless scalar force mediator. Theories containing a massless scalar have been shown to possess nontrivial asymptotic charges, which are linked with soft scalar factorization theorems [@CampigliaCoitoMizera; @Campiglia_scalars]. However, no gauge symmetry is obviously present in this context, and hence no asymptotic symmetry is available in order to explain the presence of such charges. As a possible resolution of this purported contradiction, in the case of four dimensions, we propose a matching with the asymptotic analysis of a two-form theory, to which a free on-shell scalar is linked by a duality transformation [@Campigliascoop; @twoform-io]. Finally, we discuss higher-spin asymptotic symmetries, whose analysis was initiated in [@carlo_tesi]. A first relevant feature is that, in four spacetime dimensions, these symmetries turn out to comprise an infinite-dimensional family and to underlie Weinberg’s leading soft theorem for higher-spin soft emissions [@super-io]. As in the case of lower spins, the imposition of radiation falloffs, while instrumental to the purpose of ensuring the finiteness of fluxes at infinity, actually trivializes the asymptotic symmetries in dimensions $D>4$ by reducing them to the exact Killing tensors [@Cariche-io]. The realization of infinite-dimensional asymptotic symmetries is indeed possible, but it requires to relax these falloffs to match the corresponding four-dimensional ones in any $D$, while at the same time ensuring the finiteness of relevant physical observables by means of suitable constraints involving gauge-invariant quantities. A possible way to carry out this program is offered by the following strategy. One may first consider the solution space characterized by radiation falloffs, whose physical consistency is under control, and then act upon this space by means of gauge transformations that leave the form of the equations of motion invariant, and hence map solutions to solutions, and have finite charges, but that do not preserve the falloff conditions. In this way, one generates a wider solution space, different from the previous one, that by construction exhibits infinite-dimensional asymptotic symmetries, while still ensuring the finiteness of observables [@highspinsrel-io]. A promising aspect of the higher-spin asymptotic symmetries uncovered in our approach is that they display a structure closely resembling an infinite-dimensional enhancement of (a suitable quotient of) the enveloping algebra of Poincaré, which one may interpret as the possible remnant of an underlying higher-spin algebra [@flat-algebras_1; @flat-algebras_2]. Indeed, focusing on spin three, we find three families of symmetries corresponding to (traceless projections of symmetrized) products of two Poincaré generators, namely generalized supertranslations associated to $P_{(\mu}P_{\nu)}$ and superrotations, $M_{\mu(\nu}M_{\rho)\sigma}$, together with a third family corresponding to $P_{(\mu}M_{\nu)\rho}$. More concrete steps towards this identification could be performed by discussing the non-Abelian deformation of our higher-spin asymptotic symmetries in the presence of cubic vertices, which could lead to an understanding of their commutation relations. ### Structure of this thesis {#structure-of-this-thesis .unnumbered} The material is organized in two parts, the first being devoted to the analysis of asymptotic symmetries and of their implications in spin-one theories, Maxwell and Yang-Mills, and spin-two theories, linearized gravity and general relativity, while the second part deals with extensions to the case of more exotic spins: the scalar and higher spins, $s\ge3$. We begin by reviewing the setup in which asymptotic symmetries first made their appearance, *i.e.* the study of four-dimensional asymptotically flat spacetimes, in Chapter \[chap:basics\], while also providing an overview of the calculation of the associated charges and furnishing a brief account of the analogous discussions in the case of Maxwell and Yang-Mills theory. Chapter \[chap:obs\] offers then a review of the connection between the symmetries highlighted in the previous chapter and observable effects in four dimensions: Weinberg’s soft photon and soft graviton theorem, and various types of memory effects. The remaining three chapters contain instead the original material of this thesis. In Chapter \[chap:spin1higherD\], we present the extension of the calculation of asymptotic charges and of the discussion of memory effects in connection with residual symmetries to the case of spin-one theories in higher spacetime dimensions. Chapter \[chap:scalar2form\] deals with the puzzling case of scalar asymptotic charges in four dimensions, which are tightly linked with scalar soft theorems but seemingly lack an underlying gauge symmetry. Finally, Chapter \[chap:HSP\] collects the results on the asymptotic structure of higher-spin theories in four-dimensional spacetime and a discussion of their generalization to higher dimensions. [$\ast$ $\ast$ $\ast$]{} The research work I carried out during my Ph.D., which is reviewed in this thesis, appeared in the following papers [@super-io; @Cariche-io; @proceedings-io; @twoform-io; @Memory-io]. I was additionally involved in different projects, whose content is not detailed in the present thesis, mainly focusing on foundational aspects of spontaneous symmetry breaking in quantum systems: - C. Heissenberg and F. Strocchi, “[Gauge invariance and symmetry breaking by topology and energy gap]{}”, *Mathematics* **3** (2015) 984–1000, arXiv: [1511.01757 \[math-ph\]](https://arxiv.org/abs/1511.01757). - C. Heissenberg and F. Strocchi, “[Existence of quantum time crystals]{}”, 2016, arXiv: [1605.04188 \[quant-ph\]](https://arxiv.org/abs/1605.04188). - C. M. Bender and C. Heissenberg, “[Convergent and Divergent Series in Physics]{}”, *22th Saalburg Summer School on Foundations and New Methods in Theoretical Physics (2016), Wolfersdorf, Germany*, 2017, arXiv: [1703.05164 \[math-ph\]](https://arxiv.org/abs/1703.05164). - C. Heissenberg and F. Strocchi, “[Generalized criteria of symmetry breaking. A strategy for quantum time crystals]{}”, 2019, arXiv: [1906.12293 \[cond-mat.stat-mech\]](https://arxiv.org/abs/1906.12293). In particular, in \[\], we discuss a mechanism of spontaneous symmetry breaking in which the topology of the algebra of observables allows for the presence of a gap in the energy spectrum, thus evading the main consequence of Goldstone’s theorem. The main aim of \[, \] is instead to propose a generalized criterion of spontaneous symmetry breaking that does not rely on the properties of the ground state, thereby allowing for the breaking of time translation symmetry, together with the discussion of simple models with time-independent Hamiltonian which realize this feature. The study of such systems was motivated by a pioneering proposal, due to Wilczek [@Wilczek], of quantum time crystals, namely putative systems exhibiting the spontaneous breaking of continuous time translation symmetry down to a discrete subgroup, in analogy with the situation occurring in ordinary space crystals. Asymptotic Symmetries of Lower Spins in Four Dimensions {#chap:basics} ======================================================= An interesting class of gravitational systems is the one characterized by the presence of an ideally isolated or weakly radiating mass distribution: for instance a black hole, a system of planets orbiting around a star or a pair of coalescing neutron stars. Intuitively, as one goes very far from the sources under consideration, the gravitational field induced by their presence becomes very weak, *i.e.* the underlying spacetime metric reduces to the flat, Minkowski metric. A precise description for this class of systems is provided by the definition of asymptotically flat spacetimes, first investigated in the sixties by Bondi, Metzner, van der Burg and Sachs [@BMS; @Sachs_Waves], and later reformulated in a more geometrical fashion by Penrose and Carter [@Penrose1; @Penrose2; @Carter]. A natural question that arises in this context concerns the characterization of those diffeomorphisms that leave the set of asymptotically flat spacetimes invariant, *i.e.* that map any isolated or weakly radiating system to another one. These transformations are, by definition, the asymptotic symmetries of asymptotically flat spacetimes and form the $BMS$ group, which comprises an infinite-dimensional enhancement of the Poincaré group. As is usually the case when dealing with symmetry transformations, one can then wonder whether asymptotic symmetries give rise to nontrivial conserved quantities, *i.e.* asymptotic charges. In this chapter we begin by reviewing in detail the characterization of asymptotically flat spacetimes and the structure of the $BMS$ group in Section \[sec:asymptflat\]. Section \[sec:charges\] then offers an overview of the problem of associating conserved charges to asymptotic symmetries. Although asymptotic symmetries were first introduced in the gravitational context, the problem of characterizing residual local gauge transformations that preserve assigned asymptotics is actually well-posed in any gauge theory. Indeed, as we will detail in Sections \[sec:EMD=4\] and \[sec:YMD=4\], this asymptotic analysis already furnishes nontrivial results in spin-one gauge theories, namely Maxwell and Yang-Mills. While the main focus of the discussion in this chapter is on the case of four spacetime dimensions, we will also include comments and explicit calculations concerning its higher-dimensional generalization, also in view of Chapters \[chap:spin1higherD\] and \[chap:HSP\]. Asymptotically Flat Spacetimes {#sec:asymptflat} ------------------------------ In any four-dimensional spacetime, it is always possible [@Sachs_Waves] to define locally a retarded time $u$, a radial coordinate $r$ and two angular coordinates $x^i=(\theta, \phi)$, with $i=1$, $2$, such that the spacetime metric can be written in the following form \[Sachsform\] ds\^2 = e\^[2]{}(du\^2-2dudr)+ r\^2l\_[ij]{}(dx\^i -U\^i du)(dx\^j-U\^j du), where $\beta$, $V$ and $U^i$ are independent spacetime functions, while $l_{ij}$ is subject to $\sqrt{\mathrm{det}(l_{ij})}=\sin\theta$. In total, the expression , which as we shall see can be reached independently of the Einstein equations, is specified by six functions, so that the definition of the coordinates $u$, $r$, $\theta$ and $\phi$, discussed below, has the effect of eliminating four redundant functions that are present in a generic metric, *i.e.* of performing a gauge-fixing. For instance, in the case of Minkowski space is achieved globally by letting $t=u+r$ and $\mathbf x = r\,\mathbf n$, where $\mathbf n(\theta,\phi)=(\sin\theta\cos\phi, \sin\theta\sin\phi,\cos\theta)$ is the standard parametrization of the points on the unit sphere, in terms of which $ds^2 = -du^2-2du\,dr+r^2 (d\theta^2+\sin^2\theta d\phi^2)$. Namely, $\beta=0=\mathcal U^i$ and $V=-r$, while $l_{ij}=\gamma_{ij}$ is the metric on the Euclidean two-sphere. To see that can be reached in a generic situation, consider a spacetime with a given metric $g_{ab}$ in some local coordinates $x^a$. There exists locally a function $u(x)$ such that its gradient $k_a=\partial_a u$ satisfies $k^ak_a=0$, *i.e.* the hypersurfaces of constant $u$ are null. The integral curves of $k^a$, called *rays*, are also geodesic, since $\nabla_a k_b=\nabla_a\nabla_b u=\nabla_b \nabla_a u=\nabla_b k_a$ and hence $ k^b \nabla_b k^a = k^b \nabla^a k_b = \frac{1}{2}\nabla^a(k^b k_b)=0 $. Introduce functions $\theta(x)$ and $\phi(x)$ by the condition that they be constant along rays, namely $k^a\partial_a\theta=0$ and $k^a\partial_a\phi=0$. Choosing $u$, $\theta$ and $\phi$, together with a fourth function $r$, yet to be determined, as coordinates, the components of the inverse metric satisfy $g^{uu}=k^ak_a=0$, $g^{u\theta}=k^a\partial_a\theta=0$ and $g^{u\phi}=k^a\partial_a\phi=0$, in such coordinates, and hence the determinant of the inverse metric reads \[determinanturtf\] -(g\^[ur]{})\^2\[g\^g\^-(g\^)\^2\]= -(g\^[ur]{})\^2(g\^[ij]{}). Since the the Jacobian of the transformation $x^a=x^a(u,r,\theta,\phi)$ vanishes if and only if is zero, we require $\mathrm{det}(g^{ij})\neq0$ together with $g^{ur}\neq0$. We can then define $r$ by setting $\sqrt{\mathrm{det}(g^{ij})}=(r^2\sin\theta)^{-1}$. Adopting the notation \[inverseSachs\] g\^[ab]{}=( 0 & g\^[ur]{} & 0\ g\^[ur]{} & g\^[rr]{} & g\^[ri]{}\ 0 & g\^[ri]{} & g\^[ij]{} ) =( 0 & -e\^[-2]{} & 0\ -e\^[-2]{} & -e\^[-2]{} & -U\^je\^[-2]{}\ 0 & -U\^ie\^[-2]{} & l\^[ij]{} ), one can invert $g^{ab}$ and solve for the metric components $g_{ab}$ obtaining , where $l_{ij}$ denotes the inverse of $l^{ij}$. To summarize, the above coordinates are characterized by the following properties: - the hypersurfaces of constant $u$ are tangent to the local light-cone and the integral curves of $k^a = g^{ab}\partial_bu$ are null geodesics (light rays); - the coordinates $\theta$ and $\phi$ are constant along each ray and can be interpreted as optical angles, - the radial coordinate $r$ is the luminosity distance, which means that a spherical wavefront $u=\ $constant has a total measure of $4\pi r^2$, in view of the condition $\sqrt{\mathrm{det}(l_{ij})}=\sin\theta$, and hence its luminosity (or total power) $L$ is related to its mean flux (or intensity) $F$ by $L =4\pi r^2 F$. Such coordinates are well-suited to describing gravitational radiation emitted by sources localized within a region of total radius $r_0$. A faraway detector placed at a fixed distance $r>r_0$ will characterize the properties of outflowing radiation by specifying at what (retarded) time $u$ it receives a signal and the direction $\theta$, $\phi$ from which it arrived. Although the discussion of the present chapter will mostly focus on the case of four spacetime dimensions, let us note that the above definition of retarded coordinates can be straightforwardly extended to arbitrary dimension $D$. This can be achieved by specifying $D-2$ angular coordinates $x^i$ with $i=1$, $2$, $\ldots$ , $D-2$, which generalize the ordinary azimuthal and polar angles, $\theta$ and $\phi$. This leads to provided one adopts the constraint $ \sqrt{\mathrm{det}(l_{ij})}=\sqrt{\mathrm{det}(\gamma_{ij})} $, implicitly providing the definition of the luminosity-distance $r$, where $\gamma_{ij}$ is the metric on the Euclidean $(D-2)$-unit sphere. Again quoting the example of flat space, letting \[retBondicoord\] t=u+r, x\^I=r n\^I, for $I=1,2,\ldots,D-1$, where $\mathbf n$ is a parametrization of the unit sphere in terms of the angular coordinates $x^i$, obtains \[MetricMinkD\] ds\^2 = -du\^2 - 2du dr + r\^2 \_[ij]{} dx\^idx\^j, with $\gamma_{ij}=\partial_i\mathbf n \cdot \partial_j \mathbf n$ the Euclidean metric on the sphere. For future reference, we also list the Christoffel symbols for the flat connection in this coordinate system (more details are provided in Appendix \[app:Laplaciano\]) \[christoffel\] =,=r \_[ij]{}=-,= \^[kl]{}(\_i \_[lj]{}+\_j \_[il]{}-\_l \_[ij]{} ) . In a series of pioneering papers [@BMS; @Sachs_Waves] (see *e.g.* [@Barnich_BMS/CFT] for a more recent approach), employing these coordinates, the authors have studied the space of solutions to Einstein’s equations that describes physical situations of the type under scrutiny, namely ideally isolated or weakly radiating gravitational systems, in four dimensions. The characterization of this type of solutions requires a specification of the manner in which the gravitational field becomes weak as one follows the outflowing radiation, *i.e.* $r\to\infty$ for $u$ fixed, as one approaches so-called *future null infinity*. In other words one must clarify *how* $g_{ab}$ given by reduces to the Minkowski metric $ds^2 = -du^2-2du\,dr+r^2 \gamma_{ij}\,dx^idx^j$ in this limit and the spacetime becomes asymptotically flat. The most restrictive way of assigning the *falloff conditions* characterizing asymptotically flat spacetimes that still allows for the presence of gravitational radiation is given, in four dimensions, by \[BMSFalloffs\] &=-1++O(r\^[-2]{}),\ &=+O(r\^[-3]{}),\ l\_[ij]{}&=\_[ij]{}++O(r\^[-2]{}),\ U\^i &= +O(r\^[-3]{}), where $m_B$, $\bar\beta$, $U^i$, $C_{ij}$ are functions of $u$, $\theta$ and $\phi$, and in addition $\gamma^{ij}C_{ij}=0$ in view of the requirement that $\sqrt{\mathrm{det}(l_{ij})}=\sin\theta$. The function $m_B(u,\theta,\phi)$ is called the Bondi mass aspect and, as we shall see later, is related to the notion of energy at a given retarded time. The definition of the angular momentum requires the discussion of a function appearing to subleading order in , the angular momentum aspect, and is here omitted for simplicity. In particular, the Einstein equations are satisfied to leading order provided that \[Bondieq\] U\^i = -D\_jC\^[ij]{},\_[u]{}m\_B = - N\_[ij]{}N\^[ij]{}+D\_iD\_jN\^[ij]{}, |= - C\_[ij]{}C\^[ij]{}, where $N_{ij}=\partial_u C_{ij}$ is called the Bondi news tensor and the angular indices are raised using $\gamma^{ij}$. This identifies the two independent components of the traceless tensor $C_{ij}$ as the free radiative boundary data, namely as the two independent graviton polarizations that propagate to infinity. This interpretation is further strengthened by the fact that the time dependence of $C_{ij}$ gives rise to the Bondi news and hence to energy flux carried by radiation. Indeed, a linearized perturbation on Minkowski spacetime, $h_{ab}=g_{ab}-\eta_{ab}$, satisfying the above falloffs carries a nontrivial energy flux across a sphere of retarded time $u$ and radius $r$, proportional to N\_[ij]{}N\^[ij]{} d, where $d\Omega$ is the measure element on the Euclidean unit sphere. This result can be obtained by considering the hypersurface of constant $r$ and retarded time smaller than $u$, integrating $T_{ab}n^a t^b$ on such hypersurface (where $T_{ab}$ is the stress-energy tensor for such perturbation, $n_a=\partial_a r$ is its unit normal and $t^b=(\partial_u)^b$ is the background Killing vector associated to energy) and taking the derivative with respect to $u$. Consider now a hypersurface of fixed radius $r$. Such a manifold is parametrized by $u$ and the two angles on the sphere, and can be pictured as a set of static measuring devices covering the surface of a sphere at a fixed radius $r$. Intuitively, as $r$ becomes large, the detectors are able to measure the properties of outgoing radiation very far away from the sources, thus approaching the future null “boundary” of the spacetime. This notion of a boundary, which at this level is not precise, can be made rigorous and covariant by means of a geometric definition of *future null infinity*, $\mathscr I^+$, due to Penrose and Carter (see [@Geroch_Lectures; @Ashtekar_Lectures] and [@Wald Chapter 11] for excellent reviews; a handy account is also available in [@carlo_tesi]), involving a conformal mapping that brings infinity to a finite distance. To illustrate the idea behind this construction, let us consider Minkowski spacetime in $D$ dimensions, with metric . One defines the coordinate $\Omega= 1/r$, in a region $r>r_0$, so that ds\^2 = -du\^2 + (2dud+\_[ij]{}dx\^i dx\^j), and rescales the metric by letting $d\tilde s^2 = \Omega^2 ds^2$, namely ds\^2 = - \^2 du\^2 + 2dud+\_[ij]{}dx\^i dx\^j. Future null infinity $\mathscr I^+$ is then defined as the surface $\Omega = 0$ after the conformal rescaling, which can be therefore parametrized by $u, x^i$, has a degenerate metric $ 0\cdot du^2 + \gamma_{ij}\,dx^idx^j\,, $ and represents the surface where all null trajectories of Minkowski space have their future endpoints. Following to this strategy one can in fact *define* a four-dimensional spacetime $M$ to be asymptotically flat at null infinity if, roughly speaking, there exists a conformal isometry of $M$ to a spacetime $\tilde M$ whose boundary “resembles” Minkowskian $\mathscr I^+$ (we refer again to [@Geroch_Lectures; @Ashtekar_Lectures; @Wald] for a more precise and complete discussion). It can be shown that this definition reduces to the one we have given above in terms of falloff conditions , provided one adopts a suitable set of retarded coordinates. Let us also mention that, while it has been successfully extended to spacetimes with arbitrary even dimensions [@Hollands_SI], this geometric, covariant definition of null infinity is actually precluded in the case of odd-dimensional spacetimes [@Wald_NO], in the presence of gravitational radiation, while the less formal approach adopted for instance in [@No_ST], based on taking the limit $r\to\infty$ in the given coordinate system, appears to be more flexible in this respect, although not covariant, and better suited to highlighting the similarities between gravity and other gauge theories of lower or higher spin, when it comes to discussing asymptotic symmetries. In the following, we will rely on the latter noncovariant notion of null infinity, still denoted $\mathscr I^+$, which suffices for many practical applications. According to the intuitive picture, future null infinity is therefore a hypersurface parametrized by $u$, and $x^i$ whose sections at constant $u$ are obtained by considering a sphere $S_{u,r}$ at fixed retarded time and radius $r$ and sending $r\to\infty$. The induced metric thereon is $- du^2+r^2 \gamma_{ij}dx^idx^j$ while the volume element is $r^{D-2}du\,d\Omega$, where $d\Omega$ is the measure on the unit sphere, with the limit $r\to\infty$ to be taken at the very end of the calculations of physical quantities. ### The BMS group Asymptotic symmetries are defined as the large diffeomorphisms that preserve and , and hence map the space of asymptotically flat spacetimes to itself. More precisely, they are those diffeomorphisms that respect the coordinate conditions and the falloff conditions ensuring asymptotic flatness, while still acting nontrivially on the leading field components $C_{ij}$, $U^i$, $m_B$. Furthermore, two given asymptotic symmetries should be identified if their action only differs by a small diffeomorphism or, more formally, if they induce the same transformation on $\mathscr I^+$. Anticipating a bit the discussion of charges, we may state that large diffeomorphisms are characterized by the fact that they possess a finite and nonzero charge, while small ones are those that give rise to a vanishing charge. Such symmetries are determined by solving the asymptotic Killing equation, namely imposing that the infinitesimal transformations \_g\_[ab]{}=L\_g\_[ab]{}, respect the falloffs. In our case, we can start by imposing that $g_{rr}=g_{ri}=0$ and $g^{ij}g_{ij}=0$, which follow from , namely $\mathcal L_\xi g_{rr}=\mathcal L_\xi g_{ri}=\mathcal L_\xi (g_{ij}g^{ij})=0$, which explicitly read \[akeprimetre\] g\_[ar]{}\_r\^[a]{}&=0\ g\_[ar]{}\_i \^[a]{}+g\_[ai]{}\_r\^a&=0\ g\^[ij]{}(\^a\_a g\_[ij]{}+g\_[a(i]{}\_[j)]{}\^a)&=0, where round brackets denote symmetrization of the indices. From the first equation, we read off $\partial_r\xi^u=0$, namely \[xiu\] \^u=f(u,,). The second equation in reads $ \partial_r\xi^i = e^{2\beta} g^{ij}\partial_j f $ so that \[xii\] \^i = - \_j f \_r\^e\^[2]{} g\^[ij]{}dr’ + Y\^i, with $Y^i=Y^{i}(u,\theta, \phi)$. Using now the identities $g^{ij}\partial_r g_{ij}=4r^{-1}$, $g^{ij}\partial_ug_{ij}=0$ and $g^{ij}\partial_k g_{ij}=\gamma^{ij}\partial_k\gamma_{ij}$, which follow from the condition $\mathrm{det}(g_{ij})=r^4 \mathrm{det}(\gamma_{ij})$, the third equation in gives \[xir\] \^r= U\^i \_i\^u-D\_i\^i, where $D_i$ is the covariant derivative on the unit sphere associated with the Euclidean metric $\gamma_{ij}$. Substituting the falloff conditions into , and we have \^u &= f(u,,),\ \^r &= - D\_iY\^i(u,,)+f(u,,)+O(r\^[-1]{}),\ \^i &= Y\^i(u,,)- D\^i f(u,,)+O(r\^[-2]{}), where $\Delta=D^iD_i$ and $D^i=\gamma^{ij}D_j$. Imposing that $\mathcal L_\xi g_{ur}=\mathcal O(r^{-2})$ and $\mathcal L_\xi g_{ui}=\mathcal O(1)$ gives \_u f = D\_i Y\^i,\_u Y\^i = 0, so that $Y^i(u,\theta,\phi)=Y^i(\theta,\phi)$ is actually a vector on the sphere, while \^u = T(,)+D\_i Y\^i(, )+O(r\^[-1]{}). Furthermore, $\mathcal L_\xi g_{ij}=\mathcal O(r)$ implies that $Y^i$ satisfies the conformal Killing equation on the two-sphere \[confKillingeq2\] D\_[(i]{}Y\_[j)]{}= \_[ij]{} DY, where the dot denotes contraction with $\gamma_{ij}$. Finally, the last remaining equation $\mathcal L_{\xi}g_{uu}=\mathcal O(r^{-1})$ yields the constraint \[consequenceofKilling\] (+2)DY =0. The latter equation is actually identically true, since the Ricci tensor satisfies $R_{ij}=\gamma_{ij}$ on the Euclidean two-sphere and hence, taking two divergences of , we have $\Delta D\cdot Y=D^i D^j(D_i Y_j + D_j Y_i)=2\Delta D\cdot Y + 2 D\cdot Y$. To summarize, asymptotic symmetries of four-dimensional asymptotically flat spacetimes are given by \[BMSor\] \^u&=T(, )+ DY(, ),\ \^r&=T(, )-DY(, ) + O(r\^[-1]{}),\ \^i &= Y\^i(, )-D\^i T(, )-D\^iDY(, )+ O(r\^[-2]{}), where $T(\theta, \phi)$ is an arbitrary function of the angles, while $Y^{i}$ satisfies the conformal Killing equation on the Euclidean unit sphere. Since the spacetime becomes asymptotically Minkowski, it is expectable that such a set of asymptotic symmetries contain the ordinary isometries of flat space: translations, rotations and boosts. This is indeed the case. For instance, the exact Killing vector associated to a translation in Minkowski space, $a^\mu= (a^0, \mathbf a)$ in Cartesian coordinates $x^\mu=(x^0,\mathbf x^I)=(t, \mathbf x)$ with $I=1$, $2$, $3$, reads in covariant retarded components $a_u=a_0=-a^0$, $a_r = a_0 + \mathbf n \cdot \mathbf a$, $a_i=r \partial_i\mathbf n \cdot \mathbf a$. Switching to contravariant components a\^u = a\^0-n a,a\^r = n a,a\^i = \^in a, so that, by comparison with , \[translations\] T= a\^0-n a,Y\^i = 0, once we recall that, on the Euclidean two-sphere (see Appendix \[app:Laplaciano\]), $\Delta \mathbf n = -2\mathbf n$. For an infinitesimal rotation/boost given by $l_\mu = \omega_{\mu\nu}x^\nu$ in Cartesian coordinates, with $\omega_{\mu\nu}=-\omega_{\nu\mu}$, we have $l_u=l_0=\omega_{0I}r\, n^I$ $l_r= n^{I}\omega_{I0}u$, $l_i=r \partial_i n^{I}\omega_{I0}(u+r)+r^2\partial_i n^I \omega_{IJ} n^J$, so l\^u=u n\^[I]{}\_[0I]{},l\^r=-(u+r) n\^[I]{}\_[0I]{},l\^i=-\^i n\^[I]{}\_[0I]{}+\^i n\^I \_[IJ]{} n\^J-\^i n\^[I]{}\_[0I]{}, where we recognize \[rotations/boosts\] T=0,Y\^i=-\^i n\^[I]{}\_[0I]{}+\^i n\^I \_[IJ]{} n\^J. To check that $Y^i$ is indeed a conformal Killing vector for the Euclidean two-sphere, one need recall that (see Appendix \[app:Laplaciano\]) $D_{i}D_{j}\mathbf n=-\gamma_{ij}\mathbf n$. We have checked that the set asymptotic symmetries contains the Poincaré transformations. However, it is in fact much bigger. First, it contains indeed transformations parametrized by an arbitrary function $T(\theta,\phi)$, which give an infinite-dimensional analog of translations , termed *supertranslations*. Second, one may note that, locally, the conformal Killing equation admits two infinite-dimensional families of solutions, generalizing the ordinary rotations and boosts , called *superrotations* [@Barnich_BMS/CFT; @Barnich_Revisited; @Barnich_Superrotations]. These are conveniently described by introducing the stereographic coordinates \[stereo\] z=e\^[i]{}(/2),|z=e\^[-i]{}(/2), in terms of which the Euclidean metric on the sphere takes the form $ds^2=2\gamma_{z\bar z}dz d\bar z$, with $\gamma_{z\bar z}=2/(1+z\bar z)^2$, and the conformal Killing equation reads \[holoantiholo\] \_z Y\^[|z]{}=0,\_[|z]{} Y\^z=0, so that for any holomorphic (resp. antiholomorphic) function $F(z)$ (resp. $\tilde F(\bar z)$), local solutions are given by $Y^z(z,\bar z)=F(z)$ and $Y^{\bar z}(z,\bar z)=\tilde F(\bar z)$. Globally well-defined solutions are selected by requiring that the Laurent expansion F(z)=\_[nZ]{} c\_n z\^n be nonsingular near $z=0$ and $w=\frac{1}{z}=0$, and similarly for $\tilde F(\bar z)$. This leaves us with Y\^z=c\_0 + c\_1 z + c\_2 z\^2,Y\^[|z]{}=c\_0 + c\_1 |z + c\_2 |z\^2, which can be recast in the form by noting that $\mathbf n=(z+\bar z, -i(z- \bar z), z\bar z-1)/(1+z\bar z)$, with 2c\_0&=\_[01]{}+i\_[02]{}-(\_[13]{}-i\_[23]{}),\ c\_1&=\_[03]{}+i\_[12]{},\ 2c\_2&=-(\_[01]{}+i\_[02]{})+\_[13]{}-i\_[23]{} and $\tilde c_0$, $\tilde c_1$, $\tilde c_2$ given by complex conjugation, and correspond to those infinitesimal superrotations that can be exponentiated to global rotations and boosts. Indeed, the algebra of globally well-defined conformal transformations of the Euclidean sphere is isomorphic to the Lorentz algebra $so(3,1)$. The analogous statement for the associated groups is that the group of global conformal transformations, isomorphic to $SL(2,\mathbb C)/\mathbb Z_2$, is also isomorphic to the proper, orthochronous Lorentz group $SO(3,1)$. The asymptotic symmetries induced on $\mathscr I^+$ can be obtained by projecting on a surface of fixed $r$[^1] and taking the limit $r\to\infty$, for fixed $u$, and are given by the vectors \[BMSonScri\] \^u=T+ DY, \^i= Y\^i. Such vectors are closed under Lie bracket and give rise to the following algebra: \^u = Y\_[\[1]{}D T\_[2\]]{}+ T\_[\[1]{}DY\_[2\]]{}+D,\^i = \[Y\_1, Y\_2\]\^i, so that, schematically, \[commBMS\] \[(T\_1, Y\_1),(T\_2, Y\_2)\]=(Y\_[\[1]{}D T\_[2\]]{}+ T\_[\[1]{}DY\_[2\]]{} , \[Y\_1, Y\_2\]). The vector fields , parametrized by angular functions $T$ and by conformal Killing vectors $Y^i$ on the sphere, together with the commutation relations define the Bondi-Metzner-Sachs ($BMS$) algebra. A key structural feature of this algebra is the existence of an infinite-dimensional Abelian subalgebra given by supertranslations $\xi = T(\theta, \phi)\partial_u$, which is in fact a Lie ideal since by the commutator of a supertranslation with any transformation is again a supertranslation. At the local level, the subalgebra given by superrotations is also infinite-dimensional and is in fact given by the direct sum of two Virasoro algebras (one for the holomorphic sector and one for the antiholomorphic sector ). Globally well-defined superrotations , as we have seen, instead give rise to the Lorentz algebra. At the group level, supertranslations give rise to an infinite-dimensional Abelian normal subgroup $ST$ of the full $BMS$ group. Supertranslations contain ordinary translations as a four-dimensional subgroup and enter the full asymptotic symmetry group as follows: BMS = ST SO(3,1). Denoting by $x^i=x^i(x')$ a global conformal transformation, in stereographic coordinates $x^1=z$, $x^2=\bar z$, z = where $\alpha$, $\beta$, $\gamma$ and $\delta$ are complex numbers satisfying $\alpha\delta-\beta\gamma=1$, such that \_[kl]{}=F(x’)\^2\_[ij]{}, we can express a global $BMS$ transformation as [@Schmidt_Walker_Sommers] u=u’F(x’)-T(x’),x\^i=x\^i(x’). As a final remark, let us mention that, while the translation subgroup can be characterized as the unique four-dimensional normal subgroup of the $BMS$ group, the homogeneous Lorentz transformations are not similarly unique, since any two Lorentz subgroups of $BMS$ that differ by a supertranslations are isomorphic [@Sachs_Symmetries]. ### A linearized approach From the perspective of linearized gravity (*i.e.* of a generic spin-2 massless field) the form for the metric tensor and the falloffs are equivalent to the following choice of boundary conditions: to leading order in $1/r$, \[definizione\_Bondi\_gaugerip\] h\_[ab]{}dx\^a dx\^b = du\^2 - 2U\_i du dx\^i+ r C\_[ij]{} dx\^i dx\^j+, with $\gamma^{ij}C_{ij}=0$. In particular $h_{rr}=h_{ri}=0$, while $h_{ur}=\mathcal O(r^{-2})$ and \[Bondigaugelin\] h\_[uu]{}=+O(r\^[-2]{}),h\_[ui]{}=-U\_i+O(r\^[-1]{}),h\_[ij]{}=r C\_[ij]{}+O(1). In stereographic coordinates, the trace condition takes a particularly simple form: $\gamma^{ij}h_{ij}=2h_{z\bar z}/\gamma_{z\bar z}=0$, so $h_{z\bar z}=0$. We would like to recover the $BMS$ algebra from the linearized gauge transformations $\delta_\xi h_{ab}=\nabla_{(a}\xi_{b)}$, where $\nabla_a$ denotes the background (flat) connection (see ), that preserve . Actually, we shall do so while keeping the dimension $D$ of the spacetime formally arbitrary, as this only requires a minimal modification of the calculation. From $\nabla_{(r}\xi_{r)}=0$ we obtain $\partial_r \xi_r = 0$, hence \[xiruindep\] \_r = \_r(u, x\^i), while from $\nabla_{(u}\xi_{r)}=\mathcal O(r^{-2})$ we have $\partial_r \xi_u + \partial_u \xi_r =\mathcal O(r^{-2})$ and thus \[ruuur\] \_r \_u \_u + \_u\^2 \_r =O(r\^[-2]{}). However, $\nabla_{(u}\xi_{u)}=\mathcal O(r^{-1})$ also requires $\partial_u \xi_u = \mathcal O(r^{-1})$ and hence \[uxiu\] \_r\_u \_u = O(r\^[-2]{}), which, together with , implies \^2\_u \_r = O(r\^[-2]{}). Since $\xi_r$ is actually $r$-independent , we then have \_r = - T(x\^i) - u F(x\^i). Integrating the equation $\partial_r \xi_u + \partial_u \xi_r=\mathcal O(r^{-2})$ with respect $r$ we get \_u = r F(x\^i)-S(x\^i) + O(r\^[-1]{}), where $S$ does depend on $u$, by $\partial_u\xi_u=\mathcal O(r^{-1})$. Then the equation $\nabla_{(r}\xi_{i)}=0$ takes the form \_r \_i + \_i \_r - \_i =0 and substituting the above solutions reads \_r \_i- \_i - D\_i T - u D\_i F = 0. Looking for a solution in the form of a power series in $r$, one readily sees that \_i = -r D\_i T - ru D\_i F + r\^2 Y\_i(u,z,|z). for some vector field $Y^i(u,x^j)$ (and $Y_i = \gamma_{ij}Y^j$). From the equation $\nabla_{(u}\xi_{i)}=\mathcal O(1)$, one immediately obtains \_u Y\_i = 0, so $Y_i = Y_i(x^j)$. Now we are left with the conditions $\nabla_{(i}\xi_{j)}=\mathcal O(r)$ and $\gamma^{ij}\nabla_i\xi_j=0$. The traceless projection of the former gives \[Killingeq\] D\_[(i]{}Y\_[j)]{}-\_[ij]{}DY=0, which is the conformal Killing equation, while the latter instead implies \[Ddotxi\] \^[ij]{}D\_i\_j - (D-2)r(\_u-\_r)=0. Up to now we have \_r &= -T - u F\ \_u &= r F-S+O(r\^[-1]{})\ \_i &= -r D\_iT - ur D\_iF + r\^2 Y\_i, where $T$, $S$ and $F$ are arbitrary functions on the sphere, while $Y^i$ is a conformal Killing vector. Upon substituting into , we get r\[-T + (D-2)(S-T)\]-ur(+D-2)F+r\^2\[DY-(D-2)F\]=O(1). This equation can be satisfied only if the coefficient of each independent monomial $r$, $r^2$ and $ur$ is zero: this requires S = (+D-2) T ,F = DY, while the last remaining condition, $(\Delta+D-2)D\cdot Y=0$, is identically satisfied as a consequence of the conformal Killing equation, as can be verified by taking two divergences of and recalling $[D_i, D_j]v^k=R\indices{^k_{lij}} v^l$, with $R_{ijkl}=\gamma_{ik}\gamma_{jl}-\gamma_{il}\gamma_{jk}$ (see also eq. ). To sum up, the residual gauge freedom is parametrized by \_r &= - T - DY,\ \_u &= DY - (+D-2)T + O(r\^[-1]{}),\ \_i &= - r D\_i T - D\_i DY + r\^2 Y\_i. Equivalently, in contravariant components \[BMSlinonScri\] \^u &= T + DY ,\ \^r &= T - DY + O(r\^[-1]{}),\ \^i &= -D\^iT + Y\^i - D\^i DY. In particular, these vector fields induce the same asymptotic symmetries on $\mathscr I^+$, in the relevant case $D=4$, as in (compare also with ). Let us note that, as far as the above calculation of the asymptotic symmetries in the linearized theory is concerned, it would not have been too restrictive to also impose the condition $h_{ur}=0$. With this choice, equation would simply hold sharply and not up to $\mathcal O(r^{-1})$. One might be tempted to set it to zero or conclude that it can be set to zero by means of a small gauge transformation, and indeed inspection of the equations of motion implies that the leading $\mathcal O(r^{-2})$ component $\bar\beta$ of $h_{ur}$ is actually zero in the linearized theory in four dimensions. Although the linearized theory suffices for the purpose of establishing the falloff conditions and deriving the asymptotic symmetries, by its very nature it is insensitive to self-interaction effects, such as the energy flux to null infinity, which are instead captured by the nonlinear description as we shall see explicitly in the next section. Bondi Energy and Charges {#sec:charges} ------------------------ Having introduced the concept of asymptotic symmetry in an asymptotically flat spacetime, it is natural to ask whether it is possible to find conserved quantities associated to these symmetries. This question, and more generally the subtle issue of defining conserved charges in general relativity and gauge theories, has received an extensive attention in the literature [@Barnich-Brandt; @Barnich_Charge; @Ashtekar_Lectures; @Ashtekar-Streubel], also in connection with black hole entropy [@Iyer:1994ys] and in the context of covariant phase space methods [@Wald-Zoupas]. Here, we will follow an approach based on the application of Noether’s theorem [@Avery_Schwab], commenting on the subtleties that arise in this context along the way. ### The Noether two-form One can in principle associate a conserved charge to [any]{} infinitesimal diffeomorphism according to the Noether theorem [@Avery_Schwab]. Our starting point is the Einstein-Hilbert action on a generic $D$-dimensional spacetime $M$, S = \_M R , where $k^2_D=(D-2)\Omega_{D-2}G$, with $\Omega_{D-2}$ the area of the Euclidean $(D-2)$-sphere (*i.e.* the solid angle in $D-1$ space dimensions), and $\omega$ denotes the spacetime volume form, in local coordinates $\omega=\sqrt{-g}\,dx^0\wedge dx^1 \wedge \cdots \wedge dx^{D-1}$. This action is obviously invariant under the symmetry variation $\delta_\xi g_{ab}=\nabla_{(a}\xi_{b)}$ for any vector $\xi$, since \[EHinv\] \_(R)=L\_(R) = d(R \_)=\_a (R\^a). On the other hand, under a generic variation of the (inverse) metric, we have \[Palatiniinv\] (R) &=(R\_[ab]{}-g\_[ab]{}R)g\^[ab]{}+\_a\^a,\ \^a &= \_bg\^[ab]{}-g\_[cd]{}\^ag\^[cd]{}, with $\theta^a$ the Palatini surface term. Comparing and , we obtain the current j\^a\_= (\^a\_- \^a R), which is conserved on shell, $\partial_a j^a_\xi=0$. This can be further simplified by rewriting \^a\_&= \_b\_g\^[ab]{}-g\_[cd]{}\^a\_g\^[cd]{}\ &=-\_b\^[(a]{}\^[b)]{}+2\^a\ &=\_b \^[\[b]{}\^[a\]]{} + 2 R\^[ab]{}\_b, so that j\_\^a = and, employing again the Einstein equations in the vacuum, \[Komarform\] j\_\^a=\_b \_\^[ab]{},\_\^[ab]{}= \^[\[b]{}\^[a\]]{}, where one explicitly sees that, in accordance with Noether’s second theorem, the conserved current associated to a local symmetry is equal, on shell, to the divergence of an antisymmetric rank-two tensor. The continuity equation $\partial_aj_\xi^a=0$ becomes then trivial. Equation also holds if one takes into account the introduction of the Gibbons-Hawking-York term needed in order to make the Einstein-Hilbert variational problem well-posed: S = \_M R + \_[M]{} K |, where $\bar\omega$ is the induced volume form on $\partial M$ and $K$ is the mean extrinsic curvature thereon (see [@Wald Appendix E]). Indeed, the introduction of the boundary term modifies the Lagrangian only by a total derivative (or an exact $D$-form) $R\,\omega\mapsto R\,\omega+d\alpha$ and hence does not alter the Noether current: \_(R+d)&=d\_(R+d)=\_a(R\^a)+d\_d,\ (R+d)&=( R\_[ab]{}-g\_[ab]{}R)g\^[ab]{}+\_a\^a+d, but $d\delta_\xi\alpha=d\mathcal L_\xi\alpha= d\imath_\xi d\alpha$, by Cartan’s formula, and hence the contribution due to $d\alpha$ cancels out in the calculation of $j_\xi^a$. The expression is also correct if one adds to the Einstein-Hilbert action a cosmological constant term. For a scalar field $\phi$ coupled to gravity, S = \_M , a calculation similar to the one presented above yields \_\^[ab]{}=, which reduces to for minimal coupling $\lambda=0$ (the case $\lambda= \tfrac{D-2}{4(D-1)}$, $m=0$ is also of interest due to its conformal invariance). In the case of Einstein-Maxwell theory, with Lagrangian density $R-\frac{1}{4}F_{ab}F^{ab}$, instead \_\^[ab]{}=(\^[\[b]{}\^[a\]]{}+\^cA\_c F\^[ba]{}). Formally, one may thus define a conserved quantity associated to any infinitesimal diffeomorphism $\xi$ by integrating either $j_\xi^a$ on a Cauchy hypersurface $\Sigma$ or, equivalently, $\kappa_\xi^{ab}$ on its boundary $\partial\Sigma$: \[Noethercharge\] Q\_= \_ j\_\^a d\_a = \_ \_\^[ab]{} d\_[ab]{}. The evaluation of the integral over $\Sigma$, or equivalently of the limit implicitly involved in the definition of $\partial\Sigma$, is in general a nontrivial issue. For instance, it is clear that $\mathcal Q_\xi$ is zero if $\xi$ is a *small* diffeomorphism whose action is localized in a compact region, since in that case the *surface charge* \[surfacecharge\] Q\_=\_ \_\^[ab]{} d\_[ab]{}, obtained integrating over a closed $(D-2)$-surface $\sigma$, vanishes as soon as $\sigma$ is taken outside that region. On the other hand, is obtained as the limit of where $\sigma$ is taken towards the boundary of the spacetime. ### Global charges Suppose $\xi$ is instead an exact Killing vector for a given solution of interest, $\nabla_{(a}\xi_{b)}=0$, namely a *global* symmetry. This implies $\nabla\cdot\xi=0$ and $\Box\xi^a - R_{ab} \xi^b=0$, therefore \_b \_\^[ab]{} = \_b \^[\[b]{}\^[a\]]{} = R\^[ab]{}\_b. By the Einstein equations with matter, R\_[ab]{}=k\^2\_D(T\_[ab]{}+g\_[ab]{}T), so that the charge reads \[Qstress\] Q\_= \_ (T\_[ab]{}+g\_[ab]{}T)\^a d\^b, with $T=g^{ab}T_{ab}$. In the vacuum, the integrand on the right-hand side of the previous equation vanishes, implying that the charge integral of $j^a_\xi$ over the Cauchy hypersurface $\Sigma$ only receives contributions from the regions of $\Sigma$ where stress-energy is present. Equivalently, the surface charge is actually independent of the specific closed surface on which it is performed as long as $\sigma$ is deformed without crossing any source. The simplest example where this situation occurs is provided by the Schwarzschild solution, ds\^2 = -f(r)dt\^2++r\^2\_[ij]{}dx\^idx\^j,f(r)=1-, where $t^a=(\partial_t)^a$ is an exact time translation isometry. Integrating on a sphere $S_{t,r}$ at fixed time $t$ and radius $r>C$, we have Q\_t = \_[S\_[t,r]{}]{}\^[tr]{}r\^[D-2]{}d= \_[S\_[t,r]{}]{}(\^[r]{}\_[tt]{}+f\^t\_[rt]{})r\^[D-2]{}dbut $\Gamma^{r}_{tt}=\tfrac{1}{2}f\partial_rf$ and $\Gamma^t_{rt}=\tfrac{1}{2f}\partial_rf$, which gives \[MassBH\] Q\_t = . The constant $C$ is determined by matching with the Newtonian potential in a regime of weak gravitational fields; a faraway test particle, $\frac{C}{r^{D-3}}\ll1$, moving radially in such a field at nonrelativistic speed will obey, by the geodesic equation, $\ddot r + \Gamma^{r}_{tt}=0$, hence r = - \_r(-). The nonrelativistic interaction potential between a source of mass $M$ and a test mass must match Newton’s formula $-\frac{GM}{r^{D-3}}$ and hence $C={2GM}$. Going back to , we see that the Noether charge equals \[BHoff\] Q\_t = M. We may additionally cross-check the rather awkward normalization factor by performing the static Newtonian limit directly in . In this limit, the only nonvanishing component of the stress-energy tensor is $T_{00}=\rho$ in Cartesian coordinates, $\rho$ denoting the mass density. Then, evaluated on a slice of constant time reads \[normoff\] Q\_t= dx, as expected. Equivalently, for a static linearized fluctuation $g_{00}=-1+h_{00}$, the Einstein equations give $\Delta_{\mathbb R^3} h_{00}= 2k^2_D \frac{D-3}{D-2} \rho$, and hence $h_{00}=-\frac{2GM}{r^{D-3}}$, twice the Newton potential, in the case of a particle of mass $M$ sitting in the origin $\rho(\mathbf x)=M\delta(\mathbf x)$. More generally, for any stationary spacetime, where there exists a time-like killing vector $\xi^a$, one can define the energy content as the integral \[Komartrue\] M\[\]=\_\^[\[b]{}\^[a\]]{} d\_[ab]{} =\_\_\^[ab]{}d\_[ab]{}, evaluated on any $(D-2)$-surface $\sigma$ enclosing all the sources. This quantity, whose definition takes into account and solves the above normalization issues by introducing the correct prefactor, is called the Komar mass of the spacetime. In a static spacetime, it can be understood in terms of the total force that must be exerted on a unit surface mass density distributed over a sphere enclosing all sources in order to hold it in place. Taking $\sigma=S_{u,r}$ to be a sphere of fixed retarded time $u$ and large radius $r$ in retarded coordinates, the above discussion ensures that the Komar mass evaluated on $S_{u,r}$ is independent of $r$ and of $u$. Thus, the energy can actually be calculated on a section of $\mathscr I^+$, by taking the limit $r\to\infty$, and will be observed to be constant in $u$ by asymptotic measurements on $\mathscr I^+$, as should be the case since in a stationary spacetime energy should be strictly conserved by the absence of radiation. ### Asymptotic charges The situation is potentially more interesting in the case of *asymptotic* symmetries, *i.e.* the solutions of the asymptotic Killing equation. First, being large, these diffeomorphisms are candidates to yielding nontrivial surface charges at infinity. Second, the independence on $r$ and $u$ of the corresponding surface charges is not guaranteed in general (in contrast with the case of exact isometries). This on the one hand requires to check the convergence of the limit $r\to\infty$ case by case, and on the other hand leaves open the possibility of describing charge leak due to radiation, *i.e.* a dependence of the charges on retarded time. In the case of the asymptotic symmetries of four dimensional asymptotically flat spacetimes associated to global Poincaré transformations , one is able to verify that the energy and angular momentum surface charges admit a finite and nonvanishing limit $r\to\infty$ and that they display a nontrivial dependence on retarded time $u$ due to the presence of radiation, namely due to the energy and angular momentum fluxes carried by outgoing gravitational waves. For a generic vector $\xi^a=\xi^u(\partial_u)^a+\xi^r(\partial_r)^a+\xi^i(\partial_i)^a$, the surface charge defined by the integral of $\kappa_\xi^{ur}$ on a sphere $S_{u,r}$ of fixed $u$ and $r$ reads, recalling , \[Chargexigen\] Q\_\[S\_[u,r]{}\] = \_[S\_[u,r]{}]{} e\^[-2]{} d, where we have used that the Christoffel [@Barnich_BMS/CFT] satisfy $\Gamma^{u}_{ar}=0$. Restricting to the case of time translation asymptotic symmetry, given by $T=1$, $Y^i=0$ in , namely $\xi^a=(\partial_u)^a$, this surface charge gives Q\_= \_[S\_[u,r]{}]{} e\^[-2]{} (\^r\_[ru]{}-\^u\_[uu]{}-U\^i \^u\_[iu]{})d. Evaluating the Christoffel symbols [@Barnich_BMS/CFT] and taking into account the falloffs , we see that \[Christoffelfalloff\] \^r\_[ru]{}=+O(r\^[-3]{}),\^u\_[uu]{}=-+2\_u|+O(r\^[-3]{}),\^u\_[iu]{}=O(r\^[-3]{}) and obtain the following expression \[Bondi+spur\] Q\_[t]{}\[S\_[u,r]{}\] = \_[S\_[u,r]{}]{} (m\_B-\_u|)d+ O(r\^[-1]{}), where we may use to solve for $\partial_u\bar \beta=-\frac{1}{32}\partial_u(C_{ij}C^{ij})$. The limit of this quantity as $r\to\infty$ is finite and nonzero \[Bondiin\] 8G Q\_[t]{}\[S\_[u]{}\] = \_[S\_[u]{}]{} m\_Bd+ \_u \_[S\_[u]{}]{} C\_[ij]{}C\^[ij]{} d, where $S_u$ is the spherical section of $\mathscr I^+$ at fixed $u$. The linear piece of has been given the interpretation [@BMS] of the total energy present in the spacetime at a given retarded time, also called *Bondi mass*. The proper normalization is given by the stationary limit, as in , which reduces in this case to a factor of two [@Iyer:1994ys], yielding \[Bondimass\] M\_B(u) = \_[S\_[u]{}]{} m\_Bd. The nonlinear and time-dependent term in can be eliminated by modifying the two-form $\kappa_{\xi}^{ab}$ as proposed by Tamburino and Winicour [@Tamburino_Winicour]: \[improvementTW\] \_\^[ab]{} (\^[\[b]{} \^[a\]]{}+\_c \^cm\^[\[b]{}n\^[a\]]{}) where $n^a$, to leading order, $n^a=(\partial_r)^a$ and $m^a = (\partial_u)^a-\frac{1}{2}(\partial_r)^a$. In fact, in the case of the time translation vector field $\xi^a=(\partial_u)^a$, noting that $\nabla_a\xi^a=\partial_u\log\sqrt{-g}$ and using , this additional term yields \_[S\_[u,r]{}]{} \_u d= \_[S\_[u,r]{}]{} \_u|d+O(r\^[-1]{}), and hence cancels the second term appearing in . The same result is obtained by the following procedure advocated by Geroch and Winicour [@Geroch_Winicour]. In general, the independence of the surface charge on the specific representative vector that reduces to a given asymptotic symmetry vector on $\mathscr I^+$ is not guaranteed. For instance, by inspection of equation , adding $f(\partial_r)^a$ to a vector $\xi^a$ will give the same transformation on $\mathscr I^+$ (since it projects to zero thereon) but gives rise to a difference in the surface charge given by $r^2(16\pi G)^{-1}\int e^{-2\beta}(\partial_r+\Gamma^r_{rr})fd\Omega$. Therefore, an additional condition is needed in order to resolve this ambiguity: \[Geroch-Wini\]\_a\^a=0,to be imposed on the extension of the infinitesimal symmetry vector field in the interior of the spacetime. In our case, we may extend the time translation asymptotic symmetry by introducing a $\xi^r$ component as in \^a=(\_u)\^a-(\_r)\^a+, so that the additional condition is upheld, while the projection of $\xi^a$ on a surface of constant $r$ is still $(\partial_u)^{a}$, as required by . With this modification, taking into account $\Gamma^r_{rr}=2\partial_r\beta=\mathcal O(r^{-3})$, the surface charge yields half the Komar mass with no additional terms. Notice that, in the case of exact isometries as for instance for time translations in stationary spacetimes, the Tamburino-Winicour improvement is identically zero and the Geroch-Winicour condition is identically satisfied, since $\nabla_a\xi^a=0$ follows from the Killing equation. Employing the second equation of motion , and noting that $D_iD_jN^{ij}$ integrates to zero on the sphere, we also have \[massleakD4\] \_u M\_B(u) = -\_[S\_u]{} N\_[ij]{} N\^[ij]{}d, which is the mass loss formula. Gravitational radiation, responsible for a nontrivial News tensor $N_{ij}=\partial_u C_{ij}$, gives rise to the time-dependence of the Bondi mass, which in fact always decreases as $u$ increases since the right-hand side of the previous equation is negative definite. This is interpreted as the fact that gravitational waves always carry positive amounts of energy, which leak to the null boundary of the spacetime. Under suitable regularity assumptions, it has been shown that the Bondi mass is always positive [@Reula_Tod] and that its upper bound is given by the Arnowitt-Deser-Misner [@ADM] mass $M_\mathrm{ADM}$, namely the total energy content of the spacetime: \_[u-]{}M\_B(u)=M\_, the limit being reached from below [@Schoen-Yau; @Witten_+]. These results consolidate the following picture: a gravitational system “initially” (*i.e.* for $u\to-\infty$) possesses a total energy $M_\mathrm{ADM}$, a portion of which leaks to null infinity due to the emission of gravitational waves ; this leaves behind an energy $M_B(u)$, which thus represents the amount energy *left* in the spacetime at a given retarded time $u$. Similar calculations to the one performed above are available in the case of spatial translations, $\xi^a = \mathbf n \cdot \mathbf a (\partial_u)^a$ and, taking in due account subleading terms in the expansion , of rotations and boosts . Turning our attention to a generic supertranslation $\xi^a=T(\theta,\phi)(\partial_u)^a$, as in with $Y^i=0$ and an arbitrary $T(\theta,\phi)$, we have, Q\_=\_[S\_[u,r]{}]{}e\^[-2]{}d. Substituting and , and using , the limit $r\to\infty$ gives 8G Q\_= \_[S\_u]{} m\_B T d+ D\^i C\_[ij]{}D\^j Td+ \_u \_[S\_u]{} C\_[ij]{}C\^[ij]{}Td. As in the case of the global charge, this surface charge is finite and nonvanishing. Its linear part, which can be selected by resorting to the improvement or by enforcing the condition as was done in the case of asymptotic time translations, and which must be corrected by a factor of two according to , reads \[BMSgencharge\] Q\_T(u) &= \_[S\_u]{} m\_B T d- \_[S\_u]{} D\^i D\^j C\_[ij]{} T d,\ \_u Q\_T(u) &= -\_[S\_u]{} N\_[ij]{}N\^[ij]{} Td. The term linear in $C_{ij}$ vanishes when $T$ satisfies \[DiDj1/2\] (D\_[i]{}D\_[j]{}- \_[ij]{})T=0, since $C_{ij}$ is traceless. This selects asymptotic translations, because, using that $[\Delta,D_i]T=D_i T$, the divergence of yields $D_i(\Delta+2)T=0$ and hence either $T$ is a constant or it satisfies $(\Delta+2)T=0$, whose solutions are (linear combinations of) the spherical harmonics $\cos\theta$, $\sin\theta\cos\phi$, $\sin\theta\sin\phi$ (see Appendix \[app:Laplaciano\] for more details). As we shall see in the next chapter, expressions of the type lie at the heart of the connection between $BMS$ symmetries and Weinberg’s soft graviton theorem. Let us conclude this section on asymptotic charges by a word of warning. As anticipated, we have chosen to present the calculation of surface charges without detailing subtleties that must be actually faced in order to give a general and precise definition of conserved quantities in gauge theories. Some of these issues have actually surfaced while discussing the improvement and are related to the ambiguities inherent to the definition of the canonical Noether current $j_\xi^a$ starting from a given Lagrangian. Another issue, which arises when trying to attach to conserved quantities the meaning of Hamiltonian generators is that of integrability: in this context one only calculates the *formal* variation $\slashed\delta Q_\xi$ of a given charge in field space and needs to establish whether or not this can be interpreted as an exact variation $\delta Q_\xi$. Nonintegrability of the $BMS$ charges [@Barnich_Charge], for instance, is associated with the presence of nonlinear radiation. We have glossed over such aspects in order to keep the presentation as concise as possible without renouncing on relevant formulas and physical ideas, and refer the interested reader to [@Wald-Zoupas; @Iyer:1994ys] for detailed discussions in the context of covariant phase space methods (see also [@Hajian; @Seraj]) and to the general results presented in [@Barnich-Brandt], where in particular the issue of ambiguities is resolved by appealing to the falloffs of the equations of motion. However, we will return to some of the issues raised here in Chapter \[chap:spin1higherD\], in the context of Yang-Mills theory, where they are easier to present in a self-contained and concise manner. Large Gauge Symmetries in Electromagnetism {#sec:EMD=4} ------------------------------------------ Up to this point in the discussion, we have been concerned with the symmetries of weakly radiating gravitational systems in the limit $r\to\infty$. A similar analysis can be performed in the simpler case of electromagnetism [@Strominger_QED], where once again asymptotic symmetries are identified as those residual symmetries of the gauge-fixed theory that preserve the falloff conditions assigned to the field components, while still acting nontrivially near $\mathscr I^+$. The classical action for electromagnetism coupled to a locally conserved current density $\mathcal J^a$, S = - F\_[ab]{} F\^[ab]{}- A\_a J\^a , with $\mathcal F_{ab}=\partial_a\mathcal A_b-\partial_b\mathcal A_a$, is invariant under the gauge transformation $\delta_\epsilon \mathcal A_a =\partial_a \epsilon$, since the symmetry variation of the Lagrangian is equal to the total divergence term $-\nabla_a( \mathcal J^a \epsilon )\omega$, while on the other hand for a generic variation one obtains (\^b F\_[ba]{}-J\_a)A\^a+\^b(F\_[ab]{}A\^a). This gives, on shell, the canonical current \[j\_s=11\][ j\_\^a = (F\^[ba]{}\_b+ J\^a)=\_b\_\^[ab]{}, \_\^[ab]{} = (F\^[ba]{}). ]{} In retarded Bondi coordinates, we choose the radial gauge condition A\_r=0, which is the spin-one analog of the choice we made in the gravitational setting. The falloff conditions to be assigned to currents $J^a$ depend on the type of charged matter that on is assuming to include. In particular, accounting for the possible presence of massless charged particles implies that $J^a$ behaves like J\_u= +O(r\^[-3]{}),J\_r= +O(r\^[-3]{}),J\_i = +O(r\^[-2]{}), as $r\to\infty$ for fixed $u$. Correspondingly, one finds that the equations of motion are satisfied to leading order provided one adopts the falloff conditions A\_u = A\_u(u, z, |z)/r + O(r\^[-2]{}), A\_i = A\_i(u, z, |z) + O(r\^[-1]{}), and the field components satisfy the relation \[chargedensityevM\] \_u A\_u = \_u D\^i A\_i + J\_u (compare with and respectively). Residual gauge transformations are therefore given by those gauge parameters that only depend on angular coordinates = T(z, |z), which thus exhibit a close analogy to supertranslations. The surface charge, *i.e.* the analog of , associated to this residual gauge freedom is given by Q\_= \_[r]{} r\^2\_[S\_[u,r]{}]{}F\_[ur]{}T d= \_[S\_u]{} A\_u T d. In analogy with the previous case, the global surface charge, *i.e.* the electric charge expressed as the Gauss integral, is in general independent of the specific two-sphere we choose, provided we do not cross charges flowing to $\mathscr I^+$. To see this, setting $\epsilon$ to unity, note that $\partial_b \kappa_1^{ab}=\sqrt{-g}\,\nabla_b \mathcal F^{ba}=\sqrt{-g}\,\mathcal J^a$, so that \_u Q\_1\[S\_u\]=\_[S\_u]{} J\_u d, which vanishes away from the charges. This equation is the analog, for massless electrodynamics, of the mass leak formula we encountered in the gravitational case. Note in particular that this effect, which is due to the presence of self-interactions in the case of gravity, is here caused by charges moving at the speed of light. We will encounter a similar phenomenon while discussing ordinary and null memory effects in the next chapter. For a generic function $T(z, \bar z)$, instead, the soft surface charge exhibits a nontrivial $u$-dependence. In particular, \_uQ\_=\_[S\_u]{} \_u A\_u T d=\_[S\_u]{} (\_u D\^i A\_i + J\_u) T d. This infinite-dimensional family of $u$-dependent charges plays a crucial role in the connection between asymptotic symmetries of massless electrodynamics and the soft photon theorem. We conclude by remarking that, if we want to compare the retarded radial gauge to the advanced radial gauge, we need consider that the transformation from retarded to advanced coordinates $v=u+2r$ gives (A’\_v, A’\_r, A’\_i)= (A\_u, 2 A\_u+A\_r, A\_i), so that $\mathcal A_r$ and $\mathcal A'_r$ cannot vanish simultaneously unless $\mathcal A_u$ is identically zero. This indicates that the radial gauge cannot be continued smoothly to the bulk of spacetime [@Sever]. This issue can be resolved, for instance, by performing the analysis in the Lorenz gauge; we will return on this point at the end of the next chapter. Kac-Moody Symmetry of Yang-Mills Theory {#sec:YMD=4} --------------------------------------- It is possible to perform an analysis akin to that of asymptotically flat gravitational systems also in classical Yang-Mills theory [@Strominger_YM]. Other than being of interest in its own right, this provides a convenient toy version of the corresponding gravitational case, while still encompassing genuinely nonlinear effects, in contrast with the Maxwell case. We may start from the classical action for pure Yang-Mills, S = (F\_[ab]{} F\^[ab]{}), where $\mathcal F_{ab}=\partial_a\mathcal A_b-\partial_b\mathcal A_a+[\mathcal A_a, \mathcal A_b]$. We conventionally work with anti-Hermitian fields $A_a = A_a^A X^A$ with $[X^A, X^B]=f^{ABC}X^C$, where $X^A$ and $f^{ABC}$ are the $su(N)$ generators and structure constants respectively. We normalize them according to $\mathrm{tr}(X^AX^B)=-\delta^{AB}$. This action is invariant under the gauge transformation $\delta_\epsilon \mathcal A_a =\partial_a \epsilon+[\mathcal A_a, \epsilon]$. On the other hand for a generic variation one obtains L=-(\^b F\_[ba]{}A\^a)+\^b(F\_[ba]{}A\^a). On shell, this gives the canonical current \[j\_s=12\][ j\_\^a = (F\^[ab]{}(\_b+ \[A\_b, \]) )=\_b\_\^[ab]{}, \_\^[ab]{} = (F\^[ab]{}). ]{} Adopting the retarded radial gauge condition A\_r=0, one then sees that the equations of motion are compatible with the falloffs A\_u = A\_u(u, z, |z)/r + O(r\^[-2]{}), A\_i = A\_i(u, z, |z) + O(r\^[-1]{}), provided \_u A\_u = \_u D\^i A\_i + \^[ij]{}\[A\_i, \_u A\_j\]. Comparing with , we see that, asymptotically, the nonlinearities play the same role that was played by massless charges in the previous section. This is inherent to the twofold nature of waves in nonlinear theories, where they play the role on the one hand of propagating perturbations and on the other hand of massless sources. Residual gauge transformations are therefore = T(z, |z) = T\^A(z,|z)X\^A in close analogy to supertranslations. The surface charge is thus given by Q\_T\[S\_u\]= -\_[r]{} r\^2\_[S\_[u,r]{}]{}(F\_[ur]{}T) d= -\_[S\_u]{} (A\_u T) d. The global surface charge, *i.e.* the color charge, depends on retarded time due to the nonlinearities: setting $\epsilon=X^A$, \_u Q\_A\[S\_u\]=\_[S\_u]{} \^[ij]{} \[A\_i, \_u A\_j\]\^A d. This equation is the analog, for classical Yang-Mills, of the mass leak formula we encountered in the gravitational case. For a generic parameter $T(z, \bar z)$, instead, the soft surface charge exhibits an additional nontrivial $u$-dependence that is observed to be linear in the field. In particular, \_uQ\_T\[S\_u\]=-\_[S\_u]{} (\_u D\^i A\_i + \^[ij]{}\[A\_i, \_u A\_j\] ) T d. The charges have a nontrivial algebra: =\_[T]{}Q\_[T’]{}= -\_[S\_u]{} (\[A\_u,T\]T’) d= -\_[S\_u]{} (A\_u\[T,T’\]) d=Q\_[\[T,T’\]]{}, namely a zero-level Kac-Moody algebra. Observable Effects in Four Dimensions {#chap:obs} ===================================== As we have seen, asymptotic symmetries give rise to an infinite number of independent conserved charges potentially amenable to be connected to observable quantities. Indeed, at least two broad classes of phenomena have been identified as observable effects that afford an explanation in terms of an underlying asymptotic symmetry principle: soft theorems for scattering amplitudes and memory effects [@Strominger_rev]. This chapter is devoted to reviewing these effects, together with their connection with the underlying asymptotic symmetries discussed in the previous chapter, in the well-studied cases of electromagnetic and gravity theories in four spacetime dimensions. This will also serve as preparation for our exploration of their higher-dimensional counterparts in Chapter \[chap:spin1higherD\], in the case of Maxwell and Yang-Mills theory, and of our proposal for higher-spin asymptotic symmetries and charges, and their relation to soft theorems, in Chapter \[chap:HSP\]. Soft Theorems in Particle Physics --------------------------------- Soft theorems are identities relating scattering amplitudes that differ by the emission or absorption of massless, or very light, particles with low energy [@GellMannPhysRev.96.1433; @Low; @Kazes; @Yennie:1961ad; @Weinberg_64; @Weinberg_65; @Burnett]. They have received renewed interest in the literature (see *e.g.* [@LaddhaSen:2017ygw; @Sahoo:2018lxl; @InfiniteLi:2018gnc] for some recent advances in this respect, together with the extensive list of references in [@Strominger_rev]) in the context of asymptotic symmetries, of which they have been identified as physical consequences. Indeed, although soft theorems can be directly seen to hold at the level of $S$ matrix and Feynman diagrams, they can be often derived as a consequence of the invariance of the theory under a symmetry, typically a spontaneously broken one. A textbook example of this situation is that of the spontaneously broken (approximate) symmetry $SU(2)_L\times SU(2)_R \rightarrow SU(2)_V$ of quantum chromodynamics, which lies at the heart of soft-pion techniques [@WeinbergBook2; @Ferrari_Picasso_1], dating back to the sixties, where the pions are interpreted as the (pseudo-)Goldstone bosons of said spontaneous breaking. More generally, the understanding that the dynamics of Goldstone bosons affords a universal description at low energies, dictated solely by symmetry requirements, paved the way to the formulation of effective Lagrangians and nonlinearly realized symmetries [@CCWZ1; @CCWZ2] (for a more recent approach to soft theorems in the context of effective field theories, see [@Larkoski2015; @Elvang:2016qvq]). The basic mechanism that links symmetries to scattering amplitudes can be summarized as follows. A continuous symmetry of the dynamics of a given theory is locally generated, in a canonical setup, by a conserved current $j^\mu(x)$ according to Noether’s theorem. Namely, for any local operator $A$ (which may be thought of, for instance, as $\mathcal O_1(x_1) \mathcal O_2(x_2)\cdots \mathcal O_n(x_n)$), the infinitesimal symmetry variation $\delta$ is given by \[localgen\] A = i \_[R]{} f\_R(x) \[j\^0(x),A\] dx, with a smooth cutoff function $f_R(\mathbf x)$ that equals $1$ if $|\mathbf x|<R$ and vanishes if $|\mathbf x|>R+\varepsilon$ (for a small $\varepsilon>0$). In relativistic local field theories, the right-hand side converges by the requirement that the commutator vanish when $x$ becomes spacelike with respect to the region where $A$ is localized. Furthermore, it is also independent of $t$, as a consequence of the conservation of $j^\mu$: using the Gauss theorem, \_[R]{} f\_R(x) \[\_tj\^0(x),A\] dx = \_[R]{} f\_R(x) dx =0, because $\nabla f_R(\mathbf x)$ is nonzero only for $|\mathbf x|\simeq R$, a region that becomes spacelike with respect to any point as $R\to\infty$ for fixed $x^0=t$. Taking the vacuum expectation of gives rise to Ward identities of the type \[motherWard\] 0 | A |0 =i \_[R]{} f\_R(x) 0 | \[j\^0(x),A\] |0dx = i \_[k0]{} 0 | \[j\^0(k,t),A\] |0, where $\tilde j^0$ denotes the Fourier transform of the charge density. Now, if the symmetry is unbroken, namely it admits a unitary operator $U$ that implements the symmetry in the Hilbert space of the theory, then it actually admits a canonical charge Q=\_[R]{}f\_[R]{}(x) j\^0(x) dx, such that $U=e^{iQ}$ and, restricting for simplicity to the case of internal symmetries, Q|0= 0,U|0= |0, since the vacuum $|0\rangle$ is the only translation-invariant state, so that simply reduces to 0 | A |0=0. If instead the symmetry is spontaneously broken, no generator $Q$ exists and the vacuum state is not invariant under the symmetry, namely imposes \[GBbasic\] 00 | A |0= i\_[k0]{} 0 | \[j\^0(k,t),A\] |0, which relates the vacuum expectation value of the symmetry transformation to the insertion of a *soft* operator. In fact, the intermediate states saturating the right-hand side must clearly have a gap-less dispersion relation, $\omega(\mathbf k)\to 0 $ as $\mathbf k\to0$, in order for it to be time-independent, as required by $\partial_\mu j^\mu=0$, and further inspection shows that they must be (massless) one-particle states: the Goldstone bosons. In order to make contact with scattering amplitudes, one takes the operator $A$ to be of the type $\mathcal T(\phi_1(x_1)\phi_2(x_2)\cdots \phi_n(x_n))$, where $\phi_j(x_j)$ are suitable local operators of the theory and $\mathcal T$ denotes time-ordering. The next step is to apply the LSZ reduction formulas [@LSZ1; @LSZ2] (see *e.g.* [@Schweber] for a textbook presentation), which amount to performing a Fourier transform with respect to the $x_j$ and taking the on-shell limit after amputating the propagators that arise from external particles. Restricting for simplicity to the case of a Hermitian scalar field $\varphi$ of mass $m$, with asymptotic Fock oscillators $a_\mathrm{in/out}(\mathbf q)$, the LSZ formula can be expressed as &0 | a\_(q’\_1)a\_(q’\_n) a\^\_(q\_1)a\^\_(q\_m) |0=dx’\_1dx’\_n dx\_1 dx\_m\ &e\^[i \_[j=1]{}\^n x’\_j q’\_j-i \_[k=1]{}\^m x\_k q\_k]{} (-\_[x’\_1]{}+m\^2)(-\_[x’\_n]{}+m\^2) (-\_[x\_1]{}+m\^2)(-\_[x\_m]{}+m\^2)\ & 0 | T( (x’\_1)(x’\_n) (x\_1) (x\_m) ) |0. To illustrate the above strategy in one of its simplest applications, we may derive the consequence of the (unbroken) global $U(1)$ symmetry in QED, namely the additive conservation of electric charge in all scattering processes. The Ward identities read in this case (n-m) 0 | T( (x’\_1)(x’\_n) |(x\_1) |(x\_m) A\_[\_1]{}(y\_1)A\_[\_l]{}(y\_l) ) |0= 0, so that $\langle 0 | \mathcal T( \psi(x'_1)\cdots \psi(x'_n) \bar\psi(x_1) \cdots \bar\psi(x_m) A_{\mu_1}(y_1)\cdots A_{\mu_l}(y_l) ) |0\rangle=0$ unless $n=m$. On the other hand, by the LSZ reduction formula, we see that, in terms of the numbers $N$ (resp. $N'$) of incoming (outgoing) positrons and $M$ ($M'$) of electrons, we must have $n=M+N'$ and $m=M'+N$. By comparison, this implies $M-N = M'-N'$, *i.e.* that the *in* state must have the same electric charge as the *out* state. In the case of spontaneous breaking, the so-obtained identities instead relate different amplitudes with and without the insertion of soft Goldstone bosons, according to , and hence give rise, upon LSZ reduction, to soft theorems. In fact, this strategy has been applied to a wide range of different models and symmetries (*e.g.* [@Ferrari_Picasso_1; @Hamada:2017atr]), including also non-internal symmetries such as scaling and conformal symmetry [@DiVecchia:2015jaq; @DiVecchia:2017uqn] and residual symmetries of QED, Yang-Mills theory, gravity [@Ferrari_Picasso_2; @Hamada:2018vrw]. Taking successive variations and commutators also allows one to analyze *double* soft limits, schematically 0 |\_1 \_2 A |0= - \_[k\_10]{}\_[k\_20]{} 0 | \[j\_[(1)]{}\^0(k\_1,t), \[j\_[(2)]{}\^0(k\_2,t), A\]\] |0, discussing, for instance, the dependence on the orders in which such limits are taken and its relation to the symmetry algebra [@Distler:2018rwu; @H:2018ktv]. Although the above considerations are in principle nonperturbative, since they only make reference to general properties of symmetries and of the spectrum of the theory, one must in practice face the issue of taking into account possible corrections due to loop effects in order to extend their validity to the full quantum level. A possible way out is furnished by nonrenormalization theorems that protect the commutators involving conserved currents from corrections due to ultraviolet singularities [@Symanzik]. Indeed, let us mention that some of these symmetry considerations have been extended to the quantum level [@Campiglia:2019wxe], and that soft theorems have been investigated at loop level, displaying a certain degree of stability under renormalization (see *e.g.* [@HeLoop2014; @Sen:2017nim]). Another important point that must be addressed in the discussion of loop-level results is the presence of infrared divergences. In four–dimensional QED, for instance, such infinities arise due to the masslessness of the photon and were observed to cancel out when considering cross sections that sum over emissions/absorptions of photons below a certain energy threshold, so that *soft* photons actually play a major role in the discussion of the infrared problem [@Strocchi-Erice; @Weinberg_65; @PerryStates]. At the level of semiclassical analysis, it has proven useful to investigate the relation between soft theorems in gauge theories and asymptotic or large gauge symmetries, in particular those at null infinity. This permitted in particular to interpret Weinberg’s soft photon and graviton theorems [@Weinberg_64; @Weinberg_65], which we are going to review in the next section, in terms of underlying large $U(1)$ symmetries and $BMS$ supertranslation symmetry. An analogous program has been carried out also for Yang-Mills theory [@StromingerColor] and has been extended to encompass subleading corrections to the Weinberg result [@soft_QED_Strominger; @Bern:2014vva; @Conde:2016csj]. The technical steps needed in order to perform this connection are typically a slight variation of those outlined above in the case of a standard symmetry of the quantum theory. One first identifies the asymptotic symmetry and calculates the corresponding (classical) asymptotic charge $Q$. As a consequence of Noether’s theorems, this charge takes the form of an integral over a Cauchy surface $\Sigma$ or, equivalently, over its boundary $\partial \Sigma$, Q = \_j\^d\_ = \_ \^ d\_, where $j^\mu = \partial_\nu \kappa^{\mu\nu}$ and $\kappa^{\mu\nu}=-\kappa^{\nu\mu}$, and it is actually independent of the specific Cauchy surface $\Sigma$. One then expresses the fact that the symmetry is also a symmetry of the $S$ matrix as \[Q+SSQ1\] 0=\[Q,S\]=Q\^+S-S Q\^-, where $Q^\pm$ denotes the charges evaluated on the Cauchy surfaces $\mathscr I^\pm \cup i^\pm$. Taking matrix elements of the above identity between *in* and *out* states furnishes Ward identities that lead to the soft theorems, where the piece of $Q$ that is responsible for the soft insertion is termed *soft* charge, $Q_S$, to be contrasted with the *hard* part, $Q_H$, involving the matter fields. An equivalent route, more similar to the steps presented above, is to calculate \[Q+SSQ2\] 0| A |0= i0| \[Q,A\] |0= i 0| (Q\^+ A - A Q\^-) |0, where $A$ stands for a generic product of fields, and then take the LSZ reduction of this identity in order to derive consequences at the level of scattering amplitudes. In the latter approach, which we are going to illustrate in detail in the case of Weinberg’s soft photon and graviton theorems, the hard part of the charge can usually be neglected since, in the right-hand side of the above identity, $Q$ acts on the vacuum where no stable matter is present, $Q_H|0\rangle=0$. Let us stress that the operators $Q^+$ and $Q^-$ appearing in equations and are not independent; after all, they must be the asymptotic charges corresponding to the same symmetry. Therefore, $Q^+$ and $Q^-$ must be evaluated by taking the limits of the same bulk symmetry transformation to the far future and far past: we will see a concrete example of this procedure in the case of the Maxwell theory in the Lorenz gauge. The general, $CPT$ and Lorentz-invariant way of specifying this correspondence between quantities evaluated on $\mathscr I^-$ and $\mathscr I^+$ is the antipodal map on the celestial sphere, which is obtained by the identification O\^+(n)|\_[I\^+\_-]{} = O\^-(-n)|\_[I\^-\_+]{} across spatial infinity. The antipodal identification is also instrumental in the definition of a meaningful scattering problem from $\mathscr I^-$ to $\mathscr I^+$. The process of trivial scattering is characterized by the fact that the *out* states are obtained as the antipodal images of the *in* states, as can be seen for instance by following the trajectory of a free massless particle in Minkowski space, and therefore the antipodal map provides the natural definition of “identity” (the trivial intertwining operator) between *in* and *out* Fock spaces. In the following, this identification is assumed throughout in the discussion of scattering amplitudes at null infinity. We turn now to the illustration of Weinberg’s soft theorems and to their connection with asymptotic symmetries of electromagnetism and gravity. ### Weinberg’s theorems In his celebrated $1964$ paper [@Weinberg_64], Weinberg showed that, using only the Lorentz invariance and the pole structure of the $S$ matrix, together with masslessness and spins of the photon and of the graviton, it is possible to derive the conservation of electric charge and the equality of gravitational and inertial mass. On the same grounds, he gave a possible explanation as to why we observe no macroscopic fields corresponding to massless particles of spin $3$ or higher. In particular, exploiting the (assumed) $S$-matrix pole structure and Lorentz covariance, he could prove the following two properties: - The $S$ matrix for the emission of a photon or a graviton can be written as the product of a polarization “vector” $\varepsilon^\mu$ or “tensor” $\varepsilon^\mu \varepsilon^\nu$ with a covariant vector or tensor amplitude, and it vanishes if any of the $\varepsilon^\mu$ is replaced by the photon or graviton momentum $q^\mu$; - Electric charge, defined dynamically by the strength of soft-photon interactions, is additively conserved in all reactions. Gravitational mass, defined by the strength of soft graviton interactions, is equal to inertial mass (in the nonrelativistic limit). We will now review the derivation of these results. For a few technical statements, related to the implications of covariance on the $S$-matrix structure, which are here assumed to hold, we refer to the appendices of Weinberg’s paper [@Weinberg_64]. Let us consider a process in which a massless particle is emitted with momentum $\mathbf q$ and helicity $\pm s$, limiting ourselves to integer spins. The transformation rules for $S$-matrix elements under Lorenz transformations can be inferred from the transformation law for one-particle states; using $p$ as a shorthand for a set of $p_i$, with $i=1,\ldots, n$, one has \[Strasflaw\] S\_[s]{}(q, p) = ()\^[1/2]{}e\^[i s (q, )]{}S\_[s]{}(q, p), where $\Theta$ is a function of the massless particle momentum $\mathbf q$ and of the Lorentz transformation $\Lambda$. It is always possible to write $S_{\pm s}$ as a product of a polarization “tensor” and an $M$-function in the following way: \[Spmjmathbf\] S\_[s]{}(q, p) = \_\^[\_1]{}(q) …\_\^[\_s]{}(q)M\_[\_1…\_s]{}(q, p), where $M$ is a symmetric Lorentz tensor. The polarization “vector” $\varepsilon_{\pm}^\nu$ obeys the transformation rule ( - ) \_\^(q) = e\^[i (q, )]{}\_\^(q), so that, at a closer scrutiny, $\varepsilon_{\pm}^\mu$ is not a *bona fide* Lorenz vector. An auxiliary condition will be needed, in order to make sure that $S_{\pm s}$ satisfies Lorentz invariance despite this inhomogeneous transformation rule for $\varepsilon_\pm^\mu$. The $S$-matrix transformation law then reads $$\begin{aligned}\relax S_{\pm s}(\mathbf q, p) =& \frac{1}{\sqrt{2\mathbf q}}e^{\pm i s \Theta(\mathbf q, \Lambda)} \left[ \varepsilon_{\pm}^{\mu_1}(\Lambda q)- \frac{(\Lambda q)\indices{^{\mu_1}}}{|\mathbf q|}\Lambda\indices{_\nu^0}\varepsilon_{\pm}^\nu(\Lambda q) \right]\ldots \\ &\times\left[ \varepsilon_{\pm}^{\mu_s}(\Lambda q)- \frac{(\Lambda q)\indices{^{\mu_s}}}{|\mathbf q|}\Lambda\indices{_\nu^0}\varepsilon_{\pm}^\nu(\Lambda q) \right] M_{\pm \mu_1 \ldots \mu_s}(\Lambda \mathbf q, \Lambda p). \end{aligned}$$ For an infinitesimal Lorentz transformation $\Lambda\indices{^\mu_\nu}=\delta\indices{^\mu_\nu}+\omega\indices{^\mu_\nu}$, we can use and the symmetry of $M$ to put the previous equation in the form $$\begin{aligned}\relax S_{\pm s}(\mathbf q, p) = & \left(\frac{|\Lambda \mathbf q|}{|\mathbf q|}\right)^{1/2} e^{\pm i s \Theta(\mathbf q, \Lambda)} S_{\pm s}(\Lambda \mathbf q, \Lambda p)\\ &- \frac{s}{\sqrt{2|q|^3}} \left[\omega\indices{_\nu^0}\varepsilon_{\pm}^{\nu\ast}(q)\right] q^{\mu_1}\varepsilon_{\pm}^{\mu_2\ast}(q)\ldots \varepsilon_{\pm}^{\mu_s\ast}(q) M_{\pm\mu_1\ldots\mu_s}(\mathbf q, p)\,. \end{aligned}$$ Hence the necessary and sufficient condition for this transformation law not to contradict the first one is that $S_{\pm}$ vanishes when one of the $\varepsilon_{\pm}^\mu$ is replaced with $q^\mu$: q\^[\_1]{}\_\^[\_2]{}(q)…\_\^[\_s]{}(q) M\_[\_1…\_s]{}(q, p)=0. This can be recognized, thinking in terms of fields, as a requirement of gauge invariance of the amplitude. ### Conservation of the electric charge, universality of the gravitational coupling and higher spins Considering the vertex amplitude for a very-low-energy massless particle of integer helicity $\pm s$, emitted by a particle of spin $0$ and mass $m$ (perhaps zero), and momentum $p^\mu = (\mathbf p, E)$, the only tensor which can be used to form $M_{\pm}^{\mu_1\ldots \mu_s}$ is $p^{\mu_1}\ldots p^{\mu_s}$, since terms involving $g^{\mu\mu'}$ do not contribute to the $S$ matrix because of $\varepsilon_{\pm}^\mu \varepsilon_{\pm\mu}=0$[^2]. On the other hand, terms involving the soft momentum $q^\mu$ itself would either vanish by $q_\mu \varepsilon_\pm^\mu(q)=0$ or contribute to subleading orders. Therefore, the vertex amplitude must be of the form \[da\_giustificare\] p\_[\_1]{}…p\_[\_s]{}\_\^[\_1]{}(q)…\_\^[\_s]{}(q). As we shall argue below, even for emitting particles with spin greater than $0$, the $S$-matrix elements will still be given by this expression, times $\delta_{\sigma\sigma'}$ where $\sigma$ and $\sigma'$ are respectively the initial and final helicity of the emitting particle. We define the soft photon coupling constant $e$ by the statement that the $s=1$ vertex amplitude is , and similarly for the “gravitational charge” $f$ we state that the $s=2$ vertex amplitude is . Let $S_{\beta\alpha}$ be the $S$ matrix for some process $\alpha \rightarrow \beta$, the states $\alpha$ and $\beta$ consisting of various charged and uncharged particles, perhaps including gravitons and photons. The same process can occur with emission of a very soft extra photon or graviton of momentum $\mathbf q$ and helicity $\pm 1$, or $\pm 2$, and we will denote the corresponding $S$-matrix element as $S_{\beta\alpha}^{\pm1}(\mathbf q)$ or $S_{\beta\alpha}^{\pm2}(\mathbf q)$. As illustrated in the figure below, which represents graphically the amplitude $S^\pm_{\beta\alpha}(\mathbf q)$, [MyDiagram]{} $$\begin{aligned} \begin{gathered} \begin{fmfgraph*}(90, 65) \fmfleft{i1,d1,i2,i3} \fmfv{label={\color{blue}\vdots}, label.dist=-4mm}{d1} \fmfright{o1,d2,o2,o4,o3} \fmfv{label={\color{blue}\vdots}, label.dist=-4mm}{d2} \fmf{fermion, fore=blue}{i1,v1} \fmf{fermion, fore=blue}{i2,v1} \fmf{fermion, fore=blue}{i3,v1} \fmf{fermion, fore=blue}{v1,o1} \fmf{fermion, fore=blue}{v1,o2} \fmf{fermion, fore=blue}{v1,o3} \fmf{photon, fore=red, tension=0}{v1,o4} \fmfv{d.sh=circle,d.f=empty,d.si=.25w,b=(.5,,0,,1)}{v1} \end{fmfgraph*} \end{gathered}\ = \ \begin{gathered} \begin{fmfgraph*}(90, 65) \fmfleft{i1,d1,i2,i3} \fmfv{label={\color{blue}\vdots}, label.dist=-4mm}{d1} \fmfright{o1,d2,o2,u3,o3} \fmfv{label={\color{blue}\vdots}, label.dist=-4mm}{d2} \fmf{fermion, fore=blue}{i1,v1} \fmf{fermion, fore=blue}{i2,v1} \fmf{fermion, fore=blue}{i3,v1} \fmf{fermion, fore=blue}{v1,o1} \fmf{fermion, fore=blue}{v1,u2} \fmf{photon, fore=red, tension=0}{u2,u3} \fmf{fermion, fore=blue}{u2,o2} \fmf{fermion, fore=blue}{v1,o3} \fmfv{d.sh=circle,d.f=empty,d.si=.25w,b=(.5,,0,,1)}{v1} \end{fmfgraph*} \end{gathered} \ +\ \cdots\end{aligned}$$ these emission matrix elements will have poles at $\mathbf q=0$, corresponding to the Feynman diagrams in which the extra photon or graviton is emitted by one of the incoming or outgoing external particles in states $\alpha$ or $\beta$, since then the $n$-th outgoing, respectively incoming, particle of mass $m_n$ and momentum $p_n$ gives rise to a term of the form = . These poles give the dominant contribution in this limit, while diagrams in which the soft propagator is attached to an internal line will give rise to subleading corrections. In the limit $\mathbf q \to 0$ we thus get, $$\begin{aligned}\label{WeinbergsWeinberg}\relax S_{\beta \alpha}^{\pm1}(\mathbf q) &\approx \frac{1}{(2\pi)^{3/2}\sqrt{2|\mathbf q|}}\left[ \sum_n \eta_n e_n \frac{p_n \cdot \varepsilon_{\pm}^\ast(q)}{p_n\cdot q}\right]S_{\beta\alpha}\\ S_{\beta \alpha}^{\pm2}(\mathbf q) &\approx \frac{(8\pi )^{1/2}}{(2\pi)^{3/2}\sqrt{2|\mathbf q|}}\left[ \sum_n \eta_n f_n \frac{(p_n \cdot \varepsilon_{\pm}^\ast(q))^2}{p_n\cdot q}\right]S_{\beta\alpha}, \end{aligned}$$ $\eta_n$ being $+1$ or $-1$ according to whether the particle $n$ is outgoing or incoming. As we have discussed, Lorentz invariance requires the vanishing of the vertex amplitude when a polarization is substituted with the corresponding four-momentum. This yields for $s=1$ \_n \_n e\_n=0 and for $s=2$ \_[n]{} \_n f\_n p\_n\^=0. The first one is precisely the conservation of the electric charge, whereas the second, when compared with the equation of momentum conservation $\sum \eta_n p_n=0$, yields the universality of the gravitational coupling constant $f_n=1$, for all $n$. From these calculations, Weinberg also argues that the choice of $p_{\mu_1}\ldots p_{\mu_s}$ in gives in fact the only possible form of the $M$-function, also for emitting particles with spin $1$ or higher, since any other helicity-dependent vertex amplitude could never give rise to cancellations between different poles needed to satisfy the Lorentz invariance condition. For higher helicities $s=3,4,\ldots$ one still has a factorization of the form S\_\^[s]{}(q) S\_, and the requirement \[sort\_of\_conservazione\] \_n \_n g\_n\^[(s)]{} \^[s-1]{}=0, which contradicts momentum conservation unless $g_n^{(s)}=0$. This tells us that the low-energy interaction for higher spins is trivial or, in other words, that massless higher-spin particles cannot propagate long-range forces. On the Lagrangian side, this implies that higher-spin interactions should be of multipolar type, *i.e.* the vertices should contain enough derivatives so that they vanish in the soft limit. Soft Theorems and Asymptotic Symmetries --------------------------------------- We will now make the link between asymptotic symmetries and soft theorems explicit by providing the details of the derivation of the soft photon and soft graviton theorems from the underlying large $U(1)$ and $BMS$ supertranslation symmetry. For this purpose, it is useful to recast Weinberg’s result in terms of retarded coordinates and stereographic coordinates on the sphere . To this end, consider a wave packet for a massless particle with spatial momentum centered around $\mathbf{q}$. At large $r$ (and large times $t=u+r$ for fixed retarded time $u$), this wave packet becomes localized on the sphere at null infinity near the point q = = (z + |z, -i(z-|z), 1-z|z ) , so that the momentum of massless particles may be equivalently characterized by $q^\mu$ or $(\omega, z , \bar z)$. The polarization vectors can be chosen as follows [@Choi_Shim_Song] \^+(q) &= ( |z, 1, -i, -|z) ,\ \^-(q) &= ( z, 1, i, - z) = , thus allowing to rewrite Weinberg’s soft theorem, *e.g.* for a positive-helicity emission, from the momentum space form to its position-space counterpart \[WeinbergPSPACE\] \_[0\^+]{} S\_\^[+s]{} = (-1)\^[s]{}[2\^[-1]{}]{} (1+z|z)S\_ , where, for simplicity, we have assumed that all the particles taking part in the scattering process are massless, the energy $E_i$ and the angular coordinates $(z_i,\bar z_i)$ characterizing their asymptotic states at null infinity. ### Large $U(1)$ symmetries and the soft photon theorem The action for electromagnetism coupled to a locally conserved current $\mathcal J^\mu$, S = - F\_ F\^ d\^Dx - A\_J\^d\^Dx, being invariant under $\delta \mathcal A_\mu =\partial_\mu \epsilon$ up to the boundary term, possesses the canonical current \[j\_s=1\][ j\^= F\^\_+ J\^. ]{} In Bondi coordinates, near $\scrip$, in the case $\mathcal J=0$, Q\^+ = \_ j\^r \_[z |z]{} r\^2 du d\^2z. As we discussed in Section \[sec:EMD=4\], choosing retarded radial gauge $\mathcal A_r = 0$ together with the falloff conditions A\_u =& A\_u(u, z, |z)/r + O(r\^[-2]{})\ A\_z =& A\_z(u, z, |z) + O(r\^[-1]{})\ A\_[|z]{} =& A\_[|z]{} (u, z, |z) + + O(r\^[-1]{}), and using , the charge associated to the residual gauge freedom given by angular functions $T(z, \bar z)$ computed at $\scrip$ reads , where J(u, z, |z) \_[r]{}r\^2J\^r(u, r, z, |z). Since this charge acts on matter fields by $\delta \Phi(x) = i[Q, \Phi(x)] = i e \epsilon(x) \Phi(x)$, any correlation function will satisfy \_[n=1]{}\^N \_n(x\_n) =& i 0 | ( Q\^+ \_[n=1]{}\^N \_n(x\_n) - \_[n=1]{}\^N \_n(x\_n) Q\^-) |0\ =& i \_[n=1]{}\^N e\_n (x\_n) \_[n=1]{}\^N \_n(x\_n). Performing LSZ reduction of the previous formula yields the Ward identity | (Q\^+ S - S Q\^-) |= \_[n=1]{}\^N \_[n]{} e\_n T (z\_n, |z\_n) | S |. This derivation is given in [@Avery_Schwab], where it was also observed that the antipodal identification commonly employed in the literature essentially consists in choosing the same gauge transformation for $\scrip$ and $\scrim$. This means that, in order to write down the correct Ward identity, one should take $T(z, \bar z)$ and its counterpart on $\mathscr I^-$ as limits to $\scri^\pm$ of the same bulk gauge transformation; an example of this will be given in Section \[ssec:memoryas\] in the Lorenz gauge. Furthermore, the authors also note that since the charge is computed on a surface approximating $\scri^\pm \cup i^\pm$ which necessarily cuts through time-like trajectories, the results also hold for massive fields. Using the auxiliary boundary condition $\partial_z A_{\bar z} = \partial_{\bar z} A_z$ at $\scrip_{\pm}$, which amounts to imposing the absence of long-range magnetic fields on $\mathscr I^+$, choosing $T(z, \bar z) = \frac{1}{w-z}$, where $w$ is a fixed complex parameter, and exploiting $\partial_{\bar z} \frac{1}{z-w} = 2\pi \delta^2(z-w)$ gives \[4piout\_S\] 4||= \_[n=1]{}\^N | S |, where we have used that $\mathcal J$ annihilates the vacuum, since the global $U(1)$ symmetry is unbroken. Using the free mode expansion for $A_z$ near $\scri$ and the stationary phase approximation, we obtain du e\^[iu]{} \_u A\_z=- \_0\^d\_[q]{} so that \_[-]{}\^[+]{} du \_u A\_z = - \_[0\^+]{}. Substituting this result into , together with the analogous one for $\scrim$, and using crossing symmetry yields which is Weinberg’s theorem . ### BMS symmetry and the soft graviton theorem:a linearized perspective The action for a massless Fierz-Pauli field $h_{\mu\nu}$, describing a linear perturbation of the Minkowski metric tensor, is \[S\_spin2\] S = E\^h\_d\^Dx - J\^h\_d\^Dx, where $\mathcal E^{\mu\nu}$ is the linearized Einstein tensor E\_ = h\_ - \_[(]{} h\_[)]{} - \_\_h’ + \_ (h - h’), and $J^{\mu\nu}$ is a conserved “energy-momentum tensor”, $\partial_\mu J^{\mu\nu}=0$. The action is invariant under $\delta h_{\mu\nu} = \partial_{(\mu}\xi_{\nu)}$ up to the boundary term \_d\^Dx, since $\mathcal E^{\mu\nu}$ satisfies the linearized Bianchi identity $\partial\cdot \mathcal E^\nu = 0$ and $J^{\mu\nu}$ is conserved. The equations of motion are $\mathcal E^{\mu\nu} = J^{\mu\nu}$. The symmetrized derivatives needed for the computation of the current are =&\ =& , where $H^{\mu\alpha\nu\beta}$ is defined by H\^ \^h\^ + \^ h\^ - \^ h\^ - \^ h\^ - (\^ \^ - \^ \^)h’. This tensor has the same symmetries of $R^{\mu\alpha\nu\beta}$, H\^ = -H\^ = - H\^ = H\^, satisfies the cyclic identity, H\^ + H\^ + H\^ =0, and acts as a superpotential for the linearized Einstein tensor, meaning \[superpotential\] E\^ = \_\_H\^. Defining the trace-reversed tensor $\bar h^{\mu\nu} = h^{\mu\nu} - \frac{1}{2}\eta^{\mu\nu} h'$, satisfying $\bar h'=- h'$, one gets the simpler form H\^ = \^|h\^ + \^ |h\^ - \^ |h\^ - \^ |h\^. The canonical Noether current is given by j\^= h\_[,]{} - \_ h\_ + J\^\_, where $\delta h_{\alpha\beta} = \partial_{(\alpha}\xi_{\beta)}$. The contribution $J^{\mu\nu}\xi_{\nu}$ is given by the boundary term in the variation of the action. Thus \[jmu\_spin2\] j\^= ( H\^ \_\_\_- \_H\^ (\_\_+ \_\_) ) + J\^\_, where we have used the antisymmetry of $H^{\mu\alpha\nu\beta}$ in $\nu\beta$ and the symmetry of $\partial_\nu\partial_\beta$ (or analogous considerations for similar contributions) for the first term and symmetrized in $\alpha\beta$ the second term. Let us now recover the Noether tensor $\kappa^{\mu\nu}$ satisfying $j^\mu=\partial_\nu\kappa^{\mu\nu}$, whose existence is ensured by Noether’s second theorem; for this purpose we can set $J^{\mu\nu}=0$ without loss of generality. Integrating by parts each term in , employing the equations of motion $\partial_\alpha \partial_\beta H^{\mu\alpha\nu\beta} = 0$, and renaming the indices appropriately we get j\^= { \_}, so that thanks to the cyclic identity j\^= \_\^, \^ = H\^ \_\_+ \_\_H\^. Now, we may think to have obtained these expressions in a given locally inertial frame: to covariantize them we simply replace ordinary derivatives with covariant derivatives and note that no ambiguity arises in their ordering, since the connection defining them is given by the flat background metric and hence such derivatives commute; thus $$\label{js=2}{ j^c = \frac{1}{2}\left( H^{cadb} \nabla_d \nabla_a \xi_b - \nabla_d H^{cadb} (\nabla_a \xi_b + \nabla_b \xi_a) \right)+ J^{cd}\xi_{d},}$$ and $$\label{ks=2}{ \kappa^{ca} = \frac{1}{2}H^{cadb} \nabla_d \xi_b + \xi_d \nabla_b H^{cadb}.}$$ From the perspective of linearized gravity (*i.e.* of a generic spin-2 massless field) the Bondi gauge is fixed by the following choice of boundary conditions, which as we saw stems from considerations in the nonlinear theory: \[definizione\_Bondi\_gauge\] h\_[ab]{}dx\^a dx\^b = du\^2 - 2U\_z du dz - 2 U\_[|z]{} du d|z + r C\_[zz]{} dz\^2 + r C\_[|z |z]{}d|z\^2, where $u,r,z,\bar z$ are the usual retarded coordinates. Considering now the asymptotic symmetries we have evaluated in Chapter \[chap:basics\] and restricting to supertranslations, we have the infinitesimal symmetry generators \_a dx\^a &= - (T+ D\^z D\_z T )du - T(z, |z)dr - r D\_z T dz - r D\_[|z]{}T d|z,\ \^a \_a &= T(z, |z) \_u + D\^z D\_z T \_r - (D\^zT \_z + D\^[|z]{}T \_[|z]{} ) which indeed leave the “Bondi gauge” defined by invariant. We may now compute the charge associated with this residual supertranslation gauge symmetry, starting either with the Noether tensor $k^{ab}$ or from the current $j^a$ itself. In any case, the explicit computation of the nonvanishing components of the tensor $H^{abcd}$ is quite useful: to leading order, \[useful\_table\] H\^[ur zr]{}= ,& H\^[uzrz]{} = -, H\^[rzrz]{}= ,\ &H\^[rz z|z]{}= , H\^[rzr|z]{} = , where the components with $\bar z$ and $z$ interchanged are obtained by formal conjugation of all indices. It is also convenient to compute the “commutators” $\xi_{[a;b]}=\xi_{[a,b]}$: \[tabella\_dei\_commutatori\] \_[\[u,r\]]{}=0,\_[\[u,z\]]{} = D\_z(T + D\^zD\_zT),\_[\[r,z\]]{}=0,\_[\[z,|z\]]{}=0. We start computing $\kappa^{ur}$ from , since this component is selected by the measure element of $\scrip_-$. Observe that $ \frac{1}{2}H^{urdb}\nabla_d\xi_b = \frac{1}{4}H^{urdb} \xi_{[b, d]} $ by the antisymmetry of $H^{urdb}$ in $db$; by the only potentially surviving term would be $\frac{1}{4}H^{uruz}\xi_{[u,z]}$, which vanishes anyway since $H^{uruz}$ is itself zero. The other contribution to the $\kappa^{\mu\nu}$ form from is \_d \_b H\^[urdb]{} = \_d \_b H\^[urdb]{} + \_d \^[u]{}\_[eb]{}H\^[e r db]{} + \_d \^r\_[eb]{} H\^[uedb]{} + \_d \^d\_[eb]{}H\^[ur eb]{} + \_d \^b\_[be]{}H\^[urde]{}; the fourth term on the right-hand side vanishes by symmetry/antisymmetry in the summed indices while $\Gamma^\beta_{\beta\rho}=\partial_\rho \log \sqrt{g}$ in the last term. Taking into account the nonvanishing Christoffel symbols and $H^{abcd}$ components, we get \^[ur]{}=&\ &+\ &+, were “$z\leftrightarrow\bar z$” refers to formal complex conjugation in the $z$ and $\bar z$ indices. Hence, after expanding the derivative in the second term, \^[ur]{} = 2 T + ; integrating this expression as Q\^+ = \_[\_-]{} \^[ur]{} \_[z|z]{} r\^2d\^2z, and recalling that the sphere has no boundary, we obtain Again, the factor $r^2$ from the measure element gets canceled and the charge is meaningfully expressed as an integral over the boundary of null infinity. The computation of $j^r$ from , instead, goes as follows. Note that H\^[radb]{}\_d\_a\_b = H\^[radb]{}\_a\_d\_b = H\^[radb]{}\_a \_[\[b,d\]]{}by the vanishing of the Riemann tensor and by antisymmetry in $db$. Therefore, due to , the only relevant component is $H^{rzuz}\sim 1/r^3$: this term gives a sub-leading contribution. Altogether, always taking and into account, one finds that the only leading contribution to $j^a$ comes from the following term \_u H\^[rzuz]{} \_[(z]{}\_[z)]{} + z|z = . Thus, j\^r = - - J\^[rr]{}(u,r, z, |z)T(z, |z) and where J(u, z, |z) \_[r]{}r\^[2]{}J\^[rr]{}(u, r, z, |z). Since supertranslations act on matter fields by $i T(z, \bar z) \partial_u$ at $\scrip$, we get by LSZ reduction \[Q+SSQ-\] | (Q\^+ S - S Q\^-) |= \_[n=1]{}\^N \_[n]{} f\_n E\_n T (z\_n, |z\_n) | S |, where $f_n$ is the gravitational coupling of each field. Using the auxiliary boundary condition D\^zD\^z C\_[zz]{} =D\^[|z]{} D\^[|z]{} C\_[|z |z]{}\_, we have \[T\_s=2\] Q\^+ = -2 \_ T(z, |z) \_u D\^z D\^zC\_[zz]{} \_[z|z]{} d\^2z du. Now, in order to make contact with Weinberg’s soft theorem, we choose as $T(z,\bar z)$ an angular function of the following type: \[TzzMIO\] T(z, |z) = . Then the left-hand side of , after an integration by parts in $\partial_{\bar z}$, involves computing \_[|z]{} ( ) =& -2\^2(z-w) +\ =& -2\^2(z-w) +\ =& -2\^2(z-w) + \_[z|z]{}. Therefore Q\^+ = -4du D\^w C\_[ww]{} + D\^z C\_[zz]{} \_[z|z]{} d\^2z du, where the second term is a boundary contribution on the sphere and hence gives zero. To sum up, \[Ward\_s=2\] &-4D\^z||\ &= \_[n=1]{}\^N \_[n]{} | S |. As in the spin-one case, we perform a stationary phase approximation as $r\to\infty$ to express $C_{zz}$ in terms of soft graviton creation and annihilation operators, which yields C\_[zz]{} = - \_0\^[+]{} d\_[q]{} , and du \_u C\_[zz]{} = - \_[0\^+]{}. Using crossing symmetry and the matching condition, we also have -4||= \_[0]{}|a\^\_+(x) |, and this implies, by comparison with , since \_[z|z]{}\_[|z]{} \_n \_n f\_n = \_n \_n f\_n ; note that we omitted the $\partial_{\bar z}\frac{1}{z-z_n}$ term, since here the delta multiplies a function which vanishes when $\bar z = \bar z_n$. This shows the supertranslation Ward identity to be equivalent to Weinberg’s factorization formula , without assuming from the beginning $f_n=$ constant. Notice also that our choice of $T$ is not restrictive, since we may always write f(z, |z) = f(w, |w) \_[|w]{} and then use the linearity of the Ward identity to recover the full supertranslation invariance from Weinberg’s theorem. Memory Effects -------------- By memory effects, we mean a class of observable phenomena that characterize the passage of radiation impinging on a test charge and persist after said radiation has died out. For instance, a pair of test masses may undergo a nonzero relative displacement after the passage of gravitational radiation [@Zeldovich; @Frauendiener; @memory; @Bieri:2015yia] or a small electric charge, initially at rest, may display a nonzero velocity after it is invested by electromagnetic radiation [@Bieri:2013hqa; @Mao_em; @Mao_EvenD]. Analogous Yang-Mills memory effects have also been proposed [@Jokela:2019apz] and a similar phenomenon has been identified in the context of the interaction of a two-form with a test string [@Afshar:2018sbq]. Memory effects can be also stored in the quantum states of superconducting condensates [@Susskind] and quarks [@StromingerColor]. In all these cases, such effects have the property of leaving on the test systems a permanent imprint of the radiation itself. Memory effects can be induced on a particle sitting near null infinity both by the radiation emitted by the movement of charged sources in the interior of the spacetime, or by the outflow of charged massless matter, which travels along null rays. The former case is usually referred to as *linear* or *ordinary* memory and can be regarded as a picture of the movement of the bulk charges that is stored in properties of the test particles, while the latter has been termed *nonlinear* or *null* memory, because it signals the passage of charged radiation, as occurs both in nonlinear field theories containing self-interacting massless particles and in linearized theories with charged massless sources. We will now illustrate some examples of memory effects and then turn to explaining the connection between these phenomena and asymptotic symmetries, the former being interpreted as observable consequences of the latter. ### Scalar and electromagnetic memory Although the main interest has been on memory effects in gauge theories, a simple example thereof is already provided by scalar memory. To illustrate it, in its simplest realization, it is sufficient to consider a particle, charged under a scalar field $\varphi$, that is created at rest in the origin at $t=0$. This situation is described by the equation -(t,x)=q(t)(x), (we stick to the mostly-plus convention $\Box = \eta^{\mu\nu}\partial_\mu \partial_\nu=-\partial_t^2+\nabla^2$) which is solved by convolution of the right-hand side with the retarded propagator D\_(x)= and, switching to retarded coordinates, yields (u,r)=q. This process will induce the following change in the energy of a test particle with charge $Q$ which is held at a distance $r$ from the source P\_u(u) = Q\_[-]{}\^[u]{} \_u du’ = qQ. This is just the expected variation in the Coulombic interaction energy due to the creation of the new particle in the origin. Naturally, the effect only takes place after $u=0$, namely after the world-line of the test particle crosses the wave of radiation induced by the particle’s creation in the origin, which is a spherical delta-like impulse traveling on the light-cone. Another example in the scalar case is that of a particle, initially at rest in the origin, that suddenly starts moving with nonzero velocity $\mathbf v$ at $t=0$. The corresponding solution to the field equation can be obtained by combining a time-reversal and a Lorentz boost of the previous solution, namely, by superposing the field generated by a particle that is destroyed in the origin and that of a particle that is created with velocity $\mathbf v$. The needed boost in this case is t(v)(t-v x),x x + v ((v)-1)-(v)v t or, in terms of retarded coordinates, u+O(r\^[-1]{}),tr (v)(1-nv)+O(1), where $\gamma(\mathbf v)=(1-\mathbf v^2)^{-1/2}$ and $\mathbf n = \mathbf x/r $. The solution is then given, to leading order, by (u,r,n) = q++O(r\^[-2]{}). Performing the integral of $\partial_u\varphi$ with respect to retarded time, we arrive at the memory effect P\_u =qQ (-1+ )+O(r\^[-2]{}), which is to be interpreted as the fact that, after the particle starts moving, the interaction energy becomes modified as a consequence of the relativistic length contraction. Solutions to the field equation, and hence memory effects, associated to a generic, idealized scattering process involving a number of incoming and outgoing particles and taking place near the origin can be constructed by superposition of solutions in a similar manner [@Garfinkle:2017fre], but do not introduce qualitatively new elements. While until now we have been concerned only with ordinary memory, it is also possible to provide an explicit example of null memory effect, by considering a point-like source moving at the speed of light in the $\mathbf x_0$ direction ($|\mathbf x_0|=1$) -= q(x - x\_0 t). Rewriting this equation in retarded coordinates near future null infinity, we have 2(\_r + )\_u - (\_r\^2+\_r + )= (u)(n, x\_0), where we recall that $\Delta$ is the Laplace-Beltrami operator on the Euclidean unit sphere and $\delta(\mathbf n, \mathbf x_0)$ is the normalized delta function thereon. We see that a solution is furnished by writing $\varphi$ in terms of the formal expansion \[formalsolutionmemory\] (u,r,n) = \_[k=0]{}\^r\^k\^[(k+1)]{}(u) C\^[(k)]{}(n) provided that[^3] \[C0memory\] -C\^[(0)]{}(n) = q ( (n, x\_0)- ), and the other coefficients satisfy the recursion relation \[recursionnull4\] \[+(k+1)(k+2)\]C\^[(k+1)]{}(n)=2(k+1) C\^[(k)]{}(n). Equation is solved by means of the Green’s function of the Laplace-Beltrami operator C\^[(0)]{}(n) = -(1-nx\_0), while equations can be solved recursively for $C^{(k+1)}(\mathbf n)$, up to terms proportional to the spherical harmonics, which satisfy $-\Delta Y_{k+1}^m(\mathbf n)=(k+1)(k+2)Y_{k+1}^m(\mathbf n)$ and annihilate the left-hand side. The field will then give rise to the following leading-order memory effect on a test charge $Q$: P\_i(u) = Q \_[-]{}\^u \_i du’ = qQ, provided that $u>0$, while $\Delta P_i=0$ for $u<0$, where we have made use of the fact that the all terms in the expansion of $\varphi$, except $k=0$, are multiplied by higher-order derivatives of $\delta(u)$ and hence give a (singular) contribution with support in $u=0$. Similar effects are present in the case of the electromagnetic field and can be conveniently calculated in the Lorenz gauge $\partial^\mu \mathcal A_\mu =0$, where the equations of motion reduce to a set of scalar wave equations A\_= j\_. Indeed, first considering the case of a static particle created in the origin,[^4] A\^= u\^q(t)(x), where $u^\mu=(1,0,0,0)$, hence $ \mathcal A^\mu = - u^\mu \varphi $, where $\varphi$ denotes the corresponding solution for the scalar field. Boosting this solution yields $\mathcal A^\mu=(\mathcal A^0, \mathbf A)=-\gamma(\mathbf v)(1,\mathbf v)\varphi$ and going to retarded components yields A\_u = (v),A\_r = (v)(1-n v),A\_i = -r(v) v\_i . Conjoining as before the solution corresponding to a particle destroyed in the origin and to a particle created with velocity $\mathbf v$, we obtain A\_u &= q++(r\^[-2]{}),\ A\_r &= +(r\^[-2]{})\ A\_i &= -+(r\^[-1]{}). The change four-momentum of a test charge $Q$ will be subject to \[memoryD=4\] P\_i(u) = Q \_[-]{}\^[u]{} F\_[iu]{} du’ = +(r\^[-1]{}). Let us note that the leading memory effect is proportional to the variation of the field component $\mathcal A_i$ between $u>0$ and $u<0$ (where it vanishes), which can be rewritten in the following way \[totalgradsphere\] A\_i|\_[u&gt;0]{}-A\_i|\_[u&lt;0]{} = -=q\_i (1-nv). Note that this difference takes the form of a derivative on the sphere, *i.e.* of a gauge transformation. More details on the interpretation of this equation in connection with asymptotic symmetries in the Lorenz gauge will be provided in the next section. Null memory can be instead displayed by considering A\^= q v\^(x- x\_0 t), with $v=(1,\mathbf x_0)$. Taking into account the corresponding solution for the scalar field $\varphi$, we then have $A^\mu = - v^\mu \varphi $ and, moving to retarded components, A\_u = ,A\_r = (1-n x\_0),A\_i = -r (x\_0)\_i . Consequently \[Anull4\] A\_u &= -(1-n x\_0)+,\ A\_r &= (1-n x\_0)(1-n x\_0)+,\ A\_i &= - r(1-n x\_0)+, where we have omitted terms proportional to higher-order derivatives of the delta function $\delta(u)$. The null memory formula then reads, for $u>0$, P\_i (u)= Q\_[-]{}\^u F\_[iu]{} du’ = , while $\Delta P_i=0$ for $u<0$; indeed, the terms omitted from have support in $u=0$ and hence do not contribute to the memory effect. Note that this result is formally identical to the analogous formula for ordinary memory, upon substituting $\mathbf v$ with $\mathbf x_0$. A key difference is that, in the former case, the formula is smooth over the whole celestial sphere, while in the latter it displays a puncture at the point where the sphere is pierced by the outgoing massless charge. ### Memory and asymptotic symmetries {#ssec:memoryas} Aside from being physically interesting in their own right, memory effects possess an additional piece of interest in that they can be interpreted as observable effects associated to asymptotic symmetries. More precisely, one observes that the configurations of the system before and after the passage of radiation are mapped into one another by the action of a large gauge symmetry transformation; it is this underlying transition between inequivalent radiative vacua that can be held responsible for a nontrivial memory effect. This feature is clearly exhibited by electromagnetism in radial gauge in four dimensions, where one considers solutions to the Maxwell field equations subject to $\mathcal A_r = 0$ and A\_u=O(r\^[-1]{}),A\_i=O(1), as $r\to\infty$, where $i=1$, $2$ denote two angular coordinates. The asymptotic symmetries of this system are given by gauge parameters $\epsilon = T(x^1,x^2)$ that only depend on the angular coordinates and hence generate the transformations A\_i = \_i T, while $\mathcal A_u$ is gauge-invariant. For a generic solution of the field equations, a test particle with unit electric charge, which is initially at rest at a large distance $r$ from the origin, will feel a leading-order electric field given by F\_[ui]{}=\_u A\_i+,F\_[ur]{}=O(r\^[-2]{}). Hence, assuming it is subject to a radiation train with support between retarded times $u_0$ and $u_1$, it develop a momentum kick P\_i = \_[u\_0]{}\^[u\_1]{} F\_[ui]{} du = A\_i|\_[u\_1]{}-A\_i|\_[u\_0]{}+O(r\^[-1]{}) in the direction tangent to the celestial sphere. In this step, it has been assumed that $u_1-u_0$ is sufficiently small and allows us to neglect the contribution due to the magnetic field, which will be further suppressed by the particle’s velocity. On the other hand, before and after the passage of radiation, the particle must be in a radiative vacuum configuration and hence there must exist a function $T(x^1, x^2)$ such that A\_i|\_[u\_1]{}-A\_i|\_[u\_0]{}=\_i T. We thus finally see that the momentum kick memory effect, signaled by the test particle, can be interpreted as the action of a large gauge transformation on the underlying gauge field, which connects two different “vacua” of the theory. A similar line of reasoning can be applied to the Lorenz gauge. Indeed, the memory formula strongly suggests, by , that also in this case a momentum kick is actually proportional to a gauge transformation relating the leading term of $\mathcal A_i$ before and after the passage of radiation. In order to see that this is indeed the case, one must find solutions of =0, which identifies the residual gauge freedom after imposing $\partial^\mu \mathcal A_\mu=0$, that approach asymptotically a given function $T(\mathbf n)$ on the celestial sphere. For instance, in the case of and , T(n) = q(1-n v). This can be achieved by taking the following the integral on the Euclidean unit sphere [@Hirai:2018ijc] \[epsilonLorenz4\] (x) = G(x,q) T(q)d(q), where the propagator $G$ is defined by G(x,q) = - , with $q=(1,\mathbf q)$, the limit $\varepsilon\to0^+$ being understood. The introduction of this small imaginary part in the denominator is needed in order to avoid the poles at $\mathbf n \cdot \mathbf q=t/r$, which occur when $x$ lies outside of the light-cone, $|t|<r$. In order to verify that indeed furnishes the desired result, first, we can immediately check that $\Box G=0$. Second, aligning the direction $\mathbf n=\mathbf x/r$ along the $z$ axis, which is not restrictive due to the rotational invariance of the measure $d\Omega$, we have, for any $u\neq0$, G(x,q) (q)d(q) &= \_0\^[2]{} d\_0\^ d\ &= (1+) \_0\^[2]{} \_0\^[2r/u]{}d\ & \_0\^[u ]{} T(n)==T(n). This integral gives instead zero when evaluated on the light-cone $u=0$, so that the pointwise limit $r\to\infty$ yields $T(\mathbf n)$ except for $u=0$, where it vanishes. Since this is simply a removable discontinuity on $\mathscr I^+$, we may as well ignore it for all practical purposes, and simply state \_[r]{}(u+r,r n)=T(n), which is the property we needed, together with $\Box\epsilon=0$, in order to interpret the memory effect as a large gauge transformation in Lorenz gauge. Since the explicit expression for $\epsilon(x)$ is valid everywhere in Minkowski space, it also allows us to evaluate its limit to *past* null infinity, and to address a point we raised earlier while discussing the antipodal matching condition. Indeed, performing the large $r$ limit for fixed advanced time $v=t+r=u+2r$, we find \_[r]{}(v-r,rn) = T(-n), which is precisely the antipodal matching. Gravitational Memory -------------------- We conclude this chapter on observable effects linked to asymptotic symmetries with a brief discussion of memory effects induced by the passage of a gravitational wave. Originally introduced by Zel’dovich and Polnarev [@Zeldovich] (see also [@Frauendiener] for a concise treatment within the Newman-Penrose framework [@Newman-Penrose]), gravitational memory has been the first type of memory effect to be revisited in connection with asymptotic symmetries, in this case $BMS$ supertranslation symmetry [@memory]. As in the case of electromagnetic memory, one considers a situation in which radiation crosses the sphere at large radius $r$ only between two given retarded times $u_0$ and $u_1$. In other words, the gravitational field is, asymptotically, in a radiative vacuum before $u_0$ and after $u_1$. Such radiative vacua are characterized, in terms of the $BMS$ data , by \[vacuumconditions\] \_u C\_[ij]{}=0,D\_[\[i]{}DC\_[j\]]{}=0. The first condition is the vanishing of the Bondi news $N_{ij}=\partial_u C_{ij}$, which ensures that the energy flux is zero, while the second condition guarantees in addition that no angular momentum flux be present [@Barnich_BMS/CFT; @Ashtekar_Lectures; @Ashtekar_New_Lectures]. Equations imply that $C_{ij}$ is $u$-independent and that there exists an angular function $C(x^i)$ such that [@memory] \[pureST\] C\_[ij]{}=(-2D\_i D\_j+\_[ij]{})C. Recalling , \[gravwave\] \_u m\_B = \_u DDC- N\_[ij]{}N\^[ij]{}, we see also that \[mBvacuum\] \_u m\_B=0 when equations hold. To summarize, a radiative vacuum is specified by assigning $m_B(x^i)$ and $C(x^k)$, where $C_{ij}=(-2D_i D_j+\gamma_{ij}\Delta)C$. However, the specific values attained by $C$ and $m_B$ before $u_0$ and after $u_1$ will in general differ due to the evolution induced by the passage of radiation at intermediate times and are obtained by integrating equation for a specific radiation profile: m\_B|\_[u\_1]{}-m\_B |\_[u\_0]{} = - (-2)( C|\_[u\_1]{}-C|\_[u\_0]{})-\_[u\_0]{}\^[u\_1]{} N\_[ij]{} N\^[ij]{}du On the other hand, we note that supertranslations map the space of $BMS$ vacua to itself, since an infinitesimal supertranslation parametrized by a function $T(x^i)$ (recalling and ) induces the transformation \_T m\_B=T \_u m\_B,\_T C\_[ij]{}=T \_uC\_[ij]{}+(-2D\_iD\_j+\_[ij]{})T, which leaves and invariant. Moreover, supertranslations generically map a given vacuum to an inequivalent one, since they shift the function $C$ by $T$; naturally, the corresponding shift of $C_{ij}$ vanishes precisely when $T$ parametrizes an ordinary translation (*i.e.* when $T$ is a spherical harmonic with $l=0$ or $l=1$). On the other hand, supertranslations do not alter the value of the Bondi mass aspect $m_B$, consistently with the fact that they commute with translations and hence cannot alter the Bondi four-momentum. The relevance of the above structure of radiative vacua in relation to observable effects can be seen by considering the proper distance $\delta L$ that is measured between two detectors held fixed in the positions $(r,x^i)$ and $(r,x^i+\delta x^i)$ for large $r$ and $\delta x^i=\mathcal O(r^{-1})$. Assuming for simplicity that $C_{ij}$ vanishes before $u_0$ and attains a nonzero value after $u_1$, where it takes the form of a “pure supertranslation” , we will have L|\_[u\_0]{} = r = O(r\^[-1]{}) and L |\_[u\_1]{} = L|\_[u\_0]{} . In stereographic coordinates , recalling that $C_{z\bar z}=0$ by the trace constraint, this result takes the simple explicit form L|\_[u\_0]{} = , L |\_[u\_1]{} = L|\_[u\_0]{} + . The above formulas highlight a first kind of gravitational memory effect: two static test detectors will experience a change to order $\mathcal O(r^{-1})$ in their relative proper distance after being invested by a gravitational wave burst. A related type of gravitational memory effect can be revealed by considering the geodesic deviation of two test detectors, *initially* placed in the positions $(r,x^i)$ and $(r,x^i+\delta x^i)$, which are then left free to move and exposed to a radiation train [@StromingerGravmemevD]. In this case the general equation for geodesic deviation, +R u\^b u\^c \^d=0, where $\xi^a$ denotes the coordinate separation between the two particles and $u^a$ the corresponding four-velocity, reduces to \^2\_u x\^i + Rx\^j=0 upon restricting to the $i$th angular component, up to higher-order corrections in $1/r$, since $u^a = \delta^a_u+\cdots$ . Substituting the falloffs into the expression for the Riemann tensor on the right-hand side and in the Christoffel symbols arising on the left-hand side then affords \_u\^2 x\_i = \_u\^2 C\_[ij]{}x\^j to leading order, where $\delta x_i = \gamma_{ij}\delta x^j$. This equation shows that there can be no gravitational memory *kick* to leading order, contrary to the scalar and spin-one case, and can be integrated, up to further subleading corrections, to obtain the finite displacement x\_i|\_[u\_1]{}-x\_i|\_[u\_0]{} = ( C\_[ij]{}|\_[u\_1]{}-C\_[ij]{}|\_[u\_0]{} ) x\^j|\_[u\_0]{} + . We note that this displacement effect again appears to order $\mathcal O(r^{-1})$. Similar arguments allow to prove that static or geodesic detectors also undergo a relative permanent time delay due to the passage of a gravitational radiation train [@memory], thus providing yet another example of gravitational memory effect. Maxwell and Yang-Mills Theory in Higher Dimensions {#chap:spin1higherD} ================================================== In this chapter we will explore the structure of asymptotic symmetries and charges in connection with memory effects and soft theorems in higher spacetime dimensions. We shall do so by providing an original analysis of linear and self-interacting spin-one theories, Maxwell and Yang-Mills, in $D\ge4$ [@Cariche-io; @Memory-io]. As already remarked in the four-dimensional case, the asymptotic structure of such theories shares many interesting features with the gravitational case: an infinite-dimensional asymptotic symmetry group, associated soft surface charges and soft theorems, ordinary and null memory effects, and, in the Yang-Mills case, the phenomenon of color leak. Therefore, besides being interesting in its own right, the analysis of such theories provides a conceptual laboratory for ideas and phenomena associated with the gravitational case. It also furnishes a convenient working ground in view of the higher-spin extension which we will discuss in Chapter \[chap:HSP\]. In Section \[sec:YMGlobal\] we shall address the main novel feature of the asymptotic expansion in dimensions greater than four, namely the separation into two distinct branches of Coulombic and radiation terms, in the context of a conservative analysis performed in the radial gauge. This will allow us to explicitly calculate the expressions for the power radiated by a Yang-Mills field, for the total color charge at a given retarded time, and to derive a formula expressing the time-dependence of the latter due to nonlinearities [@Cariche-io]. In particular, let us anticipate that Coulombic terms, appearing to order $\mathcal O(r^{3-D})$ in the asymptotic expansion of the gauge fields, will be responsible for finite and nonvanishing color charges. Conversely, radiation terms, which scale as $\mathcal O(r^{\frac{2-D}{2}})$ and are therefore leading compared to Coulombic order, will give no contribution to color charges, while giving rise to nonzero charge leak and energy flux. While providing satisfactory answers as far as global charges and fluxes are concerned, no infinite-dimensional asymptotic symmetry enhancement will stem from the above analysis, in striking contrast with the situation in four dimensions. This puzzling feature would seemingly leave open the question regarding the ultimate origin of the Weinberg photon theorem. The validity of this theorem is independent of the dimension of spacetime and, as we have seen in Chapter \[chap:obs\], it can be understood as a manifestation of the invariance of the $S$ matrix under asymptotic symmetries in four dimension. Another class of phenomena related to the invariance of four-dimensional electrodynamics under an infinite-dimensional family of asymptotic symmetries is provided by memory effects, which can be interpreted as signaling the transition between two inequivalent radiative vacua. Therefore, it appears natural to investigate the relation between memory effects, if any, and the classical vacuum structure of the theory in higher dimensions as well. For these reasons, Section \[sec:memoryexamples\] is then devoted to the exploration of memory effects in higher-dimensional spacetimes by means of explicit examples in the context of scalar and electromagnetic theories. Indeed, both ordinary [@Mao_EvenD] and null [@Memory-io] kick memory effects, which are precluded to radiation order, contrary to what was preliminarily conceived [@Garfinkle:2017fre; @WaldOddD], can be shown to appear to Coulombic order. We shall take the occurrence of memory effects as an additional piece of evidence indicating that the classical vacuum structure of gauge theories in higher dimension is far from the trivial picture that the analysis performed in Section \[sec:YMGlobal\] would apparently suggest. This fact, together with the very existence of Weinberg’s theorem, provides the basic motivation for performing further investigations of the asymptotic structure of electromagnetic and Yang-Mills theory in Section \[sec:em\], where we will be able to uncover a link between a suitable subclass of residual symmetries of the Lorenz gauge and memory effects. More specifically we shall see how ordinary and null kick memories can be indeed interpreted as vacuum transitions from the gauge theory perspective. This also holds for a type of memory which we did not discuss in Chapter \[chap:obs\]: phase memory in Maxwell theory and color memory in Yang-Mills, which can be interpreted as phase/color rotations induced in the Hilbert space of two asymptotic test charges [@Memory-io]. The key-points of this refined analysis are a different treatment of falloff conditions and a more flexible gauge-fixing choice, identified in the Lorenz gauge condition, which will allow us to establish a connection, on the one hand, between a suitable family of residual symmetries and memory effects, and on the other hand between *bona fide* asymptotic symmetries and nontrivial asymptotic charges responsible for the Weinberg theorem [@Memory-io]. The analysis presented in this section is partly inspired by [@StromingerGravmemevD], for the treatment of memory, and by [@StromingerQEDevenD], for the proposal of alternative falloff conditions. Asymptotic Charges for Yang-Mills {#sec:YMGlobal} --------------------------------- In this section we analyze the equations of motion for Yang-Mills theory in Minkowski spacetime, for any dimension $D$, by means of an expansion of the fields in powers of $1/r$, thereby identifying the data that contribute to color charge and to color or energy flux at null infinity [@Cariche-io]. In particular, we will complement the related discussions in [@Strominger_YM; @Strominger_QED; @StromingerKac; @StromingerColor; @Maxwelld=3Barnich; @Campiglia; @Adamo:2015fwa; @Mao_note] by providing a unified treatment of all spacetime dimensions, and those in [@Einstein-YM.Barnich; @Mao_EvenD] by explicitly checking the finiteness of asymptotic charges in any dimension while also including radiation for $D=3$. We adopt the usual retarded Bondi coordinates $u$, $r$, $x^i$, where $x^i$, for $i=1,2,\ldots,D-2$, denotes the $D-2$ angular coordinates on the sphere at null infinity. The Yang-Mills connection is denoted by $ \mathcal A_a = \mathcal A_a^A T^A\, , $ where the $T^A$ are the anti-Hermitian generators of the $su(N)$ algebra, and its gauge transformation is $ \delta_\epsilon\mathcal A_a = \nabla_{a} \epsilon + \left[ \mathcal A_a, \epsilon \right]$. The corresponding field strength reads F\_[ab]{} = \_[a]{} A\_b - \_[b]{} A\_a + , while the field equations are $\mathcal G^a=0$ with \[EEOM\] G\^b = \_a F\^[ab]{} + . In terms of retarded Bondi coordinates, we have[^5] \[eomYMD\] G\^u &=(\_r + )F\_[ur]{} + D\^i F\_[ri]{} + \[A\_r, F\_[ur]{}\]+\^[ij]{}\[A\_i, F\_[rj]{}\],\ G\^r &= \_u F\_[ru]{}+ D\^i(F\_[ir]{}-F\_[iu]{})+\[A\_u,F\_[ru]{}\]+\^[ij]{}\[A\_i, F\_[jr]{}-F\_[ju]{}\],\ G\_i &= \_u F\_[ir]{} + (\_r + ) (F\_[ri]{}-F\_[ui]{})+D\^j F\_[ji]{}\ &+ \[A\_u, F\_[ir]{}\]+\[A\_r, F\_[ri]{}-F\_[ui]{}\]+\^[jl]{}\[A\_j, F\_[li]{}\]. Furthermore, we enforce the radial gauge \[bondi1\] A\_r=0 , which completely fixes the gauge in the bulk. ### Boundary conditions For $D > 3$ we consider field configurations $\mathcal A_\mu$ whose asymptotic null behavior is captured by an asymptotic expansion in powers of $1/r$, for $r\to\infty$, while, as we shall briefly discuss in Section \[sec:3-4dim\_1\], in the three-dimensional case, we shall also consider a logarithmic dependence on $r$. In order to generalize the falloffs adopted in four dimensions (see Section \[sec:YMD=4\]) and provide a suitable guess for the form of the asymptotic expansions, we shall adhere to the following guiding principle: namely that our field configurations should furnish finite and nonzero energy flux and color energy at null infinity. The energy per unit time flowing across a section $S_u$ of $\mathscr I^+$ at constant retarded time $u$ can be calculated as follows. Starting from the Yang-Mills Lagrangian $\cL = \frac{1}{4}\mathrm{tr}(\mathcal F_{ab} \mathcal F^{ab})$, the stress-energy tensor takes the form $T_{ab}=-\mathrm{tr}\left(\mathcal F_{ac}\mathcal F\indices{_b^c}\right)+\frac{1}{4}g_{ab}\mathrm{tr}\left(\mathcal F_{cd}\mathcal F^{cd}\right)$. The energy flux across $S_u$ is then given by $-\int_{S_u} T\indices{^r_u} r^{D-2} d\Omega=\int_{S_u} (T_{uu}-T_{ur}) r^{D-2} d\Omega\, $ as $r\to\infty$. Therefore, \[power\_u\] P(u)=\_[r]{}\_[S\_u]{} \^[ij]{} (F\_[ui]{}(F\_[rj]{}-F\_[uj]{})) r\^[D-4]{} d. The request that be finite imposes that the integrand must go to zero at infinity in order to compensate for the factor of $r^{D-4}$. Furthermore, to saturate this bound and furnish a nonzero energy flux, the slowest decaying terms in the asymptotic expansion of the field strength, which we name *radiation* terms, should scale as $\mathcal O(r^{\frac{4-D}{2}})$, so as to cancel the measure factor $r^{D-4}$ exactly. On the other hand, a quick calculation of the color charge at null infinity, on which we shall provide further details in sect. \[subsec: Charges\], allows us to express it as the following integral over $S_u$, \[color\_u\] Q\^A(u)=\_[r]{}\_[S\_u]{} F\_[ur]{}\^A r\^[D-2]{} d, where we have employed the formula $\kappa_\epsilon^{ab}=\sqrt{-g}\,\mathrm{tr}(\mathcal F^{ab}\epsilon)$ derived in \[sec:YMD=4\]. We conclude, that, in order for the above expression to be nonvanishing, the asymptotic expansion of $\mathcal F_{ur}$ must contain terms scaling $\mathcal O(r^{2-D})$ (equivalently, $\mathcal O(r^{3-D})$ in the field component $\mathcal A_u$ once we adopt radial gauge), which we will refer to as *Coulombic* contributions. The above discussion of energy flux and color charge immediately highlights an interesting feature, namely that radiation terms give contributions that diverge at face value when inserted in the expression for the color charge. As we shall see below, the solution to this apparent paradox is obtained by first evaluating the charge *on-shell* on a sphere of finite $r$, and only then sending $r$ to infinity. This is in fact trivially the case as far as the linear theory is concerned, since there the equations of motion reduce to $\nabla_a \mathcal F^{ab}=0$, which ensures the independence of the flux $\int_{S_{u,r}}\mathcal F^{ab}\,d\sigma_{ab}$ both on $r$ and on $u$ by the Stokes formula. The situation is more interesting in the case of the nonlinear theory, since in this case $\nabla_a \mathcal F^{ab}=[F^{ab},\mathcal A_a]$ and the behavior of the surface charge as we vary $r$ and $u$ crucially depends on the falloffs of the field, precisely due to the nonlinear nature of the theory. This cancellation of potentially divergent terms due to the equations of motion will turn out to be possible on account of the fact that, since in higher dimensions the field strength components must decay as $r$ grows, the nonlinear terms cannot appear in the leading-order equations, provided that strictly decaying falloff conditions are also assigned to the gauge potential. This mechanism can be regarded as an *asymptotic linearization* of the equations of motion. Another interesting result of the above comparison between radiation and Coulombic terms is that the asymptotic expansion must include both integers and half-odd powers of $r$ in odd-dimensional spacetimes in order to account for both radiation and the presence of charges. This choice is forced by the fact that, as remarked above, the leading-order radiation components of the field strength scale as $\mathcal O(r^{-\frac{D}{2}})$ while Coulombic ones behave as $\mathcal O(r^{2-D})$. A bonus feature that will emerge from the resulting analysis is the description of the color flux for large $r$, namely the interplay occurring between radiation and Coulombic terms. Before moving to the explicit calculations, let us stress once again that the presence of such two distinct classes of terms in the solutions, radiation and Coulombic, is a genuinely new feature of $D>4$, while in the four-dimensional case they effectively coincide. #### Equations of motion When $D > 4$ is even, in order to saturate the bound set by , it is natural to consider an expansion of the following type in the radial gauge $\mathcal A_r=0$: \[expansion\_D\_even\] A\_u = \_[k=]{}\^,A\_i = \_[K=]{}\^, where $A_u^{(k)}$ and $A_i^{(k)}$ are functions of retarded time $u$ and of the angles $x^i$ and the index $k$ takes integer values. On the basis of the previous discussion we expect $A_u^{(\frac{D-2}{2})}$ and $\mathcal A_i^{(\frac{D-4}{2})}$ to play the role of leading radiation terms, and indeed P(u)=-\_[r]{}\_[S\_u]{} \^[ij]{} (\_u A\_i\^[()]{} \_u A\_j\^[()]{}) d. This is the expression for energy flux, which depends quadratically on the time derivative of the radiation terms. Furthermore, it is also positive definite, according to our choice of anti-Hermitian generators. The seeming discrepancy of a factor of $r$ in the $u$ and $i$ components, which is shared by the four-dimensional case, is actually due to the fact that we are employing a non-normalized, coordinate basis for the spherical components. A normalized basis is obtained by defining the noncoordinate vectors $\mathbf e_i=\frac{1}{r}\,\partial_i$ and similarly $\tilde{\mathbf e}^i=r\,dx^i$ for their duals. Correspondingly A = A\_u du + A\_i dx\^i = A\_u du + A\_i \^i, which explains the extra factor of $r$. We shall now substitute the above asymptotic expansion into the equations of motion , starting from $\mathcal G^u=0$, which gives the recursion relations \[Furr\] k(D-3-k)A\_u\^[(k)]{}-(k-1)D\^i A\_i\^[(k-1)]{}-\_l\^[ij]{}\[A\_i\^[(k-l)]{}, (l-1)A\_j\^[(l-1)]{}\]=0. This equation will be crucial for the discussion of the finiteness of charges. It also allows us to illustrate explicitly the phenomenon of asymptotic linearization. Indeed, the nonlinear term in the above equation will appear when both $l-1\ge\frac{D-4}{2}$ and $k-l\ge \frac{D-4}{2}$, which implies $l\ge \frac{D-2}{2}$ and $k\ge D-3$. The latter condition in particular tells us that the nonlinear terms do not enter the first few instances of the recursion relations, namely those with $\frac{D-2}{2}\le k \le D-4$. In other words, the nonlinear contributions are absent at radiation order and start appearing at Coulombic order. Furthermore, we extract the term proportional to $r^{2-D}$ in $\mathcal G^r=0$, which will be instrumental to calculating the time-dependence of charges, namely \[Furu\] (D-3)\_u A\_u\^[(D-3)]{}=\^[ij]{}\[A\_i\^[()]{}, \_u A\_j\^[()]{}\]+, which, remarkably, links the time dependence of the Coulombic term $A_u^{(D-3)}$ precisely to a commutator of radiation terms. The terms omitted in are of the form $D^i f_i$ and will not give contributions to the results in the next section. In the case of odd dimensions $D>4$, as anticipated, we have to include two distinct expansions in $1/r$ in order to capture both radiation and Coulombic terms: A\_u = \_[k=]{}\^ +\_[k=D-3]{}\^, A\_i = \_[k=]{}\^ +\_[k=D-4]{}\^. Aside from this observation, the discussion proceeds essentially in the same way as in the even-dimensional case, and in particular and still hold. #### Three and four spacetime dimensions {#sec:3-4dim_1} In $D = 4$, which we discussed in Section \[sec:YMD=4\], the leading radiation term and the Coulombic term coincide and the main information regarding the dynamics is contained in \[flux\_four\] \_u A\_u = \_u D\^i A\_i + \^[ij]{}. The situation for $D = 3$ is rather different with respect to the previous cases, mainly because of two features. First, the factor of $r^{-1}$ in tells us that, in order to produce a finite energy flux across $S_u$, the field components need not necessarily decay at infinity; consequently, one expects no clear distinction between radiation and Coulombic terms in the solution, because no asymptotic linearization occurs in the equations of motion. Second, the expression , and more specifically the factor of $r$ due to the line element on the circle, suggests that $\mathcal A_u$ should behave as $\log\frac{1}{r}$ in order to give a nonvanishing color charge. These considerations motivate the following leading-order ansatz in three dimensions: A\_u (u,r,) \~q +p,A\_(u,r,) \~ C, where $q$, $p$ and $C$ are $r$-independent functions. Indeed, with this choice, the color flux and the energy flux read \[QP3\] Q\^A (u) = \_[S\_u]{} q\^A d,P (u) = -\_[S\_u]{} (\[q,C\]\[q,C\]) d . Using this ansatz we find \[flux\_three\] \_u q =- , as the only relevant constraint arising from the equations of motion. This equation describes the $u$-evolution of $q$ at null infinity and hence, together with the first of , will lead to a formula for the color flux. ### Global symmetries and charges {#subsec: Charges} In this section we would like to discuss the form of the color charge at null infinity in the various dimensions. For related analyses see [@abbott-deser; @Barnich-Brandt; @Avery_Schwab; @Mao_note]. To begin with, let us discuss which large gauge symmetries are admissible at null infinity. The residual gauge symmetry within the radial gauge is parameterized by an $r$-independent gauge parameter, since 0=\_A\_r = \_[r]{} + but $\mathcal A_r=0$, hence $\partial_{r}\epsilon=0$. Then, we look for those parameters $\epsilon$ which preserve the leading falloff conditions imposed on the field $\mathcal A_a$, again distinguishing the case of $D>4$ from $D=4$ and $D=3$. When $D>4$, where radiation gives the dominant behavior at infinity, we find, to leading order r\^ \_A\_u = \_u + r\^\[A\_u, \] , which requires $\partial_u\epsilon=0$ (because $D>2$). Furthermore r\^ \_A\_i = \_i + r\^\[A\_i, \] , but, since $D>4$, this also implies $\partial_i\epsilon=0$, which reduces $\epsilon$ to a constant. Hence, in $D>4$, asymptotic symmetries coincide with the global part of the gauge group and the asymptotic charge is the ordinary color charge computed via . Now, using and recalling by the above discussion that the nonlinear terms do not contribute for $k<D-3$, we have \[ColorChargeglobale\] Q\^A(u) &= \_[r]{}\_[S\_u]{} (F\_[ur]{} T\^A) r\^[D-2]{} d\ &= (D-3)\_[S\_u]{}(A\_u\^[(D-3)]{})\^Ad, where we have used the fact that the terms of the type $D^i A_i^{(k-1)}$ give no contribution to the integral since integrate to zero on the sphere. This formula is valid for any spacetime with dimension $D>4$ of either parity. The time dependence of $\mathcal Q^A(u)$ is furnished by taking the derivative with respect to $u$ and recalling Q\^A(u) = \_[S\_u]{} \^[ij]{} \^A d. This is again consistent with the interpretation of $A_i^{(\frac{D-4}{2})}$ as the leading radiation term: this formula describes how Yang-Mills radiation, made of “classical gluons”, flowing to null infinity induces a change in the total color at successive retarded times $u$. In $D=4$, the gauge parameter must satisfy: r\^[ -1]{} \_A\_u & = \_u + r\^[-1]{}\[A\_u, \] ,\ \_A\_i & = \_i + \[A\_i, \] . The first equation again enforces $\partial_u \epsilon=0$, whereas the second allows for an $\epsilon(x^1, x^2)$ with arbitrary dependence on the angles on the celestial sphere. The corresponding asymptotic charge is therefore Q\_(u) = \_[r]{}\_[S\_u]{} (F\_[ur]{}) r\^2 d= \_[S\_u]{} (A\_u ) d. Taking into account , Q\_(u) = \_[S\_u]{} d. To complete the picture, let us now turn to the situation in $D=3$. There, neither $\mathcal A_u$ nor $\mathcal A_\varphi$ fall off at infinity, and hence any $\epsilon(u,\varphi)$ generates an allowed gauge transformation (the same result, in a slightly different setting, was already obtained in [@Maxwelld=3Barnich]). Thus, using the notation of the previous section, Q\_(u) &= \_[S\_u]{} (q ) d,\ Q\_(u) &= \_[S\_u]{} (q \_u) d- \_[S\_u]{} (\[q, p\] )d. Let us observe that these charges indeed form a representation of the underlying algebra: for $D\ge4$, since $\delta_\epsilon A_u= [A_u,\epsilon]$, \[commutation\_rel\] \[Q\_[\_1]{},Q\_[\_2]{}\]=\_[\_1]{} Q\_[\_2]{}= \_[S\_u]{} (\[A\_u, \_1\]\_2) d= \_[S\_u]{} (A\_u \[\_1,\_2\]) d= Q\_[\[\_1,\_2\]]{}. The same result holds for $D=3$, noting that $\delta_\epsilon q= [q,\epsilon]$ and $\delta_\epsilon p = \partial_u \epsilon$, but $p$ does not enter the charge formula. While holds in any dimension, it should be stressed that, when $D>4$, the corresponding charge algebra coincides with $su(N)$, whereas in $D=4$ and $D=3$ it is in fact an infinite-dimensional Kac-Moody algebra, owing to the arbitrary gauge parameters $\epsilon(x^1,x^2)$ and $\epsilon(u,\varphi)$. In particular, we note the absence of a central charge, which could however emerge by performing the analysis for the linearized theory around a nontrivial background, as pointed out in [@Avery_Schwab]. Let us conclude this section by comparing the calculation of the Noether charge with the one obtained by means of covariant phase space techniques. In particular, we will check that, for Yang-Mills theory, the quantity Q\_= \_ dx\_ (F\^ ), where $\Sigma$ is a Cauchy surface, provides not only the conserved charge as obtained by the Noether algorithm, but also the Hamiltonian generator of the gauge symmetry parameterized by $\epsilon$ on the space tangent to the surface of solutions, as calculated via covariant phase space methods. Indeed, a generic variation of the Yang-Mills Lagrangian, after integrating by parts, reads \[delta L\] L=-(G\^ A\_) + \_(F\^A\_) = -(G\^ A\_) +\_\^(A) , where we defined the symplectic potential $\theta^\mu(\delta A)=\mathrm{tr}\left(F^{\mu\nu} \delta A_\nu\right)$, while $\mathcal G^\mu$ denotes the Euler-Lagrange derivatives of $\mathcal L$, given in . The presymplectic form is then given by \^(\_1 A, \_2 A)=\_[\[1]{}\^(\_[2\]]{}A) , with square brackets denoting antisymmetrization, and correspondingly the formal variation of the Hamiltonian generator of the gauge symmetry $H_\epsilon$ is H\_= \_dx\_\^(A, \_A)=\_ dx\_ (F\^)- \_ dx\_(G\^). Noting that the last term is proportional to the linearized equations of motion, i.e. that it vanishes on the space tangent to the surface of solutions, we can write $$\slashed{\delta} H_\epsilon \approx \delta Q_\epsilon\,,$$ which explicitly shows that $\slashed{\delta} H_\epsilon$ is integrable and that we may choose to set $H_\epsilon = Q_\epsilon$ by requiring a flat connection to have zero charge. Furthermore, the Noether charge is simply \_dx\_ \^(\_A) = Q\_- \_dx\_ (G\^) Q\_, so that the two approaches agree in this case. The definition of $Q_\epsilon$ is in principle subject to ambiguities stemming from $\theta^\mu \mapsto \theta^\mu + \partial_\nu \lambda^{\mu\nu}$, where $\lambda^{\mu\nu}=-\lambda^{\nu\mu}$, which anyway do not alter . In the spirit of [@Wald-Zoupas], we may choose to set to zero the corresponding additional terms, precisely because this choice defines an integrable Hamiltonian, as shown above. Further motivation for the absence of these terms is provided by the agreement with the general analysis of [@Barnich-Brandt] and by the fact that they play no role in the generation of Ward identities for residual gauge freedom [@Avery_Schwab]. A possible way to eliminate this ambiguity could be to study its role in connection with the the addition of boundary terms to the Yang–Mills action and the well–definiteness of the corresponding variational principle [@CompereBoundaryAmb]. Classical Solutions and Memory Effects in Higher $D$ {#sec:memoryexamples} ---------------------------------------------------- As already stressed, the absence of infinite-dimensional asymptotic symmetries in $D>4$ would be at odds with the persistence of universality relations in soft amplitudes. Additional arguments in this respect actually come from a proper investigation inquiring on the presence or absence of higher-dimensional memory effects. Indeed, this latter aspect was analyzed in the literature from various angles, with emphasis on the gravitational case, leading sometimes to opposite conclusions on the status of memory effects in higher $D$ [@Garfinkle:2017fre; @WaldOddD; @Mao_EvenD; @SatishchandranWald]. Here, we shall consider the issue from the perspective of scalar and vector fields and provide explicit calculations of full-fledged memory effects in any even dimension, while also commenting on the difficulties associated with the nature of odd-dimensional memory effects. Our conclusions further motivate reconsidering the problem of asymptotic symmetries from a broader perspective. ### Scalar fields in even $D$ {#sec:spin0} Let us first consider a particle with scalar charge $q$ that is created at the origin at $t=0$. The scalar field $\varphi$ generated by this process is obtained by solving the wave equation \[scalar-theta\] -(t,x) = q (t)(x) (recalling $-\Box=-\eta^{\mu\nu}\partial_\mu\partial_\nu=\partial_t^2-\nabla^2$). The solution is given by the convolution of the source on the right-hand side with the ($D$–dimensional) retarded wave propagator $D_\text{ret}(x)$, *i.e.* in this case (t,x) = q \_[-]{}\^[t]{} D\_(, x)d. The field generated by a particle that is destroyed at the origin can then be obtained by time reversal of the above solution, while the field for a moving particle can be calculated by applying a Lorentz boost. Let us recall that, for even $D\ge2$, the retarded propagator is \[prop-even\] D\_(x)= \^[()]{}(x\^2)(x\^0) (see *e.g.* [@Friedlander]), where $\theta$ is the Heaviside distribution. Restricting to even $D\ge4$, and using the chain rule for the distribution $\delta^{(\frac{D-4}{2})}(x^2)$, the propagator can be recast as D\_(t, x)= \_[k=0]{}\^[D/2-2]{} c\_[D,k]{} , where $u=t-r$ and $r=|\mathbf x|$, while with $c_{D,k}$ we denote the coefficients c\_[D,k]{} = . In particular note that c\_[D,0]{}=,c\_[D,-3]{} = c\_[D,-2]{}= , where $\Omega_{D-2}$ is the area of the $(D-2)$-dimensional Euclidean unit sphere. The resulting scalar field is thus \[scalar-basic\] (u,r)=q \_[k=0]{}\^[D/2-2]{} c\_[D,k]{} . Notice that only the term associated to $k=D/2-2$ gives rise to a persistent field for fixed $r$, while the other terms have support localized at $u=0$, namely on the future-directed light-cone emanating from the particle’s creation event. This is a general consequence of the recursion relation , namely (D-2k-2)\_u \^[(k)]{} = \[+(k-1)(k-D+2)\]\^[(k-1)]{}, obeyed by $\varphi$ near future null infinity, which requires $[\Delta +k(k-D+3)]\varphi^{(k)}=0$ for $0<k<D-3$, and hence $\varphi^{(k)}=0$, in that range, for any stationary solution. Now, a test particle with charge $Q$, held in place at a distance $r$ from the origin, will be subject to a force $f_a =Q\,\partial_a \varphi(u,r)$ at a given retarded time $u$ due to the presence of the scalar field. Hence, its $D$–momentum $P_a$ will in general be subject to the leading-order variation P\_a|\_[u]{}-P\_|\_[u=-]{}=Q\_[-]{}\^u \_a (u’,r) du’. For this very simple example, this quantity can be calculated explicitly for any even $D$. The variations of $P_u$ and $ P_r$ in particular yield \[scalarmem0\_u\] P\_u|\_[u&gt;0]{}-P\_u|\_[u&lt;0]{}= Q\_[-]{}\^[+]{} \_u(u’, r)du’ = , and \[scalarmem0\_r\] P\_r|\_[u&gt;0]{}-P\_r|\_[u&lt;0]{}= Q\_[-]{}\^[+]{} \_r(u’, r)du’= -. Equation simply expresses the fact that the test particle will start feeling the Coulombic interaction energy with the newly created particle at the origin as soon as it crosses the light-cone subtended by the origin of spacetime. On the other hand, tells us that the test particle will feel an instantaneous, radial momentum kick, for even dimensions greater than four. Consistently with the spherical symmetry of this process, the variations of the angular components $P_i$ vanish identically. The field emitted by a particle destroyed in the origin at $t=0$ is obtained by sending $u\mapsto -u$ in . The case of a particle moving with velocity $\mathbf v$ can be instead obtained by boosting : \[standardboost\] t (v)(t-v x),x x + v ((v)-1)- (v)v t, which gives, for large $r$, denoting $\mathbf n = \mathbf x/r$, \[transf-ur\] u u (v)\^[-1]{}(1-n v)\^[-1]{}+O(r\^[-1]{}),r r (v)(1-n v)+O(1). We can then cast the boosted solution in the following form: \[boosted-scalar\] (u,r, n) = +|(u, r, n)+O(r\^[2-D]{}), where $\bar\varphi$ is a sum of terms of the type f\_(r, n)\^[()]{}(u),0, that is to say, whose support is localized on the light-cone. Let us stress that the terms in $\bar \varphi$ formally dominate the asymptotic expansion of $\varphi$ as $r\to\infty$. However, these terms will not contribute to the leading $u$-component of the momentum kick due to the presence of $\delta(u)$ and its derivatives. We can therefore conclude that P\_u|\_[u&gt;0]{}-P\_u|\_[u&lt;0]{} =Q\_[-]{}\^[+]{} \_u du’= , in any even $D$. This is, not surprisingly, just the analog of equation for the Coulombic energy in which one needs to account for the relativistic length contraction. For a more general scattering process involving a number of “in” and “out” particles destroyed or created in the origin, the result is obtained by linearly superposing solutions and therefore reads ($\eta_a=-1$ for an incoming particle and $\eta_a=+1$ for an outgoing one) P\_u|\_[u&gt;0]{}-P\_u|\_[u&lt;0]{} = \_[a]{}. Calculating radial and angular components of $P_\mu$ requires more effort, since they arise instead from the terms proportional to $\delta(u)$ in the expansion of $\bar \varphi$, whose number increases with the spacetime dimension. They have been given for any even dimension in [@Mao_EvenD] in terms of derivatives of a generating function. For our present, illustrative, purposes, it suffices to consider the first relevant case $D=6$, where the exact solution in the case of the particle created in the origin with velocity $\mathbf v$ is given by \[exactscalarD=6\] 8\^2 = +(u,r)\^[-3/2]{} with (u,r) = 1 + +. The corresponding radial and angular memory effects in $D=6$ are then, to leading order, P\_r|\_[u&gt;0]{}-P\_r|\_[u&lt;0]{} &= ,\ P\_i|\_[u&gt;0]{}-P\_i|\_[u&lt;0]{} &= , where $v_i = \partial_i \mathbf n \cdot \mathbf v$ is the component of the particle’s velocity in the $i$-th angular direction. While the above examples illustrate the phenomenon of ordinary memory, associated with the field emitted by massive charges that move in the bulk of the spacetime, we can also consider the wave equation with a source term characterizing the presence of a massless charged particle [@Memory-io], moving along a given direction $\mathbf x_0$: -= q (x - x\_0 t), with $|\mathbf x_0|=1$. This equation can be conveniently solved for any even $D\ge6$ by going to retarded coordinates, where it reads (2\_r + )\_u = (\_r\^2 + \_r + )+(u)(n, x\_0) and performing the usual asymptotic expansion $\varphi(u,r,\mathbf n)= \sum \varphi^{(k)}(u, \mathbf n)r^{-k}$, which gives \[recursionnullD\] (D-2k-2)\_u \^[(k)]{} = \[+ (k-1)(k-D+2)\]\^[(k-1)]{}+\_[k,D-3]{}(u)(n, x\_0). The latter equation is solved by setting $\varphi^{(k)}=0$ for $k\le \frac{D}{2}-2$ and for $k\ge D-3$, while, for $\frac{D}{2}-1\le k \le D-4$, \[nullmemk\] \^[(k)]{}(u, n)=\^[(D-4-k)]{}(u)C\_k(n), where the functions $C_k(\mathbf n)$ are determined recursively by (D-2k-2) C\_k(n) &= \[+ (k-1)(k-D+2)\]C\_[k-1]{}(n),\ C\_[D-4]{}(n) &= -(-D+4)\^[-1]{}(n,x\_0). Here, $(\Delta-D+4)^{-1}$ is the Green’s function for the operator $\Delta-D+4$, which is unique for $D>4$. As a consequence, the field gives rise to the null memory effect P\_i|\_[u&gt;0]{}-P\_i|\_[u&lt;0]{} = \_[-]{}\^[+]{} \_i du = -\_i(-D+4)\^[-1]{}(n,x\_0) (note that only the term with $k=D-4$ contributes), consisting in a kick along a direction tangent to the celestial sphere. ### Electromagnetic fields in even $D$ {#sec:memories} Since we shall work in the Lorenz gauge, much of the calculations are essentially the same as those of the scalar case, which we have discussed in the previous section. In Cartesian coordinates, where the gauge condition reads $\partial^\mu \mathcal A_\mu =0$, the equations of motion reduce to a set of scalar wave equations A\_= j\_. First, we consider the case of a static point-like source created at the origin,[^6] \[createdcha\] A\^= u\^q(t)(x), where $u^\mu=(1,0,\ldots,0)$, hence the retarded solution is A\^= - u\^, where $\varphi$ denotes the corresponding solution for the scalar field. Boosting this solution according to yields $\mathcal A^\mu=(\mathcal A^0, \mathbf A)=-\gamma(\mathbf v)(1,\mathbf v)\varphi$ and going to retarded coordinates gives A\_u = (v),A\_r = (v)(1-n v),A\_i = -r(v) v\_i . Focusing for simplicity on the case of $D=6$, let us consider the radiation field generated by a massive particle with charge $q$ sitting at rest in the origin for $t<0$ that starts moving with velocity $\mathbf v$ at $t=0$. Such field is obtained by matching the solution for a charge destroyed at the origin to that of a charge there created with velocity $\mathbf v$. Proceeding as before, one obtains, for large values of $r$, \[dim6vectorpot\] 8\^2A\_u &= (-1 ) +O(r\^[-3]{}) ,\ 8\^2A\_r &= (r\^[-3]{})\ 8\^2A\_i &= --+(r\^[-3]{}), where we have kept the orders relevant to the calculation of the memory effect. The change in the angular components of the momentum of a test charge $Q$, initially at rest, gives rise to a [linear]{} ([ordinary]{}) memory effect that, to leading order, reads \[memoryformulaD=6\] P\_i|\_[u&gt;0]{}-P\_i|\_[u&lt;0]{} = Q \_[-]{}\^[+]{} F\_[iu]{} du = +(r\^[-3]{}), where the magnetic force does not contribute to subleading orders, because it will be further suppressed by the powers of $1/r$ appearing in the velocity. With hindsight, having in mind in particular the results of [@Mao_EvenD], in order to interpret this leading memory effect in terms of a symmetry, it is useful to perform a further gauge fixing procedure in order to get rid of the residual symmetries affecting the radiation order. Choosing a gauge parameter of the form 8\^2= - (-1 )+O(r\^[-3]{}) allows us to cancel the leading term of $\mathcal A_u$, compatibly with the condition $\Box \epsilon=0$. The resulting field after performing this gauge transformation satisfies A\_u = O([r\^[-3]{}]{}),8\^2A\_i= - - +O(r\^[-3]{}). In particular, the $\mathcal O(r^{-2})$ component of $\mathcal A_u$ is now zero[^7], so that $\mathcal A_i$ is fully responsible for the memory formula . The effect is proportional to the variation of the latter field component between $u>0$ and $u<0$ and takes the form of a total derivative on the celestial sphere (*i.e.* a gauge transformation): \[specificexamplememory\] A\_i|\_[u&gt;0]{}-A\_i|\_[u&lt;0]{} &= -\ &=-q\_i ( + ). This result provides an explicit connection between the memory effect and a residual symmetry acting, for large values of $r$, at Coulombic order. Let us turn our attention to the case of *null* memory. Let us consider, in any even dimension $D\ge6$, the field generated by a charge moving in the $\mathbf x_0$ direction at the speed of light: in Cartesian coordinates, A\^= q v\^(x- x\_0 t), with $v^\mu=(1,\mathbf x_0)$ and $|\mathbf x_0|=1$. Taking into account the corresponding retarded solution for the scalar field $\varphi$, we then have $A^\mu = - v^\mu \varphi $ and, moving to retarded coordinates, A\_u = ,A\_r = (1-n x\_0),A\_i = -r (x\_0)\_i . Consequently \[AnullD\] A\_u &\~-(-D+4)\^[-1]{}(n, x\_0),\ A\_r &\~-(1-n x\_0)(-D+4)\^[-1]{}(n, x\_0),\ A\_i &\~r(-D+4)\^[-1]{}(n, x\_0), where $(\Delta-D+4)^{-1}$ denotes the Green function for the operator $\Delta-D+4$ and we have omitted terms proportional to higher-order derivatives of the delta function $\delta(u)$ (see ), which are *leading* with respect to those displayed in , but which do not contribute to the memory effect. The null memory formula then reads \[nullmemDeven\] P\_i|\_[u&gt;0]{}-P\_i|\_[u&lt;0]{}= Q\_[-]{}\^[+]{} F\_[iu]{} du’ = -\_i(-D+4)\^[-1]{}(n, x\_0). ### Overview of the odd-dimensional case In odd dimensions $D\ge3$ the retarded propagator is given by [@Friedlander] D\_(x)= c (-x\^2)\_+\^[1-]{}(x\^0), where $ c^{-1}=2\pi^{\frac{D}{2}-1}\Gamma(2-\frac{D}{2}) $, while $(\kappa)_+^\alpha$ is the distribution defined as ()\_+\^, () = \_0\^\^()d&gt;-1, $\chi(\kappa)$ denoting a generic test function, and analytically continued to any $\alpha\neq -1$, $-2$, $-3$, $\ldots$ by ()\_+\^, () = ()\_+\^[+n]{}, \^[(n)]{}() n&gt;-1-. A relevant feature of the wave propagator in odd dimensional spacetimes is that its support is not localized on the light-cone $|t|=r$, in contrast with the case of even dimensions, as it is nonzero also for $|t|>r$. This is to be interpreted as the fact that even an ideally sharp perturbation, $\delta(t,\mathbf x)$, will not give rise to an ideally sharp wave-front, but rather the induced radiation will display a dispersion phenomenon and nontrivial disturbances will linger on even after the first wave-front has passed. The solution to equation is then furnished by (t,x) = \^[1-D/2]{}\_+, (t-)/ . Integrating by parts, and assuming $t>r=|\mathbf x|$ (otherwise the field vanishes by causality), one obtains the following expansion \[phioddD\] &= + + +\ &+(-3)\ &+(-1)\^\_0\^[t\^2-r\^2]{}. Moving to retarded coordinates, this result can be recast as \[scalar-odd\] (u,r) = |(u,r) + (-1)\^(u) \_0\^[u(u+2r)]{} , where $\bar \varphi(u,r)$ is given by a sum of terms proportional to , with $\alpha$, $\beta$ positive and $\alpha+\beta$ half odd. In particular, it is then clear that the limit of this field as $r\to\infty$ for any fixed $u$ does not display any term with the Coulombic behavior $r^{3-D}$ and hence that there is no memory effect on $\mathscr I^+$ to that order, since \_0\^[u(u+2r)]{} \~\_0\^\~. Considering instead the limit of $\varphi$ as $t\to+\infty$ for fixed $r$, one sees that only the last term in survives and yields (-1)\^\_0\^ &= B(,)\ &= . This means that the Coulombic energy due to the newly created particle is felt by the test charge only after one has waited (for an infinite time) at a fixed distance $r$ that the perturbations due to the dispersion occurring in odd spacetime dimensions have died out. To some extent, this is to be regarded as a smeared-out memory effect, as opposed to memory effects occurring sharply at $\mathscr I^+$ near $u=0$ in even dimensions [@SatishchandranWald]. The situation does not improve if one considers a particle that is created with a nonzero velocity $\mathbf v$. Indeed, boosting the exact solution by means of , one sees that $\varphi$ goes to zero for fixed $r$ as $t\to+\infty$. The reason is that, while one waits for the dispersion to die out, the source, moving at a constant velocity, has traveled infinitely far from the test charge. Shifting our attention to the case of null memory, we see that it is possible to provide the following formal solution to the recursion relations , which hold in any dimension. We consider $\varphi= \sum \varphi^{(k)}r^{-k}$, setting $\varphi^{(k)}=0$ for $k\ge D-3$, while, for $k \le D-4$, \[nullmemkodd\] \^[(k)]{}(u, n)=\^[(D-4-k)]{}(u)C\_k(n), with the functions $C_k(\mathbf n)$ determined recursively by (D-2k-2) C\_k(n) &= \[+ (k-1)(k-D+2)\]C\_[k-1]{}(n),\ C\_[D-4]{}(n) &= -(-D+4)\^[-1]{}(n,x\_0). Thus, although the field is highly singular at $u=0$, the resulting null memory effect will be formally identical to the one occurring in even dimensions. Electromagnetic Memory and Residual Symmetries {#sec:em} ---------------------------------------------- In the previous section, we saw how to establish a connection between local symmetries acting at large $r$ and memory effects for specific matter configurations. Now we would like to investigate this connection beyond those examples by studying the general structure of the solution space in Lorenz gauge. As remarked above, the nontrivial nature of higher-dimensional memory effects suggests the existence of infinite-dimensional asymptotic symmetries, in spite of the analysis performed in Section . We will therefore analyze once more the asymptotic behavior of the fields in higher dimensions, adopting now the Lorenz gauge. First, we shall choose to adopt radiation falloff, which is sufficient to the description of the memory effect. A more general possibility will be discussed in Section \[sec:poly\], in order to explore the full structure of the asymptotic symmetry group in Lorenz gauge. The choice of Lorenz gauge is mainly motivated by the fact that it allows us to make a direct comparison with the above examples. However, as we will see, another important feature of this gauge is its greater flexibility with respect to the $r$-dependence of fields and gauge parameters, as opposed to the rigidity of radial gauge where no $r$-dependence of the gauge transformation is allowed. This will play a key role in the interpretation of memory effects as transitions between field configurations that differ by a residual symmetry acting at Coulombic order. ### Electromagnetism in the Lorenz gauge {#ssec:lorenz} The Lorenz gauge condition reads \[Lorenz\] \^a A\_a = -\_u A\_r + (\_r + )(A\_r - A\_u)+DA=0. The residual symmetry parameters then satisfy $\Box \epsilon=0$, namely \[residual\] (2\_r + )\_u = (\_r\^2 + \_r + ). The equations of motion reduce to $\Box \mathcal A_a=0$, which, component by component, reads \[eom\] &A\_u=0 ,\ &A\_r+(A\_u - A\_r)- DA=0 ,\ &A\_i-D\_i(A\_u -A\_r)=0. We assume now the expansions A\_a = \_k A\_a\^[(k)]{} r\^[-k]{}, = \_k \^[(k)]{}r\^[-k]{}, where the summation ranges are, for the moment, unspecified. Equations and then give $$\begin{aligned} \partial_u A^{(k+1)}_r&=(k-D+2)(A_u^{(k)}-A_r^{(k)})+D\cdot A^{(k-1)} \label{Lorenz_r}\, ,\\ (D-2k-2)\partial_u \epsilon^{(k)}&=[\Delta + (k-1)(k-D+2)]\epsilon^{(k-1)} \label{residual_r}\, , \end{aligned}$$ while from one obtains $$\begin{aligned} &(D-2k-2)\partial_u A_u^{(k)}=[\Delta + (k-1)(k-D+2)]A_u^{(k-1)} \label{equ_r}\, ,\\ &(D-2k-2)\partial_u A^{(k)}_r = [\Delta+k(k-D+1)]A_r^{(k-1)}\, +\, (D-2)\, A_u^{(k-1)}-2 D\cdot A^{(k-2)} \label{eqr_r}\, ,\\ &(D-2k-4)\partial_u A_i^{(k)}=[\Delta+k(k-D+3)-1]A_i^{(k-1)}-2D_i(A_u^{(k)}-A_r^{(k)}) \label{eqi_r}\,.\end{aligned}$$ Equations — appear in particular to order $r^{-k-1}$ in the asymptotic expansions of the original equations. To the purposes of analyzing the electromagnetic memory effect, we may keep the leading falloffs to match the corresponding radiation falloffs, \[radiationfall\] A\_u = O(r\^[-(D-2)/2]{}),A\_r = O(r\^[-(D-2)/2]{}),A\_i = O(r\^[-(D-4)/2]{}). More general options are possible and influence the structure of the asymptotic symmetry group. We will be concerned with these more general aspects of the discussion in Section \[sec:poly\]. The significance of the choice lies in the fact that the derivatives with respect to $u$ of the field components are unconstrained to leading order: these conditions for the asymptotic expansion are well-suited to identifying the boundary data for a radiation solution with an arbitrary wave-form. Such components also provide the energy flux at a given retarded time, as happened in the radial gauge, according to P(u) = \_[S\_u]{} \^[ij]{} \_u A\^[()]{}\_i \_u A\^[()]{}\_j d, where $S_u$ is the section of $\mathscr I^+$ at fixed $u$ and $d\Omega$ is the measure element on the unit $(D-2)$-sphere.[^8] The asymptotic behavior of radiation differs in higher dimensions with respect to the characteristic Coulombic falloff $r^{3-D}$ that, in its turn, can be identified as the leading falloff for $u$-independent solutions. (See also the discussion in Section \[ssec:em\_memory\] on this point.) As we shall see, Coulomb fields give nonvanishing contributions to the surface integral associated with the electric charge as well as to the memory effects. #### Recursive gauge fixing {#ssec:emgaugefix} The gauge variations A\_u\^[(k)]{}=\_u \^[(k)]{},A\_r\^[(k)]{}=-(k-1)\^[(k-1)]{},A\_i\^[(k)]{}=\_i \^[(k)]{} imply a number of restrictions on the allowed gauge parameters, in order to keep the falloffs of such field configurations. From the $r$-variation, we read off $\epsilon^{(k)}=0$ for $k<(D-4)/2$ and $k\neq 0$. The $u$-variation additionally requires that $\epsilon^{(\frac{D-4}{2})}$ be independent of $u$, whereas the angular variation does not give rise to further constraints at this stage. This leads to a gauge parameter of the following, provisional, form \[epsilon\] (u,r,x\^i)=1+\^[()]{}(x\^i)/r\^+. This is not the actual form of the residual gauge parameter, however, since it does not satisfy [^9]: \[noepsilonrad\] = \^[()]{}+. Thus, we need to search further down in the asymptotic expansion of $\epsilon$. We do so by employing a recursive on-shell gauge-fixing procedure [@StromingerGravmemevD]. Setting to zero , implies $\epsilon^{(\frac{D-4}{2})}=0$ since the Laplacian on the $(D-2)$-sphere is negative semidefinite. Therefore, the residual symmetry is parametrized as follows (u,r,x\^i)=1++, where we conventionally set the global part of $\epsilon$ to $1$. Equation leaves the $u$-dependence of $\epsilon^{(\frac{D-2}{2})}$ unconstrained, and therefore we may use it to set A\_u\^[()]{}=0, leaving $\epsilon^{(\frac{D-2}{2})}(x^i)$ arbitrary. We may now proceed by noting that, setting $k=D/2$, and $\eqref{residual_r}$ reduce to \_u A\_u\^[()]{}=0,\_u \^[()]{}+\^[()]{}=0, respectively. Thus, $A_u^{(\frac{D}{2})}$ is a function of the angles $x^i$ only, while $\delta A_u^{(\frac{D}{2})}=\partial_u \epsilon^{(\frac{D}{2})}$ can be expressed in terms of $\epsilon^{(\frac{D-2}{2})}$, which can be used to set $A_u^{(\frac{D}{2})}=0$, while still leaving $\epsilon^{(\frac{D}{2})}(x^i)$ arbitrary. This procedure may now be repeated recursively. Assuming A\_u\^[()]{}=A\_u\^[()]{}==A\_u\^[(q-1)]{}=0,\^[(q-1)]{}(x\^i), for some $q>D/2$. Then, for $k=q-1$, and $\eqref{residual_r}$ give \[induction-equations\] (D-2q-2)\_u A\_u\^[(q)]{}=0,(D-2q-2)\_u\^[(q)]{}=\^[(q-1)]{}. Therefore, we may employ $\epsilon^{(q-1)}(x^i)$ to set $A_u^{(q)}$ to zero provided that the differential operator on the right-hand side is invertible, which is true for any $q<D-2$. We shall now consider two options. We may first choose to truncate the recursive gauge fixing right after the step labeled by $q=D-4$, which leaves us with the asymptotic expansions \[falloffmemory1\] A\_u = \_[k=D-3]{}\^A\_u\^[(k)]{}r\^[-k]{},A\_r = \_[k=]{}\^A\_r\^[(k)]{}r\^[-k]{},A\_i = \_[k=]{}\^A\_i\^[(k)]{}r\^[-k]{}, where $A_r^{(\frac{D-2}{2})}=0$ on shell. The residual symmetry is given by \[res\_even\] (u,r,x\^i) = 1+ \^[(D-4)]{}(x\^i)r\^[4-D]{} + , whose corresponding charge, evaluated in the absence of radiation close to the past boundary $\mathscr I^+_-$ of $\mathscr I^+$, reads[^10] \[ChargeStrom2\] Q\_= \_[I\^+\_-]{} ( \_u A\^[(D-2)]{}\_r+(D-3)A\_u\^[(D-3)]{} ) \^[(D-4)]{}d. Alternatively, we may also perform the recursive gauge-fixing until the very last allowed step, $q=D-3$. In which case, \[falloffmemory2\] A\_u = \_[k=D-2]{}\^A\_u\^[(k)]{}r\^[-k]{},A\_r = \_[k=]{}\^A\_r\^[(k)]{}r\^[-k]{},A\_i = \_[k=]{}\^A\_i\^[(k)]{}r\^[-k]{} and (u,r,x\^i) = 1+ \^[(D-3)]{}(x\^i)r\^[3-D]{} + . The latter choice highlights the possibility of making the components \[gaugeinvtower\] A\_i\^[()]{},, A\_[i]{}\^[(D-5)]{}, A\_i\^[(D-4)]{} gauge-invariant, and hence in principle responsible for any observable effect due to radiation impinging on a test charge placed at a large distance $r$ from a source. Indeed, as we shall verify explicitly in Section \[sec:memories\], electromagnetic memory effects appear at Coulombic order $A_i^{(D-4)}$. ### Memory and residual symmetries {#ssec:em_memory} A test particle with charge $Q$, initially at rest at a large distance $r$ from the origin, will experience a leading-order momentum kick due to the presence of an electric field according to \[velocity\] P\_[i]{}|\_[u\_1]{}-P\_i|\_[u\_0]{} = Q\_[u\_0]{}\^[u\_1]{} F\_[iu]{} du. We are neglecting the contribution from the magnetic field by the assumption that they will always give rise to subleading contributions in $1/r$, due to their proportionality to the velocity of the test particle, as we explicitly checked in Section \[sec:memories\] in the case of an ideally sharp wavefront. Now we shall consider solutions that are stationary before $u=u_0$ and after $u=u_1$. For such solutions, the Maxwell tensor before $u_0$ and $u_1$ is not zero, in general, because static forces are present. However, it does not contain radiation, and thus all the components of the gauge potential associated to radiation are to vanish, or, more generally, are to be pure gauge. In particular, the radiation field components before $u_0$ and $u_1$ are to be identical or are to differ by a gauge transformation. Now, let us combine the information of the previous recursive gauge-fixing with the requirement that the solution be stationary before $u_0$ and after $u_1$. The asymptotic expansion of the electric field $\mathcal F_{ui}=\partial_i \mathcal A_u-\partial_u \mathcal A_i$ satisfies F\_[iu]{}\^[(k)]{} = -\_u A\^[(k)]{}\_i(u,x\^k), for $k=\frac{D-4}{2},\ldots,D-4$ since, then, $A_u^{(k)}$ is zero, thanks to the gauge-fixing procedure. Thus, \[DeltaPDeltaA\] P\_i|\_[u\_1]{}-P\_i|\_[u\_0]{} =-Q \_[k=]{}\^[D-4]{}( A\^[(k)]{}\_i|\_[u\_1]{}-A\_i\^[(k)]{}|\_[u\_0]{} )+ O(r\^[3-D]{}). With respect to our previous observation, let us notice that the components of $\mathcal A_i$ that enter the subleading terms $\mathcal O(r^{3-D})$ in are those connected with stationary properties of the field (we assume the absence of permanent long-range magnetic fields, which would induce $\mathcal O(r^{4-D})$ contributions to $\mathcal F_{ij}$), while all the leading components explicitly written enter the radiation behavior and thus their difference after $u_1$ and before $u_0$ can be at most the angular gradient of given functions. Actually, we shall immediately see that combining this information with the equations of motion will allow us to conclude that they all vanish with the exception of the last one $A_i^{(D-4)}$. Indeed, let us note that for a stationary solution, in our gauge, equations , , and read, for $k<D-2$, $$\begin{aligned} (D-k-1)(A_r^{(k-1)}-A_u^{(k-1)})+D\cdot A^{(k-2)}&=0 \label{Lorenz_ru=0}\,,\\ [\Delta + (k-1)(k-D+2)]A_u^{(k-1)}&=0 \label{equ_ru=0}\, ,\\ [\Delta+k(k-D+1)]A_r^{(k-1)}\, +\, (D-2)\, A_u^{(k-1)}-2 D\cdot A^{(k-2)}&=0 \label{eqr_ru=0}\,,\\ [\Delta+(k-1)(k-D+2)-1]A_i^{(k-2)}-2D_i(A_u^{(k-1)}-A_r^{(k-1)})&=0 \label{eqi_ru=0}\,.\end{aligned}$$ For $1<k<D-2$, equation implies $A_u^{(k-1)}=0$, compatibly with the outcome of recursive gauge-fixing. Equations and then give, for $1<k<D-2$, A\_r\^[(k-1)]{}=0 so that $A_r^{(k-1)}=0$ for $2<k<D-2$. Considering finally equation , for $2<k<D-2$, we have A\_i\^[(k-2)]{}=0 and hence $A_i^{(k-2)}=0$ provided provided that $k$ also satisfies $k_-(D)<k<k_+(D)$ with k\_(D) = ; actually, $k_-(D)<1$ and $k_+(D)>D-2$ for any $D>3$, so we conclude that stationary solutions obey $A_i^{(k-2)}=0$ for $2<k<D-2$. To summarize, $A_u^{(k)}=0$ for $0<k<D-3$, while $A_r^{(k)}=0$ for $1<k<D-3$ and A\_i\^[(k)]{}=00&lt;k&lt;D-4. By equation , the condition on $\mathcal A_i$ implies that the memory effect appears to leading order $r^{4-D}$, \[memoryform\_even\] P\_i|\_[u\_1]{}-P\_i|\_[u\_0]{} = -( A\^[(D-4)]{}\_i|\_[u\_1]{}-A\_i\^[(D-4)]{}|\_[u\_0]{} )+O(r\^[3-D]{}). In view of the discussion below , we conclude that the momentum shift must take the form \[memoryformulaf\] P\_i|\_[u\_1]{}-P\_i|\_[u\_0]{} = \_ig(x\^k)+O(r\^[3-D]{}), with $g(x^i)$ a $u$-independent function, which will depend on the shape of the radiation train and in particular on $u_0$ and $u_1$ (see for instance ). Let us note that, as it must be, this difference is not affected by the action of the residual gauge transformation , as can be understood by the fact that the latter is $u$-independent and thus does not alter the difference $A_i^{(D-4)}\big|_{u_1}-A_i^{(D-4)}\big|_{u_0}$. In this sense, whether or not one performs the last step of the recursive gauge fixing is irrelevant to the extent of calculating the electromagnetic memory. To conclude, we have established a formula that exhibits a momentum kick characterizing the transition between the initial and final vacuum configurations, parametrized by the gauge transformation $g(x^i)$, induced by the exposure to electromagnetic radiation crossing null infinity. In particular, the norm of this effect scales as $r^{3-D}$. Up to this point we have only been dealing with ordinary/linear memory effect. In order to encompass a null/nonlinear memory effect, we must modify the equations of motion by adding suitable source terms on the right-hand sides, namely a current density $J^\mu$ allowing for the outflow to future null infinity of charged massless particles. The falloff conditions on such a current can be taken as follows J\_u=O(r\^[2-D]{}),J\_r=O(r\^[2-D]{}),J\_i=O(r\^[3-D]{}). This is clearly displayed by the example of a single massless charge $q$ moving in the $\mathbf x_0$ direction, whose current reads in Minkowski components J\^0 = q (x - x\_0 t),J =q x\_0 (x - x\_0 t) and in retarded components (for $t=u+r>0$) J\_u &= - (u)(n, x\_0),\ J\_r &= -(u)(n, x\_0),\ J\_i &= (u)(n, x\_0). As far as the discussion of the previous section is concerned, namely, for the purposes of the recursive gauge fixing, the only modification is thus the introduction of a source term $J^{(D-2)}_u$ in the right-hand side of when $k=D-3$, which now actually forces us to stop the gauge fixing after the use of said equation for $k=D-4$ (the step labeled by $q=D-4$ in the previous section) and leaves us with the falloff . On the contrary, reaching is not allowed, and thus comprises a complete gauge fixing. However, also in view of the above considerations, the discussion of the memory effect and its relation to the symmetry acting at Coulombic order thus remain unaltered. ### Phase memory {#ssec:phasemem} Let us consider a pair of electric charges $q$ and $-q$ that are pinned in the positions $(r,\mathbf n_1)$ and $(r,\mathbf n_2)$, for large $r$. We will now derive the expression for an imprint that the passage of a radiation train leaves on the properties of these particles that is encoded in the phase of their states. See [@Susskind] for the discussion a very similar phenomenon in a four-dimensional setup. A quantum treatment of electromagnetic kick memory is instead given in [@Afshar:2018sbq]. To this purpose, let us assume that, as in the previous section, radiation impinges on the charges only during the interval between two given retarded times $u_0$ and $u_1$. As we have seen, this means that \[gaugevacua\] A\_i|\_[u\_1]{}-A\_i|\_[u\_0]{}=\_i g + O(r\^[3-D]{}) for a suitable angular function $g(\mathbf n)$. We will assume for simplicity that the gauge field before the onset of radiation is the trivial one. Let $|\psi_1\rangle=|q\rangle$ and $|\psi_2\rangle=|-q\rangle$ be the initial states in which the charged particles are prepared, which are uniquely labeled by their charges since translational degrees of freedom have been suppressed. Before $u_0$, the state $|\psi_2,\mathbf n_1\rangle$ obtained by the parallel transport of the second state $|\psi_2\rangle$ to the position $\mathbf n_1$ of the first the charge is |\_2,n\_1=|\_2, because $\mathcal A_\mu=0$, so that the corresponding tensor state evaluated in $\mathbf n_1$ is given by |\_1,n\_1|\_2,n\_1=|\_1|\_2. At $u_1$, instead, the same operation must be performed by calculating |\_2,n\_1= |\_2= |\_2+O(r\^[3-D]{}), where we have employed to establish the second equality. Therefore, after the passage of radiation, |\_1, n\_1|\_2,n\_1= |\_1|\_2+O(r\^[3-D]{}), which displays how the transition between two different radiative vacua, already experimentally detectable by the occurrence of a nontrivial velocity kick for a test charge, is also signaled by the variation of the relative phases in the states obtained by parallel transport of charged particles. Such a phase can be nontrivial provided that the function $g$ is not a constant, namely when there is a nontrivial memory kick . A point that should be underlined is that, in this setup, the states $|\psi_1\rangle$ and $|\psi_2\rangle$ do not evolve, since each particle is kept fixed in its position (its translational quantum numbers are *frozen*) and electromagnetic radiation cannot change its charge. The relative phase difference occurs entirely as an effect of the evolution of $\mathcal A_\mu$, which undergoes a transition between two underlying radiative vacua. We shall see in the following how this aspect is qualitatively different in a non-Abelian theory, where radiation can alter the color charge. Yang-Mills Memory {#sec:color} ----------------- We now turn to the extension of the above analysis of memory effects to the non-Abelian case. ### Yang-Mills theory in Lorenz gauge {#ssec:lorenzYM} We consider pure Yang-Mills theory of an anti-Hermitian gauge field, adopting the same conventions as in Section \[sec:YMGlobal\] (in particular, we again adopt anti-Hermitian $su(N)$ generators). We will however impose the Lorenz gauge condition $\nabla^a \mathcal A_a=0$, instead of radial gauge, which leaves as residual gauge parameters those that satisfy $ \Box \epsilon +[\mathcal A_a, \partial^a \epsilon]=0 $. Furthermore, the equations of motion reduce to $\Box \mathcal A_a + [\mathcal A^a, \nabla_a A_b+\mathcal F_{ab}]=0$. Adopting retarded Bondi coordinates, the Lorenz gauge condition reads \[Lorenz\_YM\] \_u A\_r = (\_r + )(A\_r - A\_u)+DA, while the constraint on residual transformations is \[residual\_YM\] &(\_u\^2 - 2 \_u \_r + ) + (\_r - \_u)\ &= \[\_r , A\_r - A\_u\] - \[\_u , A\_r\] + \^[ij]{}\[D\_i , A\_j\]. The equations of motion give instead \[eom\_YM\] &A\_u\ &= \[A\_u-A\_r, \_r A\_u + F\_[ru]{}\]+\[A\_r, \_u A\_u\]-\[A\_i, D\_jA\_u + F\_[ju]{}\] ,\ &A\_r+(A\_u - A\_r)- DA\ &= \[A\_u- A\_r , \_r A\_r\] + \[A\_r, \_u A\_r + F\_[ur]{}\] - \[A\_i, D\_j A\_r+ F\_[jr]{}\] ,\ &A\_i-D\_i(A\_u -A\_r)\ &= + \[A\_r, \_u A\_i + F\_[ui]{}\]-\[A\_j, D\_kA\_i+F\_[ki]{}\]. Performing the usual asymptotic expansion in inverse powers of the radial coordinate $r$, one obtains the following set of equations. Equations and give \_u A\^[(k+1)]{}\_r=(D-k-2)(A\_r\^[(k)]{}-A\_u\^[(k)]{})+DA\^[(k-1)]{} \[Lorenz\_r\_YM\] and &(D-2k-2)\_u \^[(k)]{}-\[- (k-1)(D-k-2)\]\^[(k-1)]{}\ &= \_[l+m=k]{} ( -l\[\^[(l)]{}, A\_u\^[(m)]{}-A\_r\^[(m)]{}\]+\[\_u \^[(l+1)]{}, A\_r\^[(m)]{}\]-\^[ij]{}\[D\_i \^[(l-1)]{}, A\_j\^[(m)]{}\] ) \[residual\_r\_YM\] , respectively, while from one obtains $$\begin{aligned} \label{equ_r_YM} &(D-2k-2)\partial_u A_u^{(k)}-[\Delta - (D-k-2)(k-1)]A_u^{(k-1)} \\ \nonumber = & \sum_{l+m=k} \Big( [A_r^{(m)}- A_u^{(m)}, -l A_u^{(l)}+ F_{ru}^{(l+1)}]-[A_r^{(m)}, \partial_u A_u^{(l+1)}] \\ \nonumber &+\gamma^{ij}[A_i^{(m)}, D_j A_u^{(l-1)} + F_{ju}^{(l-1)}] \Big) \, ,\\ \label{eqr_r_YM} &(D-2k-2)\partial_u A^{(k)}_r \!-\! [\Delta-k(D-k-1)]A_r^{(k-1)}\! -\! (D-2) A_u^{(k-1)}+2 D\cdot A^{(k-2)} \\ \nonumber = & \sum_{l+m=k} \Big([A_r^{(m)}-A_u^{(m)}, -l A_r^{(l)}]-[A_r^{(m)}, \partial_u A_r^{(l+1)}+F_{ur}^{(l+1)}] \\ \nonumber &+\gamma^{ij}[A_i^{(m)}, D_j A_r^{(l-1)}+F_{jr}^{(l-1)}] \Big) \, ,\\ \label{eqi_r_YM} &(D-2k-4)\partial_u A_i^{(k)}-[\Delta-k(D-k-3)-1]A_i^{(k-1)}+2D_i(A_u^{(k)}-A_r^{(k)}) \\ \nonumber = & \sum_{l+m=k} \Big([A_r^{(m)}-A_u^{(m)}, -(l+2) A_i^{(l)}+F_{ri}^{(l+1)}]-[A_r^{(m)}, \partial_u A_i^{(l+1)}+F_{ui}^{(l+1)}] \\ \nonumber &+\gamma^{jk}[A_j^{(m)}, D_k A_i^{(l-1)}+F_{ki}^{(l-1)}] \Big) \, ,\end{aligned}$$ for the corresponding components of the equations of motion. In particular, — appear to order $\mathcal O(r^{-k-1})$ in the asymptotic expansions. We choose to adopt the same radiation falloff conditions that we imposed in the linearized theory: the expansions of $\mathcal A_u$ and $\mathcal A_r$ start at order $\mathcal O(r^{-(D-2)/2})$ and that of $\mathcal A_i$ starts at order $\mathcal O(r^{-(D-4)/2})$. For completeness, in order to verify the consistency of this choice, we now calculate the color charge $\mathcal Q(u) = \mathcal Q^A(u) T^A$ at a given retarded time $u$, as we did in retarded radial gauge in Section \[sec:YMGlobal\]. This quantity is given by the surface integral Q\^A(u) =& \_[S\_u]{} F\_[ur]{}\^A r\^[D-2]{}d\ =& \_k r\^[D-2-k]{} \_[S\_u]{} (\_u A\_r\^[(k)]{} + (k-1) A\_u\^[(k-1)]{} + \_[l+m=k]{}\[A\_u\^[(m)]{}, A\_r\^[(l)]{}\] )\^A d, in the limit $r\to\infty$. Combining the Lorenz condition and the $r$-equation of motion , one obtains \[finite-charge-instr\_YM\] &(D-2-k)( \_u A\_r\^[(k)]{}+(k-1)A\_u\^[(k)]{} ) - D\^i ( D\_i A\_r\^[(k-1)]{}+(k-2)A\_i\^[(k-2)]{} )\ &= \_[l+m=k]{}( \[A\_r\^[(m)]{}-A\_u\^[(m)]{}, -l A\_r\^[(l)]{}\]-\[A\_r\^[(m)]{}, \_u A\_r\^[(l+1)]{}+F\_[ur]{}\^[(l+1)]{}\]\ & +\^[ij]{}\[A\_i\^[(m)]{}, D\_j A\_r\^[(l-1)]{}+F\_[jr]{}\^[(l-1)]{}\] ) . We see that implies $\partial_u A^{(\frac{D-2}{2})}_r=0$ and that reduces to (D-2-k)( \_u A\_r\^[(k)]{}+(k-1)A\_u\^[(k-1)]{} ) = D\^i ( D\_i A\_r\^[(k-1)]{}+(k-2)A\_i\^[(k-2)]{} ) for $k<D-2$. This allows us to conclude that the color charge is finite in the limit $r\to\infty$ and that it equals \[ColorChargeLor\] Q\^A(u) = \_[S\_u]{} ( \_u A\_r\^[(D-2)]{} +(D-3)A\_u\^[(D-3)]{} +\[A\_u\^[()]{}, A\_r\^[()]{}\] )\^A d, since all other terms either integrate to zero on the $(D-2)$-sphere or vanish in the limit $r\to\infty$. The final expression for the color charge may be equivalently rewritten in the form Q\^A(u) = \_[S\_u]{} ( A\_r\^[(D-3)]{} +(D-4)A\_u\^[(D-3)]{} +\[A\_u\^[()]{}, A\_r\^[()]{}\] )\^A d, by means of the Lorenz condition . It is amusing to note that reduces to the expression we obtained in Section \[sec:YMGlobal\], namely , for the color charge upon formally setting $\mathcal A_r=0$. Let us now investigate the dependence of $\mathcal Q^A$ on retarded time. Recalling that $\partial_u A_r^{(\frac{D-2}{2})}=0$ by the Lorenz condition, \[uder-Q\] Q\^A(u) = \_[S\_u]{} ( \_u A\_r\^[(D-3)]{} +(D-4)\_u A\_u\^[(D-3)]{} +\[\_u A\_u\^[()]{}, A\_r\^[()]{}\] )\^A d. By evaluated for $k=D-3$, we have \_u A\_r\^[(D-3)]{}+(D-4)A\_u\^[(D-4)]{} = D\^i ( D\_i A\_r\^[(D-4)]{}+(D-5)A\_i\^[(D-5)]{} ), while equation with $k=D-3$ gives \[ddotAu\] (D-4)(\_u A\_u\^[(D-3)]{}-A\_u\^[(D-4)]{}) + \[\_u A\_u\^[()]{}, A\_r\^[()]{}\] = A\_u\^[(D-4)]{} + \^[ij]{} \[A\_i\^[()]{}, \_u A\_j\^[()]{}\]. Hence, substituting into (and disregarding total divergences), we get \[GeneralColorEvolution\] Q\^A(u) = \_[S\_u]{} \^[ij]{} \[A\_i\^[()]{}, \_u A\_j\^[()]{}\]\^A d. This formula provides the flux of the total color charge due to nonlinearities of the theory, *i.e.* self-interaction effects. Note that in particular the right-hand side involves the radiation components, representing the flux of classical gluons across null infinity. We thus retrieve all the physically consistent global features of the analysis performed in \[sec:YMGlobal\]. ### Yang-Mills kick memory {#ssec:YM_memory} Starting from the radiation falloffs, we may employ the residual gauge symmetry of the theory to perform a further, recursive gauge fixing in the same spirit of Section \[ssec:emgaugefix\]. In fact, since the nonlinear corrections to equations appear to order $q=D-3$ or higher, the discussion of this gauge fixing is completely identical to that of Section \[ssec:emgaugefix\]. This is another manifestation of the mechanism of asymptotic linearization we highlighted in Section \[sec:YMGlobal\]. Moreover, in accordance with the fact that Yang-Mills theory must encompass both ordinary/linear and null/nonlinear memory, the gauge fixing stops at $q=D-4$ and cannot be performed up until $q=D-3$, as happened in the case of null electromagnetic memory. The resulting falloffs after this procedure are thus A\_u = \_[k=D-3]{}\^A\_u\^[(k)]{}r\^[-k]{},A\_r = \_[k=]{}\^A\_r\^[(k)]{}r\^[-k]{},A\_i = \_[k=]{}\^A\_i\^[(k)]{}r\^[-k]{} with residual symmetry parameter \[res\_even\_YM\] (u,r,x\^i) = c\^A T\^A+ \^[(D-4)]{}(x\^i)r\^[4-D]{} + , where $c^A$ are constant coefficients. A colored test particle with charge $Q = Q^A T^A$ interacts with the background Yang-Mills field by the Wong equations [@Wong] \[Wong\] P\^a = (Q F\^[ab]{}) x\_b,Q +x\^a \[A\_a, Q\] =0. Focusing on the region near null infinity, a test “quark”, initially at rest, subject to radiation between the retarded times $u_0$ and $u_1$ will therefore experience a leading-order momentum kick according to \[velocity\_YM\] P’\_[i]{}-P\_i = \_[u\_0]{}\^[u\_1]{} (Q F\_[iu]{}) du = -(Q [A’\_i]{}\^[(D-4)]{}) = - \[Q e\^[-\^[(D-4)]{}]{}\_i e\^[\^[(D-4)]{}]{}\], where we have chosen the vacuum configuration at $u=u_0$ to be $\mathcal A_\mu =0$. On the other hand, the color of the test quark will change, to leading order, according to Q’-Q = - \_[u\_0]{}\^[u\_1]{} \[A\_u, Q\]du = - \_[u\_0]{}\^[u\_1]{}\[A\_u\^[(D-3)]{},Q\]du, where the $u$-dependence of $\mathcal A_u$ is governed, according to equation , by \[ColorDelta\] \_u A\_u\^[(D-3)]{} = \^[ij]{}\[A\_i\^[()]{}, \_u A\_j\^[()]{}\] after recursive gauge fixing. Equation characterizes the evolution of the charge of the quark in terms of the leading outgoing radiation terms. In order to better understand the dependence of this momentum kick and color rotation on the incoming radiation near null infinity, let us further analyze the equations of motion. Combining equation and we see that \[recrad\_1\] \_u A\_r\^[(k+1)]{} = D\_k A\_r\^[(k)]{} for $k<D-3$, where we have introduced the following self-adjoint differential operators on the $(D-2)$-sphere \[operatoresfera\] D\_k = \[-(D-2-k)(k-1)\]/(D-2-2k), and \[recrad\_2\] \_u A\_r\^[(D-2)]{} = D\_[D-3]{} A\_r\^[(D-3)]{} - A\_u\^[(D-3)]{} - J, for $k=D-3$ in dimensions $D>4$, where we have defined J = 2 \^[ij]{} \[A\_i\^[()]{}, D\_j A\_r\^[()]{}\] -2 \[A\_r\^[()]{}, \_u A\_r\^[()]{}\] . Equations and , evaluated for $k=D-3$ and $k=D-2$ respectively, actually imply the following constraint on $\mathcal J$, \[constr\_J\] J = -D\^i \[D\_i A\_r\^[(D-3)]{}+(D-4)A\_i\^[(D-4)]{}\], and, taking the derivative of the previous equation with respect to $u$, \[Coul-rad\_1\] (D-4) \_u DA\^[(D-4)]{}=-\_u A\_r\^[(D-3)]{} +2 \[A\_r\^[()]{}, \_u\^2 A\_r\^[()]{}\] - 2\^[ij]{}\[\_u A\_i\^[()]{}, D\_j A\_r\^[()]{}\]. Starting from eq. and employing recursively, we find that \_u\^[()]{}A\_r\^[(D-2)]{} = \_[l=]{}\^[D-3]{}D\_l \_u A\_r\^[()]{} -\_u\^ A\_u\^[(D-3)]{} - \_u\^ J, and, by equation , \[Coul-rad\_2\] \_u\^DA\^[(D-4)]{} &= \_u\^ (\_u A\_r\^[(D-2)]{}+A\_u\^[(D-3)]{})\ &- \_u\^ A\_r\^[(D-3)]{} - \_[l=]{}\^[(D-3)]{} D\_l \_u A\_r\^[()]{}. Equations and encode the dependence of $A_i^{(D-4)}$, which as we have seen is responsible for the memory kick both in the linear theory and in the nonlinear theory, on the outflowing radiation, here encoded in particular in $A_i^{(\frac{D-4}{2})}$ and $A_r^{(\frac{D-2}{2})}$. In dimension $D=4$, the above equations can be cast in the form D\^i\_u\[A\_i\^[(0)]{}, A\_r\^[(1)]{}\]+\^[ij]{}\[A\_i\^[(0)]{},\_u\[A\_j\^[(0)]{},A\_r\^[(1)]{}\]\]=0, \_u DA\^[(0)]{} = \_u (\_u A\_r\^[(2)]{}+A\_u\^[(1)]{}). ### Color memory Let us now consider a pair of color charges that are pinned in the positions $(r,\mathbf n_1)$ and $(r,\mathbf n_2)$, for large $r$. Analyzing the effect of the passage of Yang-Mills radiation on their states will provide the non-Abelian analog of the memory effect highlighted Section , namely a relative rotation between their states in color space. This provides a generalization to even dimensions of [@StromingerColor]. We consider colored particles in the fundamental representation and, to simplify the presentation, we first focus on the case of the gauge group $SU(2)$. Assuming that radiation is nontrivial only between two given retarded times $u_0$ and $u_1$, as we have seen, the gauge field satisfies \[gaugevacua1\] A\_i|\_[u\_1]{}-A\_i|\_[u\_0]{}=e\^[-]{}\_i e\^+ O(r\^[3-D]{}) for a suitable $\alpha(\mathbf n)=\alpha^A(\mathbf n)T^A$, where $T^A$ are the $su(2)$ generators. We have assumed for simplicity that the pure gauge configuration before $u_0$ is the trivial one. Let $|\psi_1\rangle$ and $|\psi_2\rangle$ be the initial states in which the colored particles are prepared. Before $u_0$, the state $|\psi_2,\mathbf n_1\rangle$ that results from the parallel transport of the second state $|\psi_2\rangle$ to the position $\mathbf n_1$ of the first the charge is |\_2,n\_1=|\_2, because $\mathcal A_\mu=0$. In particular, one can consider as initial state $|+\rangle$ or $|-\rangle$, namely the standard eigenstates of $T^3$. In order to build a color singlet state in $\mathbf n_1$ it then suffices to prepare the superposition \[prepare-singlet\] =. We remark that, in order for the concept of singlet to be well-defined when $SU(2)$ transformations can depend on the position, it is crucial to build a singlet out of states that have been parallel-transported to the same point. A given state $|\psi\rangle$ evolves according to the covariant conservation equation [@ClassicalFTRusso][^11] \_u |= - A\_u |which means that, recalling $\mathcal A_u = \mathcal O(r^{3-D})$ for large $r$, the state after the passage of radiation differs with respect to its initial value by \[colour-evolution\] ||\_[u\_1]{}-||\_[u\_0]{}=O(r\^[3-D]{}). In particular, a state prepared in $|\pm\rangle$ before $u_0$ will evolve into $ |\pm\rangle+\mathcal O(r^{3-D}) $ at $u_1$, namely when radiation has died out. Now, at $u_1$, one must also take into account that parallel transport from $\mathbf n_2$ to $\mathbf n_1$ on the sphere is defined by |\_2,n\_1&= P|\_2\ &= |\_2 - \_[n\_2]{}\^[n\_1]{}e\^[-]{}\_i e\^dx\^i|\_2+O(r\^[3-D]{}), where $\mathcal P$ denotes path ordering and we have employed to establish the second equality. We conclude that the state obtained by parallel transport after radiation has passed is no longer the singlet , but rather: \[falsesinglet\] & =\ &- +O(r\^[3-D]{}) . Comparison between the expressions and shows that the interaction of the color charges with the external Yang-Mills radiation induces a rotation of the initial state that manifests itself to order $\mathcal O(r^{4-D})$ in the final state. In other words, a pair of particles initially prepared in a singlet will no longer be in a singlet after the passage of radiation. This color memory can be nontrivial provided that the function $\alpha$ is not constant, namely whenever there is also a nontrivial memory kick . From a technical point of view, it should be noted that the effect of radiation on each color state is to induce a time evolution to order $\mathcal O(r^{3-D})$, according to , which allows one to disentangle it from the leading effect due to the vacuum transition undergone by $\mathcal A_i$, which enters to order $\mathcal O(r^{4-D})$. A very similar derivation allows one to extend the result to more general states and gauge groups. Adopting an orthonormal basis $|n\rangle$ for the fundamental representation of $SU(N)$, we can consider the superposition \[beforenm\] \_[n,m]{} c\_[n,m]{}|n|mwith $\sum_{n,m}|c_{n,m}|^2=1$, prepared before $u_0$. Time evolution will induce modifications of $|n\rangle$ that appear to order $\mathcal O(r^{3-D})$ by $\mathcal A_u=\mathcal O(r^{3-D})$. On the other hand, the effect of parallel transport from $\mathbf n_2$ to $\mathbf n_1$ performed after $u_1$ gives rise to the leading-order correction -\_[n,m]{} c\_[n,m]{}|n\_[n\_2]{}\^[n\_1]{}e\^[-]{}\_i e\^ dx\^i | mto , due to the nontrivial configuration attained by $\mathcal A_i$. More on Asymptotic Symmetries for the Maxwell Theory {#sec:poly} ---------------------------------------------------- So far, we have focused on those symmetries directly related to the memory effects. As we saw, they act at Coulombic order and the corresponding charge displays a falloff $1/r^{D-4}$; therefore they do not comprise, strictly speaking, asymptotic symmetries at $\mathscr I^+$. In this section we shall perform a more complete analysis of the asymptotic symmetry group for the Maxwell theory in higher dimensions. As we shall see, in even dimensions, the standard power-like ansatz is not sufficient in order to highlight the presence of infinite-dimensional asymptotic symmetries with nonvanishing charge in the Lorenz gauge, which requires instead the introduction of logarithmic terms in the asymptotic expansion. Finiteness of the soft surface charges is also not manifest at face value and requires further specifications of the definition of the charges themselves. ### Polyhomogeneous expansion for $D\ge4$ In order to investigate the possible existence of large gauge symmetries acting at $\mathscr I^+$, we try to solve the wave equation with the boundary condition \[boundaryfuncI\] \_[r]{}(u,r,n) = \^[(0)]{}(u,n), for some nonconstant function $\epsilon^{(0)}(u,\mathbf n)$. As it turns out, a power-law ansatz $$\epsilon = \sum_{k=0}^\infty \frac{\epsilon^{(k)}(u,\mathbf n)}{r^k}\,,$$ effectively selects the global symmetry, namely the constant parameter, whenever $D$ is even. Indeed, considering with $k=0,1,\ldots,\frac{D-2}{2}$, we have \[no-symmetry-even\] (D-2)\_u \^[(0)]{}&=0,\ (D-4)\_u \^[(1)]{}&=\^[(0)]{},\ &\ 2\_u\^[()]{}&=\^[()]{},\ 0&=\^[()]{}. As already observed (see Appendix \[app:Laplaciano\]), the last equation sets $\epsilon^{(\frac{D-4}{2})}$ to zero, by the invertibility of the differential operator occurring on the right-hand side, for any even $D\ge6$ and to a constant for $D=4$. Then, the above equations recursively set to zero the other components $\epsilon^{(k)}$ all the way up to $\epsilon^{(0)}$ which need be a constant by $\Delta \epsilon^{(0)}=0$ and $\partial_u \epsilon^{(0)}=0$. One concludes that, in even dimensions, the power-law ansatz does not allow for an enhanced asymptotic symmetry sitting at the order $1/r^0$, as it occurs instead for instance in the radial gauge [@StromingerQEDevenD]. In order to retrieve them, one needs a more general ansatz. We find that it is sufficient, to this purpose, to consider the following type of asymptotic expansion involving also a logarithmic dependence on $r$: \[expepsilonlog\] (u,r,n)= \_[k=0]{}\^ + \_[k=]{}\^. In this fashion, the last equation becomes modified by the presence of the logarithmic branch and yields \[matching-logpot\] 2\_u \^[()]{}=\^[()]{}, an equation that determines the $u$-dependence of $\lambda^{(\frac{D-2}{2})}$. Therefore, an arbitrary $\epsilon^{(0)}(\mathbf n)$ is allowed. The need of introducing logarithms in even dimensions was also recognized in [@He-Mitra-photond+2; @HenneauxQEDanyD]. Indeed, the recursion relations expressing the equation $\Box\epsilon=0$ read \[expansioneq-with-logs\] (D-2-2k)\_u \^[(k)]{} + 2\_u \^[(k)]{} &= \[+(k-1)(k-D+2)\]\^[(k-1)]{} + (D-1-2k)\^[(k-1)]{}\ (D-2-2k)\_u \^[(k)]{} &= \[+(k-1)(k-D+2)\]\^[(k-1)]{}, and hence can be solved by direct integration with respect to $u$. Explicitly, for $k<\frac{D-2}{2}$, these equations reduce to the familiar expression (D-2-2k)\_u \^[(k)]{}=\[+(k-1)(k-D+2)\]\^[(k-1)]{}, so that the solution is given by a polynomial in $u$ with angle-dependent coefficients $C_{j,k}(\mathbf n)$, with $0\le j\le k$, and is uniquely determined by specifying the integration functions $\hat \epsilon^{(k)}(\mathbf n)$: \^[(k)]{}(u,n)=\_[l=0]{}\^k C\_[j,k]{}(n) u\^j, with C\_[j,k]{}(n) = \^[(k)]{}(n) &j=0\ \_[l=k-j+1]{}\^k D\_l \^[k-l]{}(n)&. If $k=\frac{D-2}{2}$, equation reduces to , so that \^[()]{}(u,n)=\_0\^u\^[()]{}(u’,n) du’+\^[()]{}(n) , with a suitable integration function, while $\epsilon^{(\frac{D-2}{2})}(u,\mathbf n)$ is unconstrained. For $k>\frac{D-2}{2}$, we can recast in the more suggestive form (D-2-2k)\_u \^[(k)]{} &= \[+(k-1)(k-D+2)\]\^[(k-1)]{}\ (D-2-2k)\_u \^[(k)]{} &= \[+(k-1)(k-D+2)\]\^[(k-1)]{}\ &+\^[(k-1)]{}. In this way, it becomes clear that these two equations can always be solved by first finding an integral of the first equation and then substituting it in the second one where it acts as “source” term on the right-hand side. To summarize, a solution of in even $D$ is specified by assigning a set of integration functions $\hat \epsilon^{(k)}(\mathbf n)$, for $k\ge0$, and $\hat \lambda^{(k)}(\mathbf n)$, for $k\ge\frac{D-2}{2}$, together with an arbitrary function $\epsilon^{(\frac{D-2}{2})}(u,\mathbf n)$. In particular, this achieves the boundary condition as $r\to\infty$, where $\epsilon^{(0)}(\mathbf n)$ is an arbitrary function on the celestial sphere. The discussion simplifies in odd dimensions, where the left-hand side of never vanishes for integer $k$, and hence the recursion relations can be integrated compatibly with with no need of introducing a logarithmic branch. By means of “gauge” transformations \[exact...\] A\_a A\_a + \_a identified by the gauge parameters thus obtained, we can then act on solutions to Maxwell’s equations in Lorenz gauge characterized by the radiation falloffs and generate a wider solution space. In particular, this type of solutions will generically exhibit the following asymptotic behavior as $r\to\infty$: \[dimindep-fall\] A\_u = O(r\^[-1]{}),A\_r = O(r\^[-2]{}),A\_i = O(r\^[0]{}), independently of the dimension $D$ of the spacetime, in view of , provided $D\ge5$, while \[diminde4\] A\_u = O(r\^[-1]{}r),A\_r = O(r\^[-1]{}),A\_i = O(r\^[0]{}), in dimension $D=4$. Nonetheless, such solutions will still retain all the desirable physical properties of those characterized by (also recently discussed in [@SatishchandranWald]), such as the finiteness of the energy flux at any given retarded time. This is due to the fact that they differ from this more restrictive solution space only by a transformation of the type , which does not alter the electromagnetic field, nor, by definition, any other physical observable quantity. Now, from this point of view only, it would be possible to perform with parameters $\epsilon$ that obey arbitrary asymptotics near $\mathscr I^+$; however, as we shall see explicitly below, the ones satisfying are precisely those that will give rise to finite and nonvanishing asymptotic surface charges and are therefore the true asymptotic symmetries of the theory. Going back to equation with the boundary condition , let us note that it is actually possible to provide a general solution thereof in the following more compact form in any even dimension $D$: (x) = \^[(0)]{}(q) d(q), where $q= (1,\mathbf q)$ and the limit $\varepsilon\to0^+$ is understood. The introduction of this small imaginary part is needed in order to avoid the singularities occurring in the angular integration for $|t|<|\mathbf x|$, namely outside the light-cone. Indeed, it is straightforward to verify that = 0, while, aligning $\mathbf n$ along the $(D-1)$th direction, we have & \^[(0)]{}(q) d(q)\ &=\_[0]{}\^d()\^[D-3]{}d’(q’) \^[(0)]{}(q’ , ), where $d\Omega'(\mathbf q')$ denotes the integral measure on the $(D-3)$-sphere, and letting $\tau=r(1-\cos\theta)/u$, for $u\neq0$, the previous expression becomes \[arccos\_asymptD\] \_0\^[2r/u]{} d’(q’) \^[(0)]{}( q’ , 1- ), which, as $r\to\infty$, tends to \_0\^[u]{} d’(q’) \^[(0)]{}(n)=\^[(0)]{}(n). This solution is indeed compatible with the asymptotic expansion and generalizes the expression given in [@Hirai:2018ijc]. ### Finite and nonvanishing charges on $\mathscr I$ {#ssec:Finitecharges} As is usually the case in the presence of radiation, the surface charge associated to the symmetry , namely Q\_(u) = \_[r]{} \_[S\_u]{}F\_[ur]{}(u,r,n)(u,r,n)r\^[D-2]{}d(n), is formally ill-defined, because the right-hand side contains terms of the type *e.g.* r\^\_[S\_u]{}F\_[ur]{}\^[()]{}\^[(0)]{}d, which do not vanish, even after imposing the equations of motion, precisely due to the presence of an *arbitrary* parameter $\epsilon^{(0)}(\mathbf n)$. Such difficulties are absent in the case of the *global* charge $\epsilon=1$, namely the electric charge, because the equations of motion always ensure that the surface charge is actually independent of the specific surface on which it is calculated, as long as no sources are crossed. In fact, this feature is shared by all global symmetries associated to linearized gauge theories [@Barnich-Brandt]. It should be noted, however, that the above difficulties only arise if one attempts to calculate the surface charge, in the case of the general transformation , by integrating over a sphere at a given retarded time and radius $r$ and then lets $r$ tend to infinity. Instead, the calculation of the charge on a Cauchy surface still gives a well-defined result [@HenneauxQEDanyD]. In particular, under the simplifying assumption that the electromagnetic field due to radiation vanish for $u<u_0$ in a neighborhood of future null infinity so that $S_{u}$ is the boundary of a Cauchy surface for $u<u_0$, the calculation of the surface charge then yields Q\_(u) = \_[S\_u]{} F\_[ur]{}\^[(D-2)]{}\^[(0)]{}d=\_[S\_u]{} \^[(0)]{}d, for *fixed* $u<u_0$, since indeed all radiation components $F_{ur}^{(k)}$, for $k<D-2$, vanish for a stationary solution (see the discussion in Section \[ssec:em\_memory\]). Letting then $u$ approach $-\infty$, one has Q\_(-) = \_[I\^+\_-]{} F\_[ur]{}\^[(D-2)]{}\^[(0)]{}d=\_[I\^[+]{}\_-]{} \^[(0)]{}d. For $u<u_0$, the quantity $\mathcal Q_\epsilon(u)$ must match the analogous surface integral calculated at spatial infinity because in both cases the Noether two form is integrated over the boundary of a Cauchy surface, in view of the requirement that no radiation be present in a neighborhood of $\mathscr I^+$ for $u<u_0$. The evolution of $\mathcal Q_\epsilon(u)$ *along* $\mathscr I^+$ can be then defined, even in the presence of radiation, by the equations of motion. Indeed, the Maxwell equation $\nabla\cdot \mathcal F_r=0$ gives (\_r + ) F\_[ur]{} = DF\_r, while $\nabla\cdot \mathcal F_u=0$ reads \_u F\_[ur]{}=(\_r + ) F\_[ur]{}+DF\_u \_u F\_[ur]{} = D(F\_u+ F\_r). and hence Q\_(u) = \_[S\_u]{} D(F\_u\^[(D-4)]{}-F\_r\^[(D-4)]{})\^[(0)]{}d. Analogous considerations allow one to introduce well-defined surface charges evaluated on $\mathscr I^-$. From the perspective of the analysis performed in Section \[sec:memories\], it is possible to explicitly perform the calculation of soft charges according to the above strategy. Restricting to the case of a massive charge in dimension $D=6$ that starts moving at $t=0$, the integral of $\mathcal F_{ur} \epsilon^{(0)}$ on a sphere at fixed retarded time $u$ and radius $r$ yields Q\_(r,u) &= r \^[(0)]{}(n) d\ &+\^[(0)]{}(n) d. Except for the case of the electric charge $\epsilon^{(0)}=1$, where the integral in the first line vanishes identically, the limit of this surface charge as $r\to\infty$ is ill-defined in the presence of radiation, namely on the forward light-cone $u=0$, due to the appearance of a linear divergence. However, the charge is well-defined on $\mathscr I^+$ before and after the passage of radiation, $|u|>\varepsilon$, and reads Q\_(u)=\^[(0)]{}(n) d. For $\epsilon^{(0)}=1$, this quantity reduces to the (constant) electric charge $Q=1$, while for more general parameters $\epsilon^{(0)}$, the soft charge exhibits a jump discontinuity at $u=0$, which measures the fact that the particle is no longer static for $u>0$ in a manner akin to the memory effect itself. Performing instead the limit $r\to\infty$ at fixed time $t$, it is also possible to verify the matching between the surface charge evaluated at null infinity before the onset of radiation, for $u<0$ (or, equivalently, at $\mathscr I^+_-$), and the Hamiltonian charge $\mathcal H_\epsilon(t)$, obtained by integrating on a slice at fixed time $t$. Indeed, taking into account and writing the result in terms of polar coordinates $t$, $r$ and $\mathbf n$, we have, for the scalar field, \[exactscalarD=6polar\] 8\^2 &= +(t-r,r)\^[-3/2]{}\ &-+. The corresponding electromagnetic potential is given by $\mathcal A^\mu=(\mathcal A^0,\mathbf A)=-\gamma(\mathbf v)(1,\mathbf v)\varphi$, for $t>r$, and $A^\mu=(\varphi,0)$, for $t<r$. The radial component of the electric field then yields $\mathcal F_{tr}=3 r^{-4}$ as $r\to\infty$ for fixed $t$ and hence H\_(t)= \^[(0)]{}(n) d= Q\_(u&lt;0). Similar arguments showing the finiteness of the Hamiltonian charge in higher-dimensions have been given, in the case of linearized spin-two in retarded Bondi gauge, in [@Aggarwal2018], while a renormalization procedure has been recently proposed, for Maxwell theory in the radial gauge, in [@FreidelHopfmuellerRiello]. ### Soft photon theorem {#ssec:Weinberg} The surface charge associated to , evaluated at the past boundary $\mathscr I^+_-$ of $\mathscr I^+$ reads \[Strominger1Charge\] Q\_= \_[I\^+\_-]{} ( \_u A\^[(D-2)]{}\_r+(D-3)A\_u\^[(D-3)]{} ) \^[(0)]{}d, where we have taken into account the absence of radiation terms for $u\to-\infty$. Recasting this as an integral over the whole $\mathscr I^+$, and assuming no contribution comes from $\mathscr I^+_+$, which is the case if we assume that there are no stable massive charges in the theory, \[Strominger-charge\] Q\_= -\_[I\^+]{} \_u\^2 A\_r\^[(D-2)]{} \^[(0)]{} du d, where we have used the fact that $A_u^{(D-3)}$ is independent of $u$ on shell (thanks to the recursive gauge-fixing). We would like to express in terms of the leading radiation field, which, as we shall see below, indeed contains the creation and annihilation operator of asymptotic photons. To this end, we first combine and and obtain DA\^[(k-1)]{} = A\_r\^[(k)]{}+(D-3-k)A\_u\^[(k)]{}; employing as well, \_u A\_r\^[(k+1)]{} = D\_k A\_r\^[(k)]{} - A\_u\^[(k)]{}, where the operator $\mathscr D_k$ was introduced in . Employing this relation recursively, we find \[recursive-u\] \_u\^[D/2]{} A\_r\^[(D-2)]{} = \_[l=D/2]{}\^[D-3]{} D\_l \_u DA\^[()]{}, where we have used to deduce $\partial_u A_r^{(\frac{D}{2})} = D\cdot A^{(\frac{D-4}{2})}$. In the above writing, we adopt the convention that for $D=4$ the product $\prod_l \mathscr D_l$ (which in this case has a formally ill-defined range) reduces to the identity. We can then use to recast as Q\_= -\_[-]{}\^[+]{}(\_[-]{}\^u du)\^[D/2-2]{} \_u DA\^[()]{} \_[l=D/2]{}\^[D-3]{}D\_l \^[(0)]{}dud. On the other hand, the asymptotic expansion of the free electromagnetic field operator, expressed in terms of creation and annihilation operators, yields, to leading order, A\_i(u,r,x\^k) = \_0\^[+]{} ()\^[(D-4)/2]{}e\^[-iu]{} \^\_i (x) a\_(x)d+h.c., where $\hat x^\mu = x^\mu/r$, while $\epsilon^\sigma$ are polarization tensors for the $D-2$ propagating helicities. This formula provides thus an explicit expression for $A_i^{(\frac{D-4}{2})}$ and hence allows us to make explicit the relation between the charge $\mathcal Q_\epsilon$ and the soft photon creation and annihilation operators as follows (we employ the prescription $\int_{-\infty}^{+\infty}du \int_{0}^{+\infty}d\omega\, e^{i\omega u} f(\omega)=f(0)/2$) Q\_= \_[0\^+]{} \_[S\^[D-2]{}]{} D\^i\[\_i\^(x) a\_(x) + h.c.\] \_[l=D/2]{}\^[D-3]{} D\_l \^[(0)]{}(x) d(x). Assuming that the charge $\mathcal Q$, together with its counterpart at $\mathscr I^-$, generates the residual symmetry $\delta \psi(u,r,\hat x) = i\epsilon^{(0)}(\hat x)+\cdots$ in a canonical way, and using suitable matching and crossing symmetry conditions, we have the Ward identity \[Ward-identity-per-Weinberg\] & \_[S\^[D-2]{}]{} \_i\^(x) \_[0\^+]{} D\^i|a\_(x) S |\_[l=D/2]{}\^[D-3]{} D\_l \^[(0)]{}(x) d(x)\ &= \_[n]{} e\_n \^[(0)]{}(x\_n) | S |, where the sum on the right-hand side extends to all charged external particles in the amplitude and $e_n$ is the electric charge of the $n$th particle (taking into account with a suitable sign the fact that the particle is outgoing, resp. incoming). Notably, the left-hand side contains exactly the combination $\mathcal P[\,\cdot\,] = \lim_{\omega\to 0^+}[\omega\,\cdot\,]$ that selects the pole in the amplitude with the soft insertion. On the other hand, the Weinberg theorem for an amplitude involving external massless particles with momenta $p_n=E_n(1,\hat x_n)$ and a soft photon emitted with helicity $\sigma$ pointing along the $\hat n$ direction on the celestial sphere (see Chapter \[chap:obs\]) reads \[Weinberg-massless\] \_[0\^+]{} | a\_(n) S |= \_n | S | . Multiplying this relation by $\epsilon^\sigma_i(\hat n)$ and summing over $\sigma$, we see that this is equivalent to \[identity\_i\] \^\_i(n) \_[0\^+]{}|a\_(n) S |=\_[n]{}e\_n D\_i (x\_n, n) | S |, where we have used the completeness relation for polarization vectors and defined a function (x, n) = (1-x n). This function $\alpha$ satisfies the following identity (see [@StromingerQEDevenD]) \[delta-identity\] \_[l=D/2]{}\^[D-3]{}D\_l (x, n) = (x, n), where $\hat x$ is here a treated as a constant vector on the unit sphere and $\delta(\hat x, \hat n)$ is the invariant delta function on the $(D-2)$-sphere. Now, acting with the differential operator \_[l=D/2]{}\^[D-3]{}D\_lD\^i on equation , multiplying by an arbitrary $\epsilon^{(D-4)}(\hat x)$ and integrating over the unit sphere then allows one to retrieve the Ward identity , thanks to the relation . This proves that the Weinberg factorization implies the existence of the asymptotic symmetry Ward identities. Remarkably, the charge associated to the symmetry , which is responsible for the memory effect, formally differs from only by a factor of $1/r^{D-4}$ (other than by the substitution $\epsilon^{(0)}\leftrightarrow\epsilon^{(D-4)}$), which makes it vanish on $\mathscr I^+$. However, the corresponding symmetry transformation of the matter fields would be $\delta \psi(u,r,\hat x) = i\epsilon^{(D-4)}(\hat x)/r^{D-4}+\cdots$, and hence would give rise to Ward identities completely equivalent to , with the factors of $1/r^{D-4}$ canceling each other on the two sides. This indicates that both the large gauge symmetry and residual symmetry , acting at Coulombic order, can be seen equally as consequences of the validity of Weinberg’s soft theorem. This is also reflected in the observation that the Fourier transform of the soft factor occurring in Weinberg’s theorem is strictly related to the memory formulas [@Mao_EvenD]. Scalar Soft Theorems and Two-Form Asymptotic Symmetries {#chap:scalar2form} ======================================================= We begin the part on higher spins with a chapter devoted to the scalar case. While this may seem contradictory, one can be justified to do so on account of the structural differences exhibited by this situation with respect to spin-one and spin-two theories, which we have dealt with in the previous chapters. Such differences indeed allow one to include the scalar case in the framework of “exotic” instances of asymptotic behavior. A puzzling feature of theories containing massless scalar particles, as we will briefly review in Section \[sec:scalrtheorems\], is that they exhibit a nontrivial asymptotic structure at null infinity, and in particular allow for the definition of soft charges that account for factorization theorems, while on the other hand lacking any underlying gauge symmetry of which such charges could be a manifestation. In Section \[sec:twoformss\] we propose a way in which this puzzle can be addressed by appealing to the duality that, in four spacetime dimensions, links the scalar field to a two-form gauge theory [@twoform-io]. While consistent as far as the matching between the scalar and two-form solution spaces is concerned, our choice of working in the radial gauge does not allow for a fully satisfactory matching between the scalar soft charges and nonvanishing two-form charges on $\mathscr I$, as opposed to a similar approach based on the Lorenz gauge [@Campigliascoop] which we discuss in Section \[sec:scoop\]. The reason for this shortcoming of the radial gauge can be traced back to the fact that, in order to reach such a gauge, one may need perform a large gauge transformation. As a consequence, this gauge freezes the asymptotic symmetries and effectively hides their presence. In this respect, such a failure to highlight the correct asymptotic structure of the two-form theory appears to be structurally different compared to the difficulties arising in calculation of asymptotic symmetries in higher dimensions for spin-one theories, which we discussed in the previous chapter. Indeed the latter can actually be ascribed to the adoption of too-stringent falloff conditions, namely radiation falloffs for $D>4$, while the former seems an intrinsic problem of the gauge-fixing condition, *i.e.* the radial gauge. This phenomenon begs for a deeper understanding of the gauge dependence of the nature of asymptotic symmetries or at least for a systematic criterion enabling one to know *a priori* whether a given gauge-fixing is acceptable from the point of view of the asymptotic analysis (see [@Riello] for a recent proposal in this direction). A Soft Theorem for Scalar Quanta {#sec:scalrtheorems} -------------------------------- The derivation of soft theorems carried out in Chapter \[chap:obs\] admits a rather direct extension to the case of a massless scalar mediator [@CampigliaCoitoMizera; @Campiglia_scalars; @Hamada:2017atr]. Indeed, a scattering amplitude involving, say, the emission of a very soft scalar will receive its leading contributions from the diagrams in which the soft particle is attached to an external line, schematically: [MyOtherDiagram]{} $$\begin{aligned} \begin{gathered} \begin{fmfgraph*}(90, 65) \fmfleft{i1,d1,i2,i3} \fmfv{label={\color{blue}\vdots}, label.dist=-4mm}{d1} \fmfright{o1,d2,o2,o4,o3} \fmfv{label={\color{blue}\vdots}, label.dist=-4mm}{d2} \fmf{fermion, fore=blue}{i1,v1} \fmf{fermion, fore=blue}{i2,v1} \fmf{fermion, fore=blue}{i3,v1} \fmf{fermion, fore=blue}{v1,o1} \fmf{fermion, fore=blue}{v1,o2} \fmf{fermion, fore=blue}{v1,o3} \fmf{dashes, fore=red, tension=0}{v1,o4} \fmfv{d.sh=circle,d.f=empty,d.si=.25w,b=(.5,,0,,1)}{v1} \end{fmfgraph*} \end{gathered}\ = \ \begin{gathered} \begin{fmfgraph*}(90, 65) \fmfleft{i1,d1,i2,i3} \fmfv{label={\color{blue}\vdots}, label.dist=-4mm}{d1} \fmfright{o1,d2,o2,u3,o3} \fmfv{label={\color{blue}\vdots}, label.dist=-4mm}{d2} \fmf{fermion, fore=blue}{i1,v1} \fmf{fermion, fore=blue}{i2,v1} \fmf{fermion, fore=blue}{i3,v1} \fmf{fermion, fore=blue}{v1,o1} \fmf{fermion, fore=blue}{v1,u2} \fmf{dashes, fore=red, tension=0}{u2,u3} \fmf{fermion, fore=blue}{u2,o2} \fmf{fermion, fore=blue}{v1,o3} \fmfv{d.sh=circle,d.f=empty,d.si=.25w,b=(.5,,0,,1)}{v1} \end{fmfgraph*} \end{gathered} \ +\ \cdots \end{aligned}$$ Such contributions take a factorized form analogous to Weinberg’s leading soft theorem. Explicitly, adopting the same notation employed in , S\_\^[0]{}(q) S\_, where $g_n$ denotes the coupling of the $n$th particle with the massless spin-zero mediator. To leading order and at tree level, it is also possible to recast this identity for the emission of a scalar particle in the form of a Ward identity for the corresponding $S$ matrix \[Id\_CCM\] Q\^+ S - S Q\^-=0, with $Q^\pm$ suitable operators expressed in terms of creation and annihilation operators of external physical quanta. As usual, $Q^\pm$ can be split into their hard parts $Q^\pm_h$ and soft parts $Q^\pm_s$. Adopting the usual retarded Bondi coordinates in four-dimensional Minkowski space, one can perform the standard asymptotic expansion for the free scalar field equation $\Box\varphi=0$, \[wavescalar4\] 2(\_r+)\_u= (\_r+)\_r+ , and verify that the falloff condition (u, r, z, |z)=b(u, z, |z)/r + o() satisfies to leading order, thereby correctly highlighting the free propagating mode $b(u,z,\bar z)$. The soft “scalar charges” can then be expressed in terms of the massless scalar radiative mode as follows Q\^+\_s= \_[S\^2]{} b(u, z, |z) (z, |z)\_[z|z]{}dz d|z, where $\Lambda(z,\bar z)$ is an arbitrary function of the two angular coordinates $z$ and $\bar z$ on the unit sphere while $\gamma_{z\bar z}$ is the corresponding metric. Soft Scalar and Dual Two-Form Charges {#sec:twoformss} ------------------------------------- The interpretation of the above Ward identity in terms of an underlying symmetry, however, remained elusive for some time. Indeed, differently from the analogous results holding for the case of soft particles with spin $s \geq 1$, which are the subject of chapters \[chap:basics\], \[chap:obs\], \[chap:spin1higherD\] and \[chap:HSP\], for the case of soft scalars it is not clear a priori what symmetry, if any, could be underlying the conservation of the corresponding charges. This puzzle was investigated in [@Campigliascoop; @twoform-io], where it was conjectured that the scalar soft charges $Q^\pm_s$, in four spacetime dimensions, could be identified with the Noether charges associated to the large gauge symmetries of a two-form gauge field, to be interpreted as propagating the same massless scalar degree of freedom, in a dual picture. According to this scenario would be naturally interpreted as the Ward identity arising from the large gauge symmetry of a two-form field. While this idea turned out to be essentially correct, the different gauges employed in [@Campigliascoop] and [@twoform-io] determined relevant differences in the asymptotic behavior of the corresponding charges, on which we shall comment at the end of section \[sec:scoop\]. Before detailing the procedure followed and the results obtained, let us add a few remarks on the general perspective and the possible lessons that may be learned from this exploration. The possibility to analyze the relation between asymptotic symmetries and soft theorems from the perspective of dual theories may be worth exploring in a number of additional contexts. While already approached to some extent for the case of asymptotic $U(1)$ symmetries [@Strominger-dual; @Shahin-dual] and supertranslations [@GodazgarDual1; @GodazgarDual2; @PorratiDual], in $D=4$, it would be interesting to reconsider from this vantage point the issue of higher-dimensional asymptotic symmetries for gravity and for higher spins. Some symmetries may be easier to identify in a given dual description rather than in other, on-shell equivalent, pictures, a possibility that is conceivable on account of the typically nonlocal relation that connects two dual covariant descriptions of the same degrees of freedom. On the other hand, one ought to keep in mind that dualities typically only hold at the free level, and hence may only hold between asymptotic fields. Coming back to soft scalars, let us also observe that, while the main focus here is on the four-dimensional case, the very existence of analogous duality relations between free massless scalars and $(D-2)$-forms in $D$ dimensions provides natural candidate explanations for the corresponding soft scalar charges identified in any even $D$ [@Campiglia_scalars], while also possibly indicating the existence of analogous results in odd dimensions as well. Let us also mention that factorization theorems involving soft scalars can actually be derived, according to the strategy described at the beginning of Chapter \[chap:obs\], in a specific class of models characterized by spontaneously broken scale invariance [@DiVecchia:2015jaq; @DiVecchia:2017uqn], where the soft scalar is interpreted as the corresponding Goldstone boson. ### Asymptotic symmetries for two-form gauge fields {#ssec:2formasympt} We consider the gauge field described by a two-form B=B\_[ab]{}dx\^a dx\^b, *i.e.* an antisymmetric rank-two tensor $B_{a b}= - B_{ba}$, subject to the reducible gauge transformation $\delta B = d\epsilon$, in components $\epsilon=\epsilon_a dx^a$ and B\_[ab]{} = \_[a]{} \_[b]{} - \_[b]{} \_[a]{}. In its turn, the one-form parameter $\e$ is subject to the gauge-for-gauge symmetry $\delta \epsilon = df$, *i.e.* $\d\e_{a} \, = \, \pr_{ a} f$, where $f$ is a scalar parameter. The gauge-invariant field strength is $H=dB$, equivalently H\_[a b c]{} = \_[a]{} B\_[b c]{} + \_[ c]{} B\_[a b]{} + \_[ b]{} B\_[c a]{} , while Lagrangian and equations of motion are given by = - H\_[ a b c]{} H\^[ a b c]{} , \_[ a]{} H\^[ a b c]{} = 0 , where the latter, in components of $B_{ab}$, take the form \[eomB\] B\_[ab]{} + \_[a]{} \^c B\_[ bc]{} - \_b \^c B\_[ ac]{} = 0. Our goal in this section is to investigate the asymptotic symmetries of this theory, much in the spirit of what we have done for the Maxwell theory and for (linearized) gravity. The asymptotic analysis for $p$-form fields was actually already explored in [@Shahinpforms], but in the “critical dimension” in which radiation order $\mathcal O(r^{\frac{2-D}{2}})$ and Coulombic order $\mathcal O(r^{2-D+p})$ coincide, namely $D=2(p+1)$ , while here we need to investigate a different situation where $D=4$ and $p=2$. Adopting the standard retarded coordinates, we start by exploiting the gauge-for-gauge symmetry to set $\e_r=0$, thus fixing the the scalar parameter $f$, up to an $r$-independent but otherwise arbitrary function $f (u, z, \bar{z})$. Then, we employ the gauge transformations B\_[ru]{} = \_r \_u ,B\_[ri]{} = \_r \_i, to reach the “radial gauge” B\_[ru]{}=0=B\_[ri]{} , where $x^i$, with $i=1,2$, stand for $z$, $\bar z$. This leaves a residual gauge freedom with parameters $\e_u(u,z,\bar z)$ and $\e_i(u,z,\bar z)$, and the gauge-for-gauge redundancy $f (u,z,\bar z)$. We may then further exploit the $u$-dependence of $\e (u,z,\bar z)$ to set $\e_u (u,z,\bar z)=0$. The result of this gauge-fixing strategy is the following: one is left with the gauge-field components B\_[ui]{}(u,r,z,|z),B\_[z |z]{}(u,r,z,|z), while still keeping the residual gauge parameters \[transform-gauge\] \_i (z, |z), together with the residual gauge-for-gauge symmetry encoded in f (z,|z). Expanding the equations in the above gauge yields $$\begin{aligned} & \partial_r D^j B_{ju} \, =\, 0\,,\\ & \partial_r^2 B_{ui}+\frac{1}{r^2}\partial_r D^{j} B_{ij} \, =\, 0\,,\\ & \partial_u \partial_r B_{ui} \, - \, \frac{1}{r^2}\partial_u D^j B_{ij} \, - \, \partial_r^2 B_{ui}\, - \, \frac{\Delta-1}{r^2} B_{ui} \, - \, \frac{1}{r^2}D_iD^j B_{ju} \, =\, 0\,,\\ \label{r-eq} & 2\left(\partial_r - \frac{1}{r}\right)\partial_u B_{ij} \, - \, \frac{\Delta}{r^2} B_{ij} \, + \, \left( \partial_r - \frac{2}{r} \right) \left(D_{[i}B_{j]u} \, - \, \partial_rB_{ij}\right) \, - \, \frac{1}{r^2}D_{[i}D^l B_{j]l} \, =\, 0\,,\end{aligned}$$ where we recall that $D_i$ denotes the covariant derivative on the unit sphere and $\Delta = D^i D_i$ is the Laplacian on the unit sphere. In order to impose consistent falloff conditions, we adopt two guiding criteria: we consider field configurations that radiate a finite energy per unit time across any spherical section $S_u$ of null infinity and we check compatibility with the free equations of motion to leading order as $r\to\infty$. Finiteness of the energy flux at infinity imposes that the limit \[energyflux\] (u) = \_[r]{}\_[S\_u]{}\^[ij]{}\^[jk]{} H\_[uil]{}(H\_[ujk]{}-H\_[rjk]{}) r\^[-2]{} dbe finite, hence indicating[^12] that both $B_{ij}$ and $B_{uj}$ should scale at most like $r$, as $r\to\infty$. Equation further suggests that $B_{ij}$ should scale precisely like $r$, thus saturating the energy bound, so that the leading component of $\partial_u B_{ij}$ be unconstrained on-shell. Indeed, we find that the free equations of motion are solved to leading order as $r\to\infty$ by \[falloff-radiation\] B\_[ui]{} = D\^j C\_[ij]{} r+ ,B\_[ij]{} = r C\_[ij]{}+ , where $C_{ij}(u,z,\bar z)$ is an antisymmetric tensor on the sphere. In particular, this class of asymptotic solutions highlights $C_{z\bar z}$ as the single on-shell propagating degree of freedom carried by the two-form field being the only independent function of the leading solution space. Moreover, it carries a finite amount of energy to null infinity, (u) = \_[S\_u]{} \^[ij]{}\^[jk]{} \_u C\_[il]{}\_uC\_[jk]{} d Ø, as required. The falloff conditions are invariant under any gauge transformation , which we thus identify as providing the set of residual symmetries of the theory. We can compute the corresponding surface charge [@Barnich-Brandt; @HS-charges-cov; @Avery_Schwab] \[generic-charge\] Q\^+ = \_[S\_u]{} \^[ur]{} dzd|z, where the integration is performed on a sphere $S_u$ at fixed retarded time $u$ and for a large value of the radial coordinate $r$, while the Noether two-form [@Avery_Schwab] $\kappa^{ab}$ is given by \[twoformtwoform\] \^[ab]{}= H\^[bac]{}\_[c]{} and hence satisfies \^[ur]{} = r\^[2]{}\_[z|z]{} \_H\^[ru]{} = \_[z|z]{} \_i \^[ij]{} H\_[jur]{} = -\_[z|z]{}\_i \^[ij]{}\_r B\_[uj]{}. Making use of the equations of motion we can further rewrite the charge as follows \[charge\_two\_form\] Q\^+ & = \_[S\_u]{} \^[ij]{}\^[lk]{} D\_[i]{}\_l C\_[kj]{}d\ & = \^[z|z]{}(\_z\_[|z]{}-\_[|z]{}\_z)C\_[z|z]{}dzd|z . ### Duality and scalar charges {#ssec:duality} A two-form gauge field $B_{\mu\nu}$ in $D=4$ is dual, on shell, to a scalar field $\vf$ via the relation $\ast dB = d\vf$, or equivalently $dB=\ast d\varphi$, where $d$ is the exterior derivative and $\ast$ the Hodge dual[^13] in $D=4$; in components, r\^2 \_[z|z]{} \_[abcd]{}\^a B\^[bc]{} = \_d or, equivalently, \[gaugeinvduality\] H\_[abc]{}=r\^2 \_[z|z]{} \_[abcd]{}\^d. In fact, this duality corresponds to the identification between the singlet representation $\bullet$ and the antisymmetric rank-$2$ form [$\young(\hfill,\hfill)$]{} of SO(2), [*i.e.*]{} the little group for massless particles in $D=4$. A remarkable, albeit elementary, feature of the above duality relation is that it trades the equations of motion of $\varphi$ for the Bianchi identities of $B$ and vice-versa dd=dH=0,dH=d\^2=0. This means that, if we find a two-form $B$ that satisfies the duality, it must automatically satisfy the equations of motion $d\ast H=0$, which we can use to cross check the solution obtained above. In components, the duality relation yields &\_r B\_[z|z]{} = r\^2\_[z|z]{} \_r ,\ &\_r B\_[uz]{} = -\_z ,\ &\_r B\_[u|z]{} = \_[|z]{} ,\ &\_u B\_[z|z]{}+D\_[\[z]{}B\_[|z\]u]{} - \_r B\_[z|z]{} = -r\^2 \_[z|z]{} \_u . Comparing with the falloffs for the two-form , we see that these equations are compatible to leading order with the standard falloff condition for the massless scalar \[scalarasympt4\] (u,r,z,|z) = +, which in particular satisfies the equation of motion $\Box\varphi=d\ast d\varphi=0$ identically to leading order, provided one identifies \[link\] b \_[z|z]{} = - C\_[z|z]{}. This relation furnishes the desired connection between the on-shell degree of freedom $C_{ij}$ of the two-form field and the propagating component $b$ of the massless scalar. Equation can be rewritten covariantly as $C_{ij} =i b\, \Omega_{ij}$, where $\Omega$ is the standard symplectic form on the Euclidean sphere. Let us now compare “charge” operators arising from scalar soft theorems [@CampigliaCoitoMizera] with the surface charges given by two-form asymptotic symmetries , in order to connect the former to the latter by means of the duality transformation. We recall that the soft part of the scalar charges can be expressed as \[scalar-charge4\] Q\^+\_s= \_[S\^2]{} b(u, z, |z) (z, |z)\_[z|z]{}dz d|z, where $\Lambda(z,\bar z)$ is an arbitrary function of the two angular coordinates. In view of , we may identify \[charge-link\] Q\^+\_s = r Q\^+, and correspondingly for the residual symmetry parameters \_[z|z]{} = \_z \_[|z]{} - \_[|[z]{}]{} \_[z]{}. To summarize, this picture leads to an identification between the propagating scalar mode $b (u,z,\bar z)$ and the two-form physical degree of freedom $C_{z\bar z}(u,z,\bar z)$ as $C_{z\bar z} = - \gamma_{z\bar z}b$. On the other hand, the asymptotic charge for the two-form residual symmetry takes the form Q\^+ = \_[S\^2]{} C\_[z |z]{} \^[z|z]{}(\_z \_[|z]{}- \_[|z]{}\_z) dz d |z, and are conjectured to be dual to the scalar soft charges by means of . A puzzling, although not completely unfamiliar feature of the identification is the fact that, while $Q^+_s$ is nontrivial on $\mathscr I^+$, *i.e.* even after performing the limit $r\to\infty$, our two-form charge $\tilde Q^+$ vanishes in the large-$r$ limit. This seems to be a consequence of the fact that, in radial gauge, symmetry parameters are not allowed to grow with $r$, and hence are unable to compensate for the falloff $\partial_r B_{ui}\sim 1/r$. Indeed, in order to avoid this shortcoming, one could try relaxing some of the restriction that have been imposed on the gauge parameter $\epsilon_r$ by means of gauge-for-gauge transformations. That is, one could still impose the radial gauge $B_{ar}=0$, allowing however for a generic $\epsilon_a$ satisfying $\partial_{[r}\epsilon_{a]}=0$. In fact, the matching condition between the field strength of the two-form theory and the on-shell scalar is gauge invariant and can be substituted into to obtain the charge \[chargequasi\] Q\^+=(\_[|z]{}\_z-\_z\_[|z]{})b dzd|z, whose value on $\mathscr I^+$ is nonzero provided that $\epsilon_i$ scales like $\mathcal O(r)$. However, upon performing the asymptotic expansion for $\epsilon_a$, the condition that radial gauge be preserved imposes \_[\[r]{}\_[i\]]{}=0 \^[(1)]{}\_i = \_i \_r\^[(0)]{}, which spoils the order $\mathcal O(r^{0})$ of above charge. Another possible way out of this inconvenience could be to add terms of the type \[Coulombs\] \_zf(z)r ,\_[|z]{} g(|z)r , to the two-form components $B_{uz}$, $B_{u\bar z}$ respectively. These terms are indeed allowed by the leading equations of motion and give no contribution to the energy flux at infinity. These new terms would give rise to the modified charge \[modified charge\] dzd|z - \^[z|z]{}(\_z\_[|z]{}-\_[|z]{}\_z)C\_[z|z]{}dzd|z, which no longer goes to zero as $r\to\infty$. On the other hand, in this limit, it appears to become independent of the physical degree of freedom $C_{z\bar z}$ and, in the dual interpretation, of the radiative mode $b$ of the massless scalar. Indeed, the $f(z)$ and $g(\bar z)$, appearing in the first term of , are related by duality to a scalar field $\varphi(u,r,z, \bar z) = -f(z)+g(\bar z)+\cdots$, which is static to leading order as $r\to\infty$. The terms admit a natural interpretation if one phrases the problem of studying the two-form falloffs in a spacetime of generic dimension $D$. In this setup, the asymptotic analysis of the equations of motion highlights two classes or “branches” of solutions: denoting by $x^i$ coordinates on the celestial $(D-2)$-sphere, one has a radiation branch \[radiation-branch\] B\_[ui]{} = U\_[i]{}(u,x\^k) r\^[(4-D)/2]{}+,B\_[ij]{} = C\_[ij]{}(u,x\^k) r\^[(6-D)/2]{}+, subject to $U_i=D^j C_{ij}$ (unless $D=6$, in which case only $\partial_u U_i = \partial_u D^j C_{ij}$ need be imposed) and a Coulomb-like branch \[Coulomb-branch\] B\_[ui]{} = U\_[i]{}(x\^k) r\^[5-D]{} +,B\_[ij]{} = C\_[ij]{}(u,x\^k)r\^[5-D]{}+, where $\partial_u \tilde C_{ij} = D_{[i}\tilde U_{j]}$ and $(D-5)D^j \tilde U_{j}=0$. Solutions of the first type give rise to nonzero energy flux across sections of null infinity, $\mathcal P(u)\neq 0$, and only give vanishing contributions to the (global) charges as $r\to\infty$. The second class, on the other hand, does not contribute to the energy flux, while giving nonzero contributions to charge integrals. In $D=4$, the above expressions exhibit singularities and reduces to , while gives rise to . The scalar radiative mode $b$, in four dimensions, appears thus to be dual to the a radiation solution for its two-form counterpart. From this observation, it appears natural that its soft charge may be dual to an asymptotically vanishing two-form charge. Lorenz Gauge Approach {#sec:scoop} --------------------- A similar logic can be employed, with minor modifications, to discuss the problem at stake in the Lorenz gauge [@Campigliascoop]. The main advantage of this choice, as opposed to radial gauge employed above, is that it allows for the presence of a finite nonzero asymptotic charge. One starts from the duality condition $H=\ast d\varphi$, namely , which affords \[falloffH\] H\_[urz]{}&=-+ ,\ H\_[ur|z]{}&=+,\ H\_[uz|z]{}&=r\_[z|z]{}\_ub+ ,\ H\_[rz|z]{}&=\_[z|z]{}b+, upon substituting the asymptotics for the scalar field. One then looks for a two-form $B_{ab}$ satisfying $\nabla^a B_{ab}=0$ whose field strength respects . This is achieved to leading order by \[falloffB\] B\_[ij]{}&=rB\_[ij]{}\^[(-1)]{}(u,z,|z)+ ,\ B\_[ui]{}&=B\_[ui]{}\^[(0)]{}(u,z,|z)+,\ B\_[ri]{}&=B\_[ri]{}\^[(0)]{}(z,|z)+B\_[ri]{}\^[(1)]{}(u,z,|z)+ ,\ B\_[ur]{}&=DB\_r\^[(0)]{}(z,|z)+, provided B\_[z|z]{}\^[(-1)]{}+\_[\[z]{}B\_[|z\]r]{}\^[(0)]{}=\_[z|z]{}b and \_u B\_[rz]{}\^[(1)]{}+\_z DB\_r\^[(0)]{}=\_[z|z]{}\_z b,\_u B\_[r|z]{}\^[(1)]{}+\_[|z]{} DB\_r\^[(0)]{}=-\_[z|z]{}\_[|z]{} b. Residual symmetries can be parametrized by closed two-forms $\alpha$ such that $d\alpha=0$ and $\nabla^a\alpha_{ab}=0$, which also respect the falloffs . These conditions are solved to leading order by \[falloffalpha\] \_[ij]{}&=r\_[\[i]{}\_[j\]r]{}\^[(0)]{}(z,|z)+ ,\ \_[ui]{}&=\_[ui]{}\^[(0)]{}(u,z,|z)+,\ \_[ri]{}&=\_[ri]{}\^[(0)]{}(z,|z)+\_[ri]{}\^[(1)]{}(u,z,|z) + ,\ \_[ur]{}&=D\_r\^[(0)]{}(z,|z)+, subject to \_u \_[ri]{}\^[(1)]{}+\_i D\_r\^[(0)]{}=0. Then, the expression for the charge obtained in immediately yields Q\^+ = \_[\[i]{}\_[j\]r]{}\^[(0)]{}(z,|z)b(u,z,|z) dzd|z. While this result indeed provides a more satisfactory answer to our initial question compared to the one we obtained in the radial gauge, since it allows for a matching between the soft scalar charges and *nonvanishing* two-form asymptotic charges, it shows that the calculation of asymptotic symmetries may be strongly influenced not only by the choice of falloffs, as already highlighted by the higher-dimensional spin-one and spin-two theories, but also by the more basic choice of gauge fixing. In this setup, the radial gauge effectively trivializes the asymptotic symmetries, by forbidding nontrivial $\alpha_{ir}\sim\mathcal O(1)$, and hence hides the presence of finite soft charges. One may interpret this fact by observing that reaching radial gauge from a generic field configuration (before gauge fixing) may involve a *large* gauge transformation and, therefore, that such a gauge choice is not feasible if one wants to have control on the full structure of asymptotic symmetries. Asymptotic Symmetries of Higher Spins {#chap:HSP} ===================================== The discussion of the preceding chapters allowed us to exhibit a link between asymptotic symmetries and soft theorems in the context of electrodynamics, Yang-Mills theory, gravity and, maybe less directly, in the case of scalar field theories and their duals. In these cases, the spin of the boson mediator of the interaction was thus respectively 1, 2 and 0. The goal of this chapter is to extend this type of analysis to theories containing a spin-$s$ gauge boson, namely a massless particle carrying a kind of generalized electromagnetic/gravitational force, with arbitrary integer $s$ [@carlo_tesi; @super-io; @Cariche-io]. The interest in this type of theories derives primarily from the analysis of the string theory spectrum of elementary excitations, which contains particles with arbitrary integer spin and with mass proportional to the string tension. In order to better understand the properties of these excitations in the massless limit, which is conjectured to be linked to a suitably-defined tensionless limit of the string, it is natural and worthwhile to try to set up a purely field-theoretical description for them, *i.e.* a higher-spin gauge theory. Such theories admit a consistent Lagrangian formulation on maximally symmetric backgrounds up to the level of cubic vertices. The possibility to construct a Lagrangian to all order in the vertices is subject to notorious obstructions of both technical and conceptual nature, in particular to a conflict with locality. On the other hand, the fully nonlinear construction of Vasiliev [@Vasi] (see [@BCIValgebras; @Did] for reviews) does not admit Minkowski spacetime among its vacuum solutions. Still, as we stressed in our discussion of the lower-spin cases, one may expect that the linear theory [@Fronsdal; @ML] already contains salient features related to the asymptotic symmetry transformations. In our analysis, we will thus propose a definition for the notion of asymptotic symmetries of a free spin-$s$ field theory on Minkowski background coupled to a suitable external source, representing unspecified nondynamical matter. As we shall see, in four spacetime dimensions, these asymptotic symmetries on the one hand give rise to Ward identities and allow us to make contact with the corresponding Weinberg theorem, while on the other hand they display a structure which can be regarded as interesting *per se* and in relation with the conjectured existence of an underlying non-Abelian higher-spin algebra. A Bondi-like Gauge ------------------ As repeatedly stressed, the first point we need to address, while attempting to define asymptotic symmetries in a given gauge theory, is the assignment of a suitable set of gauge-fixing conditions and falloff conditions on the gauge fields. Before tackling this issue, we will recall the basic definitions for a free spin-$s$ gauge theory in the Fronsdal formulation [@Fronsdal] (see also [@Sagnotti_e_co] for a review). ### Fronsdal formulation A spin-three gauge field is described kinematically by a tensor $\varphi_{\mu\nu\rho}$ that is completely symmetric in its indices and is subject to the gauge transformation (we adopt here Cartesian coordinates $x^\mu$ for simplicity) \[spin3-transf\] \_\_ = \_\_ +\_\_ +\_\_, where the gauge parameter $\epsilon_{\mu\nu}$ is symmetric in its two indices and has vanishing trace, \[spin3-trace\] \^\_=0. Notice that, while eq. is just the natural generalization of the gauge transformation $h_{\mu\nu}\mapsto h_{\mu\nu}+\partial_\mu \xi_\nu + \partial_\nu \xi_\mu$ for a linearized metric perturbation, the trace constraint constitutes a novel feature. In the following, we will often use a shorthand notation to denote symmetrization and contraction of indices. A round bracket enclosing a set of indices denotes their (*unweighted*) symmetrization obtained by the least possible number of terms: for instance, A\_[()]{}=A\_ + A\_ if $A$ is generic rank-two tensor, while P\_[(]{}P\_[)]{}=P\_P\_, if $[P_\mu,P_\nu]=0$. A “prime” symbol denotes a trace, *e.g.* $\varphi'_{\rho}=\eta^{\mu\nu}\varphi_{\mu\nu\rho}$. With these conventions, the spin-three gauge symmetry can be expressed as \_\_=\_[(]{}\_[)]{},’=0. Finally, a divergence, namely a contraction between a derivative and an index of the tensor on which the derivative acts, is denoted by a “dot”, for example $ \partial\cdot\varphi_{\mu\nu}=\eta^{\sigma\rho}\partial_\sigma\varphi_{\mu\nu\rho} $. The dynamics for a spin-three field freely propagating in $D$ spacetime dimensions, coupled to an external source, can be defined by means of the gauge-invariant action $S=\int \mathcal L\, d^D\!x$, \[Fronsdalacs3\] L= E\^\_ - J\^\_, where the $\mathcal E^{\mu\nu\rho}$ is given by E\^ = F\^ - \^[(]{}F’\^[)]{} and $\mathcal F^{\mu\nu\rho}$, also called the Fronsdal tensor, is defined as F\^ = \^ - \^[(]{}\^[)]{} + \^[(]{}\^’\^[)]{}. The symmetric tensor $J^{\mu\nu\rho}$ defines the external source and it is subject to the local conservation condition \[tracelesscons3\] J\^-\^J’=0, where on the left-hand side we recognize the traceless projection of $\partial\cdot J^{\mu\nu}$. In order to handle symmetric tensors with an arbitrary number of indices, it is sometimes useful to introduce a more abstract notation in which all spacetime indices are implicit [@FSagn]. In which case, the last three equations take the compact form \[FronsdalanyS\] E = F - F’, F = - + \^2 ’, J - J’=0. We will not, however, employ this notation extensively in the following. The equations of motion obtained from the variation of the action are E\^ = J\^. Under a gauge transformation, the Fronsdal tensor transforms into \_F\^ = 3 \_\_\_’, and the last term vanishes by the constraint $\epsilon'=0$. This is is sufficient, for spin three, in order to prove the gauge invariance of the equations of motion. Since this is needed for the evaluation of the Noether charge, let us explicitly verify, as an exercise, the gauge invariance of the Fronsdal action. The trace of $\mathcal F$ is given by $ \mathcal F'_\rho = 2 \Box \varphi'_\rho - 2 \partial\cdot\partial\cdot\varphi_\rho + \partial_\rho \partial\cdot\varphi' $, while the $\mathcal E$ tensor satisfies the following identities, \[anomalousBianchi\] \_E\^ = - \^F’, which may be regarded as an “anomalous” analog of the (linearized) Bianchi identities, due to the nonzero term on the right-hand side. Therefore, employing the identity , \_( E\^\_) = E\^ \_\_ = \_(E\^ \_) + F’ ’ , which is a boundary term plus a contribution that vanishes by the constraint $\epsilon'=0$. For the source term we have -\_( J\^\_ )=-3 \_( J\^ \_), by the traceless conservation condition . Thus, \_L = - (E\^\_), so that the action is indeed invariant, up to a boundary term. The standard Noether current is then given by \[jspin\] j\^\_= \_[,]{} - \_\_+ (E\^\_), where, in compact notation,[^14] \[secondderj\] =& { \^ - ( \^\^+ \^\^) + \^ - \^ ’ .\ &. +( \^\^’ + \^\^’) - ( ’\^\^+ ’\^\^) }. The formulation outlined in this section for a spin-three field can be extended to any integer spin $s$. In particular, equations hold for arbitrary $s$, the action is given by $S=\int \mathcal L\, d^D\!x$, L = E\^[\_1\_s]{}\_[\_1\_s]{} + J\^[\_1\_s]{}\_[\_1\_s]{} and the gauge transformation is given by $\delta_\epsilon \varphi_{\mu_1\cdots \mu_s}=\partial_{(\mu_1}\epsilon_{\mu_2\cdots\mu_s)}$ (or $\delta_\epsilon\varphi=\partial\epsilon$ in compact notation). An important technical point is that the consistency of the theory requires the field to be doubly traceless for spin four or higher, namely $\varphi''=0$, together with $J''=0$. Once this point is taken into account, the equations of the theory take the form for any $s$. As far as the corresponding Noether current is concerned, and the obvious generalization of remain valid for any spin. Alternatively, one may employ the Maxwell-like formulation of free higher spins [@ML; @DarioGabrieleKarapet]. In this approach, the trace constraint for the gauge parameter $\epsilon$ is traded for the transversality condition $\partial\cdot \epsilon=0$, while trace constraints may or may not be enforced, depending on the type of spectrum that one wishes to describe. It should be noted that the conditions we are going to impose on the gauge fields in the next section effectively make the two frameworks not just equivalent but actually identical as far as the asymptotic analysis is concerned. ### Boundary conditions In analogy with the lower-spin cases, we propose the following set of gauge-fixing and falloff conditions for a spin-three field, focusing for the moment on the case of four spacetime dimensions. Employing the usual retarded Bondi coordinates, with stereographic coordinates on the sphere , we impose \[radtr3\] \_[rab]{}=0=\_[a z|z]{}, for any $a$, $b$, together with the conditions that the normalized field components, namely \_[uuu]{}, ,, (and similarly for $z\leftrightarrow \bar z$), scale asymptotically as $ \mathcal O(r^{-1}) $ as $r\to\infty$ for fixed $u$. These conditions provide a natural generalization of the radial gauge $\mathcal A_r=0$ of Maxwell’s theory and the (sharp) Bondi gauge adopted in the discussion of asymptotic symmetries of asymptotically flat spaces, as we have seen in Chapter \[chap:basics\]. Notice that, in the chosen coordinates, ’\_a = -2 \_[ura]{}+\_[rra]{}+ \_[z|z a]{}, so the trace of the field is identically zero in view of the conditions . In the case of a spin-$s$ field, we impose the analogous conditions \[Bondilikegau\] \_[ra\_2a\_s]{}=0=\_[z|z a\_3a\_s]{},=O(r\^[-1]{}), where $d$ denotes the number of $z$ indices appearing in the last formula, and analogously for $z\leftrightarrow\bar z$. In the following, we shall refer to this set of conditions as the “Bondi-like gauge”. As a first consistency check, it is worthwhile to calculate the flux of energy carried by a spin-$s$ field as measured on a sphere $S_u$ at large radius $r$ and fixed retarded time $u$. The expression for this quantity in Bondi-like gauge takes the following form: P(u)=\_[S\_u]{} (\_[z|z]{})\^[-s]{} \_u \_[zzz]{} \_u \_[|z |z |z]{}\_[z|z]{}dzd|z. This is generically nonvanishing and, most importantly, finite, even in the large-$r$ limit, thus providing support to the choice of the conditions . Another important consistency check that needs to be performed on the Bondi-like gauge conditions is their compatibility with the (free) equations of motion as $r\to\infty$. This ensures that the leading field components indeed specify the free boundary data associated to radiation and charge measurements performed near null infinity. Let us elaborate on this point, focusing for the moment on the spin-three case. Imposing $\varphi_{ab r}=0$ and $\varphi_{z\bar zr}=0$, the equations of motion take the form \[Fronsdaleq\] F\_[abc]{} = \_[abc]{}-\_[(a]{}\_[bc)]{}=0 . Now, $\mathcal F_{rrr}=0$ is identically solved, and similarly for $\mathcal F_{a z \bar z}=0$. From $\mathcal F_{uur}=0$ we have & \_[uuu]{} + 2 \_r ( ) + \^2\_r \_[uuu]{}- (D\^z \_[zuu]{} + D\^[|z]{}\_[|z uu]{} )\ & - \_r =0 , and expanding $\varphi_{uuu}=B\, r^{\alpha}$, $\varphi_{zuu}=U_z\, r^{\beta}$, with $\beta=\alpha+1$, to leading order we have (+1)=0. By comparison with $\mathcal F_{uuu}=0$, which reads \_u \_[uuu]{}+ \_u \_r \_[uuu]{} - (D\^z \_[zuu]{}+D\^[|z]{}\_[|z uu]{} )=0 , and yields, upon expansion, (+4)\_u B - 3 \_u (D\^z U\_z + D\^[|z]{}U\_[|z]{})=0 , we have two possible behaviors: a “growing mode” $\alpha=2$, $\beta =3$ and a “decaying mode” $\alpha=-1$, $\beta=0$. We choose the latter, obtaining $\varphi_{uuu}=B/r$ and $\varphi_{zuu}=U_z$, together with the additional condition \[interr\_1\] \_u B= \_u (D\^z U\_z + D\^[|z]{}U\_[|z]{} ) . From $\mathcal F_{zur}=0$, we have -D\^z\_[zzu]{}+\_[zuu]{} + (\_r - )( \_r \_[zuu]{}+\_[zuu]{}-D\^z\_[zzu]{})=0 . Which is solved by setting $\varphi_{zzu}= r\, C_{zz}$, where \[interr\_2\] U\_z = D\^z C\_[zz]{}. Finally, from $\mathcal F_{zzr}=0$, we have -D\^z\_[zzz]{}+\_[zzu]{}-(\_r - )(D\^z \_[zzz]{}-\_[zzu]{}-\_r\_[zzu]{})=0 , which is solved by setting $\varphi_{zzz}=B_{zzz}\,r^{2}$, with \[interr\_3\] C\_[zz]{} = D\^zB\_[zzz]{}. This completes the consistency check for the spin-three Bondi-like gauge against the equations of motion, since the number of independent equations in four spacetime dimensions is 10, owing to the Bianchi identities. To summarize, in the Bondi-like gauge, the nonvanishing field components read, to leading order in $r$, \_[uuu]{}=,\_[uuz]{}=U\_z,\_[uzz]{}=r C\_[zz]{},\_[zzz]{}=r\^2B\_[zzz]{}, and are subject to the constraints , and . The consistency of the Bondi-like gauge for the generic spin-$s$ case can be checked in a similar manner, which we now sketch. The equations of the form F\_[rr a\_[1]{}a\_[s-2]{}]{} = 0 are identically solved once we choose $\varphi_{ra_{1}\cdots a_{s-1}}=0$ and $\varphi_{z\bar z a_{1}\cdots a_{s-2}}=0$. The equations $\mathcal F_{ru\cdots u}=0$, $\mathcal F_{uu\cdots u}=0$ and $\mathcal F_{ru\cdots uz}=0$ have the same form as the analogous equations of the spin-three case obtained by removing $s-3$ indices $u$ from them, the technical reason being that the connection coefficients $\Gamma^a_{b u}$ vanish identically. The equation $\mathcal F_{ru\cdots u z\cdots z}=0$ with a given number $1<d<s$ of $z$ indices reads explicitly &D\^z\_[z\_[d+1]{}u\_[s-d-1]{}]{}- \_[z\_d u\_[s-d]{}]{}\ &-(\_r- )(\_r\_[ z\_d u\_[s-d]{}]{}-D\^z\_[z\_[d+1]{}u\_[s-d-z]{}]{}+\_[z\_d u\_[s-d]{}]{})=0, where for brevity $\varphi_{z_du_{s-d}}$ denotes $\varphi_{u\cdots u z \cdots z}$ with $d$ indices $z$ and $s-d$ indices $u$. Altogether these equations impose \_[z\_du\_[s-d]{}]{}=B\_[z\_d]{} r\^[d-1]{}, where the tensors $B_{z_d}$ have to satisfy B\_d = D\^zB\_[z\_[d+1]{}]{}, whereas the other equations are identically satisfied to leading order. Higher-Spin Supertranslations and Superrotations ------------------------------------------------ We now turn to the illustration of the asymptotic symmetries of higher spins in four dimensions. These are determined as the large gauge symmetries that leave the Bondi-like gauge invariant, namely, restricting for the moment to spin three, as the solutions of the asymptotic Killing tensor equations $ \delta_{\epsilon}\varphi_{r ab}=0=\delta_\epsilon \varphi_{z\bar z a} $, \_\_[uuu]{}=O(r\^[-1]{}),\_\_[uuz]{}=O(r\^[0]{}),\_\_[uzz]{}=O(r\^[1]{}),\_\_[zzz]{}=O(r\^[2]{}). ### A higher-spin analog of supertranslations To begin with, we shall restrict to the case of gauge parameters that do not depend on retarded time. Then, the answer is provided by the following family of tensors, parameterized by the arbitrary function $T(z, \bar z)$: \[supertransl\_spin3\] &\_[ab]{}dx\^a dx\^b = - ( T + D\^z D\_z T + (D\^zD\_z)\^2T )du\^2 - 2 ( T + D\^zD\_z T ) du dr\ &- 2r ( D\_zT + D\_z\^2 D\^z T ) du dz - 2r ( D\_[|z]{}T + D\_[|z]{}\^2 D\^[|z]{} T ) du d|z - T dr\^2\ & - r ( D\_zT dz + D\_[|z]{}T d|z ) dr - ( D\^2\_z T dz\^2 + D\^2\_[|z]{}T d|z\^2 ) - \_[z|z]{} ( T + D\^zD\_z T ) dz d|z . We refer to Appendix \[app:spin3symm\] for the details of this derivation. Note that the corresponding contravariant tensor on $\scrip$ is given by \^[ab]{}\_a \_b = - T(z, |z) \_u\^2 , so that this residual symmetry indeed furnishes a spin-three generalization of the ordinary gravitational supertranslation symmetry $\xi^a\partial_a= T(z,\bar z)\partial_u$. The nonvanishing gauge variations generated by are: $$\begin{aligned} \delta \varphi_{uuz} &= - D_z \left( \frac{3}{4}\,T + D^z D_z T + \frac{1}{4}\,(D^zD_z)^2T \right) , \\ \delta \varphi_{uzz} &= - \frac{r}{2}\, D^2_z\left( 3\, T + D^z D_z T \right) , \\ \delta \varphi_{zzz} &= - \frac{3}{2}\, r^2 D^3_z T \,, \end{aligned}$$ together with their conjugates. Moving on to arbitrary spins, we must look for spin-$s$ gauge transformations that leave invariant our Bondi-like gauge, as summarized by \[constr\_spin-s\] \_[r a\_[s-1]{}]{}=0=\_[z |z a\_[s-2]{}]{} and \[boundary\_spin-s\] \_[u\_[s-d]{} z\_d]{} = r\^[d-1]{} B\_[z\_d]{}(u,z,|z), where we have adopted the following multi-index notation, which will be understood from now on for symmetrized indices [@massive]: a subscript attached to a spacetime index indicates the number of times that type of index appears. For instance, for a spin-five field, \_[u\_2z\_3]{}=\_[uuzzz]{},\_[ua\_4]{}=\_[uabcd]{}. As already remarked, the Bondi-like gauge conditions actually imply that the field is traceless. The solutions to the asymptotic Killing tensor condition may be conveniently labeled by the following numbers: - the number $p$ of “$u$” indices appearing, - the number $d$ of “$z$” indices appearing without $\bar z$ counterpart, - the number $c$ of pairs “$z \bar z$”, counted ignoring their order. Assuming a power-law dependence on $r$, the residual gauge symmetries that do not depend on $u$ admit then the following parametrization: $$\begin{aligned} \label{pd0} \epsilon_{u_pz_d}=&- \frac{r^d D^d_zT_p(z, \bar z)}{\prod_{k=1}^d (s-p-k)}\,,\\ \label{c+1c} \epsilon_{u_pz_{d+c+1}\bar z_{c+1}} =& - \frac{r^2}{2}\,\gamma_{z\bar z} \left( \epsilon_{u_pz_{d+c}\bar z_{c}} - 2\, \epsilon_{u_{p+1}z_{d+c}\bar z_{c}} \right),\end{aligned}$$ where $T_p(z, \bar z)$ for $p=0,\ldots,s-1$ is a set of angular functions satisfying \[p+1p\] T\_[p+1]{} =T\_p + D\^zD\_zT\_p. The details of this derivation are available in [@carlo_tesi]. We thus note that the $T_p$ are actually determined recursively in terms of only one angular function $T_0(z, \bar z) \equiv T(z,\bar z)$. The corresponding, nonvanishing, gauge variations are, for $s = p + d$, \_[u\_p z\_d]{} = d D\_[z]{}\_[u\_pz\_[d-z]{}]{} = - . This concludes the presentations of the set of “higher-spin supertranslations”, namely those asymptotic symmetries whose parameters do not depend on retarded time. To summarize, we have seen that they can be parametrized by a single angular function $T(z,\bar z)$, in analogy with gravitational supertranslations and spin-one asymptotic symmetries. As we shall see in the next section, these symmetries will play a distinguished role in connection with the leading soft theorem. Furthermore, their identification as the proper higher-spin analogs of gravitational supertranslations will receive additional support in the analysis of the general solution to the asymptotic Killing equation, which we now turn to describe. ### Higher-spin superrotations and more In the remainder of this section, we shall explore the full set of asymptotic symmetries in the case of spin three. From now on, we will adopt the notation $x^i$ for generic angular coordinates on the celestial sphere, in terms of which the Bondi-like gauge conditions take the form: \[r+traces’\] \_[rab]{}=0,\^[ij]{}\_[ija]{}=0, together with the requirement that the normalized field components \[eqWeakest’\] \_[uuu]{},,, scale as $\mathcal O(r^{-1})$ near future null infinity. As we have already seen in Chapter \[chap:spin1higherD\], this “more covariant” notation is better suited to the generalization of the present discussion to a higher-dimensional context. In fact, with this notation, it is easy and worthwhile to solve the asymptotic Killing equation dictated by requirement that and be preserved for a generic spacetime dimension D=n+2, *i.e.* when the coordinates $x^i$ take $n$ values, $i=1$, $2$,$\ldots$, $n$. However, one should keep in mind that and provide finite energy flux and specify consistent boundary data for the dynamical problem only when $D=4$, namely $n=2$. Keeping in mind these considerations, let us quote here the solution to the asymptotic Killing equation and to the trace constraint $\eta^{ab}\epsilon_{ab}=0$, whose derivation is detailed in Appendix \[app:spin3symm\]: \[gaugepars3\] \_[rr]{}&= - D+T,\ \_[ur]{}&= - + D- D + T,\ \_[uu]{}&= + D+ T,\ \_[ri]{}&= - +r\^2\_i + -D\_iD+D\_iT,\ \_[ui]{}&= DK\_i - +\ &-D\_i\[+2(n+1)\]D+\[+2(n+1)\]T,\ \_[ij]{}&= r\^4 K\_[ij]{} -+r\^3\ &+(D\_iD\_j+2\_[ij]{})DDK\ &-D+T, while the corresponding nonvanishing symmetry variations read \_\_[uui]{}&= D\_i(+n)\[+2(n+1)\]T,\ \_\_[uij]{}&= -(D\_iD\_j+\_[ij]{})\[+2(n+1)\]D\ &+( D\_iD\_j-\_[ij]{}) \[+2(n+1)\]T,\ \_\_[ijk]{}&= DDK\ &-D\ &+T. The key-point is that the full residual symmetry is parameterized by the tensors \[spin3tensors5\] T(x\^k),\^i(x\^k),K\^[ij]{}(x\^k), defined on the celestial sphere, bound to satisfy a set of constraints that we now turn to illustrate and interpret. The function $T(x^i)$ is completely arbitrary and specifies the family of asymptotic symmetries we have already identified as spin-three supertranslations above. Furthermore, the subset of these transformations that do not change the field, *i.e.* spin-three translations, is characterized by the equation (see Appendix \[app:spin3exact\]) \[l=012\] D\_[(i]{}D\_j D\_[k)]{}T-\_[(ij]{}=0, whose solutions are given by the spherical harmonics (see Appendix \[app:Laplaciano\]) with $l=0,1,2$. The solution space thus contains $1+(n+1)+\left[\tfrac{(n+2)(n+1)}{2}-1\right]=\tfrac{(n+1)(n+4)}{2}$ independent solutions. If we want to provide a tentative interpretation of this kind of symmetry in terms of generators of an underlying algebra, if any, we may draw the following analogy. In the case of the $BMS$ algebra, which can be viewed as an infinite-dimensional enhancement of the Poincaré algebra, supertranslations arise as an enlarged version of translation symmetry, associated to the Poincaré generator $P^\mu$. In the spin-three case, it is then natural to interpret these generalized supertranslations as the asymptotic symmetry analog of the exact Killing symmetry corresponding to the traceless projection of $P^{(\mu}P^{\nu)}$. Indeed, the number of these generators, $\tfrac{(n+2)(n+3)}{2}-1=\tfrac{(n+1)(n+4)}{2}$, matches the number of independent solutions of : in particular, $9$ generators in the relevant case of four dimensions. Turning our attention to the other tensors identified in , let us observe that the tensor $K^{ij}(x^k)$ must have vanishing trace \^[ij]{}K\_[ij]{}=0 and must satisfy the conformal Killing tensor equation \[confKillt2\] D\^[(i]{}K\^[jk)]{}- \^[(ij]{}DK\^[k)]{}=0. These equations take the following particularly simple form in stereographic coordinates $z$, $\bar z$ in four spacetime dimensions K\^[z|z]{}=0,\_[|z]{}K\^[zz]{}=0, \_z K\^[|z |z]{}=0, and hence locally admit two infinite-dimensional family of solutions expressed as holomorphic and anti-holomorphic functions K\^[z|z]{}=0,K\^[zz]{}=K(z),K\^[|z |z]{}=K(|z). Thus, these tensors are the natural spin-three analogs of the conformal Killing vectors parametrizing ordinary gravitational superrotations, which correspond to the enhancement of rotations and boosts generated by $M_{\mu\nu}$ at the level of finite transformations. This identification as spin-three superrotations can be further supported by counting the number of independent *global* solutions to equation . Consider for instance the Laurent expansion of $K=K^{zz}\partial_z^2$ near $z=0$ K\^[zz]{}\_z\^2=K(z)\_z\^2 = \_[nZ]{} K\_n z\^n \_z\^2. For this tensor to be regular, we need to restrict the sum to nonnegative values of $n$, namely $K_n=0$ for $n\le-1$. Performing now the change of variable $z=1/w$, we have K\^[zz]{}\_z\^2=\_[nZ]{}K\_n w\^[4-n]{}\_w\^2, from which $K_n=0$ for $n\ge5$ by regularity near $w=0$. This leaves us with five independent coefficients $K_0$, $K_1$, $K_2$, $K_3$, $K_4$, plus the additional five arising from the anti-holomorphic sector $\tilde K_0$, $\tilde K_1$, $\tilde K_2$, $\tilde K_3$, $\tilde K_4$ and hence a total of ten independent global spin-three superrotations. This is precisely the number of components of the traceless projection of $M^{\mu(\nu}M^{\rho)\sigma}$, the generators of the exact spin-three Killing symmetries corresponding to the (traceless projections of symmetrized) products of two rotations/boosts. Finally, the vector $\rho^i$ must satisfy \[confrho2\] D\^[(i]{}D\^j \^[k)]{}-=0, where the curly brackets denote the anti-commutator. Again, going to stereographic coordinates $z$, $\bar z$ in four spacetime dimensions allows us to recast this equation in the form \_[|z]{}(\^[z|z]{}\_[|z]{}\^[z]{})=0,\_[z]{}(\^[z|z]{}\_[z]{}\^[|z]{})=0, which admits solutions expressed in terms of holomorphic and anti-holomorphic functions \^[z]{}=a(z)\_z K(z,|z)+b(z),\^[|z]{}=a(|z)\_[|z]{}K(z,|z)+b(|z), where $\mathcal K(z,\bar z)$ is a Kähler potential for the metric on the unit sphere, *e.g.* $\mathcal K(z,\bar z)=2\log(1+z\bar z)$. We may identify this infinite-dimensional family as the one corresponding to “mixed” supertranslations/superrotations, namely the infinite-dimensional enhancement of the symmetries generated by the traceless part of $P^{(\mu}M^{\nu)\rho}$. Indeed, the number of independent generators of this form is 16, while on the other hand \^z\_z = \_z&= \_[nZ]{}\_z\ &= -\_[nZ]{}\_w so that the global regularity of $\rho$ allows for $a_{-1}$, $a_0$, $a_1$, $a_2$, $a_3$, $b_0$, $b_1$ and $b_2$, together with the corresponding anti-holomorphic coefficients: precisely 16 independent global transformations. To summarize, we have identified three families of spin-three asymptotic symmetries, and put them in one-to-one correspondence with elements of the universal enveloping algebra of the Poincaré algebra given by the traceless projections of the combinations $P^{(\mu}P^{\nu)}$, $P^{(\mu}M^{\nu)\rho}$ and $M^{\mu(\nu}M^{\rho)\sigma}$. These products are expected to be identified with the spin-three generators of a would-be higher-spin algebra, if any, with Poincaré subalgebra (see *e.g.* [@flat-algebras_1; @flat-algebras_2] for discussions on higher-spin algebras possibly related to four-dimensional Minkowski space). The asymptotic symmetries generated by $T$, $\r^{i}$ and $K^{ij}$ can thus be interpreted as the infinite-dimensional enhancement of the corresponding global Killing symmetries. Weinberg’s Soft Theorem as a Ward Identity for Any Spin ------------------------------------------------------- We shall now address the relation between the asymptotic symmetries of higher spins that we have described so far and Weinberg’s soft theorem, which we now briefly recall. As we discussed in Chapter \[chap:obs\], Weinberg considered the $S$-matrix element $S_{\beta\alpha} (\mathbf q)$, for arbitrary asymptotic particle states $\alpha \rightarrow \beta$, also involving an extra soft massless particle with momentum $q^{\, \mu}\equiv (\omega, \mathbf q) \to 0$ and helicity $s$. The leading contribution to this process takes a particularly simple factorized form that, in the notation of [@Weinberg_64; @Weinberg_65], can be written as \[wfact\] \_[0\^+]{} S\_\^[s]{}(q) = -\_[0\^+]{}S\_ , with $\eta_i$ being $+1$ or $-1$ according to whether the particle $i$ is incoming or outgoing ($g_n^{(s)}$ is the spin-$s$ coupling of the $n$th particle while $\varepsilon^\pm_a$ is the polarization tensor of the soft particle). In the case of higher spins, the constraints imposed by Weinberg’s theorem are in contradiction with a nontrivial interaction in the soft regime, thereby implying the absence of a long-range force associated to higher-spin quanta, if any. Nonetheless, it can be insightful to see whether or not this theorem can be associated to the presence of an underlying large gauge symmetry, as was the case for the less exotic spin $1$ and $2$ theories we considered in the previous chapters. Our aim is now to derive a Ward identity associated to spin-three supertranslation symmetry and show that it is equivalent to the corresponding Weinberg theorem, confining our attention to the case of four spacetime dimensions. In order to do this, let us first note that the only nonvanishing contribution to the Noether (surface) charge comes from $\delta\varphi_{zzz}$, and reads \[Q+spin3\] Q\^+ = \_ \_[z|z]{}\_u T(z,|z) d\^2z du - \_ \_[z|z]{} J(u, z, |z) d\^2z du , where J(u, z, |z) \_[r]{}r\^2 J\^[rrr]{}(u, z, |z) . The surface charge thus computed is in agreement, in particular, with that obtainable from the results of [@HS-charges-cov]. Under the assumption that the residual symmetry generators act on scalar matter fields as follows, = g\^[ (3)]{}\_i T(i)\^2\_u , where $g^{\, (3)}_i$ is the coupling of the corresponding matter, in the frequency domain we get | (Q\^+ S - S Q\^-) |= \_[i]{} \_[i]{}\^ g\^[(3)]{}\_i E\^2\_i T (z\_i, |z\_i) | S | . In addition, we shall impose the following auxiliary boundary condition at $\scri^\pm_\mp$ (D\^z)\^3B\_[zzz]{} = (D\^[|z]{})\^3 B\_[|z|z |z]{} . This condition can be interpreted as the higher-spin analog of the condition that no long-range magnetic fields be present on $\mathscr I^+$. We also leave aside the $J$ term, which acts trivially on the vacuum, thus obtaining Q\^+ = \_ T(z, |z) \_u (D\^z)\^3 B\_[zzz]{} \_[z|z]{} d\^2z du . An analogous result holds for $Q^-$. For the function $T(z, \bar z)$ we choose T(z, |z) = ()\^[2]{}, so that, after an integration by parts in $\partial_{\bar z}$, the computation of the charge involves \_[|z]{} ( ()\^[2]{}) = -2\^2(z-w) + \_[z|z]{} . Therefore Q\^+ = 3du D\^w D\^w B\_[www]{} - D\^zD\^z B\_[zzz]{} \_[z|z]{} d\^2z du , where in particular the last term is a vanishing boundary contribution. To sum up: \[Ward\_s=3\] &2(D\^z)\^2||\ &= \_[i]{} \_[i]{} ()\^[2]{} | S |. The large-$r$ limit performed for a free spin-three field in Fourier space gives, to leading order, B\_[zzz]{} = - \_0\^[+]{} d\_[q]{} so that du \_u B\_[zzz]{} = - \_[0\^+]{}. Thus, using crossing symmetry, we also have &-4||\ &= \_[0\^+]{}| a\^\_+(x) |, and this implies, by comparing with , \_[0\^+]{}| a\^\_+(x) |= -\_[0\^+]{} (1+z|z)\_i \_i g\_i , since (D\^z)\^2 \_i \_i g\_i = 2 \_i \_i g\_i . This shows that the Ward identity of the residual spin-three gauge symmetry implies Weinberg’s formula . We shall now sketch the generalization of these steps to the case of an arbitrary integer spin $s$. The relevant contribution to the Noether current in this case is given by \_[z…zz]{} = - D\^s\_z T . Using the auxiliary boundary condition $(D^z)^sB_{z\ldots zz} = (D^{\bar z})^sB_{\bar z\ldots\bar z \bar z}$ and integrating by parts, the charge corresponding to our family of large gauge transformation is therefore Q\^+ = (-1)\^s\_ \_[|z]{} T (D\^z)\^[s-1]{}\_u B\_[z…zz]{} d\^2z du - \_ \_[z|z]{} J(u, z, |z)d\^2z du . Choosing T(z, |z) = ()\^[s-1]{} yields &-4 (D\^z)\^[s-1]{}||\ & = \_[i]{} \_i ( )\^[s-1]{} | S | , where we have used the action = g\_i\^[(s)]{}T(i\_u)\^[s-1]{}on matter fields. The $r\to\infty$ limit gives to leading order &-4 ||\ &= \_[0\^+]{} and hence \_[0\^+]{} = (-1)\^s 2\^[s/2-1]{} (1+z|z) \_i \_i ()\^[s-1]{} , where we have employed the following identity &(D\^z)\^[s-1]{} \_i \_i ()\^[s-1]{}\ &= \_[i]{} \_i ( )\^[s-1]{} | S | . Thus, Weinberg’s factorization can be understood as a manifestation of an underlying spin-$s$ large gauge symmetry acting on the null boundary of Minkowski spacetime, namely the set of higher-spin supertranslations we discussed in the previous section. Higher Spins in Arbitrary Dimensions ------------------------------------ So far, the discussion of higher-spin asymptotic symmetries and of their link to the soft theorem has essentially focused on the case of four spacetime dimensions. This is due to the fact that, although the conditions and can be in principle adopted for any $D=n+2$, where they indeed give rise to infinite-dimensional asymptotic symmetries, they only ensure the finiteness of the energy flux and the compatibility with the equations of motion in $D=4$. In this section, we shall consider the dynamics of a higher-spin field on Minkowski background of arbitrary dimension and identify a set of gauge and boundary conditions at null infinity that are compatible with these two requirements. According to the terminology already discussed in Chapter \[chap:spin1higherD\], we will thus adopt the so-called “radiation falloffs” as the leading falloffs of our solution space, $\varphi\sim r^{-n/2}$. As we shall see, in analogy with the situation discussed in the context of Yang-Mills theory in Chapter \[chap:spin1higherD\], adopting radiation falloffs in higher dimensions effectively trivializes asymptotic symmetries. In other words, this choice selects the global Killing symmetries as solutions of the asymptotic Killing tensor equation. Therefore, the analysis will only allow us to calculate and discuss the surface charges associated to these global higher-spin Killing symmetries, and not an infinite-dimensional family of surface charges, in contrast with the four-dimensional case. However, the lower-spin examples suggest a possible way out of the seeming lack of infinite-dimensional asymptotic symmetries in higher dimensions, which would leave open the question regarding the existence of a symmetry underlying Weinberg’s theorem. According to the strategy discussed in Section \[sec:poly\], one may first establish the physical properties of the solution space characterized by radiation falloffs, by ensuring that they give rise to finite fluxes and global charges, and then act on this space by means of transformations , which preserve the equations of motion. By construction, the wider solution space thus obtained shares all observables with the former one, but it also possesses infinite-dimensional symmetries given precisely by [@highspinsrel-io]. ### Bondi-like gauge, reloaded Restricting for the moment to the spin-three case, we shall impose the conditions \[bondi3\] \_[rab]{} = 0 , \^[ij]{} \_[ija]{} = 0. Following the same logic as in the four-dimensional case, one can substitute a power-law ansatz in the equations of motion and realize that they are satisfied to leading order provided that \[boundary3\] \_[uuu]{} , , , display the asymptotic behavior of radiation, namely $\mathcal O(r^{-\frac{n}{2}})$. We shall analyze separately even and odd spacetime dimensions, due to the different nature of radiation and Coulombic terms in the asymptotic expansion in these two cases: the former are associated to half-integer powers of the radial coordinate, as already clearly displayed by the previous equation, while the latter occur at the, integer, order $r^{1-n}$. Due to their peculiarities, also the cases of $D=3$ and $D=4$ will be discussed in a separate section. Although technically and conceptually very simple, a step-by-step solution of the equations of motion would be quite lengthy. For this reason, we will only quote here the main features and outcomes of their discussion, referring to [@super-io; @Cariche-io] for a more detailed derivation. #### Even spacetime dimension When $n$ is even we consider the following ansatz for the fields in the Bondi gauge : \[ansatz3\] $$\begin{aligned} {5} \vf_{uuu} & = \sum_{l\,=\,0}^\infty r^{-\frac{n}{2}-l} B^{(l)}(u,x^m) \, , \qquad & \vf_{uui} & = \sum_{l\,=\,0}^\infty r^{-\frac{n}{2}-l+1} U^{(l)}_i(u,x^m) \, , \\ \vf_{uij} & = \sum_{l\,=\,0}^\infty r^{-\frac{n}{2}-l+2} V^{(l)}_{ij}(u,x^m) \, , \qquad & \vf_{ijk} & = \sum_{l\,=\,0}^\infty r^{-\frac{n}{2}-l+3} C^{(l)}_{ijk}(u,x^m) \, , \end{aligned}$$ with $\g^{ij}V_{ij}^{(l)} = \g^{ij} C_{ijk}^{(l)} = 0$. This choice is mainly motivated by the observation that, as already anticipated, a solution of this type gives rise to a finite and generically nonzero energy flux through null infinity: \[energy-flux3\] (u) = \_[r]{} \_[S\_u]{} (T\_[uu]{}-T\_[ur]{}) r\^n d= \_[S\_u]{} \^[i\_1j\_1]{}\^[i\_2 j\_2]{}\^[i\_3 j\_3]{}\_u[C]{}\^[(0)]{}\_[i\_1i\_2i\_3]{} \_u[C]{}\^[(0)]{}\_[j\_1 j\_2j\_3]{} d , where $T_{ab}$ denotes the energy-momentum tensor of the solution, which can be therefore interpreted as a spin-three wave reaching null infinity. Substituting the ansatz in , one obtains \[sol3\] $$\begin{aligned} {5} B^{(k)} & = \frac{2(n+2k-2)}{(n+2k)(n-2k-2)}\, \Ddot U^{(k)} & \quad \textrm{for}\ k & \neq \frac{n-2}{2} \, , \label{solB3}\\[10pt] U^{(k)}_i & = \frac{2(n+2k)}{(n+2k+2)(n-2k)}\, \Ddot V^{(k)}_i \quad & \textrm{for}\ k & \neq \frac{n}{2} \, , \label{solU3}\\[10pt] V^{(k)}_{ij} & = \frac{2(n+2k+2)}{(n+2k+4)(n-2k+2)}\, \Ddot C^{(k)}_{ij} \quad & \textrm{for}\ k & \neq \frac{n+2}{2} \, . \label{solV3} \end{aligned}$$ In the cases excluded from the previous formulas the coefficients of, respectively, $B^{(\frac{n-2}{2})}$, $U^{(\frac{n}{2})}$ and $V^{(\frac{n+2}{2})}$ vanish and the equations of motion imply instead \[constrC3\] (n-2) C\^[()]{}\_ = 0 , C\^[()]{}\_i = 0 , C\^[()]{}\_[ij]{} = 0 . Note in particular the factor $(n-2)$ in the first constraint, which shows that $C^{(0)}$ actually remains arbitrary even when $n = 2$, namely in four spacetime dimensions. Cancellations of this type are ubiquitous in this type of analysis, exhibiting the peculiarity of the $D=4$ case. Substituting the same ansatz in the equation $\cF_{ijk} = 0$ one obtains, for $l \neq \frac{n+2}{2}$, \[solC3\] \_u[C]{}\^[(l+1)]{}\_[ijk]{} & = - { C\^[(l)]{}\_[ijk]{}\ & - } . The value of $l$ excluded from shows that, from there on, the tensors $C^{(l)}$ also depend on $V^{(\frac{n+2}{2})}$, \[solC3-special\] \_u[C]{}\^[()]{}\_[ijk]{} = { D\^\_[(i]{} V\^[()]{}\_[jk)]{} - \_[(ij]{}\^ V\_[k)]{}\^[()]{} - ( + 2n - 1 ) C\^[()]{}\_[ijk]{} } . The $u$-evolution of $B^{(\frac{n-2}{2})}$, $U^{(\frac{n}{2})}$ and $V^{(\frac{n+2}{2})}$ is fixed instead by the equations $\cF_{uab} = 0$ (with $a, b \neq r$) as \[u3\] $$\begin{aligned} B^{(\frac{n-2}{2})} & = \cM - \frac{n-3}{6(n+1)n} \int_{-\infty}^u \!\!du'\!\left( \Delta - n + 2 \right) \Ddot\Ddot\Ddot C^{(\frac{n-4}{2})} ,\label{uB3} \\[10pt] U_i^{(\frac{n}{2})} & = \cN_i + \frac{u}{n+2}\, \pr_i \cM - \frac{n-1}{2(n+1)(n+2)} \int_{-\infty}^u \!\!du'\! \left( \Delta - 1 \right) \Ddot\Ddot C_i^{(\frac{n-2}{2})} \nn \\ & - \frac{n-3}{6n(n+1)(n+2)} \int_{-\infty}^u \!\!du'\! \int_{-\infty}^{u'} \!\!du''\, D_i \left( \Delta - n + 2 \right) \Ddot\Ddot\Ddot C^{(\frac{n-4}{2})} \, , \label{uU3} \\[10pt] \nonumber V_{ij}^{(\frac{n+2}{2})} & = \cP_{ij} + \frac{u}{n+3} \left( D_{(i\,} \cN_{j)} - \frac{2}{n}\, \g_{ij} \Ddot \cN \right) \\ &+\frac{u^2}{2(n+2)(n+3)} \left( D_{(i} D_{j)} \cM - \frac{2}{n}\, \g_{ij}\, \Delta \cM \right) \nn \\ & - \frac{n+1}{(n+2)(n+3)} \int_{-\infty}^u \!\!du'\! \left( \Delta + n - 2 \right) \Ddot C_{ij}^{(\frac{n}{2})} + \cdots \, ,\label{uV3} \end{aligned}$$ where $\cM$, $\cN_i$ and $\cP_{ij}$ are arbitrary functions and tensors on the celestial sphere. The omitted terms in the last equation correspond to the multiple integrals in the retarded time that one obtains by integrating the differential equation \[diff-eq-V\] \_u[V]{}\_[ij]{}\^[()]{} = { D\_[(i]{}\^ U\_[j)]{}\^[()]{} - \_[ij]{}\^ U\^[()]{} - ( + n - 2 ) C\_[ij]{}\^[()]{} } given by the equation $\cF_{uij} = 0$. At any rate, in Section \[sec:symm3\] we shall show that they do not contribute to the linearized charges. We anticipate that the calculation of the surface charges will essentially boil down to determining their precise dependence on the “integration constants” or, more precisely, “integration functions” $\cM$, $\cN_i$ and $\cP_{ij}$, which admit an arbitrary dependence on the coordinates on the sphere at null infinity and are indeed needed to specify a solution to the recursion relations dictated by the equations of motion. They all appear at order $r^{n-1}$ in the expansions and enter in combinations with a fixed polynomial dependence on the retarded time $u$. For all other powers of $1/r$, the equations $\cF_{uab} = 0$ (with $a, b \neq r$) reduce to divergences of and are therefore identically satisfied. The divergences of also imply the constraints . As a result, the latter do not impose any further condition on the $C^{(l)}$ with lower values of $l$. Let us stress once again that some of the above conclusions are only valid for $n > 2$ due to a number of cancellations that we have highlighted in the four-dimensional case. Below we will see how these results need to be emended in the four- and three-dimensional cases. #### Odd spacetime dimension When $n$ is odd one has to add further terms to the ansatz in order to obtain nontrivial asymptotic charges at null infinity. The necessity of this completion is clearly displayed by the fact that such charges are obtained by integrating a quantity linear in the fields over a sphere with measure element scaling as $r^{n}$, while the radiation ansatz only involves half-odd powers of $1/r$. In other words, one must add Coulombic contributions to the asymptotic series. We therefore consider the ansatz $$\begin{aligned} {5}\label{ansatz3-odd} \vf_{uuu} & = \vf_{uuu}[B] + \sum_{l\,=\,0}^\infty r^{1-n-l} \tilde{B}^{(l)} \, , \qquad & \vf_{uui} & = \vf_{uui}[U] + \sum_{l\,=\,0}^\infty r^{1-n-l} \tilde{U}^{(l)}_i \, , \\ \vf_{uij} & = \vf_{uij}[V] + \sum_{l\,=\,0}^\infty r^{1-n-l} \tilde{V}^{(l)}_{ij} \, , \qquad & \vf_{ijk} & = \vf_{ijk}[C] + \sum_{l\,=\,0}^\infty r^{1-n-l} \tilde{C}^{(l)}_{ijk} \, , \end{aligned}$$ where $\vf_{uuu}[B]$ and so on denote the terms introduced in , which are still necessary if one desires to describe radiation, that is if one wishes to have a nonvanishing energy flux through null infinity (which is still given by ). The new contributions to the expansion of the field components satisfy $\g^{ij} \tilde{V}_{ij}^{(l)} = \g^{ij} \tilde{C}_{ijk}^{(l)} = 0$. Since $n$ is odd, all factors entering the expansion of the equations of motion in powers of $\sqrt{r}$ are different from zero. As a result, the tensors $B^{(l)}$, $U^{(l)}$ and $V^{(l)}$ satisfy the same conditions as in , but without any constraint on the allowed values of $l$. Similarly, the tensors $C^{(l)}$ satisfy for any $l$. The tensors appearing at the leading order of the new, Coulombic branches of our ansatz must satisfy \[u3-odd\] $$\begin{aligned} \tilde{B}^{(0)} & = \cM \, , \qquad \tilde{U}_i^{(0)} = \cN_i + \frac{u}{n+2}\, \pr_i \cM \, , \\[10pt]\nonumber \tilde{V}_{ij}^{(0)} & = \cP_{ij} + \frac{u}{n+3} \left( D_{(i\,} \cN_{j)} - \frac{2}{n}\, \g_{ij} \Ddot \cN \right) \\ &+ \frac{u^2}{2(n+2)(n+3)} \left( D_{(i} D_{j)} \cM - \frac{2}{n}\, \g_{ij}\, \Delta \cM \right) , \end{aligned}$$ on account of the equations of motion $\cF_{uab} = 0$ (with $a, b \neq r$). Notice the similarity with : the only difference is that, when $n$ is odd, there is no contribution from the data of the solution that encode radiation (here stored in the $\sqrt{r}$ branch). As we shall see below, the latter terms anyway do not contribute to the linearized charges. The subleading $\cO(r^{-n})$ terms in our ansatz do not contribute as well to the linearized charges. #### Three and four spacetime dimensions {#sec:3-4dim_3} When $n = 1$ or $n = 2$, as we have already anticipated, the previous analysis must be modified. We begin by revisiting the four-dimensional case, where, in the component $\vf_{uuu}$ the leading order of the radiation and of the Coulomb-like solutions now coincide. Moreover, the equation $\cF_{ruu} = 0$ does not impose any constraint on the triple divergence of $C^{(0)}$. As a result, the equations fixing the dependence on $u$ of $B^{(0)}$, $U^{(1)}$ and $V^{(2)}$ are slightly modified as follows: \[int4D-spin3\] $$\begin{aligned} B^{(0)} & = \cM + \frac{1}{6}\, \Ddot\Ddot\Ddot C^{(0)} \, , \\[10pt] U_i{}^{(1)} & = \cN_i + \frac{u}{4}\, \pr_i \cM - \frac{1}{24} \int_{-\infty}^u \!\!du'\! \left[ \left( \Delta - 1 \right) \Ddot\Ddot C_i{}^{(0)} - 2\, D_i \Ddot\Ddot\Ddot C^{(0)} \right] , \\[10pt] V_{ij}{}^{(2)} & = \cP_{ij} + \frac{u}{5} \left( D_{(i\,} \cN_{j)} - \g_{ij} \Ddot \cN \right) + \frac{u^2}{40} \left( D_{(i} D_{j)} \cM - \g_{ij}\, \Delta \cM \right) + \cdots \, . \label{V4D} \end{aligned}$$ $\cM$, $\cN_i$ and $\cP_{ij}$ are arbitrary functions of $x^i$ as in while in we omitted the integrals that are obtained by the substitution of the previous formulas in the differential equation , which is not modified even when $n =2$. In particular, these additional terms in will be instrumental in building the charges associated to the spin-three supertranslations and superrotations identified in the previous section. When $n = 1$, namely in a three-dimensional spacetime, the radiation branch becomes subleading with respect to the Coulombic one in $\vf_{uuu}$. Moreover, fields of spin greater than one do not propagate local degrees of freedom. It is therefore natural to ignore the radiation branch and work with boundary conditions that only encompass Coulomb-type solutions of the equations of motion. The only nonvanishing components of the field in the Bondi gauge are \_[uuu]{} = () + (r\^[-1]{}) , \_[uu]{} = () + \_() +(r\^[-1]{}) , where $\theta$ denotes again the angular coordinate on the circle at null infinity and we already included the constraints on the leading terms imposed by the equations of motion. ### Asymptotic symmetries and charges for spin three {#sec:symm3} We now identify the key features of the gauge transformations preserving the Bondi-like gauge conditions and calculate the associated charges. The general expression for these charges, expressed in terms of the fields and the parameters of gauge symmetries is given by \[spin3\_cov\] Q(u) & = -\_[r ]{} \_[S\_u]{} { r\_[uuu]{} \_r \_[rr]{} + \_[rr]{} ( r\_r + 2n ) \_[uuu]{} - \_[rr]{} \_[uu]{}\ & - \^[ij]{}\ & + \^[ik]{}\^[jl]{} } , where we recall the expression for the allowed gauge parameters. However, while in four-dimensions these quantities are allowed to take an infinite number of independent values, as we shall see, the further constraints arising in higher dimensions will reduce them to a finite-dimensional family to be identified with global Killing symmetries. The tensors $K_{ij}$, $\r_i$ and $T$ that characterize the asymptotic symmetries are not arbitrary: for $n > 2$ they are bound to satisfy the differential equations (see Appendix \[app:spin3exact\]) $$\begin{aligned} \cK_{ijk} & \equiv D_{(i} K_{jk)} - \frac{2}{n+2}\, \g_{(ij} \Ddot K_{k)} = 0 \, , \qquad \g^{ij} K_{ij} = 0 \, , \label{K} \\[5pt] \cR_{ijk} & \equiv D_{(i} D_j \r_{k)} - \frac{2}{n+2} \left(\, \g_{(ij} \Delta \r_{k)} + \g_{(ij} \big\{ D_{k)} , D_l \big\} \r^l \,\right) = 0 \, , \label{R} \\[5pt] \cT_{ijk} & \equiv D_{(i} D_j D_{k)} T - \frac{1}{n+2} \left(\, \g_{(ij} \Delta D_{k)} T + \g_{(ij} \big\{ D_{k)} , D_l \big\} D^l T \,\right) = 0 \, . \label{T}\end{aligned}$$ When the dimension of spacetime is equal to four, *i.e.* when $n=2$, the last condition simply does not apply and the function $T(x^m)$ is arbitrary. In this case, the corresponding symmetry is the analogue of gravitational supertranslations, from which we derived Weinberg’s theorem in the previous section. Eq.  is the conformal Killing tensor equation of rank two on the $n$-dimensional celestial sphere. For $n > 2$ this equation admits a finite number of solutions, while when $n = 2$ it admits an infinite-dimensional family of local solutions, which we discussed, generalizing gravitational superrotations. The same is true for the less familiar eq.  satisfied by $\r_i$: when $n>2$ it admits a finite number of solutions, while for $n = 2$ locally one can build infinitely many independent solutions. All combinations appearing in the above equations are traceless, and hence they vanish identically when the dimension of spacetime is equal to three, *i.e.* $n = 1$. This implies that in three dimension $T(\theta)$ and $\r(\theta)$ are arbitrary functions, while the symmetry generated by the traceless $K_{ij}$ is absent. Substituting into the expression for the charges one obtains \[charge3-mid\] Q(u) = \_[r ]{} \_[S\_u]{} { & ( T - + K )\ + \^i & ( \_i - K\_i ) + \^[ij]{} K\_[ij]{} } , with $$\begin{aligned} \chi & = - \left( r\pr_r + 2n \right) \vf_{uuu} - \frac{n-1}{r}\, \Ddot \vf_{uu} + \frac{1}{2r}\, \pr_r \Ddot\Ddot \vf_u \, , \\ \chi_i & = 2(n+2)\,\vf_{uui} - \frac{2}{r} \left( r\pr_r - 2 \right) \Ddot \vf_{ui} \, , \\[3pt] \chi_{ij} & = \left( r\pr_r -4 \right) \vf_{uij} \, . \end{aligned}$$ The next task is to evaluate on the solutions of the equations of motion discussed above. For $n$ even and greater than two , and lead to \[all-chi\] $$\begin{aligned} \chi & = - \sum_{k\,=\,0}^{\frac{n-4}{2}} r^{-\frac{n}{2}-k} \frac{n+2k-2}{2(n-2k-2)}\, \Ddot\Ddot\Ddot C^{(k)} - r^{1-n} (n+1) \cM \nn \\ & + r^{1-n}\, \frac{n-3}{6n} \int_{-\infty}^u \!\!du'\!\left( \Delta - n + 2 \right) \Ddot\Ddot\Ddot C^{(\frac{n-4}{2})} + \mathcal O(r^{-n}) \, , \label{chi} \\[5pt] \chi_i & = \sum_{k\,=\,0}^{\frac{n-2}{2}} r^{-\frac{n}{2}-k+1}\, \frac{2(n+2k)}{n-2k}\, \Ddot\Ddot C_i{}^{\!(k)} + 2\, r^{1-n} \Big\{ (n+2)\, \cN_i + u\, \pr_i \cM \Big\} \nn \\ & - \frac{r^{1-n}}{n+1} \int_{-\infty}^u \!\!\!\!du'\! \Big\{ (n-1) \left( \Delta - 1 \right) \Ddot\Ddot C_i{}^{\!(\frac{n-2}{2})} \nn \\ &+ \frac{n-3}{3n} \int_{-\infty}^{u'} \!\!\!\!du'' D_i\! \left( \Delta - n + 2 \right)(\Ddot)^3 C^{(\frac{n-4}{2})} \Big\} , \label{chi1}+ \mathcal O(r^{-n}) \\[5pt] \chi_{ij} & = - \sum_{k\,=\,0}^{\frac{n}{2}} r^{-\frac{n}{2}-k+2} \frac{n+2k+2}{n-2k+2}\, \Ddot C_{ij}{}^{\!(k)} \label{chi2} \nn\\ & - r^{1-n} \Big\{ (n+3)\, \cP_{ij} + u \left( D_{(i\,} \cN_{j)} - \frac{2}{n}\, \g_{ij} \Ddot \cN \right) \nn\\ &+ \frac{u^2}{2(n+2)} \left( D_{(i} D_{j)} \cM - \frac{2}{n}\, \g_{ij}\,\Delta \cM \right) \Big\} + \cdots \, , \end{aligned}$$ where in , besides the terms $\mathcal O(r^{-n})$, we also omitted the integrals in the retarded time that one obtains by substituting . For the first two expressions are modified as follows, \[chi4D\] $$\begin{aligned} \chi & = - \frac{1}{r} \left\{ 3\, \cM + \frac{1}{2}\, \Ddot\Ddot\Ddot C^{(0)} \right\} + \cO(r^{-2}) \, , \\[5pt] \chi_i & = 2\,\Ddot\Ddot C_i{}^{\!(0)} \nn\\ &+ \frac{2}{r} \left\{ 4\, \cN_i + u\, \pr_i \cM + \frac{1}{6} \int_{-\infty}^u \!\!\!\!du'\! \left[ \left( \Delta - 1 \right) \Ddot\Ddot C_i{}^{\!(0)} - 2\,D_i \Ddot\Ddot\Ddot C^{(0)} \right] \right\} \nn \\ & + \cO(r^{-2}) \,, \end{aligned}$$ while $\chi_{ij}$ is instead obtained by setting $n = 2$ in and by correcting the integral terms according to . For $n$ odd the extrema of the sums become, respectively, $\frac{n-3}{2}$, $\frac{n-1}{2}$ and $\frac{n+1}{2}$, while the terms in the second and third lines of eqs.  are absent. Looking only at the $r$-dependence, the sums in the previous formulas would naively give rise to divergent contributions to the charges. This is a standard feature due to the presence of radiation fields, that roughly speaking always induce terms of the type r\^[n]{}d,n in the calculation of the surface charges. These terms vanish, however, thanks to the equations of motion and to the differential constraints on the parameters in , and, when $n > 2$, . Let us begin by exhibiting this mechanism in the simplest case: the term $\chi^{ij} K_{ij}$ in contains divergent contributions that, integrating by parts, can be cast in the form C\_[ijk]{}\^[(l)]{} D\^[(i]{} K\^[jk)]{} = C\_[ijk]{}\^[(l)]{} { \^[(ij]{} K\^[k)]{} - \^[ijk]{} } = 0 , where we recall that $\cK_{ijk}$ is the shorthand introduced in to denote the differential equation satisfied by $K_{ij}$. The next cancellation is slightly more involved: integrating by parts one obtains &\_[S\_u]{}  \^i ( \_i - K\_i )\ &\~\_[l=0]{}\^[\[\]]{} r\^[--l]{} \_[S\_u]{}   C\_[ijk]{}\^[(l)]{} ( D\^[(i]{} D\^[j]{} \^[k)]{} - D\^[(i]{} D\^[j]{} K\^[k)]{} ) + . To cancel the contribution in $\r_i$ one can use eq. , which again allows one to substitute the symmetrized gradient with a term in $\g_{ij}$. To cancel the contribution in $K_{ij}$ one can instead use the following consequence of the conformal Killing tensor equation : D\_[(i]{} D\_j K\_[k)]{} & = - 2 \_[(ij]{} K\_[k)]{} + \_[(ij]{} D\_[k)]{} K\ & - { \_[ijk]{} - D\_[(i]{} \_[jk)]{} + \_[(ij]{} \_[k)]{} + (n-3) \_[ijk]{} } . Similar considerations apply to the integral terms in the second line of . The remaining contribution in the charge formula contains three addenda whose divergent parts can be cast in the following form by integrating by parts: C\_[ijk]{}\^[(l)]{} D\^[(i]{} D\^[j]{} D\^[k)]{} T , C\_[ijk]{}\^[(l)]{} D\^[(i]{} D\^[j]{} D\^[k)]{} , C\_[ijk]{}\^[(l)]{} D\^[(i]{} D\^[j]{} D\^[k)]{} K . These terms are actually absent when $n=2$. For $n>2$ the first contribution vanishes thanks to . The other two types of terms vanish thanks to consequences of the equations satisfied by $\r_i$ and $K_{ij}$, obtained by taking symmetrized gradients thereof. We have therefore proven that, in the Bondi-like gauge , a spin-three field with the falloffs at null infinity and satisfying the field equations up to the contributions of order $r^{n-1}$ to its components admits finite asymptotic linearized charges. For any $n > 2$, these depend on the “integration functions” specifying the solution as \[finalcharge3\] Q = - \_[S\^n]{} { (n+1) T - 2(n+2) \^i \_i + (n+3) K\^[ij]{} \_[ij]{} } , where $\cM$, $\cN_i$ and $\cP_{ij}$ are the tensors on the sphere at null infinity introduced in (see for $n$ odd). As anticipated, the charges are constant along null infinity when the dimension of spacetime is greater than four. The same is true also in three dimensions: in this case both $K^{ij}$ and $\cP_{ij}$ are not present and the asymptotic charges take the form Q = - d{ T() () - 3 () () } , in agreement with the result derived in the Chern-Simons formulation [@spin3-3D_1; @3D-flat_2]. When $n = 2$, the modifications in the dependence on $u$ of the leading terms in the Coulomb-type solution recalled in (and ) lead to the following expression for the asymptotic charges: \[charge3-4D\] Q(u) = - \_[S\_u]{} { 3 T ( + C\^[(0)]{} ) - 8 \^i \_i + 5 K\^[ij]{} \_[ij]{} + } . In this formula we omitted other $u$–dependent terms in $C^{(0)}$, whose form is not particularly illuminating and can be readily obtained by substituting in . The main information is anyway that in four dimensions a dependence on the retarded time appears already in the linearized theory, thanks to the contribution to the charges of the radiation solution. As shown in the previous section, where the terms in $T$ in have been already presented, the dependence on radiation data is instrumental in deriving Weinberg’s theorem for spin-three soft quanta from the Ward identities of the supertranslation symmetry generated by the arbitrary function $T(x^i)$. Arbitrary Spin {#sec:spin-s} -------------- Without delving too much into the details, and restricting for simplicity to the higher dimensions $n>2$, we shall now illustrate the generalization of the results of the previous section to the case of an arbitrary integer spin $s$. The Bondi-like gauge in this case reads \[Bondi-s\] \_[ra\_[s-1]{}]{} = 0 , \^[ij]{} \_[ij a\_[s-2]{}]{} = 0 . together with the ansatz \[s-ansatz\_even\] \_[u\_[s-k]{}i\_k]{} = \_[l=0]{}\^r\^[-+k-l]{} C\_[i\_k]{}\^[(k,l)]{}(u,x\^m) , for even-dimensional spacetimes. We recall that we are conventionally employing a multi-index notation for symmetrized indices: for instance $\varphi_{ij a_{s-2}}$ is a shorthand notation for any component of the type $\varphi_{ij a_3 a_4 \cdots a_s}$, with two angular indices and $s-2$ arbitrary indices. As in the previous examples, the leading behavior of our ansatz is designed to give a finite flux of energy per unit time across the sphere $S_u$ at fixed $u$, a feature that we interpret as radiation flowing through null infinity. Indeed, the canonical energy-momentum tensor of the higher-spin Lagrangian in Bondi-like gauge, L = \^[a\_s]{} ( \_[a\_s]{} - \_[a]{} \_[a\_[s-1]{}]{}) , reads T\_[ab]{}= \_[a]{} \_[c\_s]{} \_[b]{} \^[c\_s]{} - s \^[c\_[s-1]{}]{} \_[b]{} \_[a c\_[s-1]{}]{}+ \_[ab]{} L . The corresponding power flowing through null infinity is then ($\gamma^{i_sj_2}=\gamma^{ij}\cdots\gamma^{kl}$ with $s$ factors) \[energy-flux\] (u) = \_[r]{} \_[S\_u]{} (T\_[uu]{}-T\_[ur]{}) r\^n d= \_[S\_u]{} \^[i\_s j\_s]{} \_u[C]{}\^[(s,0)]{}\_[i\_s]{} \_u [C]{}\^[(s,0)]{}\_[j\_s]{} d. The asymptotic analysis of the field equations shows that, for all even values of $n$, the tensors entering the ansatz are fixed in terms of the $C^{(s,l)}$, with the exception of \[def-charges\] C\_[i\_k]{}\^[(k,+k-1)]{} Q\_[i\_k]{}\^[(k)]{}  k &lt; s . The tensors $C^{(s,l)}$ are then determined (up to integrations functions) in terms of an arbitrary tensor $C^{(s,0)}(u,x^m)$, in particular via the equation $\cF_{i_s} = 0$. The remaining components of the equations of motion fix the $u$-evolution of the tensors $\mathcal Q^{(k)}$ defined in . Furthermore, on-shell, the $\mathcal Q^{(k)}$ are $n$-divergences as the other $C^{(k,l)}$, up to a set of integration functions $\cM^{(k)}(x^m)$. More specifically, $\mathcal Q^{(k)}$ depends on the integrations functions of all $\mathcal Q^{(l)}$ with $l < k$ with a precise polynomial dependence on $u$. As we have seen in the spin-three context, this is instrumental in making the independence on the retarded time of the asymptotic charges explicit. Concretely, the $Q^{(k)}$ depend on the $\cM^{(k)}$ in the following way \[finalQ\] Q\_[i\_k]{}\^[(k)]{} = \_[l=0]{}\^k u\^l \_ \_[i\_[k-l]{}]{}\^[(k-l)]{} + , where we omitted terms that will not contribute to the surface charges. When $n$ is odd, one has to consider an ansatz containing both integer and half-integer powers of $r$. Restricting to $n > 1$, we thus set \[s-ansatz\_odd\] \_[u\_[s-k]{}i\_k]{} = \_[l=0]{}\^r\^[--l+k]{} C\^[(k,l)]{}(u,x\^m) + \_[l=0]{}\^r\^[1-n-l]{} \^[(k,l)]{}(u,x\^m) , so that the leading order has the same form as in . Eq.  thus guarantees that we have a finite flux of energy per unit time across $S_u$ in this case too. The leading order of the Coulomb branch is in this case not constrained by the equations of motion, so that we can define \^[(k,0)]{} Q\^[(k)]{}  k &lt; s . The remaining $\tilde{C}^{(k,l)}$ are again fixed in terms of the $\tilde{C}^{(s,l)}$, provided that one identifies $\tilde{C}^{(k,l)} = C^{(k,\frac{n}{2}+k+l-1)}$, that is \[gen-Ctilde\] \^[(k,l+1)]{} = - \^[(k+1,l)]{} . The field equations then determine $\tilde{C}^{(s,l)}$ in terms of an arbitrary $\tilde{C}^{(s,0)}(u,x^m)$ up to integration functions. The $\mathcal Q^{(k)}$ satisfy a relation analogous to eq.  where the omitted terms, which anyway do not contribute to the charges, are actually absent. To summarize, the solution of the equations of motion up to Coulombic order can be cast into the following form \[boundary-cond-s\] \_[u\_[s-k]{}i\_k]{} = & \_[l=0]{}\^[+k-2]{} r\^[-+k-l]{} (D )\^[s-k]{} C\_[i\_k]{}\^[(s,l)]{}\ & + r\^[1-n]{} Q\_[i\_k]{}\^[(k)]{} + (r\^[--]{}) . Below, we will show that indeed leads to the cancellation of some of the potentially divergent contributions to the linearized charges, while also proving that the $\mathcal Q^{(k)}$’s give a finite contribution to them. ### Asymptotic symmetries and charges for any spin In the Bondi-like gauge the surface charges be expressed in terms of the nonvanishing field components as \[charges-s\] Q(u) & = -\_[r]{} \_[S\_u]{} \_[p=0]{}\^[s-1]{} { \_[u\_[s-p]{}i\_[p]{}]{}( r\_r + n + 2p ) \^[u\_[s-p-1]{}i\_p]{}\ & + \^[u\_[s-p-1]{}i\_p]{} } . Therefore, they only depend on the components $\e_{r_{s-k-1}i_k} = (-1)^{s-k-1} \e^{u_{s-k-1}}{}_{i_k}$ of the gauge parameters of asymptotic symmetries. Analyzing the asymptotic Killing equations obtained by enforcing that these parameters preserve the Bondi-like gauge, one obtains that these components are parametrized as \[K-u\] \_[r\_[s-k-1]{}i\_k]{} &= r\^[2k]{} ( K\_[i\_k]{}\^[(k)]{} + \_[m=1]{}\^[s-k-1]{} (D )\^m K\_[i\_k]{}\^[(k+m)]{} )\ & + (r\^[2k-1]{}) , where the tensors $K^{(k)}(x^m)$ only depend on the coordinates on the $n$-dimensional sphere at null infinity, while $(a)_n \equiv a (a+1) \cdots (a+n-1)$ is the Pochhammer symbol. Moreover, the $K^{(k)}_{i_k}$ are traceless and must satisfy suitable differential constraints, which generalize those displayed in and for the spin-three case. While substituting the explicit form of the residual symmetry parameters into the formula for the surface charge, one first has to keep track of the cancellation of all potentially divergent terms in the limit $r\to\infty$. As we have seen in the previous sections, these usually occur after integrations by parts; for instance, the coefficient of the leading power in $r$ is of the form \_[S\_u]{} K\^[(s-1)]{}\_[i\_[s-1]{}]{} C\^[(s,0)i\_[s-1]{}]{} = \_[S\_u]{} D\_i\^ K\^[(s-1)]{}\_[i\_[s-1]{}]{} C\^[(s,0)[i\_s]{}]{} , and hence vanishes because $K^{(s-1)}$ satisfies the conformal Killing tensor equation of rank $s-1$. Since this type of cancellation must occur in general, as we have verified in detail in the spin-three case, the finite contribution to the charges is determined by the Coulomb-like terms $\mathcal Q^{(k)}$ in the boundary conditions . Substituting them into , while taking into account their dependence on the integration constants in , one obtains after integration by parts $$\begin{aligned} & Q(u) = \int_{S_u} \!\frac{\sqrt{\g}}{(s-1)!} \sum_{p=0}^{s-1} \sum_{m=0}^{s-p-1} \sum_{l=0}^{p} \nn\\ &\binom{s-1}{p} \binom{s-p-1}{m} \binom{p}{l} \frac{(s+p+n-2)(s+n+p-l-2)!}{(s+n+p-2)_m (s+n+p-2)!} \nn \\ & \phantom{Q(u) = \int_{S_u}} \times (-1)^{s+m+p+l}\, u^{l+m} \cM^{(p-l)}_{i_{p-l}} (\Ddot)^{l+m} K^{(m+p)\,i_{p-l}} \label{charges-int-s} \\[5pt] & = \int_{S_u} \sum_{k,q} \frac{\sqrt{\g}\,(-1)^{s+k}(s+n+q-2)!}{(s-1)!(s+n+k+q-3)!}\nn\\ & \left[ \sum_{p=0}^{s-1} (-1)^p \binom{s-1}{p} \binom{s-p-1}{k+q-p} \binom{p}{q} \right] u^{k} \cM^{(q)}_{i_{q}} (D\cdot)^{k} K^{(k+q)\,i_{q}} \, . \nn\end{aligned}$$ The final expression has been obtained by introducing the new labels $k = l+m$ and $q = p-l$ and by swapping the sums, whose ranges precisely correspond to the values of the labels for which the integrand does not vanish (with the convention $\binom{N}{n} = 0$ for $n < 0$ and $n > N$). One can eventually verify that the sum within square brackets in the second line of vanishes for any $k > 0$, thus confirming that the charges do not depend on $u$ for any value of the spin. The $u$-independent contribution then reads \[charges-final\] Q = \_[S\^n]{} \_[q=0]{}\^[s-1]{} (-1)\^[s+q]{} (s+n+q-2) K\^[(q)]{}\_[i\_q]{} \^[(q)i\_q]{} , in full analogy with the result that we presented for $s=3$ in . Conclusions {#conclusions .unnumbered} =========== Let us briefly summarize what we achieved so far and make an attempt to identify possible directions to be explored in the future. Concerning the extension of the asymptotic analysis to higher dimensions, we were able to reproduce several aspects of the discussion of memory effects in the context of even-dimensional Maxwell and Yang-Mills theories, encompassing ordinary/linear, null/nonlinear, phase and color memories. While it is fair to say that the existence of memory in even dimensions is by now no longer disputed, still, much remains to be done in this respect. Kick, phase and color memory effects are just a few of the many more types of memory that were put forward in the four-dimensional setup. Therefore, it would be natural to understand whether also different aspects of memory formulas admit a generalization to arbitrary even spacetime dimensions: for instance, the distinction between electric and magnetic memories [@Mao_em], spin memory [@spinmemory], or refraction memory [@CompereRefraction]. Of course, an even more interesting possibility is that in higher dimensions memory effects may possess a richer zoology and may allow to define *new* types of observable phenomena. For instance, while the Coulombic order $\mathcal O(1/r^{D-3})$ is, with hindsight, a natural place where to look for memory effects, since it is associated with DC phenomena, it is conceivable that there may exist effects that scale as $\mathcal O(1/r^k)$ with $\tfrac{D-2}{2}\le k<D-3$, provided that the test probe is chosen appropriately and that suitable observables are identified. Independently of the dimension of spacetime, it would also be interesting to better understand whether proper analogs of color and phase memories in the gravitational case, *i.e.* memory effects encoded in the gravitational quantum numbers of a test particle, comprise qualitatively new phenomena or just reduce to a quantum rephrasing of the standard displacement memory (see *e.g.* [@Afshar:2018sbq] for a discussion of the quantum kick memory effect). The status of memory effects in odd spacetime dimensions is much less clear, as a consequence of the dispersion phenomena occurring in this case. These manifest themselves as the failure of the Huygens principle and, in the asymptotic expansion near null infinity, as non-analytic terms of the form $1/r^{k/2}$ with $k$ odd. While a proof of concept has been provided here for the existence in odd dimensions of null memory, which overcomes the blurring induced by dispersion since the emitting source effectively reaches the probe near null infinity, we still feel that the phenomenology of this effect is much less studied and understood compared to its odd-dimensional counterpart. The non-analyticities in $1/r$, which are also at the basis of the failure of the conformal approach to null infinity for solutions containing gravitational waves in odd dimensions, are tightly related to difficulties encountered in the discussion of soft theorems and asymptotic symmetries which were resolved in [@He-Mitra-photond+2; @He-Mitra-gauged+2; @He-Mitra-magneticd+2]. The solution proposed highlights a major drawback of the standard perturbative expansion in powers of $1/r$. In odd dimensions, the perturbative treatment of the field equations is not sufficient in order to identify constraints that link radiation $\mathcal O(1/r^{\frac{D-2}{2}})$ and Coulombic $\mathcal O(1/r^{D-3})$ components, since the corresponding recursion relations only involve perturbative orders that differ by an integer number. Such constraints only arise nonperturbatively, *e.g.* by considering exact solutions to the field equations, and are crucial for soft theorems: they are needed in order to connect the information on the propagating soft photon modes, stored in the radiation components, to the soft charges involved in the Ward identities, which are instead expressed in terms of Coulombic components. While these works focused on the discussion of field strengths and relied on the structure of soft charges for the identification of the underlying asymptotic symmetries, thus working “backwards” compared to the logic we employed in our discussion, an explicit realization of these symmetries is precluded unless one introduces the gauge fields and defines their dynamics by means of a gauge-fixing. Asymptotic symmetries can be then evaluated in any dimension according to the following proposed strategy. After gauge fixing, one first considers the set of solutions characterized by radiation falloffs, whose behavior at infinity allows one to have control over physical observable properties, but which are too restrictive and effectively forbid infinite-dimensional asymptotic symmetries. Then, one acts on this set of solutions by gauge symmetries that preserve the gauge fixing and have finite and nonzero asymptotic charge, but which do not necessarily leave radiation falloffs invariant. The wider solution space so-obtained still retains all the desirable physical properties of radiation solutions, such as finite energy and charge fluxes, but it admits nontrivial asymptotic symmetries by construction. Asymptotic surface charges, on the other hand, may first be safely evaluated on $\mathscr I^+$ (resp. $\mathscr I^-$) in regions where radiation is absent, *i.e.* where the solutions are stationary. In particular one may first consider the limit $\mathscr I^+_-$ (resp. $\mathscr I^-_+$), where, under standard assumptions, only Coulombic contributions matter. The evolution of the charges *along* $\mathscr I$ can then be defined, even in the presence of radiation, by means of the equations of motion. Combining this observation with the previous one, we may formulate the following systematic criterion to detect the existence of nontrivial asymptotic symmetries in higher dimensions: it is sufficient to find the residual subset of transformations preserving the assigned gauge-fixing and giving rise to a finite asymptotic charge when evaluated on Coulombic solutions. However, at this point, one may still wonder to which extent the nature of the asymptotic symmetry group is gauge-independent, or, if it does depend on the gauge, whether there exists a maximal asymptotic symmetry group and a systematic way in which this group allows for an embedding of the smaller ones. These are, to our knowledge, open questions. Moving to the case raised by the soft scalars, we have been able to provide a possible resolution of the purported paradox related to the existence of soft scalar charges, in the four dimensional case, by appealing to a two-form gauge symmetry giving rise to dual soft charges. Nevertheless, the analysis of this point leaves open two interesting aspects. First, again in connection with the possible gauge dependence of the asymptotic analysis, there is the problem of knowing, if possible *a priori*, when the chosen gauge fixing results in too stringent a condition and actually hides the presence of asymptotic symmetries. This seems to be the case, in this context, for the radial gauge, which trivializes the asymptotic charges, while the Lorenz gauge appears to be more flexible and allows one to overcome this difficulty. Second, the example of the scalar and the two-form in four dimensions paves the way to the possibility of discussing asymptotic structures by exploiting dualities in more general contexts, although one should bear in mind that such a possibility is essentially restricted to the free/asymptotic level. On the higher-spin side, we have seen how a strategy similar to the one employed in the context of linearized gravity allows us, upon introducing a suitable Bondi-like gauge, to perform a meaningful discussion of asymptotic symmetries for all integer spins in any dimension, for free fields. The calculation of the associated charges has also been carried out explicitly in four dimensions and, at the level of global symmetries, in any dimension. We have verified the connection of such symmetries with Weinberg’s soft theorem in four dimensions, where the infinite dimensional asymptotic symmetry enhancement occurs more naturally, but it is reasonable to envision that, once the strategy proposed above is employed in order to overcome the difficulties induced by the imposition of radiation falloffs, such a link should hold independently of the spacetime dimension as in the case of lower spins. Moving to issues less directly related to the ones we have discussed in detail, it appears natural, also in connection with cosmological observations, to wonder how far the discussion of asymptotic symmetries and their observable consequences can be carried in the presence of a cosmological constant. This issue has already attracted attention in the literature on the gravitational setting with promising results (see *e.g.* [@BieridS; @CompereLambda; @Chumemory] for recent developments in this direction), especially for de Sitter spacetime, but the fact that many of the above results on flat spacetime hold regardless of spin strongly motivates analogous investigations also for Maxwell, Yang-Mills and higher spins. Identifying similarities and differences between flat spacetime and (A)dS in this respect could prove challenging, due to their different causal structures, but it could also provide fruitful insights into the problem of flat space holography. Also the analysis of higher-spin asymptotic symmetries on Anti de Sitter spacetime may be regarded as promising, since AdS offers the only arena in which non-Abelian higher-spin gauge algebras are explicitly known so far [@BCIValgebras]. A tightly related issue is the possibility of uncovering a non-Abelian algebra that may underly the infinite-dimensional family of (commuting) asymptotic symmetries that has been put forward in this thesis for any spin on flat space. A first step in this direction would be the introduction of cubic vertices in the analysis of higher-spin asymptotic symmetries. Finally, let us mention again the tempting analogy between our higher-spin symmetry enhancement and enhanced symmetries exhibited by string amplitudes in the high-energy regime [@Gross], whose relation, if any, would provide further insight into the role played by asymptotic symmetries in the structure of string theory. A related issue would be the understanding of the role, if any, played by these symmetries of massless higher-spin states in the tensionless limit of string theory and in the mechanism of symmetry breaking that gives rise to a mass for such states. Coordinate Conventions {#app:Laplaciano} ====================== In $D$-dimensional Minkowski spacetime, retarded Bondi coordinates are defined as follows in terms of the standard Cartesian coordinates $x^\mu= (t, \mathbf x)=(x^0,x^I)$, with $I=1$, $2$,$\ldots$, $D-1$: retarded time $u=t-|\mathbf x|$, a radial coordinate $r=|\mathbf x|$, and angular coordinates $x^i$ on the Euclidean unit $(D-2)$-sphere, with metric $\gamma_{ij}$. The Minkowski metric, in such coordinates, reads $$ds^2 = -dt^2 + d\mathbf x^2= -du^2-2 du dr + r^2 \gamma_{ij} dx^i dx^j\,,$$ while the nonvanishing Christoffel symbols are $$\Gamma\indices{^i_{rj}}=\frac{1}{r}\delta\indices{^i_j}\, ,\qquad \Gamma\indices{^u_{ij}}=-\Gamma\indices{^r_{ij}}=r\gamma_{ij}\, ,\qquad \Gamma\indices{^i_{jk}}=\frac{1}{2}\gamma^{il}(\partial_j \gamma_{lk}+\partial_k \gamma_{jl}-\partial_l \gamma_{jk})\,.$$ With $D_i$ we denote the covariant derivative on the sphere associated to $\gamma_{ij}$ and $\Delta= D_iD^i$ is the corresponding Laplace-Beltrami operator. In particular, the d’Alembert operator $\Box$ acting on a scalar $\varphi$ takes the explicit form $$\Box \varphi = -\left(2\partial_r + \tfrac{D-2}{r}\right)\partial_u \varphi + \left(\partial_r^2 + \tfrac{D-2}{r}\partial_r + \tfrac{1}{r^2}\Delta\right)\varphi\,.$$ In terms of a given parametrization $\mathbf n = \mathbf n(x^i)$ of unit vectors, $|\mathbf n|=1$, identifying points on the sphere, one has $\mathbf x=r\,\mathbf n$ and $\gamma_{ij}=\partial_i\mathbf n \cdot \partial_j \mathbf n$. The Cartesian components of the radial vector $\mathbf x$ satisfy $\nabla_a \nabla_b \mathbf x=0$, since in Cartesian coordinates this reduces to $\partial_I \partial_J \mathbf x=0$. In spherical coordinates this identity gives, $$D_iD_j\mathbf n = -\gamma_{ij}\mathbf n\,,$$ which in its turn implies $$\left(D_i D_j-\tfrac{1}{D-2}\gamma_{ij}\Delta\right)\mathbf n=0\qquad\text{ and }\qquad (\Delta+D-2) \mathbf n =0\,.$$ More generally, for any homogeneous polynomial $P_l(\mathbf x)=r^l Q_l(\mathbf n)$ of degree $l\ge0$, we have that $$\delta^{IJ}\nabla_I \nabla_J P_l= r^{2-D}\partial_r(r^{D-2}\partial_r P_l)+r^{-2}\Delta P_l$$ vanishes if and only if $[\Delta + l(D+l-3)]Q_l=0$. This shows that $\Delta$ has eigenvalues $-l(D+l-3)$ for $l=0$, $1$, $2\ldots$, each with multiplicity $g_l$ equal to the dimension of the space of harmonic polynomials of degree $l$. In particular $g_0=1$, associated with the constant polynomial, and $g_1=D-1$, corresponding to the monomials given by the Cartesian components of $\mathbf x$. For $l\ge2$, we have [@libroRusso] $$g_l=\tbinom{D+l}{l}-\tbinom{D-2+l}{l}\,.$$ Spin-Three Symmetries at $\mathscr I^+$ ======================================= Spin-three Asymptotic Symmetries {#app:spin3symm} -------------------------------- In this appendix we explicitly calculate the asymptotic symmetries that preserve the following conditions for a spin-three field in $D=n+2$ dimensions: \[r+traces\] \_[rab]{}=0,\^[ij]{}\_[ija]{}=0, together with the requirement that \[eqWeakest\] \_[uuu]{},,, scale as $\mathcal O(r^{-1})$ near future null infinity. The significance of this set of conditions is that, when $D=4$, they coincide with the standard Bondi-like gauge conditions discussed in Chapter \[chap:HSP\], while, for generic dimensions, provides the strongest falloffs still allowing for the presence of infinite-dimensional higher-spins supertranslations, in complete analogy with the gravitational case. Let us start from $$\begin{aligned} \label{rrr} \delta_\epsilon\varphi_{rrr}&=3\partial_r\epsilon_{rr}=0\,,\\ \label{rru} \delta_\epsilon\varphi_{rru}&=2\partial_r\epsilon_{ur}+\partial_u\epsilon_{rr}=0\,,\\ \label{ruu} \delta_\epsilon\varphi_{ruu}&=2\partial_u\epsilon_{ur}+\partial_r\epsilon_{uu}=0\,,\\ \label{uuu} \delta_\epsilon\varphi_{uuu}&=3\partial_u\epsilon_{uu}=\mathcal O(r^{-1})\,.\end{aligned}$$ From we get \_[rr]{}=F(u,n), for some $r$-independent function $F$. Then, by , \_[ur]{}=-\_uF(u,n)+G(u,n), for an arbitrary $r$-independent function $G$. Hence, implies \_[uu]{}=\_u\^2F(u,n)-2r\_uG(u,n)+H(u,n), with $H(u,\mathbf n)$ a third arbitrary function of $u$ and the angles. Imposing , we have \[uuuvariation\] \_u\^3F - 2r\_u\^2G + \_u H = O(r\^[-1]{}), where we see that the left-hand side must actually vanish, so that \[uexpFGH\] F(u,n)&=u\^2 F\_2(n)+u F\_1(n)+T(n),\ G(u,n)&=u G\_1(n)+S(n),\ H(u,n)&=H(n), where we have introduced a number of arbitrary angular functions. Now, let us analyze $$\begin{aligned} \label{rri} \delta_\epsilon \varphi_{rri}&=2\left(\partial_r-\tfrac{2}{r}\right)\epsilon_{ri}+D_i\epsilon_{rr}=0\,,\\ \label{rui} \delta_\epsilon \varphi_{rui}&=\left(\partial_r-\tfrac{2}{r}\right)\epsilon_{ui}+\partial_u\epsilon_{ri}+D_i\epsilon_{ur}=0\,,\\ \label{uui} \delta_\epsilon \varphi_{uui}&=2\partial_u\epsilon_{ui}+D_i\epsilon_{uu}=0\,.\end{aligned}$$ Eq. requires \_[ri]{}=r\^2 I\_i(u,n)+D\_i F(u,n), while, from , \_[ui]{}=-r\^3 \_u I\_i(u,n)+r\^2 J\_i(u,n)+r D\_i G(u, n). In its turn, by , \[uuivariation\] -2r\^3 \_u\^2 I\_i+(D\_i\_u\^2 F + 4\_u J\_i)+D\_iH=O(1), which requires that the coefficients of $r^3$ and $r$ vanish: \[uexpIJ\] I\_i(u,n)&= u r\_i(n)+\_i(n),\ J\_i(u,n)&= - D\_i F\_2(n)+\_i(n). Let us now turn to the variation \_\_[rij]{}=(\_r-)\_[ij]{}+D\_[(i]{}\_[j)r]{}-2r\_[ij]{}(\_[ur]{}-\_[rr]{})=0 which gives \_[ij]{}=r\^4 K\_[ij]{}+r\^[3]{}+, where $K_{ij}(u,\mathbf n)$ is an arbitrary symmetric tensor, and to the trace constraint that must be obeyed by the gauge parameter, \^[ab]{}\_[ab]{}=\_[rr]{}-2\_[ur]{}+\^[ij]{}\_[ij]{}=0, from which we obtain three conditions, $$\begin{aligned} \label{traceK} \gamma^{ij}K_{ij}&=0\,,\\ \label{2DdotJ} 2D\cdot I + (n+1)\partial_u F &=0\,,\\ \label{Delta+2(n+1)} [\Delta + 2(n+1)]F-2(n+2)G&=0\,.\end{aligned}$$ We thus learn that $K_{ij}$ is traceless, while and using the expansions , give the constraint \[constrF2\] \[+2(n+1)\]F\_2=0 and allow to solve for the following quantities, \[interrelF1F2\] F\_2=-Dr,F\_1=-D, and \[interrelG1S\] G\_1=\[+2(n+1)\]F\_1,S=\[+2(n+1)\]T. The variation \_\_[uij]{}=\_u\_[ij]{}+D\_[(i]{}\_[j)u]{}-2r \_[ij]{}(\_[uu]{}-\_[ur]{})=O(r), namely \[uijvariation\] r\^4 \_u K\_[ij]{}+ r\^2+2r\[D\_iD\_jG-\_[ij]{}(G-H)\]=O(r), imposes \[duKij\] \_u K\_[ij]{}=0,D\_iD\_j\_uF+\_[ij]{}\_u G+D\_[(i]{}J\_[j)]{}=0, because the coefficients of $r^4$ and $r^2$ in must vanish. Furthermore the coefficient of $r$, while allowed to be nonzero, must be traceless, so that \[DeltaG+n\] G+n(G-H). More explicitly, we obtain from that $K_{ij}$ is $u$-independent and, using , , D\_iD\_j\_uF\_1+\_[ij]{}G\_1+D\_[(i]{}\_[j)]{}=0, while yields \[interrelH\] H=(+n)S, together with the constraint equation \[constrG1\] (+n)G\_1=0. The last variation to be taken into account is \_ \_[ijk]{}=D\_[(i]{}\_[jk)]{}-2r\_[(ij]{}\[\_[k)u]{}-\_[k)r]{}\]=O(r\^[2]{}), namely \[ijkvariation\] &r\^4 +r\^3{\_[(i]{}D\_j I\_[k)]{}+\_[(ij]{}\[D\_[k)]{}\_u F+2I\_[k)]{}-2J\_[k)]{}\]}\ &+r\^2=O(r\^2), which, imposing that the coefficients of $r^4$ and $r^3$ be zero, affords $$\begin{aligned} \label{protoKilling} D_{(i}K_{jk)}+2\partial_u I_{(i}\gamma_{jk)}&=0\,,\\ \label{DiDjIk} \D_{(i}D_j I_{k)}+\gamma_{(ij}[D_{k)}\partial_u F+2I_{k)}-2J_{k)}]&=0\,.\end{aligned}$$ Furthermore, the coefficient of $r^2$ in must be traceless: \[tracelessr2\] D\_iF+D\_i F + (n+2)D\_i(2F-3G)=0. The trace of , recalling that $\gamma^{ij}K_{ij}=0$ by and using , gives \[interrelri\] r\_i = -DK\_i, and substituting back into we see that $K_{ij}$ must satisfy the conformal Killing equation \[Killingtens\] D\_[(i]{}K\_[jk)]{}-\_[(ij]{}DK\_[k)]{}=0. Recalling , we obtain from D\_[(i]{}D\_j r\_[k)]{}+\_[(ij]{}=0 and D\_[(i]{}D\_j \_[k)]{} + \_[(ij]{}=0. Recalling , , and taking the trace of the last equation also gives $$\begin{aligned} \label{constrD3K} D_{(i}D_j D\cdot K_{k)}+2\gamma_{(ij}D\cdot K_{k)}+\frac{3}{n+1}\gamma_{(ij}D_{k)}D\cdot D\cdot K&=0\,,\\ \label{interrelsigmai} \rho_i+\frac{1}{n+2}(\Delta \rho_i + \{D_i, D_j\}\rho^{j})-\frac{1}{n+1}D_i D\cdot \rho&=\sigma_i\,,\\ \label{Killingrho} D_{(i}D_j \rho_{k)}-\frac{2}{n+2}\gamma_{(ij}\left[ \Delta \rho_{k)}+D\cdot D_{k)}\rho + D_{k)}D\cdot \rho \right]&=0\,.\end{aligned}$$ Finally, from we obtain three constraints \[constrFGTS\] (D\_i+D\_i)F\_2+2(n+2)D\_i F\_2&=0,\ (D\_i+D\_i)F\_1+2(n+2)D\_i F\_1-3(n+2)D\_iG\_1&=0,\ (D\_i+D\_i)T+2(n+2)D\_i T-3(n+2)D\_iS&=0. Before proceeding further, let us recall our conventions for the Riemann tensor on the Euclidean sphere, \^k = R\^l,R\_[ijkl]{}=\_[ik]{}\_[jl]{}-\_[il]{}\_[jk]{}. Repeatedly use of this relation allows one to derive a number of useful identities, which we will employ extensively in the following, for generic functions $T$, vectors $\rho^i$ and symmetric, traceless tensors $K^{ij}$ on the $n$-sphere: \[Ridentities\] \[, D\_i\]T&=(n-1)D\_iT,\ \[D\_l,D\_[(i]{}\]K&=2n K\_[ij]{},\ \[,D\_[(i]{}\]K\_[jk)]{} &= (n+3)D\_[(i]{}K\_[jk)]{}-4\_[(ij]{}DK\_[k)]{},\ \[D\_l, D\_i\]\^l &=(n-1)\_i,\ \[D\_i, \]\^i &=(n-1)D,\ D\^k\[D\_k, D\_[(i]{}\]\_[j)]{}&=D\_[(i]{}\_[j)]{}-2\_[ij]{}D,\ \[, D\_[(i]{}\]\_[j)]{} &=(n+1)D\_[(i]{}\_[j)]{}-4\_[ij]{}D,\ \[D\_l, D\_[(i]{}D\_[j)]{}\]\^l &=(2n-1)D\_[(i]{}\_[j)]{}-2\_[ij]{}D,\ \[D\_i, \^2\]\^i &=2(n-1)D+(n-1)\^2 D,\ \[D\^j, \]D\_[(i]{}\_[j)]{}&= (n+1)\_i + (n-3)D\_i D+(n\^2-1)\_i,\ \[, D\_[(i]{}\]D\_j \_[k)]{} &= (n+3)D\_[(i]{}D\_j\_[k)]{} - 4\_[(ij]{}\[D\_[k)]{}D+\_[k)]{}+(n-1)\_[k)]{}\]. We begin the analysis of the equations we obtained by noting that , , , and allow us to determine all integration functions in terms of $T(\mathbf n)$, $\rho_i(\mathbf n)$ and $K_{ij}(\mathbf n)$, namely F\_2 = ,F\_1=-,G\_1=-D, while S=T,H=T, and r\_i = -DK\_i, \_i= D\_iD+ \_i. On the other hand, $T$, $\rho_i$ and $K_{ij}$ must satisfy , and , namely $$\begin{aligned} \label{equationsK} \mathcal K_{ijk}&= D_{(i}K_{jk)}-\frac{2}{n+2}\gamma_{(ij}D\cdot K_{k)}=0\,,\qquad \gamma^{ij}K_{ij}=0\,,\\ \label{equationrho} \mathcal R_{ijk}&=D_{(i}D_j \rho_{k)}-\frac{2}{n+2}\gamma_{(ij}\left[ 2D_{k)}D\cdot \rho + (\Delta+n-1)\rho_{k)} \right]=0\,,\end{aligned}$$ plus a number of consistency conditions , , , , which take the following form: the one involving $T$ reads T-(n-1)D\_iT=0, which is actually identically satisfied on account of , the constraints on $\rho_i$ are \[rhoconstr\] \[, D\_i\]D-(n-1)D\_iD&=0,\ (+n)\[+ 2(n+1)\]D&=0,\ (+n)D\_[(i]{}\_[j)]{}+D\_i D\_j D-D-2\_[ij]{} D&=0, of which the first is again identically true by , while the consistency conditions involving $K_{ij}$ are \[Kconstr\] \[+ 2(n+1)\]DDK&=0,\ D\_i\[+ 2(n+1)\]DDK&=0,\ D\_[(i]{}D\_jDK\_[k)]{}+2\_[(ij]{}DK\_[k)]{}-\_[(ij]{}D\_[k)]{}DDK&=0. We will now prove that all these equations are actually identically satisfied, provided that $\rho_i$ and $K_{ij}$ obey and . To see this, we take divergences of $\mathcal K_{ijk}$ and $\mathcal R_{ijk}$, which, making extensive use of , read \[rhodivergences\] DR\_[ij]{}&=,\ DDR\_i &= ,\ DDDR &= (+n)\[+2(n+1)\]D, and \[kappadivergences\] DK\_[ij]{}&= + (+2n)K\_[ij]{},\ DDK\_i &= ,\ DDDK &= \[+2(n+1)\]DDK. Equations clearly show that all remaining constraints involving $\rho_i$ are actually proportional to divergences of equation , namely $D\cdot\mathcal R_{ij}$ and $D\cdot D\cdot D\cdot \mathcal R$. Furthermore, equation reduces the first two constraints in to $D\cdot D\cdot D\cdot \mathcal K$ and $D_iD\cdot D\cdot D\cdot \mathcal K$. To see that also the last equation in is identically satisfied by , we first take its trace and note that it is proportional to $D\cdot D\cdot \mathcal K_i$. The traceless projection instead reads \[DiDjDdotcalK\] D\_[(i]{}D\_jDK\_[k)]{}-\_[(ij]{}=0. We then consider D\_[(i]{}DK\_[jk)]{}= D\_[(i]{}D\_jDK\_[k)]{} +(+n-3)D\_[(i]{}K\_[jk)]{}+4\_[(ij]{}DK\_[k)]{}-\_[(ij]{}D\_[k)]{}DDK, which, upon substituting $\mathcal K_{ijk}=0$, yields \[DiDdotKjk\] D\_[(i]{}DK\_[jk)]{}={ D\_[(i]{}D\_jDK\_[k)]{}-\_[(ij]{}}=0. The traceless projection of this equation is precisely , which proves that it is identically satisfied. To reiterate, the gauge transformations that preserve the conditions and the “weak”, dimension-independent falloffs are parametrized by a conformal Killing tensor $K_{ij}(x^k)$ on the sphere, by a vector $\rho_i(x^k)$ satisfying and by an arbitrary function $T(x^i)$. Making explicit the dependence of the gauge parameters on these function affords \_[rr]{}&= - D+T,\ \_[ur]{}&= - + D- D + T,\ \_[uu]{}&= + D+ T,\ \_[ri]{}&= - +r\^2\_i + -D\_iD+D\_iT,\ \_[ui]{}&= DK\_i - +\ &-D\_i\[+2(n+1)\]D+\[+2(n+1)\]T,\ \_[ij]{}&= r\^4 K\_[ij]{} -+r\^3\ &+(D\_iD\_j+2\_[ij]{})DDK\ &-D+T. while the corresponding nonvanishing symmetry variations read \[spin3variations\] \_\_[uui]{}&= D\_i(+n)\[+2(n+1)\]T,\ \_\_[uij]{}&= -(D\_iD\_j+\_[ij]{})\[+2(n+1)\]D\ &+( D\_iD\_j-\_[ij]{}) \[+2(n+1)\]T,\ \_\_[ijk]{}&= DDK\ &-D\ &+T. Spin-three Global Symmetries {#app:spin3exact} ---------------------------- We now specialize the calculation of spin-three residual symmetries in $D=n+2$ performed above to the more conservative radiation falloff conditions. We thus require that \_\_[uui]{}, , scale as $\mathcal O(r^{-\frac{n}{2}})$. This choice will actually force us to consider, for any dimension greater than four, exact Killing tensors, namely the solutions of $\delta_\epsilon \varphi_{abc}=0$, due to the occurrence of the following additional constraints. First, $D_iH=0$ from the $\mathcal O(1)$ term in , namely \[addconstr1\] H=T would be a constant for any dimension greater than four. Second, the coefficient of $r$ in would not just be traceless, but would be actually set to zero, \[addconstr2\] D\_iD\_jG-\_[ij]{}(G-H)=0. Third, we would have the vanishing of the coefficient of $r^2$ in , \[addconstr3\] D\_[(i]{}D\_jD\_[k)]{}F+\_[(ij]{}D\_[k)]{}(2F-3G)=0. As already remarked, equations , and are equivalent to the vanishing of . We now express the additional constraints , and , or more precisely their traceless projections, since the vanishing of their traces has already been imposed above, in terms of $K_{ij}$, $\rho_i$ and $T$, thereby obtaining $$\begin{aligned} \label{DiDjDlDDK} D_{(i}D_jD_{l)}D\cdot D\cdot K -\frac{1}{n+2} \gamma_{(ij}\left[2\Delta D_{l)}D\cdot D\cdot K+ D_{l)}\Delta D\cdot D\cdot K\right]&=0\,,\\ \label{DiDjDrho} \left(D_iD_j - \tfrac{1}{n}\gamma_{ij}\Delta \right)[\Delta+2(n+1)]D\cdot \rho&=0\,,\\ \label{DiDjDlDrho} D_{(i}D_jD_{l)}D\cdot \rho -\frac{1}{n+2} \gamma_{(ij}\left[2\Delta D_{l)}D\cdot \rho+ D_{l)}\Delta D\cdot \rho\right]&=0\,,\\ \label{DiDeltaT} D_i(\Delta+n)[\Delta+2(n+1)]T&=0\,,\\ \label{DiDjT} \left(D_iD_j - \tfrac{1}{n}\gamma_{ij}\Delta \right)[\Delta+2(n+1)]T&=0\,,\end{aligned}$$ and \[constrT\] T\_[ijk]{}=D\_[(i]{}D\_jD\_[k)]{}T - \_[(ij]{}=0.\ We will now prove that equation is the only genuinely new condition, while all other constraint actually follow from $\mathcal K_{ijk}=0$, $\mathcal R_{ijk}=0$ and $\mathcal T_{ijk}=0$. The conditions and are proportional to divergences of $\mathcal T_{ijk}=0$: employing , $$\begin{aligned} D\cdot \mathcal T_{ij} &= \frac{3n}{n+2} \left(D_iD_j - \tfrac{1}{n}\gamma_{ij}\Delta \right)[\Delta+2(n+1)]T\\ D\cdot D\cdot \mathcal T_{i} &= \frac{3n(n-1)}{n+2}D_i(\Delta+n)[\Delta+2(n+1)]T\,.\end{aligned}$$ Equation then follows in the same way from . To see that the latter holds, we can start from equation and consider $D_{(i}D\cdot \mathcal R_{jk)}$, which takes the form D\_[(i]{}DR\_[jk)]{}={ (-3)D\_[(i]{}D\_j\_[k)]{} + D\_[(i]{}D\_jD\_[k)]{}D+\_[(ij]{}\[\]\_[k)]{} }, where dots in the square bracket denote a number of terms obtained from $\rho_i$ and its derivatives, with one free index. Using $\mathcal R_{ijk}=0$, given in , we can eliminate the first term on the right-hand side to cast this equation in the final form D\_[(i]{}DR\_[jk)]{}={ D\_[(i]{}D\_jD\_[k)]{}D+\_[(ij]{}\[\]\_[k)]{} }=0, whose traceless projection is . Equation can be seen to hold adopting a similar strategy. Starting from equation , we can consider $D_{(i}D_j D\cdot D\cdot \mathcal K_{k)}$, which has the form D\_[(i]{} D\_j DDK\_[k)]{}&={ (-3)D\_[(i]{}D\_jDK\_[k)]{} .\ & . + D\_[(i]{}D\_jD\_[k)]{}DDK +\_[(ij]{}\[\]\_[k)]{} }, where the square brackets now contain a suitable combination of $K_{ij}$ and its derivatives. Employing the expression for $D_{(i}D\cdot \mathcal K_{jk)}=0$ given in , we can substitute the first term on the right-hand side and obtain D\_[(i]{} D\_j DDK\_[k)]{}={ D\_[(i]{}D\_jD\_[k)]{}DDK+\_[(ij]{}\[\]\_[k)]{} }=0, which, upon taking the traceless part, yields as desired. 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[^1]: More precisely, by considering its action on $r$-independent functions. [^2]: This follows from the fact that external particles are on-shell and the corresponding tensors live in traceless representations of the stability group of $p^\mu$. [^3]: We added a uniform surface charge density $-\frac{q}{4\pi}$ on the sphere, which has no influence on $\partial_i\varphi$. [^4]: Strictly speaking, this equation is not well-posed, since the right-hand side has a nonzero divergence. However, the final process in which will be interested does not share this problem. [^5]: It can be convenient to employ $\sqrt{-g}\,\nabla_a \mathcal F^{ab} =\partial_a(\sqrt{-g}\,\mathcal F^{ab})$. [^6]: Strictly speaking, equation is not well-posed, since the right-hand side has a nonzero divergence. We shall take this aspect in due account when calculating the solution , where in particular the right-hand side of occurs just as part of a full source that respects the continuity equation. [^7]: More explicitly, $\mathcal A_u = \tfrac{q}{8\pi^2r^3}+\mathcal O(r^{-4})$. [^8]: The falloff conditions can also be heuristically justified as follows. A solution $\mathcal A_u(u,r,x^i)$ of eq. with nontrivial $u$ dependence, as required for radiation, must behave as $r^{-(D-2)/2}$ to leading order. If one then assumes the bounds $\mathcal A_r \lesssim A_r r^{-(D-6)/2}$ and $\mathcal A_i \lesssim A_i r^{-(D-4)/2}$, which provide a sufficient conditions for the finiteness of the energy flux, then equations and actually impose $\mathcal A_r \sim A_r r^{-(D-2)/2}$ and $\mathcal A_i \sim A_i r^{-(D-4)/2}$. [^9]: Interestingly, the corresponding putative charge \_= \_[r]{}\_[S\_u]{} F\_[ur]{} r\^[D-2]{} d = DA\^[()]{} \^[()]{} d is finite and nonvanishing as $r\to\infty$. Unfortunately, projects $\epsilon^{(\frac{D-4}{2})}$ to zero. [^10]: A similar procedure is adopted in Section \[sec:poly\]. [^11]: The covariant conservation equation for the color states, which in general reads $\left(\tfrac{d}{d\tau}+\dot x^a \mathcal A_a\right)|\psi\rangle=0$, implies the Wong equation for the color charge $\tfrac{d}{d\tau}Q+[\dot x^a \mathcal A_a,Q]=0$, where $Q=Q^AT^A$ and $Q^A=\langle \psi|T^A|\psi\rangle$. [^12]: Following the discussion of the previous chapter, it is worth keeping in mind that one may weaken these radiation falloffs, if needed, provided one does so while still ensuring the finiteness of the gauge-invariant quantity . [^13]: We recall that, given the $p$-form $$\alpha= \frac{1}{p!}\,\alpha_{a_1\cdots a_p}dx^{a_1}\wedge\cdots\wedge dx^{a_p}\,,$$ its Hodge dual is defined in terms of the metric tensor $g_{ab}$ as $$\ast\alpha = \frac{\sqrt{-g}}{p!(D-p)!}\,\alpha^{a_1\cdots a_p} \epsilon_{a_1\cdots a_p b_1 \cdots b_{D-p}}dx^{b_1}\wedge\cdots\wedge dx^{b_{D-p}}$$ with indices raised by $g^{ab}$. [^14]: I am grateful to Dario Francia for sharing his unpublished notes on higher-spin conserved currents.

--- abstract: 'We study the growth rate of the number of maximal arithmetic subgroups of bounded covolumes in a semi-simple Lie group using an extension of the method due to Borel and Prasad.' address: - ' M. Belolipetsky Department of Mathematical Sciences, Durham University, Durham DH1 3LE, U.K. Institute of Mathematics, Koptyuga 4, 630090 Novosibirsk, Russia ' - ' J. Ellenberg Department of Mathematics, Princeton University, Princeton, NJ 08544, U.S.A. ' - ' A. Venkatesh Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139-4307, U.S.A. ' author: - | Mikhail Belolipetsky\ \ with an appendix by Jordan Ellenberg and Akshay Venkatesh title: Counting maximal arithmetic subgroups --- Introduction ============ A classical theorem of Wang [@W] states that a simple Lie group not locally isomorphic to $\SL_2(\R)$ or $\SL_2(\C)$ contains only finitely many conjugacy classes of discrete subgroups of bounded covolumes. This theorem describes the distribution of lattices in the higher rank Lie groups, it also brings an attention to the quantitative side of the distribution picture. By now several attempts have been made towards a quantitative analogue of Wang’s Theorem but still very little is known. The problem of determining the number of discrete subgroups of bounded covolumes naturally splits into two parts: the first part is to count maximal lattices and the second, to count subgroups of bounded index in a given lattice. A recent project initiated by A. Lubotzky in [@L] resulted in a significant progress towards understanding the subgroup growth of lattices, which also allowed to formulate general conjectures on the asymptotic of the number of lattices of bounded covolumes in semi-simple Lie groups (see [@BGLM], [@GLP], [@LN] and [@LS]). In this paper we consider another part of the problem: we count maximal lattices in a given semi-simple Lie group. Let $H$ be a product of groups $H_s(k_s)$, $s\in S$, where $S$ is a finite set, each $k_s$ is an archimedean local field (i.e. $k_s = \R$ or $\C$) and $H_s(k_s)$ is an absolutely almost simple $k_s$-group. Then $H$ is a semi-simple Lie group. Throughout this paper we shall consider only semi-simple Lie groups of this form. We shall assume, moreover, that $H$ is connected and none of the factors $H_s(k_s)$ is compact or has type $\An_1$. In particular, $H$ can be a non-compact simple Lie group (real or complex) not locally isomorphic to $\SL_2(\R)$ or $\SL_2(\C)$. Let $m^u_H(x)$ and $m^{nu}_H(x)$ denote the number of conjugacy classes of maximal cocompact irreducible arithmetic subgroups and maximal non-cocompact irreducible arithmetic subgroups in $H$ of covolume less than $x$, respectively. If the real rank of $H$ is greater than $1$, then by Margulis’ theorem [@M Theorem 1, p. 2] these numbers are equal to the numbers of the conjugacy classes of maximal uniform and non-uniform irreducible lattices in $H$ of covolume less than $x$. In the real rank $1$ case there may also exist non-arithmetic lattices which we do not consider here. [**A.**]{} If $H$ contains an irreducible cocompact arithmetic subgroup (or, equivalently, $H$ is isotypic), then there exist effectively computable positive constants $A$ and $B$ which depend only on the type of almost simple factors of $H$, such that for sufficiently large $x$ $$x^A \le m^u_H(x) \le x^{B\beta(x)},$$ where $\beta(x)$ is a function which we define for an arbitrary $\epsilon > 0$ as $\beta(x) = C(\log x)^\epsilon$, $C = C(\epsilon)$ is a constant which depends only on $\epsilon$. [**B.**]{} If $H$ contains a non-cocompact irreducible arithmetic subgroup then there exist effectively computable positive constants $A'$, which depends only on the type of almost simple factors of $H$, and $B'$ depending on $H$, such that for sufficiently large $x$ $$x^{A'} \le m^{nu}_H(x) \le x^{B'}.$$ Conjecturally, the function $\beta(x)$ in part A can be also replaced by a constant and the constant $B'$ in part B depends only on the type of almost simple factors of $H$. This would require, in particular, a polynomial bound on the number of fields with a bounded discriminant. The existence of such a bound is an old conjecture in number theory which is possibly due to Linnik, it appears in a stronger form as Conjecture 9.3.5 in H. Cohen’s book [@Coh]. In fact, as a corollary from the proof of Theorem 1 we can show an equivalence of the conjecture “$\beta(x)= {\rm const}$” to Linnik’s conjecture (see Sect. \[s65\]). Theorem 1 is motivated by the problem of distribution of lattices in semi-simple Lie groups. An application of this results and its corollaries in Section 6 to the problem is a part of a joint work with A. Lubotzky [@BL]. Concluding the introduction let us briefly outline the proof of the theorem. Our method is based on the work of Borel and Prasad [@BP]. What makes our task different from the results in [@BP] is that besides proving the finiteness of the number of arithmetic subgroups of bounded covolumes we want to give bounds or, at least, [asym]{}ptotic bounds for the number. This requires certain modifications to the method on one side and some special number theoretic results on the other. In Proposition \[prop:BP\] we improve a number theoretic result from [@BP Sect. 6] which enables us to effectively count the possible fields of definition of the maximal arithmetic subgroups. The proof of this proposition is technical and can be safely skipped in the first reading. The key ingredient for a good upper bound for the number of fields is an extension of a recent work of J. Ellenberg and A. Venkatesh [@EV], which we formulate in Proposition \[prop:ND\] and for which the proof is given in Appendix. After bounding the number of possible fields of definition $k$, we count admissible $k$-forms, corresponding collections of local factors and conjugacy classes of arithmetic subgroups. Here we use some Galois cohomology techniques, the Hasse principle and basic number theoretic Proposition \[prop:NP\]. Altogether these lead to the proof of the upper bounds in Theorem 1 given in Section 4. The lower bounds are easier, they are established in Section 5. [**Acknowledgement.**]{} I am grateful to Alex Lubotzky for his lecture, which inspired this project, and for many helpful discussions. I wish to thank Tsachik Gelander, Alireza Salehi Golsefidy, Gopal Prasad and the anonymous referees for the critical remarks and suggestions. Jordan Ellenberg and Akshay Venkatesh contributed to this project valuable number theoretic support. This research was carried out while I held a Golda–Meir Postdoctoral Fellowship at the Hebrew University of Jerusalem. I also would like to thank for hospitality the MPIM in Bonn where a part of this work was completed. Preliminaries on arithmetic subgroups ===================================== The aim of this section is to give a short account of the fundamental results of Borel and Prasad ([@BP], [@P]) which will be used in this paper. We would like to encourage the reader to look into the original articles cited above for a better understanding of the subject. Our modest purpose here is to settle down the notation and recall some formulas for the future reference. Throughout this paper $k$ is a number field, $\cO_k$ its ring of integers, $V = V(k)$ the set of places (valuations) of $k$ which is the union of $V_\infty(k)$ archimedean and $V_f(k)$ nonarchimedean places, and $\A = \A_k$ is the ring of adèles of $k$. The number of archimedean places of $k$ is denoted by $a(k) = \#V_\infty(k)$. Let $r_1(k)$, $r_2(k)$ denote the number of real and complex places of $k$, respectively, so $a(k) = r_1(k) + r_2(k)$. As usual, $\D_k$ and $h_k$ stay for the absolute value of the discriminant of $k /\Q$ and the class number of $k$. For a finite extension $l$ of $k$, $\D_{l/k}$ denotes the $\Q$-norm of the relative discriminant of $l$ over $k$. All logarithms in the paper are in base $2$ unless stated otherwise. {#s21} Let $\G/k$ be an algebraic group defined over a number field $k$ such that there exists a continuous surjective homomorphism $\phi: \G(k\otimes_\Q\R)^o\to H$ with a compact kernel. We shall call such fields $k$ and $k$-groups $\G$ [*admissible*]{}. If $S\subset V_\infty(k)$ is the set of archimedean places of $k$ over which $\G$ is isotropic (i.e., non-compact), then $\phi$ induces an epimorphism $\G_S = \prod_{v\in S} \G(k_v)^o\to H$ whose kernel is a finite subgroup of $\G_S$. We consider $\G$ as a $k$-subgroup of $\GL(n)$ for large enough $n$ and define a subgroup $\Gamma$ of $\G(k)$ to be [*arithmetic*]{} if it is commensurable with the subgroup of $k$-integral points $\G(k)\cap \GL(n,\cO_k)$, that is, the intersection $\Gamma\cap\GL(n,\cO_k)$ is of finite index in both $\Gamma$ and $\G(k)\cap\GL(n,\cO_k)$. The subgroups of $H$ which are commensurable with $\phi(\Gamma)$ for some admissible $\G/k$ are called [*arithmetic subgroups*]{} of $H$ defined over the field $k$. We restrict ourselves to the irreducible lattices which implies that in the definition of the arithmetic subgroups it is enough to consider only simply connected absolutely almost simple algebraic groups $\G$ (see [@M Chapter 9.1]). {#s22} A semi-simple Lie group contains irreducible lattices if and only if all its almost simple factors have the same type ($H$ is [*isotypic*]{}). For example, we can take $H = \SL(2,\R)^a\times\SL(2,\C)^b$ or $H = \SO(p_1,q_1)\times\SO(p_2,q_2)$ ($p_1+q_1 = p_2+q_2$), but in $\SL(2,\R)\times\SL(3,\R)$ all lattices are reducible. Sufficiency of this condition is provided by the Borel–Harder theorem [@BH] and its necessity is discussed e.g. in [@M Chapter 9.4]. Note that in general the assumption that $H$ is isotypic does not imply that $H$ contains non-uniform irreducible lattices as is shown by an example due to G. Prasad (see Prop. 12.31 in D. Witte Morris, [*Introduction to Arithmetic Groups*]{}, preliminary version of the book available from the author’s website at http://people.uleth.ca/$\tilde{\ }$dave.morris). This is the reason why we have to impose an additional assumption concerning the existence of non-uniform irreducible lattices in part B of the theorem. {#s24} The methods of Borel–Prasad require a considerable amount of the Bruhat–Tits theory of reductive groups over local fields. We assume familiarity with the theory and recall only some basic definitions. An extensive account of what we need can be found in Tits’ survey article [@T]. Let $K$ be a nonarchimedean local field of characteristic zero (a finite extension of the $p$-adic field $\Q_p$) and let $\G$ be an absolutely almost simple simply connected $K$-group. The Bruhat–Tits theory associates to $\G/K$ a simplicial complex $\B = \B(\G/K)$ on which $\G(K)$ acts by simplicial automorphisms which are special (which implies, in particular, that if an element of $\G(K)$ leaves a simplex of $\B$ stable, then it fixes the simplex pointwise). The complex $\B$ is called the [*affine building*]{} of $\G/K$. A [*parahoric subgroup*]{} $P$ of $\G(K)$ is defined as a stabilizer of a simplex of $\B$. Every parahoric subgroup is compact and open in $\G(K)$ in the $p$-adic topology. Minimal parahoric subgroups are called [*Iwahori*]{}, they are defined as subgroups of $\G(K)$ fixing chambers (i.e., maximal simplexes) in $\B$. All Iwahori subgroups are conjugate in $\G(K)$. Maximal parahoric subgroups are the maximal compact subgroups of $\G(K)$, they are characterised by the property of being stabilizers of the vertices of $\B$. A maximal parahoric subgroup is called [*special*]{} if it fixes a [*special vertex*]{} of $\B$. A vertex $x\in\B$ is special if the affine Weyl group $W$ of $\G(K)$ is a semidirect product of the translation subgroup by the isotropy group $W_x$ of $x$ in $W$. In this case $W_x$ is canonically isomorphic to the (finite) Weyl group of the $K$-root system of $\G$. If $\G$ is quasi-split over $K$ and splits over an unramified extension of $K$, then $G(K)$ contains [*hyperspecial*]{} parahoric subgroups. We refer to [@T 1.10] for the definition of hyperspecial parahorics, an important property of these subgroups is that they have maximal volumes among all parahoric subgroups [@T 3.8.2]. {#s25} We shall now define a Haar measure $\mu$ on $H$ with respect to which the volumes of arithmetic quotients will be computed. Of course, the final result then will hold for any other normalization of the Haar measure on $H$. The definition and most of the subsequent facts come from [@P], [@BP]. Let $\G$ be a semi-simple algebraic group defined over a number field $k$. Let $v\in V_\infty(k)$ be an archimedean place of $k$. First consider the case when $k_v = \R$. There exists a unique anisotropic $\R$-form $\G_{cpt}$ of $\G$ which has a natural Haar measure giving the group volume $1$. This measure can be transferred to $\G(k_v)$ in a standard way and we define $\mu_v$ as its image. It is a canonical Haar measure on $\G(\R)$. In case $k_v = \C$ we have $\G(k_v) = \G_1(\R)$ with $\G_1 = {\rm Res}_{\C/\R}\G$, and we define $\mu_v$ to be equal to the canonical measure on $\G_1(\R)$. For a subset $S$ of $V_\infty(k)$ the Haar measure $\mu_S$ on $\G_S$ is defined as a product of $\mu_v$, $v\in S$. This also gives us a measure $\mu$ on a semi-simple Lie group $H$. {#s26} A collection $P = (P_v)_{v\in V_f}$ of parahoric subgroups $P_v$ of a simply connected $k$-group $\G$ is called [*coherent*]{} if $\prod_{v\in V_\infty}\G(k_v)\cdot\prod_{v\in V_f} P_v$ is an open subgroup of the adèle group $\G(\A_k)$. A coherent collection of parahoric subgroups $P = (P_v)_{v\in V_f}$ defines an arithmetic subgroup $\Lambda = \G(k)\cap\prod_{v\in V_f} P_v$ of $\G(k)$ which will be called [*principal arithmetic subgroup*]{} associated to $P$. The corresponding arithmetic subgroup $\Lambda' = \phi(\Lambda)\subset H$ shall be also called [*principal*]{}. The covolume of a principal arithmetic subgroup $\Lambda$ of $\G_S$ (in our case $S\subset V_\infty$) with respect to the measure $\mu$ defined as above is given by Prasad’s formula [@P Theorem 3.7]: $$\mu_S(\G_S/\Lambda) = \D_k^{\d}(\D_l/\D_k^{[l:k]})^{\frac12s} \left(\prod_{i=1}^{r}\frac{m_i!}{(2\pi)^{m_i+1}}\right)^{[k:\Q]} \tau_k(\G)\:\E(P),$$ where - ${\rm dim (\G)}$ and $m_i$ denote the dimension and Lie exponents of $\G$; - $l$ is a Galois extension of k defined as in [@P 0.2] (if $\G$ is not a $k$-form of type $^6\Dn_4$, then $l$ is the split field of the quasi-split inner $k$-form of $\G$, and if $\G$ is of type $^6\Dn_4$, then $l$ is a fixed cubic extension of $k$ contained in the corresponding split field; in all the cases $[l:k]\le 3$); - $s = s(\G)$ is an integer defined in [@P 0.4], in particular, $s=0$ if $\G$ is an inner form of a split group and $s\ge 5$ if $\G$ is an outer form; - $\tau_k(\G)$ is the Tamagawa number of $\G$ over $k$, since $\G$ is simply connected and $k$ is a number field $\tau_k(\G) = 1$; - $\E(P) = \prod_{v\in V_f} e_v$ is an Euler product of the local factors $e_v = e(P_v)$; for $v\in V_f$, $e_v$ is the inverse of the volume of $P_v$ with respect to the Haar measure $\gamma_v\omega_v^*$ defined in [@P 1.3, 2.1]. These factors can be computed using the Bruhat–Tits theory, in particular, $e_v > 1$ for every $v\in V_f$ [@P Prop. 2.10(iv)]. {#1:BP} Any maximal arithmetic subgroup $\Gamma$ of $H$ can be obtained as a normalizer in $H$ of the image $\Lambda'$ of some principal arithmetic subgroup of $\G(k)$ (see [@BP Prop. 1.4(iv)]). Moreover, the collections of parahoric subgroups which are associated to the maximal arithmetic subgroups have maximal types in a sense of Rohlfs [@R] (see also [@CR]). So in order to compute the covolume of a maximal arithmetic subgroup we need to be able to compute the index of a principal arithmetic subgroup in its normalizer. In a general setting the upper bound for the index was obtained in [@BP Sect. 2]: $$[\Gamma : \Lambda'] \le n^{\epsilon\# S}\cdot \#\H^1(k,{\rm C})_\xi \cdot \prod_{v\in V_f}\#\Xi_{\Theta_v}.$$ Here $n$ and $\epsilon$ are constants to be defined below, so $n^{\epsilon\# S}$ depends only on $H$ and does not depend on the choice of $\G(k)$ and $\Lambda$. $\H^1(k,{\rm C})_\xi$ is a finite subgroup of the first Galois cohomology group of $k$ with coefficients in the center of $\G$ defined in [@BP 2.10]. The order of $\H^1(k,{\rm C})_\xi$ can be further estimated (see [@BP Sect. 5]), the following bound is a combination of Propositions 5.1 and 5.6 \[loc. cit.\]: $$\# \H^1(k,{\rm C})_\xi \le 2h_l^{\epsilon'} n^{\epsilon a(k) + \epsilon' a(l) +\epsilon \# T} (\D_l/\D_k^{[l:k]})^{\epsilon''},$$ where - $n = r+1$ if $\G$ is of type $\An_r$; $n = 2$ if $\G$ is of type $\Bn_r$, $\Cn_r$ ($r$ arbitrary), $\Dn_r$ (with $r$ even), or $\En_7$; $n = 3$ if $\G$ is of type $\En_6$; $n = 4$ if $\G$ is of type $\Dn_r$ with $r$ odd; $n = 1$ if $\G$ is of type $\En_8$, $\Fn$ or $\Gn$; - $\epsilon = 2$ if $\G$ is of type $\Dn_r$ with $r$ even, $\epsilon = 1$ otherwise (so the center ${\rm C} = {\rm C}(\G)$ is isomorphic to $(\Z/n\Z)^\epsilon$ and $\#{\rm C} = n^{\epsilon}$); - $\epsilon' = \epsilon$ if $\G$ is an inner form of a $k$-split group, $\epsilon' = 1$ otherwise; - $\epsilon'' = 1$ if $\G/k$ is an outer form of type $\Dn_r$ ($r$ even) and $=0$ otherwise; - $T$ is the set of places $v\in V_f$ for which $\G$ splits over an unramified extension of $k_v$ but is not quasi-split over $k_v$. Finally, $\Xi_{\Theta_v}$ is a subgroup of the automorphism group of the affine Dynkin diagram which is coming from the adjoint group and preserving the type $\Theta_v$ of $P_v$. In particular, $\#\Xi_{\Theta_v}\le r+1$ and $\#\Xi_{\Theta_v} = 1$ if $P_v$ is special. {#2:BP} As a result we have the following lower bound for the covolume of $\Gamma$: $$\mu(H/\Gamma) \ge (2n^{\epsilon a(k)+\epsilon' a(l)}h_l^{\epsilon'})^{-1} \D_k^{\d}\left(\frac{\D_l}{\D_k^{[l:k]}}\right)^{\hspace*{-.4em}s'}\hspace*{-.6em} \left(\prod_{i=1}^{r}\frac{m_i!}{(2\pi)^{m_i+1}}\right)^{[k:\Q]}\hspace*{-1em}\tau_k(\G)\F,$$ where - $s' = s/2 - 1$ if $\G/k$ is an outer form of type $\Dn_r$, $r$ even, and $s' = s/2$ otherwise; - $\F = \prod_{v\in V_f} f_v$, with $f_v = e_v(\#\Xi_{\Theta_v})^{-1} = e_v$ if $\G$ is quasi-split over $k_v$ and $P_v$ is hyperspecial (which is true for almost all $v$), $f_v = e_vn^{-\epsilon}(\#\Xi_{\Theta_v})^{-1}$ if $\G$ splits over an unramified extension of $k_v$ but is not quasi-split over $k_v$, and $f_v = e_v(\#\Xi_{\Theta_v})^{-1}$ in the rest of the cases. Using the computations in [@BP Appendixes A,C] it is not hard to check that $f_v > 1$ for every $v\in V_f$. We refer to [@BP Sections 3,5] for more details about this formula. Number theoretic results ======================== {#prop:ND} Let $N_{k,d}(x)$ be the number of $k$-isomorphism classes of extensions $l$ of $k$ such that $[l:k] = d$, $\D_{l/k} < x$, and let $N(x)$ be the number of isomorphism classes of number fields with discriminant less than $x$. For large enough positive $x$ we have - given a number field $k$ and a fixed degree $d$, there exist absolute constants $c$, $b_1$, $b_2 > 0$ such that $N_{k,d}(x)\le c\D_k^{b_1}x^{b_2}$; - for every $\epsilon > 0$ there exists a constant $C = C(\epsilon) > 0$ such that\ $N(x) \le x^{\beta(x)}$, $\beta(x) = C(\log x)^{\epsilon}$. $(i)$ follows e.g. from [@EV Theorem 1.1]. $(ii)$ follows from the method of [@EV] but requires some extra work, namely, we have to know how the implicit constants in [@EV Theorem 1.1] depend on the degree of the extensions in order to be sure that this does not change the expected upper bound. This will be carried out in detail in Appendix provided by J. Ellenberg and A. Venkatesh. {#prop:NP} Let $Q_k(x)$ be the number of squarefree ideals of $k$ of norm $\le x$. - $\displaystyle Q_k(x) = \frac{{\rm Res}_{s=1}(\zeta_k)}{\zeta_k(2)}\ x + o(x)$ for $x\to\infty$; - There exist absolute constants $b_3$, $b_4$ (not depending on $k$) such that\ $Q_k(x) \le \D_k^{b_3}x^{b_4}$. $(i)$ is a known fact from the analytic number theory. For a short and conceptual proof we refer to [@S Theorem 14]. $(ii)$ As far as we do not claim that $b_4 = 1$ the proof is easy. Consider the Dedekind zeta function of $k$: $$\zeta_k(s) = \sum_{n=1}^{\infty}\frac{a_n}{n^s},$$ $a_n$ is the number of ideals of $k$ of norm $n$, $s>1$. Let $I_k(x)$ denote the number of ideals of $k$ of norm less than $x$. We have $$\begin{gathered} I_k(x) = a_1 + a_2 + \ldots + a_{[x]};\\ \zeta_k(s)\cdot x^s \ge I_k(x).\end{gathered}$$ Taking $s = 2$ we obtain $$Q_k(x) \le I_k(x) \le \zeta_k(2)\cdot x^2 \le \left(\frac{\pi^2}{6}\right)^{[k:\Q]} x^2 \le c_1^{\log \D_k} x^2 = \D_k^{b_3}x^{b_4}.$$ Here we used inequalities $\zeta_k(2)\le\zeta(2)^{[k:\Q]}$ and $[k:\Q] \le c\log \D_k$. The first of them follows from the definition of the functions $\zeta$ and $\zeta_k$, and the second is a well known corollary of Minkowski’s discriminant bound. {#prop:BP} Finally we shall need an improved version of a number theoretic result from [@BP Sect. 6]. The main idea is that instead of using $\D_k^\d$ to absorb the small factors in the volume formula we shall use only part of it saving the rest for a later occasion. This is easy to achieve for the groups of a large enough absolute rank while when the rank becomes small the estimates become much more delicate. Let $\G/k$ be an absolutely almost simple, simply connected algebraic group of absolute rank $r\ge2$, so the numbers $n$, $\epsilon$, $\epsilon'$, $s'$ and $m_1\le\ldots\le m_r$ are fixed and defined as in Sect. 2. Let $$\begin{aligned} B(\G/k) = \D_k^{\d} n^{-\epsilon a(k)-\epsilon' a(l)}h_l^{-\epsilon'} (\D_l/\D_k^{[l:k]})^{s'}\left(\prod_{i=1}^{r}\frac{m_i!}{(2\pi)^{m_i+1}}\right)^{[k:\Q]}\!\!\!\!\!\!\!\!\!.\end{aligned}$$ Then by \[2:BP\] we have $\mu(H/\Gamma) \ge \frac12 B(\G/k)\tau_k(\G)\F \ge \frac12 B(\G/k)$ for every arithmetic subgroup $\Gamma$ of $H$ which is associated to $\G/k$. There exist positive constants $\delta_1$, $\delta_2$ depending only on the absolute type of $\G$, such that $B(\G/k) \ge \D_k^{\delta_1} \D_{l/k}^{\delta_2}$ for almost all number fields $k$. Given an absolutely almost simple, simply connected algebraic group $\G$ of an absolute type $T$ and rank $r$, we shall show that for almost all $k$: - $B(\G/k) \ge \D_k^{\d-2} \D_{l/k}$ if $r\ge 30$; - $B(\G/k) \ge \D_k \D_{l/k}$ if $r<30$ and $T$ is not $\An_2,\ \An_3,\ \Bn_2$; - $B(\G/k) \ge \D_k^{0.1} \D_{l/k}$  if $T$ is $\An_3$ or $\Bn_2$; - $B(\G/k) \ge \D_k^{0.01} \D_{l/k}^{0.5}$ if $T$ is $\An_2$. Clearly, altogether these four inequalities imply the proposition. Assume first that $\G$ is not a $k$-form of type $^6\Dn_4$. We have $$\begin{gathered} [l:k]\le 2;\\ n^{-\epsilon a(k)-\epsilon' a(l)} \ge n^{-\epsilon( a(k)+ a(l))} \ge (r+1)^{-3[k:\Q]}.\end{gathered}$$ It is known that $$\label{31} h_l \le 10^2\left(\frac{\pi}{12}\right)^{[l:\Q]}\D_l$$ (cf. [@BP proof of 6.1], let us point out that this bound holds without any assumption on the degree of the field $l$); $$\label{32} \D_l/\D_k^{[l:k]} = \D_{l/k} \ge 1.$$ Combining the above inequalities we obtain $$\begin{aligned} \lefteqn{ B(\G/k) = \D_k^{\d}n^{-\epsilon a(k)-\epsilon' a(l)}h_l^{-\epsilon'} (\D_l/\D_k^{[l:k]})^{s'}\left(\prod_{i=1}^{r}\frac{m_i!}{(2\pi)^{m_i+1}}\right)^{[k:\Q]}} \\ &\quad \ge 10^{-2\epsilon'}\D_k^{\d - 2}\left(\frac{\pi}{12}\right)^{-\epsilon'[l:\Q]} \D_{l/k}\left(\frac{1}{(r+1)^3}\prod_{i=1}^{r}\frac{m_i!}{(2\pi)^{m_i+1}}\right)^{[k:\Q]} \\ &\quad \ge 10^{-2\epsilon'}\D_k^{\d - 2}\D_{l/k} \left(\frac{1}{(\pi/12)(r+1)^3}\prod_{i=1}^{r}\frac{m_i!}{(2\pi)^{m_i+1}}\right)^{[k:\Q]}\end{aligned}$$ (if $\G$ is $k$-split, then $s'=0$, $l = k$, $\D_{l/k} = 1$; in the non-split case we use the fact that $s'>2$). Since for $i$ large enough $m_i!\gg (2\pi)^{m_i+1}$ it is clear that for large enough $r$ $$\frac{1}{(\pi/12)(r+1)^3}\prod_{i=1}^{r}\frac{m_i!}{(2\pi)^{m_i+1}} > 1.$$ An easy direct computation shows that starting from $r=30$, $$10^{-2\epsilon'}\D_k^{\d - 2}\D_{l/k} \left(\frac{1}{(\pi/12)(r+1)^3}\prod_{i=1}^{r}\frac{m_i!}{(2\pi)^{m_i+1}}\right)^{[k:\Q]} \ge \D_k^{\d-2}\D_{l/k}.$$ So for $r\ge 30$, $\delta = \d - 2$ and any field $k$ we have $B(\G/k) \ge \D_k^\delta \D_{l/k}$, the finite set of the exceptional fields is empty and $(i)$ is proved. To proceed with the argument let us remark that $$B(\G/k) \ge \D_k^{\d-2}\D_{l/k}c,$$ where $c>0$ depends only on the absolute type of $\G$ and degree $d = [k:\Q]$. So, if the degree $d$ is fixed, then for any $z>0$ which will be chosen later we have $$\label{325} B(\G/k) \ge \D_k^{\d-2-z}\D_{l/k} \D_k^z c \ge \D_k^{\d-2-z}\D_{l/k},$$ for all $k$ with $\D_k\ge c^{-z}$. Since there are only finitely many number fields with a bounded discriminant, (\[325\]) holds for all but finitely many $k$ of degree $d$. Since we always have $\d > 2$, this allows to assume (at least when $\G$ is not $^6\Dn_4$) that the degree of $k$ is large enough. We now come to the case $(ii)$. Let $\G$ be still not of type $^6\Dn_4$. By the previous remark we can suppose that $[k:\Q]$ is large enough. We have the following lower bound for $\D_k$ due to A. Odlyzko [@Odlyzko Theorem 1]: $$\label{33} {\rm If\ } [k:Q] > 10^5,{\rm\ then\ } \D_k \ge 55^{r_1(k)}21^{2r_2(k)}.$$ So for $[k:Q] > 10^5$, $$B(\G/k) \ge 10^{-2\epsilon'}\left(\frac{21^{\d - 2 - \delta}}{(\pi/12)(r+1)^3} \prod_{i=1}^{r}\frac{m_i!}{(2\pi)^{m_i+1}}\right)^{[k:\Q]} \D_k^\delta \D_{l/k}.$$ A direct case by case verification shows that for $\delta=1$ the latter expression is $\ge \D_k^\delta \D_{l/k}$. So if we put $z = \d - 3$ in (\[325\]), then we obtain that for all but finitely many $k$, $B(\G/k) \ge \D_k \D_{l/k}$. Let now $\G/k$ be a triality form of type $^6\Dn_4$. We have $$\begin{gathered} \epsilon = 2,\ \epsilon' = 1,\ n = 2,\ s' = 2.5,\ \{m_i\} = \{1,3,5,3\};\\ [l:k] = 3 {\rm\ and\ } a(l)\le 3[k:\Q].\end{gathered}$$ So if $[k:Q] > 10^5$, $$B(\G/k) \ge 10^{-2}\left(\frac{21^{14-3 -\delta}}{(\pi/12)\cdot2^3} \cdot\frac{6\cdot120\cdot6}{(2\pi)^{16}}\right)^{[k:\Q]} \D_k^\delta \D_{l/k}.$$ For $\delta = 1$ it is $\ge \D_k \D_{l/k}$. If $[k:\Q] \le 10^5$ we still have the inequality (\[325\]) (the precise formula for the constant $c$ would be different but it is not essential), so for all but finitely many $k$, again $B(\G/k)\ge \D_k \D_{l/k}$. The case $(ii)$ is now settled completely. Let $\G/k$ be of type $\An_3$ or $\Bn_2$. As before we can assume $[k:Q] > 10^5$. We have $$B(\G/k) \ge 10^{-2}\left(\frac{\pi}{12}\right)^{-[l:\Q]} \left(\frac{21^{\d - [l:k] - \delta}}{n^{1+[l:k]}} \prod_{i=1}^{r}\frac{m_i!}{(2\pi)^{m_i+1}}\right)^{[k:\Q]} \D_k^\delta \D_{l/k}.$$ Now, $n = 4$ and $n = 2$ for the types $\An_3$ and $\Bn_2$, respectively; if $l\neq k$, then $[l:\Q] = 2[k:\Q]$. Using this it is easy to check that if $\delta = 0.1$, then $B(\G/k)\ge \D_k^\delta \D_{l/k}$ in each of the possible cases. It remains to consider $(iv)$. This is the most difficult case but the proof almost repeats the argument of [@BP Prop. 6.1(vi)]. With the notations of [@BP], for $[l:\Q] > 10^5$ we have $$\begin{aligned} \lefteqn{ B(\G/k) = \D_k^4 \cdot 3^{- a(k)- a(l)}h_l^{-1} (\D_l/\D_k^{[l:k]})^{5/2}\left(\frac{1}{2^4\pi^5}\right)^{[k:\Q]}} \\ &\quad = \D_k^{\delta} \D_l^{2-\delta/2} (\D_{l/k})^{1/2} 3^{- a(k)- a(l)}h_l^{-1} \left(\frac{1}{2^4\pi^5}\right)^{[k:\Q]} \\ &\quad \ge \D_k^{\delta}(\D_{l/k})^{1/2} \frac{0.02}{s(s-1)} \left(\frac{55^{(4-\delta-s)/2}}{2\cdot3^{3/2}\cdot\pi^{(6-s)/2}}\right)^{r_1(l)} \left(\frac{21^{(4-\delta-s)}}{2^{(4-s)}\cdot3^{2}\cdot\pi^{(5-s)}}\right)^{r_2(l)} \\ &\quad\quad \cdot\exp\big((3-\delta-s)Z_l(s) - (4-\delta-s)(\frac{c_1}{2} + (s-1)^{-1})\\ &\phantom{xxxxxxxxxxxxxxxxxxxx}+ (0.1-(c_3 + c_4)(s-1))a(l)\Big).\end{aligned}$$ Now let $\delta = 0.01$. Since $$\begin{aligned} & 55^{2.99/2}(2\cdot3^{3/2}\cdot\pi^{5/2})^{-1} > 2.19,\quad 21^{2.99}(2^{3}\cdot3^{2}\cdot\pi^{4})^{-1} > 1.28,\\ {\rm and}\ & \exp((3-\delta-s)Z_l(s))\ge 1\ {\rm if}\ s<2-\delta,\end{aligned}$$ by choosing $s>1$ sufficiently close to $1$, we obtain that there is an absolute constant $c_6$ such that $$\D_k^4 \cdot 3^{- a(k)- a(l)}h_l^{-1} (\D_l/\D_k^{[l:k]})^{5/2}\left(\frac{1}{2^4\pi^5}\right)^{[k:\Q]} \ge \D_k^{0.01}(\D_{l/k})^{1/2} 2.19^{r_1(l)} 1.28^{r_2(l)} c_6.$$ The right-hand side is $\ge \D_k^{0.01}(\D_{l/k})^{1/2}$ if $[l:Q]$ is large enough, say, $[l:Q] > d_l$ (and $d_l \ge 10^5$). If $[l:\Q] < d_l$, then $[k:\Q] < d_l$ and by $(3)$ for all but finitely many fields $k$ we have $$\begin{aligned} B(\G/k) \ge \D_k \D_{l/k} \ge \D_k^{0.01} \D_{l/k}^{0.5}.\end{aligned}$$ The proof provides explicit values of $\delta_1$, $\delta_2$ for each of the types, however in many cases the bound for $B(\G/k)$ can be improved. This will require more careful argument and can be useful for particular applications. Proof of the theorem. The upper bound ===================================== As before $H$ denotes a connected semi-simple Lie group whose almost simple factors are all non-compact and have the same type different from $\An_1$, $\G$ is an absolutely almost simple simply connected $k$-group admissible in the sense that there exists a continuous surjective homomorphism with a compact kernel. Counting number fields {#s41} ---------------------- For a (maximal) arithmetic subgroup $\Gamma$ of $H$ we have (see \[2:BP\]) $$\label{41} \mu(H/\Gamma) \ge \frac12 B(\G/k)\tau_k(\G)\F$$ where $k$ is the field of definition of $\Gamma$, $$\begin{gathered} \intertext{\quad\quad\quad\ $\G/k$ is a $k$-form from which $\Gamma$ is induced (see~\ref{s21});} B(\G/k) = \D_k^{\d} n^{-\epsilon a(k)-\epsilon' a(l)}h_l^{-\epsilon'} (\D_l/\D_k^{[l:k]})^{s'}\left(\prod_{i=1}^{r}\frac{m_i!}{(2\pi)^{m_i+1}}\right)^{[k:\Q]} \!\!\!\!\!\!\!\!\!;\\ \tau_k(\G) = 1;\\ \F = \prod_{v\in V_f} f_v > 1 {\rm\ \ will\ be\ considered\ later}.\\\end{gathered}$$ By Proposition \[prop:BP\] for all but finitely many number fields $k$ $$\mu(H/\Gamma) \ge c_1 \D_k^{\delta_1} \D_{l/k}^{\delta_2},$$ where $\delta_1$, $\delta_2$ are the constants determined by the absolute type of $\G$ (which is the type of almost simple factors of $H$). So for large enough $x$, if $\mu_S(H/\Gamma) < x$, then $\D_k < (x/c_1)^{1/\delta_1}$, $\D_{l/k} < (x/c_1)^{1/\delta_2}$. By Proposition \[prop:ND\](ii), the number of such fields $k$ is at most $$(x/c_1)^{\beta((x/c_1)^{1/{\delta_1}})}\le x^{c_2\beta(x)},$$ and by Proposition \[prop:ND\](i), for each $k$ the number of such extensions $l$ is at most $$c\D_k^{b_1}(x/c_1)^{b_2/{\delta_2}} \le c(x/c_1)^{b_1/\delta_1}(x/c_1)^{b_2/\delta_2}\le x^{c_3}.$$ It follows that the number of all admissible pairs $(k,l)$ is bounded by $$x^{c_2\beta(x)+c_3},$$ and, moreover, since $k\neq\Q$ implies $[k:\Q]\le c\log\D_k$, for all admissible $k$ we have $$a(k) \le c_4\log x.$$ Non-cocompact case {#s42} ------------------ If $\Gamma$ is non-cocompact, the degree of the field of definition of $\Gamma$ is bounded. Indeed, the non-cocompactness of $\Gamma$ implies that the corresponding algebraic group $\G$ is $k$-isotropic, so $\G/k_v$ is non-compact for every $v\in V$. It follows that the number of infinite places of $k$ is equal to the number $\# S$ of almost simple factors of $H$, so $[k:\Q]\le 2\# S$. Now in \[s41\] we can consider only the number fields $k$ with $[k:\Q]\le 2\# S$ and the number fields $l$ with $[l:\Q]\le 3[k:\Q]\le 6\# S$. By Proposition \[prop:ND\](i), for large enough $x$ the number of admissible pairs $(k,l)$ is at most $$x^{c_5},\ c_5 = c_5(\# S)$$ (in fact, here we could use a weaker result by W. M. Schmidt, [*Astérisque*]{}, [**228**]{}(1995), 189–195, who showed that the number of degree $n$ extensions $l$ of $k$ with $\D_{l/k} < x$ is bounded by $C(n)x^{(n+2)/4}$). Counting $k$-forms {#s43} ------------------ Given an admissible pair $(k,l)$ of number fields, there exists a unique quasi-split $k$-form $\Gqs$ for which $l$ is the splitting field (or a certain subfield of the splitting field if $\Gqs$ is of type $^6\Dn_4$ and $[l:\Q] = 3$). So we have an upper bound for the number of quasi-split groups for which there can exist an inner form that defines an arithmetic subgroup of covolume less than $x$. We now fix a quasi-split $k$-form $\Gqs$ and estimate the number of admissible inner forms. Since every inner equivalence class of $k$-forms contains a unique quasi-split form, this will give us a bound on the total number of admissible $\G/k$. By the assumption, $\prod_{v\in V_\infty(k)} \G(k_v)$ is isogenous to $H\times K$ ($K$ is a compact Lie group), so the $k_v$-form of $\G$ is almost fixed at the infinite places of $k$. More precisely, let $c_h$ be the number of non-isomorphic almost simple factors of $H$. For each $v\in V_\infty(k)$, $\G(k_v)$ is isomorphic to one of $c_h$ non-compact (simply connected) groups or is compact, and the number of places $v$ at which $\G(k_v)$ is non-compact $n_h = \# S$. This implies that the number of variants for $\G(k_v)$ at the infinite places of $k$ is bounded by $$c_h^{n_h} \binom{a(k)}{n_h} < (c_h a(k))^{n_h} \le (\log x)^{c_6}$$ ($\binom{\cdot}{\cdot}$ denotes the binomial coefficient). Let now $v$ be a finite place of $k$. The inner $k_v$-forms of $\G$ correspond to the elements of the first Galois cohomology set $\H^1(k_v, {\overline \G})$, ${\overline \G}$ is the adjoint group of $\G$. The order of $\H^1(k_v, {\overline \G})$ can be computed from the cohomological exact sequence $$\H^1(k_v, {\G}) \to \H^1(k_v, {\overline \G}) \stackrel{\delta}{\to} \H^2(k_v, {\rm C}),$$ which corresponds to the universal $k_v$-covering sequence of groups $$1 \to {\rm C} \to {\G} \to {\overline \G} \to 1.$$ For a simply connected $k_v$-group ${\G}$ the first cohomology $\H^1(k_v, {\G})$ is trivial by a theorem of Kneser [@Kn], so $\delta$ is injective. Furthermore, the group $\H^2(k_v, {\rm C})$ can be identified with a subgroup of the Brauer group of $k_v$ and then explicitly computed using results from the local class field theory. We refer to [@PlR Chapter 6] for details and explanations. As a corollary here we have that the number of inner $k_v$-forms is bounded by $n^\epsilon$ in the notation of \[1:BP\] (recall that $n^{\epsilon} = \# {\rm C}$ is the order of the center of $\G$). Let $T_1\subset V_f(k)$ be a (finite) subset of the nonarchimedean places of $k$ such that $\G$ is not quasi-split over $k_v$ for $v\in T_1$. It follows from [@P Prop. 2.10] that there exists a constant $\delta > 0$, which depends only on the absolute type of $\G$, such that for every $v\in T_1$, $$\label{43} f_v \ge n^{-\epsilon}(\#\Xi_{\Theta_v})^{-1}e_v \ge q_v^\delta$$ ($q_v$ denotes the order of the residue field of $k$ at $v$). Indeed, we can take $\delta = \log(2^{r_v}n^{-\epsilon})$ if the absolute type of $\G$ is not $\An_2$ and $\delta = \log(2^2\cdot3^{-1}) = 0.415\dots$ for the type $\An_2$, and then check that $\delta>0$ and inequality (\[43\]) holds going through the case-by-case consideration in [@BP Appendix C.2]. To a set $T\subset V_f(k)$ we can assign an ideal $\I_T$ of $\cO_k$ given by the product of prime ideals defining the places in $T$. Reversely, each squarefree ideal of $\cO_k$ uniquely defines a subset $T$ in $V_f(k)$ corresponding to its prime decomposition. Note also that $\prod_{v\in T}q_v = {\rm Norm}(\I_T)$. Now for an arithmetic subgroup $\Gamma$ induced from $\G$ we have $$\mu(H/\Gamma) = \frac12 B(\G/k)\tau_k(\G)\F \ge c_7\prod_{v\in T_1} q_v^\delta.$$ This implies that if $\mu(H/\Gamma) \le x$, then ${\rm Norm}(\I_{T_1}) = \prod_{v\in T_1}q_v \le x^{c_8}$. By Proposition \[prop:NP\](ii) the number of variants for $T_1$ is bounded by $x^{c_9}$, moreover, since for every $v\in V_f$, $q_v\ge 2$, for every such a set $T_1$ we have $\# T_1 \le c_{10}\log x$. Now the Hasse principle implies that a $k$-form of $\G$ is uniquely determined by $(\G(k_v))_{v\in V(k)}$. The Hasse principle for semi-simple groups is valid due to the work of Kneser, Harder, Chernousov (cf. [@PlR Chapter 6]). So the number of the admissible $k$-forms is at most $$(\log x)^{c_6} x^{c_8} n^{\epsilon c_{10}\log x} \le x^{c_{11}}.$$ Counting collections of parahorics {#s44} ---------------------------------- For a given large enough $x$ we have defined a collection of $\G/k$ for which there exists a (centrally) $k$-isogenous group $\G'$ which may give rise to the arithmetic subgroups $\Gamma\subset H$ with $\mu(H/\Gamma) < x$. The number of such $k$-groups $\G$ is finite and can be bounded as in \[s43\], but still each $\G/k$ defines countably many maximal arithmetic subgroups. We shall now fix a group $\G/k$ and estimate the number of coherent collections of parahoric subgroups of $\G$ which can give rise to the maximal arithmetic subgroups with covolumes less than $x$. In the classical language, what we are going to do in this section is to count the number of admissible genera. We use again the local to global approach. Let us fix a central $k$-isogeny $i: \G\to \G'$ with $\G'$ such that $\G'_S$ projects onto $H$. Every maximal arithmetic subgroup $\Gamma\subset \G'_S$ is associated to some coherent collection $P = (P_v)_{v\in V_f}$ of parahoric subgroups of $\G$ (see [@BP Prop. 1.4]): $$\Gamma = N_{\G'}(i(\Lambda)),\ \Lambda = \G(k)\ \cap \!\!\prod_{v\in V_f(k)} P_v.$$ The image of $\Gamma$ in $H$ is an arithmetic subgroup and every maximal arithmetic subgroup of $H$ can be obtained as a projection of some such $\Gamma$. For almost all finite places $v$ of $k$, $\G$ is quasi-split over $k_v$ and splits over an unramified extension of $k_v$. Moreover, for almost all such $v$, $P_v$ is hyperspecial. Any two hyperspecial parahoric subgroups of $\G(k_v)$ are conjugate under the action of the adjoint group $\Gb(k_v)$ [@T Sect. 2.5], so $P$ is determined up to the action of $\Gb(\A_f)$ by the types of $P_v$ at the remaining places. Using this we shall now count the number of $P$’s. Let as in \[s43\], $T_1$ denote the set of places of $k$ for which $\G$ is not quasi-split. By the previous argument we have $$\# T_1 \le c_{10}\log x,\ \ \# ({\rm variants\ for}\ T_1) \le x^{c_9} \ \ \rm{(see\ \ref{s43})}.$$ Let $R$ denote the set of places for which $\G$ is quasi-split but is not split over an unramified extension of $k_v$. For such places $v\in V_f$, $l_v = l\otimes_k k_v$ is a ramified extension of $k_v$ and so by the formula from [@P Appendix], each of such places contributes to $\D_{l/k}$ a power of $q_v$. Using again Proposition \[prop:NP\](ii) and the inequality $\D_{l/k}\le x^c$ from \[s41\], we obtain $$\# R \le c_{12}\log x,\ \ \# ({\rm variants\ for}\ R) \le x^{c_{13}}.$$ Finally, let $T_2\subset V_f\backslash(T_1\cup R)$ be the set of places for which $P_v$ is not hyperspecial. If $v\in T_2$, then by [@P Prop. 2.10(iv)] $$e_v \ge (q_v+1)^{-1}q_v^{r_v+1}.$$ Similarly to (\[43\]) it implies $$f_v \ge q_v^\delta.$$ By Proposition \[prop:NP\](ii) and the volume formula, $$\# T_2 \le c_{14}\log x,\ \ \# ({\rm variants\ for}\ T_2) \le x^{c_{15}}.$$ Now for a given $v\in V_f$ the number of the possible types of parahoric subgroups (parametrized by the subsets of the set of simple roots) is bounded by a constant $c_t$ which depends only on the absolute type of $\G$. We conclude that for a given $\G$ the number of $P$’s, up to the action of $\Gb(\A_f)$, is at most $$c_t^{\#(T_1 \cup R \cup T_2)}\# ({\rm variants\ for}\ T_1 \cup R \cup T_2) \le c_t^{(c_{10}+c_{12}+c_{14})\log x} x^{c_9+c_{13}+c_{15}} = x^{c_{16}}.$$ Counting conjugacy classes {#s45} -------------------------- In this final step we give an upper bound for the number of conjugacy classes of arithmetic subgroups associated to a fixed group $\G'/k$ and a given $\Gb(\A_f)$-orbit of collections of parahoric subgroups $P$ of a simply connected group $\G$ centrally $k$-isogenous to $\G'$. We are interested in the $\Gb(k)$-conjugacy classes of maximal subgroups associated to $P$ which are indexed by the double cosets $\Gb(k)\backslash\Gb(\A)/\Gb_\infty\Pb$, where $\Gb_\infty = \prod_{v\in V_\infty} \Gb(k_v)$, $\Pb_v$ is the stabilizer of $P_v$ in $\Gb(k_v)$ and $\Pb = \prod_{v\in V_f} \Pb_v$ is a compact open subgroup of $\Gb_f = \prod_{v\in V_f}\Gb(k_v)$ (see [@BP Prop. 3.10]). The number $c(\Pb)$ of the double cosets is called the [*class number*]{} of $\Gb$ with respect to $\Pb$. The argument is similar to the proof of Proposition 3.9 in [@BP] except that we need to get an explicit upper bound for $c(\Pb)$. Let $\omega$ be a non-zero invariant exterior $k$-form of top degree on $\Gb$; such a form is unique up to multiplication by an element of $k^*$ and is called a Tamagawa form. We denote by $|\omega|$ the Haar measure on the adèle group $\Gb(\A)$ determined by $\omega$. The natural embedding of $k$ into $\A$ gives an embedding of $\Gb(k)$ in $\Gb(\A)$ and it is well known that the image of $\Gb(k)$ is a lattice in $\Gb(\A)$. By the product formula its covolume with respect to the measure $|\omega|$ does not depend on the choice of the form $\omega$, thus the number $\tau_k(\Gb) := \D_k^{-\dim(\Gb)/2} |\omega|(\Gb(k)\backslash\Gb(\A))$ is correctly defined. It is called the [*Tamagawa number*]{} of $\Gb/k$. By a theorem of T. Ono [@Ono], $\tau_k(\Gb)$ is bounded by a constant multiple of the order of the center of the simply connected covering group $\G$ multiplied by $\tau_k(\G)$. The Tamagawa number of a simply connected group is equal to $1$ according to the Weil conjecture which has been proved completely for the groups over number fields due to the work of many people (see [@P 3.3] for a short discussion). Therefore we have $$\label{eq_tau} |\omega|(\Gb(k)\backslash\Gb(\A)) = \tau_k(\Gb) \D_k^{\dim(\Gb)/2}\le c_{17} \D_k^{\dim(\Gb)/2}$$ where $c_{17}$ depends only on the absolute type of $\G$. Coming back to the problem of bounding the class number $c(\Pb)$, we recall that the double cosets $\Gb(k)\backslash\Gb(\A)/\Gb_\infty\Pb$ correspond bijectively to the orbits of $\Gb_\infty\Pb$ on $\Gb(k)\backslash\Gb(\A)$ which are open. Given an upper bound for $|\omega|(\Gb(k)\backslash\Gb(\A))$, in order to give a bound for $c(\Pb)$ it is enough to obtain a uniform lower bound for the $|\omega|$-volumes of these orbits. The double cosets are represented by elements of $\Gb_f$, so it is sufficient to consider the orbit of the image of $a\in\Gb_f$ which is isomorphic to $\Gamma_a\backslash\Gb_\infty a\Pb a^{-1}$, $\Gamma_a = \Gb(k)\cap\Gb_\infty a\Pb a^{-1}$. Let $\Gamma_a'$ be the projection of $\Gamma_a$ to $\Gb_\infty$ with respect to the decomposition $\Gb(\A) = \Gb_\infty\times\Gb_f$. As $a\Pb a^{-1}$ is a compact open subgroup of $\Gb_f$, $\Gamma_a'$ is an arithmetic subgroup of $\Gb_\infty$. We have $$|\omega|(\Gamma_a\backslash\Gb_\infty a\Pb a^{-1}) = |\omega|_\infty(\Gamma_a'\backslash\Gb_\infty) |\omega|_f(\Pb),$$ where $|\omega|_\infty$, $|\omega|_f$ denote the product measures on $\Gb_\infty$, $\Gb_f$ corresponding to $\omega$. In order to estimate the factors in the right-hand side of the formula, for each $v\in V(k)$ we shall relate the measure $|\omega|$ to the canonical measure $|\omega_{\Gb_v}|$ on $\Gb(k_v)$ defined in [@Gross Sections 4, 11]. In particular, if $\G$ is simply connected the measure $|\omega_{\G_v}|$ coincides with the measure $\gamma_v\omega_v^*$ which is used for the local computations in [@P]; for $v\in V_\infty(k)$, $|\omega_{\Gb_v}|$ is equal to the measure $\mu$ on $\Gb(k_v)$ defined as in \[s25\]; and for all but finitely many $v$, $|\omega_{\Gb_v}| = |\omega|_v$. Let $\gamma_v$ denote the ratio $|\omega_{\Gb_v}|/|\omega|_v$ which by the previous remark is equal to $1$ for all but finitely many places $v$. Hence $$|\omega|(\Gamma_a\backslash\Gb_\infty a\Pb a^{-1}) = \mu_\infty(\Gamma_a'\backslash\Gb_\infty) \prod_{v\in V_f} |\omega_{\Gb_v}|(\Pb_v) / \prod_{v\in V}\gamma_v.$$ We now recall the main result of [@BP], which implies that covolumes of arithmetic subgroups of $\Gb_\infty$ with respect to the measure $\mu$ are bounded from below by a universal constant and thus $\mu_\infty(\Gamma_a'\backslash\Gb_\infty) \ge \mu_0$. The crucial ingredient which allows us to carry out the required estimates is the product formula for $\gamma_v$. It was obtained in [@P Theorem 1.6] for the simply connected groups and later extended by B. Gross to arbitrary reductive groups defined over number fields (see also [@Ku] for the groups over global function fields). Thus by [@Gross Theorem 11.5] we have $$\prod_{v\in V} \gamma_v = (\D_l/\D_k^{[l:k]})^{\frac12s} \left(\prod_{i=1}^{r}\frac{m_i!}{(2\pi)^{m_i+1}}\right)^{[k:\Q]}\!\!\!\!\!\!\!\!\!.$$ Finally, we shall make use of the following inequality: [*Claim.*]{} $|\omega_{\Gb_v}|(\Pb_v) \ge |\omega_{\G_v}|(P_v) = e(P_v)^{-1}$. The proof of this claim which is given below is quite technical but not conceptually new, related questions were studied in detail and full generality in [@Gross], [@Ku]. The argument falls into several steps. Let $K = k_v$ be a nonarchimedean local field, $\cO$ its ring of integers, $\G$ a simply connected semi-simple $K$-group, $i: \G \to \G'$ a central $K$-isogeny (we actually need only the case $\G'=\Gb$) and let $X = X(\G)$ denote the Bruhat-Tits building of $\G/K$. [**1.**]{} We shall assume first that the groups $\G$ and $\G'$ are quasi-split over $K$. Let $x\in X$ be a special vertex in $X$ chosen as in \[Gr, Sect. 4\] (see also \[P, 1.2\]). The Bruhat-Tits theory assigns to $\G'/K$ and $x\in X(\G)$ a smooth affine group scheme $\Gix$ over $\cO$. Its generic fiber is isomorphic to $\G'/K$ and its special fiber $\Gixs$ is connected. Let $P_x = \Gx(\cO) (= \Gx^0(\cO))$, $P'_x = \Gix(\cO)$. Then $P_x$ (resp. $P'_x$) is an open compact subgroup of $\G(K)$ (resp. $\G'(K)$), $P_x$ is the stabilizer of $x$ in $\G(K)$ and $P'_x$ is contained in the stabilizer of $x$ in $\G'(K)$ with finite index. Recall also that the measure $\og$ (resp. $\ogi$) corresponds to a differential $\omega_\G$ (resp. $\omega_{\G'}$) of top degree on $\G$ (resp. $\G'$) over $K$ which has good reduction (see \[Gr, Sect. 4\]). This brings us to the conditions of Proposition I.2.5 of \[Oe\], which implies $$\og(P_x) = \#\Gxs(\F_q) q^{-\mathrm{dim}\:\G},\ \ogi(P'_x) = \#\Gixs(\F_q) q^{-\mathrm{dim}\:\G'},$$ ($\F_q$ denotes the residue field of $K$). Since $\G$ and $\G'$ are isogenous, $\Gxs$ and $\overline{\mathrm G}'_x$ are isogenous. Hence, $\mathrm{dim}\:\G = \mathrm{dim}\:\G'$, and by Lang’s theorem, $\#\Gxs(\F_q) = \#\Gixs(\F_q)$. Thus we obtain $\og(P_x) = \ogi(P'_x)$. [**2.**]{} Let now $C$ be a chamber of $X$ which contains $x$ and let $\Omega$ be a subset of $C$. Denote by $I_C$ (resp. $I'_C$) the Iwahori subgroup of $\G(K)$ (resp. $\G'(K)$) corresponding to $C$. Note that by definition $I'_C$ is the preimage in $\G'(K)$ of a Borel subgroup $\Bb'$ of $\Gixs(\F_q)$. Let $P_\Omega$ (resp. $P'_\Omega$) be the parahoric subgroup of $\G(K)$ (resp. $\G'(K)$) associated to $\Omega$; so $P_\Omega = \Go(\cO)$, $P'_\Omega = \Gio(\cO)$ and any parahoric subgroup of $\G(K)$ is conjugate to some $P_\Omega$. The inclusion $\Omega\subset C$ induces a group scheme homomorphism $\rho_{\Omega C}: {\underline{\mathrm G}'}^0_C \to \Gio$ whose reduction maps the group ${\overline{\mathrm G}'}^0_C$ onto a Borel subgroup $\Bb'$ of $\Gios$. Therefore we have $$[P'_\Omega:I'_C] = [\Gios(\F_q):\Bb'] = [\Gos(\F_q):\Bb] = [P_\Omega:I_C],$$ as $\Gios$ is isogenous to $\Gos$, $\Bb'$ is isogenous to $\Bb$ and all the groups are connected. It follows that $|\omega_{\G'}|(P'_\Omega) = |\omega_{\G}|(P_\Omega)$. [**3.**]{} We finally note that $P'_\Omega \subset \overline{P}'_\Omega$ ($\overline{P}'_\Omega$ denotes the stabilizer of $\Omega$ in $\G'(K)$) and thus $|\omega_{\G'}|(\overline{P}'_\Omega) \ge |\omega_{\G'}|(P'_\Omega) = |\omega_{\G}|(P_\Omega)$, which implies the desired inequality in the quasi-split case. [**4.**]{} In order to extend this result to the general case we have to recall the definition of the canonical measure $\og$ for the general $\G$ by pull-back from the quasi-split inner form (see [@Gross p. 294]) and its interpretation in terms of the volume form $\nu_G$ associated to an Iwahori subgroup of $\G(K)$ described in [@Gross pp. 294–295]. The latter allows us to apply the argument similar to step 1 to Iwahori subgroups $I_C$ and $I'_C$ corresponding to a chamber $C$ of $X(\G)$, proving $\og(I_C) = \ogi(I'_C)$. All the rest of the proof does not depend on the quasi-split assumption and the claim follows. Let us collect together the results of this section. We obtain: $$\begin{aligned} \label{eq_cl} \lefteqn{ c(\Pb) \le \frac{|\omega|(\Gb(k)\backslash\Gb(\A)) \prod_{v\in V}\gamma_v} {\mu_0 \prod_{v\in V_f}e(P_v)^{-1}} } \notag\\ &\quad \le \frac{1}{\mu_0}\D_k^{\dim(\Gb)/2}(\D_l/\D_k^{[l:k]})^{\frac12s} \left(\prod_{i=1}^{r}\frac{m_i!}{(2\pi)^{m_i+1}}\right)^{[k:\Q]}\tau_k(\Gb) \E(P).\end{aligned}$$ This formula can be viewed as an extension of the upper bound for the class number from [@P Theorem 4.3]. We now bound the right-hand side of (\[eq\_cl\]). By \[s41\], \[s42\] we have $\D_k < (x/c_1)^{1/\delta_1}$, $\D_{l/k} < (x/c_1)^{1/\delta_2}$ and $[k:\Q]\le c\log\D_k$. By (\[eq\_tau\]), $\tau_k(\Gb)\le c_{17}$. From \[s43\] and \[s44\] it follows that if $\mu(H/\Gamma) \le x$, then $\prod_{v\in V_f} e_v \le x^{c_{18}}$ for some constant $c_{18}$ which depends only on the type of almost simple factors of $H$. Hence it follows from (\[eq\_cl\]) that there exists a constant $c_{19}$ such that $$c(\Pb) \le x^{c_{19}}.$$ The upper bounds ---------------- It remains to combine the results of the previous sections to get the upper bounds. By \[s41\], \[s43\], \[s44\], \[s45\] $$m^u_H(x) \le x^{c_2\beta(x)+c_3} x^{c_{11}} x^{c_{16}} x^{c_{19}} \le x^{B\beta(x)},$$ and constant $B$ depends only on the type of almost simple factors of $H$. By \[s42\], \[s43\], \[s44\], \[s45\] $$m^{nu}_H(x) \le x^{c_5} x^{c_{11}} x^{c_{16}} x^{c_{19}} \le x^{B'},$$ constant $B'$ depends on the type and the number of almost simple factors of $H$. Proof of the theorem. The lower bound ===================================== Cocompact case. {#s51} --------------- A theorem of Borel and Harder [@BH] implies that a semi-simple group over a local field of characteristic $0$ contains cocompact arithmetic lattices. The method of [@BH] actually proves the existence of such lattices defined over a given filed $k$, which satisfies a natural admissibility condition, for any isotypic semi-simple Lie group. So if $H$ has $a_1$ real and $a_2$ complex almost simple factors (all of the same type) and $k$ is a number field with $>a_1$ real and precisely $a_2$ complex places, then $H$ contains a cocompact arithmetic subgroup $\Gamma_1$ defined over $k$. Let $\Gamma_0$ be a maximal arithmetic subgroup of $H$ which contains $\Gamma_1$. There exists an absolutely almost simple simply connected $k$-group $\G$ and a principal arithmetic subgroup $\Lambda_0$ of $\G$ such that $\Gamma_0 = N_H(\phi(\Lambda_0))$. We assume $x$ is large enough and estimate the number of principal arithmetic subgroups $\Lambda\subset \G(k)$ which are associated to the coherent collections of parahoric subgroups of $\cO$-maximal types ([@R], [@CR]) and such that $\mu_S(\G_S/\Lambda) < x$. Then for $\Gamma = N_H(\phi(\Lambda))$ we also have $\mu_S(\G_S/\Gamma) < x$. Moreover, by Rohlfs’ theorem each such $\Gamma$ is a maximal arithmetic subgroup of $H$ and all maximal arithmetic subgroups of $H$ are obtained as the normalizers of the images of the principal arithmetic subgroups corresponding to $\cO$-maximal collections of parahorics. The condition of maximality for the type of a collection of parahoric subgroups $P = (P_v)_{v\in V_f}$ is a local condition on the types of $P_v$ at each $v\in V_f$, while $\cO$-maximality requires an additional global restriction which is needed to further narrow down the set of admissible collections of parahoric subgroups of maximal types. We shall not give precise definitions here referring the reader to the above cited papers. What is important for our argument is that given $P_0 = (P_{0,v})_{v\in V_f}$, a collection of parahoric subgroups of $\cO$-maximal type, for every $v_0\in V_f$ there exists another $\cO$-maximal collection $P = (P_v)_{v\in V_f}$ such that for $v\neq v_0$, $P_v = P_{0,v}$ and $P_{v_0}\not\cong P_{0,v_0}$. This is clearly true: for the groups of the absolute rank greater than one (which is our standing assumption) it is enough to consider the maximal types corresponding to single vertices of the affine Dynkin diagram and for such types $\cO$-maximality can be easily checked. We have $$\begin{aligned} \mu_S(\G_S/\Lambda) & = \D_k^{\d}(\D_l/\D_k^{[l:k]})^{\frac12s} \left(\prod_{i=1}^{r}\frac{m_i!}{(2\pi)^{m_i+1}}\right)^{[k:\Q]} \tau_k(\G)\:\E(P) \\ & = c_1\prod_{v\in T} e(P_v)/e(P_{0,v}) \\ & \le c_1 \prod_{v\in T} e(P_v), \ \ c_1 = \mu_S(\G_S/\Lambda_0),\end{aligned}$$ where $P_v$ (resp. $P_{0,v}$) is the closure of $\Lambda$ (resp. $\Lambda_0$) in $\G(k_v)$, $v\in V_f$; $T$ is a finite subset of the nonarchimedean places of $k$ for which $P_v\not\cong P_{0,v}$; the constant $c_1$ depends on $\G/k$ and $\Lambda_0$ but does not depend on the choice of $\Lambda$. If $\prod_{v\in T} e(P_v) < x/c_1$, then $\mu_S(\G_S/\Lambda) < x$. There exists a constant $\delta$ determined by the absolute type of $\G$ such that for every $v\in V_f$ and every parahoric subgroup $P_v\subset\G(k_v)$, $e(P_v) \le q_v^{\delta}$ (e.g., take $\delta = {\rm dim}(\G)$). This implies $$\prod_{v\in T} e(P_v) \le \prod_{v\in T} q_v^{\delta}.$$ Hence $\prod_{v\in T} q_v < (x/c_1)^{1/\delta}$ is sufficient for $\mu_S(\G/\Lambda) < x$. The number of variants for such sets $T$ is controlled via Proposition \[prop:NP\](i) (note that the field $k$ is fixed). We obtain that for large enough $x$ there are at least $$c_2(x/c_1)^{1/\delta} \ge x^A$$ variants for $T$, where the constant $A > 0$ is determined by $\delta$ and thus depends only on the absolute type of $\G$. It remains to recall that for each $T$ there exists a collection of parahoric subgroups $P = (P_v)_{v\in V_f}$ such that $P_v = P_{0,v}$ for $v\in V_f\setminus T$, $P_v \not\cong P_{0,v}$ for $v\in T$ and $P$ has $\cO$-maximal type. Each such collection defines a maximal arithmetic subgroup of $H$ of covolume less than $x$ and subgroups corresponding to different $T$’s are not conjugate. The number of maximal arithmetic subgroups obtained this way is at least $x^A$ with $A>0$, a constant depending only on the absolute type of $\G$. This proves the lower bound for $m^u_H(x)$. {#section-1} Let us point out that all the maximal arithmetic subgroups constructed in \[s51\] are commensurable. It is also possible to construct different commensurability classes which contain arithmetic subgroups of covolumes less than $x$. This may enlarge the constant $A$ in the asymptotic inequality but, as it follows from the first part of the proof and the conjecture on the number of isomorphism classes of fields with discriminant less than $x$, would hardly change the type of the asymptotic. Non-cocompact case. {#s52} ------------------- Let now $\Gamma_1$ be a non-uniform irreducible lattice in $H$ which exists by the assumption of part B of the theorem, and let $\G$ be a corresponding algebraic $k$-group. Arithmetic subgroups of $H$ which are induced from $\G(k)$ will be all non-cocompact (they are all commensurable with $\Gamma_1$). To provide a lower bound for $m^{nu}_H(x)$ it remains to repeat the argument of \[s51\] for the group $\G$. Note that in contrary to the cocompact case the existence of non-cocompact arithmetic subgroups in $H$ is in general not guaranteed by the condition that $H$ is isotypic (a counterexample was already mentioned in \[s22\]). Still the conditions under which such examples can be constructed are rather exceptional and in most of the cases isotypic groups contain both cocompact and non-cocompact arithmetic subgroups. The theorem is now proven. Corollaries, conjectures, remarks ================================= {#s61} There exists a constant $C_1$ which depends only on the type of almost simple factors of $H$ such that if $\Lambda$ is a principal arithmetic subgroup of $H$ and $\Gamma = N_H(\Lambda)$ has covolume less than $x$, then $[\Gamma:\Lambda] \le x^{C_1}$. By [@BP] (see Sect. \[1:BP\] for the notations and precise references): $$\label{e61} [\Gamma : \Lambda] \le n^{\epsilon\# S}\cdot 2h_l^{\epsilon'} n^{\epsilon a(k) + \epsilon' a(l) +\epsilon \# T} (\D_l/\D_k^{[l:k]})^{\epsilon''} \cdot \prod_{v\in V_f}\#\Xi_{\Theta_v}.$$ Now, since $\mu(H/\Gamma)\le x$, the group $\Gamma$ has to satisfy the conditions on the subgroups of covolume less than $x$ obtained in the proof of the upper bound of Theorem 1 (in Theorem 1 only maximal arithmetic subgroups are considered but the proof of the upper bound applies without a change to arbitrary principal arithmetic subgroups and their normalizers thus providing a somewhat stronger result to which we appeal here). We have: $$\begin{gathered} \D_l/\D_k^{[l:k]} = \D_{l/k} \le x^{c_1},\ \ a(k) \le c_2\log x {\rm\ \ (Sect.~\ref{s41})};\\ \# T \le \# T_1 \le c_{3}\log x {\rm\ (Sect.~\ref{s43})};\\ \{v\in V_f,\ \#\Xi_{\Theta_v}\neq 1\}\subset T_1\cup R\cup T_2 {\rm\ as\ for\ the\ rest\ of\ } v,\ P_v {\rm \ is\ special,\ so}\\ \#\{v\in V_f,\ \#\Xi_{\Theta_v}\neq 1\} \le \#(T_1\cup R\cup T_2) \le c_4\log x {\rm\ (Sect.~\ref{s44}).}\end{gathered}$$ Also recall that $\#\Xi_{\Theta_v} \le r+1$, $r$ is the absolute rank of $\G$; $h_l \le c^{[l:\Q]}\D_l\le x^{c_5}$ (see e.g. proof of Prop. \[prop:BP\]) and $a(l) \le 3a(k)$ (as $[l:k]\le 3$). Altogether these imply the proposition. {#s62} For some particular cases the bound in Corollary \[s61\] can be improved. Let us assume that the degrees of the fields of definition of the arithmetic subgroups are bounded: $$\label{e62} [k:\Q] \le d,$$ which is the case, for example, if we consider only non-uniform lattices in $H$. Assumption (\[e62\]) implies that the number $m$ of different prime ideals $\cP_1, \ldots \cP_m$ of $\cO_k$ such that ${\rm Norm} (\cP_1\ldots\cP_m) \le x$ is bounded by $c\log x/\log\log x$, $c=c(d)$ (instead of the bound $\log x$ which we used for the general case). Indeed, for $k = \Q$ it follows from the Prime Number Theorem, and the case of arbitrary $k$ of bounded degree can be easily reduced to the rational case. Therefore assumption (\[e62\]) implies that most of the terms in (\[e61\]) are $\le c^{\log x/\log\log x}$ with $c=c(d)$. What remains is $\D_{l/k} = \D_l/\D_k^{[l:k]}$ (for type $\Dn_r$, $r$ even) and $h_l$ (or $h_k$ if $l=k$). The former, in fact, appears in the formula as an upper bound for $2^{\# R}$ (see [@BP Sect. 5.5]), which again can be improved to $c^{\log x/\log\log x}$ by the same argument. What remains is the class number. Going back to [@BP Sect. 5 and Prop. 0.12], we see that what occurs in the formula is not $h_l$ but the order of the group $C_n(l)$, which consists of the elements of the class group $C(l)$ whose orders divide $n$ (as before $n$ is a constant determined by the type of $H$). Instead of using the trivial bound $\#C_n(l)\le h_l$, let us keep it as it is. We now come to the following formula: $$\label{e63} [\Gamma:\Lambda] \le c^{\log x/\log\log x} \#C_n(l),\ x\ge\mu(H/\Gamma).$$ If $n = p$ is a prime, let $\rho_p(l)$ denote the $p$-rank of $C(l)$. Then, clearly, $\#C_p(l) \le p^{\rho_p(l)}$, and in general for $n = p_1^{\alpha_1}\ldots p_m^{\alpha_m}$, $\#C_n(l) \le p_1^{\alpha_1\rho_{p_1}(l)} \ldots p_m^{\alpha_m\rho_{p_m}(l)}$. So we are interested in the upper bounds for $p$-ranks of the class groups. Apparently, even though this and related questions were much studied, there are very few results beyond the celebrated Gauss’ theorem which can be applied in our case. We have: - if $[l:\Q] = 2$, then $\rho_2(l) \le t_l - 1$ (by Gauss); - if $[k:\Q] = 2$ and $[l:k] = 2$, then $\rho_2(l) \le 2(t_l + t_k - 1)$ (by [@Cor Theorem 2]); where $t_k$ (resp. $t_l$) denotes the number of primes ramified in $k/\Q$ (resp. $l/k$). From this we obtain $$\label{e64} \#C_n(l) \le n^{c \log x/\log\log x},$$ if $n$ is a power of $2$ and $l$ is as in $(i)$ or $(ii)$. Similar results for other $n$ and other fields can be only conjectured: Even $\rho_3(k)$ for quadratic fields $k$ seems to be out of reach with the currently available methods. Still estimates (\[e63\]) and (\[e64\]) imply the following: {#s63} Let $H$ be a simple Lie group of type $\An_{2^\alpha-1}$ $(\alpha > 1)$, $\Bn_r$, $\Cn_r$, $\Dn_r$ $( r\neq 4 )$, $\En_7$, $\En_8$, $\Fn$ or $\Gn$. There exists a constant $C_2$ which depends only on the type of $H$ such that if $\Lambda$ is a non-cocompact principal arithmetic subgroup of $H$ and $\Gamma = N_H(\Lambda)$ has covolume less than $x$, then $[\Gamma:\Lambda] \le C_2^{\log x/\log\log x}$. Indeed, the assumption that $H$ has one of the given types implies that $n$ is a power of $2$ (see the definition of $n$ in Sect. \[1:BP\]). Since $H$ is simple and the arithmetic subgroup is non-compact, its field of definition $k$ is either $\Q$ or an imaginary quadratic extension of $\Q$, depending on $H$ being real or complex Lie group (see also Sect. \[s42\]). Finally, the type of $H$ is not $\Dn_4$ implies $[l:k]\le 2$. The corollary now follows from the discussion in \[s62\]. We expect similar estimates to be valid for the non-uniform lattices in other groups but we do not know how to prove it. {#section-2} Concerning the general case, let us point out that if the degrees of the fields are a priori not bounded then we can not expect a $\log x/\log\log x$–bound for the $p$-rank of the class group. An example of a sequence of fields $k_i$ for which $\rho_2(k_i)$ grows as $\log \D_{k_i}$ was constructed by F. Hajir (On the growth of $p$-class groups in $p$-class field towers, [*J. Algebra*]{} [**188**]{} (1997), 256–271), the fields $k_i$ in Hajir’s example form an infinite class field tower. This remark together with the previous estimates motivates the following question: [*Is the estimate in Corollary \[s61\] sharp, i.e. given a group $H$ is there a constant $C_0 = C_0(H) > 0$ such that there exists an infinite sequence of pairwise non-conjugate principal arithmetic subgroups $\Lambda_i$ in $H$ for which $[\Gamma_i:\Lambda_i] \ge \mu(H/\Gamma_i)^{C_0}$, where $\Gamma_i = N_H(\Lambda_i)$ and $\mu$ is a Haar measure on $H$?*]{} Corollaries \[s61\] and \[s63\] are important for [@BL] where we study the growth rate of the number of irreducible lattices in semi-simple Lie groups. {#s64} All along the line the groups of type $\An_1$ were excluded. The reason for this is that we can hardly hope to use the formula from Sect. \[2:BP\] combined with an analogue of Proposition \[prop:BP\] for this case even to prove a finiteness result. Still one can follow another method, also due to Borel, and use geometric bounds for the index of a principal arithmetic subgroup in a maximal arithmetic. This indeed allows to establish the finiteness of the number of arithmetic subgroups of bounded covolume in $\SL(2,\R)^a\times\SL(2,\C)^b$ [@B]. Now the problem is that the quantitative bounds which can be obtained this way are only exponential. We suppose that the true bounds should be similar to the general case (and conjecturally polynomial), although we do not know how to prove this conjecture and leave it as an open problem: [*Find the growth rate of the number of maximal arithmetic subgroups for the semi-simple Lie groups whose almost simple factors have type $\An_1$ or obtain a better than exponential upper bound for the growth.*]{} {#s65} Two conjectures were mentioned in the introduction: There exists an absolute constant $B$ such that for large enough $x$ the number of isomorphism classes of number fields with discriminants less than $x$ is at most $x^B$. Given a connected semi-simple Lie group $H$ without almost simple factors of type $\An_1$ and compact factors, there exists a constant $B_H > 0$ which depends only on the type of almost simple factors of $H$ such that for large enough $x$ the number of conjugacy classes of maximal irreducible arithmetic subgroups of $H$ of covolumes less than $x$ is at most $x^{B_H}$. We now prove: Conjectures $1$ and $2$ are equivalent. $'1\to 2'$ follows directly from the proof of the upper bounds in Theorem 1. $'2\to 1'.$ Assume that Conjecture 2 is true but Conjecture 1 is false, i.e. $m^u_H(x) + m^{nu}_H(x) \le x^{B_H}$ for every $x > x_0$, and for an arbitrary $C$ there exists $x>x_0$ such that $N(x) > x^C$. We shall need some additional assumption on $x_0$ which will become clear later but could be imposed from the beginning. So let us fix $C>1$ and let $x > x_0$ be such that $N(x) > x^C$. Let $N_{i,j}(x)$ denote the number of extensions of $\Q$ of discriminant less than $x$ which have precisely $i$ real and $j$ complex places. We have $$N(x) = \sum_{\substack{i=1,\ldots,n\\j=1,\ldots,m}}N_{i,j}(x).$$ The condition that the discriminants of the fields are less than $x$ implies by Minkowski’s theorem that the degrees of the extensions are bounded by $c\log x$ for an absolute constant $c$, and so the number of summands is less than $(c\log x)^2$. By Dirichlet’s box principle there exists a pair $(i,j)$ such that $N_{i,j}(x) > x^C/(c\log x)^2 \ge x^{C-1}$ (this inequality requires $(c\log x)^2 \le x$ which is true for large enough x and gives a first condition on $x_0$). Let $\mathcal K$ be the set of such number fields, $\#\mathcal K = N_{i,j}(x) > x^{C-1}$. Consider a simply connected semi-simple Lie group $H$ which has $i$ split real simple factors and $j$ complex simple factors all of the same type. For each $k\in\mathcal K$, let $\G/k$ be a simply connected, absolutely simple split group of the same absolute type as the simple factors of $H$ and defined over $k$. Let $P = (P_v)_{v\in V_f}$ be a coherent collection of parahoric subgroups of $\G$ all of which are hyperspecial (such a collection exists since $\G$ splits over $k$), and let $\Lambda$ be the principal arithmetic subgroup of $H$ defined by $P$. We have: - for $S=V_\infty(k)$, $\G_S\cong H$; - $\mu(H/\Lambda) = \D_k^{\d}\left(\prod_{i=1}^{r}\frac{m_i!}{(2\pi)^{m_i+1}}\right)^{[k:\Q]}\E(P)$  by Prasad’s formula. Using the orders of finite groups of Lie type the Euler product $\E(P)$ can be expressed as a product of the Dedekind zeta function of $k$ and certain Dirichlet $\rm L$-functions at the integers $m_i+1$, $m_i$ are the Lie exponents of $\G$ (see [@P Rem. 3.11]). Obvious inequalities $L(s,\chi)\le\zeta_k(s)$ and $\zeta_k(s)\le \zeta(s)^{[k:\Q]}$, for $s \ge 2$, imply that there exists a constant $c_1$ which depends only on the type of simple factors of $H$ and such that each zeta or $\rm L$-function in the product is bounded from above by $c_1^{[k:\Q]}$. Since $[k:\Q]\le c\log x$, we have $\E(P) \le (c_1^{c\log x})^r$. By definition of the set $\mathcal K$, $k\in \mathcal K$ implies $\D_k\le x$. Therefore we obtain $$\mu(H/\Lambda) \le x^{\d} c_2^{c\log x} (c_1^{c\log x})^r \le x^\delta,$$ where $\delta$ is $>1$ and depends only on the type of simple factors of $H$. The latter inequality may require that x is larger than certain value which depends on the type of simple factors of $H$ and gives us the second condition on $x_0$. Clearly, both conditions do not depend on $C$ and so could be imposed from the beginning. For each $k\in\mathcal K$ we have at least one maximal arithmetic subgroup of $H$ of covolume less then $x^\delta$. Now if we take $C = \delta B_H + 1$ we arrive to a contradiction with Conjecture 2 for $H$ and $x^\delta > x_0$. Let us remark that in the proof Conjecture 2 is used only for non-cocompact arithmetic subgroups of semi-simple groups $H$ which have simple factors of a fixed type, it then implies Conjecture 1 which in turn implies Conjecture 2 in the whole generality. It is possible to specify further the relation between two conjectures but we shall not go into details. What we would like to emphasize is that our result provides a new geometric interpretation for a classical number theoretic problem. An optimistic expectation would be that study of the distributions of lattices in semi-simple Lie groups can give a new insight on the number fields and their discriminants. Appendix {#appendix .unnumbered} ======== *by J. Ellenberg and A. Venkatesh* {#thmone} Let $N(X)$ denote the number of isomorphism classes of number fields with discriminant less than $X$. For every $\epsilon > 0$ there is a constant $C(\epsilon)$ such that $\log N(X) \leq C(\epsilon) (\log X)^{1+\epsilon}$, for every $X \geq 2$. In fact we prove the more precise upper bound that $$\log N(X) \leq C_6 \log X \exp(C_7 \sqrt{\log \log X})$$for absolute constants $C_6,C_7$. This theorem (almost) follows from [@EV Theorem 1.1], the only point being to control the dependence of implicit constants on the degree of the number field. We refer to [@EV] for further information and for some motivational comments about the method. In the proof $C_1, C_2, \dots$ will denote certain [*absolute*]{} constants. {#section-3} Let $K$ be an extension of $\Q$ of degree $d \geq 200$. Denote by $\Sigma(K)$ the set of embeddings of $K$ into $\C$ ($\#\Sigma(K) = d$), and by $\overline{\Sigma}(K)$ a set of representatives for $\Sigma(K)$ modulo complex conjugation (in the notations of the paper $\overline{\Sigma}(K) = V_\infty (K)$). We regard the ring of integers ${\mathcal{O}}_K$ as a lattice in $K \otimes_{\Q} \R = \prod_{\sigma \in \Sigma(K)} K_{\sigma}$. We endow the real vector space $K \otimes_{\Q} \R$ with the supremum norm, i.e. $\|(x_\sigma)\| = \sup_{\sigma} |x_{\sigma}|$. Here $|\cdot|$ denotes the standard absolute value on $\C$. In particular, we obtain a “norm” on ${\mathcal{O}}_K$ by restriction. Explicitly, for $z \in {\mathcal{O}}_K$, we have $\|z\| = \sup_{\sigma \in \Sigma(K)} |\sigma(z)|$. We denote by $\M_d(\Z)$ (resp. $\M_d(\Q)$) the algebra of $d$ by $d$ matrices over $\Z$ (resp. $\Q$). By the “trace form” we mean the pairing $(x,y) \mapsto {\mathrm{Tr}_{K/\Q}}(xy)$. It is a symmetric nondegenerate $\Q$-bilinear pairing on $K^2$. Let $s$ be a positive integer which we shall specify later. We denote by ${\mathbf{y}}= (y_1, y_2,\ldots,y_s)$ an ordered $s$-tuple of elements of ${\mathcal{O}}_K$ and write $\|{\mathbf{y}}\| :=$ $\max(\|y_1\|,\ldots, \|y_s\|)$. For ${\mathbf{y}}= (y_1, \ldots, y_s) \in {\mathcal{O}}_K^s$ and $l \geq 1$, we shall set $$\begin{gathered} S(l) = \{(k_1,\ldots,k_s) \in \Z^s:k_1 + \ldots+ k_s \leq l,\ k_1, \ldots, k_s \geq 0\}, \\ S({\mathbf{y}}, l) = \{y_1^{k_1} y_2^{k_2} \ldots y_s^{k_s} : (k_1, \ldots, k_s) \in S(l)\} \subset {\mathcal{O}}_K.\end{gathered}$$ If $S$ is a subset of $S(l)$ we denote by $S({\mathbf{y}})$ the set $\{y_1^{k_1} y_2^{k_2} \ldots y_s^{k_s} : (k_1, \ldots, k_s) \in S$.} {#section-4} \[lem:span\] Let $S$ be a subset of $S(l)$ such that $S({\mathbf{y}})$ spans a $\Q$-linear subspace of $K$ with dimension strictly greater than $d/2$. Let $S+S$ be the set of sums of two elements of $S$. Then $(S+S)({\mathbf{y}})$ spans $K$ over $\Q$. (cf. [@EV Lemma 2.1].) Suppose that there existed $z \in K$ which was perpendicular, w.r.t. the trace form, to the $\Q$-span of $(S+S)({\mathbf{y}}).$ Since $(S+S)({\mathbf{y}})$ consists precisely of all products $\alpha \beta$, with $\alpha, \beta \in S({\mathbf{y}})$, it follows that $$\label{eqn:bart}{\mathrm{Tr}_{K/\Q}}(z \alpha \beta) = 0, \ \ (\alpha, \beta \in S({\mathbf{y}})).$$ Call $W \subset K$ the $\Q$-linear span of $S({\mathbf{y}})$. Then (\[eqn:bart\]) implies that $z W$ is perpendicular to $W$ w.r.t. the trace form, contradicting $\dim(W) > d/2$. {#section-5} \[lem:traces\] Let $\mathcal{C} \subset {\mathcal{O}}_K$ be a finite subset containing $1$ and generating $K$ as a field over $\Q$. Let $z_1, z_2, \dots, z_d$ be a $\Q$-linear basis for $K$. For each $u \in \mathcal{C}$, let $ M(u) = \left(\mathrm{Tr}_{K/\Q}(uz_i z_j)\right)_{1 \leq i,j\leq d} \in \M_d(\mathbb{Q}).$ Then the $\Q$-subalgebra of $\M_d(\Q)$ generated by $M(u) M(1)^{-1}$, as $u$ ranges over $\mathcal{C}$, is isomorphic to $K$. (cf. [@EV Lemma 2.2].) In fact, $M(u) M(1)^{-1}$ gives the matrix of “multiplication by $u$,” in the basis $\{z_i\}$. {#section-6} \[lem:redtheory\] There is an absolute constant $C_1 \in \mathbb{R}$ such that, for any $K$ as above, there exists a basis $\gamma_1, \gamma_2, \dots, \gamma_d$ for ${\mathcal{O}}_K$ over $\Z$ such that $$\label{eqn:red} \|\gamma_j\| \leq \|\gamma_{j+1}\|, \ \ \prod_{i=1}^{d} \|\gamma_d\| \leq {\mathcal{D}_K}^{1/2} C_1^{d}, \ \ \|\gamma_i\| \leq (C_1^d {\mathcal{D}_K}^{1/2})^{\frac{1}{d-i}} \ (i<d).$$ (${\mathcal{D}_K}$ denotes the absolute value of the discriminant of $K$.) This is reduction theory (cf. [@EV Prop. 2.5]). The final statement of (\[eqn:red\]) follows from the preceding statements, in view of the fact that $\|\gamma_j\| \geq 1$ for each $j$. {#section-7} Let $r,l$ be integers such that $d/2 < r \leq |S(l)| = \binom{l+s}{s}$. \[lem:dimcount\] Suppose $W \subset K$ is a $\Q$-linear subspace of dimension $r$, and let $S \subset S(l)$ be a subset of size $r$. Then there exists ${\mathbf{y}}= (y_1, y_2, \ldots y_s) \in W^s$ such that the elements of $S({\mathbf{y}})$ are $\Q$-linearly independent. This is precisely [@EV Lemma 2.3]. {#section-8} \[lem:lambda\] Let $\Lambda = \Z \gamma_1 + \Z \gamma_2 + \dots + \Z \gamma_{r}$ and let $S \subset S(l)$ be a subset of size $r$. Then there is ${\mathbf{y}}= (y_1, y_2, \ldots, y_s) \in \Lambda^s$ such that the elements of $S({\mathbf{y}})$ are linearly independent over $\Q$, and $\|{\mathbf{y}}\| \leq r^2 l (C_1^d {\mathcal{D}_K}^{1/2})^{\frac{1}{d-r}}$. Considering $\Lambda^s$ as a $\Z$-module of rank $rs$, the proof of [@EV Lemma 2.3] shows that there is a polynomial $F$ of degree at most $rl$ in the $rs$ variables so that the elements of $S({\mathbf{y}})$ are linearly independent over $\Q$ whenever $F({\mathbf{y}}) \neq 0$. Lemma 2.4 of [@EV] then shows that we can choose such a ${\mathbf{y}}$ whose coefficients are at most $(1/2)(rl+1) \leq rl$. It follows that y\_i r\^2 l (C\_1\^d [\_K]{}\^[1/2]{})\^ for $i = 1,2,\ldots s$. {#section-9} \[lem:final\] The number of number fields with degree $d \geq 200$ and discriminant of absolute value at most $X$ is at most (C\_3 d)\^[d (C\_4 )]{} X\^[(C\_5 )]{}. Fix once and for all a total ordering of $S(2l)$. We denote the order relation as $(k_1,\ldots,k_s) \prec (k_1', \ldots, k_s')$. Choose $S \subset S(l)$ of cardinality $r$ as above. Let $K$ have degree $d$ over $\Q$ and satisfy ${\mathcal{D}_K}< X$. Chose ${\mathbf{y}}$ as in Lemma \[lem:lambda\]. By Lemma \[lem:span\], $S(2l)({\mathbf{y}})$ spans $K$ over $\Q$. It follows that there exists a subset $\Pi \subset S(2l)$ of size $d$ such that $\{z_1, \dots, z_d\} := \{y_1^{k_1} y_2^{k_2} \ldots y_s^{k_s}: (k_1, k_2, \ldots, k_s) \in \Pi\}$ forms a $\Q$-basis for $K$, and such that the ordering $z_1,\ldots, z_d$ conforms with the specified ordering on $\Pi \subset S(2l)$. We apply Lemma \[lem:traces\] to $\{z_1, \dots, z_d\}$ and $\mathcal{C} = (1, y_1, y_2, \ldots, y_s)$. Then each product $u z_i z_j \, (u \in \mathcal{C}, 1 \leq i,j \leq d)$ is contained in $S(4l+1)$. Put ${\mathbf{A}}= ({\mathrm{Tr}_{K/\Q}}(y_1^{k_1} y_2^{k_2} \ldots y_s^{k_s}))_{(k_1, k_2, \ldots k_s) \in S(4l+1)}.$ For each $K$, the collection of matrices $M(u)$ is determined by ${\mathbf{A}}$ and $\Pi$. Since $|{\mathrm{Tr}_{K/\Q}}(z)| \leq d \|{\mathbf{y}}\|^{4l+1}$ for any $z \in S({\mathbf{y}},4l+1)$, the number of possibilities for ${\mathbf{A}}$ is at most $(d \|{\mathbf{y}}\|^{4l+1})^{|S(4l+1)|}$; since $\Pi$ is a subset of $|S(2l)|$, the number of possibilities for $\Pi$ is at most $2^{|S(2l)|}$. Lemma \[lem:traces\] now yields that the number of possibilities for the isomorphism class of $K$ is at most $2^{|S(2l)|} (d \|{\mathbf{y}}\|^{4l+1})^{|S(4l+1)|}$. By our bound on $\|{\mathbf{y}}\|$ we now have that the number of possibilities for $K$ is at most $$2^{|S(2l)|} (d (r^2 l (C_1^d {\mathcal{D}_K}^{1/2})^{\frac{1}{d-r}})^{4l + 1})^{|S(4l+1)|}. \label{eq:bigineq}$$ Note that $|S(4l+1)| = \binom{s + 4l+1}{s}$. Now, just as in the paragraph following (2.6) of [@EV], we choose $s$ to be the greatest integer less than $\sqrt{\log{d}}$ and $l$ to be the least integer greater than $(ds!)^{1/s}$. Note that $l < \exp(C_2 \sqrt{\log d})$. Now $|S(l)| = \binom{s+l}{s}$ is at least $d$, so we may choose $r$ between $d/2$ and $3d/4$. In particular, $r^2 l < d^3.$ Also, $\binom{s+4l+1}{s}$ is at most $10^s d$ and $|S(2l)| = \binom{s+2l}{s} \leq 6^s d$. Finally, $s < 2 \sqrt{\log d}$. Substituting these values into we get that the number of possible $K$ is at most 2\^[6\^s d]{} (d (d\^3 (C\_1\^d X\^[1/2]{})\^[4/d]{})\^[5 (C\_2 ]{})\^[10\^s d]{} which is in turn at most (C\_3 d)\^[d (C\_4 )]{} X\^[(C\_5 )]{}. {#section-10} There are absolute constants $C_6,C_7$ with N(X) C\_6 X (C\_7 ). By Minkowski’s discriminant bound, there is an absolute constant $C_6 > 1$ such that ${\mathcal{D}_K}> C_6^{[K:\Q]}$ for any extension $K/\Q$; so we may take $d$ to be bounded by a constant multiple of $\log X$. From Lemma \[lem:final\] it now follows that the logarithm of the number of extensions $K/\Q$ with ${\mathcal{D}_K}< X$ and $[K:\Q] \geq 200$ is bounded by $C_6 \log X \exp(C_7 \sqrt{\log \log X})$. 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--- abstract: | The non-classical, nonmonotonic inference relation associated with the answer set semantics for logic programs gives rise to a relationship of [*strong equivalence*]{} between logical programs that can be verified in 3-valued Gödel logic, [**G3**]{}, the strongest non-classical intermediate propositional logic [@lpv]. In this paper we will show that [**KC**]{} (the logic obtained by adding axiom $\neg A\vee\neg\neg A$ to intuitionistic logic), is the weakest intermediate logic for which strongly equivalent logic programs, in a language allowing negations, are logically equivalent. KEY WORDS: answer set semantics, strongly equivalent programs, propositional intermediate logics. author: - | DICK DE JONGH and LEX HENDRIKS\ Institute of Logic, Language and Computation, University of Amsterdam\ Plantage Muidergracht 24\ 1018 TV Amsterdam\ The Netherlands\ , title: Characterization of Strongly Equivalent Logic Programs in Intermediate Logics --- \[theorem\][Corollary]{} \[theorem\][Definition]{} \[theorem\][Fact]{} \[theorem\][Lemma]{} =.25em Introduction ============ In logic programming certain fragments of first-order logic are given a computational meaning. The first and best known example of such a fragment is that of the [*Horn clauses*]{}, quantifier-free formulas of the form $B$ or ${\mbox{\raisebox{.4ex}{$\bigwedge$}}}A_i{\to}B$, where the $A_i$ and $B$ are atomic formulas, the basis for the programming language Prolog [@kow]. A logic program is a finite set of such formulas (called [*rules*]{}). A logic program $\Pi$ in the language $L$ can be interpreted in first-order logic as a set of sentences in $L$, by taking the universal closure $\forall\vec{x} R$ of each of the rules $R$ in $\Pi$. An alternative method of eliminating the free variables in the rules, is the substitution of [*ground terms*]{}, i.e.terms built from the constants and functions in $L$. The set of ground terms of $L$, the [*Herbrand Universe*]{} of $L$, may be used as the domain for models of theories in $L$, the [*Herbrand models*]{}. Replacing each $R$ by the set of all possible substitutions with ground terms, $\Pi$ is now replaced by a set of quantifier-free sentences in $\Pi^H$ in $L$. The rationale behind this is given by the following well-known fact (for a proof see for example [@doets]). Let $\Pi$ be a set of universal sentences. The following are equivalent: 1. $\Pi$ has a model 2. $\Pi$ has a Herbrand model 3. $\Pi^H$ is satisfiable in propositional logic In this paper we are interested in logic programs as (possibly infinite) sets of propositional formulas. In general, our program rules may include negations and disjunctions and hence the results in this paper extend to disjunctive logic programming as well. We will denote a fragment of the language of propositional logic by enumerating between square brackets the connectives and constants that are allowed in formulas of the fragment. So $[\wedge,\neg]$ will denote the set of formulas built from atomic formulas, using only conjunction and negation. And $[\wedge, \vee,\bot,\top]$ will be the fragment of formulas built with conjunction and disjunction from atomic formulas and the constants $\bot$ and $\top$. As long as we remain in classical propositional logic, a model $w$ for a program $\Pi$ can be identified by the set of atoms $X$ valid in $w$, in our notation $w=\langle X\rangle$. In logic programming one is interested in constructing a most ’general’ model for a program $\Pi$ (in the language $L$ based on the Herbrand Universe of $L$). Such a ’most general’ model should not identify terms, for example, unless such an identity is implied by $\Pi$. For the propositional analogue of programs with Horn clauses as rules, the [*minimal Herbrand model*]{} is such a ’most general’ model. In propositional logic an obvious candidate for the most general model of $\Pi$ is the intersection of all sets of atoms $X$ such that $\langle X\rangle\models\Pi$ (i.e., $\Pi$ is true in the valuation that makes exactly the atoms in $X$ true). For programs with Horn clauses as rules this works fine as can be seen from the following fact. Let $\Pi\subseteq \{A{\to}B\mid A, B\in [\wedge,\bot,\top]\}$ and $X=\bigcap\{Y\mid \langle Y\rangle\models\Pi\}$. Then $\langle X\rangle\models\Pi$. Although the above fact introduces a fragment slightly richer than the language of Horn clauses, it is still easy to prove. A simple example, like $p\vee q$, shows that there may not be a unique minimal model for $\Pi$ if disjunctions are allowed in the rules of $\Pi$. And even more serious problems arise for the notion of [*most general model*]{}, when negations in the head or body of rules of $\Pi$ are allowed. Several solutions have been proposed for the semantics of logic programs with disjunctions and negations. The answer set (or ’stable model’) semantics we use in this paper was introduced by Gelfond and Lifschitz in . The main idea[^1] is that $X$ is an [*answer set*]{} of $\Pi$ if $\langle X\rangle\models\Pi^X$ where $\Pi^X$ is the program that arises if we replace all negations $\neg A$ in $\Pi$ by either $\bot$ or $\top$ according to whether $\langle X\rangle\models A$ or not, and that $X$ is minimal in this respect. The behavior of negation in this semantics resembles that of the [*negation as failure*]{} (or [*negation by default*]{}) in many Prolog implementations. We define a logic [**L**]{} to be [*sound for stable models*]{} or [*sound for stable inference*]{} if, whenever $\Pi\vdash_{\bf L} A$, then the answer sets for $\Pi\cup\{A\}$ are the same as for $\Pi$. Classical propositional logic, [**CPL**]{}, turns out to be too strong to have this property. For example, the program $\neg\neg p$ does not have answer sets at all, whereas $\neg\neg p\vdash_{\bf CPL} p$ and $\{\neg\neg p, p\}$ has $\{p\}$ as its answer set. Logics weaker than classical logic [**CPL**]{} can give a solution to this problem, in particular [*intermediate logics*]{}, i.e. logics derived from intuitionistic propositional logic [**IPL**]{} by adding axioms that are valid in [**CPL**]{}, do provide a sound basis for stable inference. David Pearce used in [@pearce] 3-valued Gödel logic [**G3**]{} to prove logic programs strongly equivalent. The notion of [*strong equivalence*]{} of logic programs was introduced in [@lpv]. Logic programs $\Pi_1$ and $\Pi_2$ are said to be [*strongly equivalent*]{} in the sense of stable model semantics if for every logic program $\Pi$, $\Pi_1\cup\Pi$ and $\Pi_2\cup\Pi$ have the same answer sets. For programs in the language $\{A{\to}B\mid A, B\in [\wedge,\vee,\neg]\}$ it was shown in [@lpv] that $\Pi_1$ and $\Pi_2$ are strongly equivalent precisely if they are equivalent in [**G3**]{}. As pointed out in [@lpv], the notion of strong equivalence may be of interest in showing that a part of a program can be replaced by a simpler equivalent part, without affecting the behavior of the whole program or its extensions. Replacing nonclassical, nonmonotonic stable inference by a well-understood monotonic intermediate logic like [**G3**]{}, will simplify the verification of such strong equivalence between logic programs. The logic [**G3**]{}, also known as the Smetanich logic of [*here-and-there*]{}, is the intermediate logic whose models are based on the partially ordered frame $\langle h, t\rangle$ with $h\leq t$ ([@pearce], [@cz], and see section \[prel\]). In this paper we will consider the problem of the ’weakest’ intermediate logic [**L**]{} for which provable equivalence is the same as strong equivalence in the sense of stable models. In other words, which [**L**]{} has the property that logic programs $\Pi_1\cup\Pi$ and $\Pi_2\cup\Pi$ have the same answer sets for all $\Pi$ iff $\Pi_1$ and $\Pi_2$ are equivalent in [**L**]{}, but this property does not hold for any strictly weaker logic. Note that this will depend on the language one allows for the programs. Our main result is that for programs in the language $\{A{\to}B\mid A, B\in [\wedge,\vee,\neg]\}$ the weakest intermediate logic for which equivalence of programs equals strong equivalence on stable models is the logic [**KC**]{}, axiomatized by adding axiom $\neg A\vee\neg\neg A$ to [**IPL**]{}. This logic (also known as [*Jankov’s logic*]{} or the [*logic of the weak law of excluded middle*]{}) was introduced in [@ja]. Our main result remains true if we restrict the language of programs to $\{A{\to}B\mid A, B\in [\vee,\neg]\}$ or $\{A{\to}B\mid A, B\in [\wedge,\neg]\}$. But in the language $\{A{\to}B\mid A, B\in [\wedge,\vee,\bot,\top]\}$ strong equivalence of programs coincides with equivalence in [**IPL**]{} itself. Let us note that [**G3**]{} is easier to implement than [**KC**]{}. This is witnessed by the fact that satisfiability in [**G3**]{} is $NP$ and satisfiability in [**KC**]{} is $PSPACE$. However, in many particular cases it is easy to see that certain formulas are not derivable in [**KC**]{} whereas this is a complex matter for [**G3**]{}. This point also shows up when one wants to prove that the disjunctive rule $p\vee q$ is not strongly equivalent to any nondisjunctive rule. It is not clear how this could be done using the characterization of strong equivalence as provable equivalence in [**G3**]{} since $\vee$ is definable from the other connectives in [**G3**]{}. But with our characterization of strong equivalence as provable equivalence in [**KC**]{} it is a rather simple corollary which we will prove at the end of the paper. Acknowledgements {#acknowledgements .unnumbered} ================ We would like to thank David Pearce who challenged us to find the weakest intermediate logic for which equivalent programs are strongly equivalent. We are also obliged to him and V. Lifschitz for diligently explaining some of the subtleties in answer set semantics. Finally the anonymous referees should be acknowledged for their corrections and valuable suggestions for improving the presentation. Preliminaries {#prel} ============= In the language of propositional logic formulas are built from atoms (plus possibly constants $\top$ and $\bot$) using $\wedge,\vee,{\to}$, $\neg$. Fragments of propositional logic are obtained by restricting the use of atoms, constants and/or the use of the connectives. The Kripke semantics in this paper is fairly standard. A [*Kripke frame*]{} $\langle W, \leq\rangle$ is a set of worlds (or nodes) $W$ with a partial ordering $\leq$. A [*model*]{} $M$ will be such a frame together with a function $\mbox{atom}(w)$ mapping each world $w\in W$ to a set of atomic formulas, such that if $w\leq v$ then $\mbox{atom}(w)\subseteq\mbox{atom}(v)$. Note that Kripke models are not necessarily rooted. A [*maximal world*]{} $w\in W$ (i.e. such that for all $v\in W$, $w\leq v$ implies $v=w$) will be called a [*terminal node*]{} of $W$ (or $M$). For the language of propositional logic the interpretation in a world $w$ of a model $M$ (by which we mean $w\in W$ if $M=\langle W,\leq\rangle$) is given by the usual rules. $w\models_M A$ ($A$ is true in $M$ at $w$) is defined by recursion on the length of $A$. 1. $w\models_M p{\mbox{$\quad\Leftrightarrow\quad$}}p\in {atom}(w)$, 2. $w\models_M A\wedge B{\mbox{$\quad\Leftrightarrow\quad$}}w\models_M A$ and $w\models_M B$, 3. $w\models_M A\vee B{\mbox{$\quad\Leftrightarrow\quad$}}w\models_M A$ or $w\models_M B$, 4. $w\models_M A{\to}B{\mbox{$\quad\Leftrightarrow\quad$}}\forall v\geq w\, (v\not\models_M A$ or $v\models_M B)$. 5. $w\models_M\top$, 6. $w\not\models_M\bot$, If it is clear from the context which model is meant in $w\models_M A$, we will omit the subscript (and simply write $w\models A$). If $T$ is a set of formulas and $w$ a world in a Kripke model $M$, then $w\models T$ iff $w\models A$ for all $A\in T$. We will write $M\models A$ (or $M\models T$) if for all $w$ in $M$ it is true that $w\models A$ (or $w\models T$). A well-known fact about Kripke models is that if $w\models A$ and $w\leq v$ then $v\models A$, which is true for atomic formulas $A$ by the monotonicity of the function $\mbox{\em atom}$ but extends to all formulas $A$. Intuitionistic propositional logic ([**IPL**]{}) is sound and complete for the set of finite Kripke models. Thus, $\vdash_{\bf IPL} A$ iff $M\models A$ for each finite $M$. Classical propositional logic ([**CPL**]{}) is sound and complete for the set of Kripke models where the partial ordering is identity. Hence, a classical model consists of a world $w$ that may be identified with $\mbox{\em atom}(w)$, the set of atomic formulas valid in $w$. If $\mbox{\em atom}(w)=X$ we will denote $w\models A$ by $\langle X\rangle\models A$. In the case where $w$ is a node in a Kripke model $M$, $\langle w\rangle$ will denote the classical world with the same set of atoms as the node $w$ (so $\langle w\rangle = \langle\mbox{\em atom}(w)\rangle$). An [*intermediate logic*]{} is a logic obtained by adding formulas valid in [**CPL**]{}, to [**IPL**]{} as schemes. 3-valued Gödel logic [**G3**]{} can be defined as the logic sound and complete for models based on the frame $\langle\{ h, t\},\leq\rangle$ with $h\leq t$ (in a short notation: $\langle h, t\rangle$). We will call these models [*here-and-there models*]{}. [**G3**]{} traditionally is introduced by giving the (3-valued) truth tables for the connectives. The three values correspond in the context of Kripke models of course to the three sets of nodes that a formula can be true in: $\emptyset,\{t\},\{h,t\}$. Alternatively [**G3**]{} may be obtained by adding e.g. one of the following axioms to [**IPL**]{}: 1. \[luk\] $(\neg A{\to}B){\to}(((B{\to}A){\to}B){\to}B)$ 2. $(A\leftrightarrow B)\vee(A\leftrightarrow C)\vee(A\leftrightarrow D)\vee(B\leftrightarrow C) \vee(B\leftrightarrow D)\vee(C\leftrightarrow D)$ 3. \[hos\] $A\vee(A{\to}B)\vee\neg B$ 4. $(((A{\to}(((B{\to}C){\to}B){\to}B)){\to}A){\to}A)\wedge(\neg A \vee\neg\neg A)$ Łukasiewicz [@lu] seems to have been the first to axiomatize [**G3**]{}, using axiom \[luk\]. The second axiom is Gödel’s [@go] formula expressing that there are only three truth values. The third is a simplified version of Hosoi’s axiom $A\vee\neg A\vee(A{\to}B)\vee(B{\to}C)$ [@ho]. The last axiom is a combination of the iterated Peirce formula (the substitution of the Peirce formula $((B{\to}C){\to}B){\to}B$ for $B$ in $((A{\to}B){\to}A){\to}A$ ) and the axiom for [**KC**]{} (see below), together expressing that the logic will be complete with respect to frames of maximal depth 2 and a single terminal node. Clearly $\neg A\vee\neg\neg A$ can also easily be derived from \[hos\] (take $B=\neg A$ and use that $A{\to}\neg A$ and $\neg A$ are equivalent and $A\vee\neg\neg A$ is equivalent to $\neg\neg A$) or the other axioms. For more details see [@cz]. We will use the notation $\langle Y, X\rangle$ for the Kripke model $\langle h, t\rangle$, with $X=\mbox{\em atom}(t)$ and $Y=\mbox{\em atom}(h)$. The intermediate logic [**KC**]{} is given by the rules and axioms of intuitionistic propositional logic [**IPL**]{} plus the axiom $\neg A\vee \neg\neg A$. [**KC**]{} is sound and complete with respect to the finite (rooted) Kripke models with a single terminal node ([@ja], see [@cz]). The Kripke models of [**G3**]{} are a special kind of [**KC**]{}-Kripke models, hence by the soundness and completeness theorems for [**G3**]{} and [**KC**]{}, provability (from a set of formulas $T$) in [**KC**]{} implies provability (from $T$) in [**G3**]{}: $T\vdash_{\bf KC}~A$ implies $T\vdash_{\bf G3} A$. Answer sets and stable models ============================= In this section we recall some of the definitions and results from  and [@ltt] for programs in the language $\{A{\to}B\mid A, B\in [\wedge,\vee,\neg]\}$. As in [@lpv] our language allows more complex rules than the usual $A_1\wedge\ldots\wedge A_m\wedge\neg A_{m+1}\ldots\wedge\neg A_n{\to}B_1\vee\ldots\vee B_k$ (conjunctions, disjunctions and negations can be nested). We will try to state and prove the results for as large a class of formulas as possible. We will start with some results for programs in the language $\{A{\to}B\mid A,B\in [\wedge, \vee, \bot,\top]\}$. Examples of rules in this language are: $p$ (i.e. $\top{\to}p$), $p\wedge q\vee r{\to}p\wedge r$ and $p{\to}\bot$. Note that also negations of formulas $A$ in $[\wedge, \vee]$ are allowed, as rules, if $\neg A$ is written as $A{\to}\bot$. \[as1\] Let $\Pi$ be a program in $\{A{\to}B\mid A,B\in [\wedge, \vee, \bot,\top]\}$. A set of atoms $X$ is an [*answer set*]{} of $\Pi$ if for all $Y\subseteq X$ it is true that $\langle Y\rangle\models\Pi {\mbox{$\quad\Leftrightarrow\quad$}}Y=X$. A program in $\{A{\to}B\mid A,B\in [\wedge, \vee, \bot, \top]\}$ may have several answer sets (like for example the program $p\vee q$) and (logically) different programs may have the same answer sets (for example $p{\to}q$ and $q{\to}p$ both have the empty set as their only answer set). Programs $\Pi_1$ and $\Pi_2$ in $L$ are called [*strongly equivalent*]{} (in $L$) if for every program $\Pi$ in $L$ the programs $\Pi_1\cup\Pi$ and $\Pi_2\cup\Pi$ have the same answer sets. Logic programs in $\{A{\to}B\mid A,B\in [\wedge, \vee, \bot, \top]\}$ are strongly equivalent if and only if (viewed as sets of propositional formulas) they are equivalent in classical propositional logic. \[t1\] Let $L=\{A{\to}B\mid A,B\in [\wedge, \vee, \bot,\top]\}$ and let $\Pi_1$ and $\Pi_2$ be programs in $L$. $\Pi_1$ and $\Pi_2$ are strongly equivalent if they are equivalent in [**CPL**]{}, i.e. $\Pi_1\equiv_{\bf CPL} \Pi_2$. First assume $\Pi_1\equiv_{\bf CPL} \Pi_2$ and let $X$ be an answer set for $\Pi_1\cup\Pi$. Then for $Y\subseteq X$ with $\langle Y\rangle\models \Pi_2\cup\Pi$ we may infer that $\langle Y\rangle\models \Pi_1\cup\Pi$ and hence $Y=X$. Which proves $X$ is also an answer set for $\Pi_2\cup\Pi$. Likewise, every answer set for $\Pi_2\cup\Pi$ can be proven to be an answer set for $\Pi_1\cup\Pi$ and hence $\Pi_1$ and $\Pi_2$ are strongly equivalent. For the other direction, let $\langle X\rangle\models\Pi_1$ and let $\Pi=X$. Observe that $X$ is an answer set for $\Pi_1\cup\Pi$ and, as $\Pi_2$ is strongly equivalent to $\Pi_1$, $X$ is also an answer set for $\Pi_2\cup\Pi$. Which proves $\langle X\rangle\models\Pi_2$. Likewise, every model of $\Pi_2$ will be a model of $\Pi_1$, which proves $\Pi_1\equiv_{\bf CPL}\Pi_2$. For a more general treatment of negations in logic programs the following [*reduction*]{} of a program was introduced in ,[@ltt]. \[ax\] Let $X$ be a set of atomic formulas and $A$ a formula. $A^X$ is defined recursively as: $\begin{array}{lll} p^X & = p & \mbox{if}\; p\;\mbox{is atomic}\\ (A\circ B)^X & = A^X\circ B^X & \mbox{for}\;\circ\in\{\wedge,\vee,{\to}\}\\ (\neg A)^X & = \left\{\begin{array}{ll} \bot & \mbox{if}\; \langle X\rangle\models A\\ \top & \mbox{otherwise}\\ \end{array}\right. \end{array}$ For a program $\Pi\subseteq\{A{\to}B\mid A, B\in [\wedge,\vee,\neg]\}$ the reduction $\Pi^X$ will be a program in $\{A{\to}B\mid A,B\in [\wedge, \vee,\bot,\top]\}$. \[as2\] Let $\Pi\subseteq \{A{\to}B\mid A, B\in [\wedge,\vee,\neg]\}$. A set $X$ of atomic formulas is called an [*answer set*]{} for $\Pi$ if for all $Y\subseteq X$ we have $\langle Y\rangle\models\Pi^X{\mbox{$\quad\Leftrightarrow\quad$}}Y=X$. If we restrict the language to $\{A{\to}B\mid A, B\in [\wedge,\vee,\bot,\top]\}$, we have $\Pi^X=\Pi$ and definition \[as2\] coincides with definition \[as1\]. To find a theorem similar to theorem \[t1\] for strong equivalence in $\{A{\to}B\mid A,B\in [\wedge, \vee, \neg]\}$, we will use the characterization of answer sets in [@pearce], based on Kripke models for the intermediate logic [**G3**]{}. The following lemma is not only useful in this case but also will have applications in the next section. Recall that for a world $w$ in a Kripke model $M$, $\langle w\rangle$ denotes the classical model $\langle\mbox{\em atom}(w)\rangle$. \[wv\] Let $w$ be a node in a Kripke model $M$. For $A, B\in [\wedge,\vee,\bot,\top]$, $w\models A$ iff $\langle w\rangle\models A$ and, if $w\models A{\to}B$ then $\langle w\rangle\models A{\to}B$. First we prove for $A\in [\wedge,\vee,\bot,\top]$ that $w\models A$ iff $\langle w\rangle\models A$. If $A$ is atomic, $\bot$ or $\top$, this is obvious. By induction on the complexity of $A$, the proof for the cases of conjunction and disjunction is straightforward. For the second part of the proof, let both $A$ and $B$ be in $[\wedge,\vee,\bot,\top]$. If $\langle w\rangle\not\models A{\to}B$ then $\langle w\rangle\models A$ and $\langle w\rangle \not\models B$. By the first part of the lemma then $w\not\models A{\to}B$. As an immediate consequence of lemma \[wv\] we have the following lemma for models of [**G3**]{}. \[01\] For $A, B\in [\wedge,\vee,\bot,\top]$, $\langle Y, X\rangle\models A{\to}B$ iff $\langle X\rangle\models A{\to}B$ and $\langle Y\rangle\models A{\to}B$ Assume $\langle Y, X\rangle\models A{\to}B$. By lemma \[wv\] we may conclude that $\langle X\rangle\models A{\to}B$ and $\langle Y\rangle\models A{\to}B$. For the other direction, assume $\langle X\rangle\models A{\to}B$ and $\langle Y\rangle\models A{\to}B$. If $\langle Y, X\rangle\models A$, then $\langle Y\rangle\models A$ and hence $\langle Y\rangle\models B$, which implies $\langle Y, X\rangle\models B$, so $\langle Y, X\rangle\models A{\to}B$. On the other hand if $\langle Y, X\rangle\not\models A$ then $\langle X\rangle\models A{\to}B$ immediately implies $\langle Y, X\rangle\models A{\to}B$. The next lemma is true for all propositional formulas. \[x3\] For all sets of atoms $X$ and $Y$ such that $Y\subseteq X$ it is true that $\langle Y, X\rangle\models A {\mbox{$\quad\Leftrightarrow\quad$}}\langle Y, X\rangle\models A^X$. Observe that $\langle Y,X\rangle\models\neg A {\mbox{$\quad\Leftrightarrow\quad$}}\langle X\rangle\not\models A$. As a consequence we have $\langle Y,X\rangle\models\neg A {\mbox{$\quad\Leftrightarrow\quad$}}(\neg A)^X =\top {\mbox{$\quad\Leftrightarrow\quad$}}\langle Y,X\rangle\models (\neg A)^X$. Hence for all $A$ it is true that $\langle Y,X\rangle\models \neg A{\mbox{$\leftrightarrow$}}(\neg A)^X$. This implies, using the definition \[ax\], that for all $A\in[\wedge,\vee,{\to},\neg]$ it is true that $\langle Y,X\rangle\models A{\mbox{$\leftrightarrow$}}A^X$, from which the lemma immediately follows. Theorem \[p1\], theorem \[sim\] and corollary \[lpv\] restate the main result of [@lpv]. \[p1\] Let $\Pi\subseteq \{A{\to}B\mid A, B\in [\wedge,\vee,\neg]\}$ and $X$ a set of atomic formulas. $X$ is an an answer set of $\Pi$ if and only if for all $Y\subseteq X$ it is true that $\langle Y,X\rangle\models\Pi{\mbox{$\quad\Leftrightarrow\quad$}}X=Y$. \[sim\] Let $\Pi_1$ and $\Pi_2$ be programs in $\{A{\to}B\mid A, B\in [\wedge,\vee,\neg]\}$. $\Pi_1$ and $\Pi_2$ are strongly equivalent if and only if they are equivalent in [**G3**]{}, i.e. $\Pi_1\equiv_{\bf G3}\Pi_2$. \[lpv\] Let $\Pi_1$ and $\Pi_2$ be programs in $\{A{\to}B\mid A, B\in [\wedge,\vee,\neg]\}$. $\Pi_1$ and $\Pi_2$ are strongly equivalent if and only if for all $\Pi\subseteq \{p{\to}q\mid p \;\mbox{atomic or}\; p=\top, q \;\mbox{atomic}\}$, $\Pi_1\cup\Pi$ and $\Pi_2\cup\Pi$ have the same answer sets. According to the corollary above, the notion of strong equivalence of logic programs may depend on the language for the programs $\Pi_1$ and $\Pi_2$, but in all sublanguages $L$ of $\{A{\to}B\mid A, B\in [\wedge,\vee,\neg]\}$, we may use theorem \[sim\], as long as rules of the form $p{\to}q$ (with $p$ and $q$ atomic) are in $L$. Stable inference in intermediate logics ======================================= The previous section linked strong equivalence of logic programs in stable inference with equivalence in [**CPL**]{} (for programs without negations in the head or the body of the rules) or in [**G3**]{}. In this section we will determine for several fragments of propositional logic the weakest intermediate logic for which equivalence of programs is implied by strong equivalence in stable inference. For the fragment $\{A{\to}B\mid A, B\in [\wedge,\vee,\bot,\top]\}$ we have the following lemma. \[wv2\] Let $L=\{A{\to}B\mid A, B\in [\wedge,\vee,\bot,\top]\}$ and $T\subseteq L$, $C\in L$. Then $T\vdash_{\bf CPL} C{\mbox{$\quad\Leftrightarrow\quad$}}T\vdash_{\bf IPL} C$. The direction from [**IPL**]{} to [**CPL**]{} is trivial. So let us assume $T\vdash_{\bf CPL} C$. Let $M$ be a Kripke model and $w$ a node in $M$ such that $w\models T$. Using lemma \[wv\] we may conclude $\langle w\rangle\models T$ and hence $\langle w\rangle\models C$. Again, use lemma \[wv\] to prove $w\models C$. This proves that $T$ implies $C$ in Kripke models in general and hence $T\vdash_{\bf IPL} C$. Let $L=\{A{\to}B\mid A, B\in [\wedge,\vee,\bot,\top]\}$ and $\Pi_1,\Pi_2 \subseteq L$. Then $\Pi_1$ and $\Pi_2$ are strongly equivalent in $L$ iff $\Pi_1\equiv_{\bf IPL}\Pi_2$. Is an immediate consequence of theorem \[t1\] and lemma \[wv2\]. Strong equivalence between programs in the language $\{A{\to}B\mid A, B\in [\wedge,\vee,\neg]\}$ will not be the same as equivalence in [**IPL**]{}, as for example $\neg p\vee\neg\neg p$ is strongly equivalent to $\top$ (it is a derivable formula in [**G3**]{}) and is not derivable in [**IPL**]{}. The intermediate logic [**KC**]{}, which has $\neg A\vee\neg\neg A$ as its axiom, will, in the following, turn out to be the weakest intermediate logic for which equivalence of programs is implied by strong equivalence in answer set semantics. Let $L=\{A{\to}B\mid A, B\in [\wedge,\vee,\neg]\}$, $T\subseteq L$ and $C\in L$. Then $T\vdash_{\bf KC} C{\mbox{$\quad\Leftrightarrow\quad$}}T\vdash_{\bf G3} C$. Again the direction from [**KC**]{} to [**G3**]{} is trivial. For the other direction, let $T{\mbox{$\;\not\vdash\;$}}_{\bf KC} A{\to}B$ (where $A, B\in[\wedge,\vee,\neg]$). Then for some Kripke model $M$ with a single terminal (i.e. maximal) node $t$, there is a $w\in M$ such that $w\models T, w\models A$ and $w\not\models B$. We will prove that for the [**G3**]{}-model $\langle w, t\rangle$ we have for all formulas $C\in[\wedge,\vee,\neg]$ that $w\models C{\mbox{$\quad\Leftrightarrow\quad$}}\langle w,t\rangle\models C$ and for $C, D\in[\wedge,\vee,\neg]$ that $w\models C{\to}D{\mbox{$\quad\Rightarrow\quad$}}\langle w,t\rangle\models C{\to}D$. As a consequence, $\langle w,t\rangle\models T, \langle w, t\rangle\models A$ and $\langle w,t\rangle\not\models B$, which proves $T{\mbox{$\;\not\vdash\;$}}_{\bf G3} A{\to}B$. The proof that for $C\in[\wedge,\vee,\neg]$ we have $w\models C{\mbox{$\quad\Leftrightarrow\quad$}}\langle w,t\rangle\models C$ is by structural induction. For atomic formulas it is obvious and the cases for conjunctions and disjunctions are trivial. For the case of negation, observe that $w\models\neg C{\mbox{$\quad\Leftrightarrow\quad$}}t\not\models C$, and $t\not\models C{\mbox{$\quad\Leftrightarrow\quad$}}\langle t\rangle\not\models C{\mbox{$\quad\Leftrightarrow\quad$}}\langle w,t\rangle\models\neg C$. Now let $C, D\in[\wedge,\vee,\neg]$ and $w\models C{\to}D$. Since $w\leq t$, $\langle t\rangle \models C{\to}D$. So, if $w\not\models C$, we have (by the above part of the proof) $\langle w, t\rangle \not\models C$ and hence $\langle w,t\rangle\models C{\to}D$. On the other hand, if $w\models C$, then also $w\models D$ and by the above part of the proof, also $\langle w,t\rangle\models C{\to}D$. Which proves that $w\models C{\to}D$ implies $\langle w,t\rangle\models C{\to}D$. \[colkc\] Let $L=\{A{\to}B\mid A, B\in [\wedge,\vee,\neg]\}$ and $\Pi_1,\Pi_2 \subseteq L$. Then $\Pi_1$ and $\Pi_2$ are strongly equivalent in $L$ iff $\Pi_1\equiv_{\bf KC}\Pi_2$. [**KC**]{} is the weakest intermediate logic [**L**]{} such that $\Pi_1,\Pi_2\subseteq\{A{\to}B\mid A, B\in [\wedge,\vee,\neg]\}$ are strongly equivalent iff $\Pi_1\equiv_{\bf L}\Pi_2$. Note that the [**KC**]{} axiom $\neg A\vee\neg\neg A$ can be expressed in the language $\{A{\to}B\mid A, B\in [\wedge,\vee,\neg]\}$. That [**KC**]{} is the weakest intermediate logic for strong equivalence in almost any language with negation (where negation is taken to be a [*negation by default*]{} and strong equivalence defined according to the answer set semantics) can be seen from the following corollary. Let $L=\{A{\to}B\mid A,B\in[\neg]\}$ and $\Pi_1,\Pi_2 \subseteq L$. Then $\Pi_1$ and $\Pi_2$ are strongly equivalent in $L$ iff $\Pi_1\equiv_{\bf KC}\Pi_2$. [**KC**]{} can alternatively be axiomatized as [**IPL**]{} plus $((\neg A{\to}B)\wedge (\neg\neg A{\to}B){\to}B$. In one direction this is clear from the fact that this axiom immediately follows from $\neg A\vee\neg\neg A$, for the other direction substitute $\neg A\vee\neg\neg A$ for $B$ in the axiom, and $\neg A\vee\neg\neg A$ follows. So, the programs $\{q\}$ and $\{\neg p{\to}q,\neg\neg p{\to}q\}$ are strongly equivalent and any logic making such programs equivalent will be as strong as [**KC**]{}. As a consequence also strong equivalence in for example $\{A{\to}B\mid A, B\in [\wedge,\neg]\}$ and $\{A{\to}B\mid A, B\in [\vee, \neg]\}$ will coincide with equivalence in [**KC**]{}. Even if we restrict the language further, allowing in the body only atoms or negated atoms and in the head only atoms (apart from simple statements of atoms and negation of atoms), [**KC**]{} is still the weakest intermediate logic [**L**]{} such that equivalence of programs in [**L**]{} corresponds with strong equivalence. In logic programming the programs in this restricted language are known as [*normal*]{} programs and have historically been most important. Most Prolog implementations of [*negation by default*]{} are restricted to this kind of programs, often called [*general programs*]{} in this context (see [@doets]). A [*normal*]{} logic program is a finite set of rules ${\mbox{\raisebox{.4ex}{$\bigwedge$}}}l_i{\to}p$, where the $l_i$ are literals (so either atomic or a negation of an atomic formula) and $p$ is atomic. First we will prove that an alternative axiomatization of [**KC**]{}, in the language of normal programs, is possible. \[qlem\] $A\wedge C{\to}D, \neg A{\to}B, \neg C{\to}B\vdash_{\bf KC} \neg D{\to}B$. Of course, this can be automatically checked in a tableau system as in [@afm], but let us do it from scratch. By [**KC**]{} we have $\neg A$ or $\neg\neg A$. If $\neg A$, $B$ and hence $\neg D{\to}B$, is immediate from $\neg A{\to}B$. So, we can assume $\neg\neg A$. Similarly, we can assume $\neg\neg C$. By [**IPL**]{}, $\neg\neg(A\wedge C)$ follows. Again by [**IPL**]{}, $A\wedge C{\to}D$ now implies $\neg\neg D$, from which again $\neg D{\to}B$. Let $L$ be the set of formulas coding normal logic programs. As derivability in [**KC**]{} implies derivability in [**G3**]{}, we can use lemma \[qlem\] to prove that $\Pi_1=\{p\wedge r{\to}s, \neg p{\to}q, \neg r{\to}q\}$ and $\Pi_2=\{p\wedge r{\to}s, \neg p{\to}q, \neg r{\to}q, \neg s{\to}q\}$ are strongly equivalent programs in $L$. On the other hand it is easily seen that for each intermediate logic [**L**]{} that proves $\Pi_1$ and $\Pi_2$ equivalent, we have $A\wedge C{\to}D, \neg A{\to}B, \neg C{\to}B\vdash_{\bf L} \neg D{\to}B$. The following lemma shows that such an [**L**]{} has to contain [**KC**]{}. \[normkc\] If [**L**]{} is the intermediate logic with, apart from the axioms of [**IPL**]{}, the axiom $(A\wedge C{\to}D)\wedge(\neg A{\to}B)\wedge(\neg C{\to}B){\to}(\neg D{\to}B)$, then [**L**]{} is equivalent with [**KC**]{}. That $\vdash_{\bf L} A$ implies $\vdash_{\bf KC} A$ is a simple consequence of lemma \[qlem\]. For the other direction, let $A:= p$, $B:= \neg p\vee\neg\neg p$, $C:=\neg p$ and $D:=\bot$ in $(A\wedge C{\to}D)\wedge(\neg A{\to}B)\wedge(\neg C{\to}B){\to}(\neg D{\to}B)$. All the antecedents as well as $\neg D$ are then derivable, so $\neg p\vee\neg\neg p$ follows. The result of the above discussion is summarized in the next corollary. Let $\Pi_1$ and $\Pi_2$ be normal logical programs. Then $\Pi_1$ and $\Pi_2$ are strongly equivalent iff $\Pi_1\equiv_{\bf KC}\Pi_2$. Moreover, [**KC**]{} is the weakest intermediate logic for which provable equivalence in the logic and strong equivalence of normal logic programs coincide. The first part immediately follows by corollary \[colkc\]. Lemma \[normkc\] implies that [**KC**]{} is the weakest intermediate logic for which equivalent normal programs are strongly equivalent. No program in the language $\{A{\to}B\mid A,B\in [\wedge,\neg]\}$ is strongly equivalent to the program $\{p\vee q\}$. Recall that [**KC**]{} is sound and complete with respect to the finite Kripke models with a single terminal node. Let the model $M$ be as pictured below, where $\mbox{\em atom}(t) = \{p, q \}$, $\mbox{\em atom}(u) =\{p\}$, $\mbox{\em atom}(v) =\{q\}$ and $\mbox{\em atom}(w) = \emptyset$. (25, 30)(-55, 0) (15,5)(0,20)[2]{} (15,5)(10,10)[2]{}[(-1,1)[10]{}]{} (5,15)(20,0)[2]{} (15,5)(-10,10)[2]{}[(1,1)[10]{}]{} (12,25)[(0,0)[$t$]{}]{} (20,25)[(0,0)[$p, q$]{}]{} (2,15)[(0,0)[$u$]{}]{} (9,15)[(0,0)[$p$]{}]{} (22,15)[(0,0)[$v$]{}]{} (29,15)[(0,0)[$q$]{}]{} (15,2)[(0,0)[$w$]{}]{} Clearly $u\models p\vee q$ and $v\models p\vee q$, but $w\not\models p\vee q$. By induction on the complexity of formulas $A\in [\wedge,{\to},\neg]$ one easily proves that $$w\models A {\mbox{$\quad\Leftrightarrow\quad$}}u\models A\;\mbox{and}\; v\models A$$ Hence if $\Pi\subseteq \{A{\to}B\mid A,B\in [\wedge,\neg]\}$ and $\Pi\equiv_{\bf KC} \{p\vee q\}$, we would have $u\models\Pi$, $v\models\Pi$, which would imply $w\models\Pi$, a contradiction. Observe that the type of model we need for the proof above is not a [**G3**]{} model (not of the form $\langle h,t\rangle$). In fact, in the full language of [**G3**]{} we can define disjunction using $p\vee q = ((p{\to}q){\to}q)\wedge ((q{\to}p){\to}p)$. The simple proof that this is not possible (in [**G3**]{}) if one restricts the language of the programs to $\{A{\to}B\mid A,B\in [\wedge,\neg]\}$ indicates that the proof of certain properties of answer set programs may benefit from a detour in the logic [**KC**]{}. [99]{} Avellone, A., Ferrari, M. and Miglioli, P. (1999) Duplication-free tableau calculi and related cut-free sequent calculi for the interpolable propositional intermediate logics. *Logic Journal of the IGPL*, 7(4): pp. 447–480. Chagrov, A. and Zakharyaschev, A. M. (1997) Modal Logic. Clarendon Press. Doets, K. (1994) From Logic to Logic Programming. MIT Press. Gelfond, M. and Lifschitz, V. (1988) The stable model semantics for logic programs. In R.A. Kowalski and K.A. Bowen (editors), *Proceedings of the Fifth International Conference and Symposium on Logic Programming 2*, pp. 1070–1080. MIT Press. Gödel, K. (1932) Zum intuitionistischen Aussagenkalkül, *Anzeiger der Akademie der Wissenschaften in Wien*, 69: pp. 65–66. Hendriks, A. (1996) Computations in Propositional Logic, PhD Thesis University of Amsterdam. Hosoi, T. (1966) The axiomatization of the intermediate propositional systems $S_n$ of Gödel, *Journal of the Faculty of Science of the University of Tokio*, 13: pp. 183–187. Jankov, V. A. (1968) The calculus of the weak “law of excluded middle”. *Mathematics of the USSR, Izv.*, 2: pp. 997–1004. Kowalski, R. A. (1974) Predicate logic as a programming language. *Proceedings IFIP’74*, pp. 569–574. North-Holland. Lifschitz, V., Pearce, D. and Valverde, A. (2001) Strongly equivalent logic programs. [*ACM Transactions on Computational Logic*]{}, 2(4): pp. 526–541. Lifschitz, V., Tang, L. R. and Turner, H. (1999) Nested expressions in logic programs. *Annals of Mathematics and Artificial Intelligence*, 25: pp. 369–389. Łukasiewicz, J. (1938) Die Logik und das Grundlagenproblem. *Les Entretiens de Zürich sur les Fondaments et la Méthode des Sciences Mathématiques 6–9*, 12: pp. 82–100. Pearce, D. (1997) A new logical characterization of stable models and answer sets. In J. Dix, L. Pereira, T. Przymusinski (editors), *Non-Monotonic Extensions of Logic Programming, Lecture Notes in Artificial Intelligence, 1216*, pp. 57–70. Springer-Verlag. [^1]: A more precise definition will be given in the sequel.

--- abstract: 'Quantum simulation of complex quantum systems and their properties often requires the ability to prepare initial states in an eigenstate of the Hamiltonian to be simulated. In addition, to compute the eigenvalues of a Hamiltonian is in general a non-trivial problem. Here, we propose a hybrid quantum-classical probabilistic method to compute eigenvalues and prepare eigenstates of Hamiltonians which are simulatable with a trapped-ion quantum processor.' author: - 'Jing-Ning Zhang$^1$' - Iñigo Arrazola$^2$ - Jorge Casanova$^2$ - Lucas Lamata$^2$ - Kihwan Kim$^1$ - 'Enrique Solano$^{2,3,4}$' title: 'Probabilistic Eigensolver with a Trapped-Ion Quantum Processor' --- Introduction {#sec:introduction} ============ Simulating complex quantum systems is known to be an intractable task if accomplished with classical computing resources. This is because the dimension of the Hilbert space needed to describe a quantum system increases exponentially with its number of constituents. Quantum simulation [@Feynman82], i.e. the use of a fully controlled quantum system that simulates the dynamics of another one, was originally proposed to overcome this problem, and has led to plenty of developments in different quantum platforms [@Georgescu14]. On the other hand, quantum computing is a more general approach that allows not only to simulate quantum dynamics, but also to solve systems of linear equations [@Harrow09], linear differential equations [@Berry17; @Xin18; @Arrazola18], or the eigenvalue problem of complex Hamiltonians [@Abrams99; @Nielsen00], with an exponential speedup. Regarding the latter, it is important to remark that the knowledge of the energy spectrum of a system is crucial in different fields such as quantum chemistry and condensed matter physics. The [*phase estimation algorithm*]{} [@Abrams99] is a prominent method for finding eigenvalues and preparing eigenstates of target Hamiltonians, however other approaches have been also designed. Among them, we can mention the use of adiabatic evolutions [@Aspuru05], algorithmic cooling [@Xu14] or variational quantum eigensolvers [@Peruzzo14; @Yung14; @McClean16]. The latter are designed to prepare the ground state (or excited state [@Shen17]) of a complex Hamiltonian, which interestingly combine a quantum processor and a classical optimization algorithm. The variational eigensolvers have been implemented experimentally in state-of-the-art quantum platforms like photonics [@Peruzzo14], superconducting circuits [@Kandala17], or trapped ions [@Hempel18], and their accuracy in eigenstates preparation, as well as in the computing of the associated eigenvalues, highly depends on the flexibility of the ansatz [@Malley16; @Shen17; @Kandala17]. In this respect, having a more complex and flexible input state, or ansatz, implies that more variational parameters have to be optimized simultaneously, which could be haunted by the presence of local minima. In addition, the preparation of the ansatz may involve non-trivial entanglement operations among the available quantum registers. In this manner, it is important to explore alternative algorithms to be applied in near-future quantum processors [@Debnath16]. In this article, we propose a hybrid classical-quantum eigensolver that uses projective measurements on a single ancilla qubit to probabilistically prepare the eigenstates of target Hamiltonians in trapped ions. Unlike the quantum variational solver or adiabatic state preparation, in our method the only requirement for the initial state is to have a non-negligible overlap with the desired target eigenstate. In Sec. \[sec:general\], we first introduce the general method for preparing an arbitrary target eigenstate, given that the overlap between the target and the initial state is nonzero. Afterwards, we present how the method can be combined with a classical optimization algorithm, resulting in a significant reduction on the number of steps required for the preparation, while maintaining acceptable success probability. In Sec. \[sec:toolkit\], we present the toolkit that trapped-ion quantum simulators offer to implement our method. Finally, in Sec. \[sec:examples\], we present some models in which our probabilistic eigenstate preparation could be applied using a trapped-ion quantum processor, and demonstrate its performance with numerical simulations. General framework {#sec:general} ================= Consider a target quantum system described by the Hamiltonian $\hat H$ such that $$\begin{aligned} \hat H{\left| j \right\rangle}=E_j{\left| j \right\rangle}.\end{aligned}$$ Here, ${\left| j \right\rangle}$ ($E_j$) is the $j$-th eigenstate (eigenenergy) and the energy spectrum $\left\{E_j\right\}$ is bounded from below and sorted in ascending order. To prepare one of the eigenstates of $\hat H$, we consider a quantum system isomorphic to the target system and an additional ancillary qubit. The evolution of the whole system is described by a unitary operator acting on the product Hilbert space ${\mathscr H}_{\rm S}\otimes{\mathscr H}_{\rm A}$, with ${\mathscr H}_{\rm S}$ (${\mathscr H}_{\rm A}$) being the Hilbert space of the target system (the ancilla qubit). For preparing the $j$-th eigenstate ${\left| j \right\rangle}$, our method requires the knowledge of all eigenenergies below $E_j$. As, in general, this is not the case, we have to start with the ground state preparation. In this manner, this first step can be considered as a cooling protocol. To this end, we apply the following unitary operator on the whole system $$\begin{aligned} \label{eq:U_tau} \hat W_\gamma(\tau) = \exp\left[-i\left(\hat H_{\rm S}+\gamma\right)\hat\sigma_{\rm A}^{\rm x}\tau\right].\end{aligned}$$ Here, $\hat\sigma_{\rm A}^{\rm x}$ is the x Pauli matrix acting on the ancillary qubit and the parameters $\gamma$ and $\tau$ are real numbers to be determined. $\hat W_\gamma(\tau)$ can be viewed as an evolution operator of the whole system with $\tau$ being the effective evolution time. We use $\gamma$ to shift the energy spectrum of $\hat H_{\rm S}$ such that $E_0+\gamma\geq0$ (i.e. the shifted spectrum is always positive), where $E_0$ is the ground state energy of the target system. Our protocol is an iterative method where each repetition involves, first, initializing the ancillary qubit to a reference state ${\left| 0 \right\rangle}_{\rm A}$ such that $\hat\sigma^z_{\rm A}{\left| 0 \right\rangle}_{\rm A}=-{\left| 0 \right\rangle}_{\rm A}$, and second, applying $\hat W_\gamma(\tau)$. In the $k$-th iteration we will obtain a state ${\left| \Psi_{k} \right\rangle}$, which is $$\begin{aligned} {\left| \Psi_{k} \right\rangle}&=&\hat W_\gamma(\tau){\left| \psi_{k-1} \right\rangle}{\left| 0 \right\rangle}_{\rm A}\nonumber\\ &=&\hat{\mathcal C}_\gamma(\tau){\left| \psi_{k-1} \right\rangle}{\left| 0 \right\rangle}_{\rm A}-i\hat{\mathcal S}_\gamma(\tau){\left| \psi_{k-1} \right\rangle}{\left| 1 \right\rangle}_{\rm A}.\end{aligned}$$ Here, ${\left| \psi_{k-1} \right\rangle}$ is the normalized state vector in ${\mathscr H}_{\rm S}$ that results from the $k-1$ iteration, and the non-unitary operators are defined as $\hat{\mathcal C}_\gamma(\tau)=\cos\left[\left(\hat H+\gamma\right)\tau\right]$ and $\hat{\mathcal S}_\gamma(\tau)=\sin\left[\left(\hat H+\gamma\right)\tau\right]$. Finally, we perform a projective measurement on the ancillary qubit and make post-selection on the measurement results to select certain the output state in ${\mathcal H}_{\rm S}$. The two possible normalized output states, conditioned on the measurement results, are ${\left| \psi_k \right\rangle}_0=\hat{\mathcal C}_\gamma(\tau)/{\sqrt{P^{0}_{k}}}{\left| \psi_{k-1} \right\rangle}$ and ${\left| \psi_k \right\rangle}_1=\hat{\mathcal S}_\gamma(\tau)/{\sqrt{P^{1}_k}}{\left| \psi_{k-1} \right\rangle}$, and their probabilities are $P^{0}_k =\sum_j\left|{\left\langle j | \psi_{k-1} \right\rangle}\right|^2\cos^2\left[\left(E_j+\gamma\right)\tau\right]$ and $P_k^{1}=\sum_j\left|{\left\langle j | \psi_{k-1} \right\rangle}\right|^2\sin^2\left[\left(E_j+\gamma\right)\tau\right]$, respectively. We would like to point out that, although we use a pure input state ${\left| \psi_{k-1} \right\rangle}$ to illustrate the process, our method can be straightforwardly applied to mixed input states without any modification. Now, if we compare the average energy of the two possible output states with the average energy of the input state, we find that, for positive and small $\tau$, the following relation holds (See Appendix A) $$\begin{aligned} \label{eq:average_energy} {\left\langle \hat H \right\rangle}^{(0)}_{k}\leq{\left\langle \hat H \right\rangle}_{k-1}^{(0)}<{\left\langle \hat H \right\rangle}^{(1)}_{k}.\end{aligned}$$ Here ${\left\langle \cdot \right\rangle}_k^{(i)}\equiv{\left\langle \psi_k^{(i)} \left| \hat H \right| \psi_k^{(i)} \right\rangle}$ and the superscript $i = 0, 1$ denotes the outcome of the projective measurement on the ancillary qubit. In other words, the average energy of the system is lowered compared to the one of the input state, i.e. the post-selected state from the previous round, if we successfully project the ancillary qubit in ${\left| 0 \right\rangle}_{\rm A}$. Here, it is noteworthy to comment that the probability of projecting on the reference ancilla state ${\left| 0 \right\rangle}_{\rm A}$ (i.e. on the state that leads to an effective average energy reduction) is close to one for a sufficiently small $\tau$ (note that $P_k^0=1$ if $\tau=0$). An intuitive way to understand this energy reduction for small $\tau$ is provided when studying the transformation of the state. If ${\left| \psi_{k-1} \right\rangle}=\sum_{j}c^{k-1}_{j}{\left| j \right\rangle}$ is the system input state before the $k$-th iteration, the output state after the application of $\hat W_\gamma(\tau)$ and post-selection of ${\left| 0 \right\rangle}_{\rm A}$ will be ${\left| \psi_{k} \right\rangle}={\left| \psi_k \right\rangle}_0\equiv\sum_{j}c_j^{k}{\left| j \right\rangle}$, where the probability distribution of the state have changed as $$\begin{aligned} \label{eq:prob_dist} \left|c_j^k\right|^2=\frac{1}{P_{k}^{0}} \cos^2{[(E_j+\gamma)\tau]}\left|c_j^{k-1}\right|^2.\end{aligned}$$ As $\cos^2(\cdot)$ is a monotonic decreasing function in the vicinity of $\tau=0$, the probability amplitudes of those eigenstates with lower eigenenergies $E_j$ will be enhanced (note that a larger value of $E_j$ implies a lower value for the corresponding $\cos^2(\cdot)$ function) compared to those with higher eigenenergies, leading to the decrease of the average energy, an effective [*cooling*]{} effect. After the $k$-th iteration, we have to measure the average energy of the resulting state $\bar E^{k}\equiv {\left\langle \hat H \right\rangle}^{(0)}_{k}$, and compare it with the average energy of the previous stage $\bar E^{k-1}$. If $ |\bar E^{k-1}-\bar E^{k}|<\epsilon$ holds (with $\epsilon$ being a small positive value), we conclude that that the ground-state within a certain precision $\epsilon$ has been reached and stop the protocol. If the state does not converge at the $k$-th iteration, saying $ |\bar E^{k-1}-\bar E^{k}|>\epsilon$, the output state ${\left| \psi_{k} \right\rangle}_0$ will be sent through another iteration for further reducing its energy. An important feature of our protocol is that if the initial state ${\left| \psi_0 \right\rangle}$ has no overlap with the desired eigenstate, it is not possible to prepare it. It can be seen from Eq. (\[eq:prob\_dist\]) that if the $(k-1)$-th input state has no overlap with a given eigenstate ${\left| j \right\rangle}$ (i.e. $c_{j}^{k-1}=0$), the resulting $k$-th state will also share the same condition $c_{j}^{k}=0$. Although this may seem a disadvantage, it can actually be used to prepare excited eigenstates. To this end, we start from an arbitrary initial state ${\left| \phi_0 \right\rangle}$ and apply the unitary operation $$\begin{aligned} \hat U_{s}=\exp\left[-i\left(\frac{\pi}{2 E_{s}}\right)\hat H\hat\sigma_{\rm A}^{\rm x}\right],\end{aligned}$$ and measure the ancilla qubit. The action of the unitary operator $\hat U_{s}$ on a general state ${\left| \phi \right\rangle}$ is to conditionally produce a state ${\left| \psi^{(0)} \right\rangle}$ where the contribution of the basis state ${\left| s \right\rangle}$ goes to zero ($\langle s |\psi^{(0)}\rangle=0$), as long as the ancilla qubit is projected on ${\left| 0 \right\rangle}_{\rm A}$. As a result, the output state after post-selection does not have any overlap with the eigenstate ${\left| s \right\rangle}$. For example, to prepare the first excited state of $\hat H$ starting from an arbitrary initial state ${\left| \phi_0 \right\rangle}$, one starts with applying the unitary $\hat U_{0}$ and projecting the ancilla on the ${\left| 0 \right\rangle}_{\rm A}$ state. If the projection succeeds, one will obtain a state ${\left| \psi_0 \right\rangle}$ that has zero overlap with the ground state of $\hat H$, which serves as input state for the [*cooling stage*]{}. Then, one applies the previously described protocol (See Fig. \[fig:Fig1\]), until the state converges to the lowest energy eigenstate that has a non-zero overlap with the initial state, which in this case is the first excited state. To prepare an arbitrary eigenstate ${\left| j \right\rangle}$ starting from ${\left| \phi_0 \right\rangle}$, one first removes any overlap with lower-energy eigenstates, by sequentially applying $\hat U_{j'}$ and the following projective measurement with $j' = 0, 1, \ldots, j-1$. If all of the post-selection operations succeed, the resulting state is sent to the [*cooling stage*]{}, which would probabilistically converge to the desired eigenstate ${\left| j \right\rangle}$. Note that once a post-selection operation fails, the whole procedure should be started all over again. ![Scheme for preparing the first excited state of an $N$ dimensional Hamiltonian. First, we apply $\hat{U}_0$ and measure the ancilla qubit. Then we use post-selection to remove the ground-state contribution from the initial state ${\left| \phi_0 \right\rangle}$. After that, we apply the cooling unitary $\hat{W}_\gamma(\tau_k)$ and subsequent ancilla measurement $k$ times, until we reach convergence.[]{data-label="fig:Fig1"}](Fig1.pdf "fig:"){width="45.00000%"}\ We consider now two different variants for the [*cooling stage*]{}, namely the [*fixed-step*]{} and [*variational*]{} approaches. [*The fixed-step approach*]{} treats $\tau$ as a small fixed parameter. The operational procedure is as follows: After each implementation of $\hat W_\gamma(\tau)$, we perform a projective measurement on the ancilla qubit. We continue with the next iteration if the outcome is ${\left| 0 \right\rangle}_{\rm A}$, otherwise we restart the whole process. At the $k$-th stage of the process, we estimate the average energy of the output state $\bar E^{k}$. The process stops if $|\bar E^{k-1}-\bar E^{k}| \leq \epsilon$, where the precision $\epsilon$ is a positive small value. Otherwise, we continue the protocol by moving to the $(k+1)$-th stage. In the [*variational approach*]{}, the value of the parameter $\tau$ is optimized in each stage such that it minimizes the average energy. This hybrid optimization is done by feeding the average energy obtained from the quantum platform to a classical optimization algorithm, which provides new trial value for $\tau$ to the quantum platform based on the given information (see Fig. \[fig:variational\]). After several optimization steps (trial steps), the classical algorithm will converge to $\tau_k$, the optimal value of $\tau$ at the $k$-th stage, and the quantum platform will estimate the average energy $\bar E^k$. As before, the process stops if $|\bar E^{k-1}-\bar E^{k}| \leq \epsilon$. At the end, we obtain a set of optimized values $\left\{\tau_1, \tau_2, \ldots, \tau_{k}\right\}$. Compared to the [*fixed-step*]{} approach, the [*variational*]{} approach allows the probabilistic preparation of the desired eigenstate with a substantial reduction on the circuit depth, i.e. with a qualitatively less amount of gates and post-selections. Both the [*fixed-step*]{} and the [*variational*]{} approaches are probabilistic because we need to restart the whole procedure depending on the results of the projective measurements on the ancilla qubit, and provide an estimation of the eigenenergy of the target eigenstate within the preset precision $\epsilon$. However, the variational protocol additionally provides a recipe for shallow-depth probabilistic preparation of the target state, described by the set of optimized parameters $\{\tau_1,\ldots,\tau_k\}$. The overall success probability $P_{\rm suc}$ of both protocols is ultimately bounded by the overlap between the initial state and the desired eigenstate, i.e, $P_{\rm suc} \leq |\langle\psi_0|j'\rangle|^2$. On the other hand, to maximize $P_{\rm suc}$ for the preparation of eigenstate $|j'\rangle$, one should choose the value of $\gamma$ to be the closest possible to $-E_{j'}$. However, it is possible that we do not know the exact value of $E_{j'}$, in which case one can always choose $\gamma$ to be minus the lower bound of the norm of the Hamiltonian, which can be analytically approximated in general. Finally, we comment that the unitary operator $\hat W_\gamma(\tau)$ required in our method can be decomposed as $\hat W_\gamma(\tau)\equiv\hat W(\tau)R_{\rm x}^{\rm A}\left(2\gamma\tau\right)$, with $\hat W(\tau)=\exp\left(-\hat H\hat\sigma_{\rm A}^{\rm x}\tau\right)$ and $R_\alpha^{\rm A}\left(\theta\right)\equiv\exp\left(-\frac{i\theta}{2}\hat\sigma^\alpha_{\rm A}\right)$, $\alpha = {\rm x, y, z}$, being the single qubit rotation acting on the ancilla qubit. Now, the $\hat W(\tau)$ operator that entangles the system and the ancilla qubit, can be trotterized into a set of basic operations [@Lloyd96]. More specifically, suppose that the system Hamiltonian is composed of $M$ components $\hat H_m$ that don’t commute with each other, namely $\hat H = \sum_{m=1}^M\hat H_m$. Then, the unitary evolution in Eq. (\[eq:U\_tau\]) can be approximately written using the second-order Trotter-Suzuki expansion, $$\begin{aligned} \hat W(\tau)&=&\left[\hat W_1\left(\frac{\tau}{2r}\right)\ldots\hat W_{M-1}\left(\frac{\tau}{2r}\right)\hat W_M\left(\frac{\tau}{r}\right)\right.\nonumber\\ &&\left.\times\hat W_{M-1}\left(\frac{\tau}{2r}\right)\ldots\hat W_1\left(\frac{\tau}{2r}\right)\right]^r+{\mathcal O}\left(\frac{\tau^3}{r^2}\right),\end{aligned}$$ where $r$ is the number of Trotter steps and $\hat W_m(\tau) = \exp\left(-i\hat H_m\hat\sigma_{\rm A}^{\rm x}\tau\right)$ with $m= 1,2,\ldots M$. Note that by increasing the number of Trotter steps, arbitrary precision can be achieved in principle. Toolkit in trapped-ion platforms {#sec:toolkit} ================================ Trapped ions have been demonstrated to be suitable for digital [@Lanyon11] and analog [@Zhang17; @Safari17] quantum simulations, having access not only to the manipulation of the ion-qubits, but also to the coherent control of their collective vibrational modes [@Toyoda15]. The digital-analog approaches which combines both resources could enhance the computational power of the quantum platform for solving hard problems [@Shen14], for example, non-trivial many-body models or even problems in quantum field theories [@Casanova11]. We envision four types of primitive system-ancilla couplings in the toolkit of trapped-ion systems, including: 1) $\hat\sigma_i^{\rm z}\hat\sigma_{\rm A}^{\rm x}$, 2) $\hat a_m^\dag\hat a_m\hat\sigma_{\rm A}^{\rm x}$, 3) $\hat\sigma_{i_1}^{\rm x}\hat\sigma_{i_2}^{\rm x}\hat\sigma_{\rm A}^{\rm x}$, 4) $\left(\hat a_m^\dag+\hat a_m\right)\hat\sigma_i^{\rm x}\hat\sigma_{\rm A}^{\rm x}$. Using these interactions and the measurement on the ancilla, the protocol described in the previous section could be implemented in trapped-ion systems for a wide variety of models. The free Hamiltonian $\hat H_0$ of an array of $N$ ions trapped in a linear Paul trap is written as follows, $$\begin{aligned} \hat H_0 = \sum_{n=1}^N\frac{\hbar\omega_{\rm 0}}{2}\hat\sigma^{\rm z}_n+\sum_{m = 1}^N\hbar\omega_m\left(\hat a_m^\dag\hat a_m+\frac{1}{2}\right),\end{aligned}$$ where $\hat\sigma^{\rm x, y, z}_n$ are the Pauli matrices acting on the $n$-th ion with $\omega_{\rm 0}$ being the frequency splitting of the two involved internal levels, and $\hat a_m$ ($\hat a_m^\dag$) is the annihilation (creation) operator of the $m$-th collective motional mode with the mode frequency $\omega_m$. For a better readability, from now on we will omit the subscript $n$ ($m$) of $\hat\sigma_n^{\rm x,y,z}$ ($\hat a_m$) when there is no ambiguity. In the following, we provide a brief explanation of how to engineer the four types of system-ancilla couplings in a trapped-ion quantum simulator: First, spin-spin couplings can be implemented by the Mølmer-Sørensen (MS) type of interaction, which takes the form of $\sigma_i^{\rm x}\sigma_j^{\rm x}$ and can be engineered between any pair of ions embedded in a linear chain of ions [@Leung18]. With the appropriate single qubit rotations on the first qubit, we can extend the coupling to the form $\hat\sigma^{\mathbf n}\sigma_A^{\rm x}$, that would correspond to a simulated local spin-1/2 operator $\hat H_{\rm S}\propto\hat\sigma_{\mathbf n}$, where ${\mathbf n}=\left(n_{\rm x}, n_{\rm y}, n_{\rm z}\right)^{\rm T}$ is a unit vector. Second, the $\hat a^\dag\hat a\hat\sigma_{\rm A}^{\rm x}$ interaction term involving the $m$-th mode could be realized with a detuned red-sideband coupling, for which the Hamiltonian is written, in the rotating frame with respect $\hat H_0$ and after the rotating wave approximation, as $$\begin{aligned} \hat H_{\rm I}&=&\frac{\hbar\Omega_{\rm A}}{2}\left[\left(\hat\sigma^+e^{i\Delta t}+\hat\sigma^-e^{-i\Delta t}\right)\right]\nonumber\\ &&+\frac{i\hbar\eta\Omega_{\rm A}}{2}\left(\hat a\hat\sigma^+e^{-i\delta t}-\hat a^\dag\hat\sigma^-e^{i\delta t}\right),\end{aligned}$$ where $\Omega_{\rm A}$ is the ancilla Rabi frequency, $\eta\equiv\eta_m^{\rm A} = \delta k\sqrt{\frac{\hbar}{2M\omega_m}}$ is the Lamb-Dicke (LD) parameter associated to the $m$-th mode and the ancilla qubit, with $\delta k$ being the transferred wave-vector of the laser and $M$ being the mass of the ion, and $\Delta$ and $\delta$ are the detunings to the carrier and the first red-sideband transitions, respectively. In the dispersive regime, i.e. $\Omega_{\rm A}\ll\Delta$ and $\eta\Omega_{\rm A}\ll\delta$, the effective Hamiltonian can be written as follows [@Solano05], $$\begin{aligned} \hat H_{\rm eff} = \frac{\hbar\Omega_{\rm A}^2}{4\Delta}\hat\sigma^{\rm z}+\frac{\hbar\eta^2\Omega_{\rm A}^2}{4\delta}\left(\hat a^\dag\hat a+\frac{1}{2}\right)\hat\sigma^{\rm z},\end{aligned}$$ where the AC-Stark shift term can be absorbed into the free Hamiltonian, and the term that depends on the bosonic operators can be transformed into $\hat a^\dag\hat a\hat\sigma_{\rm A}^{\rm x}$ by appropriate single-qubit rotations on the ancilla. The third term, $\hat\sigma^{\rm z}_1\hat\sigma^{\rm z}_2\hat\sigma^{\rm z}_3$, can be implemented by combining the controlled-not (C-NOT) gates [@Nielsen00] or the MS gates [@Muller11] with single-qubit z-rotations. Fig. \[fig:Fig2\](a) shows a recipe to construct this nonlocal spin operator with C-NOT gates. Finally, similarly to the previous quantum circuit, we show that the dynamics governed by $\left({\hat a}^\dagger+\hat a\right)\hat\sigma_i^{\rm x}\hat\sigma_{\rm A}^{\rm x}$ can be implemented by combining C-NOT gates (or MS gates [@Casanova11_2]) with analog blocks that involve the dipolar coupling between the $m$-th motional mode with the ancilla qubit. In Fig. \[fig:Fig2\](b) we show the complete circuit to achieve the desired interaction. The central part of the circuit is a unitary operator produced by a simulated dipolar coupling that can be implemented by simultaneously driving the red-sideband and blue-sideband transitions of the ancilla qubit [@Solano02]. The associated evolution operator $\hat U_{\rm R}(\phi)$ in the rotating frame defined by the free Hamiltonian $\hat H_0$ is written as follows, $$\begin{aligned} \hat U_{\rm R}(\phi) = \exp\left[-\frac{i\phi}{2}\left(\hat a+\hat a^\dag\right)\hat\sigma_{\rm A}^{\rm x}\right],\end{aligned}$$ where $\phi$ is proportional to the Rabi frequency $\Omega_{\rm A}$, the Lamb-Dicke parameter $\eta_m^{\rm A}$, and the interaction time. Another qubit will be involved by a pair of C-NOT gates and all the other single-qubit rotations are used for changing the basis of the Pauli matrices. ![Quantum circuits for the trapped-ion toolkit. (a) Scheme for implementing the unitary $\exp{[-\frac{i\phi}{2}\hat\sigma_{i_1}^{\rm x}\hat\sigma_{i_2}^{\rm x}\hat\sigma_{\rm A}^{\rm x}]}$ using C-NOT gates and single-qubit rotations. Here, $\hat R_{\rm y}$ represents the action of a single-qubit rotation $\hat R_{\rm y}\equiv\exp{(i\pi \hat\sigma_{\rm y}/4)}$. (b) Scheme for implementing the unitary operator $\exp{[-\frac{i\phi}{2}\left(\hat a+\hat a^\dag\right)\hat\sigma_i^{\rm x}\hat\sigma_{\rm A}^{\rm x}]}$ using C-NOT gates, single-qubit gates and the analog block $\hat U_{\rm R}(\phi)$ involving the $m$-th motional mode.[]{data-label="fig:Fig2"}](Fig2.pdf "fig:"){width="45.00000%"}\ With the described toolkit available on trapped-ion quantum platforms, the probabilistic eigensolver can be extended to a wide range of models involving both qubits and bosonic modes. The latter include spin-spin interaction models like the Ising, XY or Heisenberg models [@Arrazola16], the quantum Rabi model [@Pedernales15; @Puebla17; @Puebla16], the Dicke models [@Aedo18], or second-quantization Hamiltonians like the Holstein model [@Mezzacapo12]. Also, more general methods like the ones for computing $n$-time correlation functions [@Pedernales14] or to simulate dissipative processes [@DiCandia15] could benefit from the presented eigensolver as the latter can be used to prepare arbitrary eigenstates. In the next section, we numerically investigate the performance of the probabilistic eigensolver with several examples, including a single harmonic oscillator, the quantum Rabi model, and the Hubbard model. Examples {#sec:examples} ======== Harmonic Oscillator {#subsec:ho} ------------------- One of the simplest and most important models of quantum mechanics is the quantum harmonic oscillator, which gives a mathematical description of various physical phenomena, including mechanical oscillators in harmonic potentials and electromagnetic fields [@Sakurai11]. The Hamiltonian of a single harmonic oscillator is written as follows, $$\begin{aligned} \label{eq:h_harmonic} \hat H=\hbar\omega_{\rm h.o.}\hat a^\dag\hat a,\end{aligned}$$ where $\hat a$ ($\hat a^\dag$) is the annihilation (creation) operator of the mode with the mode angular frequency $\omega_{\rm h.o.}$. The Hamiltonian in Eq. (\[eq:h\_harmonic\]) provides an equal-spacing spectrum with an infinite number of levels. One could think on a simple example of the protocol presented in Sec. \[sec:general\], where the application of a unitary of the form $\hat W(\tau)=\exp{[-i\tau\hat a^\dag \hat a \hat \sigma_{\rm A}^{\rm x}]}$ (see Sec. \[sec:toolkit\]) in a single ion, combined with measurements on the ancilla, could lead to a probabilistic preparation of the motional ground state. Note that the ground state energy for a harmonic oscillator is positive, thus $\gamma$ is set to zero. Starting from, for example, a thermal state, each time the ancilla is measured to be in the state ${\left| 0 \right\rangle}_{\rm A}$ after applying $\hat W(\tau)$, the motional state is effectively “cooled down" (i.e. the average energy is lowered). ![Probabilistic ground-state preparation of a single harmonic mode. (a) Average energy ${\left\langle \hat H \right\rangle}$ (black dots) and the overall success probability $P_{\rm suc}(k)$ (blue dots) as functions of the number of iterations $k$ for the fixed-step protocol. The normalized step size $\tau$ is fixed to be $0.3$. (b) Average energy ${\left\langle \hat H \right\rangle}$ (red dots over the black curve) and the overall success probability $P_{\rm suc}(k)$ (red dots over the blue curve) as functions of the number of iterations $k$ for the optimized variational protocol. Black dots represent results for all the different trial steps that are needed until reaching convergence for the optimum $\tau_k$ which are $\{0.8487, 0.5044, 0.9919, 0.9919, 0.9910, 0.7430, 0.4194, 0.9881\}$.[]{data-label="fig:cooling_harmonic"}](harmonic_oscillator "fig:"){width="40.00000%"}\ In Fig. \[fig:cooling\_harmonic\], we show the results for the probabilistic ground-state preparation of a harmonic oscillator with $\omega_{\rm h.o.}=1$. The initial state is naturally chosen as a thermal equilibrium state characterized by the thermal average number $\bar n_{\rm th} = 0.5$. Both the fixed-step cooling with $\tau = 0.3$ (Fig. \[fig:cooling\_harmonic\] (a)) and the variational cooling (Fig. \[fig:cooling\_harmonic\] (b)) show a monotonically decreasing average energy ${\left\langle \hat H \right\rangle}$ with respect to the number of iterations $k$. The overall success probability $P_{\rm suc}(k)$ up to the $k$-th stage is defined as $$\begin{aligned} P_{\rm suc}(k)= \prod_{k' = 1}^kP_{k'}^0,\end{aligned}$$ where $P_{k'}^0$ is the probability of projecting the ancilla to ${\left| 0 \right\rangle}_{\rm A}$ at the $k'$-th iteration. It is shown in Fig. \[fig:cooling\_harmonic\] that both in the fixed-step and variational cases, $P_{\rm suc}(k)$ decreases towards a saturate value around $0.65$. In this example, the fixed-step and the variational protocols reach convergence in $18$ and $8$ steps respectively, where a precision of $\epsilon=10^{-3}$ is pursued. In Fig. \[fig:cooling\_harmonic\] (b), it is shown that the total number of trial steps (black dots) needed to obtain the optimized variational protocol are around $80$ ($\sim10$ trials per step). However, once that you have the set of optimized values $\{\tau_k\}$, the variational approach requires fewer steps than the fixed-step approach, which reduces the circuit depth and alleviates the requirement for long-term quantum coherence. Quantum Rabi model ------------------ The quantum Rabi model [@Rabi36] describes the dipolar interaction between a magnetic dipole and an oscillating magnetic field. Recently, its quantum simulation has been realized experimentally with highly tunable parameters in many physical platforms, including superconducting circuits [@Braumuller17; @Langford17] and trapped ions [@Lv18]. The Hamiltonian of the quantum Rabi model can be separated into two non-commuting parts, $\hat H = \hat H_1+\hat H_2$, with $$\begin{aligned} \label{eq:h_rabi} \hat H_1 &=& \frac{\hbar\omega_0}{2}\hat\sigma^{\rm z}+\hbar\omega\hat a^\dag\hat a,\nonumber\\ \hat H_2 &=& \hbar g\left(\hat a+\hat a^\dag\right)\hat\sigma^{\rm x},\end{aligned}$$ where $\hat\sigma^{\rm x, z}$ are the Pauli matrices acting on the two-level system with the energy splitting $\omega_0$, $\hat a$ ($\hat a^\dag$) is the annihilation (creation) operator of the harmonic mode with frequency $\omega$, and $g$ is the coupling strength of the dipolar interaction. We implement $\hat W(\tau)$ through the second-order Trotter-Suzuki expansion, $$\begin{aligned} \hat W_\gamma(\tau) = \left[\hat W_1\left(\frac{\delta\tau}{2}\right)\hat W_2\left(\delta\tau\right)\hat W_1\left(\frac{\delta\tau}{2}\right)\right]^r,\end{aligned}$$ where $\delta\tau=\tau/r$ is the size of the Trotter step and $\hat W_j(\tau)=\exp\left(-i\hat H_j\hat\sigma_{\rm A}^{\rm x}\tau\right)$ with $j = 1$ and $2$. $\hat W_1$ could be implemented combining the first and second interactions described in Sec. \[sec:toolkit\], while $\hat W_2$ requires the implementation of the fourth interaction. Overall, the simulation requires two qubits and a motional mode. Note that depending on the step size $\tau$, more than one Trotter step may be needed. ![Probabilistic ground-state preparation of the quantum Rabi model with $\omega_0 = 1.2$, $\omega = 0.8$ and $g = 1.0$. The energy is in units of $g$ and the number of stages is denoted by $k$. The number of Trotter steps is chosen as $r = 3$. (a) Fixed-step probabilistic state preparation. The step size is chosen as $\tau_{\rm fix}=0.3$. (b) Variational probabilistic state preparation. The total number of trial steps is denoted by $n$ (black dots). The trial steps for different stages are discriminated by different background colours. Red dots represent the energy and overall probability for different stages, using the optimized $\tau$ values, $\{0.4762, 0.9839, 0.9032, 0.5575\}$.[]{data-label="fig:cooling_Rabi"}](rabi_model "fig:"){width="45.00000%"}\ Fig. \[fig:cooling\_Rabi\] shows the results for the ground state preparation of the quantum Rabi model in the deep strong coupling regime [@Casanova10]. We choose the state ${\left| \downarrow \right\rangle}{\left| 0 \right\rangle}$ as the initial state, for it can be easily prepared by optical pumping and sideband cooling. Comparing both cases we clearly see how the variational approach offers a faster route to the ground state preparation, with an overall success probability around $60\%$, very similar to the fixed-step approach. Hubbard Model ------------- The Hubbard model [@Hubbard63] is one of the most important models in solid-state physics. In spite of the simplicity in form, it describes the quantum phase transition between superconducting and Mott-insulator phases. The analytical solution has not been available for arbitrary dimensions and the exact numerical treatment is believed to be hard for classical computers because of the sign problem. In order to illustrate the performance of our protocol in a system with only qubits, we consider the one-dimensional Hubbard model with $L$ sites and open boundary conditions, whose Hamiltonian is written as follows, $$\begin{aligned} \hat H&=&-t\sum_{i=1}^{L-1}\sum_\sigma\left(\hat c_{i,\sigma}^\dag\hat c_{i+1,\sigma}+{\rm H.c.}\right)\nonumber\\ &&+U\sum_{i=1}^L\hat c_{i,\uparrow}^\dag\hat c_{i,\uparrow}\hat c_{i,\downarrow}^\dag\hat c_{i,\downarrow},\end{aligned}$$ where $t$ and $U$ are the nearest-neighbour hopping strength and the on-site interaction strength, respectively. This model can be mapped to a neighrest-neighbour spin chain of $2N$ spins through the Jordan-Wigner transformation [@Jordan28], which establishes a mapping between a fermionic operator and a set of spin operators. We choose the one-dimensional Hubbard model as an example, while the extension to 2D and 3D models can be done in trapped ions using nonlocal interactions [@Casanova11]. The operator in Eq. (\[eq:U\_tau\]) can be constructed using the three body spin operators described in Sec. \[sec:toolkit\] and the Suzuki-Trotter expansion. For the simulations, we choose $t=1$ and $U=2$ if not explicitly specified otherwise. ![Probabilistic ground state preparation of 2-site (a) and 3-site (b) Hubbard models. The black dots indicate the average energy of the trial states after successfully projecting the ancilla qubit to the reference state ${\left| 0 \right\rangle}_{\rm A}$, while the blue dots represent the overall success probability. Red dots represent the average energy and overall success probability after the $k$-th stage, which is done by applying the $\hat W_{\gamma}(\tau)$ unitary with the optimized $\tau_k$, using $r=3$ Trotter steps. The optimized taus are $\{0.3692, 0.3959, 0.3684, 0.3959\}$ and $\{0.2694, 0.3661, 0.3247, 0.3973, 0.2726, 0.3959\}$ for the 2-site and 3-site models, respectively.[]{data-label="fig:cooling_Hubbard"}](hubbard_model "fig:"){width="45.00000%"}\ Fig. \[fig:cooling\_Hubbard\] shows the numerical simulations for the ground state preparation of the 2-site and 3-site Hubbard models, using the variational approach for probabilistic state preparation. As initial states, we choose ${\left| \uparrow\downarrow\uparrow\downarrow \right\rangle}$ and ${\left| \downarrow\uparrow\downarrow\downarrow\uparrow\downarrow \right\rangle}$ for the 2-site and 3-site model respectively. In addition, the $\hat W_{\gamma}(\tau)$ operation is implemented using the symmetric Trotter expansion with $r=3$ Trotter steps. The results show that with less than $10$ steps, one can obtain the ground state for both models. The overall success probability is in this case around $15\%$ which is still an acceptable value. Conclusions {#sec:conclusions} =========== We have presented a probabilistic eigensolver which is capable of preparing arbitrary eigenstates of Hamiltonians that are implementable with a trapped-ion quantum processor. The method, applicable to a digital or digital-analog quantum simulator, requires control operations and measurements on an extra ancilla qubit. Moreover, we provide a recipe to enhance the performance of the probabilistic eigenstate preparation by means of a hybrid classical-quantum optimization algorithm. We describe a basic toolbox natural in trapped-ion quantum platforms, which can be used as building blocks to implement the method for complex Hamiltonian models. Finally, we numerically simulate the method for some interesting examples that could be implemented in state-of-the-art trapped-ion setups. This work was supported by the National Key Research and Development Program of China under Grants No. 2016YFA0301900 and No. 2016YFA0301901 and the National Natural Science Foundation of China Grants No. 11374178, No. 11574002, and No. 11504197. I. A. acknowledges support from Basque Government PhD Grant PRE-2015-1-0394, J. C. acknowledges support from Juan de la Cierva Grant IJCI-2016-29681, and L. L. acknowledges support from Ramón y Cajal Grant RYC-2012-11391. We also acknowledge funding from Spanish MINECO/FEDER FIS2015-69983-P and Basque Government IT986-16. Probabilistic Cooling Effect ============================ For an arbitrary input state ${\left| \psi \right\rangle}=\sum_jc_j{\left| j \right\rangle}$, the state for the system plus the ancilla qubit after applying $\hat W_\gamma(\tau)$ is denoted as ${\left| \Psi \right\rangle}\equiv\hat W_\gamma(\tau){\left| \psi \right\rangle}{\left| 0_{\rm A} \right\rangle}$. The values of the average energy with respect to the input state and conditioned output states are $$\begin{aligned} {\left\langle \hat H \right\rangle}&=&{\left\langle \psi \left| \hat H_{\rm S} \right| \psi \right\rangle}=\sum_jE_j\left|c_j\right|^2,\\ {\left\langle \hat H \right\rangle}^{(0)}&=&{\left\langle \Psi \left| \hat H_{\rm S}\otimes|0_{\rm A}\rangle\langle0_{\rm A}| \right| \Psi \right\rangle}\nonumber\\ &=&\frac{1}{P_0}\sum_jE_j\left|c_j\right|^2\cos^2\left[\left(E_j+\gamma\right)\tau\right],\nonumber\\ {\left\langle \hat H \right\rangle}^{(1)}&=&{\left\langle \Psi \left| \hat H_{\rm S} \otimes |1_{\rm A}\rangle\langle1_{\rm A}| \right| \Psi \right\rangle}\nonumber\\ &=&\frac{1}{P_1}\sum_jE_j\left|c_j\right|^2\sin^2\left[\left(E_j+\gamma\right)\tau\right],\nonumber\end{aligned}$$ where $P_{0} =\sum_j |c_j|^2 \cos^2\left[\left(E_j+\gamma\right)\tau\right]$ and $P_{1}=\sum_j|c_j|^2\sin^2\left[\left(E_j+\gamma\right)\tau\right]$. For small $\tau$, i.e. $(E_j+\gamma)\tau \ll 1$, we expand ${\left\langle \hat H \right\rangle}^{(0)}$ to the second order of $\tau$, and calculate the difference between ${\left\langle \hat H \right\rangle}^{(0)}$ and ${\left\langle \hat H \right\rangle}$ as follows, $$\begin{aligned} \label{eq:quasi_cooling} {\left\langle \hat H_{\rm S} \right\rangle}-{\left\langle \hat H_{\rm S} \right\rangle}^{(0)}=\sum_{j}(|c_j|^2-|c'_j|^2)E_j,\end{aligned}$$ where the new probability distribution is related with the old one as $$\begin{aligned} \label{eq:quasi_cooling2} \frac{c'_j}{c_j}&=&\frac{1}{P_{0}} \cos{[(E_j+\gamma)\tau]} \\ &\approx&\frac{1}{P_{0}}(1-\frac{1}{2}(E_j+\gamma)^2\tau^2)+{\mathcal O}\left(\tau^4\right). \nonumber\end{aligned}$$ As $E_0+\gamma\geq0$, the factor $(1-\frac{1}{2}(E_j+\gamma)^2\tau^2)$ decreases for higher excited states, meaning that the higher the energy of the state, the lower the probability amplitude at the output state. Similarly, we can calculate the difference between ${\left\langle \hat H_{\rm S} \right\rangle}^{(1)}$ and ${\left\langle \hat H_{\rm S} \right\rangle}$ to the leading order and obtain the following expression, $$\begin{aligned} {\left\langle \hat H_{\rm S} \right\rangle}^{(1)}-{\left\langle \hat H_{\rm S} \right\rangle}=\sum_{j}(|c''_j|^2-|c_j|^2)E_j,\end{aligned}$$ where now the new probability distribution is related with the old one as $$\begin{aligned} \frac{c''_j}{c_j}&=&\frac{1}{P_{1}} \sin{[(E_j+\gamma)\tau]} \\ &\approx&\frac{1}{P_{1}}(E_j+\gamma)\tau+{\mathcal O}\left(\tau^3\right). \nonumber\end{aligned}$$ Notice that in this case the lower the energy of the state, the lower the ratio between the new and the old probability amplitude. [99]{} R. P. Feynman, Simulating Physics with Computers, Int. J. Theor. Phys. [**21**]{}, 467 (1982). I. M. Georgescu, S. Ashhab, and F. Nori, Quantum Simulation, Rev. Mod. Phys. [**86**]{}, 153 (2014). A. W. Harrow, A. Hassidim, and S. 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--- abstract: 'A. Mukhopadhyay, M. R. Murty and K. Srinivas have recently studied various arithmetic properties of the discriminant $\Delta_n(a,b)$ of the trinomial $f_{n,a,b}(t) = t^n + at + b$, where $n \ge 5$ is a fixed integer. In particular, it is shown that, under the $abc$-conjecture, for every $n \equiv 1 \pmod 4$, the quadratic fields $\Q\(\sqrt{\Delta_n(a,b)}\)$ are pairwise distinct for a positive proportion of such discriminants with integers $a$ and $b$ such that $f_{n,a,b}$ is irreducible over $\Q$ and $|\Delta_n(a,b)|\le X$, as $X\to \infty$. We use the square-sieve and bounds of character sums to obtain a weaker but unconditional version of this result.' author: - | [Igor E. Shparlinski[^1]]{}\ Department of Computing, Macquarie University\ Sydney, NSW 2109, Australia\ [[email protected]]{} title: On Quadratic Fields Generated by Discriminants of Irreducible Trinomials --- Ł[[L]{}]{} \ ${\left(} \def$[)]{} \#1[\#1]{} \#1[\#1]{} *[\_m]{}* #### Mathematical Subject Classification (2000): Primary 11R11; Secondary 11L40, 11N36, 11R09 #### Keywords: Irreducible trinomials, quadratic fields, square sieve, character sums Introduction ============ For a fixed integer $n\ge 2$, we use $\Delta_n(a,b)$ to denote the discriminant of the trinomial $$f_{n,a,b}(t) = t^n + at + b.$$ A. Mukhopadhyay, M. R. Murty and K. Srinivas [@MMS] have recently studied the arithmetic structure of $\Delta_n(a,b)$. In particular, it is shown in [@MMS], under the $abc$-conjecture, that if $n \equiv 1 \pmod 4$ then for a sufficiently large positive $A$ and $B$ such that $B \ge A^{1+\delta}$ with some fixed $\delta> 0$, there are at least $\gamma AB$ integers $a,b$ with $$A \le |a| \le 2A \qquad \text{and}\qquad B \le |b| \le 2B$$ and such that $f_{n,a,b}$ is irreducible and $\Delta_n(a,b)$ is square-free, where $\gamma > 0$ depends only on $n$ and $\delta$. Then this result is used to derive (still under the $abc$-conjecture) that the quadratic fields $\Q\(\sqrt{\Delta_n(a,b)}\)$ are pairwise distinct for a positive proportion of such discriminants with integers $a$ and $b$ such that $f_{n,a,b}$ is irreducible over $\Q$. More precisely, for a real $X \ge 1$, let $Q_n(X)$ be the number of distinct fields $\Q\(\sqrt{\Delta_n(a,b)}\)$ taken for all of pairs of integers $a, b$ such that $f_{n,a,b}$ is irreducible over $\Q$ and $|\Delta_n(a,b)|\le X$. Throughout the paper, we use $U = O(V)$, $U \ll V$, and $V \gg U$ as equivalents of the inequality $|U| \le c V$ with some constant $c> 0$, which may depend only on $n$. It is shown in [@MMS] that for a fixed $n \equiv 1 \pmod 4$, $$\label{eq:Q MMS bound} Q(X) \gg X^{\kappa_n},$$ where $$\kappa_n=\frac{1}{n} + \frac{1}{n-1}$$ and $c_0 > 0$ is a constant depending only on $n$. It is also noted in [@MMS] that the Galois groups of irreducible trinomials $f_{n,a,b}$ have some interesting properties, see also [@CMS; @HeSa; @PlVi]. We remark that, since $$\label{eq:Dab} \Delta_n(a,b) = (n-1)^{n-1}a^n + n^n b^{n-1}$$ for $n \equiv 1 \pmod 4$, there are $O\(X^{\frac{1}{n} + \frac{1}{n-1}}\)$ integers $a$ and $b$ with $|\Delta_n(a,b)|\le X$ and thus indeed  means that $$Q(X) \gg \#\{(a,b) \in \Z^2~:~ |\Delta_n(a,b)|\le X\}.$$ We use the square-sieve and bounds of character sums to obtain a weaker but unconditional version of this result. We note that, without the irreducibility of $f_{n,a,b}$ condition, the problem of estimating $Q(X)$ can be viewed as a bivariate analogue of the question, considered in [@LuSh], on the number of distinct quadratic fields of the form $\Q\(\sqrt{F(n)}\)$ for $n=M+1,\ldots, M+N$, for a nonconstant polynomial $F(T) \in \Z[T]$. Accordingly we use similar ideas, however we also exploit the specific shape of the polynomial $\Delta_n(a,b)$ given by . Main result =========== In fact as in [@LuSh] we consider a more general quantity than $Q(X)$. Namely, for real positive $A$, $B$, $C$ and $D$ and a square-free integer $s$, we denote by $T_n(A,B,C,D;s)$ the number of pairs of integers $$(a,b) \in [C,C+A]\times [D,D+B]$$ such that $\Delta_n(a,b) = sr^2$ for some integer $r$. We write $\log x$ for the maximum of the natural logarithm of $x$ and 1, thus we always have $\log x \ge 1$. \[thm: TABCDs\] For real $A\ge 1$, $B\ge 1$, $C\ge 0$ and $D\ge 0$ and a square-free $s$, we have $$\begin{aligned} T_n(A,B,C,D;s) &\ll &(AB)^{2/3} \log (AB) + A \log (AB) + B \log (AB)\\ & & \qquad\qquad\qquad +~(AB)^{1/3} \(\frac{\log (ABCD) \log (AB)}{\log \log (ABCD)}\)^2. \end{aligned}$$. Now, for real $A$, $B$, $C$ and $D$ we denote by $S_n(A,B,C,D)$ the number of distinct quadratic fields $\Q\(\sqrt{\Delta_n(a,b)}\)$ taken for all pairs of integers $$(a,b) \in [C,C+A]\times [D,D+B]$$ such that $f_{n,a,b}$ is irreducible over $\Q$. Using that the bound of Theorem \[thm: TABCDs\] is uniform in $s$, we derive \[thm: SABCD\] For real $A\ge 1$, $B\ge 1$, $C\ge 0$ and $D\ge 0$ , we have $$\begin{split} S_n(A,B,C,D) \gg \min\Biggl\{\frac{(AB)^{1/3}}{\log (AB)},\, \frac{A}{\log (AB)},\, & \frac{B}{\log (AB)} ,\\ (AB)^{2/3} & \(\frac{\log \log (ABCD)}{\log (ABCD) \log (AB)}\)^2\Biggr\}. \end{split}$$. The results of Theorems \[thm: TABCDs\] and \[thm: SABCD\] are nontrivial in a very wide range of parameters $A$, $B$, $C$ and $D$ and apply to very short intervals. In particular, $AB$ could be logarithmically small compared to $CD$. Furthermore, taking $$A = C = \frac{1}{4 (n-1)^{n-1}}X^{1/n} \qquad \text{and} \qquad B = D = \frac{1}{4n^n}X^{1/(n-1)}$$ we see that $$Q(X) \gg X^{\kappa_n/3} (\log X)^{-1},$$ which, although is weaker than , does not depend on any unproven conjectures. Character Sums with the Discriminant ==================================== Our proofs rest on some bounds for character sums. For an odd integer $m$ we use $(w/m)$ to denote, as usual, the Jacobi symbol of $w$ modulo $m$. We also put $$\em(w)= \exp(2 \pi i w/m).$$ Given an odd integer $m\ge 3$ and arbitrary integers $\lambda, \mu$, we consider the double character sums $$S_n(m; \lambda, \mu) = \sum_{u,v=1}^{m} \(\frac{\Delta_n(u,v)}{m}\) \em\(\lambda u + \mu v\).$$ We need bounds of these sums in the case of $m =\ell_1\ell_2$ being a product of two primes $\ell_1> \ell_2\ge n$. However, using the multiplicative property of character sums (see [@IwKow Equation (12.21)] for single sums, double sums behaves exactly the same way) we see that it is enough to estimate $S_n(\ell; \lambda, \mu)$ for primes $\ell$. We start with evaluating these sums in the special case of $\lambda = \mu=0$ where we define $$S_n(\ell) = S_n(\ell; 0, 0) .$$ \[lem:Sm00\] For $n \equiv 1 \pmod 4$ and a prime $\ell$, we have $$S_n(\ell) \ll \ell.$$ We can certainly assume that $\ell\ge n$ as otherwise the bound is trivial. Recalling , we derive $$\begin{aligned} S_n(\ell) & = & \sum_{u,v=1}^{\ell} \(\frac{(n-1)^{n-1}u^n + n^n v^{n-1}}{\ell}\) \\ & = & \sum_{u, v=1}^{\ell-1} \(\frac{(n-1)^{n-1}u^n + n^n v^{n-1}}{\ell}\) + O(\ell). \end{aligned}$$ Substituting $uv$ instead of $u$, we obtain $$\begin{aligned} S_n(\ell) & = & \sum_{v=1}^{\ell-1} \sum_{u=1}^{\ell-1} \(\frac{(n-1)^{n-1}(uv)^n + n^n v^{n-1}}{\ell}\) + O(\ell) \\ & = & \sum_{u, v=1}^{\ell-1} \(\frac{\( (n-1)^{n-1}u^nv + n^n\) v^{n-1}}{\ell}\) + O(\ell)\\ & = & \sum_{u, v=1}^{\ell-1} \(\frac{(n-1)^{n-1}u^nv + n^n }{\ell}\)+ O(\ell) \end{aligned}$$ since $n-1$ is even. We now rewrite it in a slightly more convenient form as $$S_n(\ell) = \sum_{u=1}^{\ell-1} \sum_{v=1}^{\ell} \(\frac{(n-1)^{n-1}u^nv + n^n }{\ell}\)+ O(\ell).$$ As $\gcd(\ell,n-1)=1$, making the change of variables $(n-1)^{n-1}u^nv + n^n= w$, we note that for every $u= 1, \ldots, \ell-1$ if $v =1, \ldots, \ell$, then $w$ runs through the complete residue system modulo $\ell$. Hence, $$S_n(\ell)= (\ell -1) \sum_{w=1}^{\ell} \(\frac{w }{\ell}\) + O(\ell) = O(\ell),$$ which concludes the proof. The following result can be derived from [@FK Theorem 1.1], however we give a self-contained and more elementary proof. \[lem:Sm arb\] For $n \equiv 1 \pmod 4$, a prime $\ell$ and arbitrary integers $\lambda,\mu$ with $\gcd(\lambda,\mu, \ell)=1$, we have $$|S_n(\ell;\lambda, \mu)| \ll \ell.$$ As in the proof of Lemma \[lem:Sm00\], we can certainly assume that $\ell\ge n$ as otherwise the bound is trivial. Also as in the proof of Lemma \[lem:Sm00\], we obtain $$S_n(\ell;\lambda, \mu) = \sum_{u=1}^{\ell-1} \sum_{v=1}^{\ell} \(\frac{(n-1)^{n-1}u^nv + n^n }{\ell}\)\el\((\lambda u + \mu) v\) + O(\ell).$$ As $\gcd(\ell,n-1)=1$, making the change of variables $(n-1)^{n-1}u^nv + n^n = w$, we obtain $$\begin{aligned} S_n(\ell;\lambda, \mu) & = &\sum_{u=1}^{\ell-1} \sum_{w=1}^{\ell} \(\frac{w}{\ell}\)\el\((n-1)^{-n+1}u^{-n}(\lambda u + \mu) (w -n^n)\) + O(\ell)\\ & = &\sum_{u=1}^{\ell-1} \el\(-(n-1)^{-n+1}n^nu^{-n}(\lambda u + \mu) \) \\ & &\qquad \qquad \quad \sum_{w=1}^{\ell} \(\frac{w}{\ell}\)\el\((n-1)^{-n+1}u^{-n}(\lambda u + \mu) w\) + O(\ell).\end{aligned}$$ The sums over $w$ is the Gauss sum, thus $$\begin{aligned} \lefteqn{ \sum_{w=1}^{\ell} \(\frac{w}{\ell}\)\el\((n-1)^{-n+1}u^{-n}(\lambda u + \mu) w\) }\\ & & \qquad \qquad \qquad \qquad \qquad = \(\frac{(n-1)^{-n+1}u^{-n}(\lambda u + \mu)}{\ell}\) \vartheta_\ell \ell^{1/2}, \end{aligned}$$ for some complex $ \vartheta_\ell$ with $| \vartheta_\ell|=1$ (which depends only on the residue class of $\ell$ modulo $4$), we refer to [@IwKow; @LN] for details. Since $n\equiv 1 \pmod 4$ we have $$\(\frac{u^{-n}}{\ell}\) = \(\frac{u^{-1}}{\ell}\) .$$ Thus, combining the above identities we obtain $$\begin{aligned} S_n(\ell;\lambda, \mu) & = & \vartheta_\ell \ell^{1/2}\sum_{u=1}^{\ell-1} \(\frac{(n-1)^{-n+1}(\lambda + \mu u^{-1})}{\ell}\)\\ & &\qquad \qquad \qquad \el\(-(n-1)^{-n+1}n^nu^{-n}(\lambda u + \mu) \) + O(\ell).\end{aligned}$$ Since $\gcd(\lambda,\mu, \ell)=1$, the Weil bound (see [@IwKow Bound (12.23)]) applies and implies that the sum over $u$ is $O(\ell^{1/2})$ which concludes the proof. Combining Lemmas \[lem:Sm00\] and \[lem:Sm arb\], and using the aforementioned multiplicativity property, we obtain \[lem:Sm compl\] For $n \equiv 1 \pmod 4$, an integer $m = \ell_1\ell_2$ which is a product of two distinct primes $\ell_1 \ne \ell_2$ and arbitrary integers $\lambda,\mu$, we have $$|S_n(m; \lambda, \mu)| \ll m.$$ Finally, using the standard reduction between complete and incomplete sums (see [@IwKow Section 12.2]), we derive from Lemma \[lem:Sm compl\] \[lem:Sm incompl\] For $n \equiv 1 \pmod 4$, an integer $m = \ell_1\ell_2$ which is a product of two distinct primes $\ell_1 \ne \ell_2$ and real positive $A$, $B$, $C$ and $D$, we have $$\sum_{C \le a \le C+A}\, \sum_{D \le b \le D+B} \(\frac{\Delta_n(a,b)}{m}\) \ll \(\frac{A}{m} + 1\)\(\frac{B}{m}+1\) m \log m.$$ Irreducibility ============== As in [@MMS] we recall a very special case of a results of S. D. Cohen [@Coh] about the distribution of irreducible polynomials over a finite field $\F_q$ of $q$ elements. \[lem:Irred Trinom\] For any prime $p$, there are $p^2/n + O(p^{3/2})$ irreducible trinomials $t^n +\alpha t + \beta \in \F_p[t]$. Proof of Theorem \[thm: TABCDs\] ================================ For a real number $z \ge 1$ we let $\cL_z$ be the set of primes $\ell \in [z, 2z]$. For a positive integer $k$ we write $\omega(k)$ for the number of prime factors of $k$. We note that if $k\ge 1$ is a perfect square, then for $z \ge 3$, $$\sum_{\ell \in \cL_z} \(\frac{k}{\ell}\) \ge \# \cL_z - \omega(k),$$ For each pair $(a,b)$, counted in $T_n(A,B,C,D;s)$, we see t hat $s\Delta_n(a,b)$ is a perfect square and that $s\mid \Delta_n(a,b)$. Hence, $$\omega(s\Delta_n(a,b)) = \omega(\Delta_n(a,b)).$$ Thus, for such $(a,b)$ we have $$\sum_{\ell \in \cL_z} \(\frac{s\Delta_n(a,b)}{\ell}\) \ge \# \cL_z - \omega_z(s\Delta_n(a,b)) = \# \cL_z - \omega(\Delta_n(a,b)) .$$ Since $\omega(k)! \le k$, we see from the Stirling formula that $$\omega(k) \ll \frac{\log k}{\log \log k}.$$ Thus $$\omega(\Delta_n(a,b)) \ll \frac{\log (A+B+C+D)}{\log \log (A+B+C+D)} \ll \frac{\log (ABCD)}{\log \log (ABCD)}.$$ In particular, by the Cauchy inequality, $$\begin{aligned} \lefteqn{\(\# \cL_z\)^2T_n(A,B,C,D;s) }\\ & & \qquad \ll \sum_{C \le a \le C+A}\, \sum_{D \le b \le D+B} \(\sum_{\ell \in \cL_z} \(\frac{s\Delta_n(a,b)}{\ell}\) + \omega(\Delta_n(a,b)) \)^2\\ & & \qquad \ll \sum_{C \le a \le C+A}\, \sum_{D \le b \le D+B} \(\sum_{\ell \in \cL_z} \(\frac{s\Delta_n(a,b)}{\ell}\) \)^2 \\ & & \qquad \qquad \qquad\qquad\qquad\qquad\qquad +~AB \( \frac{\log (ABCD)}{\log \log (ABCD)} \)^2. \end{aligned}$$ We note that $$\begin{aligned} \lefteqn{ \sum_{C \le a \le C+A}\, \sum_{D \le b \le D+B} \(\sum_{\ell \in \cL_z} \(\frac{s\Delta_n(a,b)}{\ell}\) \)^2 }\\ & & \qquad = \sum_{C \le a \le C+A}\, \sum_{D \le b \le D+B} \(\(\frac{s}{\ell}\) \sum_{\ell \in \cL_z} \(\frac{\Delta_n(a,b)}{\ell}\) \)^2\\ & & \qquad \le \sum_{C \le a \le C+A}\, \sum_{D \le b \le D+B} \(\sum_{\ell \in \cL_z} \(\frac{\Delta_n(a,b)}{\ell}\) \)^2 .\end{aligned}$$ Squaring out and changing the order of summation, we obtain $$\begin{aligned} \lefteqn{\(\# \cL_z\)^2T_n(A,B,C,D;s) }\\ & & \qquad \ll \sum_{\ell_1, \ell_2 \in \cL_z} \sum_{C \le a \le C+A}\, \sum_{D \le b \le D+B} \(\frac{\Delta_n(a,b)}{\ell_1\ell_2}\) \\ & & \qquad \qquad \qquad\qquad\qquad+ AB \( \frac{\log (ABCD)}{\log \log (ABCD)}\)^2. \end{aligned}$$ We now estimate the double sum over $a$ and $b$ trivially as $O(AB)$ on the “diagonal” $\ell_1 = \ell_2$ and use Lemma \[lem:Sm incompl\] otherwise, getting $$\label{eq:Prelim} \begin{split} \(\# \cL_z\)^2T_n(A,B,C,D;s) \ll \# \cL_z AB + \(\# \cL_z\)^2 & \(\frac{A}{z^2} + 1\)\(\frac{B}{z^2}+1\) z^2 \log z\\ + AB & \( \frac{\log (ABCD)}{\log \log (ABCD)}\)^2. \end{split}$$ By the prime number theorem we have $\# \cL_z \gg z/\log z$ so we derive from  that $$\begin{aligned} T_n(A,B,C,D;s) & \ll & ABz^{-1} \log z + AB z^{-2} + A \log z + B \log z\\ & & \qquad \qquad +~z^2 \log z+ABz^{-2} \( \frac{\log (ABCD) \log z}{\log \log (ABCD)}\)^2. \end{aligned}$$ Clear the first term always dominates the second one, so the second term can be simply dropped. Thus taking $z = (AB)^{1/3}$ to balance the terms $ABz^{-1} \log z $ and $ z^2 \log z $, we obtain the desired estimate. Proof of Theorem \[thm: SABCD\] =============================== Let $p_0$ be smallest prime for which there exists an irreducible trinomial $$t^n +\alpha_0 t + \beta_0 \in \F_{p_0}[t]$$ ($p_0$ exists by Lemma \[lem:Irred Trinom\]). We now define the sets of integers $$\label{eq:Sets AB} \begin{split} \cA & =~\{a \in [C,C+A]\cap \Z~:~ a \equiv \alpha_0 \pmod {p_0}\};\\ \cB & =~\{b \in [D,D+B]\cap \Z~:~ b \equiv \beta_0 \pmod {p_0}\} \end{split}$$ Clearly $$\label{eq:CardAB} \begin{split} \# \cA \gg A \mand \# \cB \gg B \end{split}$$ and every trinomials $t^n + at + b$ with $a\in \cA$, $b \in \cB$ is irreducible over $\Z$. Using Theorem \[thm: TABCDs\] to estimate the number of pairs $(a,b) \in \cA\times \cB$ for which $\Q\(\sqrt{\Delta_n(a,b)}\)$ is a given quadratic field, we obtain the desired result. Remarks ======= Similar results can be obtained for more general trinomials $t^n +at^m + b$ with fixed integers $n > m \ge 1$. Some properties of the Galois group of these trinomials have been studied in [@CMS; @HeSa; @PlVi] where one can also find an explicit formula for their discriminant (which generalises ). In the case of $a=b=1$ it becomes $(-1)^{n(n-1)/2} \(n^n - (-1)^{n}m^m (n-m)^{n-m}\)$. Studying arithmetic properties of this expression, for example, its square-free part, when $n$ and $m$ vary in the region $N \ge n > m \ge 1$ for a sufficiently large $N$, is a very challenging question. [99]{} S. D. Cohen, ‘The distribution of polynomials over finite fields’, [*Acta Arith.*]{}, [**17**]{} (1970), 255–271. S. D. Cohen, A. Movahhedi and A. Salinier, ‘Galois groups of trinomials’, [*J. Algebra*]{}, [**222**]{} (1999), 561–573. E. Fouvry and N. Katz, ‘A general stratification theorem for exponential sums, and applications’, [*J. Reine Angew. Math.*]{}, [**540**]{} (2001), 115-166. D. R. Heath-Brown, ‘The square sieve and consecutive squarefree numbers’, [*Math. Ann.*]{}, [**266**]{} (1984), 251–259. A. Hermez and A. Salinier, ‘Rational trinomials with the alternating group as Galois group’, [*J. Number Theory*]{}, [**90**]{} (2001), 113–129. H. Iwaniec and E. Kowalski, [*Analytic number theory*]{}, Amer. Math. Soc., Providence, RI, 2004. R. Lidl and H. Niederreiter, [*Finite fields*]{}, Cambridge University Press, Cambridge, 1997. F. Luca and I. E. Shparlinski, ‘Quadratic fields generated by polynomials’, [*Archiv Math.*]{}, (to appear). A. Mukhopadhyay, M. R. Murty and K. Srinivas, ‘Counting squarefree discriminants of trinomials under $abc$’, [*Preprint*]{}, 2008, (available from [http://arxiv.org/abs/0808.0418]{}). B. Plans and N. Vila, ‘Trinomial extensions of $\mathbb Q$ with ramification conditions’, [*J. Number Theory*]{}, [**105**]{} (2004), 387–400. [^1]: This work was supported in part by ARC Grant DP0556431

--- abstract: 'The mesoscopic characteristics of a quantum dot (QD), which make the dipole approximation (DA) break down, provide a new dimension to manipulate light-matter interaction \[M. L. Andersen, *et al.*, Nat. Phys. **7**, 215 (2011)\]. Here we investigate the power spectrum and the second-order correlation property of the fluorescence from a resonantly driven QD placed on a planar metal. It is revealed that due to the pronounced QD spatial extension and the dramatic variation of the triggered surface plasmon near the metal, the fluorescence has a notable contribution from the quadrupole moment. The $\pi$-rotation symmetry of the fluorescence to the QD orientation under the DA is broken. By manipulating the QD orientation and quadrupole moment, the spectrum can be switched between the Mollow triplet and a single peak, and the fluorescence characterized by the antibunching in the second-order correlation function can be changed from the weak to the strong radiation regime. Our result is instructive for utilizing the unique mesoscopic effects to develop nanophotonic devices.' author: - 'Chun-Jie Yang' - 'Jun-Hong An' title: Resonance fluorescence beyond the dipole approximation of a quantum dot in a plasmonic nanostructure --- introduction ============ Quantum optics has advanced to the stage of experimental measurement and manipulation of individual quantum systems in single quanta level [@Haroche2013a; @Wineland2013; @Koelemeij2014; @Bloom2014; @Wang2015; @Kato2015], where the light-matter interaction plays an important role. Considerable interest has been generated in exploring new mechanisms that enable efficient control of the light-matter interaction. In past years, with the sufficient reduction of the effective mode volume for photons, strong and even ultrastrong light-matter interaction have been experimentally realized [@Wallraff2004; @Guebrou2012; @Niemczyk2010; @Scalari2012; @Tudela2013; @Cacciola2014; @Torma2015]. Recently, a scheme exploiting the mesoscopic characteristics of quantum dots (QDs) was proposed [@Andersen2011], by means of which the plasmon-matter interaction [@Akimov2007] can be strongly modified. The light-matter interaction is generally described under the dipole approximation (DA), which works well in atomic systems where the variation of the field is negligible within the atomic spatial extension [@Cronin2009; @Haroche2013b; @Muller2007; @Monniello2013]. However, once the spatial variation of the field becomes pronounced, such as the surface plasmons triggered by the radiation field of the quantum emitter [@Zayatsa2005], and the emitter is spatially extended, such as a QD several tens of nanometers in size [@Yoshie2004; @Lodahl2015], the validity of the DA is not clear *a priori*. Experimentally, a large deviation from the dipole theory was observed for QDs in close proximity to a silver mirror [@Andersen2011]. The optical response of quantum nanosystems beyond the long-wavelength approximation (equivalent to the DA) has been studied semiclassically [@Ajiki2002; @Cho2003; @Bamba2008; @Iida2009], which indicated that the nonlocal spatial interplay between the wave functions of the QD exciton and the electromagnetic field makes the DA invalid. Unconventional phenomena exceeding the DA have been found, such as the selection-rule breakdown of an isolated single-walled carbon nanotube in a nanogap [@Iida2011; @Takase2013], entangled-photon generation from biexcitons in a semiconductor film [@Bamba2010], and enhanced up-conversion of entangled photons in nanostructures [@Osaka2012]. In the fully quantum theory, making Taylor expansion of the field spatial distribution function to the first order, it is found that the nonlocal interaction is described by the quadrupole moment, and a microscopic picture of it from a circular quantum current density flowing along a curved path inside the QD has been provided [@Stobbe2012; @Tighineanu2015]. Furthermore, as the quadrupole moment can be tuned by controlling the size and shape of the QD [@Johansen2008; @Leistikow2009], it has potential applications in the development of a nanophotonic devices. For the requirement of developing nanoplasmonic single-photon source, a study on the fluorescence from a resonantly driven QD, especially the second-order correlation property of the fluorescence, modified by the mesoscopic characteristics is necessary and important. In this work, we study the resonance fluorescence of a mesoscopic QD in different spatial orientations placed near a plasmonic nanostructure. Going beyond the DA, a microscopic description to the power spectrum and the second-order correlation property of the QD fluorescence is established. The substantial deviations from the dipole theory are found when the QD is positioned within the penetration depth of the plasmons into the dielectric. It is revealed that the spatial rotation symmetry for the resonance fluorescence spectrum over the QD orientation is changed from $\pi$ under the DA to $2\pi$ due to the interference of the emission from the dipole and the quadrupole moments. The widths and intensities of the spectral peaks differ dramatically from those under the DA due to the cooperative actions of the dipole and quadrupole moments. Explicitly, by exploiting the QD mesoscopic effects, the spectrum can be switched between a single peak and the Mollow triplet. The analysis on the second-order correlation property indicates that, keeping the nonclassical antibunching nature, the fluorescence can be changed from the weak to the strong emission regime by increasing the quadrupole moment. This opens an avenue to develop nanophotonic single-photon devices by use of the QD mesoscopic characters. Our parameter values are all experimentally attainable. Our paper is organized as follows. In Sec. \[model\], we show the model and establish a microscopic description to the QD–surface-plasmon interaction in arbitrary QD orientations beyond the DA. In Sec. \[fluorescence\], the fluorescence spectrum and the second-order correlation property are numerically studied. In Sec. \[con\], a summary is given. QD–surface-plasmon interaction beyond the DA {#model} ============================================ System and Green’s tensor ------------------------- ![Diagram of a QD with characteristic frequency $\omega_0$ embedded in gallium arsenide at a distance $\Delta z$ above a metal. $\varepsilon_1$ and $\varepsilon_m$ are dielectric constants of the media. $z$ and $z^\prime$ represent different orientations of the QD.[]{data-label="Fig1"}](Fig1.eps){width="0.9\columnwidth"} Our system is depicted in Fig. \[Fig1\]: a QD with frequency $\omega_0$ embedded in gallium arsenide (GaAs) media is placed on a dissipative metal. The metal is characterized by a complex Drude dielectric function $\varepsilon_m (\omega )=\varepsilon _{\infty }[1-\frac{\omega _{p}^{2}}{\omega (\omega +i\gamma _{p})}]$, where $\omega _{p}$ is the bulk plasma frequency, $\varepsilon _{\infty}$ is the high-frequency limit of the metal dielectric function, and $\gamma _{p}$ represents the Ohmic loss responsible for the dissipation of the electromagnetic field in the metal. Here the metal is chosen as silver with the parameters $\omega_{p}=3.76$ eV, $\varepsilon_{\infty}=9.6$, $\gamma_{p}=0.03\omega_{p}$ in the interested frequency range, and the dielectric permittivity of GaAs is $\varepsilon_1=12.25$ [@Johnson1972; @Tudela2010]. We assume that the layered media are linear, isotropic and nonmagnetic ($\mu_{1}=\mu_{\text{m}}=1$). The electromagnetic field in dispersive and absorbing dielectrics is described by the Green’s tensor $\mathbf{G}(\mathbf{r},\mathbf{r}^{\prime };\omega )$ [@Dung1998; @Sondergaard2001], which is rendered as the field in frequency $\omega$ evaluated at $\mathbf{r}$ due to a point source at $\mathbf{r}^{\prime}$. It can be obtained by solving the Maxwell-Helmholtz wave equation $[{\pmb\nabla}\times{\pmb\nabla}\times-\frac{\omega^{2}}{c^{2}}\varepsilon(\omega)]\mathbf{G}(\mathbf{r},\mathbf{r}^{\prime };\omega )=\mathbf{I}\delta (\mathbf{r}-\mathbf{r}^{\prime })$, where $\mathbf{I}$ is identity matrix. For general geometries, the solving needs some numerical methods, such as the finite difference time domain and the finite element methods [@Chen2010b; @Cano2010]. For symmetric geometries such as spheres, cylinders, or planes, its analytical solution is achievable [@Novotny2006; @Dzsotjan2010; @Zubairy2014]. In our configuration, $\mathbf{G}(\mathbf{r},\mathbf{r}^{\prime };\omega )$ in the upper half-space of the metal-dielectric interface is calculated as the sum of the free-space and reflected Green’s tensors $\mathbf{G}(\mathbf{r},\mathbf{r}^{\prime },\omega )=\mathbf{G}_{0}(\mathbf{r},\mathbf{r}^{\prime },\omega )+\mathbf{G}_{\text{R}}(\mathbf{r},\mathbf{r}^{\prime},\omega )$. See more details in Appendix \[Green\]. Three distinct modes are triggered by the emission of the QD. The first one is the radiative modes propagating into the free space. The second one is the damped non-radiative mode due to the Ohmic loss in the metal. The last one is the tightly confined field called surface plasmon propagating along the metal surface [@Pitarke2007]. The electromagnetic modes associated with the surface plasmon enable strong confinement of light on the surface and thus enhance the light-matter interaction, which has inspired great interests in studying surface plasmon subwavelength optics and quantum plasmonics [@Barnes2003; @Garcia2010; @Tame2013]. In addition, due to the exponential decay of the intensity of the electromagnetic field perpendicular to the metal surface, its variation along this direction is pronounced within the spatial extension of the QDs. It causes the breakdown of the DA. Thus a new theory in describing light-matter interaction beyond the DA is necessary. QD–surface-plasmon interaction beyond the DA {#int-b-da} -------------------------------------------- The QD-field interaction is described by the minimal coupling Hamiltonian $\hat{H}_{\text{int}}(\mathbf{r},t)=-\frac{q}{m}\mathbf{A}(\mathbf{r},t)\cdot \mathbf{\hat{p}}$, where $\mathbf{\hat{p}}$ is the momentum operator, $q$ and $m$ are the electronic charge and mass, respectively, and $\mathbf{A}(\mathbf{r},t)$ is the vector potential of the field [@Stobbe2012]. In quantization, $\mathbf{A}(\mathbf{r},t)$ is expanded as $\mathbf{A}(\mathbf{r},t)=\sum_{l}\sqrt{\frac{\hbar }{2\omega_{l}\varepsilon _{0}}}[\mathbf{A}_{l}(\mathbf{r})\hat{a}_{l}e^{-i\omega_{l}t}+\text{h.c.}]$, where $\mathbf{A}_l(\mathbf{r})$ relevant to the Green’s tensor is the field spatial distribution function, $\hat{a}_{l}$ is the annihilation operator with frequency $\omega_{l}$, $\varepsilon_0$ is the vacuum dielectric function, and $l=(\mathbf{k},s)$ is the combined index of the wave vector $\mathbf{k}$ and polarization $s\in(1,2)$. To go beyond the DA, we make a Taylor expansion of $\mathbf{A}_{l}\mathbf{(r)}$ to the first order around the QD center $$\mathbf{A}_{l}(\mathbf{r})\simeq \mathbf{A}_{l}(\mathbf{r}_{0})+(\mathbf{r}-\mathbf{r}_{0})\cdot \pmb{\mathfrak{J}}\mathbf{A}_{l}(\mathbf{r})|_{\mathbf{r}=\mathbf{r}_{0}},$$ where $\pmb{\mathfrak{J}}\mathbf{A}_{l}(\mathbf{r})$ is the Jacobian matrix of partial derivatives of $\mathbf{A}_{l}(\mathbf{r})$. The QD in the strong confinement regime can be well described by a two-band model with states $| c\rangle$ and $| v\rangle$ representing an electron and a hole in the conduction and heavy valence band, respectively [@Stobbe2009]. Employing the rotating wave approximation, we arrive at the interaction Hamiltonian beyond the DA in the interaction picture $$\hat{H}_{\text{I}}(t)=\hbar \sum_{l}(g_{l}e^{i\Delta _{l}t}\hat{\sigma}_{-}\hat{a}_{l}^{\dag }+\text{H.c.}), \label{H-inter0}$$ where $\hat{\sigma}_{-}=|v\rangle \langle c |$, and $\Delta_l=\omega_{l}-\omega_{0}$ is the frequency detuning. The QD-field coupling strength is $$g_{l}=-\frac{q}{m}\sum_{j,k}(\frac{1}{2\hbar \epsilon _{0}\omega _{l}})^{1/2}[(\mu _{j}+\Lambda _{j,k}\nabla _{k})A_{lj}^{\ast }(\mathbf{r})]_{\mathbf{r=r}_{0}}, \label{couple}$$ where $j$ and $k$ index the three Cartesian coordinates $x,y,z$, $\mu_j=\langle v|\hat{p}_j|c\rangle $ and $\Lambda_{j,k}=\langle v|\hat{p}_{j}r_k|c\rangle $ denote the dipole and the quadrupole moments of the QD, respectively, and $\nabla _{k}$ represents the differential of $A_{lj}^{\ast }(\mathbf{r})$ to the $k$th coordinate component. One can see from Eq. (\[couple\]) that both the dipole and quadrupole moments contribute to its interaction with the radiation field. The former couples to the field distribution function $\mathbf{A}_l(\mathbf{r})$, while the latter couples to the gradient of $\mathbf{A}_l(\mathbf{r})$. In atomic systems, the atom is much smaller than the wavelength and the typical length of the spatial distribution of the field, i.e., $\nabla_k A_{l,j}^{\ast }(\mathbf{r})|_{\mathbf{r}=\mathbf{r}_0} \simeq 0$. Thus the contributions from the quadrupole moment can be safely abandoned and the DA is applicable. However, in the QD system, as the QD is large in size and the spatial variation of the field is pronounced, they cannot be ignored and the DA is inapplicable. The interaction between the QD and the field is further characterized by the spectral density $J(\omega )=\sum_{l}g_{l}^{2}\delta (\omega -\omega _{l})$. Combined with Eq. (\[couple\]), it takes the form $$\begin{aligned} \label{spectral} \begin{split} J (\omega) &=\frac{q^{2}}{\pi c^{2}\hbar ^{2}\varepsilon_{0}m^{2}}\sum_{j,n,j^{\prime },n^{\prime }}\{(\mu _{j}+\Lambda_{j,n}\nabla _{n})\\ &\times (\mu _{j^{\prime }}^{\ast }+\Lambda _{j^{\prime },n^{\prime }}^{\ast }\nabla_{n^{\prime }}^{\prime })\text{Im}[G_{j,j^{\prime }}(\mathbf{r},\mathbf{r}^{\prime };\omega )]\}_{\mathbf{r=r}^{\prime }=\mathbf{r}_{0}} , \end{split}\end{aligned}$$ where $G_{j,j^{\prime }}(\mathbf{r},\mathbf{r}^{\prime };\omega )$ is the $(j,j^\prime)$ element of the Green’s tensor and the relation $\textrm{Im}[G_{j,j^{\prime }}(\mathbf{r,r}^{\prime };\omega )]=\frac{\pi c^{2}}{2\omega }\sum_{l}A_{l,j}^{\ast }(\mathbf{r})A_{l,j^{\prime }}(\mathbf{r}^{\prime })\delta (\omega -\omega _{l})$ has been utilized [@Stobbe2012]. In the past decade, a division of the Green’s tensor into the surface plasmons bounded on the surface and the out-of-plane waves propagating away from the surface has been studied [@Sondergaard2004; @Siahpoush2012; @Siahpoush2014]. In the following, we shall study the spectral density for different orientations of the QD. Spectral density in arbitrary QD orientations --------------------------------------------- Consider first the special case that the QD orientates in $z$ axis. According to the symmetry of the electron and hole wavefunctions, we can calculate the two moments $\vec{\mu}=\bar{\mu}\left(\begin{array}{ccc}1 &i &0\end{array}\right)^T$ and $\mathbf{\Lambda }=\bar{\Lambda}\left(\begin{array}{ccc}0 & 0 & 0 \\0 & 0 & 0 \\1 & i & 0\end{array}\right)$, where $\bar{\mu}$ and $\bar{\Lambda}$ can be fitted experimentally [@Andersen2011]. When the QD orientates in $z^\prime$ shown in Fig. \[Fig1\], which can be expressed as a $\phi$-rotation of the QD along the $x$ axis from the $z$ direction, it can be proved that the QD-field interaction Hamiltonian (\[H-inter0\]) is unchanged except that the moments change into $\vec{\tilde{\mu}}(\phi)=\langle c|\hat{U}_{x}^{\dag }(\phi)\mathbf{\hat{p}}\hat{U}_{x}(\phi )|v\rangle $ and $\tilde{\mathbf{\Lambda}}(\phi)=\langle c|\hat{U}_{x}^{\dag }(\phi)\mathbf{\hat{p}r}\hat{U}_{x}(\phi )|v\rangle $, where $\hat{U}_{x}(\phi )=e^{-(i/\hbar)\hat{L}_{x}\phi}$ with $\hat{L}_{x}$ the QD angular momentum and $\phi$ the angle between $z$ and $z^\prime$. We thus have the moments $$\begin{aligned} \vec{\tilde{\mu}}(\phi) &=&\bar{\mu}\left( \begin{array}{ccc} 1 & i\cos \phi & i\sin \phi \end{array}% \right)^T , \label{m1}\\ \tilde{\mathbf{\Lambda}}(\phi) &=&\bar{\Lambda}\left( \begin{array}{ccc} 0 & 0 & 0 \\ -\sin \phi & -i\sin \phi \cos \phi & -i\sin \phi \sin \phi \\ \cos \phi & i\cos \phi \cos \phi & i\cos \phi \sin \phi \end{array}% \right) .\label{m2}\end{aligned}$$ Inserting Eqs. (\[m1\]) and (\[m2\]) into Eq. (\[spectral\]), we obtain the spectral density $$J(\omega)=J _{0}(\omega)+J _{\text{R}}(\omega),$$ where $J _{0}(\omega)$ and $J _\text{R}(\omega)$ contain the contributions from the free-space field $\mathbf{G}_{0}(\mathbf{r},\mathbf{r}^{\prime },\omega )$ and the reflected field $\mathbf{G}_\text{R}(\mathbf{r},\mathbf{r}^{\prime },\omega )$, respectively. Their forms in the cylindrical coordinate are $$\begin{aligned} &J_{0}(\omega) =\frac{\omega }{\Phi }\int ds\textrm{Re}[A_{1}\bar{\mu}^{2}+B_{1}\bar{\Lambda}^{\prime 2}],& \\ &J_{\text{R}}(\omega ) =\frac{\omega }{\Phi }\int ds\textrm{Re}\{[A_{2}\bar{\mu}^{2}+B_{2}\bar{\Lambda}^{\prime 2}+B_{3}\bar{\mu}\bar{\Lambda}^{\prime }]e^{2ik_{z_1}\Delta z}\}.~~~&\label{ref}\end{aligned}$$ where $\Phi =8\pi^2 \varepsilon _{0}m^{2}\hbar^{2}c^{3}/(q^{2}n_{1})$, $\bar{\Lambda}^{\prime}=k_{1}\bar{\Lambda}$ with $k_1=n_1\omega/c$, $s=k_{\rho}/k_{1}$, and $s_{z}\equiv\sqrt{1-s^{2}}=k_{z_{1}}/k_{1}$. The coefficients are given as $$\begin{aligned} A_{1} &=&\frac{s}{s_{z}}[(2-s^{2})(1+\cos ^{2}\phi )+2s^{2}\sin^{2}\phi ], \label{aa}\\ A_{2}&=&\frac{s}{s_{z}}[(r^{\text{s}}-r^{\text{p}}s_{z}^{2})(1+\cos ^{2}\phi)+2 r^{\text{p}}s^{2}\sin ^{2}\phi], \\ B_{1} &=&\frac{s}{s_{z}}\{(2-10s^2+{35s^4\over 4})\sin ^{4}\phi+2 s^2(4-5s^2)\nonumber\\ &&\times\sin^{2}\phi+2s^4\}, \label{bb}\\ B_{2}&=&\frac{s}{s_{z}}\{[r^\text{p}-r^\text{s}+\frac{3}{4}(r^\text{p}+r^\text{s}-r^\text{p}s^2)s^2]\sin^4\phi\nonumber\\ &&+s^2(r^\text{s}-3r^\text{p})\sin^2\phi+2r^\text{p}s^4 \},\\ B_{3} &=&2is[2 r^{\text{p}}s^{2}+(r^{\text{p}}s^{2}+r^{\text{p}}-r^{\text{s}})\sin ^{2}\phi]\cos \phi,\end{aligned}$$ where $r^{\text{s}}=\frac{s_{z}-\sqrt{n_{m1}^{2}-s^{2}}}{s_{z}+\sqrt{n_{m1}^{2}-s^{2}}}$ and $r^{\text{p}}=\frac{\varepsilon (\omega )s_{z}-\varepsilon _{1}\sqrt{n_{m1}^{2}-s^{2}}}{\varepsilon (\omega )s_{z}-\varepsilon _{1}\sqrt{n_{m1}^{2}-s^{2}}}$ are the Fresnel reflection coefficients for s- and p-polarized lights with the relative dielectric function $n_{m1}=\sqrt{\varepsilon_m(\omega)/\varepsilon_1}$. Up to now, going beyond the DA, we have analytically established the microscopic description to the interaction between the QD and the radiation field propagating near a plasmonic nanostructure. It can be seen that $J_{\text{R}}(\omega )$ is contributed from two types of field modes along the $z$-direction by dividing the integration range $[0,\infty]$ of $s$ in Eq. (\[ref\]) into two intervals $[0,1]$ and $[1,\infty]$. The former has a real $k_{z_1}$ and is associated with the reflected plane waves by the metal-dielectric interface, while the latter has a complex $k_{z_1}$ and is associated with the surface plasmons and the damped non-radiative mode [@Chang2006; @Tudela2014]. Furthermore, the cooperative effect of the dipole and the quadrupole moments are self-consistently contained in $J_{\text{R}}(\omega )$, where the $B_3$ term characterizes the interference between the two moments. Just due to this interference, the decoherence of the QD shows significant differences from the one under the DA. Under the DA, the spectral density contributed uniquely from the dipole moment has a $\pi$-rotation symmetry over the QD orientation. When the quadrupole moment is taken into account, the symmetry is changed into $2\pi$ because of the presence of the $B_{3}$ term. It is noted that in the special case $\phi=0$ or $\pi$, our result reduces exactly to the one in Ref. [@Andersen2011]. Fluorescence modified by the QD mesoscopic effects {#fluorescence} ================================================== We consider explicitly that the QD is resonantly driven by a laser so that the resonance fluorescence of the QD near the metal surface beyond the DA can be measured. In a frame rotating at the laser frequency $\omega_0$, the master equation under the Born-Markovian approximation reads $$\begin{aligned} \begin{split}\label{master} \dot{\rho}(t)=&-i\frac{\Omega}{2} [\hat{\sigma}_{+}+\hat{\sigma}_{-},\rho (t)]\\ &+\frac{\Gamma}{2}[2\hat{\sigma}_{-}\rho (t)\hat{\sigma}_{+}-\hat{% \sigma}_{+}\hat{\sigma}_{-}\rho (t)-\rho (t)\hat{\sigma}_{+}\hat{\sigma}_{-}], \end{split}\end{aligned}$$ where $\Omega$ is the Rabi frequency denoting the laser pumping strength and $\Gamma =2\pi J(\omega_0 )$ is the QD spontaneous emission rate. ![Orientation dependence of $\Gamma$ for different dimensionless separation $\Delta \bar{z}$ under (a) and beyond (b) the DA. (c) A cross section of (a) and (b) at $\Delta \bar{z}=0.3$ \[blue solid and green dot-dashed lines from (a) and (b), respectively\]. (d): $\Gamma$ as a function of $\bar {\Lambda}^\prime/\bar{\mu}$ when $\phi=0$ (purple solid line) and $\phi=\pi$ (orange dashed line). Parameters are $\omega_0=1.2$ eV, $\Phi=3.0\times10^{9}\bar{\mu}^2$, and $\bar{\Lambda}^{\prime }/\bar{\mu}=0.5$, which are obtained by fitting the experimental result in Ref. [@Andersen2011].[]{data-label="Fig2"}](Fig2.eps){width="\columnwidth"} In Figs. \[Fig2\](a) and \[Fig2\](b) we plot $\Gamma$ in different QD orientations as a function of the QD-interface separation $\Delta \bar{z}=\Delta z\omega _{p}/c$ under and beyond the DA, respectively. It can be seen that $\Gamma$ attenuates rapidly at small $\Delta\bar{z}$ and tends to a persistent oscillation with the increase of $\Delta\bar{z}$. This is understandable based on Eq. (\[ref\]), which reveals that the contribution from the surface plasmons only dominates the small $\Delta \bar{z}$ regime. Beyond this regime, the spectral density originates mainly from reflected plane waves, which shows a lossless oscillation with the increase of $\Delta \bar{z}$. In addition, the significant deviations to the result under DA can be seen at small $\Delta \bar{z}$, where the surface plasmons bounded around the metal surface play a significant role. The typical distance where this deviation is observable is the penetration depth of the plasmons into the dielectric, which takes $\Delta \bar{z}_\text{c}\thicksim2$ for our parameters. Therefore, the DA is inapplicable especially when the QD is positioned within $\Delta \bar{z}_\text{c}$. This has been verified experimentally [@Andersen2011]. A cross section view at $\Delta \bar{z}=0.3$ is plotted in Fig. \[Fig2\](c). It indicates clearly that $\Gamma$ experiences a $\pi$ rotation symmetry over the QD orientation under the DA, while it is changed to $2\pi$ once the mesoscopic effects are taken into account. This agrees with our analytical expectation. To evaluate explicitly the mesoscopic effects, we plot in Fig. \[Fig2\](d) $\Gamma$ as a function of $\bar{\Lambda}^\prime/\bar{\mu}$ at $\phi=0$ and $\pi$. It indicates that the interference between the dipole and quadrupole moments can cause a constructive increase or a destructive decrease of the decay rate under the DA (i.e., the value when $\bar{\Lambda}^\prime=0$). It demonstrates that we can control the QD decay by manipulating the mesoscopic characteristics of the QD, such as its spatial orientation and quadrupole moment. The resonance fluorescence spectrum of the driven QD is defined as $S(\omega )=\frac{I_0}{\pi }\textrm{Re}[\int_{0}^{\infty }d\tau e^{i\omega \tau }\langle \hat{\sigma}_{+}(t) \hat{\sigma}_{-}(t+\tau )\rangle _{\text{ss}}]$, where “ss" denotes the steady state and $I_0(\mathbf{r})$ depending on the distance between the detector and the QD is a constant. From the master equation (\[master\]) and with the use of the quantum regression theorem, the spectrum admits an analytical expression [@Carmichael2000]. It consists of the coherent (Rayleigh scattering) and incoherent (inelastic scattering) components. The coherent one is a delta function and ignored here, while the incoherent one takes the form $$\begin{aligned} \begin{split}\label{spectrum} &S(\delta\omega ) =\frac{Y^{2}}{8(1+Y^{2})}\Big[\frac{\Gamma }{\delta \omega^{2}+\frac{\Gamma ^{2}}{4}}\\ &~~~+\frac{\frac{3\Gamma }{4}P-(\delta \omega -\alpha )Q}{(\delta\omega -\alpha)^{2}+(\frac{3\Gamma }{4})^{2}}+\frac{\frac{3\Gamma }{4}P+(\delta\omega +\alpha )Q}{(\delta\omega +\alpha)^{2}+(\frac{3\Gamma }{4})^{2}}\Big], \end{split}\end{aligned}$$ where $\delta\omega=\omega -\omega_{0}$, $Y=\frac{\sqrt{2}\Omega }{\Gamma }$, $i\alpha =\frac{\Gamma }{4}\sqrt{1-8Y^{2}}$, $P=\frac{Y^{2}-1}{Y^{2}+1}$, and $Q=\frac{\Gamma }{4\delta }\frac{1-5Y^{2}}{1+Y^{2}}$. In the strong-driving and weak-radiation situation ($\Omega>\Gamma/4$), the spectrum constitutes of a sum of three Lorentzian components centered at $\omega_0$ and $\omega_0\pm \Omega$, respectively. This is the typical feature of the Mollow’s triplet structure. In the weak-driving and strong-radiation situation ($\Omega<\Gamma/4$), the two sideband peaks disappear. Furthermore, as shown in Eq. (\[spectrum\]), the positions, the widths, and the intensities of the three peaks of the spectrum are all associated with the decay rate $\Gamma$. Therefore, the spectrum can be greatly influenced by the mesoscopic effects of the QD via $\Gamma$.  except for $\Omega=5$ ns$^{-1}$.[]{data-label="Fig3"}](Fig3.eps){width="\columnwidth"}  except for $\Omega=5$ ns$^{-1}$.[]{data-label="Fig4"}](Fig4.eps){width="\columnwidth"} Figures \[Fig3\](a) and \[Fig3\](b) show the spectrum in different QD orientations under and beyond the DA, respectively. The spectrum under the DA has a $\pi$ rotation symmetry over the QD orientation, while beyond the DA, it has $2\pi$ symmetry. It agrees with the behavior of $\Gamma$, Fig. \[Fig2\](c). Another interesting observation is that although the spectrum keeps the Mollow triplet structure in the whole range of $\phi$ when the DA is applied \[see Fig. \[Fig3\](a)\], it can be switched from the Mollow triplet to a single peak centered at $\delta\omega=0$ by adjusting the QD orientation when the mesoscopic effect is considered \[see Fig. \[Fig3\](b)\]. This result manifests clearly the anisotropy of the spontaneous emission of the QD placed on the metal surface. Figure \[Fig4\] plots the spectrum with the change of $\bar{\Lambda}^\prime/\bar{\mu}$ for $\phi=0$ (a) and $\pi$ (b), respectively. When $\phi=0$, the Mollow triplet can be either strengthened or weakened with the increasing of $\bar{\Lambda}^\prime/\bar{\mu}$, while at $\phi=\pi$, it gradually switches to a single peak with the increase of $\bar{\Lambda}^\prime/\bar{\mu}$, which characterizes a strong radiation of the QD. Thus, for the spectrum to be more evident in experiment, a proper designation of the QD orientation and the quadrupole moment is needed. ![$g^{(2)}(\tau )$ versus $\phi$ at $\bar{\Lambda}^\prime/\bar{\mu}=0.5$ (a) and versus $\bar{\Lambda}^\prime/\bar{\mu}$ at $\phi=0$ (b). Parameters are the same as in Fig. \[Fig3\] and Figs. \[Fig4\].[]{data-label="Fig5"}](Fig5.eps){width="\columnwidth"} Another magnitude of the experimental interest is the statistical properties of the emitted light from the QD [@Wrigge2008; @Ates2009], which is measured by the second-order correlation function $g^{(2)}(\tau )=\frac{G^{(2)}(\tau )}{\lim_{\tau \rightarrow \infty}G^{(2)}(\tau )}$ with $G^{(2)}(\tau )=\langle \hat{\sigma}_{+}(t)\hat{\sigma}_{+}(t+\tau )\hat{\sigma}_{-}(t+\tau )\hat{\sigma}_{-}(t)\rangle _\text{ss}$. From Eq. (\[master\]) and with the use of the quantum regression theorem, $g^{(2)}(\tau )$ can be analytically obtained as $$g^{(2)}(\tau )=1-\Big[\cos (\alpha \tau )+\frac{3\Gamma }{4\alpha }\sin (\alpha \tau)\Big]e^{-3\Gamma \tau /4}.$$ One can find that $g^{(2)}(\tau)>g^{(2)}(0)$, which indicates that the probability to detect two emitted photons with time delay $\tau$ is larger than the one without time delay. This is a typical nonclassical property of light, i.e., the antibunching character. It ensures the single-photon nature of the emitted fluorescence. We plot $g^{(2)}(\tau )$ in different QD orientations for $\bar{\Lambda}^\prime/\bar{\mu}=0.5$ in Fig. \[Fig5\](a). The $2\pi$ symmetry over the QD orientation is also kept by $g^{(2)}(\tau )$. In addition, $g^{(2)}(\tau )$ experiences from monotonically increase to oscillatory increase in different $\phi$. The oscillation is a manifestation of the laser-driven Rabi oscillation, while the damping of its amplitude is caused by the QD spontaneous emission. The oscillation of $g^{(2)}(\tau )$ in fixed QD orientation can be dramatically suppressed with the increase of the quadrupole moment \[see Fig. \[Fig5\](b)\]. Such type of transition can be viewed as a manifestation of the fluorescence changed from the weak radiation to the strong radiation regimes due to the presence of the QD mesoscopic effects [@Tudela2010]. Both the spectrum and the second-order correlation function of the fluorescence indicate that one can control the single-photon emission of the QD by its mesoscopic effects, e.g., the spatial orientation and the quadrupole moment. Conclusions {#con} =========== In summary, we have studied the resonance fluorescence of a mesoscopic QD placed in plasmonic nanostructure. Going beyond the DA, a microscopic description of the decoherence dynamics of the QD has been established. It is revealed that, modified by the QD mesoscopic effects, the spectrum and its statistical property of the resonance fluorescence exhibit different rotation symmetry over the QD orientation and shows significant deviation from those under the DA. The results demonstrate that one can control the interaction between the QD and the surface plasmons by manipulating its mesoscopic effects, which offers a dimension to control the radiation properties of the QD. Our studies are within the present experimental state of the art and instructive for the utilization of the QD mesoscopic characteristics in the nanophotonic device developments. Acknowledgments {#acknowledgments .unnumbered} =============== This work is supported by the Specialized Research Fund for the Doctoral Program of Higher Education, by the Program for New Century Excellent Talents in University, and by the National Natural Science Foundation of China (Grant No. 11474139). Green’s tensor of planar interface {#Green} ================================== The Green’s tensor in our system is $$\mathbf{G}(\mathbf{r},\mathbf{r}^{\prime };\omega )=\mathbf{G}_{0}(\mathbf{r},\mathbf{r}^{\prime };\omega )+\mathbf{G}_{\text{R}}(\mathbf{r},\mathbf{r}^{\prime};\omega ),$$ where $\mathbf{G}_{0}(\mathbf{r},\mathbf{r}^{\prime };\omega )$ and $\mathbf{G}_{\text{R}}(\mathbf{r},\mathbf{r}^{\prime};\omega )$ denote the contributions of the field propagating in the dielectric and reflected by the metal-dielectric interface, respectively. Under the angular spectrum representation, they take the form [@Novotny2006] $$\begin{aligned} \mathbf{G}_{0}(\mathbf{r},\mathbf{r}^{\prime };\omega ) &=&\frac{i}{8\pi ^{2}% } \int_{-\infty }^{\infty }dk_{x}dk_{y}\frac{e^{i[k_{x}(x-x^{\prime })+k_{y}(y-y^\prime )+k_{z_1}|z-z^\prime|]}}{k_{1}^{2}k_{z_1}}\left( \begin{array}{ccc} k_1^{2}-k_{x}^{2} & -k_{x}k_{y} & \mp k_{x}k_{z_1} \\ -k_{x}k_{y} & k_1^{2}-k_{y}^{2} & \mp k_{y}k_{z_1} \\ \mp k_{x}k_{z_1} & \mp k_{y}k_{z_1} & k_1^{2}-k_{z_1}^{2}% \end{array}% \right),\label{free-green0} \\ \mathbf{G}_{\text{R}}(\mathbf{r},\mathbf{r}^{\prime };\omega ) &=&\frac{i}{8\pi ^{2}% } \int_{-\infty }^{\infty }dk_{x}dk_{y}\frac{e^{i[k_{x}(x-x^\prime )+k_{y}(y-y^\prime )+k_{z_1}(z+z^\prime )]}}{k_{x}^{2}+k_{y}^{2}} \left( \mathbf{M}^{\text{s}}+\mathbf{M}^{\text{p}}\right),\label{ref-green0}\end{aligned}$$ where $\mathbf{k}_1=(k_x,k_y,k_{z_1})$ is the wavevector in the dielectric. The two different signs in $\mathbf{G}_{0}(\mathbf{r},\mathbf{r}^{\prime };\omega )$ are determined by the absolute value of $|z-z^\prime|$, where the upper (lower) sign is applied when $z>z^\prime$ ($z<z^\prime$). In Eq. (\[ref-green0\]), $\mathbf{G}_{\text{R}}(\mathbf{r},\mathbf{r}^{\prime };\omega ) $ has been splitted into the s-polarized part and p-polarized parts $$\mathbf{M}^{\text{s}}=\frac{r^{\text{s}}(k_x,k_y)}{k_{z_1}}\left( \begin{array}{ccc} k_{y}^{2} & -k_{x}k_{y} & 0 \\ -k_{x}k_{y} & k_{x}^{2} & 0 \\ 0 & 0 & 0% \end{array}% \right) ,~\mathbf{M}^{\text{p}}=\frac{-r^{\text{p}}(k_x,k_y)}{k_{1}^2}\left( \begin{array}{ccc} k_{x}^{2}k_{z_1} & k_{x}k_{y}k_{z_1} & k_{x}(k_{x}^{2}+k_{y}^{2}) \\ k_{x}k_{y}k_{z_1} & k_{y}^{2}k_{z_1} & k_{y}(k_{x}^{2}+k_{y}^{2}) \\ -k_{x}(k_{x}^{2}+k_{y}^{2}) & -k_{y}(k_{x}^{2}+k_{y}^{2}) & -(k_{x}^{2}+k_{y}^{2})^{2}/k_{z_1}\end{array}\right).$$ where $r^{\text{s}}(k_{x},k_{y})=\frac{\mu_{m}k_{z_{1}}-\mu_{1}k_{z_{\text{m}}}}{\mu_{m}k_{z_{1}}+\mu_{1}k_{z_{\text{m}}}}$ and $r^{\text{p}}(k_{x},k_{y})=\frac{\varepsilon_{\text{m}}k_{z_{1}}-\varepsilon _{1}k_{z_{\text{m}}}}{\varepsilon_{\text{m}}k_{z_{1}}+\varepsilon _{1}k_{z_{\text{m}}}}$ with $k_{z_m}$ being the $z$ component of the wave vector in the metal are the normal Fresnel reflection coefficients for the s-polarized and p-polarized light in the metal-dielectric interface, respectively. 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--- abstract: 'Abdo and Dimitrov defined the total irregularity of a simple undirected graph $G$ to be $irr_t(G) =\frac{1}{2}\sum \limits_{u,v \in V(G)}|d(u) - d(v)|.$ In this study we allocate the *Fibonacci weight*, $f_i$ to a vertex $v_j$ of a simple connected graph $G,$ if and only if $d(v_j) = i$ and define the *total fibonaccian irregularity* or $f_t$-irregularity as $firr_t (G) = \sum \limits_{i=1}^{n-1} \sum \limits_{j=i+1}^{n}|f_i - f_j|.$ The concept of an *edge-joint* denoted $G\rightsquigarrow_{vu}H$ is also introduced This paper presents results for the undirected underlying graph $J^*_n(x)$ of a Jaco Graphs, $J_n(x),$ $n,x \in \Bbb N.$' --- [**Keywords:** Total irregularity, Fibonacci weight, Total $f$-irregularity, Fibonaccian irregularity, Jaco graphs, Jaconian vertices, Fisher algorithm, Edge-joint.]{}\ \ [**AMS Classification Numbers:** 11B39, 05C07, 05C20, 05C22, 05C75]{} Introduction ============ For a general reference to notation and concepts of graph theory see \[3\]. Unless mentioned otherwise, only simple undirected graphs or the underlying graph of directed graphs will be considered. Abdo and Dimitrov \[1, 2\] defined total irregularity of a simple undirected graph $G$ to be $irr_t(G) =\frac{1}{2}\sum \limits_{u,v \in V(G)}|d(u) - d(v)|.$ If the vertices of a simple undirected graph $G$ on $n$ vertices are labeled $v_i, i = 1, 2, 3, \dots, n$ then the definition may be $irr_t(G) = \frac{1}{2}\sum \limits_{i=1}^{n} \sum \limits_{j=1}^{n}|d(v_i) - d(v_j)| = \sum \limits_{i=1}^{n} \sum \limits_{j=i+1}^{n}|d(v_i) - d(v_j)|$ or $\sum \limits_{i=1}^{n-1} \sum \limits_{j=i+1}^{n}|d(v_i) - d(v_j)|.$ For a simple graph on a singular vertex (*1-empty* graph), we define $irr_t(G) = 0$ .\ \ A new notion of vertex labeling is inherent to the definition of total irregularity. That is, let $g:V(G) \mapsto \Bbb N$ with $g(v) = d_G(v),$ $\forall v \in V(G).$ In Section 2 the total irregularity of finite linear Jaco graphs is discussed. This is followed by the introduction of *total fibonaccian irregularity* of graphs. This new irregularity or vertex labeling is applied to finite linear Jaco graphs in Section 3.\ \ The content is fairly straight forward in that it demonstrates *constructive counting technique* only. The real contribution is that the approach demonstrates the graphical embodiment of mainly, a number theoretical problem stemming from well-defined graphs in terms of their structure. Essentially, we see that total irregularity presents a sum of differences between all pairs of natural numbers in a subset $X\subset \Bbb N$ for $X$ having an even number of odd numbers. Similarly, total fibonaccian irregularity presents a sum of differences between all pairs of fibonacci numbers in a subset $X\subset \Bbb F$ for $X$ having an even number of fibonacci numbers with odd subscripts. Total Irregularity of Finite Linear Jaco Graphs =============================================== The concept of linear Jaco graphs was introduced Kok et al. \[4, 5\]. In the initial studies the concepts of *order 1* and *order $a$* Jaco graphs, denoted $J_n(1)$, $J_n(a)$ respectively, were reported on. In a more recent study (see \[5\]) a unifying definition was adopted and the generalised family called, linear Jaco graphs was defined.A particular family of finite directed graphs called Jaco Graphs and denoted by $J_n(x),$ $n,x \in \Bbb N$ are derived from a particular well-defined infinite directed graph, called the *x*-root digraph. The *x*-root digraph has four fundamental properties which are; $V(J_\infty(x)) = \{v_i:i \in \Bbb N\}$ and, if $v_j$ is the head of an arc then the tail is always a vertex $v_i, i<j$ and, if $v_k,$ for smallest $k \in \Bbb N$ is a tail vertex then all vertices $v_ \ell, k< \ell<j$ are tails of arcs to $v_j$ and finally, the degree of vertex $k$ is $d(v_k) = k.$ The family of finite directed graphs are those limited to $n \in \Bbb N$ vertices by lobbing off all vertices (and arcs) $v_t, t > n.$ Hence, trivially we have $d(v_i) \leq i$ for $i \in \Bbb N.$ When the context is clear we refer to the Jaco graph $J_n(x)$, the underlying Jaco graph $J^*_n(x)$, arcs $A(J_n(x))$ and edges $E(J_n(x))$, the degree $d_{J_n(x)}(v_i) = d^+_{J_n(x)}(v_i) + d^-_{J_n(x)}(v_i) = d^+_{J^*_n(x)}(v_i) + d^-_{J^*_n(x)}(v_i) = d(v_i),$ interchangeably. $[6]$ The infinite Jaco Graph $J_\infty(x)$ is defined by $V(J_\infty(x)) = \{v_i: i \in \Bbb N\}$, $A(J_\infty(x)) \subseteq \{(v_i, v_j): i, j \in \Bbb N, i< j\}$ and $(v_i,v_ j) \in A(J_\infty(x))$ if and only if $2i - d^-(v_i) \geq j.$ $[6]$ The family of finite Jaco Graphs are defined by $\{J_n(x) \subseteq J_\infty(x):n,x\in \Bbb {N}\}.$ A member of the family is referred to as the Jaco Graph, $J_n(x).$ For illustration the adapted table below follows from the Fisher algorithm $[4]$ for $J_n(x),$ $n,x \in \Bbb N,$ $n\leq 12.$ The degree sequence of $J^*_n(x)$ is denoted $\Bbb D(J^*_n(x)).$ Note that for the underlying graphs $J^*_n(x),$ the values $irr_t(J^*_n(x))$ have been calculated manually, as it is not provided for in the Fisher algorithm.\ \ Table 1.\ $i\in{\Bbb{N}}$ $d^-(v_i)$ $d^+(v_i) = i - d^-(v_i)$ $\Bbb D(J^*_i(x))$ $irr_t(J^*_i(x))$ ----------------- ------------ --------------------------- -------------------------------------- ------------------- 1 0 1 (0) 0 2 1 1 (1, 1) 0 3 1 2 (1, 2, 1) 2 4 1 3 (1, 2, 2, 1) 4 5 2 3 (1, 2, 3, 2, 2) 8 6 2 4 (1, 2, 3, 3, 3, 2) 14 7 3 4 (1, 2, 3, 4, 4, 3, 3) 26 8 3 5 (1, 2, 3, 4, 5, 4, 4, 3) 42 9 3 6 (1, 2, 3, 4, 5, 5, 5, 4, 3) 60 10 4 6 (1, 2, 3, 4, 5, 6, 6, 5, 4, 4) 86 11 4 7 (1, 2, 3, 4, 5, 6, 7, 6, 5, 5, 4) 116 12 4 8 (1, 2, 3, 4, 5, 6, 7, 7, 6, 6, 5, 4) 149 \ \ \ Note that the Fisher Algorithm determines $d^+(v_i)$ on the assumtion that the Jaco Graph is always sufficiently large, so at least $J_n(x), n \geq i+ d^+(v_i).$ For a smaller graph the degree of vertex $v_i$ is given by $d(v_i) = d^-(v_i) + (n-i).$ In $[4, 5],$ Bettina’s theorem describes an arguably, closed formula to determine $d^+(v_i)$. Since $d^-(v_i) = n - d^+(v_i)$ it is then easy to determine $d(v_i)$ in a smaller graph $J_n(1), n< i + d^+(v_i).$\ \ The next result presents a *partially* recursive formula to determine $irr_t(J_{n+1}(x))$ if $irr_t(J_n(x)),\\ n\geq 1$ is known. Consider the underlying Jaco Graph, $J^*_n(x),$ $n,x \in \Bbb N$ with $\Delta(J^*_n(x))= k$ and $irr_t(J^*_n(x))$ known. Let $d(v_i),$ $d^*(v_i)$ denote the degree of vertex $v_i$ in $J^*_n(x)$ and $J^*_{n+1}(x)$, respectively. Then for the underlying Jaco graph $J^*_{n+1}(x)$ we have that:\ \ $irr_t(J^*_{n+1}(x)) = irr_t(J^*_n(x)) + \sum \limits_{i=1}^{\ell_1} i - \sum \limits_{i=1}^{\ell_2} i + \sum\limits_{i=1}^{n}|(n-k) - d^*(v_i)|,$\ \ with $\ell_1$ the number of vertices $v_i$ with $d(v_i) \leq d(v_{k+j}), j \in \{1, 2, \dots, n-k\},$ and $\ell_2$ the number of vertices $v_i$ with $d(v_i) > d(v_{k+j}), j \in \{1, 2, \dots, n-k\}.$ Let $J^*_n(x)$ have the prime Jaconian vertex, $v_k,$ hence $d(v_k) = \Delta (J^*_n(x))$ as defined in $[4].$ It is also true that $d(v_k) = \Delta (J^*_n(x)) = d^*(v_k).$ By adding vertex $v_{n+1}$ to construct $J_{n+1}(x),$ the vertex $v_{n+1}$ obtains degree, $d^*(v_{n+1}) = n - k.$ Each vertex $v_{k+j}, j= 1,2, \dots, n-k$ obtains an additional edge, $v_{k+j}v_{n+1}$ as well.\ \ So clearly $d^*(v_{k+j}) = d(v_{k+j}) + 1$ for $j = 1,2, \dots, n-k.$ It implies that $|d^*(v_{k+1}) - d(v_i)_{i \leq k}| = \\ \\ |d(v_{k+1}) - d(v_i)_{i \leq k}| + 1$ iff $d(v_i)_{i \leq k} \leq d(v_{k+1}).$ It follow that for the cases $d(v_i)_{i \leq k} > d(v_{k+1}),$\ \ we have $|d^*(v_{k+1}) - d(v_i)_{i \leq k}| = |d(v_{k+1}) - d(v_i)_{i \leq k}| - 1.$ The “split-result” follows similarly for $|d^*(v_{k+j}) - d(v_i)_{i \leq k}|, j = 2,3, \dots, n-k.$ Therefore the terms, $+ \sum \limits_{i=1}^{\ell_1} i - \sum \limits_{i=1}^{\ell_2} i$ follow easily.\ \ The terms, $+ \sum \limits_{i=1}^{k}|(n-k) - d(v_i) | + \sum\limits_{i=k+1}^{n}|(n-k) - d^*(v_i)|$ follow directly from the definition of *total irregularity* and since it is true that $d(v_i) = d^*(v_i) \forall i \leq k,$ we have that:\ \ $\sum \limits_{i=1}^{k}|(n-k) - d(v_i) | + \sum\limits_{i=k+1}^{n}|(n-k) - d^*(v_i)| = \sum\limits_{i=1}^{n}|(n-k) - d^*(v_i)|.$\ \ So in conclusion we have:\ \ $irr_t(J^*_{n+1}(x)) = irr_t(J^*_n(x)) + \sum \limits_{i=1}^{\ell_1} i - \sum \limits_{i=1}^{\ell_2} i + \sum\limits_{i=1}^{n}|(n-k) - d^*(v_i)|.$ $f_t$-Irregularity of Finite Linear Jaco Graphs =============================================== Let $\Bbb{F} = \{f_0=0, f_1=1,f_2=1,f_3=2, \dots, f_n=f_{n-1} + f_{n-2}, \dots\}$ be the set of Fibonacci numbers.\ \ Consider $g:V(G) \mapsto \Bbb F$ defined as folows. Allocate the *Fibonacci weight*, $f_i$ to a vertex $v_j$ of a simple connected graph $G,$ if and only if $d(v_j) = i.$ Define the *total fibonaccian irregularity* as, $firr_t (G) = \sum \limits_{i=1}^{n-1} \sum \limits_{j=i+1}^{n}|f_i - f_j|.$ For a simple graph on a singular vertex (*1-empty* graph), define $firr_t(G) = 0.$\ \ If all vertices of a graph carry equal fibonacci weight the graph is said to be $f$-regular. It follows not surprisingly that a regular graph $G$ is *f-regular*, hence $firr_t(G) = 0.$ Note that a connected graph need not be regular, to be $f$-regular. The path $P_n,$ $n\in \Bbb N$ is the only example of such non-regular graph which is $f$-regular. Determining $firr_t(G)$ is generally complex but certain graphs provide simple results. One example is for a star i.e., $firr_t(S_{1,n}) = n(f_n -1).$ Equally straight forward is that for a complete bipartite graph $K_{n,m},$ $n,m \in \Bbb N,$ $n\geq m$ we have $firr_t(K_{n,m}) = nm(f_n - f_m).$\ \ For illustration the adapted table below follows from the Fisher algorithm $[3]$ for $J_n(x), n\leq 12.$ The $f_i$-sequence of $J^*_n(x)$ is denoted $\Bbb F(J^*_n(x)).$ Note that for the underlying graphs $J^*_n(x),$ the values $firr_t(J^*_n(x))$ have been calculated manually, as it is not provided for in the Fisher algorithm.\ \ Table 2.\ $i\in{\Bbb{N}}$ $d^-(v_i)$ $d^+(v_i) = i - d^-(v_i)$ $\Bbb F(J^*_i(x))$ $firr_t(J^*_i(x))$ ----------------- ------------ --------------------------- ---------------------------------------- -------------------- 1 0 1 (0) 0 2 1 1 (1, 1) 0 3 1 2 (1, 1, 1) 0 4 1 3 (1, 1, 1, 1) 0 5 2 3 (1, 1, 2, 1, 1) 4 6 2 4 (1, 1, 2, 2, 2, 1) 9 7 3 4 (1, 1, 2, 3, 3, 2, 2) 20 8 3 5 (1, 1, 2, 3, 5, 3, 3, 2) 54 9 3 6 (1, 1, 2, 3, 5, 5, 5, 3, 2) 70 10 4 6 (1, 1, 2, 3, 5, 8, 8, 5, 3, 3) 133 11 4 7 (1, 1, 2, 3, 5, 8, 13, 8, 5, 5, 3) 224 12 4 8 (1, 1, 2, 3, 5, 8, 13, 13, 8, 8, 5, 3) 322 \ \ \ The next result presents a *partially* recursive formula to determine $firr_t(J^*_{n+1}(x))$ if $firr_t(J^*_n(x)),$ $n\geq 1$ is known. (Lumin’s Theorem)[^1] Consider the underlying Jaco Graph, $J^*_n(x),$ $n,x \in \Bbb N$ with $\Delta(J^*_n(x))= k$ and $firr_t(J^*_n(x))$ known. Let $d(v_i),$ $d^*(v_i)$ denote the degree of vertex $v_i$ in $J^*_n(x)$ and $J^*_{n+1}(x)$, respectively. Then for the underlying Jaco graph $J^*_{n+1}(x)$ we have that:\ \ $firr_t(J^*_{n+1}(x)) = firr_t(J^*_n(x)) + \sum \limits_{i=1}^{n}|f_{n-k} - f_{d^*(v_i)}| + \sum \limits_{i\in \{k+1, k+2, \dots, n\}}\ell_{(1,i)}|f_{d(v_i)+1} - f_{d(v_i)}| - \\ \\ \sum \limits_{i\in \{k+1, k+2, \dots, n\}}\ell_{(2,i)}|f_{d(v_i)+1} - f_{d(v_i)}| + \sum \limits_{j=k+1}^{n-1} \sum \limits_{j=i+1}^{n}||f_{d(v_i)} - f_{d(v_j)}| - |f_{d(v_i)+1} - f_{d(v_j)+1}||,$\ \ with $\ell_{(1,i)}$ the number of vertices $v_j, j \in \{1,2,3, \dots, k\},$ with $d^*(v_i) > d(v_j), i \in \{k+1, k+2, \dots, n\}$\ and $\ell_{(2,i)}$ the number of vertices $v_j, j \in \{1,2,3, \dots, k\},$ with $d^*(v_i) \leq d(v_j), i \in \{k+1, k+2, \dots, n\}.$ Let $J^*_n(x)$ have the prime Jaconian vertex, $v_k,$ hence $d(v_k) = \Delta (J^*_n(x))$ as defined in $[3].$ It is also true that $d(v_k) = \Delta (J^*_n(x)) = d^*(v_k).$ By adding vertex $v_{n+1}$ to construct $J_{n+1}(x),$ the vertex $v_{n+1}$ obtains degree, $d^*(v_{n+1}) = n - k.$ Each vertex $v_{k+j}, j= 1,2, \dots, n-k$ obtains an additional edge, $v_{k+j}v_{n+1}$ as well. So clearly $d^*(v_{k+j}) = d(v_{k+j}) + 1$ for $j = 1,2, \dots, n-k.$ We also have that $d^*(v_i) = d(v_i), i = 1, 2, \dots, k.$ It implies that to calculate $firr_t(J^*_{n+1}(1)),$ the term $\sum \limits_{i=1}^{n}|f_{n-k} - f_{d^*(v_i)}|$ must be added.\ \ For each vertex $v_i,(k+1) \leq i \leq n$ the *fibonacci weight* increases by $|f_{d^*(v_i)} -f_{d(v_i)}| = |f_{d(v_i) + 1} - f_{d(v_i)}|.$ It implies that to calculate $firr_t(J^*_{n+1}(x)),$ the term $k(\sum \limits_{i= k+1}^{n}|f_{d(v_i)+1} - f_{d(v_i)}|)$ must be added as well.\ \ Finally, the increase in the $f_t$-irregularity between $J^*_n(x)$ and $J^*_{n+1}(x)$ from amongst vertices, $v_{k+1}, v_{k+2}, \dots, v_n$ is given by the term, $\sum \limits_{j=k+1}^{n-1} \sum \limits_{j=i+1}^{n}||f_{d(v_i)} - f_{d(v_j)}| - |f_{d^*(v_i)} - f_{d^*(v_j)}||.$\ \ Hence, the result:\ \ $firr_t(J^*_{n+1}(x)) = firr_t(J^*_n(x)) + \sum \limits_{i=1}^{n}|f_{n-k} - f_{d^*(v_i)}| + k(\sum \limits_{i=k+1}^{n}|f_{d(v_i)+1} - f_{d(v_i)}|) + \sum \limits_{j=k+1}^{n-1} \sum \limits_{j=i+1}^{n}||f_{d(v_i)} - f_{d(v_j)}| - |f_{d(v_i)+1} - f_{d(v_j)+1}||,$ follows. $firr_t$ Resulting from Edge-joint between Jaco Graphs ------------------------------------------------------ Abdo and Dimitrov $[2]$ observed that $irr_t(G \cup H) \geq irr(t(G) + irr_t(H)).$ We present a result for $irr_t(J^*_n(x) \cup J^*_m(x))$ followed by a corollary in respect of $firr_t.$ (Lumin’s 2nd Theorem) For the Jaco Graphs $J^*_n(x)$ and $J^*_m(x),$ we have that: $$irr_t(J^*_n(x) \cup J^*_m(x)) \begin{cases} \leq 2(irr_t(J^*_n(x) + irr_t(J^*_m(x))) + \sum\limits_{i=\ell+1}^{n}\sum\limits_{j= n+(\ell+1)}^{m} |d(v_i)- d(v_j)|, &\text {if $n > m,$}\\ \\ = 4(irr_t(J^*_n(x))), &\text {if $n = m$,} \end{cases}$$\ with $\ell = \Delta J^*_m(x).$ Case 1: Consider the Jaco Graphs $J^*_n(x)$ and $J^*_m(x), n > m.$ Label the vertices $\underbrace{v_1, v_2, \dots, v_n,}_{vertices-in-J^*_n(x)}\\ \underbrace {v_{n+1}, v_{n+2}, \dots, v_{n+m}}_{vertices-in-J^*_m(x)}.$ Let us expand the definition of $irr_t(J^*_n(x) \cup J^*_m(x))$ into three parts.\ \ Part(i): In respect of $J^*_n(x)$ itself, we have the partial sum,\ \ $|d(v_1) - d(v_2)| + |d(v_1) - d(v_3)| + \dots + |d(v_1) - d(v_{n-2})| + |d(v_1) - d(v_{n-1})| + |d(v_1) - d(v_n)| +\\ |d(v_2) - d(v_3)| + |d(v_2) - d(v_4)|+ \dots + |d(v_2) - d(v_{n-1})|+ |d(v_1) - d(v_n)| +\\ |d(v_3) - d(v_4)| + |d(v_3) - d(v_5)| + \dots + |d(v_3) - d(v_n)| + \\ .\\ .\\ .\\ |d(v_{n-2}) - d(v_{n-1})| + |d(v_{n-2}) - d(v_n)| +\\ |d(v_{n-1}) - d(v_n)|\\ =irr_t(J^*_n(x)).$\ \ Part (ii): In respect of $J^*_m(x)$ itself, we have the partial sum,\ \ $|d(v_{n+1}) - d(v_{n+2})| + \dots + |d(v_{n+1}) - d(v_{(n+m)-2})| + |d(v_{n+1}) - d(v_{(n+m)-1})| + |d(v_{n+1}) - d(v_{n+m})| +\\ |d(v_{n+2}) - d(v_{n+3})| + \dots + |d(v_{n+2}) - d(v_{(n+m)-1})|+ |d(v_{n+2}) - d(v_{n+m})| +\\ |d(v_{n+3}) - d(v_{n+4})| + \dots + |d(v_{n+3}) - d(v_{n+m})| + \\ .\\ .\\ .\\ |d(v_{(n+m)-2}) - d(v_{(n+m)-1})| + |d(v_{(n+m)-2}) - d(v_{n+m})| +\\ |d(v_{(n+m)-1}) - d(v_{n+m})|\\ =irr_t(J^*_m(x)).$\ \ Part (iii): In respect of $J^*_n(x)$ towards $J^*_m(x)$ we have the partial sum,\ \ $|d(v_1) - d(v_{n+1})| + |d(v_1) - d(v_{n+2})| + \dots + |d(v_1) - d(v_{n+m})| +\\ |d(v_2) - d(v_{n+1})| + |d(v_2) - d(v_{n+2})| + \dots + |d(v_2) - d(v_{n+m})| +\\ .\\ .\\ .\\ |d(v_n) - d(v_{n+1})| + |d(v_n) - d(v_{n+2})| + \dots + |d(v_n) - d(v_{n+m})| =\\ \\ 0 + |d(v_1) - d(v_{n+2})|+ \dots + |d(v_1) - d(v_{n+m})| +\\ |d(v_2) - d(v_{n+1})| + 0 + |d(v_2) - d(v_{n+3})|+ \dots + |d(v_2) - d(v_{n+m})|+\\ |d(v_3) - d(v_{n+1})| + |d(v_3) - d(v_{n+2})| + 0 + |d(v_3) - d(v_{n+4})| + \dots + |d(v_3) - d(v_{n+m})|+\\ .\\ .\\ .\\ \underbrace{|d(v_{\ell}) - d(v_{n+1})| + \dots + \underbrace{0}_{\ell^{th}-term} + |d(v_{\ell}) - d(v_{n+(\ell + 1)})| + \dots + |d(v_{\ell}) - d(v_{n+m})|}_{\ell^{th}-row} + \\ \\ |d(v_{\ell+1}) - d(v_{n+1})| + |d(v_{\ell+1}) - d(v_{n+2})| + \dots + |d(v_{\ell +1}) - d(v_{n+m})| +\\ .\\ .\\ .\\ |d(v_n) - d(v_{n+1})| + |d(v_n) - d(v_{n+2})| + \dots + |d(v_n) - d(v_{n+m})|.$\ \ By observing that a term $|d(v_i) - d(v_j)|,1 \leq i \leq \ell$ and $i \leq j \leq (n+ i) -1$ can be converted to $|d(v_i) - d(v_j)| = |d(v_{j-n}) - d(v_i)|,$ with $|d(v_{n-j}) - d(v_i)|$ a term of $\sum \limits_{i=1}^{n-1} \sum \limits_{j=i+1}^{n}|d(v_i) - d(v_j)|_{v_i, v_j \in J^*_n(x)}.$ It is also noted that a term $|d(v_i) - d(v_j)|,\ell+1 \leq i \leq n$ and $n+1 \leq j \leq n+ \ell$ can be converted to $|d(v_i) - d(v_j)| = |d(v_{j-n}) - d(v_i)|,$ with $|d(v_{n-j}) - d(v_i)|$ a term of $\sum \limits_{i=1}^{n-1} \sum \limits_{j=i+1}^{n}|d(v_i) - d(v_j)|_{v_i, v_j \in J^*_n(x)}.$\ \ Similarly, by observing that a term $|d(v_i) - d(v_j)|, 1 \leq i \leq \ell-1$ and $n+ 2 \leq j \leq n+ \ell$ can be converted to $|d(v_i) - d(v_j)| = |d(v_{n+i}) - d(v_j)|,$ with $|d(v_{n+i}) - d(v_j)|$ a term of $\sum \limits_{i=1}^{n-1} \sum \limits_{j=i+1}^{n}|d(v_i) - d(v_j)|_{v_i, v_j \in J^*_m(x)}.$ It is also noted that a term $|d(v_i) - d(v_j)|,1 \leq i \leq \ell$ and $(n+ \ell)+1 \leq j \leq n+ m$ can be converted to $|d(v_i) - d(v_j)| = |d(v_{j-n}) - d(v_i)|,$ with $|d(v_{n+i}) - d(v_j)|$ a term of $\sum \limits_{i=1}^{n-1} \sum \limits_{j=i+1}^{n}|d(v_i) - d(v_j)|_{v_i, v_j \in J^*_m(x)}.$\ \ Finally it is observed that the terms $|d(v_i) - d(v_j)| \geq 0, \ell+1 \leq i \leq n+m$ and $(n+\ell) + 1 \leq j \leq n+m$ cannot be converted.\ \ Hence, the result:\ \ $irr_t(J^*_n(x) \cup J^*_m(x)) \leq 2(irr_t(J^*_n(x)) + irr_t(J^*_m(x))) + \sum\limits_{i=\ell+1}^{n}\sum\limits_{j= n+(\ell+1)}^{m} |d(v_i)- d(v_j)|,$ follows.\ \ Case 2: Parts (i) and (ii) follow similarly to that of Case 1.\ \ Part (iii): In respect of $J^*_n(x)$ towards $J^*_n(x)$ we have the partial sum,\ \ $|d(v_1) - d(v_{n+1})| + |d(v_1) - d(v_{n+2})| + \dots + |d(v_1) - d(v_{2n})| +\\ |d(v_2) - d(v_{n+1})| + |d(v_2) - d(v_{n+2})| + \dots + |d(v_2) - d(v_{2n})| +\\ .\\ .\\ .\\ |d(v_n) - d(v_{n+1})| + |d(v_n) - d(v_{n+2})| + \dots + |d(v_n) - d(v_{2n})| =\\ \\ 0 + |d(v_1) - d(v_{n+2})|+ \dots + |d(v_1) - d(v_{2n})| +\\ |d(v_2) - d(v_{n+1})| + 0 + |d(v_2) - d(v_{n+3})|+ \dots + |d(v_2) - d(v_{2n})|+\\ |d(v_3) - d(v_{n+1})| + |d(v_3) - d(v_{n+2})| + 0 + |d(v_3) - d(v_{n+4})| + \dots + |d(v_3) - d(v_{2n})|+\\ .\\ .\\ .\\ |d(v_n) - d(v_{n+1})| + |d(v_n) - d(v_{n+2})| + \dots + |d(v_n) - d(v_{2n-1})| + \underbrace{0}_{n^{th}-term}.$\ \ So similary to the *term conversion rules* of Part (iii) above in Case 1 we calculate exactly another $irr_t(J^*_n(x))$ on the *left under of 0-entries* and another $irr_t(J^*_n(x))$ on the *right upper of 0-entries*. So, Parts (i), (ii) and (iii) added together gives the result:\ \ $irr_t(J^*_n(x) \cup J^*_n(x)) = 4(irr_t(J^*_n(x))).$ For the Jaco Graphs $J^*_n(x)$ and $J^*_m(x),$ we have that: $$firr_t(J^*_n(x) \cup J^*_m(x)) \begin{cases} \leq 2(firr_t(J^*_n(x) + firr_t(J^*_m(x))) + \sum\limits_{i=\ell+1}^{n}\sum\limits_{j= n+(\ell+1)}^{m} |f_{d(v_i)}- f_{d(v_j)}| , &\text {if $n > m$,}\\ \\ = 4(firr_t(J^*_n(x))), &\text {if $n = m$.} \end{cases}$$\ Similar to the proof of Theorem 2.2. The edge-joint of two simple undirected graphs $G$ and $H$ is the graph obtained by linking the edge $vu,$ $v \in V(G), u \in V(H)$ and denoted, $G\rightsquigarrow_{vu}H.$ Consider the underlying Jaco graphs $J^*_n(x)$ and $J^*_m(x)$ on the vertice $v_1, v_2, v_3, \dots, v_n$ and $u_1, u_2, u_3, \dots, u_m$, respectively, then $firr_t(J^*_n(x) \cup J^*_m(x)) = firr_t(J^*_n(x) \rightsquigarrow_{v_1u_1} J^*_m(x)).$ Since $d(v_1) =1 $ and $d(u_1) = 1$ in $J^*_n(x)$ and $J^*_m(x)$ respectively, the *fibonacci weight* of $v_1,$ $u_1$ equals $f_1 = 1,$ respectively. In the graph $J^*_n(x)\rightsquigarrow_{v_1u_1}J^*_m(x)$, we have that $d(v_1) = 2, d(u_1) = 2$ with both fibonacci weights remaining 1, so the result follows. Consider the underlying graphs $J^*_n(x),$ $n \geq 3$ and $J^*_m(x),$ $m\geq 1$ on the vertices $v_1, v_2, v_3, \dots, v_n$ and $u_1, u_2, u_3, \dots, u_m$, respectively. Without loss of generality choose any vertex $v_i, i\neq 1$ from $V(J^*_n(x)).$ Let $V_1 = \{v_x| f_{d(v_x)} \leq f_{d(v_i)}\}, |V_1| = a;$ $V_2 = \{v_y| f_{d(v_y)} > f_{d(v_i)}\}, |V_2| = b;$ $V_3 = \{u_x|f_{d(u_x)} \leq f_{d(v_i)}\}, |V_3| = a^*$ and $V_4 = \{u_y| f_{d(u_y)} > f_{d(v_i)}\}, |V_4| = b^*.$ For the simple connected graph $G' = J^*_n(x)\rightsquigarrow_{v_iu_1}J^*_m(x)$ we have that:\ \ $firr_t(G') = firr_t(J^*_n(x)) + firr_t(J^*_m(x)) + \sum \limits_{j=1}^{n} \sum\limits_{k=1}^{m}|f_{d(v_j)}- f_{d(u_k)}|_{v_j \in V(J^*_n(x)), u_k \in V(J^*_m(x))} + \sum\limits_{j=1}^a|f_{d(v_i)} -f_{d(v_j)}|_{v_j \in V_1} - \sum\limits_{j=1}^b|f_{d(v_i)} -f_{d(v_j)}|_{v_j \in V_2} + \sum\limits_{j=1}^{a^*}|f_{d(v_i)} -f_{d(v_j)}|_{v_j \in V_3} - \sum\limits_{j=1}^{b^*}|f_{d(v_i)} -f_{d(v_j)}|_{v_j \in V_4}.$ Clearly for $G = J^*_n(x) \cup J^*_m(x)$ we have that $firr_t(G) = firr_t(J^*_n(x)) + firr_t(J^*_m(x)) + \sum \limits_{j=1}^{n} \sum\limits_{k=1}^{m}|f_{d(v_j)}- f_{d(u_k)}|_{v_j \in V(J^*_n(x)), u_k \in V(J^*_m(x))}.$\ \ Since $f_1 = f_2 =1$, increasing $d(u_1)$ by 1 has no effect on the value of $firr_t(J^*_n(x)\rightsquigarrow_{v_iu_1}J^*_m(x)).$ By increasing $d(v_i)$ by 1 we increase the value of $firr_t(J^*_n(x)\rightsquigarrow_{v_iu_1}J^*_m(x))$ by exactly $|f_{d(v_i)} - f_{d(v_j)}|_{1 \leq j \leq a}$ in respect of $J^*_n(x).$ So the total partial increase is given by sum $(\sum \limits_{j=1}^{a}|f_{d(v_i)} - f_{d(v_j)}|)_{v_j \in V_1}.$ It also reduces the value of $firr_t(J^*_n(x)\rightsquigarrow_{v_iu_1}J^*_m(x))$ by exactly $|f_{d(v_i)} - f_{d(v_j)}|_{1 \leq j \leq b}.$ So the total partial decrease is given by $(\sum \limits_{j=1}^{b}|f_{d(v_i)} - f_{d(v_j)}|)_{v_j \in V_2}.$\ \ In respect of $J^*_m(x)$ it also increases the the value of $firr_t(J^*_n(x)\rightsquigarrow_{v_iu_1}J^*_m(x))$ by exactly $|f_{d(v_i)} - f_{d(v_j)}|_{1 \leq j \leq a^*}.$ So the total partial increase is given by $(\sum \limits_{j=1}^{a^*}|f_{d(v_i) - d(v_j)}|)_{v_j \in V_3}.$ It also reduces the value of $firr_t(J^*_n(x)\rightsquigarrow_{v_iu_1}J^*_m(x))$ by exactly $|f_{d(v_i)} - f_{d(v_j)}|_{1 \leq j \leq b^*}.$ So the total partial decrease is given by $(\sum \limits_{j=1}^{b^*}|f_{d(v_i)} - f_{d(v_j)}|)_{v_j \in V_4}.$\ \ Hence, the result:\ \ $firr_t(G') = firr_t(J^*_n(x)) + firr_t(J^*_m(x)) + \sum \limits_{j=1}^{n} \sum\limits_{k=1}^{m}|f_{d(v_j)}- f_{d(u_k)}|_{v_j \in V(J^*_n(x)), u_k \in V(J^*_m(x))} + \sum\limits_{j=1}^a|f_{d(v_i)} -f_{d(v_j)}|_{v_j \in V_1} - \sum\limits_{j=1}^b|f_{d(v_i)} -f_{d(v_j)}|_{v_j \in V_2} + \sum\limits_{j=1}^{a^*}|f_{d(v_i)} -f_{d(v_j)}|_{v_j \in V_3} - \sum\limits_{j=1}^{b^*}|f_{d(v_i)} -f_{d(v_j)}|_{v_j \in V_4},$ follows. Conclusion ========== The allocation of Fibonacci weights to the vertices of graphs as a function of vertex-degree is a variation of graph labeling. Deriving a useful edge labeling from the primary vertex labeling is still open. From this study the following can be pursued:\ \ (i) Describe a well-defined algorithm that determines the result of Theorem 1.1,\ (ii) Describe a well-defined algorithm that determines the result of Theorem 2.1.\ \ $f$-Irregularity can be studied for a number of classes of graphs such as paths, cycles, trees and for graph operations. Furthermore, we propose the following vertex labeling study. Define the $\pm$*Fibonacci weight*, $f_i^\pm$ of a vertex $v_j$ to be $-f_{d(v_j)},$ if $d(v_j)$ is odd and, $f_{d(v_j)},$ if $d(v_j)$ is even. Determine $firr_t^\pm(G) = \sum \limits_{i=1}^{n-1} \sum \limits_{j=i+1}^{n}|f_i^\pm - f_j^\pm|$ in general or for some special classes of graphs. For example, $firr_t^\pm(P_n) = 4(n-2)$ and $firr_t^\pm(C_n) = firr_t(C_n) = irr_t(C_n) = 0.$ The latter result holds for all regular graphs.\ \ Because total irregularity presents a sum of differences between all pairs of natural numbers in a subset $X\subset \Bbb N$ for $X$ having an even number of odd numbers we see that $f_t$-irregularity is the specialisation thereof by mapping on Fibonnaci numbers. This allows for studies where mapping on complex numbers or other number classes or families of *number abstractions* which have the notions of *even abstractions* and *odd abstractions* imbedded, to be considered. Clearly such number abstraction could be a graph. For example let a path on $n$ vertices be $v_1e_1v_2e_2v_3e_3\dots e_{n-1}v_n.$ For a vertex $v\in V(G)$, $d_G(v) = k$ add the path $P_k$ as an *ear* to vertex $v$ by adding the edges $vv_1$, $vv_n$. An *path-eared* graph is denoted, $G^\mathcal{P}$. Assume the chromatic number of paths is the invariant. Define total $\chi$-irregularity or $\chi_t$-irregularity as $\chi_t(G^\mathcal{P}) =\frac{1}{2}\sum \limits_{u,v \in V(G)}|\chi(P_{d(u)}) - \chi(P_{d(v)})|.$ Furthering a study for certain classes of graphs $\mathcal{H}$ with $H\in \mathcal{H},$ $|V(H)| \geq 2$ to be *H-eared* to the vertices of $G$ and selecting any invariant $\mu$ to define $\mu_t$-irregularity is worthy. Furthemore, if a meaningful definition for edge labeling can be found for a $H$-eared graph, a new field of total graph labeling may be researched.\ \ ***Open access:*** This paper is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution and reproduction in any medium, provided the original author(s) and the source are credited.\ \ References (Limited)\ \ $[1]$ H. Abdo, S. Brandt, H. Dimitrov, *The total irregularity of a graph,* Discrete Mathematics and Theoretical Computer Science, Vol 16(1), (2014), 201-206.\ $[2]$ H. Abdo, D. Dimitrov, *The total irregularity of a graph under graph operations,* Miskolc Mathematical Notes, Vol 15 (1) (2014) 3-17.\ $[3]$ J.A. Bondy, U.S.R. Murty, **Graph Theory with Applications,** Macmillan Press, London, (1976).\ $[4]$ J. Kok, P. Fisher, B. Wilkens, M. Mabula, V. Mukungunugwa, *Characteristics of Finite Jaco Graphs, $J_n(1), n \in \Bbb N$*, arXiv: 1404.0484v1 \[math.CO\], 2 April 2014.\ $[5]$ J. Kok, P. Fisher, B. Wilkens, M. Mabula, V. Mukungunugwa, *Characteristics of Jaco Graphs, $J_\infty(a), a \in \Bbb N$*, arXiv: 1404.1714v1 \[math.CO\], 7 April 2014.\ $[6]$ J. Kok, C. Susanth, S.J. Kalayathankal, *A Study on Linear Jaco Graphs*, Journal of Informatics and Mathematical Sciences, Vol 7(2), (2015), 69-80.\ $[7]$ Y. Zhu, L. You, J. Yang, *The Minimal Total Irregularity of Graphs,* arXiv: 1404.0931v1 \[math.CO\], 3 April 2014.\ [^1]: Named after the young lady, Lumin Bruyns from Klitsgras who it is hoped will grow up with a deep fondness for mathematics.

--- abstract: 'In the surface acoustic wave quantum computer, the spin state of an electron trapped in a moving quantum dot comprises the physical qubit of the scheme. Via detailed analytic and numerical modeling of the qubit dynamics, we discuss the effect of excitations into higher-energy orbital states of the quantum dot that occur when the qubits pass through magnetic fields. We describe how single-qubit quantum operations, such as single-qubit rotations and single-qubit measurements, can be performed using only localized static magnetic fields. The models provide useful parameter regimes to be explored experimentally when the requirements on semiconductor gate fabrication and the nanomagnetics technology are met in the future.' author: - 'S. Furuta' - 'C. H. W. Barnes' - 'C. J. L. Doran' title: 'Single-qubit gates and measurements in the surface acoustic wave quantum computer' --- Introduction ============ Quantum computation promises enormous technological advances in the field of information processing and the quest for its realization has attracted many strong contenders in the field of physics and engineering. This paper is concerned with a scheme for quantum computation put forward by Barnes, Shilton and Robinson,[@Barnes:PRB:2000] which falls into the semiconductor quantum dot category.[@Loss:PRA:1998; @Burkard:PRB:99; @Loss:Proc:1999; @Troiani:PRB:2000; @DiVincenzo:Nat:2000; @Elzerman:PRB:2003; @Calarco:PRA:2003] The proposal for quantum computation is based on the results of ongoing experiments that have demonstrated the capture and transport of single electrons in moving quantum dots. The dots are formed when a surface acoustic wave (SAW) travels along the surface of a piezoelectric semiconductor containing a two-dimensional electron gas (2DEG). See Fig. \[sawdevice\] for a schematic diagram of the device. When the SAW is made to pass through a constriction in the form of a quasi-one-dimensional channel (Q1DC), the induced piezoelectric potential drags electrons into and along the Q1DC. In certain parameter regimes the device transports one electron per potential minimum of the SAW.[@Talyanskii:97] The spin on the trapped electron represents the physical qubit. Quantum computation involves performing qubit operations on the trapped electrons as they move with the speed of the SAW. Many schemes for quantum computation, such as conventional quantum dots, doped silicon,[@Kane:1998] superconducting boxes,[@Makhlin:2001] and ion traps,[@Wineland:2003] involve static qubits. The surface acoustic wave quantum computer on the other hand is of the “flying qubit” type, which include linear optics schemes,[@knill:2001] some ion trap schemes with ion shuttling[@Cirac:Nature:2000] and schemes based on coherent electron transport in quantum wires.[@Bertoni:PRL:1999; @Barnes:PRB:2000; @APopescu:2003] All these have in common that the carriers of quantum information physically move through space during the computation. Flying qubits have the advantage of being able to distribute information quickly over large distances across the quantum circuit when decoherence times are short and to interface with quantum memory registers at fixed locations. Another advantage of the SAW quantum computation scheme is its ensemble nature. It intrinsically performs time-ensemble computation, in much the same way NMR quantum computation performs molecular-ensemble computations.[@Cory:Proc:1997; @chuang:1998; @Jones:Proc:2003] Time-ensemble computation alleviates the demand for single-shot spin measurements and has the advantage of being robust against small random errors. ![ (Color online) Schematic diagram of an experimental device for producing quantized acoustoelectric currents through a narrow Q1DC constriction. \[sawdevice\] ](ExpDevice.eps){width="8.6cm"} This paper considers proposals[@Barnes:PRB:2000] for implementing quantum gates on single SAW qubits using only static magnetic fields generated by surface magnetic gates. Detailed modeling of the gate operation has been accomplished by means of both analytic solutions and numerical simulations of the Pauli equation. We show how electrostatically confined moving electrons behave under the influence of various magnetic fields and discuss the implications for quantum computing with surface acoustic wave electrons. The physics of the SAW-guided qubit is explained in more detail in section \[Section:Qubit\]. In sections \[Section:Unitary\] and \[Section:Readout\], we present results on single-qubit unitary gates and single-qubit readout gates based on the Stern-Gerlach effect. Section \[Section:Neglect\] discusses some of the decoherence processes involved in quantum-dot based schemes. Section \[Section:Summary\] is a summary of the results with parameter regimes of interest for future experiments. Scheme for quantum computation {#Section:Qubit} ============================== We begin by summarizing the quantum computation scheme proposed by Barnes, Shilton and Robinson.[@Barnes:PRB:2000] Figure \[sawdevice\] shows a schematic diagram of the experimental setup originally designed to demonstrate quantized currents in semiconductors. A NiCr/Al interdigitated transducer is patterned on a GaAs/AlGaAs heterostructure. A narrow depleted Q1DC splits the 2DEG into two regions, the source and drain. When a high frequency AC signal is applied to the transducer, a SAW propagates through the 2DEG, producing a periodic piezoelectric potential across the 2DEG. The potential drags electrons in the source region through the narrow Q1DC constriction into the drain. It has been shown experimentally that over a range of SAW power and gate voltages, the current passing through the Q1DC is quantized in units of $ef$, where $e$ is the electronic charge and $f$ is the frequency of the SAW.[@Shilton:96; @Talyanskii:97; @Talyanskii:98] The smallest quantized current observed corresponds to the transport of a single electron in each SAW minimum. Typically, the SAW in GaAs moves at $2700{\ \mathrm{ms}}^{-1}$ at a frequency of around $2.7{\ \mathrm{GHz}}$, with an applied power of $3-7{\ \mathrm{dBm}}$.[@Talyanskii:97] These parameters produce currents in the range of nanoamps. Given the ability to trap single electrons in the SAW minima, the scheme for quantum computation is as follows. It is possible for an array of $N$ Q1DCs in parallel to capture $N$ qubits in every $M$th minimum, with a single electron in each Q1DC, producing a qubit register along the SAW wavefront. $M$ can be chosen sufficiently large to ensure that the Coulomb interaction between successive qubit registers do not interfere with each separate computation. The qubits move with the minima of the SAW, passing through a sequence of static one- and two-qubit gates before arriving at an array of spin readout devices. One-qubit gates may be operated by nanoscale electromagnetic fields. Where two-qubit gates are needed, neighboring Q1DCs are allowed into a tunnel contact controlled by a potential on a surface gate.[@Barnes:PRB:2000] The use of the Coulomb coupling between neighboring Q1DCs is a common tool in spintronics which can be used to generate entangled states in dual-rail qubit representations.[@Ionicioiu:PRA:2001; @Kitagawa:PRL:1991] Figure \[sawnetwork\] illustrates the network of Q1DCs and qubit gates envisaged for performing a particular quantum computation. ![ (Color online) Schematic diagram of a quantum gate network in a SAW quantum computer: (a) Two-qubit tunneling gates; (b) One-qubit magnetic gates (in various orientations); (c) Gate network for quantum computation with SAW electron spins. Gray lines running horizontally represent Q1DCs; blackened regions indicate the SAW minima where the qubits reside; arrows represent spin polarization; rings represent magnetic surface gates; white squares represent readout gates. \[sawnetwork\] ](Network.eps){width="8.6cm"} The SAW-trapped electron is well confined in all three spatial directions. In this paper we define Cartesian coordinates such that the SAW propagates along the $x$ axis with the $z$ axis normal to the 2DEG. The 2DEG is produced by a band-energy mismatch at the GaAs/AlGaAs interface which gives rise to a confining potential in the $z$ direction. The energy level spacings in the 2DEG well are on the order of $50-100{\ \mathrm{meV}}$.[@Kelly:Book:95] Further confinement in the $y$ direction is provided by an extended Q1DC, etched into the surface so as to avoid screening the SAW-induced potential with metallic surface gates.[@Utko:2003] Finally, the confinement in the $x$ direction is due to the SAW potential minimum which is approximately sinusoidal. The SAW amplitude is typically 40 meV,[@Robinson:PRB:2001] which is sufficiently large to prevent qubits being lost via tunneling into neighboring SAW minima. There are two important aspects of the SAW quantum computation scheme that distinguishes it from other similar quantum dot schemes. First, the scheme provides repetitions of the same quantum computation with each passing of a single wavefront of the SAW. Therefore, a statistical time-ensemble of identical computations can be read out at the end of the Q1DCs as a measurable current, alleviating the need for single-electron measurements. Two sources of noise in the measured current may be estimated as follows: The shot noise is largely determined by how well the current is quantized to $I=e f_\mathrm{SAW}$ and precisions of $< 0.1\%$ can be experimentally achieved.[@Robinson:Proc:2003] Johnson noise arises from the resistance of the ohmic contacts and the 2DEG which are on the order of $\sim 100-1000{\ \mathrm{\Omega}}$. At temperatures of $1{\ \mathrm{K}}$ they produce a rms voltage noise spectral density on the order of $10^{-10}{\ \mathrm{V/\sqrt{Hz}}}$ at most. This cannot drive a current noise through the SAW device since its effective intrinsic impedance of $10{\ \mathrm{M\Omega}}$ is comparatively very large. See experimental papers[@Shilton:96; @Talyanskii:97; @Talyanskii:98; @Robinson:PRB:2002; @Robinson:Proc:2003] for more detailed discussions. The second key aspect of the scheme is the static nature of the gate components of the quantum circuit. This alleviates the need for strong, targeted and carefully-timed electromagnetic pulses that can be difficult and expensive to implement. The requirement of such expensive control resources often limit the scalability of most quantum computing implementations. It would certainly be convenient, though not essential, to have a means of preparing a pure fiducial qubit state. In NMR quantum computing, operations are carried out on ensembles of replica qubits which remain close to a highly mixed state of thermal equilibrium. Nevertheless, a successful readout of the computation is obtained because the sum over many identical computations provides a measurable signal. Similarly in the SAW quantum computing scheme, states close to the maximally-mixed state are still useful because of the time-ensemble nature of the scheme. However, in contrast to NMR schemes, it is in principle possible to read out single electrons in the SAW scheme. We will therefore begin with nonensemble quantum computation in mind and only later exploit the advantages of ensemble computation to deal with noise. Of course, the more pure the qubit states remain, the faster the computation will converge to the result. For these reasons, we describe in this paper two simple methods for preparing pure fiducial qubit states. In the original proposal for the SAW quantum computer,[@Barnes:PRB:2000] it was noted that the application of an external magnetic field of about $1{\ \mathrm{T}}$ will influence which spin states of electrons are favored in the capture process.[@Robinson:PRB:2001] This polarized capture process can be summarized as follows. The SAW is strongly screened in the bulk 2DEG until it is raised above the Fermi energy and the quantum dot begins to form. When the higher-energy minority quantum dot forms for the higher-energy sub-band of polarized electrons, the probability of capturing minority electrons is small (see Fig. \[BCapture\]), whilst the probability of capturing electrons from the lower-energy sub-band is large. Once a cloud of approximately polarized electrons is captured, the exchange interaction will generally entangle electrons in the same dot together, so that the subsequent loss of electrons into the Fermi sea will lead to a decoherent process that could relax the remaining electrons into the low-energy polarized state. A more detailed multiparticle analysis will be needed to determine the final state of the remaining single electron, but the combination of the above two processes is likely to lead to a high level of polarization. ![ (Color online) Energy diagram showing polarized electron capture by means of Zeeman band splitting in the presence of a uniform magnetic field: Arrows on electrons indicate spin polarizations; $E_f$ is the Fermi energy. The SAW is strongly screened in the bulk 2DEG below the Fermi energy. As the SAW enters the Q1DC constriction, the confining potential begins to form. The probability of capturing spin-down is small at the point the minority spin-type dot forms (upper curve). In contrast, the probability of capturing spin-up electrons is high when the majority spin-type dot forms (lower curve). \[BCapture\] ](b_field_capture.eps){width="8.6cm"} The above method is conceptually simple and would be easy to implement in the laboratory. However, the macroscopic magnetic field required in the capture region will need to be shielded from the rest of the device where the quantum computation is to be carried out, and this may present a nontrivial problem. If we chose to drive spin flips using microwave pulses, then the macroscopic field is actually required across the whole device. However, the problem with using microwaves is their relatively long wavelength which tends to affect every part the computation. Alternatively, we may use local, static magnetic fields to initialize, rotate and read out single qubits without a global magnetic field. In Section \[Section:Readout\], it will be demonstrated that spin-polarized electrons can be prepared and measured using a gate driven by the Stern-Gerlach effect. First we turn to the implementation of single qubit rotations using local static magnetic gates. Single-qubit unitary gates {#Section:Unitary} ========================== Single-qubit unitary operations may be carried out, in principle, by allowing the trapped single electrons to pass through regions of uniform magnetic field. If there is no spin-orbit coupling effect, the spin state of the qubit will evolve according to the Zeeman term in the Hamiltonian: ${\frac{1}{2}}g {\mu_\mathrm{B}}\,\bm{B}\cdot \bm{\sigma}\,\psi$, where $\bm{B}$ is the magnetic field, $\bm{\sigma}$ is the vector form of the 3 Pauli operators and $\psi=(\alpha,\beta)$ is a spinor. The Bloch vector $n=\left(-2\,\mathrm{Im}[\alpha\beta^*],2\, \mathrm{Re}[\alpha\beta^*],|\alpha|^2-|\beta|^2\right)$ precesses about the direction of the magnetic field with angular frequency $g {\mu_\mathrm{B}}|B|/\hbar$. A local static magnet may be produced by a magnetic force microscope or by evaporative deposition of a ferromagnetic material such as Cobalt, or a permalloy such as NiFe. Inevitably, different samples will produce different strengths of magnetic field. But there are methods to vary the strength and pattern of the field once the sample has been fabricated. It has been demonstrated that ferromagnetic properties of thin-film 3d transition metals can be modified via ion irradiation.[@Kaminsky:2001] Another method would be to use oxidation techniques with the atomic force microscope. Only two independent directions of the B-field are necessary to produce an arbitrary single-qubit manipulation and we will choose these to be perpendicular to the direction of the SAW, one aligned with and the other perpendicular to the 2DEG. For idealized qubits with no spatial degree of freedom, the above model for single-qubit rotations is complete. However, the trapped electron is a charged particle with a spatial distribution within the dot. The fields couple to both the spatial and spin degrees of freedom, causing the electron to experience the Lorentz force as well as spin-precession. It is therefore clear that one cannot increase magnetic fields arbitrarily, since the Lorentz force will upset the confinement properties of the electron. Nor can the direction of the field be chosen arbitrarily without consequences for the robustness of the gate. To address these concerns, we analyze the behavior of the full wavefunction: a two-component spinor field defined over space and time, $\psi_\sigma(\bm{x},t)$, under the action of gates operated by static magnets. Pauli Hamiltonian with uniform magnetic fields ---------------------------------------------- Assuming a uniform magnetic field in the lab frame, we solve the Pauli equation for the spin field from which the probability density field and the Bloch vector field can be obtained. The qubit is trapped in a net electrostatic potential with contributions from the Q1DC split gates, the 2DEG confining potential and the SAW piezoelectric potential. The parameter regimes we consider allow us to neglect motion out of the 2DEG plane.[^1] The net confining potential in the $xy$ plane is modeled by [@Robinson:PRB:2001] $$\begin{aligned} V &=& V_\mathrm{Q1DC} + V_\mathrm{SAW} \nonumber \\ &=& V_0 \frac{y^2}{2 w^2} + A\, \big\{1-\cos\left[2\pi(x/\lambda - f t)\right]\big\}. \label{V}\end{aligned}$$ The Q1DC split gate voltages are such that typically $V_0 \sim 2800 {\ \mathrm{meV}}$. The width of the Q1DC $w$ is typically between $1$ and $2{\ \mathrm{\mu m}}$. The amplitude of the SAW is $A \sim 40{\ \mathrm{meV}}$ with wavelength $\lambda \sim 1{\ \mathrm{\mu m}}$ and frequency $f \sim 2.7{\ \mathrm{GHz}}$. The appropriate nonrelativistic equation for the two-component spinor field is the Schrödinger equation with a Pauli Hamiltonian. For an electron moving in an arbitrary vector potential $A$ and potential energy $V$, the Pauli Hamiltonian $H$ is $$\label{PauliEq} \frac{1}{2m^*} \left( \hat{p}^2 + e\,(\hat{p}\cdot A) + 2 e A\cdot \hat{p} + e^2A^2\right) + {\frac{1}{2}}\, g {\mu_\mathrm{B}}\, \sigma \cdot B + V,$$ where $e>0$ is the electronic charge, $g \simeq 0.44$ is the Landé g-factor and $m^* \simeq 0.067 m_e$ is the effective mass in GaAs. By simulating in the plane of the 2DEG we simplify to a 2+1 dimensional model, with variables $(x,y,t)$. If the particle were chargeless but with an anomalous magnetic moment, terms in $H$ that explicitly involve $e$ can be dropped. In such a case, there is no Lorentz force acting on the particle, the Schrödinger equation simplifies significantly and there would be no need to concern ourselves with the spatial behavior of the qubit. Before proceeding further, it is convenient to transform into the rest frame of the electron moving with the SAW speed $v$. We may use simple Galilean transformations $ x' = x - vt,\ y' = y,\ t' = t$, since the SAW velocity is non-relativistic. The potentials in the electron rest frame become, in the harmonic oscillator approximation, $$\begin{aligned} A(x',t') &=& -\beta(x'+ v t') \\ V(x',y') &=& V_0 \frac{{y'}^2}{2 w^2}+ A \left[1-\cos\left( \frac{2\pi x'}{\lambda}\right)\right]\\ &\simeq& \frac{V_0}{2 w^2}\, {y'}^2 + \frac{A}{2}\, k^2 {x'}^2, \label{V'}\end{aligned}$$ with $k=2\pi/\lambda$. A further convenience is to use a system of natural units such that $\hbar,m^*,v,e$ are unity. The units of length, time and energy become $\hbar/m^*v=0.640{\ \mathrm{\mu m}}$, $\hbar/m^*v^2=0.237\,\mathrm{ns}$ and $m^*v^2=2.78\,\mathrm{\mu eV}$, respectively. The natural unit of magnetic field is $1.61 {\ \mathrm{mT}}$. Parameters can now be assigned with dimensionless values with respect to the above units. In the electron rest frame, $$\label{PauliEq2} \begin{split} -{\frac{1}{2}}\nabla^2 \psi - i\left[\,A\cdot\nabla+{\frac{1}{2}}(\nabla\cdot A)\,\right] \psi + {\frac{1}{2}}A^2 \psi \\ + {\frac{1}{2}}\, g {\mu_\mathrm{B}}\, \sigma\cdot B \psi + V \psi = i {\frac{\partial }{\partial t}}\psi, \end{split}$$ where the primes will subsequently be dropped from the coordinates. We will further simplify the Hamiltonian by adopting the Coulomb gauge $\nabla\cdot A=0$. Qubit rotation: Uniform transverse magnetic field ------------------------------------------------- For a magnetic field in the $y$ direction in the plane of the 2DEG and transverse to the Q1DC, we model the magnetic field in the region of the single-qubit gate with a vector potential of the form $$\label{Az} A_z(x) = -\beta\,x,$$ with $A_x=A_y=0$. Clearly this does not vanish at infinity, but we only consider interactions over regions of finite extent. This potential generates a uniform magnetic field of strength $\beta$ in the $y$ direction. With this potential Eq. (\[PauliEq2\]) becomes $$\label{PauliEq3} -{\frac{1}{2}}\nabla^2 \psi + {\frac{1}{2}}\, \beta^2(x+t)^2 \psi + {\frac{1}{2}}\, g {\mu_\mathrm{B}}\beta\,\sigma_y\psi + V\psi = i{\frac{\partial }{\partial t}}\psi,$$ in which the potential $V$ is given by (\[V’\]). An effective potential $V + A^2/2$ can be identified, which resembles a harmonic oscillator but is time-dependent. Anticipating an analytic solution by separation of variables we apply the ansatz $$\psi_\pm(x,y,t) = \chi(x,t)\,\phi_n(y)\,e^{-iE_nt}|s_y\rangle\,e^{- i s_y \Delta E\, t/2}, \label{ansatz}$$ where $\phi_n$ are the harmonic oscillator eigenstates with energies $E_n=\omega_y (n+{\frac{1}{2}})$, and $|s_y\rangle$ are the eigenstates of $\sigma_y$ with eigenvalues $s_y=\pm 1$. We have introduced the oscillator frequency $\omega_y = \sqrt{V_0}/w$ and the Zeeman energy gap $\Delta E = g {\mu_\mathrm{B}}\beta$. The energy eigenstates in the $y$ direction are exactly the harmonic oscillator modes. The wavefunction in the $x$ direction is time-dependent and it is the solution of $\chi(x,t)$ to which we now turn. On substituting (\[ansatz\]) into (\[PauliEq3\]), we obtain a PDE for $\chi(x,t)$: $$\label{chiEquation} -{\frac{1}{2}}\, \partial^2_x\, \chi + \left[{\frac{1}{2}}(c_0+c_1)\,x^2 + c_1\,xt\right]\chi = i\, \partial_t\, \chi,$$ with $c_0= A\,k^2$ and $c_1= \beta^2$. Terms which depend only on $t$ have been dropped, as they merely contribute global, time-dependent phases that do not affect the dynamics. A Gaussian solution of (\[chiEquation\]) can be found with a further ansatz $$\label{GaussAnsatz} \chi(x,t) = \exp\left[f_1(t)\,x^2 + f_2(t)\, x + f_3(t)\right].$$ This system of time-dependent functions can be determined self-consistently, assuming a Gaussian groundstate of the SAW dot for the initial condition at $t=0$. The resulting expressions for $f_1,f_2,f_3$ are complicated, but by noting a few general properties of the solution we can understand all the important features of the dynamics. $f_3(t)$ takes account of the normalization but is otherwise of no more interest. $f_1(t)$ and $f_2(t)$ together describe a groundstate Gaussian wavefunction evolving in the time-dependent vector potential. The solution is particularly simple in that it remains Gaussian throughout, so we will only need to keep track of the position of the central peak $\mu$ and the width (standard deviation) $\delta$ of the probability distribution $|\chi(x,t)|^2$: $$\begin{aligned} \mu(t) &=& -{\frac{1}{2}}\frac{\mathrm{Re}[f_2(t)]}{\mathrm{Re}[f_1(t)]} \\ \delta(t) &=& {\frac{1}{2}}\, \sqrt{-\mathrm{Re}[f_1(t)]}.\end{aligned}$$ The typical energy scales for the SAW electron encountered in experiment puts the system in the regime where $s \ll 1$ and $c_1 \ll c_0$. Expanding $f_1$ and $f_2$ to first order in $s$ and $c_1/c_0$, we obtain the asymptotic behavior for the position of the peak: $$\mu(t)\rightarrow \frac{-c_1}{c_0+c_1}\,t.$$ From this result we see that the magnetic field introduces a constant drift velocity of the peak in the $-x$ direction. On exiting the interaction region, the peak will be off-center with respect to the SAW dot and the probability distribution will subsequently oscillate in the $x$ direction, perhaps exciting higher-energy orbital states of the dot. If the charge distribution is pulled too far off-center, it is likely to escape the SAW quantum dot. We could ask how long the electron can remain in the field before it is dragged a distance $\lambda$ away from the center of the dot in the $x$ direction: $$T_\mathrm{max} \simeq \frac{c_0+c_1}{2 c_1}\lambda.$$ When this is compared with the time required for the Bloch vector to rotate by some appreciable angle such as a $\pi$ rotation, we obtain the ratio $$T_\mathrm{max}/T_\pi= g{\mu_\mathrm{B}}\left(\frac{A\,k^2+\beta^2}{k}\right),$$ which is about $2000$ for a $1{\ \mathrm{T}}$ field. This means that the Bloch vector is allowed to precess approximately $1000$ times before the drag effect destabilizes the qubit confinement. In the limit of weak fields, this ratio is no smaller than $g{\mu_\mathrm{B}}\,A\,k\sim 800$. Therefore, whilst it is possible that higher orbital states are excited as a consequence of the interaction, there is little danger of the electron escaping the SAW quantum dot during the operation of the gate. We now turn to the behavior of $\delta(t)$, the width of the Gaussian wavefunction, which in the limit $c_1 \ll c_0$ oscillates according to the simple expression $$\delta(t) = \sqrt{\frac{\sqrt{c_0}}{2(c_0+c_1)}} \left[1+\frac{c_1}{c_0}\,\cos^2(\sqrt{c_0+c_1}\ t)\right]^{1/2},$$ with $c_0 = A\,k^2$ and $c_1 = \beta^2$. The frequency of the oscillation increases with the energy of the dot and is typically much faster than the Zeeman spin precession frequency, i.e., rate of precession of the qubit Bloch vector. These results are plotted for a specific case in Fig. \[GaussianSolution\]. The top two plots show the evolution of $\mu$ and $\delta$. As the Bloch vector rotates about the field along the $y$-axis, the Gaussian translates in the $x$ direction with a rapidly oscillating width. The bottom plot of Fig. \[GaussianSolution\] shows probability amplitudes of excitation into higher-energy simple harmonic oscillator (SHO) modes in the $x$ direction of the quantum dot. ![ Gaussian evolution of the probability distribution $|\chi(x,t)|^2$ in a constant magnetic field $B_y$. The top plot shows peak position $\mu$ and the middle plot shows the width $\delta$. The bottom plot shows the probability amplitudes $|C_n|^2$ of the $n$th SHO mode in the $x$ direction: $|C_1|^2$ (long dash), $|C_2|^2$ (short dash). The plot also shows $1-|C_0|^2$ (solid). Typical values for the SAW dot are taken: $A = 40{\ \mathrm{meV}}$; $\lambda = 1.0{\ \mathrm{\mu m}}$; magnetic field of $1{\ \mathrm{T}}$. $T= \pi/g{\mu_\mathrm{B}}\beta\approx 0.08{\ \mathrm{ns}}$ is the time taken for a $\pi$ rotation of the Bloch vector about the $y$-axis. The initial Gaussian wavefunction was taken to be the SAW quantum dot groundstate $(C_0=1)$. In the limit $c_1/c_0 \ll 1$ (weak field), $\mu(t)$ moves approximately linearly and $\delta(t)$ undergoes bounded oscillations about $\delta(0)=s/\sqrt{2}$. \[GaussianSolution\] ](GaussSol_StateProb.eps){width="8.4cm"} Our conclusions are as follows. In addition to rotating the Bloch vector as required, the gate has the effect of displacing the center of the Gaussian wavefunction which will increase the energy of the bound state by an amount depending on both the gate time and the field strength. Higher-energy orbital states are likely to be excited, which could lead to decoherence via spin-orbit couplings or dipole coupling to phonons and photons. According to Fig. \[GaussianSolution\], the electron is almost completely out of the groundstate after a $\pi/2$ rotation of the Bloch vector. In the extreme case after many spin precessions (typically thousands), the qubit will leave the SAW quantum dot. However, for quantum computations the gate time need only be long enough to conduct a single orbit of the Bloch sphere, in which case the excitation into higher orbital states is negligible. These observations reveal some important features of the gate operation that are not revealed by an idealized spin-only model of the qubit. Qubit rotation: Uniform perpendicular magnetic field ---------------------------------------------------- To move the qubit state to an arbitrary point on the Bloch sphere a second axis of rotation on the Bloch sphere is needed, and to this end we consider a gate implemented by a uniform magnetic field in the $z$ direction. This will not be just a trivial extension of the preceding analysis, since the 3D rotation symmetry is broken by the motion of the SAW in the $x$ direction. In practice, the $y$ magnetic field may be easier to fabricate than the $z$ magnetic field, since the latter passes perpendicularly through the 2DEG structure. It could feasibly be produced by layering oppositely aligned thin-film magnets just beneath and just above the 2DEG, or alternatively by applying a global $z$ magnetic field which is shielded in regions where it is not needed. A vector potential generating the uniform $B_z$ magnetic field is $$\label{Ax} A_x(y)=-\beta y,$$ with $A_y=A_z=0$. It should be noted that although the field is uniform and static in the laboratory frame, the electron sees a moving uniform field. In the electron rest frame, $A_x$ is time-independent and $z$-independent, allowing a Hamiltonian in $x$ and $y$ only. The chosen gauge facilitates numerical simulations which follow shortly. An effective scalar potential that is quadratic in $y$ arises from the $A^2$ term in the Hamiltonian (\[PauliEq2\]). It is interesting to compare the strength of this confining potential with the Q1DC potential, which is also approximately quadratic in y: $$\label{CompareEnergies} \frac{\textrm{Energy of $A^2$ term}}{\textrm{Energy of Q1DC term}} = \frac{\beta^2}{V_0/w^2}.$$ With magnetic fields of order 1 Tesla and typical Q1DC energies, this ratio is of the order unity. This means that for the parameter values being considered the effective scalar potential arising form the $A^2$ term is comparable to the Q1DC confinement potential. A stronger field would start to significantly deform the shape of the dot. This will not be a problem as long as the evolution has occurred adiabatically during the deformation. However, as we shall discuss in a moment, the probability of excitation into higher orbital states due to the $B_z$ field is not negligible. The other interesting term in the Hamiltonian is the asymmetric coupling $ -i\beta y\, \partial_x$ between the $x$ and $y$ variables. This is expected to introduce rotational behavior, as we would intuitively anticipate some form of Landau orbital motion due to the Lorentz force. A numerical simulation implementing a Crank-Nicholson, [@CrankNicolson] finite difference algorithm (alternating direction method) was used to simulate the operation of the gate. We started with an initial Gaussian groundstate of the dot with spin state $|\!\uparrow_x\rangle$ and subjected it to $1{\ \mathrm{T}}$ of magnetic field for a duration of $T_{\pi/2} = \pi/2 g {\mu_\mathrm{B}}\beta$, which is the gate time required for a $\pi/2$ rotation of the Bloch vector about the $z$ axis. The evolution of the Bloch vector is simple due to the uniformity of the magnetic field: $n_x(t)=\cos(g{\mu_\mathrm{B}}\beta t)$, $n_y(t)=\sin(g{\mu_\mathrm{B}}\beta t)$, and $n_z(t)=0$. All other parameters were assigned those values given just after Eq. (\[V\]). The result of the simulation is shown in Fig. \[rotation:z\], which shows time-shots of the probability distribution in the 2DEG ($xy$) plane. Losing its initial elliptic contours, the distribution develops two lobes which rotate about its midpoint. The density at the center increases due to the spatial squeezing from the $A^2$ term in the Hamiltonian. This clearly shows excitation into higher-energy orbital states. Using perturbation theory to second order in the strength of the field $\beta$, with harmonic oscillator modes as the basis, we found the following facts: (i) To second order of perturbation theory, only the second excited state with SHO quantum numbers $n_x=n_y=1$ becomes populated; (ii) The probability ratio with respect to the groundstate is $$\label{OscPert} \left|\frac{C_{11}}{C_{00}}\right|^2 = {\beta }^2\, \frac{\omega_x \sin^2[t\,(\omega_x + \omega_y)/2]}{\omega_y( {\omega_x} +{\omega_y})^{2}},$$ where $\omega_x$ and $\omega_y$ are frequencies arising from the harmonic oscillator approximation; (iii) The amplitude of the ratio approaches unity when the field approaches $1{\ \mathrm{T}}$; (iv) The ratio of the spin precession frequency to the frequency of the $|C_{11}|$ oscillation is about $1.6\times 10^{-5}$ for typical SAW parameters (see just after Eq. (\[V\])). The above model is useful in assessing the robustness of the qubit and its susceptibility to decoherence due to orbital motion. Population into this excited state is a problem for decoherence, since the oscillating charge in the dot couples via dipole interactions to phonons and other charges outside the dot. However, provided we have sufficient control of the gate time, we can use (\[OscPert\]) to make the electron exit from the gate in its groundstate. Otherwise, a gate driven by radiofrequency pulses in the presence of a global magnetic field could provide an alternative means to implement spin rotations about the $z$ axis. ![ Contour plots of electron probability density during the operation of a qubit rotation gate. This is a numerical simulation in the $xy$ (2DEG) plane of a SAW electron undergoing a $\pi/2$ rotation gate about the $z$ axis, under a uniform magnetic field $B_z$ of strength $1{\ \mathrm{T}}$. Snap-shots are shown at the following times (ps): (a) 0, (b) 2.89, (c) 11.6, (d) 20.3, (e) 28.9, (f) 37.6. The initial probability distribution is a Gaussian groundstate with standard widths $\sigma_x = 26.0$ and $\sigma_y = 20.6$. The probability density rotates about the $z$ axis under the influence of the Lorentz force acting in the $xy$ plane. Moreover, an effective scalar potential contributed by the $A^2$ term in the Hamiltonian is quadratic in $y$, which spatially squeezes the initial Gaussian distribution into the center of the dot. \[rotation:z\] ](ConfSpinRotWithA_UniFieldz1.eps){width="3.3in"} Single-qubit initialization and measurements {#Section:Readout} ============================================ In addition to single-qubit rotation gates, we require the ability to initialize and measure qubits at the beginning and end of the computation. Spin-polarized electrons can be obtained from injection through a ferromagnetic contact.[@Datta:APL:1990; @Ohno:Nat:1999] There is also a method to polarize spin using nondispersive phases (Aharonov-Bohm and Rashba) without the need for ferromagnetic contacts.[@Ionicioiu:PRB:2003] In the field of quantum computing, a well-known method for achieving readout of solid state spin qubits is to convert spin information into charge information[@Loss:PRA:1998] and subsequently use single-electron transistors or point contacts to detect charge displacements or Rabi oscillations. However, recent theoretical results by Stace and Barrett [@Stace:PRL:2004] argue the absence of coherent oscillations in a continuously measured current noise, contrary to previous results and assumptions,[@Goan:PRB:2001; @Korotkov:PRB:2001] and therefore raise concerns about the measurability of charge oscillations in similar scenarios. In any case, it is difficult to apply these methods to qubits in moving quantum dots. We therefore turn towards a quite different approach, one which enables the initialization and readout of SAW electron spin qubits solely with the aid of nanomagnets and ohmic contacts.[@Barnes:PRB:2000] The readout gate we consider is based on the Stern-Gerlach effect. In the 1920s Bohr and Pauli asserted that a Stern-Gerlach measurement on free electrons was impossible,[@bohr'sArg; @Pauli:6thSolvay] using arguments which combined the concept of classical trajectories and the uncertainty principle. This subsequently led physicists to analyze single-electron Stern-Gerlach measurements within increasingly more rigorous quantum settings, ultimately ending the debate by showing that the measurement can indeed be done, albeit with certain caveats. Thus Stern-Gerlach measurements on *free* electrons have been extensively investigated, but little attention has been given to such measurements on confined electrons.[@dehmelt2] An interesting semiclassical analysis of a Stern-Gerlach type experiment with conduction electrons has been reported,[@Fabian:2002] in which the authors justifiably neglect the Lorentz force effects. In contrast, we analyze a single-electron Stern-Gerlach device, providing a full quantum mechanical treatment and including all Lorentz force effects. In the SAW electron system, single electrons are transported in moving quantum dots. Can this system undergo a Stern-Gerlach type measurement, and would it be of any use in a scheme for quantum computation using qubits trapped in surface acoustic waves? The electron confinement to low dimensions allows us to guide the electron through the magnetic field in ways that enhance the spin measurement and suppress the deleterious effects of delocalization and the Lorentz force. Surface magnets can be arranged in such a way as to produce a local magnetic field inhomogeneity. For example, two north poles placed on either side of the Q1DC will produce a region of intense magnetic field gradient inbetween (see Fig. \[SGDevice\]). Via the Zeeman interaction term $\propto\sigma\cdot B$ in the Pauli Hamiltonian, the field inhomogeneity has the effect of correlating the spatial location of a wavepacket to its spin state.[@Peres:QTCM] ![ Schematic diagram of the spin readout/polarizing device based on the Stern-Gerlach effect: SAW propagates from left to right transporting a single electron in a moving quantum dot. Q1DC relaxes with a gradient $\tan(\theta)$ to partially delocalize the particle during the gate operation; $C_\uparrow$($C_\downarrow$) labels the Q1DC receiving electron flux in the spin up(down) state of the $\sigma_y$ operator; Magnets (of any geometry and polarity) need to produce a localized, inhomogeneous distribution of magnetic field. \[SGDevice\] ](SGDevice.eps){width="8.6cm"} In most situations it is necessary to continue confining the qubit during the operation of the gate, because the spreading time[^2] for a free Gaussian wavefunction is comparatively short – on the order of $1\,\mathrm{ns}$. But the gate must cause a wavepacket splitting in order for the spin states to be resolved, hence the Q1DC must relax to allow for motion in the $y$ direction. This is achieved by patterning the Q1DC in a funnel shape, with an opening angle $\theta$ (see Fig. \[SGDevice\]), such that in the electron rest frame the potential looks like $$\label{VQ1DC} V_\textrm{Q1DC} = V_0\, y^2 / 2\left[w+\tan(\theta)u(t)t\right]^2,$$ where $u(t)$ is the step function: $u(t<0)=0$ and $u(t\ge 0)=1$. A negatively biased surface gate placed on the $x$ axis can be used to guide the electron into the channels. If necessary, both the position of the electrode and the opening angle $\theta$ could be optimized for a particular sample device under low-temperature and high-vacuum conditions by erasable electrostatic lithography.[@Cook:2003] We will study the quality of the readout obtained from the gate for two different magnetic field configurations, both sufficiently simple so as to be realizable in the near future: the linearly inhomogeneous field and the 2D dipole field. Stern-Gerlach gate using a linearly inhomogeneous field ------------------------------------------------------- In the first model we will analyze a simple, unidirectional and linearly inhomogeneous field pointing in the $y$ direction: $$\label{UniformInhomo} B_y = - \beta\, y,$$ with $B_x=B_z=0$. A wedge-shaped single domain surface magnet of appropriate dimensions can produce an inhomogeneous magnetic field of this form near its center of symmetry. A vector potential for this field is $A_x=-\beta\, y z$, with $A_y=A_z=0$. This field exerts a spin-dependent force in the $y$ direction and it is the simplest field that induces the Stern-Gerlach effect. Although a $z$ dependence enters into the vector potential, by considering motion only in the $z=0$ plane we may avoid contributions from terms involving $A$ in the Hamiltonian, as well as the $z$ component of magnetic field. The absence of $x$ in the potential immediately allows us to write down harmonic oscillator modes for the $x$ dependence. The remaining $(y,t)$ dependent part obeys $$\label{diffeq} \left( -{\frac{1}{2}}\nabla^2 - {\frac{1}{2}}\, g {\mu_\mathrm{B}}\beta s_y\, y + V_\mathrm{Q1DC}(y,t)- i\,\partial_t \right) \psi(y,t) = 0$$ where $s_y$ is the eigenvalue $\pm 1$ of the spinor $|s_y\rangle$, and $V_\mathrm{Q1DC}$ is given by (\[VQ1DC\]). The initial state is again the Gaussian ground state of the Q1DC. A solution to (\[diffeq\]) is obtained by a time-dependent Gaussian ansatz (\[GaussAnsatz\]) $$\psi(y,t) = \exp\left[f_1(t)\,y^2 + f_2(t)\, y + f_3(t)\right].$$ In a similar way as before, we solve the system of coupled ordinary differential equations and derive the time dependence of the standard deviation $\delta(t)$ and the position $\mu(t)$ of the probability distribution. Let $c_0=V_0/w^2$ and $c_1=g {\mu_\mathrm{B}}\beta s_y$. The width of the initial Gaussian groundstate wavefunction is determined by $s= c_0^{-1/4}$. The behavior of the Gaussian solution is then characterized by the following time-dependent parameters $$\begin{aligned} \delta(\tau)^2 &=& \frac{\tau }{2\,\gamma\,\sqrt{c_0}} \, \big[ -4\,c_0 + {\alpha }^2\,\cosh (\sqrt{\gamma}\,\ln(\tau)/\alpha)\nonumber \\ &&- \alpha \,{\sqrt{\gamma }}\,\sinh (\sqrt{\gamma}\,\ln(\tau)/\alpha) \big] \\ \mu(\tau) &=& \frac{\sqrt{\tau}\,c_1}{2\,(2\,\alpha^2+c_0)} \big[ \tau^{3/2} - \cosh (\sqrt{\gamma}\,\ln(\tau)/\alpha) \nonumber\\ &&- \frac{3\alpha}{\sqrt{\gamma}} \sinh ({\sqrt{\gamma}\,\ln(\tau)/\alpha}) \big], \label{MuEvolution}\end{aligned}$$ where we have introduced ancillary variables $\alpha= \tan(\theta)/w$, $\gamma = \alpha^2 - 4 \,c_0$ and $\tau = 1+ \alpha \,t$. The parameter $\gamma$ is useful in determining when the trigonometric functions become oscillatory. If the angle $\theta$ is critical such that $\tan(\theta)=2\sqrt{V_0}$, then $\gamma=0$ and the width has an especially simple behavior: $$\delta(t)^2= \frac{ \tau }{\alpha} \,\left[ 1 - \ln\tau + {\frac{1}{2}}\,(\ln\tau)^2 \right].$$ This model predicts the width and position of the qubit wavefunction after a time $t$ in the linearly inhomogeneous field with field gradient $\beta$. In practice we cannot produce arbitrary magnetic fields in arbitrary configurations. For a weak magnetic field such that $c_0 \gg c_1 \gg \alpha$, $$\begin{aligned} \delta(\tau) &=& \sqrt{\frac{\tau}{2 \sqrt{c_0}}} = \delta(0)\,\sqrt{ 1 + \tan(\theta)\, \frac{t}{w} } \\ \mu(\tau) &=& \frac{c_1}{2\,c_0} \left[ \tau^2 - \sqrt{\tau}\,\cos\left(\frac{\sqrt{c_0}}{\alpha}\ln\,\tau\right) \right].\end{aligned}$$ The width increases as $\sim\sqrt{t}$ and the position moves as $\sim t^2$ as expected. We may compare $\delta(t)$ with the dispersion of a Gaussian in free space $$\delta(t) = \sqrt{\delta(0)^2 + \frac{t^2}{4\,\delta(0)^2}}\ ,$$ which is linear in $t$ at large times. The broadening of the width is suppressed due to the confinement potentials. The effect of the Stern-Gerlach can be pictured as follows. Consider a qubit in the state $|\!\uparrow_y\rangle$ entering the gate. The Q1DC broadens as the qubit enters the field, allowing the Stern-Gerlach effect to produce a spin-dependent shift in the center of mass towards the channel $C_\uparrow$. However, the initial localized distribution will deloclize due to the Q1DC broadening, allowing a small current to be detected in the wrong channel $C_\downarrow$. Thus this gate is a spin polarizing filter with some intrinsic error rate, independent of decoherence effects, shot noise and Johnson noise. It remains for us to judge how well it performs, and to this end we introduce a fidelity measure of the gate. Fidelity of the Stern-Gerlach gate with linearly inhomogeneous field -------------------------------------------------------------------- As an input to the Stern-Gerlach gate, we consider a density matrix of the form $$\rho(y,y') = g(y)\,g(y')^* \otimes \sigma,$$ where $g(y)$ is the Gaussian groundstate of the dot and $\sigma$ is the initial spin density matrix. In the inhomogeneous field the spatial degrees of freedom couple to the spin degrees of freedom, leading to $$\rho(y,y',t) = \sum_{s,s'=\uparrow,\downarrow} g_s(y,t)\,g_{s'}(y',t)^* \, \sigma_{s\,s'}|s\rangle\langle s'|.$$ After a gate time $t$, we post-select the state in, say, the $C_\uparrow$ channel and renormalize: $$\begin{aligned} \tilde{\rho}(y,y',t) = \frac{1}{N(t)}\, \sum_{s,s'} \sigma_{s\,s'} |s\rangle\langle s'| \times \nonumber\\ \int dz\,dz'\, E_\uparrow(y,z)\,g_s(z,t)\,g_{s'}(z',t)^*\,E_\uparrow(z',y'),\end{aligned}$$ The probability yield in the channel is the normalization $N(t)$ of the post-selected state. The simplest projection kernel projecting into the support of $C_\uparrow$ is $E_\uparrow(y,y') = \delta(y-y')\,u(y)$, where $u(y)$ is the step-function defined in (\[VQ1DC\]). We finally define our fidelity as $$\begin{aligned} F_\uparrow &=& \int dy\, \langle \uparrow | \tilde{\rho}(y,y,t) | \uparrow\rangle \\ &=& \frac{ \sigma_{ \uparrow \uparrow} }{N(t)}\int_0^\infty |g_\uparrow(y,t)|^2 dy.\end{aligned}$$ The yield $N(t)$ can be calculated without explicit knowledge of $g_\downarrow$, since the Gaussian solutions are related by reflection symmetry: $g_\uparrow(y,t)=g_\downarrow(-y,t)$. We choose various magnetic field gradients ranging from $0.1{\ \mathrm{T\,\mu m^{-1}}}$ to $100{\ \mathrm{T\,\mu m^{-1}}}$ and plot the output fidelity of a maximally-mixed input state as a function of the Q1DC angle $\theta$ and the gate time $t$ (see Fig. \[fidelitycontour\]). Perfect spin filtering corresponds to a fidelity $F=1$, and the worst-case fidelity is $F=1/2$. The density plots show how the optimal parameter regions increase with larger magnetic field gradients. This model is useful in determining the appropriate gate geometry and field gradients required to achieve good fidelities. ![ Fidelity $F_\uparrow$ in the $C_\uparrow$ channel for a Stern-Gerlach gate operated by a linearly inhomogeneous and unidirectional magnetic field in the $y$ direction. White regions $\Rightarrow$ $F_\uparrow=1$. $\theta$ is the angle between the Q1DC and SAW direction. $t$ is the gate time. Input state is a maximally-mixed spin state in the groundstate of the SAW quantum dot. Various magnetic field gradients have been chosen to show the increase in the optimal parameter region (white) in the $(t,\theta)$ plane with increasing field gradient. $t_\mathrm{max}\approx 100{\ \mathrm{ns}}$ is a crude order of magnitude estimate for the GaAs spin relaxation lifetime. \[fidelitycontour\] ](FidelityDensityPlot.eps){width="3.4in"} The technology of nanoscale single domain magnets is not yet capable of producing large fields in arbitrary configurations. The above simple linearly inhomogeneous magnetic field is therefore a simple approximation to the real magnetic field configuration on any particular device. In the next subsection we consider the effect of a different (perhaps more realistic) field configuration, namely, the field near one of the poles of a dipole magnet. Stern-Gerlach gate with dipole field {#DipoleField} ------------------------------------ In this subsection we study the quality of the readout obtained from a Stern-Gerlach type gate driven by a simple 2D dipole field lying in the 2DEG plane, as shown in Fig. \[VectorField\]. In Appendix \[DeriveDipoleField\], we derive the magnetic vector potential $$\label{potential1} A_z = \frac{\beta x}{x^2 +(y-d)^2},$$ where $\beta$ is a strength parameter and $d$ is the distance of the 2D dipole from the Q1DC. ![ Vector field plot in the $xy$ (2DEG) plane showing the magnetic field due to an infinite string of dipole moments running parallel to the $z$-axis at a distance $d$ from the Q1DC. \[VectorField\] ](DipoleSketch.eps){width="3.6in"} Figure \[VectorField\] shows that the gradient of the field in the $y$ direction reverses direction at two points, $x=\pm d$. In order to prevent the moving qubit from experiencing opposite field gradients, we will restrict the gate time to $T=2\,d$ so that the qubit moves in the region $ -T/2<x<T/2$ in the laboratory frame. The question of how we are to shield the extraneous field from the qubit does not have a simple solution. It has been suggested that a superconducting material such as Niobium could be used to shield the magnetic field where it is not needed.[@Barnes:PRB:2000] The results of a 1D simulation of the wavefunction along the $x=z=0$ direction are shown in Fig. \[DipoleFieldSimulation\]. The numerical method used was a Crank-Nicholson algorithm adapted to the Pauli Equation. The plot shows three time-shots of the probability density $|\psi(y)^2|$, normalized to peak. The initial state is a Gaussian groundstate with spin state $|\!\uparrow_y\rangle$. The simulation shows no significant spatial displacement of the electron density in the $y$ directions, in contrast to the behavior exhibited under the linearly inhomogeneous field (c.f. Eq. (\[MuEvolution\])). This is partially due to the effect of the Lorentz force generated by the $A^2$ term in the Hamiltonian, which contributes an effective potential that is highly confining in the $y$ direction. The $A^2$ confinement is so strong that the Q1DC opening angle $\theta$ becomes irrelevant. Bohr and Pauli’s claim that the Lorentz force washes out the Stern-Gerlach effect are upheld in this particular scenario. Another source of problems for this field configuration is that the $|\!\uparrow_y\rangle,\,|\!\downarrow_y\rangle$ states are not eigenstates of the field as they were with the linearly inhomogeneous field. Figure \[BlochVecOsc\] shows the evolution of the Bloch vector in a spin-only model of the electron spin qubit. It demonstrates how the Bloch vector begins precessing about the $x$ component of the magnetic field, causing the Stern-Gerlach force to act in different directions. Although careful timing of the gate and possible compensation mechanisms could be put in place, this gate is essentially not robust. We draw the conclusion that in parameter regimes relevant to single-electron transport by SAW in GaAs, a field of this type will not be suitable for a Stern-Gerlach measurement gate, and that field unidirectionality is generally an important requirement for quantum gates driven by static magnetic fields. ![ Three time-shots (see legend) showing the spatial electron probability density (normalized to peak). $y$ is the direction transverse to the Q1DC. This is a Stern-Gerlach type readout gate using the 2D dipole field ($1{\ \mathrm{T\mu m^{-1}}}$ at $x=y=0$) shown in Fig. \[VectorField\]. Initial condition is a Gaussian wavepacket with spin $|\!\uparrow_\mathrm{y}\rangle$. Sideways translation of the probability density is suppressed due to both the strongly confining effective potential arising from the $A^2$ term and the precessional motion of the Bloch vector in the non-unidirectional field. \[DipoleFieldSimulation\] ](DipoleProbDist1.eps){width="8.6cm"} ![ (Color online) Evolution of the Bloch vector in a 2D dipole field ($1{\ \mathrm{T\mu m^{-1}}}$ at $x=y=0$): $\langle\sigma_x\rangle$ (long-short dash), $\langle\sigma_y\rangle$ (solid), $\langle\sigma_z\rangle$ (short dash). Time $t=0$ corresponds to the starting point $x=-d$ (see Fig. \[VectorField\]). We have used a spin-only model of the electron spin qubit to illustrate spin precession in the non-unidirectional field. The Bloch vector changes direction rapidly, which in turn causes the Stern-Gerlach force to fluctuate during the operation of the gate. \[BlochVecOsc\] ](PointParticleModel.eps){width="8cm"} Decoherence effects \[Section:Neglect\] ======================================= The major effects of decoherence on this SAW quantum computing system have already been considered qualitatively in the original proposal and subsequent papers,[@Barnes:PRB:2000; @Robinson:PRB:2001] and more specifically by a number of authors. These effects include: The interaction of the electron qubit with other electrons in the 2DEG, surface gates and donor impurities;[@Robinson:PRB:2001; @Barrett:PRB:2003] the coupling of qubits to phonons,[@Golovach:2003; @Semenov:PRL:2004] nuclear spins[@Burkard:PRB:99] and radio-frequency photons[@Yamamoto:Book:1999]. The models we have presented here give rise to behavior which was previously neglected in the original proposal for the quantum computation scheme.[@Barnes:PRB:2000] They allow us to predict the contribution to decoherence from excitation into higher orbital states and give their probabilities of occupation. Any potential decoherence due to tunneling between neighboring dots is negligible, and if needed the spatial separation of qubits can be increased by either introducing higher harmonics of the SAW fundamental frequency or by active gating at the entrance to each Q1DC. A major source of error is the current fluctuations in the quantized single-electron SAW current. In recent experiments,[@Robinson:PRB:2002; @Robinson:Proc:2003] error in the current quantization (including shot and Johnson noise) of less than $0.1\%$ were observed. The relevant decoherence timescales that underpin our proposals is the spin lifetime ($T_1$ and $T_2$) of single electrons in quantum dots. For n-type bulk semiconductors, $T_1$ spin lifetimes in GaAs of $100{\ \mathrm{ns}}$ have been reported, which gives a very crude ball-park figure.[@Kikkawa:98] An estimate that is more relevant to the SAW electron system is the spin relaxation lifetime of a few nanoseconds, which was obtained from spin-resolved microphotoluminescence spectroscopy measurements on photoexcited electrons.[@Sogawa:PRL:2001] However, this is the worst-case scenario because the method of electron capture we propose[@Barnes:PRB:2000] produces a conduction-band hole that is more extended, short-lived and mixed than the photoexcited hole. Indeed, recent experiments on static GaAs quantum dots report a lower bound on $T_1$ as long as $50{\ \mathrm{\mu s}}$,[@Hanson:PRL:2003] which makes an estimate of $100{\ \mathrm{ns}}$ for the $T_1$ lifetime of SAW electrons quite reasonable in the light of current technology. Summary {#Section:Summary} ======= We have analyzed in detail a proposal[@Barnes:PRB:2000] for implementing quantum computation on electron spin qubits trapped in SAW-Q1DC electrostatically-defined dots, using only static magnetic gates to perform single-qubit initializations, rotations and readouts. Applying the full Pauli Hamiltonian, we described the quantum dynamics of both the spin and orbital states of the qubit for various parameter regimes relevant in SAW single-electron transport experiments. In the analysis of single-qubit unitary operations with localized uniform magnetic fields, we showed that the effect of the Lorentz force puts upper bounds on field strengths and gate times. Moreover, simulations showed that field directions normal to the 2DEG excite rotational states in the dot. Probabilities of excitation into higher-energy orbitals were given. In terms of feasibility, the models indicate that a field strength of about $80{\ \mathrm{mT}}$ will be sufficient to conduct a $\pi$ rotation in about $1{\ \mathrm{ns}}$, without compromising the confinement properties of the trapped qubit. Since $T_1$ spin lifetimes of microseconds have been reported,[@Hanson:PRL:2003] it is feasible for hundreds of single-qubits gates to operate before all coherence is lost in the computation. We also studied a device for single-qubit measurement and initialization based on the Stern-Gerlach effect. The problematic Lorentz force can be partly suppressed by virtue of the geometry of the 2DEG system, and for a unidirectional, linearly inhomogeneous field, the correlation between the spin states and spatial location of the qubit leads to a good quantum measurement of its spin. For a 2D dipole field, the vector potential has a deleterious effect. Namely, it contributes an effective confining potential via the $A^2$ term in the Pauli Hamiltonian which suppresses the transverse motion in the $y$ direction. Furthermore, the component of magnetic field in the $x$ direction causes the Bloch vector to rotate in undesired directions. The latter problem could be overcome by using unitary gates prior to the Stern-Gerlach gate to correct any spurious rotations, but it is unlikely that such a finely-tuned arrangement can be made robust. Magnetic fields that are good approximations to a unidirectional and linearly inhomogeneous field can provide a source of polarized electron qubit states in channels $C_\uparrow$ and $C_\downarrow$ with high yield. The very same gate can be used to measure the ratio $|\alpha|^2/|\beta|^2$ for an input qubit state $|\psi\rangle=\alpha|\!\uparrow_y\rangle+\beta|\!\downarrow_y\rangle$. The advantages of this readout method are that the averaging time of the measured current can be made long enough to render most noise sources (e.g. shot noise, input-referred noise of the current pre-amp) unimportant, and that missing electrons from SAW minima do not contribute any error since we are measuring the ratio of currents out of the two channels. Field gradients in the range $0.1-100{\ \mathrm{T\mu m^{-1}}}$ lead to gate times that are easily within the range of $T_1$ spin lifetimes in GaAs. These parameter regimes will need to be probed by experiment if we are to make progress towards feasible quantum computation with nanomagnets in the SAW quantum computer. Acknowledgements {#acknowledgements .unnumbered} ================ The authors would like to thank Sean Barrett, Masaya Kataoka, Alexander Moroz, Andy Robinson and Tom Stace for their ideas and helpful discussions. The authors gratefully acknowledge the support of the EPSRC. This work was partly funded by the Cambridge-MIT institute. Derivation of the 2D dipole field vector potential {#DeriveDipoleField} ================================================== We adopt cylindrical coordinates $(r,\phi,y)$ about the $y$ axis, with $r=\sqrt{x^2+z^2}$. We consider the field due to an infinite string of dipoles passing parallel to the $z$-axis through $y = d$ as shown in Fig. \[VectorField\]. The resulting magnetic field has planar symmetry with respect to the $xy$ (2DEG) plane. The Q1DC runs along the $x$ axis. A vector potential for this field can be constructed as follows. A magnetic dipole at $y=z=0$ pointing in the $+y$ direction has a vector potential[@Jackson:EM] $$A'_{\phi}(r,y) = \frac{\mu_0 m_B dz}{4\pi} \frac{r}{(r^2+y^2)^{3/2}},$$ where $A'_r= A'_y=0$ and $m_B$ is the dipole moment per unit length in the $z$ direction. We now displace this potential by $d$ in the $y$ direction and integrate over all $z$, obtaining $$A_z = \frac{\beta x}{x^2 +(y-d)^2},$$ with $A_x=A_y=0$ and strength parameter $\beta=\mu_0 m'_B/2\pi$. The magnetic field can be derived by applying the curl in cylindrical coordinates: $B_r = -r^{-1}\partial_y (r A_\phi)$ and $B_y = r^{-1}\partial_r (rA_\phi)$. Figure \[VectorField\] shows the vector field plot of the resulting magnetic field, which falls off as $r^{-2}$ for large $r$. 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(), . , , , , ****, (). , , , ****, (). , ****, (). , , , ****, (). , ****, (). , ** (, ). , , , , ****, (). , ****, (). , , , , , , in ** (), vol. . , , , , , , ****, (). , , , , , , , , ****, (). , , , , , , , ****, (). , , , , , ****, (). , , , , , , ****, (). , in ** (), vol. , p. . , ****, (). , , , , , , ****, (). , ****, (). , , , , , , , , ****, (). , , , , ****, (), . , ****, (), . , ****, (). , ****, (). , ****, (). , , , , ****, (). , , , ****, (). , ** (, ), chap. . , in ** (, ), pp. . , , , ****, (). , ****, (). , , , ****, (). , ****, (). , ****, (), . , ** (, ). , , , , , , ****, (). , ****, (). , , (), . , ****, (). , ** (, ), ISBN . , ****, (). , , , , , , ****, (). , , , , , , ****, (). , ** (, ), p. , ed. [^1]: In other words, we ignore the $z$ dependence of the wavefunction. This is partly justified if $e\,\beta\,\Delta x/\sqrt{\Delta E\,m^*} \ll 1$, where $\Delta E$ is the characteristic energy gap between the 2DEG energy bands and $\Delta x$ is the length of the gate. This is indeed the case for parameter regimes that we consider. [^2]: The spreading time for a Gaussian state is the time taken for its standard deviation to double.

--- abstract: 'We consider a quantum circuit model describing the evaporation process of black holes. We specifically examine the behavior of the multipartite entanglement represented by this model, and find that the entanglement structure depends on the black hole mass $M$ and the frequency of the Hawking radiation $\omega$. For sufficiently small values of $M\omega$, the black hole and the radiation system becomes a separable state after the Page time and a firewall-like structure appears. On the contrary, for larger values of $M\omega$, the entanglement between the black hole and the radiation is not lost. These behaviors imply that owing to the monogamy property of the multipartite entanglement, low frequency modes of the Hawking radiation destroys the quantum correlation between the black hole and the emitted Hawking radiation.' author: - Tomoro Tokusumi - Akira Matsumura - Yasusada Nambu date: 'October 1, 2018' title: Quantum Circuit Model of Black Hole Evaporation --- Introduction ============ By investigating the quantum field in black hole spacetimes, Hawking predicted that the black hole spontaneously creates entangled particle pairs in the vicinity of the black hole horizon [@Hawking_particle_creation]. One particle of the created pair is thermally radiated away to the spatially far region, and the other particle with negative energy falls into the black hole. As the result, the horizon shrinks and the black hole evaporates. In 1976, Hawking pointed out that the information paradox exists in the evaporation process of the black hole [@Hawking_info_loss]. Assuming that the initial state of the black hole is pure and keeping in mind that the thermal radiation is expressed as the mixed state such as the Gibbs state, it turns out that the black hole becomes a mixed state as the evaporation proceeds. However, in theories such as the quantum mechanics, the quantum field theory and the string theory, time evolution is unitary. Thus a pure state evolves with unitary operators and the state remains as a pure state after the evaporation. Consequently, evaporation process of the black hole seems to be inconsistent with the quantum theory. This inconsistency is also shown by considering the entanglement. The amount of the entanglement between systems A and B can be evaluated by the von Neumann entropy of the reduced density operator $S_\text{A} = -\Tr \rho_\text{A} \log \rho_\text{A}$. The von Neumann entropy $ S_\text{A} $ is zero if and only if the system A is in pure state. In other words, by measuring the amount of entanglement, we can investigate the presence or absence of information paradox because the existence of entanglement is connected to mixedness of the considering system. According to Hawking’s calculation, the amount of the entanglement between a particle falling into the black hole and a particle radiated away is zero at the initial stage of evaporation and monotonically increases as the evaporation proceeds. In other words, particles radiated away from the black hole are in pure states at the beginning, then become mixed after evaporation completes. This is why Hawking concluded the information paradox exists. For this paradox to be solved, the von Neumann entropy should begin to decrease in the middle stage of the evaporation and finally it must become zero. Assuming that the black hole and the Hawking radiation are in a random state and the dynamics of the black hole is chaotic, Page showed that the von Neumann entropy of the Hawking radiation increases from zero, and it has a maximum value at the middle stage of the evaporation and then decreases monotonously. This behavior of the von Neuman entropy is called the Page curve  [@Page_curve; @page_theorem] and the time attaining the maximum of the entropy is called the Page time. It is believed that the Page curve represents the general feature of the evaporation process of the black hole and the information of the black hole is carried away by the Hawking radiation. In studies of AdS/CFT correspondences [@AdSCFT], it is partially shown that the information paradox of the black hole evaporation does not exist if the assumption of the black hole complementarity [@BH_complementarity] is imposed: Postulate 1: : The process of formation and evaporation of a black hole, as viewed by a distant observer, can be described entirely within the context of standard quantum theory. In particular, there exists an unitary S-matrix which describes the evolution from infalling matter to outgoing Hawking-like radiation. Postulate 2: : Outside the stretched horizon of a massive black hole, physics can be described to good approximation by a set of semi-classical field equations. Postulate 3: : To a distant observer, a black hole appears to be a quantum system with discrete energy levels. The dimension of the states of a black hole with mass $M$ is the exponential of the Bekenstein-Hawking entropy $$\label{eq:BHent} S_\text{BH}(M)=\frac{\mathcal{A}}{4G},\quad G=m_\text{pl}^{-2},$$ where $\mathcal{A}$ is the surface area of the black hole. Hence, it is widely believed that there is no information paradox with these assumptions, and a consistent evaporation process occurs in the quantum theory. However, considering the black hole after the Page time, A. Almheiri, D. Marolf, J. Polchinski and J. Sully (AMPS) [@AMPS] argued that the postulates of the black hole complementarity are still inconsistent and the following postulate should be added: Postulate 4: : A freely falling observer experiences nothing out of the ordinary when crossing the horizon. To make these four postulates consistent, as the most conservative solution, an observer falling through the horizon burns out before it passes through. Therefore, AMPS proposed that the horizon is covered by a firewall. The “monogamy” relation which is the property of multipartite entanglement plays an important role to show this inconsistency. The monogamy relation states that one system can not entangle with any other system when it is already entangled strongly with another system. Let us divide a system of the evaporation process into three subsystems as follows: (early radiation) is a previously emitted Hawking particle, is a Hawking particle just emitted to exterior, is a partner of the Hawking radiation just created and falls into the black hole. After the Page time (when the Page curve is decreasing), is entangled strongly with due to Page’s theorem, and is also entangled with due to purity of the Hawking radiation. Owing to the postulate 4, there is no drama at the black hole horizon, so there must be very strong entanglement between and . If the monogamy relation holds, these appears inconsistency. In order to resolve this inconsistency, the firewall must be introduced to cut the entanglement between and . To justify the existence of the firewall, it is important to investigate the feature of the multipartite entanglement between the radiated Hawking particles and the particles falling to the black hole. In particular, it is necessary to show whether entanglement is really broken or not. However, it is not easy to evaluate multipartite entanglement in the context of the quantum field theory and there are several works investigating more general quantum information structure by modeling the evaporation process of black holes with qubit systems [@Mathur:correlation; @Braunstein_qubit1; @Braunstein_qubit2; @Hwang; @Albrecht_multi]. A burning paper model is well known as a physical model to explain the Page curve [@Mathur:correlation]. When a paper is burning, photons are emitted from the surface of the paper. The photon can be entangled with the spin of atoms composing the paper. The entanglement entropy between photons and the paper monotonically increases at the beginning. Then, atoms entangled with photon also become ash and are emitted to the outside and the entanglement entropy may start to decrease at a certain time. When all the paper burns out, ash is released and the entanglement entropy becomes zero. Due to the very complex interaction of the atom with the emitted photons during the burning process, the paper’s information is fully encoded into radiated photons and ashes. Although the burning paper model is adequate to explain the Page curve, it has no horizon structure. This is because atoms constituting the paper are discharged to the outside in the form of ashes. Since the horizon is an important structure of spacetimes to characterize the black hole , this difference is unacceptable to investigate the evaporation of the black hole using a toy models with qubits. Qubit models have been applied to various evaporation scenarios. J. Hwang *et al.* [@Hwang] investigated a qubit model from the viewpoint of the remnant scenarios which are based on a different postulate from AMPS’s one, and revealed restrictions of various scenarios of remnant picture. In the research by S. Luo *et al.* [@Albrecht_multi], properties of multipartite entanglement in the evaporation process were investigated. During the entire history of evaporation, they found entanglement between and differs from AMPS’s expectation and they pointed out the firewall may not be necessary to resolve the information loss problem . However, the time evolution of evaporation in their model is different from that of the Hawking radiation. In this paper, we propose a quantum circuit model with qubits representing the Hawking radiation with the horizon structure, and investigate behavior of the multipartite entanglement. We aim to discuss emergence of the firewall in our model. The structure of this paper is as follows. In Sec. II we introduce our quantum circuit model representing the black hole evaporation. In Sec. III, we present results of numerical simulation of our model. In Sec. IV, we analytically obtain the state after the Page time and investigate the mutual information and the negativity between the black hole and the Hawking radiation. Sec. V is devoted to summary and discussion. Quantum Circuit Model ===================== We introduce our quantum circuit model of the black hole evaporation. Setup ----- We describe the Hawking radiation using qubits in our model. A qubit has two distinct internal states $ \ket{0}$ and $ \ket{1} $. In order to formulate the black hole evaporation, we separate the Hilbert space of the total system to that of the black hole, the Hawking radiation and the Hawking radiation that had been radiated previously. Hence, prepare the Hilbert space of black hole (), just radiation (), early radiation () (Fig. \[suffix\]) $$\begin{aligned} \mathcal{H}_\text{tot} = \mathcal{H}_\textsf{BH}\otimes \mathcal{H}_\textsf{JR}\otimes\mathcal{H}_\textsf{ER}.\end{aligned}$$ Here, we regard the state $ \ket{0} $ of and as the “vacuum state” with no particles. ![Defition of , and in our model.[]{data-label="suffix"}](./figs/signature.pdf){width="0.8\hsize"} We treat particles in , that are paired with the emitted Hawking particles and fall into the black hole, constitute the black hole’s degrees of freedom. This is because particles falling into the black hole can be regarded as a part of the black hole owing to the black hole complementary postulate 4. Evaporation ----------- Particles come out from the black hole in the evaporation process. For example, a particle in will belong to at the next time step. This process can be realized using the SWAP operation, which exchanges inputted two qubits and is represented by the unitary quantum gate shown in Fig. \[fig:SWAP\]. ![The SWAP gate.[]{data-label="fig:SWAP"}](./figs/swap.pdf){width="0.2\hsize"} In the evaporation process, the black hole shrinks by emitting particles and finally disappears. Furthermore, the information of the black hole is transferred to radiated particles. Qubits in gradually change to $\ket{0}$ state during the evaporation process. When the evaporation finishes and the black hole disappears completely, all the qubits in become $\ket{0}$ state. On the other hand, at the initial stage of evaporation, most of the radiated particles are in $\ket{0}$ state (no particles). Thus, radiated particles state will becomes very complicated because information of the black hole is carried away by the radiated particles. We expect that evaporation can be modeled by swapping the information (qubit) of the black hole with a part of radiation’s degrees of freedom. In our model, the SWAP are performed twice between and , and between and . As a demonstration, we look at the case of the 5 qubit total system and show how the SWAP procedure transfer the information of to radiations. The total system is $$\begin{aligned} \ket{\psi}=\ket{\textsf{BH}}\otimes \ket{\textsf{JR}}\otimes \ket{\textsf{ER}},\end{aligned}$$ with $|\textsf{BH}|=2^2$ (2 qubit), $|\textsf{JR}|=2^1$ (1 qubit), $|\textsf{ER}|=2^2$ (2 qubit). The initial state of each subsystem is assumed to be $${\textcolor{black}{\ket{\textsf{BH}}=\frac{\ket{0}_{\textsf{BH}_1}\ket{1}_{\textsf{BH}_2}+\ket{1}_{\textsf{BH}_1}\ket{0}_{\textsf{BH}_2}}{\sqrt{2}}}},\quad \ket{\textsf{JR}}=\ket{0}_\textsf{JR},\quad \ket{\textsf{ER}}=\ket{0}_{\textsf{ER}_1}\ket{0}_{\textsf{ER}_2},$$ and the initial total state $\ket{\psi_0}$ is $$\begin{aligned} \ket{\psi_0}={\textcolor{black}{\frac{1}{\sqrt{2}}}}\Bigl[\ket{10}+\ket{01}\Bigr]_{\textsf{BH}_{12}}\ket{0}_\textsf{JR}\ket{00}_{\textsf{ER}_{12}}.\end{aligned}$$ Although the Page curve is the average of entanglement entropy, we focus here on a specific state and compute the non-averaged entanglement entropy. The quantum circuit we introduce is shown in Fig. \[fig:SWAP-circuit\]. ![The SWAP circuit for 5 qubit system.[]{data-label="fig:SWAP-circuit"}](./figs/SWAP-circuit.pdf){width="0.4\hsize"} Let us follow the time steps realized by this circuit. 1. Apply the SWAP between the first qubit of and , $$\begin{aligned} \ket{\psi_0}\longmapsto\ket{\psi_{0'}}={\textcolor{black}{\frac{1}{\sqrt{2}}}}\ket{0}_{\textsf{BH}_1}\Bigl[\ket{01}+\ket{10}\Bigr]_{\textsf{BH}_2,\textsf{JR}}\ket{00}_{\textsf{ER}_{12}}. \end{aligned}$$ 2. Apply the SWAP between and the first qubit of , $$\begin{aligned} \ket{\psi_{0'}}\longmapsto\ket{\psi_{1}}={\textcolor{black}{\frac{1}{\sqrt{2}}}}\ket{0}_{\textsf{BH}_1}\ket{0}_\textsf{JR}\Bigl[\ket{01}+\ket{10}\Bigr]_{\textsf{BH}_2,\textsf{ER}_1}\ket{0}_{\textsf{ER}_2}. \end{aligned}$$ <!-- --> 1. Apply the SWAP between the second qubit of and , $$\begin{aligned} \ket{\psi_1}\longmapsto\ket{\psi_{1'}}={\textcolor{black}{\frac{1}{\sqrt{2}}}}\ket{00}_{\textsf{BH}_{12}}\Bigl[\ket{01}+\ket{10}\Bigr]_{\textsf{JR},\textsf{ER}_1}\ket{0}_{\textsf{ER}_2}. \end{aligned}$$ 2. Apply the SWAP between and the second qubit of , $$\begin{aligned} \ket{\psi_{1'}}\longmapsto\ket{\psi_{2}}={\textcolor{black}{\frac{1}{\sqrt{2}}}}\ket{00}_{\textsf{BH}_{12}}\ket{0}_\textsf{JR}\Bigl[\ket{10} +\ket{01}\Bigr]_{\textsf{ER}_{12}}.\end{aligned}$$ To check this process reflects the property of evaporating models, we compute the von Neumann entropy of radiations $:=$$\cup$, $S_\textsf{R} = - \Tr \rho_\textsf{R} \log_2 \rho_\textsf{R}$ at each step of the SWAP (Fig. \[fig:swap0\]). ![The Page curve obtained by the SWAP circuit in Fig. \[fig:SWAP-circuit\].[]{data-label="fig:swap0"}](./figs/page.pdf){width="0.45\hsize"} The typical shape of the Page curve is obtained and the information of is transferred to the radiation by the SWAP operation. The effective size of ’s Hilbert space at step $n$ is defined by the amount of information $$|\textsf{BH}_n^{\text{eff}}| := |\textsf{BH}|*2^{-(n-1)}\quad (n\geq 1),\qquad |\textsf{BH}_0^{\text{eff}}| := |\textsf{BH}|$$ where $|\textsf{BH}|$ is the dimension of ’s Hilbert space determined by the number of qubits. Particles are radiated from by the first SWAP at each step. Thus, this quantum circuit does not have the black hole horizon and the entanglement structure of this circuit is determined only by the initial state. Qubits in are entangled with each other at the initial step. Then, a part of entangled qubit is moved to become and the entanglement between and increase. However, the entanglement of the Hawking radiation is originated from the entangled particle pairs created in the vicinity of the black hole horizon and this SWAP circuit is the model of evaporation process without the horizon. Hawking radiation ----------------- The quantum state of the Hawking radiation is expressed as $\prod_{i}\sum_{n_i=0}^\infty e^{-n_i\beta \omega_i/2} \ket{n_i}_\textsf{BH}\otimes\ket{n_i}_\textsf{JR}$ where $\beta=8\pi M$ is the inverse temperature of the black hole and $ \omega $ is the frequency of the radiation [@Hawking_particle_creation]. The state $\ket{n_i}_\textsf{BH}\otimes\ket{n_i}_\textsf{JR}$ represents pairs of the Hawking particles. For $M\omega\gg 0.1 $, the state is almost same as the vacuum state without particle excitation and for $ M\omega < 0.1$, $\ket{1}_\textsf{BH}\otimes\ket{1}_\textsf{JR}$ particle state mainly contributes to the total state of the Hawking radiation. Therefore, the essence of the entanglement structure can be revealed even if two or more particle number states are ignored. Thus, we assume the following entangled state represents the Hawking radiation $$\begin{aligned} \frac{1}{\sqrt{1+\exp(-8\pi M\omega)}}\biggl[\ket{0}_{\textsf{BH}}\ket{0}_\textsf{JR} +\exp(-4\pi M\omega)\ket{1}_{\textsf{BH}}\ket{1}_\textsf{JR}\biggr]. \label{eq:haw} \end{aligned}$$ From now on, we focus on a specific $\omega$ mode of the Hawking radiation. We consider a quantum gate mimicking this state (Fig. \[fig:squeezing-gate\]). The gate consists of two well-known quantum gates and outputs the entangled state. ![The CNOT-U gate.[]{data-label="fig:squeezing-gate"}](./figs/CNOT-U.pdf){width="0.7\hsize"} The gate of the right part in the circuit is an unitary gate acting on two qubit called the CNOT gate and works as $\ket{x} \ket{y} \to \ket{x} \ket{x\oplus y}$. The gate U in the left part is an unitary gate acting on a single qubit. In terms of the matrix representation, it is expressed as $$\begin{aligned} U := \begin{bmatrix} \cos{\gamma} & \sin{\gamma}\\ \sin{\gamma} & - \cos{\gamma} \label{rotation} \end{bmatrix},\quad \tan\gamma=\exp(-4\pi M\omega),\end{aligned}$$ where $\gamma$ is the squeezing parameter. For $\gamma = \pi/4$, U corresponds to the Hadamard gate H, and the CNOT-U gate will output the maximally entangled EPR state. The gate in Fig. \[fig:squeezing-gate\] can output an entangled state and the amount of entanglement depends on the parameter $ \gamma $. The strength of the entanglement between the created Hawking particle pairs is determined by the mass of the black hole and we must determine the relation between the parameter $ \gamma $ and the black hole mass. For this purpose, we consider the correspondence between the dimension of the Hilbert space ( qubit number) and the black hole mass $ M $. From the formula of the Bekenstein-Hawking entropy $$S_\text{BH}(M)=4\pi \left(\frac{M}{m_\text{pl}}\right)^2 = \log_2 |\textsf{BH}^\text{eff}|,$$ we have $$\begin{aligned} M_n = \frac{m_\text{pl}}{2}\sqrt{\frac{\log_2{|\textsf{BH}_n^\text{eff}|}}{\pi}}. \end{aligned}$$ Note that $ | \textsf{BH}_n^\text{eff} | $ is just the effective size of Hilbert space and not the actual number of the states. Using these relations, in the natural unit, we obtain $$\begin{aligned} \gamma_n = \mathrm{Arctan}\left[\exp\left(-2\omega \sqrt{\pi \log_2 |\textsf{BH}_n^\text{eff}|}\right)\right]. \label{gamma}\end{aligned}$$ With this squeezing parameter $ \gamma_n $, the CNOT-U gate in Fig. \[fig:squeezing-gate\] mimics the the Hawking radiation (\[eq:haw\]). We combine the SWAP gate and the CNOT-U gate to construct a model of the evaporation process of the Hawking radiation. We have already confirmed that the initial state of is transferred to and by the SWAP circuit and the Page curve is obtained. In the following analysis, we focus on the property of the entanglement caused by the Hawking radiation to clarify how the dynamics of the evaporation is connected with the quantum information process and the firewall argument. Hence, the initial state is assumed to be non-entangled state $$\ket{\text{init}} = \ket{000 \cdots 000}_\textsf{BH} \ket{0}_\textsf{JR} \ket{000 \cdots 000}_\textsf{ER}.$$ The quantum circuit of our model is presented in Fig. \[fig,haw-circuit\]. ![A quantum circuit of the black hole evaporation. The CNOT-U gate acts on and to generate a Hawking particle pair. The SWAP gate acts on two places between and , and to move a qubit of to and move a qubit in to . We note that the SWAP step is just after creation of the Hawking pair, so particles do not directly escape from to .[]{data-label="fig,haw-circuit"}](./figs/haw-circuit.pdf){width="\hsize"} In this model, the initial mass of the black hole corresponds to qubit number of . The qubit number can not make so large due to increase of the simulation time. We denote the black hole mass at a certain step $n$ as $M_{n} \omega$. Because $M_n\propto \sqrt{\log_2 |\textsf{BH}_n|}$, the black hole mass at arbitrary step $n$ is $$\begin{aligned} M_n = M_{\text{init}}\sqrt{\frac{\log_2|\textsf{BH}_n|}{\log_2|\textsf{BH}_{\text{init}}|}} = M_{\text{init}} \sqrt{1-\frac{n-1}{N_\textsf{BH}}}, \end{aligned}$$ where $M_\text{init}$ is the initial mass of the black hole and $N_\textsf{BH}$ is the qubit number of used for analysis. The only parameters contained in this model are the value of $ M_{n} \omega $ and $n$. In our analysis, we choose $n$ as the step at the Page time. Numerical Result ================ In this section, we will analyze evolution of the state in our circuit model (Fig. \[fig,haw-circuit\]) focusing on the entanglement structure. In our analysis, the size of is 6 qubit ($|\textsf{BH}|=2^6, N_\textsf{BH}=6$), is 1 qubit ($|\textsf{JR}|=2^1$) and is 6 qubit ($|\textsf{ER}|=2^6$). Page curve ---------- As in Sec \[evaporation\], we check the amount of information transferred from the to the emitted particles and investigate whether the black hole in our model evaporates. For this purpose, we evaluate the entanglement entropy between and =$\cup$. The evolution of this quantity provides the Page curve, which is an indicator of the information transfer from to radiation. The result is shown for $ M_4\, \omega = 0.1$ and $0.01$ in Fig. \[fig:page\_result\]. Here, $ M_4 $ denotes the mass of the black hole at step 4. For $ M_4 \omega=0.01 $ (small $M_4 \omega$), the Page curve has a symmetric shape with respect to the middle point of the whole time steps. On the other hand, for $ M_4 \omega=0.1$ (large $M_4\omega$), the entanglement entropy between and is smaller compared to the small $M_4\omega$ case during whole period of the evaporation process. For large $M_4\omega$, the black hole radiates Hawking particles with weak entanglement (see Eq. ). Thus the evaporation process is not random enough as Page assumed. However, our model results in Page curves for both values of $M_4\omega$ because the degree of freedom of decreases as the time step proceeds due to the action of the SWAP gate. The entanglement entropy becomes maximum at step 4 and this time step is the Page time of our model. Expectation value of particle number of JR {#number_ev} ------------------------------------------ For the state $\ket{\psi} = \sum_{ijk} C_{ijk} \ket{i}_\textsf{BH}\ket{j}_\textsf{JR}\ket{k}_\textsf{ER}$, the expectation value of the particle number of is expressed as $$\begin{aligned} \ev{n_\textsf{JR}} = \sum_{ik} |C_{i1k}|^2. \label{eq:expect-b}\end{aligned}$$ We checked these formulas for $ M_4 \omega =1,~0.1,~0.01,~0.001$ (Fig. \[fig:JR\_num\]). As the black hole evaporates, the mass decreases and the time step advances from right to left. In our model, $\expval{n_\textsf{JR}}$ well coincides with the formula except at the last step with $M_4\omega\geq 0.1$. For $M_4\omega\geq 0.1$, $\expval{n_\textsf{JR}}<0.3$ and this means the emitted pair of Hawking particles is weakly entangled. On the other hand, for $M_4\omega\leq0.01$, $\expval{n_\textsf{JR}}\approx 0.5$ and this means the Hawking pair is nearly maximally entangled. At the last step, the both formula coincides for $ M_4 \omega\leq 0.01 $ but differs much for $ M_4 \omega\geq 0.1 $. This disagreement is caused by the different treatment of the Hawking radiation at the last step in our circuit model. In the evaporation process, $M\rightarrow 0$ is realized at the last step of evaporation. On the other hand, in our model, there is no CNOT-U gate at the last step (see Fig. \[fig,haw-circuit\]), so sufficient radiation will not be created at the last step. This is the reason why large disagreement appeared at the last step for $M_4\omega\geq 0.1$. However, if the number of qubits is increased and make the interval of the time step sufficiently small, this difference is expected to be reduced. Entanglement structure {#entstructre} ---------------------- Let us analyze the entanglement measure in our model. As we have already introduced the entanglement entropy as the measure of entanglement in Sec. \[evaporation\], we introduce here other two measures needed for our analysis. The mutual information of the bipartite system is defined as $$\begin{aligned} I(\text{A}\!:\!\text{B})= S_\text{A} + S_\text{B} - S_{\text{A}\cup\text{B}}. \label{mutual-info}\end{aligned}$$ This quantity evaluates how the system is close to the product state. If the mutual information is zero, the state is the product state without classical and quantum correlations. The negativity [@negativity_Vidal; @negativity_He] is defined by $$\begin{aligned} \mathcal{N}(\text{A}\!:\!\text{B})= \frac{1}{2}\left(\sum_i|\lambda_i^T|-1\right),\quad\sum_i\lambda_i^T=1,\label{def:negativity} \end{aligned}$$ where $\lambda_i^T$ is the eigenvalue of the partially transposed state $\rho^{T_\text{A}}_\text{AB}$. The partial transposition $T_\text{A}$ for the state $\rho = \sum C_{ab:a'b'}\ket{a}\bra{a'}\otimes \ket{b}\bra{b'}$ is defined by $\rho^{T_\text{A}}:= \sum C_{ab:a'b'}\ket{a'}\bra{a}\otimes \ket{b}\bra{b'}$. We calculate these entanglement measures between and , and , and for $ M_4\, \omega = 0.1,~0.01 $. The result is shown in Fig. \[fig:result\]. ![Evolution of the mutual information and the negativity. Upper panels: Evolution of the mutual information $I(\textsf{BH}\!:\!\textsf{JR})$ and the negativity $\mathcal{N}(\textsf{BH}\!:\!\textsf{JR})$. For $M_4\omega=0.01$, after step 5, the negativity becomes zero and the mutual information has very small but non-zero values. This implies the firewall-like structure appears after the Page time. Middle panels: Information of emitted radiation, which show how much information is extracted from at each step. Bottom panels: Entanglement of $\textsf{BH}\cup\textsf{ER}$ subsystem shows nearly the same behavior as the Page curve. []{data-label="fig:result"}](./figs/result-a.pdf "fig:"){width="0.45\linewidth"} ![Evolution of the mutual information and the negativity. Upper panels: Evolution of the mutual information $I(\textsf{BH}\!:\!\textsf{JR})$ and the negativity $\mathcal{N}(\textsf{BH}\!:\!\textsf{JR})$. For $M_4\omega=0.01$, after step 5, the negativity becomes zero and the mutual information has very small but non-zero values. This implies the firewall-like structure appears after the Page time. Middle panels: Information of emitted radiation, which show how much information is extracted from at each step. Bottom panels: Entanglement of $\textsf{BH}\cup\textsf{ER}$ subsystem shows nearly the same behavior as the Page curve. []{data-label="fig:result"}](./figs/result-b.pdf "fig:"){width="0.45\linewidth"}\ ![Evolution of the mutual information and the negativity. Upper panels: Evolution of the mutual information $I(\textsf{BH}\!:\!\textsf{JR})$ and the negativity $\mathcal{N}(\textsf{BH}\!:\!\textsf{JR})$. For $M_4\omega=0.01$, after step 5, the negativity becomes zero and the mutual information has very small but non-zero values. This implies the firewall-like structure appears after the Page time. Middle panels: Information of emitted radiation, which show how much information is extracted from at each step. Bottom panels: Entanglement of $\textsf{BH}\cup\textsf{ER}$ subsystem shows nearly the same behavior as the Page curve. []{data-label="fig:result"}](./figs/result-c.pdf "fig:"){width="0.45\linewidth"} ![Evolution of the mutual information and the negativity. Upper panels: Evolution of the mutual information $I(\textsf{BH}\!:\!\textsf{JR})$ and the negativity $\mathcal{N}(\textsf{BH}\!:\!\textsf{JR})$. For $M_4\omega=0.01$, after step 5, the negativity becomes zero and the mutual information has very small but non-zero values. This implies the firewall-like structure appears after the Page time. Middle panels: Information of emitted radiation, which show how much information is extracted from at each step. Bottom panels: Entanglement of $\textsf{BH}\cup\textsf{ER}$ subsystem shows nearly the same behavior as the Page curve. []{data-label="fig:result"}](./figs/result-d.pdf "fig:"){width="0.45\linewidth"}\ ![Evolution of the mutual information and the negativity. Upper panels: Evolution of the mutual information $I(\textsf{BH}\!:\!\textsf{JR})$ and the negativity $\mathcal{N}(\textsf{BH}\!:\!\textsf{JR})$. For $M_4\omega=0.01$, after step 5, the negativity becomes zero and the mutual information has very small but non-zero values. This implies the firewall-like structure appears after the Page time. Middle panels: Information of emitted radiation, which show how much information is extracted from at each step. Bottom panels: Entanglement of $\textsf{BH}\cup\textsf{ER}$ subsystem shows nearly the same behavior as the Page curve. []{data-label="fig:result"}](./figs/result-e.pdf "fig:"){width="0.45\linewidth"} ![Evolution of the mutual information and the negativity. Upper panels: Evolution of the mutual information $I(\textsf{BH}\!:\!\textsf{JR})$ and the negativity $\mathcal{N}(\textsf{BH}\!:\!\textsf{JR})$. For $M_4\omega=0.01$, after step 5, the negativity becomes zero and the mutual information has very small but non-zero values. This implies the firewall-like structure appears after the Page time. Middle panels: Information of emitted radiation, which show how much information is extracted from at each step. Bottom panels: Entanglement of $\textsf{BH}\cup\textsf{ER}$ subsystem shows nearly the same behavior as the Page curve. []{data-label="fig:result"}](./figs/result-f.pdf "fig:"){width="0.45\linewidth"} For $ M_4 \omega=0.01 $, the negativity between and becomes zero at step 5 after the Page time and and are separable. In other words, for the Hawking radiation with sufficiently low frequencies, a structure similar to the firewall appears between and . On the other hand, for $ M_4 \omega=0.1 $, the values of mutual information and negativity between and remain nonzero, and and are entangled until the end of the evaporation. For sufficiently small $ M_4 \omega $, nearly maximally entangled pair of the Hawking particles is created by the CNOT-U gate and the qubit strongly entangled with other qubit goes through the CNOT-U gate again after the Page time. However, due to the limited amount of entanglement between qubits, which is known as the monogamy property of the entanglement, it is impossible to add more entanglement and no entangled pairs can be created when passing through the second CNOT-U gate after the Page time. If this reasoning for small $ M_4 \omega $ is correct, the emergence of the firewall-like structure can be explained. Analytic Evaluation of State after Page time {#after-pagetime} ============================================ From the results obtained by the numerical calculation, it turned out that the firewall-like structure appears after the Page time if $ M_4 \omega $ is sufficiently small. We intend to explain why our model shows the firewall-like behavior. Structure of State around Page Time ----------------------------------- According to our numerical calculation, the firewall-like behavior becomes remarkable as $M_4\omega$ becomes small. To obtain comprehensive understanding of this structure, we analytically evaluate the state around the Page time. We introduce the matrix representation of the U gate and the CNOT gate, $$U_n= \begin{bmatrix} \cos\gamma_n & \sin\gamma_n \\ \sin\gamma_n & -\cos\gamma_n \end{bmatrix},\quad \mathrm{CNOT}= \begin{bmatrix} \mathbb{I}_2 & 0 \\ 0 & X \end{bmatrix},$$ where $\mathbb{I}_2$ is the $2\times 2$ identity matrix, $X=\begin{bmatrix} 0 & 1 \\ 1 & 0\end{bmatrix}$ and $$\gamma_n=\mathrm{Arctan}\left[\exp\left(-4\pi M_4\omega\sqrt{\frac{7-n}{3}}\right)\right],\quad 1\le n\le 7.$$ Then the matrix representation of the CNOT-U gate is $$\begin{aligned} \text{CNOT-U}=\begin{bmatrix} \mathbb{I}_2 & 0 \\ 0 & X\end{bmatrix}(\mathrm{U}_n\otimes\mathbb{I}_2) =\begin{bmatrix} \cos\gamma_n & 0 & \sin\gamma_n & 0 \\ 0 & \cos\gamma_n & 0 & \sin\gamma_n \\ 0 & \sin\gamma_n & 0 & -\cos\gamma_n \\ \sin\gamma_n & 0 & -\cos\gamma_n & 0 \end{bmatrix}.\end{aligned}$$ We prepare $\ket{\text{init}} = \ket{000000}_\textsf{BH}\ket{0}_\textsf{JR}\ket{000000}_\textsf{ER}$ as the initial state and consider the action of the quantum circuit on this state. **Before the Page time:** The Page time corresponds to step 4 and the circuit up to just before step 4 is equivalent to the following circuit diagram (Fig. \[fig,before4\]). We rearranged order of qubit and changed the number of labels from those defined in the original circuit in Fig. \[fig,haw-circuit\]. The state of the total system is $$\begin{aligned} \ket{\psi^{\text{before step 4}}} = \ket{\text{squ}}_{\textsf{BH}_1,\textsf{ER}_1}\ket{\text{squ}}_{\textsf{BH}_2,\textsf{ER}_2} \ket{\text{squ}}_{\textsf{BH}_3,\textsf{ER}_3} \ket{0}_{\textsf{BH}_{456},\textsf{JR},\textsf{ER}_{456}}.\end{aligned}$$ **At step 4:** The circuit just before the Page time to step 4 (the Page time) can be drawn as Fig. \[fig:just4\]. Only $\textsf{BH}_1$, $\textsf{JR}$ and $\textsf{ER}_1$ get involved. The state at step 4 is obtained as $$\begin{aligned} \ket{\psi^4}_{\textsf{BH}_1,\textsf{JR},\textsf{ER}_1} &= \text{CNOT}_{\textsf{BH}_1,\textsf{JR}}\, U_{\textsf{BH}_1}\ket{\text{squ}}_{\textsf{BH}_1,\textsf{ER}_1}\ket{0}_{\textsf{JR}}\notag\\ &= \text{CNOT}_{\textsf{BH}_1,\textsf{JR}}\biggl(\cos\gamma_1\Bigl[\cos\gamma_4\ket{0}+\sin\gamma_4\ket{1}\Bigr]_{\textsf{BH}_1}\ket{0}_{\textsf{ER}_1} \notag \\ &\qquad\qquad\qquad\qquad +\sin\gamma_1\Bigl[\sin\gamma_4\ket{0}+\cos\gamma_4\ket{1}\Bigr]_{\textsf{BH}_1}\ket{1}_{\textsf{ER}_1} \biggr)\ket{0}_\textsf{JR} \notag \\ &= \cos\gamma_1\Bigl[\cos\gamma_4\ket{000}+\sin\gamma_4\ket{101}\Bigr]_{\textsf{BH}_1, \textsf{JR}, \textsf{ER}_1} \notag \\ &\qquad +\sin\gamma_1\Bigl[\sin\gamma_4\ket{010}-\cos\gamma_4\ket{111}\Bigr]_{\textsf{BH}_1, \textsf{JR}, \textsf{ER}_1}. \label{step4}\end{aligned}$$ In the low frequency limit $M_4\omega\rightarrow 0$, squeezing parameters are $\gamma_1,\gamma_4\rightarrow \pi/4$ and $$\ket{\psi^4}_{\textsf{BH}_1, \textsf{JR}, \textsf{ER}_1}\longrightarrow H_\textsf{JR}\frac{1}{\sqrt{2}}\Bigl[\ket{000}+\ket{111}\Bigr]_{\textsf{BH}_1, \textsf{JR}, \textsf{ER}_1},$$ where $H_\textsf{JR}:=\begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}$ acts on the qubit of . The appeared state $\qty(\ket{000}+\ket{111})/\sqrt{2}$ is the GHZ state, which is the maximally entangled triqubit state. If one qubit in the GHZ state is traced out, the remaining subsystem with two qubits becomes separable. Since an unitary operator acting only on a single qubit does not affect the entanglement structure, and become separable at step 4 in the low frequency limit if is traced out. As we will see, this leads to the firewall-like structure at step 5. **At before step 5:** The circuit diagram is Fig. \[fig:before5\]. Qubits pass through the SWAP gate from step 4 to step 5. At step 4, one qubit in and two qubits in and are entangled. At just before step 5, one qubit in is $\ket{0}$ state, and one qubit in and two qubit in are entangled. **At step 5:** The quantum circuit up to step 5 is shown in Fig. \[fig:just5\]. The state at step 5 is obtained as $$\begin{aligned} \ket{\psi^5}_{\textsf{BH}_{12},\textsf{JR},\textsf{ER}_{124}} = \ket{0}_{\textsf{BH}_1}\ket{\psi^5}_{\textsf{BH}_2,\textsf{JR},\textsf{ER}_{124}},\end{aligned}$$ where $$\begin{aligned} \ket{\psi^5}_{\textsf{BH}_2,\textsf{JR},\textsf{ER}_{124}} &=\mathrm{CNOT}_{\textsf{BH}_2, \textsf{JR}}\,U_{\textsf{BH}_2}\ket{\text{squ}}_{\textsf{BH}_2, \textsf{ER}_2}\ket{\psi^4}_{\textsf{JR}, \textsf{ER}_{14}} \notag \\ &=\mathrm{CNOT}_{\textsf{BH}_2, \textsf{JR}}\biggl[\cos\gamma_2\left(\cos\gamma_5\ket{00}+\sin\gamma_5\ket{10}\right) \notag \\ &\quad\qquad\qquad\qquad +\sin\gamma_2\left(\sin\gamma_5\ket{01}-\cos\gamma_5\ket{11}\right)\biggr]_{\textsf{BH}_2, \textsf{ER}_2}\ket{\psi^4}_{\textsf{JR}, \textsf{ER}_{14}} \notag \\ &=\cos\gamma_1\cos\gamma_2\cos\gamma_4\cos\gamma_5\biggl( \ket{0}_{\textsf{BH}_2}\Bigl[\ket{0}+\tan\gamma_2\tan\gamma_5\ket{1}\Bigr]_{\textsf{ER}_2} \notag \\ &\qquad\times\Bigl[\ket{000}+\tan\gamma_4\ket{101}+\tan\gamma_1\tan\gamma_4\ket{010}-\tan\gamma_1\ket{111}\Bigr]_{\textsf{JR}, \textsf{ER}_{14}} \notag \\ &\qquad +\ket{1}_{\textsf{BH}_2}\Bigl[\tan\gamma_5\ket{0}-\tan\gamma_5\ket{1}\Bigr]_{\textsf{ER}_2} \notag \\ &\qquad\times \Bigl[\ket{100}+\tan\gamma_4\ket{001}+\tan\gamma_1\gamma_4\ket{110}-\tan\gamma_1\ket{011}\Bigr]_{\textsf{JR}, \textsf{ER}_{14}} \biggr).\end{aligned}$$ Negativity and Mutual Information at step 4 and 5 ------------------------------------------------- As we have obtain the state at step 4 and step 5, it is possible to investigate the entanglement structure by evaluating the mutual information and the negativity between and . ### At step 4 The reduced density matrix $\rho^4_{\textsf{BH}_1\cup\textsf{JR}}$ at step 4 is $$\begin{aligned} \rho^4_{\textsf{BH}_1\cup\textsf{JR}}&=\Tr_{\textsf{ER}}\left[\ket{\psi^4}\bra{\psi^4}_{\textsf{BH}_1, \textsf{JR}, \textsf{ER}_1}\right] \notag \\ &=\left(\cos^2\gamma_1\cos^2\gamma_4+\sin^2\gamma_1\sin^2\gamma_4\right) \ket{00}\bra{00}+\frac{1}{2}\cos 2\gamma_1\sin 2\gamma_4\ket{00}\bra{11} \notag \\ &\quad +\frac{1}{2}\cos 2\gamma_1\sin 2\gamma_4\ket{11}\bra{00}+\left( \cos^2\gamma_1\sin^2\gamma_4+\sin^2\gamma_1\cos^2\gamma_4\right)\ket{11}\bra{11} \notag \\ &= \frac{1}{2}\begin{bmatrix} 1+\cos2\gamma_1\cos2\gamma_4 & 0 & 0 & \cos 2\gamma_1\sin 2\gamma_4 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \cos 2\gamma_1\sin 2\gamma_4 & 0 & 0 & 1-\cos 2\gamma_1\cos 2\gamma_4 \end{bmatrix}.\end{aligned}$$ To obtain the mutual information $I(\textsf{BH}_1\!:\!\textsf{JR})$, we prepare the reduce states $$\begin{aligned} \rho^4_{\textsf{BH}_1}&=\Tr_{\textsf{JR}}\left[\rho^4_{\textsf{BH}_1\cup\textsf{JR}}\right]=\frac{1}{2}\begin{bmatrix}1+\cos 2\gamma_1 \cos 2\gamma_4 & 0 \\ 0 & 1-\cos 2\gamma_1 \cos 2\gamma_4 \end{bmatrix},\notag \\ \rho^4_{\textsf{JR}}&=\Tr_{\textsf{BH}_1}\left[\rho^4_{\textsf{BH}_1\cup\textsf{JR}}\right]=\frac{1}{2}\begin{bmatrix}1+\cos 2\gamma_1 \cos 2\gamma_4 & 0 \\ 0 & 1-\cos 2\gamma_1 \cos 2\gamma_4 \end{bmatrix}.\end{aligned}$$ Eigenvalues of these states are $$\begin{aligned} &\lambda_{\textsf{BH}_1\cup\textsf{JR}}=\frac{1}{2}\left(1\pm\cos 2\gamma_1\right),\quad \lambda_{\textsf{BH}_1}=\lambda_{\textsf{JR}}=\frac{1}{2}(1\pm\cos 2\gamma_1\cos 2\gamma_4),\end{aligned}$$ and the mutual information is $$I(\textsf{BH}_1\!:\!\textsf{JR})= -\sum_{i=\textsf{BH}_1, \textsf{JR}}\lambda_i\log_2\lambda_i +\sum_{i=\textsf{BH}_1\cup\textsf{JR}}\lambda_i\log_2\lambda_i.$$ The red line in the left panel of Fig. \[fig:step5neg\] shows the mutual information as a function of $M_4\omega$. It behaves as $$I\sim \begin{cases} 1+16\pi^2(M_4\omega)^2 &\quad \text{for} \quad M_4\omega\ll 0.1\\ 16\sqrt{2}\pi (M_4\omega)e^{-8\pi M_4\omega} &\quad \text{for} \quad M_4\omega\gg 0.1 \end{cases}$$ The negativity is obtained using eigenvalues of the partial transposed state $\rho^{4T_\textsf{JR}}_{\textsf{BH}\cup\textsf{JR}}$ $$\frac{1}{2}\left(1\pm\cos 2\gamma_1\cos 2\gamma_4\right),\quad\pm\frac{1}{2}\cos 2\gamma_1\sin 2\gamma_4,$$ and $$\mathcal{N}(\textsf{BH}_1\!:\!\textsf{JR})=\frac{1}{2}\cos 2\gamma_1\sin 2\gamma_4.$$ The red line in the right panel of Fig. \[fig:step5neg\] shows the negativity as a function of $M_4\omega$. The negativity at step 4 has a peak at $M_4\omega\approx 0.1$ and behaves as $$\mathcal{N}\sim \begin{cases} 2\sqrt{2}\pi M_4\omega &\quad \text{for} \quad M_4\omega\ll 0.1\\ e^{-4\pi M_4\omega} &\quad \text{for} \quad M_4\omega\gg 0.1 \end{cases}$$ As we have mentioned, $\ket{\psi^4}_{\textsf{BH}_1, \textsf{JR}, \textsf{ER}_1}$ becomes the GHZ state for $M_4\omega\rightarrow 0$. Corresponding to this, the state $\rho^4_{\textsf{BH}_1\cup\textsf{JR}}$ becomes separable in this limit. The von Neuman entropies for this state are $S(\textsf{BH}_1)=S(\textsf{JR})=S(\textsf{BH}_1\!\cup\!\textsf{JR})=1$ (bit). Thus the mutual information has the value 1 (bit), which means $\textsf{BH}_1$ and have the perfect classical correlation. ### At step 5 The reduced density matrix $\rho^5_{\textsf{BH}_2\cup\textsf{JR}}$ is $$\rho^5_{\textsf{BH}_2\cup \textsf{JR}}=\Tr_\textsf{ER}\Bigl[\ket{\psi^5}\bra{\psi^5}_{\textsf{BH}_2, \textsf{JR}, \textsf{ER}_{14}}\Bigr]= \begin{bmatrix} CA & 0 & 0 & EA \\ 0 & CB & EB & 0 \\ 0 & EB & DB & 0 \\ EA & 0 & 0 & DA \end{bmatrix},$$ where $$\begin{aligned} &A=\frac{1}{2}(1+\cos 2\gamma_1\cos 2\gamma_4),\quad B=\frac{1}{2}(1-\cos 2\gamma_1\cos 2\gamma_4),\quad C=\frac{1}{2}(1+\cos 2\gamma_2\cos 2\gamma_5), \\ &D=\frac{1}{2}(1-\cos 2\gamma_2\cos 2\gamma_5),\quad E=\frac{1}{2}\cos 2\gamma_2\sin 2\gamma_5.\end{aligned}$$ To obtain the mutual information between $\textsf{BH}_2$ and , we calculate eigenvalues of the states $ \rho_\textsf{JR}, \rho_{\textsf{BH}_2}$ and $\rho_{\textsf{BH}_2\cup\textsf{JR}} $. Eigenvalues $\lambda_i$ ($i=$ , $\textsf{BH}_2$, $\textsf{BH}_2\!\cup\!\textsf{JR}$) are $$\begin{aligned} &\lambda_\textsf{JR}=\frac{1}{2}(1\pm\cos 2\gamma_1\cos 2\gamma_2\cos 2\gamma_4\cos 2\gamma_5),\quad \lambda_\textsf{BH}=\frac{1}{2}(1\pm\cos 2\gamma_2\cos 2\gamma_5),\notag \\ &\lambda_{\textsf{BH}_2\cup\textsf{JR}}=\frac{1}{2}(1\pm\cos 2\gamma_1\cos 2\gamma_4)\cos^2 \gamma_2,~ \frac{1}{2}(1\pm\cos 2\gamma_1\cos 2\gamma_4)\sin^2 \gamma_2.\end{aligned}$$ The mutual information between $\textsf{BH}_2$ and as a function of $M_4\omega$ is shown in Fig. \[fig:step5neg\] (the blue line in the left panel). It behaves as $I\approx (40\pi^2/3)(M_4\omega)^2$ for $ M_4\omega\ll0.1 $ and $I\rightarrow 0$ for $M_4\omega\rightarrow 0$ limit. To evaluate the negativity, we obtain the eigenvalues of the partially transposed state $\rho^{T_\textsf{JR}}_{\textsf{BH}_2,\textsf{JR}}$ as $$\begin{aligned} \lambda_i&=\frac{(C+D)A\pm\sqrt{(C-D)^2A^2+4E^2B^2}}{2},\quad \frac{(C+D)B\pm\sqrt{(C-D)^2B^2+4E^2A^2}}{2},\end{aligned}$$ and the negativity is given by $\mathcal{N}=\left(\sum_i|\lambda_i|-1\right)/2$. The result is shown in the right panel in Fig. \[fig:step5neg\] (the blue line). The negativity becomes exactly zero for $M_4\omega\leq 0.041$. Thus, at step 5 (next step to the Page time), our model shows $\textsf{BH}_2$ and become separable for low frequency modes satisfying $M_4\omega < 0.041$ and have only the classical correlation. This separable state corresponds to the firewall. In the $M_4\omega\rightarrow 0$ limit ($\gamma\rightarrow\pi/4$), the reduced density matrix at step 5 is $$\begin{aligned} \rho_{\textsf{BH}_2\cup\textsf{JR}} = &\frac{1}{4}\qty(\op{0}+\op{1})_\textsf{JR}\otimes \qty(\op{0}+\op{1})_{\textsf{BH}_2}=\frac{1}{4}\,\mathbb{I}_{4}.\end{aligned}$$ Thus, $\textsf{BH}_2$ and are in a product state with no classical correlation (random). For $0<M_4\omega\ll 0.041$, the state is separable with weak classical correlations. The schematic diagram representing the quantum states at step 4 and step 5 is shown in Fig. \[fig\_result\_state\]. Three qubits in , and form the GHZ state at step 4, which is nearly maximally entangled. At step 5, the CNOT-U gate acts on and . However, as and are already maximally entangled with other qubits, and no entanglement can be shared between and by the monogamy property of the multipartite entanglement. This is the reason why the entanglement between and is lost and the firewall-like structure arises for $M_4\omega\leq 0.041$. Summary and Discussion ====================== We constructed a quantum circuit model that realizes the evaporation of the black hole. Using the formula of the Bekenstein-Hawking entropy, the black hole mass is introduced to our model. Then, analysis was performed assuming that a single frequency mode passes independently through the quantum circuit. We revealed that the negativity between and becomes zero after the Page time for $M_4\omega\leq 0.041$ and the firewall-like structure appears. This separable state has the small non-zero mutual information, which becomes zero in $M_4\omega\rightarrow 0$ limit. On the other hand, for $ M_4 \omega>0.041$, the entanglement does not become zero and we do not have the firewall-like structure. One notice concerning our model is that it contains the structure of horizon. In the burning paper model, particles’ information is released to the outside as the ash, hence it is a model without horizon. In the model introduced in this paper, qubits are discharged to the outside through CNOT-U gate, which squeezes inputted qubits states and mimics the Hawking radiation. This means the effect of the horizon is taken into our model. Another notice is that we assume $\ket{0}_\textsf{BH}$ as the initial state of . It may be possible to adopt the initial state with some excitation of particles determined by temperature of the initial mass of the black hole. However, we expect that such a generalization does not alter the main feature of entanglement obtained in this paper because the non-vacuum weakly entangled initial state of will not change the entanglement structure after the Page time so much. Related to our investigation in this paper, the result of S. Luo *et al.* [@Albrecht_multi] may correspond to the high frequency mode of our model and they found that the firewall is not necessary. However, our conclusion is that by considering low frequency modes, the black hole always shows the firewall behavior after the Page time. AMPS pointed out necessity of some mechanism to affect the function of horizon to keep the theoretical consistency. Our result can be interpreted that the long wavelength mode influences the horizon so as to form the firewall. In particular, for the low frequency mode with $ M_4 \omega\leq 0.041$, the Hawking particles emitted earlier than the Page time influences the function of the horizon after the Page time. They form the GHZ state with newly created Hawking particle pairs. Therefore, and can only possess the classical correlation without entanglement. The impact of the low frequency mode (soft mode) of quantum fluctuations on the entanglement structure is also discussed in different contexts and different systems. In the paper [@Hotta2018], if the zero-energy soft mode emission is involved with the evaporation process, the entanglement entropy between the black hole and the radiation becomes much larger than the black hole’s thermal entropy as opposed to the Page curve prediction. In cosmological situations, soft modes of which wavelength is larger than the Hubble horizon scale, becomes separable and shows the similar entanglement structure to the firewall [@Matsumura2018]. It may be interesting to obtain unified understanding of emergence of the separable state from the viewpoint of the horizon structure (geometry) and the monogamy property (entanglement). [10]{} S. W. Hawking, “Particle creation by black holes”, *Comm. Math. Phys.* **43**, (1975) 199–220. S. W. 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Pati, “[Quantum information cannot be completely hidden in correlations: Implications for the black-hole information paradox]{}”, *Phys. Rev. Lett.* **98**, (2007) 080502. S. L. Braunstein, H.-J. Sommers, and K. Zyczkowski, “[Entangled black holes as ciphers of hidden information]{}”, *arXiv:0907.0739* . J. Hwang, D. S. Lee, D. Nho, J. Oh, H. Park, D.-h. Yeom, and H. Zoe, “[Page curves for tripartite systems]{}”, *Class. Quant. Grav.* **34**, (2017) 145004. S. Luo, H. Stoltenberg, and A. Albrecht, “[Multipartite Entanglement and Firewalls]{}”, *Phys. Rev.* **D95**, (2017) 064039. G. Vidal and R. F. Werner, “Computable measure of entanglement”, *Phys. Rev. A* **65**, (2002) 032314. H. He and G. Vidal, “Disentangling theorem and monogamy for entanglement negativity”, *Phys. Rev. A* **91**, (2015) 012339. A. Peres, “Separability Criterion for Density Matrices”, *Phys. Rev. Lett.* **77**, (1996) 1413. 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--- abstract: 'We compare the degrees of enumerability and the closed Medvedev degrees and find that many situations occur. There are nonzero closed degrees that do not bound nonzero degrees of enumerability, there are nonzero degrees of enumerability that do not bound nonzero closed degrees, and there are degrees that are nontrivially both degrees of enumerability and closed degrees. We also show that the compact degrees of enumerability exactly correspond to the cototal enumeration degrees.' address: - | School of Mathematics\ University of Leeds\ Leeds, LS2 9JT, United Kingdom - | Dipartimento di Ingegneria Informatica e Scienze Matematiche\ Università Degli Studi di Siena\ I-53100 Siena, Italy author: - Paul Shafer - Andrea Sorbi bibliography: - 'ShaferSorbiEnumVsClosed.bib' title: Comparing the degrees of enumerability and the closed Medvedev degrees --- Introduction {#introduction .unnumbered} ============ The purpose of this work is to explore the distribution of the so-called *degrees of enumerability* with respect to the closed degrees within the Medvedev degrees. Both the enumeration degrees and the Turing degrees embed into the Medvedev degrees. The Medvedev degrees corresponding to enumeration degrees are called *degrees of enumerability*, and the Medvedev degrees corresponding to Turing degrees are called *degrees of solvability*. The embedding of the Turing degrees into the Medvedev degrees is particularly nice. The degrees of solvability are all closed (being the degrees of singleton sets), and the collection of all degrees of solvability is definable in the Medvedev degrees. On the other hand, whether the degrees of enumerability are definable in the Medvedev degrees is a longstanding open question of Rogers [@Rogers:Agenda; @Rogers:Book]. In light of Roger’s question and the nice definability and topological properties of the degrees of solvability, we find it natural to investigate the behavior of the degrees of enumerability with respect to the closed degrees. Together, our main results show that the relation between the degrees of enumerability and the closed degrees is considerably more nuanced than the relation between the degrees of solvability and the closed degrees. - There are nonzero closed degrees that do not bound nonzero degrees of enumerability. In fact, there are nonzero degrees that are closed, uncountable, and meet-irreducible that do not bound nonzero degrees of enumerability (Proposition \[prop-ClosedNotEnum\]). - There are nonzero closed (indeed, compact) degrees of enumerability that do not bound nonzero degrees of solvability (Theorem \[thm:qm-compact\]). Moreover, the compact degrees of enumerability exactly correspond to the cototal enumeration degrees (Theorem \[thm:cototal=compact\]). - There are nonzero degrees of enumerability that do not bound nonzero closed degrees (Theorem \[thm:no-bounding\]). We work in Baire space and interpret an arbitrary set ${\ensuremath{\mathcal{A}}} \subseteq \omega^\omega$ as representing an abstract mathematical problem, namely the problem of finding (or, computing) a member of ${\ensuremath{\mathcal{A}}}$. For this reason, we refer to subsets of Baire space as *mass problems*. For sets ${\ensuremath{\mathcal{A}}}, {\ensuremath{\mathcal{B}}} \subseteq \omega^\omega$, we say that *${\ensuremath{\mathcal{A}}}$ Medvedev (or strongly) reduces to ${\ensuremath{\mathcal{B}}}$*, and we write ${\ensuremath{\mathcal{A}}} {\leq_{\mathrm s}}{\ensuremath{\mathcal{B}}}$, if there is a Turing functional $\Phi$ such that $\Phi({\ensuremath{\mathcal{B}}}) \subseteq {\ensuremath{\mathcal{A}}}$, meaning that $\Phi(f)$ is total and is in ${\ensuremath{\mathcal{A}}}$ for every $f \in {\ensuremath{\mathcal{B}}}$. Under the interpretation of subsets of Baire space as mathematical problems, ${\ensuremath{\mathcal{A}}} {\leq_{\mathrm s}}{\ensuremath{\mathcal{B}}}$ means that problem ${\ensuremath{\mathcal{B}}}$ is at least as hard as problem ${\ensuremath{\mathcal{A}}}$ in a computational sense because every solution to problem ${\ensuremath{\mathcal{B}}}$ can be converted into a solution to problem ${\ensuremath{\mathcal{A}}}$ by a uniform computational procedure. Medvedev reducibility induces an equivalence relation called *Medvedev (or strong) equivalence* in the usual way: ${\ensuremath{\mathcal{A}}} {\equiv_{\mathrm s}}{\ensuremath{\mathcal{B}}}$ if and only if ${\ensuremath{\mathcal{A}}} {\leq_{\mathrm s}}{\ensuremath{\mathcal{B}}}$ and ${\ensuremath{\mathcal{B}}} {\leq_{\mathrm s}}{\ensuremath{\mathcal{A}}}$. The ${\equiv_{\mathrm s}}$-equivalence class ${\operatorname{deg}_{\mathrm s}}({\ensuremath{\mathcal{A}}}) = \{{\ensuremath{\mathcal{B}}} : {\ensuremath{\mathcal{B}}} {\equiv_{\mathrm s}}{\ensuremath{\mathcal{A}}}\}$ of a mass problem ${\ensuremath{\mathcal{A}}}$ is called its *Medvedev (or strong) degree*, and the collection of all such equivalence classes, ordered by Medvedev reducibility, is a structure called the *Medvedev degrees*. The Medvedev degrees form a bounded distributive lattice (in fact, a Brouwer algebra), with least element ${\ensuremath{\mathbf{0}}} = \{{\ensuremath{\mathcal{A}}} : \text{${\ensuremath{\mathcal{A}}}$ has a recursive member}\}$ and greatest element ${\ensuremath{\mathbf{1}}} = \{\emptyset\}$. Joins and meets in the Medvedev degrees are computed as follows: $$\begin{aligned} {\operatorname{deg}_{\mathrm s}}({\ensuremath{\mathcal{A}}}) {\vee}{\operatorname{deg}_{\mathrm s}}({\ensuremath{\mathcal{B}}}) &= {\operatorname{deg}_{\mathrm s}}({\ensuremath{\mathcal{A}}} \oplus {\ensuremath{\mathcal{B}}})\\ {\operatorname{deg}_{\mathrm s}}({\ensuremath{\mathcal{A}}}) {\wedge}{\operatorname{deg}_{\mathrm s}}({\ensuremath{\mathcal{B}}}) &= {\operatorname{deg}_{\mathrm s}}(0^{\smallfrown}{\ensuremath{\mathcal{A}}} \cup 1^{\smallfrown}{\ensuremath{\mathcal{B}}}).\end{aligned}$$ For joins, ${\ensuremath{\mathcal{A}}} \oplus {\ensuremath{\mathcal{B}}} = \{f \oplus g : f \in {\ensuremath{\mathcal{A}}} {\text{ and }}g \in {\ensuremath{\mathcal{B}}}\}$, where $f \oplus g$ is the usual Turing join of $f$ and $g$: $(f \oplus g)(2n) = f(n)$ and $(f \oplus g)(2n+1) = g(n)$. For meets, $0^{\smallfrown}{\ensuremath{\mathcal{A}}} \cup 1^{\smallfrown}{\ensuremath{\mathcal{B}}}$ is the set obtained by prepending $0$ to every function in ${\ensuremath{\mathcal{A}}}$, prepending $1$ to every function in ${\ensuremath{\mathcal{B}}}$, and taking the union of the resulting sets. Under the interpretation of mass problems as mathematical problems, problem ${\ensuremath{\mathcal{A}}} \oplus {\ensuremath{\mathcal{B}}}$ corresponds to the problem of solving problem ${\ensuremath{\mathcal{A}}}$ *and* solving problem ${\ensuremath{\mathcal{B}}}$, and problem $0^{\smallfrown}{\ensuremath{\mathcal{A}}} \cup 1^{\smallfrown}{\ensuremath{\mathcal{B}}}$ corresponds to the problem of solving problem ${\ensuremath{\mathcal{A}}}$ *or* solving problem ${\ensuremath{\mathcal{B}}}$. Medvedev introduced the structure that now bears his name in [@Medvedev1955]. For an introduction to the Medvedev degrees, including its origins and motivation, see [@Rogers:Book]\*[Chapter 13.7]{}. For surveys on the Medvedev degrees and related topics, see [@SorbiSurvey; @HinmanSurvey]. For recursive aspects of the Medvedev degrees, see [@Dyment1976Rus]. For algebraic aspects of the Medvedev degrees and applications to intermediate logics, see for instance [@Skvortsova:intuitionismEng; @Sorbi:Brouwer; @Kuyper-Medvedev]. Notation {#notation .unnumbered} -------- We use the following notation and terminology regarding strings and trees. Denote by $\omega^{<\omega}$ the set of all finite strings of natural numbers, and denote by $2^{<\omega}$ the set of all finite binary strings. For $\sigma \in \omega^{<\omega}$, $|\sigma|$ denotes the length of $\sigma$. We denote the empty string by $\emptyset$. For $\sigma, \tau \in \omega^{<\omega}$, $\sigma \subseteq \tau$ means that $\sigma$ is an initial segment of $ \tau$, and $\sigma^{\smallfrown}\tau$ denotes the concatenation of $\sigma$ and $\tau$. Similarly, for $ \sigma \in \omega^{<\omega}$ and $f \in \omega^\omega$, $\sigma \subset f$ means that $ \sigma$ is an initial segment of $f$, i.e., $(\forall n < |\sigma|)(f(n) = \sigma(n))$, and $\sigma^{\smallfrown}f$ denotes the concatenation of $\sigma$ and $f$: $$\begin{aligned} (\sigma^{\smallfrown}f)(n) = \begin{cases} \sigma(n) & \text{if $n < |\sigma|$}\\ f(n-|\sigma|) & \text{if $n \geq |\sigma|$}. \end{cases}\end{aligned}$$ If $\sigma \in \omega^{<\omega}$ and $f \in \omega^\omega$, $\sigma \# f$ denotes the result of replacing the initial segment of $f$ of length $|\sigma|$ by $\sigma$: $$\begin{aligned} (\sigma \# f)(n) = \begin{cases} \sigma(n) & \text{if $n < |\sigma|$}\\ f(n) & \text{if $n \geq |\sigma|$}. \end{cases}\end{aligned}$$ For $\sigma \in \omega^{<\omega}$ and ${\ensuremath{\mathcal{A}}} \subseteq \omega^\omega$, we define $\sigma^{\smallfrown}{\ensuremath{\mathcal{A}}} = \{\sigma^{\smallfrown}f : f \in {\ensuremath{\mathcal{A}}}\}$ and $\sigma\#{\ensuremath{\mathcal{A}}} = \{\sigma \# f : f \in {\ensuremath{\mathcal{A}}}\}$. Finally, for $\sigma \in \omega^{<\omega}$ and $n \leq |\sigma|$, $\sigma\!\restriction\!n$ denotes the initial segment ${\langle}\sigma(0), \dots, \sigma(n-1) {\rangle}$ of $\sigma$ of length $n$. Similarly, for $f \in \omega^\omega$ and $n \in \omega$, $f\!\restriction\!n$ denotes the initial segment ${\langle}f(0), \dots, f(n-1) {\rangle}$ of $f$ of length $n$. A *tree* is a set $T \subseteq \omega^{<\omega}$ that is closed under initial segments: $(\forall \sigma, \tau \in \omega^{<\omega})((\sigma \subseteq \tau {\text{ and }}\tau \in T) {\rightarrow}\sigma \in T)$. A node $\sigma$ in a tree $T$ is a *leaf* if there is no $\tau \supset \sigma$ with $\tau \in T$. A tree $T$ is *finitely branching* if for every $\sigma \in T$ there are at most finitely many strings $\tau \in T$ with $|\tau| = |\sigma| + 1$. A string $\sigma \in \omega^{<\omega}$ is *bounded* by an $h \in \omega^\omega$ (or *$h$-bounded*) if $(\forall n < |\sigma|)(\sigma(n) < h(n))$. Likewise, a tree $T$ is *$h$-bounded* if $(\forall \sigma \in T)(\text{$\sigma$ is $h$-bounded})$. For $b \in \omega$, $b$-bounded means bounded by the function that is constantly $b$. An $f \in \omega^\omega$ is an infinite path through a tree $T$ if $\forall n (f\!\restriction\!n \in T)$. The subset of Baire space consisting of all infinite paths through a tree $T$ is denoted by $[T]$. The closed subsets of Baire space are exactly those of the form $[T]$ for a tree $T$, and the compact subsets of Baire space are exactly those of the form $[T]$ for a finitely branching tree $T$. Throughout, we refer to a standard listing $(\Phi_e: e \in \omega)$ of all Turing functionals on Baire space. If $\Phi$ is a Turing functional and $\sigma$ is a finite string of natural numbers, then $\Phi(\sigma)$ denotes the longest string $\tau$ such that $(\forall m < |\tau|)(\tau(m) = \Phi(\sigma)(m){{\downarrow}})$). We also refer to a standard listing $(\Psi_e: e \in \omega)$ of all enumeration operators. If $\Psi$ is an enumeration operator and $\phi$ is a partial function, then $\Psi(\phi)$ stands for $\Psi(\operatorname{graph}(\phi))$. Recall that ${\langle}\cdot, \cdot {\rangle}\colon \omega^2 {\rightarrow}\omega$ is the usual recursive Cantor pairing function and that $\operatorname{graph}(\phi) = \{{\langle}n, y {\rangle}: n \in \operatorname{dom}(\phi) {\text{ and }}\phi(n) = y\}$. For further background concerning recursion theory, trees, and the topology of Baire space, we refer the reader to standard textbooks such as [@Rogers:Book; @Moschovakis:Descriptive]. Degrees of solvability and degrees of enumerability {#degrees-of-solvability-and-degrees-of-enumerability .unnumbered} --------------------------------------------------- As discussed in the introduction, part of the interest in the Medvedev degrees comes from the fact that the structure embeds both the Turing degrees and the enumeration degrees. Singleton subsets of Baire space are called *problems of solvability*, and their corresponding Medvedev degrees are called *degrees of solvability*. It is easy to see that the assignment ${\operatorname{deg}_{\mathrm T}}(f) \mapsto {\operatorname{deg}_{\mathrm s}}(\{f\})$ embeds the Turing degrees into the Medvedev degrees, preserving joins and the least element, and that the range of this embedding is exactly the degrees of solvability. Moreover, the degrees of solvability are definable in the Medvedev degrees [@Medvedev1955; @Dyment1976Rus] (see also [@SorbiSurvey; @Rogers:Book]). To embed the enumeration degrees into the Medvedev degrees, given a nonempty $A \subseteq \omega$, let $$\begin{aligned} {\ensuremath{\mathcal{E}}}_A = \{f : \operatorname{ran}(f) = A\}.\end{aligned}$$ ${\ensuremath{\mathcal{E}}}_A$ is called the *problem of enumerability of $A$*, and it represents the problem of enumerating the set $A$. The corresponding Medvedev degree ${\ensuremath{\mathbf{E}}}_A = {\operatorname{deg}_{\mathrm s}}({\ensuremath{\mathcal{E}}}_A)$ is called the *degree of enumerability of $A$*. For nonempty $A, B \subseteq \omega$, it is easy to see that $A {\leq_\mathrm{e}}B$ if and only if ${\ensuremath{\mathcal{E}}}_A {\leq_{\mathrm s}}{\ensuremath{\mathcal{E}}}_B$. This gives rise to an embedding ${\operatorname{deg}_{\mathrm{e}}}(A) \mapsto {\ensuremath{\mathbf{E}}}_A$ of the enumeration degrees into the Medvedev degrees. The embedding preserves joins and the least element, and the range of the embedding is exactly the degrees of enumerability [@Medvedev1955] (see also [@SorbiSurvey; @Rogers:Book]). Again we mention that, contrary to definability of the degrees of solvability, it is still an open question (see Rogers [@Rogers:Agenda; @Rogers:Book]) whether the degrees of enumerability are definable, or at least invariant under automorphisms, in the Medvedev degrees. The following lemma (which we state and prove for later reference) is well-known. It corresponds to the fact that the Turing degrees embed (again via an embedding that preserves joins and the least element) into the enumeration degrees of total functions. \[lem-EnumTotGraph\] If $f \colon \omega {\rightarrow}\omega$ is total, then ${\ensuremath{\mathcal{E}}}_{\operatorname{graph}(f)} {\equiv_{\mathrm s}}\{f\}$. Clearly ${\ensuremath{\mathcal{E}}}_{\operatorname{graph}(f)} {\leq_{\mathrm s}}\{f\}$ via the Turing functional $\Phi(f)(n)=\langle n, f(n)\rangle$. To see that $\{f\} {\leq_{\mathrm s}}{\ensuremath{\mathcal{E}}}_{\operatorname{graph}(f)}$, let $\Gamma$ be the Turing functional such that, for every total $g \colon \omega {\rightarrow}\omega$ and $n \in \omega$, $\Gamma(g)(n)$ searches for the least $k$ such that $g(k) = {\langle}n, y {\rangle}$ for some $y$, and outputs $y$. Then $\Gamma(g) = f$ whenever $\operatorname{ran}(g) = \operatorname{graph}(f)$, so $\{f\} {\leq_{\mathrm s}}{\ensuremath{\mathcal{E}}}_{\operatorname{graph}(f)}$. In analogy with the common terminology used in the enumeration degrees, we say that a problem of enumerability ${\ensuremath{\mathcal{E}}}$ is *total* if ${\ensuremath{\mathcal{E}}} {\equiv_{\mathrm s}}\{f\}$ for some total $f$. That is, a problem of enumerability is total if it is Medvedev-equivalent to a problem of solvability. Likewise, we say that a degree of enumerability is *total* if it is the Medvedev degree of a total problem of enumerability. Now recall that an $A \subseteq \omega$ is *quasiminimal* if $A$ is not r.e. and there is no nonrecursive total $f$ with $f {\leq_\mathrm{e}}A$ (meaning, as usual, that there is no nonrecursive total $f$ with $\operatorname{graph}(f) {\leq_\mathrm{e}}A$). We say that a problem of enumerability ${\ensuremath{\mathcal{E}}}$ is *quasiminimal* if ${\ensuremath{\mathcal{E}}} {\equiv_{\mathrm s}}{\ensuremath{\mathcal{E}}}_A$ for a quasiminimal $A$. Likewise, we say that a degree of enumerability is *quasiminimal* if it is the Medvedev degree of a quasiminimal problem of enumerability. Lemma \[lem-EnumTotGraph\] implies the following lemma. \[lem:withoutproof\] If ${\ensuremath{\mathcal{E}}}_A$ is a quasiminimal problem of enumerability, then ${\ensuremath{\mathbf{0}}} {<_{\mathrm s}}{\ensuremath{\mathbf{E}}}_A$ and ${\ensuremath{\mathcal{E}}}_A {\not\equiv_{\mathrm s}}\{f\}$ for every total $f$ (in fact $\{f\} {\nleq_{\mathrm s}}{\ensuremath{\mathcal{E}}}_A$ for every nonrecursive total $f$). Both the degrees of solvability and the degrees of enumerability enjoy the algebraic property of *meet-irreducibility*. Recall that an element $a$ of a lattice $L$ is called *meet-reducible* if it is the meet of a pair of strictly larger elements: $(\exists b,c \in L)(b > a {\text{ and }}c > a {\text{ and }}a = b {\wedge}c)$. An element of a lattice is called *meet-irreducible* if it is not meet-reducible. It is well-known that, in a distributive lattice $L$ such as the Medvedev degrees, an element $a$ is meet-irreducible if and only if $(\forall b,c \in L)((a \geq b {\wedge}c) {\rightarrow}(a \geq b {\text{ or }}a \geq c))$ (see [@BalbesDwinger]\*[Section III.2]{}). We now recall some helpful terminology and a lemma before proving that the degrees of solvability and the degrees of enumerability are meet-irreducible. These facts are known in the literature, but we include proofs for the sake of completeness. For a mass problem ${\ensuremath{\mathcal{A}}}$ and a $\sigma \in \omega^{<\omega}$, let ${\ensuremath{\mathcal{A}}}_\sigma=\{f \in {\ensuremath{\mathcal{A}}} : \sigma \subset f\}$. Call a mass problem ${\ensuremath{\mathcal{A}}}$ *uniform* if ${\ensuremath{\mathcal{A}}}_\sigma {\leq_{\mathrm s}}{\ensuremath{\mathcal{A}}}$ whenever $\sigma \in \omega^{<\omega}$ is such that $\sigma \subset f$ for some $f \in {\ensuremath{\mathcal{A}}}$. [@Dyment1976Rus]\*[Corollary 2.8]{}\[lem:uniform-meet-irr\] Every uniform mass problem has meet-irreducible Medvedev degree. Suppose that ${\ensuremath{\mathcal{A}}}$ is a uniform mass problem and that ${\ensuremath{\mathcal{B}}}$ and ${\ensuremath{\mathcal{C}}}$ are arbitrary mass problems such that $0^{\smallfrown}{\ensuremath{\mathcal{B}}} \cup 1^{\smallfrown}{\ensuremath{\mathcal{C}}} {\leq_{\mathrm s}}{\ensuremath{\mathcal{A}}}$. We may assume that ${\ensuremath{\mathcal{A}}} \neq \emptyset$ as clearly ${\ensuremath{\mathbf{1}}}$ is meet-irreducible. Let $\Phi$ be such that $\Phi({\ensuremath{\mathcal{A}}}) \subseteq 0^{\smallfrown}{\ensuremath{\mathcal{B}}} \cup 1^{\smallfrown}{\ensuremath{\mathcal{C}}}$. Choose any $f \in {\ensuremath{\mathcal{A}}}$, and let $ \sigma \subset f$ be such that $\Phi(\sigma)(0){{\downarrow}}$. Let $b = \Phi(\sigma)(0)$, and observe that $b \in \{0,1\}$. Suppose for the sake of argument that $b = 0$. Then, as every $f \in {\ensuremath{\mathcal{A}}}_\sigma$ begins with $\sigma$ and is in ${\ensuremath{\mathcal{A}}}$, we have that $\Phi(f)(0)=0$ for every $f \in {\ensuremath{\mathcal{A}}}_\sigma$, thus yielding ${\ensuremath{\mathcal{B}}} {\leq_{\mathrm s}}0^{\smallfrown}{\ensuremath{\mathcal{B}}} {\leq_{\mathrm s}}{\ensuremath{\mathcal{A}}}_\sigma {\leq_{\mathrm s}}{\ensuremath{\mathcal{A}}}$. Similarly, if $b = 1$, then ${\ensuremath{\mathcal{C}}} {\leq_{\mathrm s}}{\ensuremath{\mathcal{A}}}$. Thus either ${\ensuremath{\mathcal{B}}} {\leq_{\mathrm s}}{\ensuremath{\mathcal{A}}}$ or ${\ensuremath{\mathcal{C}}} {\leq_{\mathrm s}}{\ensuremath{\mathcal{A}}}$. So ${\ensuremath{\mathcal{A}}}$ has meet-irreducible degree. \[prop-MeetIrred\] In the Medvedev degrees, every degree of solvability is meet-irreducible, and every degree of enumerability is meet-irreducible. It suffices to prove that the degrees of enumerability are meet-irreducible (see [@Sorbi:Someremarks]\*[Theorem 4.5]{}) because every degree of solvability is also a degree of enumerability. (It is also easy to simply observe that if $0^{\smallfrown}{\ensuremath{\mathcal{B}}} \cup 1^{\smallfrown}{\ensuremath{\mathcal{C}}} {\leq_{\mathrm s}}\{f\}$, then either ${\ensuremath{\mathcal{B}}} {\leq_{\mathrm s}}\{f\}$ or ${\ensuremath{\mathcal{C}}} {\leq_{\mathrm s}}\{f\}$.) Let ${\ensuremath{\mathbf{E}}}_A$ be the degree of enumerability of $A \subseteq \omega$. The proposition follows from Lemma \[lem:uniform-meet-irr\], as it is easy to see that ${\ensuremath{\mathcal{A}}}={\ensuremath{\mathcal{E}}}_A$ is uniform: if $\sigma \subset f$ for some $f \in {\ensuremath{\mathcal{E}}}_A$, just consider the reduction ${\ensuremath{\mathcal{A}}}_\sigma {\leq_{\mathrm s}}{\ensuremath{\mathcal{A}}}$ given by $\Phi(f)=\sigma^{\smallfrown}f$. In a similar spirit, Dyment proved that if ${\ensuremath{\mathcal{B}}}$ is a countable (or finite) mass problem, if $ {\ensuremath{\mathcal{E}}}_A$ is the problem of enumerability of $A \subseteq \omega$, and if ${\ensuremath{\mathcal{B}}} {\leq_{\mathrm s}}{\ensuremath{\mathcal{E}}} _A$, then there is a $g \in {\ensuremath{\mathcal{B}}}$ such that $g {\leq_\mathrm{e}}A$ [@Dyment1976Rus]\*[Theorem 3.4]{}. Call a Medvedev degree *countable* if it is the degree of a countable (or finite) mass problem, and call it *uncountable* otherwise. Dyment’s result implies that if ${\ensuremath{\mathbf{E}}}$ is a nontotal degree of enumerability, then ${\ensuremath{\mathbf{E}}}$ is uncountable [@Dyment1976Rus]\*[Corollary 3.14]{}. Comparing degrees of enumerability and closed degrees {#comparing-degrees-of-enumerability-and-closed-degrees .unnumbered} ===================================================== A Medvedev degree is called *closed* if it is of the form ${\operatorname{deg}_{\mathrm s}}({\ensuremath{\mathcal{C}}})$ for a closed ${\ensuremath{\mathcal{C}}} \subseteq \omega^\omega$. Every degree of solvability is closed because singletons are closed. Thus there are closed degrees of enumerability because every degree of solvability is also a degree of enumerability. It is, however, easy to produce examples of closed degrees that are not degrees of enumerability. Let $f,g \in \omega^\omega$ be such that $f {\mid_{\mathrm T}}g$. Then $ {\operatorname{deg}_{\mathrm s}}(\{f,g\})$ is closed, but it is not a degree of enumerability because it is meet-reducible (as $ {\operatorname{deg}_{\mathrm s}}(\{f,g\}) = {\operatorname{deg}_{\mathrm s}}(\{f\}) {\wedge}{\operatorname{deg}_{\mathrm s}}(\{g\})$), whereas all degrees of enumerability are meet- irreducible by Proposition \[prop-MeetIrred\]. In fact, by the discussion following Proposition \[prop-MeetIrred\], we know that a degree of enumerability must be meet-irreducible and either total (i.e., a degree of solvability) or uncountable. This begs the question of whether there are Medvedev degrees that are closed, meet-irreducible, and uncountable, yet not degrees of enumerability. We show that the Medvedev degree of the $\{0,1\} $-valued diagonally nonrecursive functions is such a degree. Recall that $f \in \omega^\omega$ is *diagonally nonrecursive* (DNR for short) if $\forall e(\Phi_e(e){{\downarrow}}{\rightarrow}f(e) \neq \Phi_e(e))$. Let ${\mathrm{DNR}}_2 = \{f \in 2^\omega : \text{$f$ is DNR}\}$. \[lem-TreeEnum\] Let $T \subseteq \omega^{<\omega}$ be an infinite $h$-bounded tree for some $h \in \omega^\omega$. If $A \subseteq \omega$ is such that ${\ensuremath{\mathcal{E}}}_A {\leq_{\mathrm s}}[T]$, then $A$ is r.e. in $T \oplus h$. Let $\Phi$ be such that $\Phi([T]) \subseteq {\ensuremath{\mathcal{E}}}_A$. Using $T \oplus h$ as an oracle, enumerate the set $$\begin{aligned} B = \{n : \exists k(\forall\,\text{$h$-bounded $\sigma$ with $|\sigma|=k$})(\sigma \in T {\rightarrow}n \in \operatorname{ran}(\Phi(\sigma)))\}.\end{aligned}$$ We show that $B = A$, thus showing that $A$ is r.e. in $T \oplus h$. Suppose that $n \in B$. Let $k$ be such that $n \in \operatorname{ran}(\Phi(\sigma))$ whenever $\sigma \in T$ has length $k$. Let $f \in [T]$. Then $f\!\restriction\!k \in T$, so $n \in \operatorname{ran}(\Phi(f\!\restriction\!k))$. However, $\operatorname{ran}(\Phi(f)) = A$ because $\Phi(f) \in {\ensuremath{\mathcal{E}}}_A$, so it must be that $n \in A$. Hence $B \subseteq A$. Now suppose that $n \notin B$. Then for every $k$ there is an $h$-bounded $\sigma$ of length $k$ with $\sigma \in T$ but $n \notin \operatorname{ran}(\Phi(\sigma))$. So the subtree $S \subseteq T$ given by $S = \{\sigma \in T : n \notin \operatorname{ran}(\Phi(\sigma))\}$ is infinite. By König’s lemma, there is a path $f \in [S] \subseteq [T]$. However, $n \notin \operatorname{ran}(\Phi(f))=A$, giving $n \notin A$ as desired. In the next proposition, our proof that ${\operatorname{deg}_{\mathrm s}}({\mathrm{DNR}}_2)$ is uncountable relies on the following fact. If ${\ensuremath{\mathcal{A}}} \subseteq 2^\omega$ is a nonempty $\Pi^0_1$ class with no recursive member and ${\ensuremath{\mathcal{B}}}$ is a countable mass problem with no recursive member, then ${\ensuremath{\mathcal{B}}} {\nleq_{\mathrm s}}{\ensuremath{\mathcal{A}}}$. This fact follows immediately from [@Jockusch-; @Soare:Pi01]\*[Theorem 2.5]{}, which essentially states that such an ${\ensuremath{\mathcal{A}}}$ must in fact have continuum-many members that are all pairwise Turing incomparable and also all Turing incomparable with all members of ${\ensuremath{\mathcal{B}}}$. So in fact ${\ensuremath{\mathcal{B}}} {\nleq_{\mathrm w}}{\ensuremath{\mathcal{A}}}$, where ${\leq_{\mathrm w}}$ is *Muchnik reducibility*: ${\ensuremath{\mathcal{X}}} {\leq_{\mathrm w}}{\ensuremath{\mathcal{Y}}}$ if $(\forall g \in {\ensuremath{\mathcal{Y}}})(\exists f \in {\ensuremath{\mathcal{X}}})(f {\leq_{\mathrm T}}g)$. That ${\ensuremath{\mathcal{B}}} {\nleq_{\mathrm s}}{\ensuremath{\mathcal{A}}}$ can also be deduced from the well-known fact that the image of a recursively bounded $\Pi^0_1$ class under a Turing functional is another recursively bounded $\Pi^0_1$ class (see [@Simpson]\*[Theorem 4.7]{}), which is easier to prove than [@Jockusch-Soare:Pi01]\*[Theorem 2.5]{}. Suppose for a contradiction that ${\ensuremath{\mathcal{B}}} {\leq_{\mathrm s}}{\ensuremath{\mathcal{A}}}$ via the Turing functional $\Phi$. Then ${\ensuremath{\mathcal{B}}}_0 = \Phi({\ensuremath{\mathcal{A}}}) \subseteq {\ensuremath{\mathcal{B}}}$ is a countable recursively bounded $\Pi^0_1$ class and therefore must have a recursive member, contradicting that ${\ensuremath{\mathcal{B}}}$ has no recursive member. This argument can also be used to show that ${\ensuremath{\mathcal{B}}} {\nleq_{\mathrm w}}{\ensuremath{\mathcal{A}}}$ because if ${\ensuremath{\mathcal{B}}} {\leq_{\mathrm w}}{\ensuremath{\mathcal{A}}}$, then (see [@Simpson]\*[Lemma 6.9]{}) there is a nonempty $\Pi^0_1$ class ${\ensuremath{\mathcal{A}}}_0 \subseteq {\ensuremath{\mathcal{A}}}$ such that ${\ensuremath{\mathcal{B}}} {\leq_{\mathrm s}}{\ensuremath{\mathcal{A}}}_0$, and then the argument can be repeated with ${\ensuremath{\mathcal{A}}}_0$ in place of ${\ensuremath{\mathcal{A}}}$. \[prop-ClosedNotEnum\] The Medvedev degree ${\operatorname{deg}_{\mathrm s}}({\mathrm{DNR}}_2)$ is closed, meet-irreducible, and uncountable, yet also not a degree of enumerability (in fact, it does not bound any nonzero degree of enumerability). It is well-known that ${\mathrm{DNR}}_2$ is a $\Pi^0_1$ class because ${\mathrm{DNR}}_2 = [T]$ for the recursive tree $$\begin{aligned} T = \{\sigma \in 2^{<\omega} : (\forall e < |\sigma|)(\text{$\Phi_e(e)$ halts within $|\sigma|$ steps} {\rightarrow}\sigma(e) \neq \Phi_e(e))\}.\end{aligned}$$ By the above discussion, if ${\ensuremath{\mathcal{B}}}$ is a countable mass problem with no recursive member, then ${\ensuremath{\mathcal{B}}} {\nleq_{\mathrm s}}{\mathrm{DNR}}_2$. Hence ${\operatorname{deg}_{\mathrm s}}({\mathrm{DNR}}_2)$ is uncountable. That ${\operatorname{deg}_{\mathrm s}}({\mathrm{DNR}}_2)$ is meet-irreducible follows from Lemma \[lem:uniform-meet-irr\], as it is easy to see that ${\ensuremath{\mathcal{A}}} = {\mathrm{DNR}}_2$ is uniform. If $\sigma \subset f$ for an $f$ in ${\mathrm{DNR}}_2$, consider the reduction procedure ${\ensuremath{\mathcal{A}}}_\sigma {\leq_{\mathrm s}}{\ensuremath{\mathcal{A}}}$ given by $\Phi(f)=\sigma \# f$. That ${\operatorname{deg}_{\mathrm s}}({\mathrm{DNR}}_2)$ is not a degree of enumerability follows from Lemma \[lem-TreeEnum\]. We know that ${\mathrm{DNR}}_2 = [T]$ for a recursive tree $T \subseteq 2^{<\omega}$. Thus if ${\ensuremath{\mathcal{E}}}_A {\leq_{\mathrm s}}{\mathrm{DNR}}_2$ for some $A \subseteq \omega$, then $A$ would have to be r.e. by Lemma \[lem-TreeEnum\]. However, if $A$ is r.e., then ${\ensuremath{\mathcal{E}}}_A$ would have a recursive member, in which case ${\mathrm{DNR}}_2 {\nleq_{\mathrm s}}{\ensuremath{\mathcal{E}}}_A$. Thus there is no $A$ such that ${\mathrm{DNR}}_2 {\equiv_{\mathrm s}}{\ensuremath{\mathcal{E}}}_A$. In fact ${\mathrm{DNR}}_2$ does not bound any nonzero degree of enumerability. If ${\ensuremath{\mathbf{E}}}_A\ne \mathbf{0}$ and ${\ensuremath{\mathbf{E}}}_A$ is not quasiminimal, then there are nonrecursive functions $f$ such that $\{f\} {\leq_{\mathrm s}}{\ensuremath{\mathcal{E}}}_{A}$, so ${\ensuremath{\mathbf{E}}}_A$ bounds some nonzero closed degree. As observed in [@Lewis-Shore-Sorbi], there are also quasiminimal degrees of enumerability ${\ensuremath{\mathbf{E}}}_A$ that bound nonzero closed degrees. Given an infinite set $A$, consider the mass problem $$\begin{aligned} {\ensuremath{\mathcal{C}}}_A = \{f : \text{$f$ is one-to-one and $\operatorname{ran}(f) \subseteq A$}\}.\end{aligned}$$ As observed in [@BianchiniSorbi], ${\ensuremath{\mathcal{C}}}_A$ is closed, ${\operatorname{deg}_{\mathrm s}}({\ensuremath{\mathcal{C}}}_A) {\leq_{\mathrm s}}{\ensuremath{\mathbf{E}}}_A$, and, if $A$ is immune (meaning that $A$ has no infinite r.e. subset), then ${\operatorname{deg}_{\mathrm s}}({\ensuremath{\mathcal{C}}}_A) \neq {\ensuremath{\mathbf{0}}}$. So, if $A$ is immune and of quasiminimal e-degree (which is the case, for instance, if $A$ is a $1$-generic set, see [@Copestake:1-generic]), then we have a quasiminimal degree of enumerability which bounds a nonzero closed degree. On the other hand, if $A$ contains an infinite set $B$ such that $A {\nleq_\mathrm{e}}B$, then ${\ensuremath{\mathbf{E}}}_A {\nleq_{\mathrm s}}{\operatorname{deg}_{\mathrm s}}({\ensuremath{\mathcal{C}}}_A)$ because in this case ${\ensuremath{\mathcal{C}}}_A {\leq_{\mathrm s}}{\ensuremath{\mathcal{E}}}_B$ (as ${\ensuremath{\mathcal{C}}}_B \subseteq {\ensuremath{\mathcal{C}}}_A$) but ${\ensuremath{\mathcal{E}}}_A {\nleq_{\mathrm s}}{\ensuremath{\mathcal{E}}}_B$. This gives examples of sets $A$, even of total e-degree, for which $\mathbf{0}<_\textrm{s} {\operatorname{deg}_{\mathrm s}}({\ensuremath{\mathcal{C}}}_A) <_\textrm{s} {\ensuremath{\mathbf{E}}}_A$. There is a total $f \colon \omega {\rightarrow}\omega$ such that ${\operatorname{deg}_{\mathrm s}}{({\ensuremath{\mathcal{C}}}_{\operatorname{graph}(f)})} \neq \mathbf{0}$ and ${\ensuremath{\mathcal{C}}}_{\operatorname{graph}(f)} <_\textrm{s} {\ensuremath{\mathcal{E}}}_{\operatorname{graph}(f)}$. By the above remarks and by Lemma \[lem-EnumTotGraph\], consider two biimmune sets $A,B$ with $A {\mid_{\mathrm T}}B$, and let $f = \chi_A \oplus \chi_B$ (where $\chi_Z$ denotes the characteristic function of $Z$). Then $\operatorname{graph}(f)$ is immune, and it contains an infinite subset (for instance $\{\langle 2x, f(2x)\rangle : x \in \omega\}$) to which it does not Turing-reduce, and hence, by totality, to which it does not e-reduce. However, if $f$ is total, then ${\ensuremath{\mathcal{C}}}_{\operatorname{graph}(f)} {\equiv_{\mathrm s}}{\ensuremath{\mathcal{E}}}_{\operatorname{graph}(f)}$ is almost true, as argued in the following proposition. If $f \colon \omega {\rightarrow}\omega$ is total, then there is a set $B {\equiv_\mathrm{e}}\operatorname{graph}{(f)}$ such that $ {\ensuremath{\mathcal{C}}}_B {\equiv_{\mathrm s}}{\ensuremath{\mathcal{E}}}_{\operatorname{graph}(f)}$. Given $f$ total, let $B=\{\sigma\in \omega^{<\omega} \colon \sigma \subset f\}$. It is easy to see that $f {\equiv_\mathrm{e}}B$, so ${\ensuremath{\mathcal{E}}}_{\operatorname{graph}(f)} {\equiv_{\mathrm s}}{\ensuremath{\mathcal{E}}}_{B}$. To see that ${\ensuremath{\mathcal{E}}}_{B} {\leq_{\mathrm s}}{\ensuremath{\mathcal{C}}}_{B}$, let $\Phi$ be a Turing functional such that, for every $g$ and $n$, $\Phi(g)(n)$ searches for an $m$ such that $g(m)$ is a string $ \sigma$ with $|\sigma| \geq n$ and then outputs $\sigma\!\restriction\!n$. Then $\operatorname{ran}(g)$ is an infinite subset of $B$ whenever $g \in {\ensuremath{\mathcal{C}}}_B$, in which case $\operatorname{ran}(\Phi(g)) = B$. Hence $ \Phi$ witnesses that ${\ensuremath{\mathcal{E}}}_{B} {\leq_{\mathrm s}}{\ensuremath{\mathcal{C}}}_{B}$. While it is true that every total degree of enumerability bounds (and in fact is equivalent to) a closed mass problem, if we move away from totality, then all possibilities may occur. That is, there are nontotal (in fact quasiminimal) degrees of enumerability that are closed (in fact compact, see Theorem \[thm:qm-compact\] below), and there are nonzero degrees of enumerability that do not bound nonzero closed degrees (see Theorem \[thm:no-bounding\] below). Compactness and cototality {#compactness-and-cototality .unnumbered} -------------------------- We make use of *uniformly e-pointed trees*. This notion was originally introduced by Montalbán [@Montalban:Comm] in the context of computable structure theory (see also [@Montalban:Book]), and it has since been studied by McCarthy in the context of the enumeration degrees [@Mccarthy]. Montalbán’s uniformly e-pointed trees are subtrees of $2^{<\omega}$, which we refer to as *uniformly e-pointed trees w.r.t.sets*. We find it convenient to work with finitely branching subtrees of $\omega^{<\omega}$ instead, so we define *uniformly e-pointed trees w.r.t. functions*.[^1] \[def-plus\] For a function $g \in 2^\omega$, let $g^+ = \{n : g(n) = 1\}$ denote the set of which $g$ is the characteristic function. [ ]{} - A *uniformly e-pointed tree with respect to sets* is a tree $T \subseteq 2^{<\omega}$ with no leaves for which there is an enumeration operator $\Psi$ such that $(\forall g \in [T])(\Psi(g^+) = T)$. - A *uniformly e-pointed tree with respect to functions* is a finitely branching tree $T \subseteq \omega^{<\omega}$ with no leaves for which there is an enumeration operator $\Psi$ such that $(\forall g \in [T])(\Psi(g) = T)$. (Recall that, as $\Psi$ is an enumeration operator, $\Psi(g)$ means $\Psi(\operatorname{graph}(g))$.) We show that the two notions of uniform e-pointedness coincide up to e-equivalence. \[prop:from-sets-to-functions\] Every uniformly e-pointed tree w.r.t. sets is a uniformly e-pointed tree w.r.t. functions. Let $T \subseteq 2^{<\omega}$ be a uniformly e-pointed tree w.r.t. sets. Let $\Psi$ be an enumeration operator such that $(\forall g \in [T])(\Psi(g^{+}) = T)$. Fix an enumeration operator $\Gamma$ such that $(\forall A \subseteq \omega)(\Gamma(\chi_A) = A)$. By composing $\Psi$ and $\Gamma$, we get an enumeration operator $\Theta$ such that $(\forall g \in [T])(\Theta(g) = T)$. Thus $T$ is a uniformly e-pointed tree w.r.t. functions. \[prop:link\] Let $T \subseteq \omega^{<\omega}$ be a uniformly e-pointed tree w.r.t.functions. Then there is a uniformly e-pointed tree $S \subseteq 2^{<\omega}$ w.r.t. sets such that $S {\equiv_\mathrm{e}}T$. (In fact we may choose $S$ so that $[S]$ consists of exactly the characteristic functions of the graphs of elements of $[T]$.) Let $T \subseteq \omega^{<\omega}$ be a uniformly e-pointed tree w.r.t.functions. Say that $ \gamma \in 2^{<\omega}$ is *consistent with $T$* if there is a $\sigma \in T$ such that $$\begin{aligned} (\forall {\langle}i, n {\rangle}< |\gamma|)(i < |\sigma| {\text{ and }}(\gamma({\langle}i,n {\rangle}) = 1 {\leftrightarrow}\sigma(i)=n)).\end{aligned}$$ Notice that if $\eta \subseteq \gamma \in 2^{<\omega}$ and $\gamma$ is consistent with $T$, then $\eta$ is also consistent with $T$. Let $$\begin{aligned} S = \{\gamma \in 2^{<\omega} : \text{$\gamma$ is consistent with $T$}\}.\end{aligned}$$ Then $S$ is a tree, $S$ has no leaves because $T$ has no leaves, and it is immediate to check that $S {\leq_\mathrm{e}}T$. To see that $T {\leq_\mathrm{e}}S$, observe that $$\begin{aligned} T = \{\sigma \in \omega^{<\omega} : (\exists \gamma \in S)(\forall i < |\sigma|)({\langle}i, \sigma(i) {\rangle}\in \operatorname{dom}(\gamma) {\text{ and }}\gamma({\langle}i, \sigma(i) {\rangle}) = 1)\}.\end{aligned}$$ Furthermore, $[S] = \{\chi_{\operatorname{graph}(f)} : f \in [T]\}$. If $f \in [T]$, then $\chi_{\operatorname{graph}(f)}\!\restriction\!n$ is consistent with $T$ for every $n$ (as witnessed by $f\!\restriction\!n$), thus $\chi_{\operatorname{graph}(f)}\!\restriction\!n \in S$ for every $n$, thus $\chi_{\operatorname{graph}(f)} \in [S]$. Conversely, suppose that $f \notin [T]$. Then there is an $n$ such that $f\!\restriction\!n \notin T$. We want to find an $m$ such that $\chi_{\operatorname{graph}(f)}\!\restriction\!m \notin S$ in order to conclude that $\chi_{\operatorname{graph}(f)} \notin [S]$. By the fact that $T$ is finitely branching, let $k$ be large enough so that $(\forall i < | \sigma|)(\sigma(i) < k)$ whenever $\sigma \in T$ has length $\leq n$. Let $m > {\langle}n, k {\rangle}$. Suppose for a contradiction that $\chi_{\operatorname{graph}(f)}\!\restriction\!m$ is consistent with $T$, and let $ \sigma$ witness this. Then it must be that $|\sigma| \geq n$ and $(\forall i < n)(\sigma(i) = f(i))$. Thus $\sigma \supseteq f\!\restriction\!n$, contradicting that $f\!\restriction\!n \notin T$. Thus $ \chi_{\operatorname{graph}(f)}\!\restriction\!m$ is not consistent with $T$, so $\chi_{\operatorname{graph}(f)}\!\restriction\!m \notin S$. To finish, we need to find an enumeration operator $\Psi$ such that $(\forall g \in [S])(\Psi(g^{+}) = S)$. So let $\Theta$ be an enumeration operator such that $ (\forall f \in [T])(\Theta(f) = T)$, and let $\Gamma$ be an enumeration operator witnessing that $S {\leq_\mathrm{e}}T$. By composing $\Gamma$ and $\Theta$, we get an enumeration operator $\Psi$ such that $(\forall f \in [T])(\Psi(f) = S)$. However, this is exactly what we want because we have shown that if $g \in [S]$, then $g = \chi_{\operatorname{graph}(f)}$ for some $f \in [T]$ and therefore that if $g \in [S]$ then $g^+ = \operatorname{graph}(f)$ for some $f \in [T]$. Thus $(\forall g \in [S]) (\Psi(g^+) = S)$, as desired (recall that $\Theta(f)=\Theta(\operatorname{graph}(f))$). A set $A$ is called *cototal* if $A {\leq_\mathrm{e}}\overline{A}$, and an e-degree is called *cototal* if it contains a cototal set [@Andrews-et-al]. Every uniformly e-pointed tree w.r.t. sets is cototal by [@Mccarthy]\*[Theorem 4.7]{}, and, by [@Mccarthy]\*[Corollary 4.9.1]{}, an e-degree is cototal if and only if it contains a uniformly e-pointed tree w.r.t. sets. \[prop-ePointedCototal\] An enumeration degree is cototal if and only if it contains a uniformly e-pointed tree w.r.t. functions. An e-degree is cototal if and only if it it contains a uniformly e-pointed tree w.r.t. sets by [@Mccarthy]\*[Theorem 4.7]{} if and only if it contains a uniformly e-pointed tree w.r.t. functions by Proposition \[prop:from-sets-to-functions\] and Proposition \[prop:link\]. We also find it interesting to give a more direct proof that every uniformly e-pointed tree w.r.t. functions has cototal enumeration degree. This can be accomplished via the easy characterization of the cototal enumeration degrees in terms of the *skip operator* from [@Andrews-et-al]. Recall that $(\Psi_e : e \in \omega)$ is a standard list of all enumeration operators, and recall the following definitions. - For an $A \subseteq \omega$, $K_A = \{{\langle}e, x {\rangle}: x \in \Psi_e(A)\}$. - For an $A \subseteq \omega$, $A^{\diamond}= \overline{K_A}$ is called the *skip* of $A$. By [@Andrews-et-al]\*[Proposition 1.1]{}, a set $A \subseteq \omega$ has cototal enumeration degree if and only if $A {\leq_\mathrm{e}}A^{\diamond}$. Let $T$ be a uniformly e-pointed tree w.r.t. functions. We show that $T {\leq_\mathrm{e}}T^{\diamond}$ and therefore that $T$ has cototal enumeration degree. Let $\Psi$ be an enumeration operator such that $(\forall f \in [T])(\Psi(f) = T)$. For each $n \in \omega$, let $T^n = \{\sigma \in T : |\sigma| = n\}$ denote level $n$ of $T$. For $b, n \in \omega$, let $b^n = \{\sigma \in \omega^{<\omega} : |\sigma| = n {\text{ and }}(\forall i < |\sigma|)(\sigma(i) < b)\}$ denote the set of all $b$-bounded strings of length $n$. Let $B = \{{\langle}n, b {\rangle}: T^n \smallsetminus b^n \neq \emptyset\}$. That is, $B$ is the set of all pairs ${\langle}n, b {\rangle}$ where $b$ is not big enough to bound every entry of every string in $T^n$. We have $B {\leq_\mathrm{e}}T$, thus $B\le_1 K_T$, and therefore $\overline B {\leq_\mathrm{e}}T^{\diamond}$. The point is that if ${\langle}n, b {\rangle}\in \overline B$, then $T^n \subseteq b^n$, which allows us enumerate $T$ from an enumeration of $\overline{T} \oplus \overline{B}$. Indeed, $$\begin{aligned} T = \{\sigma : (\exists {\langle}n, b {\rangle}\in \overline{B})(\exists L \subseteq b^n \cap \overline{T})(\forall \tau \in b^n \smallsetminus L)(\sigma \in \Psi(\tau))\}.\end{aligned}$$ That is, we know that $\sigma \in T$ when we see a bound $T^n \subseteq b^n$ and a set of strings $L \subseteq b^n$ that are *not* in $T$ such that the remaining $\tau \in b^n \smallsetminus L$ all satisfy $\sigma \in \Psi(\tau)$. Thus $T {\leq_\mathrm{e}}\overline{T} \oplus \overline{B} {\leq_\mathrm{e}}T^{\diamond}$, so $T$ has cototal enumeration degree. We extend the cototal terminology to the degrees of enumerability by saying that ${\ensuremath{\mathbf{E}}}_{A}$ is *cototal* if $A$ has cototal enumeration degree. To conclude this section, we show that cototality and compactness are equivalent properties of a degree of enumerability. \[lem-EnumTree\] Let $A \subseteq \omega$ be nonempty, and let ${\ensuremath{\mathcal{C}}} \subseteq \omega^\omega$ be closed such that ${\ensuremath{\mathcal{C}}} {\leq_{\mathrm s}}{\ensuremath{\mathcal{E}}}_A$. Then there is a tree $T \subseteq \omega^{<\omega}$ with no leaves such that $T {\leq_\mathrm{e}}A$ and $[T] \subseteq {\ensuremath{\mathcal{C}}}$. Furthermore, if ${\ensuremath{\mathcal{C}}}$ is compact, then $T$ is finitely branching. Let $\Phi$ be a Turing functional such that $\Phi({\ensuremath{\mathcal{E}}}_A) \subseteq {\ensuremath{\mathcal{C}}}$, with ${\ensuremath{\mathcal{C}}}$ closed. Let $$\begin{aligned} T = \{\sigma : \exists \alpha(\operatorname{ran}(\alpha) \subseteq A {\text{ and }}\sigma \subseteq \Phi(\alpha))\}.\end{aligned}$$ Then $T$ is a tree and $T {\leq_\mathrm{e}}A$. To see that $T$ has no leaves, let $\sigma \in T$, and let $\alpha$ be such that $\operatorname{ran}(\alpha) \subseteq A$ and $\sigma \subseteq \Phi(\alpha)$. Let $f \colon \omega {\rightarrow}\omega$ be such that $\alpha \subset f$ and $\operatorname{ran}(f) = A$. Let $\beta$ be such that $\alpha \subseteq \beta \subset f$ and $\Phi(\beta)(|\sigma|){{\downarrow}}$. Then $\sigma \subsetneq \Phi(\beta) \in T$, so $\sigma$ is not a leaf. To see that $[T] \subseteq {\ensuremath{\mathcal{C}}}$, we consider a $g \in [T]$ and show that $g$ is in the closure of ${\ensuremath{\mathcal{C}}}$. To this end, let $n \in \omega$, let $\alpha$ be such that $\operatorname{ran}(\alpha) \subseteq A$ and $g\!\restriction\!n \subseteq \Phi(\alpha)$, and let $f \colon \omega {\rightarrow}\omega$ be such that $\alpha \subset f$ and $\operatorname{ran}(f) = A$. Then $\Phi(f) \in {\ensuremath{\mathcal{C}}}$ and $\Phi(f)\!\restriction\!n = g\!\restriction\!n$. Hence $g$ is in the closure of ${\ensuremath{\mathcal{C}}}$, so $g \in {\ensuremath{\mathcal{C}}}$. Lastly, if ${\ensuremath{\mathcal{C}}}$ is compact, then $[T]$ is compact because $[T] \subseteq {\ensuremath{\mathcal{C}}}$. This means that $T$ must be finitely branching because $T$ has no leaves. \[lem-EnumCompact\] Let $A \subseteq \omega$ be nonempty. Then ${\ensuremath{\mathbf{E}}}_A$ is compact if and only if there is a uniformly e-pointed tree $T \subseteq \omega^{<\omega}$ w.r.t. functions such that $T {\equiv_\mathrm{e}}A$. Suppose that ${\ensuremath{\mathcal{E}}}_A {\equiv_{\mathrm s}}{\ensuremath{\mathcal{C}}}$, where ${\ensuremath{\mathcal{C}}}$ is compact. Let $\Phi$ be a Turing functional such that $\Phi({\ensuremath{\mathcal{C}}}) \subseteq {\ensuremath{\mathcal{E}}}_A$. ${\ensuremath{\mathcal{C}}}$ is compact, so its image ${\ensuremath{\mathcal{D}}} = \Phi({\ensuremath{\mathcal{C}}})$ is also compact by the continuity of the Turing functional $\Phi$. By Lemma \[lem-EnumTree\], there is a finitely branching tree $T \subseteq \omega^{<\omega}$ with no leaves such that $T {\leq_\mathrm{e}}A$ and $[T] \subseteq {\ensuremath{\mathcal{D}}} \subseteq {\ensuremath{\mathcal{E}}}_A$. Furthermore, $A = \bigcup_{\sigma \in T} \operatorname{ran}(\sigma)$ because $[T] \subseteq {\ensuremath{\mathcal{E}}}_A$, which implies that $A {\leq_\mathrm{e}}T$. Hence $T {\equiv_\mathrm{e}}A$. Also, if $g \in [T]$, then $\operatorname{ran}(g)=A$, thus there is a uniform procedure enumerating $A$ and hence $T$ from any enumeration of $g$, which shows that $T$ is uniformly e-pointed w.r.t. functions. Conversely, suppose that there is a uniformly e-pointed tree $T \subseteq \omega^{<\omega}$ w.r.t. functions such that $T {\equiv_\mathrm{e}}A$. Then ${\ensuremath{\mathcal{E}}}_A {\leq_{\mathrm s}}[T]$, as one can uniformly transform any function $g \in [T]$ into a function that enumerates $A$ because $T{\leq_\mathrm{e}}g$ uniformly and $A{\leq_\mathrm{e}}T$. To see that $[T] {\leq_{\mathrm s}}{\ensuremath{\mathcal{E}}}_A$, consider the Turing functional $\Phi$ which, on an $f \in {\ensuremath{\mathcal{E}}}_A$, uses $\operatorname{ran}(f)$ to simultaneously enumerate $T$ (via the reduction $T {\leq_\mathrm{e}}A$) and a path through $T$ (which is possible because $T$ has no leaves). Thus ${\ensuremath{\mathcal{E}}}_A {\equiv_{\mathrm s}}[T]$, and $[T]$ is compact because $T$ is finitely branching. Observe that the proof of Lemma \[lem-EnumCompact\] also proves the following fact, which we record for posterity. Let $A \subseteq \omega$ be nonempty. If $T \subseteq \omega^{<\omega}$ is a finitely branching tree with no leaves such that $T {\leq_\mathrm{e}}A$ and $[T] \subseteq {\ensuremath{\mathcal{E}}}_A$, then $T {\equiv_\mathrm{e}}A$, $[T] {\equiv_{\mathrm s}}{\ensuremath{\mathcal{E}}}_A$, and $T$ is uniformly e-pointed w.r.t. functions. \[thm:cototal=compact\] ${\ensuremath{\mathbf{E}}}_A$ is a compact degree of enumerability if and only if $A$ has cototal enumeration degree. Hence a degree of enumerability is compact if and only if it is cototal. The degree of enumerability ${\ensuremath{\mathbf{E}}}_A$ is compact if and only if $A {\equiv_\mathrm{e}}T$ for some uniformly e-pointed tree $T \subseteq \omega^{<\omega}$ w.r.t.functions by Lemma \[lem-EnumCompact\], which is the case if and only if $A$ has cototal enumeration degree by Proposition \[prop-ePointedCototal\]. A quasiminimal degree of enumerability that is compact. {#a-quasiminimal-degree-of-enumerability-that-is-compact. .unnumbered} ------------------------------------------------------- The existence of quasiminimal problems of enumerability that are equivalent to compact mass problems is a consequence of Theorem \[thm:cototal=compact\] and the fact that there are cototal quasiminimal e-degrees [@Andrews-et-al]. We think, however, that it is instructive to directly construct a quasiminimal uniformly e-pointed tree w.r.t. functions. The corresponding degree of enumerability is then quasiminimal by definition and compact by Lemma \[lem-EnumCompact\]. Recall that ${\langle}\cdot, \cdot {\rangle}\colon \omega^2 {\rightarrow}\omega$ is the recursive pairing function. Let $\pi_0, \pi_1 \colon \omega {\rightarrow}\omega$ denote the projection functions $\pi_0({\langle}m, n {\rangle}) = m$ and $\pi_1({\langle}m, n {\rangle}) = n$. \[lem-SelfEnumTree\] There is a finitely branching tree $A \subseteq \omega^{<\omega}$ such that - $A$ has no leaves, - $A$ is quasiminimal, and - $\operatorname{ran}(\pi_1 \circ f) = A$ for every $f \in [A]$. Notice that such a tree is uniformly e-pointed w.r.t. functions. For the purposes of this proof, we make the following definitions for finite trees $T, S \subseteq \omega^{<\omega}$: - $\operatorname{leaves}(T) = \{\sigma \in T : \text{$\sigma$ is a leaf of $T$}\}$. - $S$ *leaf-extends* $T$ if $T \subseteq S$ and $(\forall \tau \in S \smallsetminus T)(\exists \sigma \in \operatorname{leaves}(T))(\sigma \subseteq \tau)$; - $S$ *properly leaf-extends* $T$ if $S$ leaf-extends $T$ and $(\forall \sigma \in T)(\exists \tau \in S)(\sigma \subsetneq \tau)$. We build a sequence of finite trees $A_0 \subseteq A_1 \subseteq A_2 \subseteq \dots$, where $A_{s+1}$ properly leaf-extends $A_s$ for each $s \in \omega$. This way, $A = \bigcup_{s \in \omega} A_s$ has no leaves and is finitely branching. Furthermore, we build the sequence so that $$\begin{aligned} (\forall s \in \omega)(\forall \sigma \in \operatorname{leaves}(A_{s+1}))(A_s \subseteq \operatorname{ran}(\pi_1 \circ \sigma) \subseteq A_{s+1}).\end{aligned}$$ This ensures that $\operatorname{ran}(\pi_1 \circ f) = A$ for every $f \in [A]$. To help ensure that $A {>_\mathrm{e}}\emptyset$, we also maintain a sequence of finite sets of strings $O_0 \subseteq O_1 \subseteq O_2 \subseteq \dots$ such that $\forall s(A_s \cap O_s = \emptyset)$. We satisfy the requirements $$\begin{aligned} {\ensuremath{\mathcal{Q}}}_e\colon& A \neq W_e\\ {\ensuremath{\mathcal{R}}}_e\colon& \text{if $\Psi_e(A)$ is the graph of a total function $f$, then $f$ is recursive.}\end{aligned}$$ *Stage $0$*: set $A_0 = \{\emptyset\}$, and set $O_0 = \emptyset$. *Stage $s+1 = 2e+1$*: We satisfy ${\ensuremath{\mathcal{Q}}}_e$. If $W_e$ is finite, then set $O_{s+1} = O_s$. If $W_e$ is infinite, then choose any $\sigma \in W_e \smallsetminus A_s$, and set $O_{s+1} = O_s \cup \{\sigma\}$. To extend $A_s$ to $A_{s+1}$, first choose $n$ greater than (the code of) every element in $O_{s+1}$. Then choose any enumeration $(\alpha_i)_{i < k}$ of $A_s$. Then let $\beta$ be the string ${\langle}{\langle}n, \alpha_0 {\rangle}, {\langle}n, \alpha_1 {\rangle}, \dots, {\langle}n, \alpha_{k-1} {\rangle}{\rangle}$. Now let $A_{s+1}$ be the tree obtained by extending each leaf of $A_s$ by $\beta$: $$\begin{aligned} A_{s+1} = \{\sigma : (\exists \tau \in \operatorname{leaves}(A_s))(\sigma \subseteq \tau^{\smallfrown}\beta)\}.\end{aligned}$$ Having chosen $n$ big enough, we have guaranteed that $A_{s+1}$ is disjoint from $O_{s+1}$. *Stage $s+1 = 2e+2$*: We satisfy ${\ensuremath{\mathcal{R}}}_e$. Set $O_{s+1} = O_s$. For finite trees $T, S \subseteq \omega^{<\omega}$, call $S$ a *good* extension of $T$ if $S$ leaf-extends $T$, $S \cap O_{s+1} = \emptyset$, and $(\forall \sigma \in S)(\operatorname{ran}(\pi_1 \circ \sigma) \subseteq S)$. Ask if there is a good extension $R$ of $A_s$ such that $$\begin{aligned} (\exists m, n, o)(n \neq o {\text{ and }}{\langle}m, n {\rangle}\in \Psi_e(R) {\text{ and }}{\langle}m, o {\rangle}\in \Psi_e(R)).\end{aligned}$$ If there is such an $R$, let $\widehat{A}_s = R$. Otherwise, let $\widehat{A}_s = A_s$. Now extend $\widehat{A}_s$ to $A_{s+1}$ the same way that we extend $A_s$ to $A_{s+1}$ during the odd stages. The fact that $(\forall \sigma \in \widehat{A}_s)(\operatorname{ran}(\pi_1 \circ \sigma) \subseteq \widehat{A}_s)$ ensures that $(\forall \sigma \in A_{s+1})(\operatorname{ran}(\pi_1 \circ \sigma) \subseteq A_{s+1})$. This completes the construction. Let $A = \bigcup_{s \in \omega} A_s$. We show that all requirements are satisfied. For requirement ${\ensuremath{\mathcal{Q}}}_e$, consider stage $s+1 = 2e+1$. If $W_e$ is finite, then $A \neq W_e$ because $A$ is infinite. If $W_e$ is infinite, then at stage $s+1$ we chose a $\sigma \in W_e \smallsetminus A_s$ and put $\sigma$ in $O_{s+1}$. Thus $\forall t (\sigma \notin A_t)$, so $\sigma \notin A$. Hence $A \neq W_e$. For requirement ${\ensuremath{\mathcal{R}}}_e$, suppose that $\Psi_e(A)$ is the graph of a total function $f$, and consider stage $s+1 = 2e+2$. We show that $\operatorname{graph}(f)$ is r.e., which implies that $f$ is recursive. Let $$\begin{aligned} X = \{{\langle}m, n {\rangle}: \text{there is a good extension $B$ of $A_s$ with ${\langle}m, n {\rangle}\in \Psi_e(B)$}\}\end{aligned}$$ (where here ‘good’ means with respect to the $O_{s+1}$ at stage $s+1$). Clearly $X$ is r.e. We show that $X = \operatorname{graph}(f)$. For $\operatorname{graph}(f) \subseteq X$, suppose that ${\langle}m, n {\rangle}\in \operatorname{graph}(f) = \Psi_e(A)$. Let $t \geq s+1$ be such that ${\langle}m, n {\rangle}\in \Psi_e(A_t)$. Then $A_t$ is a good extension of $A_s$ with ${\langle}m, n {\rangle}\in \Psi_e(A_t)$, so ${\langle}m, n {\rangle}\in X$. Conversely, suppose that ${\langle}m, n {\rangle}\in X$, and let $B$ be a good extension of $A_s$ with ${\langle}m, n {\rangle}\in \Psi_e(B)$. If ${\langle}m, n {\rangle}\notin \operatorname{graph}(f)$, then ${\langle}m, o {\rangle}\in \operatorname{graph}(f) = \Phi_e(A)$, where $o = f(m) \neq n$. Let $t \geq s+1$ be such that ${\langle}m, o {\rangle}\in \Psi_e(A_t)$. Then $A_t$ is a good extension of $A_s$, and, moreover, $A_t \cup B$ is also a good extension of $A_s$. Thus there is a good extension $R = A_t \cup B$ of $A_s$ such that $n \neq o {\text{ and }}{\langle}m, n {\rangle}\in \Psi_e(R) {\text{ and }}{\langle}m, o {\rangle}\in \Psi_e(R)$, for some $m, n, o \in \omega$. Therefore, at stage $s+1$, we extended $A_s$ to an $A_{s+1}$ such that $$\begin{aligned} (\exists m, n, o)(n \neq o {\text{ and }}{\langle}m, n {\rangle}\in \Psi_e(A_{s+1}) {\text{ and }}{\langle}m, o {\rangle}\in \Psi_e(A_{s+1})).\end{aligned}$$ This contradicts that $\Psi_e(A)$ is the graph of a function. All together, we have that $A$ has no leaves, that $\operatorname{ran}(\pi_1 \circ f) = A$ for every $f \in [A]$ by construction, and that $A$ is quasiminimal by the ${\ensuremath{\mathcal{Q}}}_e$ requirements and the ${\ensuremath{\mathcal{R}}}_e$ requirements. \[thm:qm-compact\] There is a degree of enumerability ${\ensuremath{\mathbf{E}}}_A$ that is both quasiminimal and compact. Hence ${\ensuremath{\mathbf{E}}}_A$ is closed, nonzero, and does not bound any nonzero degree of solvability. Let $A$ be the tree from Lemma \[lem-SelfEnumTree\]. Then $A$ has quasiminimal e-degree, so ${\ensuremath{\mathbf{E}}}_A$ is quasiminimal by definition. Furthermore, $A$ is uniformly e-pointed w.r.t. functions, so ${\ensuremath{\mathbf{E}}}_A$ is compact by Lemma \[lem-EnumCompact\]. In Lemma \[lem-SelfEnumTree\], one can make the tree $A$ be not cototal by a small modification to the proof. Thus although every uniformly e-pointed tree w.r.t. functions has cototal *e-degree* by Proposition \[prop-ePointedCototal\], it is not the case that every such tree is cototal as a set. To modify the proof, replace each old ${\ensuremath{\mathcal{Q}}}_e$ requirement $A \neq W_e$ with the new requirement $A \neq \Psi_{e}(\overline{A})$. (Notice that a set $A$ satisfying all of the new ${\ensuremath{\mathcal{Q}}}_e$ requirements is still not r.e., which is required in order for a set $A$ to be quasiminimal.) To satisfy the new ${\ensuremath{\mathcal{Q}}}_e$, modify stage $s+1 = 2e+1$ as follows. If there are a finite set $D \subseteq \overline{A_s}$ and a string $\sigma \in \overline{A_s}$ with $\sigma \in \Psi_e(D)$, then choose such a $D$ and $\sigma$, and set $O_{s+1} = O_s \cup D \cup \{\sigma\}$. Otherwise simply set $O_{s+1} = O_s$. Then choose $n$ greater than (the code of) every element in $O_{s+1}$, and extend $A_s$ to $A_{s+1}$ as before. To verify that ${\ensuremath{\mathcal{Q}}}_e$ is satisfied, suppose for a contradiction that $\Psi_e(\overline{A}) = A$. As $A$ is infinite and $A_s$ is finite, fix some $ \sigma \in A \setminus A_s$. Let $D \subseteq \overline{A} \subseteq \overline{A_s}$ be a finite set such that $\sigma \in \Phi_e(D)$. Then, at stage $s+1$, we were able to choose a $D$ and $\sigma$, ensuring that $\Phi_e(\overline{A}) \neq A$. This is a contradiction. A degree of enumerability that does not bound any nonzero closed degree {#a-degree-of-enumerability-that-does-not-bound-any-nonzero-closed-degree .unnumbered} ----------------------------------------------------------------------- Finally, we show that there are examples of nonzero degrees of enumerability that do not bound nonzero closed degrees. Such examples are of course quasiminimal, and indeed the property of being nonzero but not above any nonzero closed degree can be viewed as an interesting generalization of quasiminimality. Theorem \[thm:no-bounding\] below can also be phrased by saying that there are nonzero degrees of enumerability that do not lie in the filter generated by the nonzero closed degrees, which coincides with the collection of all Medvedev degrees bounding nonzero closed degrees (see [@Sorbi-filters]). \[lem-Prob9Helper\] There is a set $A {>_\mathrm{e}}\emptyset$ such that, for all $T {\leq_\mathrm{e}}A$, if $T$ is a subtree of $\omega^{<\omega}$ with no leaves, then $T$ has an r.e. subtree with no leaves. For the purposes of this proof, we assume that if $\Psi$ is an enumeration operator, $X \subseteq \omega$, and $\Psi(X)$ enumerates some $\sigma \in \omega^{<\omega}$ (i.e., $\sigma \in \Psi(X)$), then it also enumerates all $\tau \subseteq \sigma$. In fact, from any enumeration operator $\Gamma$, one can effectively produce an enumeration operator $\Psi$ such that, for all $X$, - $\Psi(X)$ is a tree, and - if $\Gamma(X)$ is a tree, then $\Psi(X) = \Gamma(X)$. To accomplish this, just take $\Psi=\{{\langle}\tau, D {\rangle}: (\exists \sigma)(\tau \subseteq \sigma {\text{ and }}{\langle}\sigma , D {\rangle}\in \Gamma)\}$. Therefore, we can define an effective list $(\Psi_e : e \in \omega)$ of enumeration operators such that - $\Psi_e(X)$ is a tree for every $e$ and $X$, and - if $T {\leq_\mathrm{e}}X$ for a tree $T$ and set a $X$, then there is an $e$ such that $\Psi_e(X) = T$. Also, recall the notation $g^+ = \{n : g(n) = 1\}$ from Definition \[def-plus\]. We extend this notation to strings $\alpha \in 2^{< \omega}$ by defining $\alpha^+ = \{i < |\alpha| : \alpha(i)=1\}$. Additionally, if $A \subseteq \omega$ and $\alpha \in 2^{<\omega}$, we write $A \subseteq^+ \alpha^+$ to mean that $(\forall n < |\alpha|)(n \in A {\rightarrow}\alpha(n) = 1)$ (i.e., $\{n \in A: n < |\alpha|\} \subseteq \alpha^+$). We satisfy the requirements $$\begin{aligned} {\ensuremath{\mathcal{Q}}}_e\colon& A \neq W_e\\ {\ensuremath{\mathcal{R}}}_e\colon& \text{either $\Psi_e(A)$ contains a leaf or there is an r.e.\ $T \subseteq \Psi_e(A)$ with no leaves.}\end{aligned}$$ We build a sequence of binary strings $\alpha_0 \subseteq \alpha_1 \subseteq \alpha_2 \subseteq \dots$ along with sequences of recursive sets $I_0 \supseteq I_1 \supseteq I_2 \supseteq \dots$ and $J_0 \subseteq J_1 \subseteq J_2 \subseteq \dots$ such that, for every $s \in \omega$, $I_s \smallsetminus J_s$ is infinite and $J_s \subseteq^+ \alpha_s^+ \subseteq I_s$. In the end, we let $A = \bigcup_{s \in \omega}\alpha_s^+$, and we have $\bigcup_{s \in \omega} J_s \subseteq A \subseteq \bigcap_{s \in \omega} I_s$. *Stage $0$*: Set $\alpha_0 = \emptyset$, set $I_0 = \omega$, and set $J_0 = \emptyset$. *Stage $s+1 = 2e+1$*: We satisfy ${\ensuremath{\mathcal{Q}}}_e$. Let $n \in I_s \smallsetminus J_s$ be least such that $n > |\alpha_s|$. If $n \in W_e$, set $I_{s+1} = I_s \smallsetminus \{n\}$, set $J_{s+1} = J_s$, and extend $\alpha_s$ to an $\alpha_{s+1}$ with $J_s \subseteq^+ \alpha_{s+1}^+ \subseteq I_s$ and $\alpha_{s+1}(n) = 0$. If $n \notin W_e$, set $I_{s+1} = I_s$, set $J_{s+1} = J_s$, and extend $\alpha_s$ to an $\alpha_{s+1}$ with $J_s \subseteq^+ \alpha_{s+1}^+ \subseteq I_s$ and $\alpha_{s+1}(n) = 1$. *Stage $s+1 = 2e+2$*: We satisfy ${\ensuremath{\mathcal{R}}}_e$. Ask if there is a $\beta \supseteq \alpha_s$ and a recursive set $R$ such that - $J_s \subseteq^+ \beta^+ \subseteq R \subseteq I_s$, - $R \smallsetminus J_s$ is infinite, and - there is a $\sigma \in \Psi_e(\beta^+)$ that is a leaf in $\Psi_e(R)$. If there are such $\beta$ and $R$, set $\alpha_{s+1} = \beta$, set $I_{s+1} = R$, and set $J_{s+1} = J_s$. If there are no such $\beta$ and $R$, then set $\alpha_{s+1} = \alpha_s$, set $I_{s+1} = I_s$, and choose any recursive $J_{s+1}$ whose characteristic function extends $\alpha_{s+1}$, $J_s \subseteq J_{s+1} \subseteq I_{s+1}$, and $J_{s+1} \smallsetminus J_s$ and $I_{s+1} \smallsetminus J_{s+1}$ are both infinite. This completes the construction. Let $A = \bigcup_s \alpha_s^+$. The ${\ensuremath{\mathcal{Q}}}_e$ requirements are clearly satisfied, and together they ensure that $A$ is not r.e. Hence $A {>_\mathrm{e}}\emptyset$. Now suppose that $T {\leq_\mathrm{e}}A$ is a tree with no leaves, and let $\Psi_e$ be such that $T = \Psi_e(A)$. At stage $s+1 = 2e+2$, there must not have been a $\beta$ and an $R$ because if there were, then we would have $\beta = \alpha_{s+1}$ and $\beta^+ \subseteq A \subseteq R = I_{s+1}$, so there would be a leaf $\sigma \in \Psi_e(A) = T$. It must therefore be that $\Psi_e(J_{s+1})$ is a tree with no leaves. To see this, suppose instead that $\Psi_e(J_{s+1})$ has a leaf $\sigma$. Let $\beta$ be such that $\alpha_{s+1} \subseteq \beta$, $\beta^+ \subseteq J_{s+1}$, and $\sigma \in \Psi_e(\beta^+)$. Then at stage $s+1$, we could have taken $\beta$ and $R = J_{s+1}$, which is a contradiction. This finishes the proof because $\Psi_e(J_{s+1}) \subseteq T$ since $J_{s+1} \subseteq A$, and $\Psi_e(J_{s+1})$ is r.e. since $J_{s+1}$ is recursive. \[thm:no-bounding\] There is a nonzero degree of enumerability that does not bound a nonzero closed degree. Let $A$ be as in Lemma \[lem-Prob9Helper\]. Consider a closed ${\ensuremath{\mathcal{C}}} {\leq_{\mathrm s}}{\ensuremath{\mathcal{E}}}_A$. By Lemma \[lem-EnumTree\], there is a tree $T {\leq_\mathrm{e}}A$ with no leaves such that $[T] \subseteq {\ensuremath{\mathcal{C}}}$. By Lemma \[lem-Prob9Helper\], $T$ has an r.e. subtree $S$ with no leaves. Thus $[S] \subseteq [T] \subseteq {\ensuremath{\mathcal{C}}}$. However, being a tree with no leaf, $S$ has a recursive path, so ${\ensuremath{\mathcal{C}}}$ has a recursive member, so ${\operatorname{deg}_{\mathrm s}}({\ensuremath{\mathcal{C}}}) = {\ensuremath{\mathbf{0}}}$. Acknowledgements {#acknowledgements .unnumbered} ================ We thank Douglas Cenzer, Antonio Montalbán, Alexandra A. Soskova, and Mariya I. Soskova for helpful comments and discussions. [^1]: The authors are thankful to Alexandra A.Soskova and Mariya I. Soskova for bringing to their attention, after a first draft of this paper was completed, the notion of uniformly e-pointed tree w.r.t. sets, called simply *uniformly e-pointed* by Montalbán and McCarthy.

--- abstract: 'We study $3$-coloring properties of triangle-free planar graphs $G$ with two precolored $4$-cycles $C_1$ and $C_2$ that are far apart. We prove that either every precoloring of $C_1\cup C_2$ extends to a $3$-coloring of $G$, or $G$ contains one of two special substructures which uniquely determine which $3$-colorings of $C_1\cup C_2$ extend. As a corollary, we prove that there exists a constant $D>0$ such that if $H$ is a planar triangle-free graph and $S\subseteq V(H)$ consists of vertices at pairwise distances at least $D$, then every precoloring of $S$ extends to a $3$-coloring of $H$. This gives a positive answer to a conjecture of Dvořák, Král’ and Thomas, and implies an exponential lower bound on the number of $3$-colorings of triangle-free planar graphs of bounded maximum degree.' author: - 'Zdeněk Dvořák[^1]' - 'Bernard Lidický[^2]' bibliography: - 'cylgen.bib' title: 'Fine structure of $4$-critical triangle-free graphs II. Planar triangle-free graphs with two precolored $4$-cycles' --- Introduction ============ The interest in the $3$-coloring properties of planar graphs was started by a celebrated theorem of Grötzsch [@grotzsch1959], who proved that every planar triangle-free graph is $3$-colorable. This result was later generalized and strengthened in many different ways. The one relevant to the topic of this paper concerns graphs embedded in surfaces. While the direct analogue of Grötzsch’s theorem is false for any surface other than the sphere, $3$-colorability of triangle-free graphs embedded in a fixed surface is nowadays quite well understood. A first step in this direction was taken by Thomassen [@thom-torus], who proved that every graph of girth at least five embedded in the projective plane or torus is $3$-colorable. Thomas and Walls [@tw-klein] extended this result to graphs embedded in the Klein bottle. More generally, Thomassen [@thomassen-surf] proved that for any fixed surface $\Sigma$, there are only finitely many $4$-critical graphs of girth at least $5$ that can be drawn in $\Sigma$ (a graph $G$ is *$4$-critical* if it is not $3$-colorable, but every proper subgraph of $G$ is $3$-colorable). In other words, there exists a constant $c_\Sigma$ such that if a graph of girth at least $5$ drawn in $\Sigma$ is not $3$-colorable, then it contains a subgraph with at most $c_\Sigma$ vertices that is not $3$-colorable. Thomassen’s bound on $c_\Sigma$ is double exponential in the genus of $\Sigma$. This was improved by Dvořák, Král and Thomas [@trfree3], who gave a bound on $c_\Sigma$ linear in the genus of $\Sigma$. In follow-up papers [@trfree4; @trfree6], they also described structure of $4$-critical triangle-free graphs on surfaces. Essentially, they show that such a graph can be cut into a bounded number of pieces, each of them satisfying one of the following: every $3$-coloring of the boundary of the piece $H$ extends to a $3$-coloring of $H$; or a $3$-coloring of the boundary of $H$ extends to a $3$-coloring of $H$ if and only if it satisfies a specific condition (“winding number constraint”); or the piece is homeomorphic to the cylinder and either all the faces of the piece have length $4$, or both boundary cycles of the piece have length $4$ (the *cylinder* is the sphere with two holes). While the first two possibilities for the pieces of the structure give all the information about their $3$-colorings, in the last subcase the information is much more limited. The main aim of this series of papers is to fix this shortcoming. Let us remark that we cannot eliminate this subcase entirely—based on the construction of Thomas and Walls [@tw-klein], one can find $4$-critical triangle-free graphs $G_1$, $G_2$, …drawn in a fixed surface $\Sigma$, such that for any $i\ge 1$, $G_i$ contains two non-contractible $4$-faces bounding a subset $\Pi$ of $\Sigma$ homeomorphic to the cylinder, such that at least $i$ $5$-faces of $G$ are contained in $\Pi$. Such a class of graphs cannot be described using only the first two subcases of the structure theorem. Hence, the main problem studied in this paper concerns describing subgraphs of $4$-critical triangle-free graphs embedded in the cylinder with rings of length $4$. The exact description of such subgraphs under the additional assumption that all $(\le\!4)$-cycles are non-contractible was given by Dvořák and Lidický [@dvolid]. Even this special case is rather involved (in addition to one infinite class of such graphs mentioned in the previous paragraph, there are more than 40 exceptional graphs) and extending it to the general triangle-free case would be difficult. However, in the applications it is mostly sufficient to deal with the case that the boundary $4$-cycles of the cylinder are far apart, and this is the case considered in this paper. With this restriction, it turns out that there are only two infinite classes of critical graphs. Before stating the result precisely, let us mention one application of this characterization. \[lemma-fourext\] There exists a constant $D\ge 0$ with the following property. Let $G$ be a plane triangle-free graph, let $C$ be a $4$-cycle bounding a face of $G$ and let $v$ be a vertex of $G$. Let $\psi$ be a $3$-coloring of $C+v$. If the distance between $C$ and $v$ is at least $D$, then $\psi$ extends to a $3$-coloring of $G$. This confirms Conjecture 1.5 of Dvořák et al. [@trfree5], who also proved that this implies the following more interesting result (stated as Conjecture 1.4 in [@trfree5]). \[cor-farsv\] There exists a constant $D\ge 0$ with the following property. Let $G$ be a planar triangle-free graph, let $S$ be a set of vertices of $G$ and let $\psi:S\to \{1,2,3\}$ be an arbitrary function. If the distance between every two vertices of $S$ is at least $D$, then $\psi$ extends to a $3$-coloring of $G$. Thomassen [@thom-many] conjectured that all triangle-free planar graphs have exponentially many 3-colorings. If $G$ is an $n$-vertex graph of maximum degree at most $\Delta$, then there exists a set $S\subseteq V(G)$ of size at least $n/\Delta^D$ such that the distance between every two vertices of $S$ is at least $D$. Hence, Corollary \[cor-farsv\] implies that this conjecture holds for triangle-free planar graphs of bounded maximum degree. \[cor-many\] Let $D$ be the constant of Corollary \[cor-farsv\]. If $G$ is an $n$-vertex planar triangle-free graph of maximum degree at most $\Delta$, then $G$ has at least $\left(3^{1/\Delta^D}\right)^n$ distinct $3$-colorings. Let us now introduce a few definitions. The *disk* is the sphere with a hole. In this paper, we generally consider graphs embedded in the sphere, the disk, or the cylinder. Furthermore, we always assume that each face of the embedding contains at most one hole, and if a face contains a hole, then the face is bounded by a cycle, which we call a *ring*. Note that the rings do not have to be disjoint. Let $C$ be the union of the rings of such a graph $G$ ($C$ is empty when $G$ is embedded in the sphere without holes). We say that $G$ is *critical* if $G\neq C$ and for every proper subgraph $G'$ of $G$ such that $C\subseteq G'$, there exists a $3$-coloring of $C$ that extends to a $3$-coloring of $G'$, but not to a $3$-coloring of $G$; that is, removing any edge or vertex not belonging to $C$ affects the set of precolorings of $C$ that extend to a $3$-coloring of the graph. We construct a sequence of graphs $T_1$, $T_2$, …, which we call *Thomas-Walls graphs* (Thomas and Walls [@tw-klein] proved that they are exactly the $4$-critical graphs that can be drawn in the Klein bottle without contractible cycles of length at most $4$). Let $T_1$ be equal to $K_4$. For $n\ge 1$, let $uv$ be any edge of $T_n$ that belongs to two triangles and let $T_{n+1}$ be obtained from $T_n-uv$ by adding vertices $x$, $y$ and $z$ and edges $ux$, $xy$, $xz$, $vy$, $vz$ and $yz$. The first few graphs of this sequence are drawn in Figure \[fig-thomaswalls\]. ![Some Thomas-Walls graphs.[]{data-label="fig-thomaswalls"}](fig-thomaswalls) For $n\ge 2$, note that $T_n$ contains unique $4$-cycles $C_1=u_1u_2u_3u_4$ and $C_2=v_1v_2v_3v_4$ such that $u_1u_3,v_1v_3\in E(G)$. Let $T'_n=T_n-\{u_1u_3,v_1v_3\}$. We also define $T'_1$ to be a $4$-cycle. We call the graphs $T'_1$, $T'_2$, …*reduced Thomas-Walls graphs*, and we say that $u_1u_3$ and $v_1v_3$ are their *interface pairs*. Note that $T'_n$ has an embedding in the cylinder with rings $C_1$ and $C_2$. A *patch* is a graph drawn in the disk with ring $C$ of length $6$, such that $C$ has no chords and every face of the patch other than the one bounded by $C$ has length $4$. Let $G$ be a graph embedded in the sphere, possibly with holes. Let $G'$ be any graph which can be obtained from $G$ as follows. Let $S$ be an independent set in $G$ such that every vertex of $S$ has degree $3$. For each vertex $v\in S$ with neighbors $x$, $y$ and $z$, remove $v$, add new vertices $a$, $b$ and $c$ and a $6$-cycle $C=xaybzc$, and draw any patch with ring $C$ in the disk bounded by $C$. We say that any such graph $G'$ is obtained from $G$ by *patching*. This operation was introduced by Borodin et al. [@4c4t] in the context of describing planar $4$-critical graphs with exactly $4$ triangles. Consider a reduced Thomas-Walls graph $G=T'_n$ for some $n\ge 1$, with interface pairs $u_1u_3$ and $v_1v_3$. A *patched Thomas-Walls graph* is any graph obtained from such a graph $G$ by patching, and $u_1u_3$ and $v_1v_3$ are its interface pairs (note that $u_1$, $u_3$, $v_1$, and $v_3$ have degree two in $G$, and thus they are not affected by patching). Let $G$ be a graph embedded in the cylinder, with the rings $C_i=x_iy_iz_iw_i$ of length $4$, for $i=1,2$. Let $y'_i$ be either a new vertex or $y_i$, and let $w'_i$ be either a new vertex or $w_i$. Let $G'$ be obtained from $G$ by adding $4$-cycles $x_iy'_iz_iw'_i$ forming the new rings. We say that $G'$ is obtained by *framing on pairs $x_1z_1$ and $x_2z_2$*. Let $G$ be a graph embedded in the cylinder with rings $C_1$ and $C_2$ of length $3$, such that every other face of $G$ has length $4$. We say that such a graph $G$ is a *$3,3$-quadrangulation*. Let $G'$ be obtained from $G$ by subdividing at most one edge in each of $C_1$ and $C_2$. We say that such a graph $G'$ is a *near $3,3$-quadrangulation*. We say that a graph $G$ embedded in the cylinder is *tame* if $G$ contains no contractible triangles, and all triangles of $G$ are pairwise vertex-disjoint. The main result of this paper is the following. \[thm-mfar\] There exists a constant $D\ge 0$ such that the following holds. Let $G$ be a tame graph embedded in the cylinder with rings $C_1$ and $C_2$ of length at most $4$. If the distance between $C_1$ and $C_2$ is at least $D$ and $G$ is critical, then either - $G$ is obtained from a patched Thomas-Walls graph by framing on its interface pairs, or - $G$ is a near $3,3$-quadrangulation. This clearly implies the previously mentioned result on precolored $4$-cycle and vertex. Suppose for a contradiction that there exists a precoloring $\psi$ of $C+v$ that does not extend to a $3$-coloring of $G$. Let $G'$ be obtained from $G$ by adding three new vertices $u_1$, $u_2$ and $u_3$ and edges $vu_1$, $u_1u_2$, $u_2u_3$ and $u_3v$, with the $4$-cycle $C'=vu_1u_2u_3$ drawn so that it forms a face. Drill a hole inside $C'$, turning it into a ring. Let $\psi'$ be any $3$-coloring of $C\cup C'$ that extends $\psi$, and note that $\psi'$ does not extend to a $3$-coloring of $G'$. Hence, $G'$ has a critical subgraph $G''$. By Theorem \[thm-many\], either $G''$ is obtained from a patched Thomas-Walls graph by framing on its interface pairs, or $G''$ is a near $3,3$-quadrangulation. However, this is a contradiction, since in either of the cases, each ring contains at most two vertices of degree two, while $C'$ contains three vertices of degree two. Let us remark that which $3$-colorings of the rings of a framed patched Thomas-Walls graph or a near $3,3$-quadrangulation extend to a $3$-coloring of the whole graph was determined in the first paper of this series (Dvořák and Lidický [@cylgen-part1], Lemmas 2.5 and 2.9). In Section \[sec-cutting\], we prove a lemma based on an idea of [@trfree5] that allows us to introduce new non-contractible $(\le\!4)$-cycles. This enables us to apply the result of the first paper of the series [@cylgen-part1], where we considered graphs in cylinder containing many non-contractible $(\le\!4)$-cycles, and thus prove Theorem \[thm-mfar\] (Section \[sec-mfar\]). Cutting a cylinder {#sec-cutting} ================== Let us start by introducing several previous results. The following is a special case of the main result of Dvořák et al. [@trfree4]. \[thm-critsize\] For every integer $k\ge 0$, there exists a constant $\beta$ with the following property. Let $G$ be a graph embedded either in the disk or the cylinder such that the sum of the lengths of the rings is at most $k$. Suppose that $G$ contains no contractible triangles, and that every non-contractible $(\le\!4)$-cycle in $G$ is equal to one of the rings. Let $F$ be the set of faces of $G$. If $G$ is critical, then $$\begin{aligned} \sum_{f\in F} (|f|-4)\le \beta. \label{eq:beta}\end{aligned}$$ Notice that gives an upper bound $\beta$ on the number of $(\ge\!5)$-faces as well as upper bound $5\beta$ on number of vertices incident to $(\ge\!5)$-faces in $G$. Furthermore, we need a result of Gimbel and Thomassen [@gimbel], which essentially states that in a plane triangle-free graph, every precoloring of a facial cycle $C$ of length at most $6$ extends to a $3$-coloring, unless $|C|=6$ and $G$ contains a subgraph such that every face other than the one bounded by $C$ has length $4$. \[thm-6cyc\] Let $G$ be a triangle-free graph drawn in the disk with the ring $C$ of length at most $6$. If $G$ is critical, then $|C|=6$ and all other faces of $G$ have length $4$. Furthermore, a precoloring $\psi$ of $C=v_1v_2v_3v_4v_5v_6$ does not extend to a $3$-coloring of $G$ if and only if either - $C$ has a chord $v_iv_{i+3}$ for some $i\in\{1,2,3\}$ and $\psi(v_i)=\psi(v_{i+3})$, or - $C$ is an induced cycle and $\psi(v_1)=\psi(v_4)$, $\psi(v_2)=\psi(v_5)$, and $\psi(v_3)=\psi(v_6)$. Let us give an observation about critical graphs that is often useful. \[lemma-fr\] Let $G$ be a graph drawn in the sphere with holes so that every triangle is non-contractible. If $G$ is critical, then - every vertex $v\in V(G)$ that does not belong to the rings has degree at least three, - every contractible $(\le\!5)$-cycle in $G$ bounds a face, and - if $K$ is a closed walk in $G$ forming the boundary of an open disk $\Lambda$, then either $\Lambda$ is a face of $G$, or all faces of $G$ contained in $\Lambda$ have length $4$. Let $C$ be the union of the rings of $G$, and consider any vertex $v\in V(G)\setminus V(C)$. Suppose for a contradiction that $v$ has degree at most $2$. Let $\psi$ be any $3$-coloring of $C$ that extends to a $3$-coloring $\varphi$ of $G-v$. Then $\psi$ also extends to a $3$-coloring of $G$, by giving the vertices of $V(G)\setminus\{v\}$ the same color as in the coloring $\varphi$ and by choosing a color of $v$ distinct from the colors of its neighbors. This contradicts the assumption that $G$ is critical. The other two conclusions of the lemma follow by a similar argument using Theorem \[thm-6cyc\]. Let $G_1$ and $G_2$ be two graphs embedded in the cylinder, with the same rings. We say that $G_1$ *dominates* $G_2$ if every precoloring $\psi$ of the rings that extends to a $3$-coloring of $G_1$ also extends to a $3$-coloring of $G_2$. For two subgraphs $H_1$ and $H_2$ of a graph $G$, let $d(H_1, H_2)$ denote the length of the shortest path in $G$ with one end in $H_1$ and the other end in $H_2$. We now prove the key result of this section. \[lemma-cut\] There exists an integer $d_0\ge 3$ with the following property. Let $G$ be a critical graph embedded in the cylinder with rings $C_1$ and $C_2$ of length at most $4$, such that the distance between $C_1$ and $C_2$ is $d\ge d_0$, $G$ contains no contractible triangles, and every non-contractible $(\le\!4)$-cycle of $G$ is equal to one of the rings. There exists a tame graph $G'$ in the cylinder with the same rings, such that - $G'$ dominates $G$, - the distance between $C_1$ and $C_2$ in $G'$ is at least $d-2$, - $G'$ contains a non-contractible $(\le\!4)$-cycle distinct from $C_1$ and $C_2$, - there exists a vertex $z\in V(G')$ such that the distance between $C_1\cup C_2$ and $z$ is at least three, and $z$ is contained in all non-contractible $(\le\!4)$-cycles of $G'$ distinct from the rings, and - if $H'$ is a near $3,3$-quadrangulation with rings $C_1$ and $C_2$ and $H'\subseteq G'$, then $G$ is a near $3,3$-quadrangulation with the same $5$-faces as $H'$. Let $\beta$ be the constant of Theorem \[thm-critsize\]. Let $d_0=(5\beta+1)(5(\beta+4)+2)+12$. We prove the claim by induction on the number of vertices of $G$; hence, assume that the claim holds for all graphs with fewer than $|V(G)|$ vertices. Let $f=v_1v_2v_3v_4$ be a $4$-face of $G$ at distance at least $3$ from $C_1\cup C_2$ such that no vertex of $f$ is incident with a face of length greater than $4$. Let $G_1$ be the graph obtained from $G$ by identifying $v_1$ with $v_3$ to a new vertex $z_1$, and let $G_2$ be the graph obtained from $G$ by identifying $v_2$ with $v_4$ to a new vertex $z_2$. If $G_1$ contained a contractible triangle $z_1xy$, then $G$ would contain a contractible $5$-cycle $v_1v_2v_3xy$. Since $G$ is critical, Lemma \[lemma-fr\] implies that $v_1v_2v_3xy$ bounds a face, contrary to the assumption that all faces incident with $v_2$ have length $4$. We conclude that every triangle in $G_1$ is non-contractible. By the assumption on the distance between $f$ and $C_1\cup C_2$, the triangles of $G_1$ that contain $z_1$ are disjoint with $C_1\cup C_2$. Let us first discuss the case that the distance between $C_1$ and $C_2$ in $G_1$ is at least $d$. Observe that every $3$-coloring of $G_1$ extends to a $3$-coloring of $G$ (by giving $v_1$ and $v_3$ the color of $z_1$), and thus $G_1$ dominates $G$. Let $G'_1$ be a maximal critical subgraph of $G_1$, and note that $G'_1$ also dominates $G$. If $G'_1$ is tame, then let $G''_1=G'_1$. Otherwise, let $T_1$ and $T_2$ be triangles of $G'_1$ containing $z_1$, chosen so that the part $\Sigma$ of the cylinder between $T_1$ and $T_2$ is maximal. Let $G''_1$ be the graph obtained from $G'_1$ by removing all vertices and edges contained in the interior of $\Sigma$ and by identifying the corresponding vertices of $T_1$ and $T_2$. In the latter case, Theorem \[thm-6cyc\] implies that every $3$-coloring of $G''_1$ extends to a $3$-coloring of $G'_1$, and thus $G''_1$ also dominates $G$. Note that $G''_1$ is tame and all non-contractible $(\le\!4)$-cycles in $G''_1$ distinct from the rings contain $z_1$. Let us consider the situation that a near $3,3$-quadrangulation $H'$ with rings $C_1$ and $C_2$ is a subgraph of $G''_1$. Consider any face $h$ of $H'$. Then $h$ either corresponds to a face in $G$, or $h$ is incident with $z_1$ and corresponds to a contractible $(|h|+2)$-cycle $K$ in $G$ containing the path $v_1v_2v_3$. Suppose the latter. Since the distance between $z_1$ and $C_1\cup C_2$ in $H'$ is at least three, it follows that $h$ has length $4$. Since $v_2$ is not incident with a $6$-face in $G$, all the faces of $G$ contained in the closed disk bounded by $K$ have length $4$ by Lemma \[lemma-fr\]. Since $v_1$ is not incident with a $6$-face in $G$, the same argument shows that if $G''_1\neq G'_1$, then all faces of $G$ contained in the disk bounded by the closed walk corresponding to $T_1\cup T_2$ have length $4$. Note that one of these cases accounts for all faces of $G$, and thus $G$ is a near $3,3$-quadrangulation with the same $5$-faces as $H'$. If $G''_1$ contains a non-contractible $(\le\!4)$-cycle distinct from the rings, then the distance between the rings of $G''_1$ is at least $d-2$ (due to the possible identification of $T_1$ with $T_2$ during the construction of $G''_1$) and we can set $G'=G''_1$ in the conclusions of Lemma \[lemma-cut\]. Otherwise, $G''_1=G'_1$, and thus $G''_1$ is critical and the distance between the rings of $G''_1$ is at least $d$. By the induction hypothesis, there exists a graph $G'$ dominating $G'_1$ (and thus also $G$) that satisfies the conclusions of Lemma \[lemma-cut\]. Therefore, we can assume that the distance between $C_1$ and $C_2$ in $G_1$ is less than $d$, and by symmetry, the distance between $C_1$ and $C_2$ in $G_2$ is also less than $d$. Thus, without loss of generality, there exist paths $P_1$ and $P_2$ joining $v_1$ and $v_2$, respectively, to $C_1$, and paths $P_3$ and $P_4$ joining $v_3$ and $v_4$, respectively, to $C_2$, such that $|P_1|+|P_3|\le d-1$ and $|P_2|+|P_4|\le d-1$. Since the distance between $C_1$ and $C_2$ in $G$ is at least $d$, we have $|P_1|+|P_4|\ge d-1$ and $|P_2|+|P_3|\ge d-1$. Summing the inequalities, we have $2d-2\le |P_1|+|P_2|+|P_3|+|P_4|\le 2d-2$, and thus all the inequalities hold with equality. Consequently, $|P_1|=|P_2|$ and $|P_3|=|P_4|$. Therefore, we can assume that for every $4$-face $f$ of $G$ at distance at least $3$ from $C_1\cup C_2$ that shares no vertex with a $(\ge\!5)$-face, there exists a labelling $v_1v_2v_3v_4$ of the vertices of $f$ such that $d(C_1,v_1)=d(C_1,v_2)$ and $d(C_1,v_3)=d(C_1,v_4)=d(C_1,v_1)+1$. For an integer $a\ge 0$, let $A_a$ denote the set of vertices of $G$ at distance exactly $a$ from $C_1$. Suppose that $5\le a\le d_0-5$ and that all vertices of $G$ in $A_{a-1}\cup A_a\cup A_{a+1}$ are only incident with $4$-faces. Observe that if two vertices of $A_a$ are incident with the same face, then they are adjacent, and that $C_1$ and $C_2$ are in different components of $G-A_a$. Therefore, there exists a non-contractible cycle in $G$ with all vertices in $A_a$; let $Q_a$ denote the shortest such cycle (in particular, $Q_a$ is induced). Let $Q_a=v_1\ldots v_k$. For $1\le i\le k$, let $P_i$ denote a path of length $a$ from $v_i$ to $C_1$. Observe that we can choose the paths so that for any $i,j\in \{1,\ldots, k\}$, if $P_i$ and $P_j$ intersect, then $P_i\cap P_j$ is a path with an end in $C_1$. Let $P_{k+1}=P_1$ and $v_{k+1}=v_1$. For $1\le i\le k$, if $P_i$ and $P_{i+1}$ intersect, then let $K_i$ be the cycle contained in $P_i\cup P_{i+1}\cup \{v_iv_{i+1}\}$. Observe that since $C_1$ has at most $4$ edges, the cycle $K_i$ exists for at least $k-4$ values of $i$. On the other hand, $K_i$ has odd length, and thus an odd face of $G$ is contained in the open disk bounded by $K_i$. Therefore, the cycle $K_i$ exists for at most $\beta$ values of $i$. We conclude that $k\le\beta+4$. By Theorem \[thm-critsize\], at most $5\beta$ vertices of $G$ are incident with a face of length greater than $4$. By the choice of $d_0$, there exists an integer $a\ge 5$ such that all vertices of $G$ at distance at least $a-1$ and at most $a+5(\beta+4)+1\le d_0-4$ from $C_1$ are incident only with $4$-faces. Since $|Q_a|\le \beta+4$, there exists an integer $b$ such that $a\le b\le a+5(\beta+4)-5$ and $|Q_b|\le |Q_j|$ for all $b\le j\le b+5$. Let $Q_b=v_1\ldots v_k$ and consider any $i\in\{1,\ldots,k\}$. Since no $4$-face has three vertices at distance $b$ from $C_1$, there exists an edge $v_iu_i\in E(G)$ such that $Q_b$ separates $u_i$ from $C_1$. If there existed another such edge incident with $v_i$, then $G$ would contain a $4$-face $u_iv_iu'_iw$ with $d(u_i,C_1)=d(u'_i,C_1)=b+1$, which is a contradiction. It follows that there exists exactly one such vertex $u_i$ for each $i=1,\ldots, k$. Since all faces incident with $Q_b$ have length $4$, $u_1u_2\ldots u_k$ is a non-contractible closed walk in $G$. By the choice of $Q_b$, $u_1u_2\ldots u_k$ is a cycle, and we can choose it as $Q_{b+1}$. ![Situation at the end of Lemma \[lemma-cut\] for $k=6$.[]{data-label="fig-cut"}](fig-cut) The same argument shows that we can choose $Q_{b+2}$, …, $Q_{b+5}$ so that the subgraph of $G$ drawn between $Q_b$ and $Q_{b+5}$ is a $6\times k$ cylindrical grid, see Figure \[fig-cut\]. Let $Q_{b+2}=x_1x_2\ldots x_k$ and $Q_{b+3}=y_1y_2\ldots y_k$. If $k$ is even, then let $R=\{x_1,y_2,x_3,y_4,\ldots, x_{k-3}\}$. If $k$ is odd, then let $R=\{x_1,y_2,x_3,\ldots, x_{k-2}\}$. Let $G'$ be the graph obtained from $G$ by identifying the vertices of $R$ to a single vertex $r$. Observe that $G'$ dominates $G$ and that $r$ is contained in a unique non-contractible $(\le\!4)$-cycle in $G'$. Furthermore, since we only contracted $4$-faces inside a quadrangulated cylinder, observe that if $G'$ contains a near $3,3$-quadrangulation with rings $C_1$ and $C_2$ as a subgraph, then $G$ is a near $3,3$-quadrangulation as well. Hence, $G'$ satisfies the conclusions of Lemma \[lemma-cut\]. We need an iterated version of this lemma. Let $G$ be a tame graph embedded in the cylinder with rings of length at most $4$. We say that $G$ is a *chain* of graphs $G_1$, …, $G_n$, if there exist non-contractible $(\le\!4)$-cycles $C_0$, …, $C_n$ in $G$ such that - the cycles are pairwise vertex-disjoint except that for $(i,j) \in \{(0,1),(n,n-1)\}$, $C_i$ can intersect $C_j$ if $C_i$ is a 4-cycle and $C_j$ is a triangle, - for $0\le i<j<k\le n$ the cycle $C_j$ separates $C_i$ from $C_k$, - the cycles $C_0$ and $C_n$ are the rings of $G$, - every triangle of $G$ is equal to one of $C_0$, …, $C_n$, and - for $1\le i\le n$, the subgraph of $G$ drawn between $C_{i-1}$ and $C_i$ is isomorphic to $G_i$. We say that $C_0$, …, $C_n$ are the *cutting cycles* of the chain. \[lemma-manycuts\] For every integer $c>0$, there exists an integer $d_1\ge 0$ with the following property. Let $G$ be a tame graph embedded in the cylinder with rings $C_1$ and $C_2$ of length at most $4$, such that the distance between $C_1$ and $C_2$ is $d\ge d_1$. If $G$ is not a chain of at least $c$ graphs, then there exists a tame graph $G'$ in the cylinder with the same rings, such that - $G'$ dominates $G$, - the distance between $C_1$ and $C_2$ in $G$ is at least $d-6c$, - $G'$ is a chain of $c$ graphs $G_1$, …, $G_c$, - every non-contractible $(\le\!4)$-cycle in $G'$ intersects a ring of one of $G_1$, …, $G_c$, and - if $G'$ contains a near $3,3$-quadrangulation with rings $C_1$ and $C_2$ as a subgraph, then $G$ contains a near $3,3$-quadrangulation with rings $C_1$ and $C_2$ as a subgraph as well. Let $d_0$ be the constant from Lemma \[lemma-cut\], and let $d_1=c(d_0+7)+6c$. Let $\mathcal{C}=K_0, K_1, \ldots, K_t$ be a sequence of cutting cycles of a chain in $G$ with $t<c$, chosen so that every non-contractible $(\le\!4)$-cycle in $G$ intersects one of the cutting cycles. We prove that the conclusions of the lemma hold if the distance between $C_1$ and $C_2$ is at least $c(d_0+7)+6(c-t)$. The proof is by induction on decreasing $t$; hence, we assume that the claim holds for all chains of length greater than $t$. Since the distance between $K_0$ and $K_t$ is greater than $c(d_0+7)$, there exists $i$ such that the distance between $K_i$ and $K_{i+1}$ is at least $d_0+4$. Let $K'_i$ and $K'_{i+1}$ be non-contractible $(\le\!4)$-cycles in $G$ intersecting $K_i$ and $K_{i+1}$, respectively, such that the subgraph $F$ of $G$ drawn between $K'_i$ and $K'_{i+1}$ is minimal. Let $F_0$ be a maximal critical subgraph of $F$. Note that the distance between the rings of $F_0$ is at least $d_0$, and that $F_0$ contains no non-contractible $(\le\!4)$-cycles distinct from the rings by the choice of $\mathcal{C}$. Let $F'$ be the graph obtained from $F_0$ by applying Lemma \[lemma-cut\]. Let $Q$ be the graph obtained from $G$ by replacing the subgraph drawn between $K'_i$ and $K'_{i+1}$ by $F'$. If $Q$ contains a near $3,3$-quadrangulation $H$ with rings $C_1$ and $C_2$ as a subgraph, then $H\cap F'$ is also a near $3,3$-quadrangulation, and thus $F_0$ is a near $3,3$-quadrangulation with the same $5$-faces as $H\cap F'$ by the conclusions of Lemma \[lemma-cut\]; and thus the subgraph of $G$ obtained from $H$ by replacing $H\cap F'$ by $F_0$ is a near $3,3$-quadrangulation. Since the distance between $K'_i$ and $K'_{i+1}$ in $F'$ is smaller than the distance in $F$ by at most two, it follows that the distance between $C_1$ and $C_2$ in $Q$ is smaller than the distance in $G$ by at most $6$. Let $\mathcal{C}'$ be the sequence obtained from $\mathcal{C}$ by adding one of the $(\le\!4)$-cycles of $F'$ distinct from its rings. Observe that every non-contractible $(\le\!4)$-cycle in $Q$ intersects one of the cycles of $\mathcal{C}'$. If $t=c-1$, then Lemma \[lemma-manycuts\] holds with $G'=Q$. Otherwise, Lemma \[lemma-manycuts\] follows by the induction hypothesis for $Q$ and $\mathcal{C}'$. Graphs in cylinder with rings far apart {#sec-mfar} ======================================= In [@cylgen-part1], we proved the following. \[thm-many\] There exists a constant $c\ge 0$ such that the following holds. Let $G$ be a tame graph embedded in the cylinder with the rings $C_1$ and $C_2$ of length at most $4$. If $G$ is a chain of at least $c$ graphs, then one of the following holds: - every precoloring of $C_1\cup C_2$ extends to a $3$-coloring of $G$, or - $G$ contains a subgraph obtained from a patched Thomas-Walls graph by framing on its interface pairs, with rings $C_1$ and $C_2$, or - $G$ contains a near $3,3$-quadrangulation with rings $C_1$ and $C_2$ as a subgraph. It is now straightforward to prove our main result by combining Theorem \[thm-many\] and Lemma \[lemma-manycuts\]. Let $c$ be the constant of Theorem \[thm-many\], and let $d_1$ be the constant of Lemma \[lemma-manycuts\] with this $c$. Let $D=\max(d_1, 15c)$. Since $G$ is critical, there exists a precoloring of $C_1\cup C_2$ that does not extend to a $3$-coloring of $G$. If $G$ is a chain of at least $c$ graphs, then the conclusions of Theorem \[thm-mfar\] follow from Theorem \[thm-many\], since all contractible cycles of length at most $5$ in $G$ bound faces by Lemma \[lemma-fr\]. Hence, assume that $G$ is not a chain of at least $c$ graphs, and let $G'$ be the graph obtained by Lemma \[lemma-manycuts\]. Since $G'$ dominates $G$, there exists a $3$-coloring of $C_1\cup C_2$ that does not extend to a $3$-coloring of $G'$. The graph $G'$ is a chain of graphs $G_1$, …, $G_c$, such that every every non-contractible $(\le\!4)$-cycle in $G'$ intersects one of their rings. By Theorem \[thm-many\], there exists a subgraph $H$ of $G'$ with rings $C_1$ and $C_2$ such that either $H$ is obtained from a patched Thomas-Walls graph by framing on its interface pairs, or $H$ is a near $3,3$-quadrangulation. In the latter case, the last condition in the conclusions of Lemma \[lemma-manycuts\] implies that $G$ is a near $3,3$-quadrangulation. Hence, assume the former. However, since the distance between $C_1$ and $C_2$ in $G'$ is at least $9c$, there exists $i\in\{1,\ldots,c\}$ such that the distance between the rings of $G_i$ is at least $5$. But since $H\subseteq G$, it follows that $G_i$ contains a non-contractible $4$-cycle disjoint from its rings, which is a contradiction. [^1]: Computer Science Institute (CSI) of Charles University, Malostransk[é]{} n[á]{}m[ě]{}st[í]{} 25, 118 00 Prague, Czech Republic. E-mail: [](mailto:[email protected]). Supported by project 14-19503S (Graph coloring and structure) of Czech Science Foundation. [^2]: Iowa State University, Ames IA, USA. E-mail: [](mailto:[email protected]). Supported by NSF grant DMS-1266016.

--- abstract: 'We give here a result of diophantine approximation between ${\mathcal{O}}_N$, the ring of power series in several variables, and the completion of the valuation ring that dominates ${\mathcal{O}}_N$ for the ${\mathfrak{m}}$-adic topology. We deduce from this that the Artin function of a homogenous polynomial in two variables is bounded by an affine function, which can be interpreted in term of a [Ł]{}ojasiewicz inequality on the wedges space.' address: | Department of Mathematics\ University of Toronto\ Toronto, Ontario M5S 2E4\ Canada\ () author: - Guillaume Rond title: Approximation diophantienne dans les corps de séries en plusieurs variables --- Introduction ============ La motivation de cet article est d’étudier le problème de la divisibilité dans l’anneau des séries formelles ou convergentes en plusieurs variables, problème qui apparait dans l’étude des singularités d’un point de vue analytique. Si $x$ et $y$ sont deux éléments de ${\Bbbk}[[T]]$, nous avons toujours $x/y\in{\Bbbk}[[T]]$ ou $y/x\in{\Bbbk}[[T]]$. Cela vient du fait que ${\Bbbk}[[T]]$ est un anneau de valuation. La valuation considérée ici est la valuation ${\mathfrak{m}}$-adique, où ${\mathfrak{m}}$ est l’idéal maximal de l’anneau ${\Bbbk}[[T]]$. Dans le cas des séries en plusieurs variables, si $x$ et $y$ sont deux éléments de ${\Bbbk}[[T_1,...,\,T_N]]$, il est en général faux que $x/y\in{\Bbbk}[[T_1,...,\,T_N]]$ ou $y/x\in{\Bbbk}[[T_1,...,\,T_N]]$. Nous pouvons donner deux exemples des conséquences de ce problème de “manque de divisiblité” :\ \ Le premier est lié au problème des coins introduit par M. Lejeune-Jalabert ([@LJ], voir aussi [@Re]), que l’on peut résumer comme suit. Soit $(X,\,0)$ un germe de singularité de surface et soit $(\mathfrak{X},\,0)$ sa désingularisation minimale. Un arc sur $(X,\,0)$ est la donnée d’un morphisme $\varphi$ de $({\text{Spec}\,}{\Bbbk}[[T]],\,0)$ vers $(X,\,0)$, et un coin sur $(X,\,0)$ est la donnée d’un morphisme $\varphi$ de $({\text{Spec}\,}{\Bbbk}[[T_1,\,T_2]],\,0)$ vers $(X,\,0)$. Un arc peut toujours se relever sur un éclaté de $(X,\,0)$ de centre régulier si son point générique n’est pas inclu dans le centre de l’éclatement (c’est-à-dire si son point générique peut se relever). Ceci découle du critère de propreté des morphismes propres, mais nous pouvons le voir aussi “à la main”. Si l’anneau des fonctions sur $(X,\,0)$ est ${\Bbbk}\{X_1,...,\,X_n\}/(f_1,...,\,f_p)$, un arc $\varphi$ est la donnée de $n$ séries $x_1(T)$,..., $x_n(T)$ qui vérifient $f_i(x(T))=0$ pour $1\leq i\leq p$. Si $I=(x_1,...,\,x_q)$ est le centre de l’éclatement et qu’il existe $j\leq q$ tel que $x_j(T)\neq 0$, nous pouvons, quitte à réordonner les $x_j$, supposer que $${\text{ord}}(x_q(T)))\geq{\text{ord}}(x_{q-1}(T))\geq\cdots\geq{\text{ord}}(x_1(T))\neq +\infty$$ où ${\text{ord}}$ est la valuation ${\mathfrak{m}}$-adique sur ${\Bbbk}[[T]]$. En l’origine de la carte $x_1\neq 0$, un relevé de $\varphi$ est donc donné par les séries $$x_1(T),\,x_2(T)/x_1(T),...,\,x_q(T)/x_1(T),\,x_{q+1}(T),..., \,x_n(T)$$ Le problème des coins consiste à savoir si un coin en position “suffisamment générale” peut toujours se relever en un coin sur $(\mathfrak{X},\,0)$. Un coin en position “suffisamment générale” est un coin $\varphi(T_1,\,T_2)$ tel que l’arc $\varphi(T_1,\,0)$ soit transversal au lieu singulier de $(X,\,0)$. En général un coin ne se relève pas sur un éclaté, du fait que si l’on peut ordonner les $x_j(T_1,\,T_2)$ à l’aide de ${\text{ord}}$, la valuation ${\mathfrak{m}}$-adique sur ${\Bbbk}[[T_1,\,T_2]]$, nous ne pouvons effectuer la division comme précédemment pour les séries en une variable.\ \ Le second exemple réside dans le comportement de la fonction de Artin d’un idéal $I$ de ${\Bbbk}[[T_1,...,\,T_N]][X_1,...,\,X_n]$. Si $I=(f_1(X),...,\,f_p(X))$, cette fonction est la plus petite fonction numérique de ${\mathbb{N}}$ dans ${\mathbb{N}}$ qui vérifie la propriété suivante :\ $$\forall i\in{\mathbb{N}}\ \forall x\in {\Bbbk}[[T_1,...,\,T_N]]^n\text{ tels que }$$ $$\left(\forall l \ \ f_l(x)\in {\mathfrak{m}}^{{\beta}(i)+1}\ \right)\,\Longrightarrow \left(\exists \,{\overline}{x}\in x+{\mathfrak{m}}^{i+1} \text{ tel que } \forall l \ \ f_l({\overline}{x})=0\ \right)$$ où ${\mathfrak{m}}$ est l’idéal maximal de ${\Bbbk}[[T_1,...,_,T_N]]$. Son existence découle de [@Ar].\ Si $N=1$, la fonction de Artin de $I$ est toujours bornée par une fonction affine [@Gr] et cela peut s’interpréter en terme d’inégalité de type [Ł]{}ojasiewicz sur l’espace des arcs (cf. [@Hi] proposition 3.1 par exemple). La preuve de ce résultat découle du fait que nous avons une inégalité ${\beta}(i)\leq 2{\beta}'(i)$ pour tout $i\in{\mathbb{N}}$, où ${\beta}$ (resp. ${\beta}'$) est la fonction de Artin de $I$ (resp. de l’idéal jacobien de $I$ vu comme un idéal de polynômes à coefficients dans ${\Bbbk}[[T_1,...,\,T_N]]$). Là encore, cette majoration se montre en utilisant le fait que $x$ divise $y$ dans ${\Bbbk}[[T]]$ si ${\text{ord}}(x)\leq {\text{ord}}(y)$.\ Si $N\geq 2$, l’inégalité précédente est fausse, et la fonction de Artin d’un idéal $I$ comme précédemment n’est pas toujours bornée par une fonction affine [@Ro].\ \ Une question qui se pose alors naturellement est de savoir se qui “se passe” quand on divise dans ${\mathcal{O}}_N:=\Bbbk[[T_1,...,T_N]]$ (ou ${\mathcal{O}}_N:=\Bbbk\{T_1,...,T_N\}$ si ${\Bbbk}$ est muni d’une norme). Plus précisément nous allons comparer l’anneau ${\mathcal{O}}_N$ avec le complété de l’anneau de valuation qui domine ${\mathcal{O}}_N$ pour la valuation ${\mathfrak{m}}$-adique, qui lui est de la forme ${\mathbb{K}}[[T]]$, avec ${\mathbb{K}}$ un corps. Nous allons donner ici une manière d’appréhender ce problème de “manque de divisibilité”, à l’aide d’un théorème d’approximation diophantienne entre ${\mathbb{K}}_N$, le corps des séries en plusieurs variables, et ${\widehat}{{\mathbb{K}}}_N$, son complété pour la topologie ${\mathfrak{m}}$-adique.\ Dans [@Ro], nous avons montré qu’étudier le comportement de la fonction de Artin des polynômes homogènes, à coefficients dans l’anneau de séries formelles ou convergentes en plusieurs variables sur un corps ${\Bbbk}$, dont le seul zéro est $(0,...,\,0)$, est équivalent à un tel problème d’approximation diophantienne.\ \ Nous montrons ici le théorème suivant (les notations sont données à la suite) :\ \[applin\] Soit $z\in{\widehat}{{\mathbb{K}}}_N$ algébrique sur ${\mathbb{K}}_N$ tel que $z\notin{\mathbb{K}}_N$. Alors il existe $a\geq 1$ et $K\geq 0$ tels que $$\label{dioplin}\left|z-\frac{x}{y}\right|\geq K|y|^{a},\ \forall x,\,y\in {\mathcal{O}}_N.$$\ Nous utilisons alors ce résultat pour montrer le théorème suivant qui donne une réponse à une question qui nous a été posée par M. Hickel : Soit $P(X,\,Y)$ un polynôme homogène en $X$ et $Y$ à coefficients dans ${\mathcal{O}}_N$. Alors $P$ admet une fonction de Artin bornée par une fonction affine. Notations ---------- Soient $N$ un entier positif non nul et ${\Bbbk}$ un corps ; nous utiliserons les notations suivantes : - ${\mathcal{O}}_N$ est l’anneau des séries, en $N$ variables, formelles $\Bbbk[[T_1,...,T_N]]$ ou convergentes $\Bbbk\{T_1,...,T_N\}$ (si ${\Bbbk}$ est muni d’une norme) selon le cas et indifféremment. - ${\mathfrak{m}}$ est l’idéal maximal de ${\mathcal{O}}_N$ et ${\text{ord}}$ est la valuation ${\mathfrak{m}}$-adique sur ${\mathcal{O}}_N$. Cette valuation définit une norme $|\ |$ sur ${\mathcal{O}}_N$ en posant $|x|=e^{-{\text{ord}}(x)}$ et cette norme induit une topologie appelée topologie ${\mathfrak{m}}$-adique. - $V_N:=\left\{\frac{x}{y}\,/\,x,\,y\in{\mathcal{O}}_N \text{ et } {\text{ord}}(x)\geq{\text{ord}}(y)\right\}$, l’anneau de valuation discrète qui domine ${\mathcal{O}}_N$ pour ${\text{ord}}$. Nous noterons ${\mathfrak{m}}'$ son idéal maximal. - Notons ${\widehat}{V}_N$ le complété pour la topologie ${\mathfrak{m}}$-adique de $V_N$. Nous avons ${\widehat}{V}_N={\Bbbk}\left(\frac{T_1}{T_N},...,\,\frac{T_{N-1}}{T_N}\right)[[T_N]]$. En effet cet anneau correspond au complété de l’éclatement le long de ${\mathfrak{m}}$ de ${\mathcal{O}}_N$. Nous noterons ${\widehat}{{\mathfrak{m}}}$ l’idéal maximal de cet anneau, ${\text{ord}}$ l’extension de la valuation ${\mathfrak{m}}$-adique et $|\ |$ l’extension de la norme associée. - ${\mathbb{K}}_N$ et ${\widehat}{{\mathbb{K}}}_N$ sont respectivement les corps de fractions de ${\mathcal{O}}_N$ et de ${\widehat}{V}_N$. On peut remarquer que ${\widehat}{{\mathbb{K}}}_N$ est le complété de ${\mathbb{K}}$ pour la norme $|\ |$. Il existe une théorie de l’approximation diophantienne entre le corps des polynômes en une variable et le corps des séries en 1 variables (cf. [@La] pour une introduction), trés similaire à celle entre ${\mathbb{Q}}$ et ${\mathbb{R}}$. La différence fondamentale entre l’approximation diophantienne que nous traitons ici et celle entre ${\mathbb{Q}}$ et ${\mathbb{R}}$ réside dans le fait que les éléments non nuls de ${\mathbb{Z}}$ sont de norme supérieure ou égale à 1, alors que les éléments non nuls de ${\mathcal{O}}_N$ sont tous de norme inférieure ou égale à 1. En particulier il n’existe pas de version du théorème de Liouville dans notre cadre (cf. remarque \[liouv\]). \[liouv\] Dans [@Ro], nous avons donné une suite d’éléments $x_p\in{\widehat}{{\mathbb{K}}}_N$, pour $p\in{\mathbb{N}}\backslash{\{0,\,1,\,2\}}$, de degré 2 sur ${\mathbb{K}}_N$, pour lesquels il existe $u_{p,\,k}$ et $v_{p,\,k}$ tels que $\left|x_p-\frac{u_{k,\,p}}{v_{k,\,p}}\right|=C_p|v_k|^{\frac{p}{2}-1}$ et ${\text{ord}}(v_{p,\,k})$ tend vers $+\infty$ avec $k$, où $C_p$ est une constante qui dépend de $p$. Nous voyons donc que, contrairement au cas des nombres réels algébriques, la meilleure borne $a$ telle qu’il existe $K$ avec $$\left|x-\frac{u}{v}\right|>K|v|^a,\ \forall u,\,v\in R$$ ne peut pas être bornée par le degré de l’extension de ${\mathbb{K}}_N$ par $x$. Il n’existe donc pas de version du théorème de Liouville pour les extensions finies de ${\mathbb{K}}_N$ dans ${\widehat}{{\mathbb{K}}}_N$.\ Je tiens à remercier ici M. Hickel et M. Spivakovsky pour leurs conseils et commentaires à propos de ce travail. Preuve du théorème principal ============================ Nous nous sommes inspiré là de la preuve de la proposition 5.1 de [@I2]. Nous utilisons en particulier le théorème d’Izumi sur les Inégalités Complémentaires Linéaires associées à l’ordre ${\mathfrak{m}}$-adique sur un anneau local analytiquement irréductible (cf. [@I2] et [@Re1]). Pour cela, nous allons tout d’abord introduire les notations suivantes :\ Soit $z\in{\widehat}{{\mathbb{K}}}_N\backslash{\mathbb{K}}_N$ algébrique sur ${\mathbb{K}}_N$. Soit $Q(Z)$ un polynôme irréductible de ${\mathcal{O}}_N[Z]$ annulant $z$. Ce polynôme engendre l’idéal des polynômes de ${\mathbb{K}}_N[Z]$ s’annulant en $z$. Nous pouvons écrire $Q(Z)=a_0+a_1Z+\cdots+a_dZ^d$ avec $d\geq 2$. Nous noterons $P(X,\,Y):=a_0Y^d+a_1XY^{d-1}+\cdots+a_dX^d$. Considérons les extensions suivantes : $${\mathcal{O}}_N{\longrightarrow}{\mathcal{O}}_{N+1}{\longrightarrow}{\mathcal{O}}_{N+1}/(Q(Z))={\mathcal{O}}$$ où ${\mathcal{O}}_{N+1}={\Bbbk}[[T_1,...,\,T_N,\,Z]]$. Notons ${\text{ord}}$ la valuation $(T_1,...,\,T_N,\,Z)$-adique sur ${\mathcal{O}}_{N+1}$ qui étend la valuation ${\text{ord}}$ définie précédemment et ${\text{ord}}_{{\mathcal{O}}}$ l’ordre $(T_1,...,\,T_N,\,Z)$-adique sur ${\mathcal{O}}$. Considérons ${\overline}{Q(Z)}$, le terme initial de $Q(Z)$ pour la valuation ${\text{ord}}$ sur ${\mathcal{O}}_{N+1}$. Nous avons alors $$Gr_{{\mathfrak{m}}_{{\mathcal{O}}}}{\mathcal{O}}=\frac{\left(Gr_{{\mathfrak{m}}}{\mathcal{O}}_N\otimes_{{\Bbbk}}\left({\Bbbk}\oplus (Z)/(Z)^2\oplus(Z)^2/(Z)^3\oplus\cdots\right)\right)}{({\overline}{Q(Z)})}.$$\ Tout d’abord, nous allons nous ramener au cas où ${\mathcal{O}}$ est intègre et ${\overline}{Q(Z)}=Z^d$. Nous serons alors sous les hypothèses du théorème d’Izumi que nous pourrons donc utiliser.\ Pour tout $u\in{\mathcal{O}}_N$, nous notons $Q_u(Z)=u^da_d^{d-1}Q(Z/ua_d)$. Nous avons $$Q_u(Z)=a_0u^da_d^{d-1}+a_1u^{d-1}a_d^{d-2}Z+\cdots+Z^d.$$ Le polynôme $Q_u(Z)$ est dans ${\mathcal{O}}_N[Z]$, unitaire, et irréductible comme polynôme de ${\mathbb{K}}_N[Z]$ car $Q(Z)$ est irréductible. Donc $Q_u(Z)$ est irréductible dans ${\mathcal{O}}_N[Z]$. Fixons $u$ de telle manière à ce que ${\overline}{Q_u(Z)}=Z^d$ et tel que ${\text{ord}}(u)\geq1$ (il suffit de choisir $u$ d’ordre grand). Le polynôme $Q_u(Z)$ est un polynôme distingué car ${\text{ord}}(u)\geq1$. Donc $Q_u(Z)$ est irréductible dans ${\mathcal{O}}_{N+1}$ ([@To], lemme 1.7) et ${\mathcal{O}}$ est intègre. Les zéros de $Q_u(Z)$ dans ${\widehat}{{\mathbb{K}}}_N$ sont les $ua_dz_i$ où les $z_i$ sont les zéros de $Q(Z)$ dans ${\widehat}{{\mathbb{K}}}_N$. Soit $z$ un zéro de $Q(Z)$ dans ${\widehat}{{\mathbb{K}}}_N$. Si nous avons $\left|\frac{x}{y}-ua_dz\right|\geq K|y|^{a}$, $\forall x,\,y\in{\mathcal{O}}_N$, alors nous avons $\left|\frac{x}{y}-z\right|\geq \frac{K}{|ua_d|}|y|^{a}$, $\forall x,\,y\in{\mathcal{O}}_N$.\ Nous pouvons donc supposer que ${\mathcal{O}}$ est intègre et que ${\overline}{Q(Z)}=Z^d$, ce que nous ferons à partir de maintenant.\ \ Notons $z_1,...,z_p$ les différentes solutions de $Q(Z)$ autres que $z$ dans ${\widehat}{{\mathbb{K}}}_N$, et fixons $x$ et $y$ dans ${\mathcal{O}}_N$. Notons ${\overline}{Z}$ l’image de $Z$ dans ${\mathcal{O}}$.\ Notons $r:=\max\{{\text{ord}}(z),\,{\text{ord}}(z-z_k),\, k=1,..,\,p\}$ si $p\neq 0$ et $r={\text{ord}}(z)$ sinon. Supposons que ${\text{ord}}((x/y)-z)> r$. En particulier ${\text{ord}}(x)-{\text{ord}}(y)={\text{ord}}(z)$. Notons $C:={\text{ord}}(x)-{\text{ord}}(y)={\text{ord}}(z)$.\ Nous avons $Y^dQ\left(\frac{X}{Y}\right)=P(X,\,Y)$. Or $Q({\overline}{Z})=0$ dans l’anneau ${\mathcal{O}}$ qui est intègre, donc $Q(T)=(T-{\overline}{Z})(b_{d-1}T^{d-1}+b_{d-2}T^{d-2}+\cdots+b_0)$ dans ${\mathcal{O}}[T]$, et donc $P(X,\,Y)=(X-{\overline}{Z}Y)(b_{d-1}X^{d-1}+b_{d-2}X^{d-2}Y+\cdots+b_0Y^{d-1})$ dans ${\mathcal{O}}[X,\,Y]$. En développant l’expression précédente nous voyons que $$\begin{array}{c} b_{d-1}=a_d\qquad\qquad\qquad\qquad\qquad\\ b_{i}=a_{i+1}+{\overline}{Z}b_{i+1}, \ 1\leq i\leq d-2\\ b_0=-a_0/{\overline}{Z}\qquad\qquad\qquad\qquad\quad\end{array}$$ D’où $$b_i:={\overline}{Z}^{d-i-1}a_d+{\overline}{Z}^{d-i-2}a_{d-1}+\cdots+a_{i+1}.$$ Notons $$h:=b_{d-1}x^{d-1}+b_{d-2}x^{d-2}y+\cdots+b_0y^{d-1}.$$ Nous avons alors $(x-{\overline}{Z}y)h=P(x,\,y)$ dans ${\mathcal{O}}$. Nous pouvons écrire $$h=\frac{P(x,\,y)-a_0y^d}{x}+\frac{P(x,\,y)-a_0y^d-a_1xy^{d-1}}{x^2}y{\overline}{Z}+\cdots\qquad\qquad$$ $$\qquad\qquad+\frac{P(x,\,y)-a_0y^d-a_1xy^{d-1}-\cdots-a_{d-1}x^{d-1}y}{x^d}(y{\overline}{Z})^{d-1}.$$ Notons $$f_i:=\frac{P(x,\,y)-a_0y^d-\cdots-a_{i}x^{i}y^{d-i}}{x^{i+1}}y^{i}$$ Nous avons alors $h=f_0+f_1{\overline}{Z}+\cdots+f_{d-1}{\overline}{Z}^{d-1}$ et les $f_i$ sont dans ${\mathcal{O}}_N$.\ \ L’anneau ${\mathcal{O}}$ étant intègre, d’après le théorème d’Izumi [@I2], il existe deux constantes $A\geq1$ et $B\geq 0$ telles que $$\label{izumi}A({\text{ord}}_{{\mathcal{O}}}(x-{\overline}{Z}y)+{\text{ord}}_{{\mathcal{O}}}(h))+B\geq {\text{ord}}_{{\mathcal{O}}}(P(x,\,y))\geq{\text{ord}}(P(x,\,y)).$$ Deux cas se présentent alors :\ **Cas 1 :** Soit ${\text{ord}}(P(x,\,y))\leq {\text{ord}}(a_0y^d)$, et alors dans ce cas $$\label{3}{\text{ord}}(P(x,\,y))\leq d\,{\text{ord}}(y)+{\text{ord}}(a_0).$$ **Cas 2 :** Soit ${\text{ord}}(P(x,\,y))> {\text{ord}}(a_0y^d)$. Soit $g=g_0+g_1{\overline}{Z}+\cdots+g_{d-1}{\overline}{Z}^{d-1}$ avec les $g_i\in {\mathcal{O}}_N$. Alors l’image de $g$ dans $Gr_{{\mathfrak{m}}_{{\mathcal{O}}}}$ est non nulle puisque ${\overline}{Q(Z)}$ est de degré $d$ en $Z$. Donc ${\text{ord}}_{{\mathcal{O}}}(g)=\min_i\{{\text{ord}}(g_i)+i\}$. En particulier, $${\text{ord}}_{{\mathcal{O}}}(h)\leq {\text{ord}}\left(\frac{P(x,\,y)-a_0y^d}{x}\right)=d\,{\text{ord}}(y)-{\text{ord}}(x)+{\text{ord}}(a_0).$$ D’autre part ${\text{ord}}_{{\mathcal{O}}}(x-{\overline}{Z}y)= \min\{{\text{ord}}(x),\,{\text{ord}}(y)+1\}$. Donc, d’après l’équation (\[izumi\]), nous obtenons $${\text{ord}}(P(x,\,y))\leq Ad\,{\text{ord}}(y)-A\,{\text{ord}}(x)+A\,\min\{{\text{ord}}(x),\,{\text{ord}}(y)+1\}+A{\text{ord}}(a_0)+B.$$ Nous avons alors $$\label{4}{\text{ord}}(P(x,\,y))\leq Ad\,{\text{ord}}(y)+A{\text{ord}}(a_0)+B$$\ **Conclusion :** Si ${\text{ord}}((x/y)-z)> r$, alors il existe deux constantes, $a$ et $b$ telles que $${\text{ord}}(P(x,\,y))\leq a\,{\text{ord}}(y)+b.$$ Or $P(x,\,y)=y^dQ(x/y)$. Nous pouvons écrire $Q(Z)$ sous la forme $$Q(Z)=R(Z)\prod_{k=1}^p(Z-z_k)^{n_k}.(Z-z)^n$$ avec $R$ un polynôme de degré $q$ en $Z$ qui n’admet pas de zéro dans ${\widehat}{{\mathbb{K}}}_N$. Nous avons alors ${\text{ord}}(R(x/y))\geq c$ pour une constante $c$ indépendante de $x$ et de $y$ car ${\text{ord}}(x/y)=C$ est fixé. Comme ${\text{ord}}((x/y)-z)> r$, nous avons ${\text{ord}}((x/y)-z_k)\leq r$. D’où $${\text{ord}}(Q(x/y))\geq c+\sum_{k=1}^p n_k\min\{{\text{ord}}(x/y),\,{\text{ord}}(z_k)\}+n\,{\text{ord}}((x/y)-z)$$ $$\geq c+\sum_{k=1}^p n_k\min\{C,\,{\text{ord}}(z_k)\}+n\,{\text{ord}}((x/y)-z).$$ Notons $D=c+\sum_{k=1}^p n_k\min\{C,\,{\text{ord}}(z_k)\}$. Nous obtenons alors $$(a-d){\text{ord}}(y)+b\geq n\,{\text{ord}}((x/y)-z)+D$$ où encore $$\left|\frac{x}{y}-z\right|\geq K|y|^{{\alpha}}$$ avec $K:=e^{(D-b)/n}$ et ${\alpha}:=(a-d)/n$. Si ${\text{ord}}((x/y)-z)\leq r$, nous avons alors $\left|\frac{x}{y}-z\right|\geq K'$ avec $K':=e^{-r}$. Dans tous les cas nous avons l’inégalité voulue.$\quad\Box$\ \ Application à l’étude de fonctions de Artin =========================================== Nous pouvons alors donner le résultat suivant : Soit $P(X,\,Y)$ un polynôme homogène en $X$ et $Y$ à coefficients dans ${\mathcal{O}}_N$. Alors $P$ admet une fonction de Artin bornée par une fonction affine.\ Soit $Q(Z)\in{\mathcal{O}}_N[Z]$ un polynôme unitaire n’ayant aucune racine dans ${\mathcal{O}}_N$. Par exemple $Q(Z)=Z^d-T_1^{d+1}$ ou $Q(Z)=Z^d-(T_1^d+T_2^{d+1})$. Définissons $P(X,\,Y):=Y^dQ(X/Y)$. Le polynôme $P$ est homogène en $X$ et $Y$ et n’admet pas d’autre solution dans ${\mathcal{O}}_N$ que $(0,\,0)$. Un tel polynôme admet donc une fonction de Artin bornée par une fonction affine. Ceci peut s’énoncer sous la forme suivante : il existe $a$ et $b$ deux constantes telles que $$\forall x, y\in{\mathcal{O}}_N,\ \ {\text{ord}}(P(x,\,y))\leq a\min\{{\text{ord}}(x),\,{\text{ord}}(y)\}+b.$$ En notant $\vert (x,\,y)\vert:=\max\{\vert x\vert,\,\vert y\vert\}$, ceci peut encore se réénoncer en terme d’inégalité de type [Ł]{}ojasiewicz : c’est-à-dire qu’il existe $K>0$ et $a\geq 1$ tels que $$\vert P(x,\,y)\vert\geq K\vert (x,\,y)\vert^a,\ \ \forall x,y\in{\mathcal{O}}_N.$$\ La preuve du théorème nous permet de dire que l’on peut choisir ${\alpha}$ égal à $d$ dans le premier cas (car $Q(Z)$ n’a pas de zéro dans ${\widehat}{V}_N$). Dans le second cas ${\alpha}$ est strictement plus grand que $d$ car $Q(Z)$ admet des zéro dans ${\widehat}{V}_N$. **Preuve :** Soit $P$ comme dans l’énoncé et $x,\,y\in {\mathcal{O}}_N$. Quitte à renommer les variables, nous pouvons supposer que ${\text{ord}}(y)\leq{\text{ord}}(x)$. Nous avons alors $$P(x,\,y)=y^dP\left(1,\frac{x}{y}\right) .$$ Notons alors $Q(Y)$ le polynôme $P(1,\,Y)$.\ i) Supposons que $Q$ n’a pas de racine dans ${\widehat}{V}_N$. Comme ${\widehat}{V}_N$ est de la forme ${\mathbb{K}}[[T]]$ où ${\mathbb{K}}$ est un corps et $T$ une variable formelle, d’après le théorème de Greenberg (cf. [@Gr]), $Q$ admet une fonction de Artin bornée par une constante $c$ et dans ce cas nous obtenons alors ${\text{ord}}\left(P\left(1,\frac{x}{y}\right)\right)< c+1$. Donc $${\text{ord}}\left(P(x,\,y)\right)<d\,{\text{ord}}(y)+c+1\ .$$ ii) Si $Q$ a des racines dans ${\widehat}{V}_N$, toujours d’après [@Gr], $Q$ admet une fonction de Artin bornée par une fonction affine $i{\longmapsto}{\lambda}i+\mu$ où ${\lambda}\leq d$. Remarquons que $Q$ n’a qu’un nombre fini de racines car ${\widehat}{V}_N$ est intègre. Soit $z$ un zéro de $Q$ dans ${\widehat}{{\mathbb{K}}}_N$, plus proche de $x/y$ que tous les autres zéros de $Q$ pour la topologie ${\mathfrak{m}}$-adique. Comme $i{\longmapsto}{\lambda}i+\mu$ majore la fonction de Artin de $Q$, nous avons $${\text{ord}}\left(Q\left(\frac{x}{y}\right)\right)\leq {\lambda}\, {\text{ord}}\left(z-\frac{x}{y}\right)+\mu.$$ $\bullet$ Si $z\in V_N$, alors $z=u/v$ avec $u$ et $v$ dans ${\mathcal{O}}_N$ premiers entre eux, et $${\text{ord}}\left(z-\frac{x}{y}\right)={\text{ord}}\left(\frac{uy-vx}{vy}\right).$$ Donc $${\text{ord}}\left(P(x,\,y)\right)\leq (d-{\lambda}){\text{ord}}(y)+{\lambda}{\text{ord}}(uy-vx)+(\mu-{\lambda}{\text{ord}}(v)).$$ D’après le lemme d’Artin-Rees, il existe $i_0\geq 0$ ne dépendant que de $u$ et $v$, tel que $$\label{AR}(u,\,v)\cap{\mathfrak{m}}^{i+i_0}\subset (u,\,v){\mathfrak{m}}^{i}$$ pour tout entier $i$ positif. Donc si ${\text{ord}}(uy-vx)\geq i+i_0$, alors il existe ${\varepsilon}_1$ et ${\varepsilon}_2$ dans ${\mathfrak{m}}^{i}$ tels que $uy-vx=u{\varepsilon}_1-v{\varepsilon}_2$. En posant ${\overline}{x}=x-{\varepsilon}_1$ et ${\overline}{y}=y-{\varepsilon}_1$, alors $u{\overline}{y}-v{\overline}{x}=0$ et ${\overline}{x}-x,\,{\overline}{y}-y\in{\mathfrak{m}}^i$. Nous choisirons dorénavant une constante $i_0$ pour laquelle l’inclusion (\[AR\]) ci-dessus est vérifiée pour tout $(u,\,v)$, où $u/v$ est une racine de $Q$ et $u$ et $v$ sont premiers entre eux. Comme $Q$ n’a qu’un nombre fini de racines, une telle constante existe.\ $\bullet$ Si $z\notin V_N$, d’après le théorème \[applin\], il existe $a$ et $b$ tels que $${\text{ord}}\left(z-\frac{x}{y}\right)\leq a\,{\text{ord}}{y}+b.$$ Donc nous avons $${\text{ord}}(P(x,\,y))\leq {\lambda}\, {\text{ord}}\left(z-\frac{x}{y}\right)+d{\text{ord}}(y)+\mu\leq (a{\lambda}+d){\text{ord}}{y}+{\lambda}b+\mu.$$ Nous pouvons faire le même raisonnement si ${\text{ord}}(y)\geq{\text{ord}}(x)$. Donc dans tous les cas nous avons $${\text{ord}}\left(P(x,\,y)\right)\leq A\min\left\{{\text{ord}}(x),\,{\text{ord}}(y)\right\}+B\max_{u/v\text{ z\'ero de } Q}{\text{ord}}(uy-vx)+C$$ où $A$, $B$ et $C$ sont des constantes, et $u/v$ est un zéro de $Q$ dans ${\mathbb{K}}_N$ avec $u$ et $v$ premiers entre eux dans ${\mathcal{O}}_N$.\ Supposons que nous ayons ${\text{ord}}(P(x,\,y)\geq 2\max\{A,\,B\}(i+i_0)+C$, alors nous avons soit $\min\left\{{\text{ord}}(x),\,{\text{ord}}(y)\right\}\geq i+i_0\geq i$, soit ${\text{ord}}(uy-vx)\geq i+i_0$. Dans le premier cas, nous posons ${\overline}{x}={\overline}{y}=0$. Dans le second cas, d’après le lemme d’Artin-Rees, il existe ${\overline}{x}$ et ${\overline}{y}$ tels que $u{\overline}{y}-v{\overline}{x}=0$ et ${\overline}{x}-x,\,{\overline}{y}-y\in{\mathfrak{m}}^i$. Dans tous les cas $P({\overline}{x},{\overline}{y})=0$ et ${\overline}{x}-x,\,{\overline}{y}-y\in{\mathfrak{m}}^i$. C’est-à-dire que $P$ admet une fonction de Artin bornée par la fonction affine $i{\longmapsto}2\max\{A,\,B\}(i+i_0)+C$.$\quad\Box$\ M. Artin, Algebraic approximation of structures over complete local rings, *Publ. Math. IHES*, **36**, (1969), 23-58. M. J. Greenberg, Rational points in henselian discrete valuation rings, *Publ. Math. IHES*, **31**, (1966), 59-64. M. Hickel, Calcul de la fonction d’Artin-Greenberg d’une branche plane, *Pacific J. of Math.*, **213**, (2004), 37-47. S. Izumi, A measure of integrity for local analytic algebras, *Publ. RIMS, Kyoto Univ.*, **21**, (1985), 719-736. A. Lasjaunias, A survey of Diophantine approximation in fields of power series, *Monatsh. Math.*, **130**, (2000), no. 3, 211-229. M. Lejeune-Jalabert, Arcs analytiques et résolution minimales des singularités des surfaces quasihomogènes, *Lectures Notes in Math.*, **777**, (1980), 303-336. J. Nash, Arcs structure of singularities, *Duke Math. J.*, **81**, (1995), 31-38. D. Rees, Izumi’s theorem, Commutative algebra (Berkeley, CA, 1987), 407–416, *Math. Sci. Res. Inst. Publ.*, **15**, Springer, New York, (1989). A. Reguera, Image of the Nash map in terms of wedges, *C. R. Math. Acad. Sci. Paris*, **338**, (2004), no. 5, 385-390. G. Rond, Contre-exemple à la linéarité de la fonction de Artin, prépublication ArXiv, (2004). J.-C. Tougeron, *Idéaux de fonctions différentiables*, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 71. Springer-Verlag, Berlin-New York, (1972). J. J. Wavrik, A theorem on solutions of analytic equations with applications to deformations of complex structures, *Math. Ann.*, **216**, (1975), 127-142.

--- abstract: 'We investigate the thermal photon production-rates using one dimensional boost-invariant second order relativistic hydrodynamics to find proper time evolution of the energy density and the temperature. The effect of bulk-viscosity and non-ideal equation of state are taken into account in a manner consistent with recent lattice QCD estimates. It is shown that the *non-ideal* gas equation of state i.e $\varepsilon-3\,P\,\neq 0$ behaviour of the expanding plasma, which is important near the phase-transition point, can significantly slow down the hydrodynamic expansion and thereby increase the photon production-rates. Inclusion of the bulk viscosity may also have similar effect on the hydrodynamic evolution. However the effect of bulk viscosity is shown to be significantly lower than the *non-ideal* gas equation of state. We also analyze the interesting phenomenon of bulk viscosity induced cavitation making the hydrodynamical description invalid. It is shown that ignoring the cavitiation phenomenon can lead to a very significant over estimation of the photon flux. It is argued that this feature could be relevant in studying signature of cavitation in relativistic heavy ion collisions.' author: - 'Jitesh R. Bhatt' - Hiranmaya Mishra - 'V. Sreekanth' title: Cavitation and thermal photon production in relativistic heavy ion collisions --- INTRODUCTION ============ Thermal photons emitted from the hot fireball created in relativistic heavy-ion collisions is a promising tool for providing a signature of quark-gluon plasma [@kapu91; @baier92; @Ruus92; @thoma95; @traxler95; @arnoldJHEP] (see [@Alam01R; @Peitz-Thoma:02R; @Gale:03R] for recent reviews). Since they participate only in electromagnetic interactions, they have a larger mean free path compared to the transverse size of the hot and dense matter created in nuclear collisions [@kapusta]. Therefore these photons were proposed to verify the existence of the QGP phase [@Feinberg76; @Shyryak78]. Spectra of thermal photons depend upon the fireball temperature and they can be calculated from the scattering cross-section of the processes like $q\bar q\rightarrow g\gamma$, *bremsstrahlung* etc. Time evolution of the temperature can be calculated using hydrodynamics with appropriate initial conditions. Thus the spectra depend upon the equation of state (EoS) of the medium and they may be useful in finding a signature of the quark-gluon plasma[@dk99:EPJC; @thoma07; @dkdir08; @Liu09]. Recently the thermal photons are proposed as a tool to measure the shear viscosity of the strongly interacting matter produced in the collisions[@skv-photon; @dusling09]. Understanding shear viscosity of QGP is one of the most intriguing aspects of the experiments at Relativistic Heavy Ion Collider (RHIC). Analysis of the experimental data collected from RHIC show that the strongly coupled matter produced in the collisions is not too much above the phase transition temperature $T_c$ and it may have extremely small value of shear viscosity $\eta$. In fact the ratio of the shear viscosity $\eta$ to the entropy density $s$ i.e. $\eta/s$ is around $1/4\pi$ which is the smallest for any known liquid in the nature[@Schf-Teaney-09R]. In fact the arguments based on AdS-CFT suggest that the values of $\eta/s$ can not become lower than $1/4\pi$. This is now known as Kovtun-Son-Starinets or ’KSS- bound’ [@kss05]. Thus the quark-gluon plasma produced in RHIC experiments is believed to be in a form of the most perfect liquid[@Hirano:2005wx]. No wonder ideal hydrodynamic appears to be best description of such matter as suggested by comparison between the experimental data[@RHIC] and the calculations done using second-order relativistic hydrodynamics [@BR07; @rom:PRL; @DT08; @hs08; @LR08; @Molnar08; @Song:2008hj; @LR09]. However there remains uncertainties in understanding the application and validity of the hydrodynamical procedure in the relativistic heavy-ion collision experiments. It is only very recently realized that the effect of bulk viscosity can bring complications in the hydrodynamical description of the heavy-ion collisions. Generally it was believed that the bulk viscosity does not play a significant role in the hydrodynamics of relativistic heavy-ion collisions. It was argued that that since $\zeta$ scaled like $\varepsilon -3P$ at very high energy the bulk viscosity may not play a significant role because the matter might be following the ideal gas type equation of state[@wein]. But during its course of expansion the fireball temperature can approach values close to $T_c$. Recent lattice QCD results may not have ideal gas EoS and the ratio $\zeta/s$ show a strong peak around $T_c$ [@Bazavov:2009; @Meyer:2008]. The bulk viscosity contribution in this regime can be much larger than that of the the shear viscosity. Recently the role of bulk viscosity in heating and expansion of the fireball was analyzed using one dimensional hydrodynamics[@fms08]. Another complication that bulk viscosity brings in hydrodynamics of heavy-ion collisions is phenomenon of cavitation[@kr:2010cv]. Cavitation arises when the fluid pressure becomes smaller than the vapour pressure. Since the bulk viscosity (and also shear viscosity) contributes to the pressure gradient with a negative contribution, it may be possible for the effective fluid pressure to become zero. Once the cavitation sets in the hydrodynamical description breaks down. It was shown in Ref.[@kr:2010cv] that cavitation may happen in RHIC experiments when the effect of bulk viscosity is included in manner consistent with the lattice results. It was shown that the cavitation may significantly reduce the time of hydrodynamical evolution. One of the main objectives of this paper is to study the photon spectra with the effect of bulk viscosity and cavitation. Finite $\zeta$ effect can either significantly reduce the time for the hydrodynamical evolution (by onset of cavitation) or it can increase the time by which the system reaches $T_c$! Moreover the *non-ideal* gas EoS can also significantly influence the hydrodynamics (see below). In what follows, we use equations of relativistic second order hydrodynamics to incorporate the effects of finite viscosity. We take the value of $\zeta/s$ same as that in Ref.[@kr:2010cv] and keep $\eta/s=1/4\pi$. Further we use one dimensional boost invariant hydrodynamics in the same spirit as in Refs.[@fms08; @kr:2010cv]. One of the limitations of this approach is that the effects of transverse flow cannot be incorporated. As the boost-invariant hydrodynamics is known to lead to underestimation of the effects of bulk viscosity[@fms08], we believe that our study of the photon spectra will provide a conservative estimate of the effect. FORMALISM ========= Viscous Hydrodynamics --------------------- We represent the energy momentum tensor of the dissipative QGP formed in high energy nuclear collisions as $$T^{\mu\nu}=\varepsilon \, u^\mu\,u^\nu\, - P\, \Delta^{\mu\nu} + \Pi^{\mu\nu} \label{Tmunu}$$ where $\varepsilon$, $P$ and $u^\mu$ are the energy density, pressure and four velocity of the fluid element respectively. The operator $\Delta^{\mu\nu}~=~g^{\mu\nu} - u^\mu\,u^\nu$ acts as a projection perpendicular to four velocity. The viscous contributions to $T^{\mu\nu}$ are represented by $$\Pi^{\mu\nu} = \pi^{\mu\nu}-\,\Delta^{\mu\nu}\, \Pi$$ where $\pi^{\mu\nu}$, the traceless part of $\Pi^{\mu\nu}$; gives the contribution of shear viscosity and $\Pi$ gives the bulk contribution. The corresponding equations of motion are given by, $$\begin{aligned} D \varepsilon + (\varepsilon+P)\, \theta-\Pi^{\mu\nu}\nabla_{(\mu}\,u_{\nu)}=0\label{edot}\\ (\varepsilon+P)\,Du^{\alpha}-\nabla^{\alpha}P\,+\,\Delta_{\alpha\nu}\,\partial_\mu \Pi^{\mu\nu}=0\end{aligned}$$ where $D\equiv u^\mu\partial_\mu$, $\theta\equiv\partial_{\mu}\,u^\mu$, $\nabla_{\alpha}=\Delta_{\mu\alpha}\partial^{\mu}$ and $A_{(\mu}\,B_{\nu)}=\frac{1}{2} [A_\mu\,B_\nu+A_\nu\,B_\mu]$ gives the symmetrization.\ We employ Bjorken’s prescription[@bjor] to describe the one dimensional boost invariant expanding flow, were we use the convenient parametrization of the coordinates using the proper time $\tau = \sqrt{t^2-z^2}$ and space-time rapidity $y=\frac{1}{2}\,ln[\frac{t+z}{t-z}]$; $t=\tau$ cosh$\,y$ and $z=\tau$ sinh$\,y$. Then the four velocity is given by, $$u^\mu=(\rm{cosh}\,y,0,0,\rm{sinh}\,y).$$ We note that with this transformation of the coordinates, $D=\frac{\partial\,}{\partial\tau}$ and $\theta=1/\tau$. Form of the energy momentum tensor in the local rest frame of the fireball is then given by[@Teaney03; @SHC06; @AM07; @AM-prl02] $$T^{\mu\nu} = \left( \begin{array}{cccc} \varepsilon & 0 & 0 & 0 \\ 0 & P_{\perp} & 0 & 0 \\ 0 & 0 & P_{\perp} & 0 \\ 0 & 0 & 0 & P_{z} \end{array} \right) \label{Tmunu}$$ where the effective pressure of the expanding fluid in the transverse and longitudinal directions are respectively given by $$\begin{aligned} P_{\perp} &=& P + \Pi + \frac{1}{2} \Phi \nonumber \\ P_{z} &=& P + \Pi - \Phi \label{pressures}\end{aligned}$$ Here $\Phi$ and $\Pi$ are the non-equilibrium contributions to the equilibrium pressure $P$ coming from shear and bulk viscosities. Respecting the symmetries in the transverse directions the traceless shear tensor has the form $\pi^{ij} = \mathrm{diag}(\Phi/2, \Phi/2,-\Phi)$. In the first order Navier-Stokes dissipative hydrodynamics $$\label{NS} \Pi = -\zeta \partial_\mu u^\mu \quad \rm{and} \quad \pi^{\mu\nu} = 2\eta \nabla^{\langle\mu} u^{\nu\rangle} \, ,$$ with $\zeta,\eta>0$ and $\nabla_{\langle\mu} u_{\nu\rangle}=2\nabla_{(\mu}\,u_{\nu)} -\frac{2}{3}\,\Delta_{\mu\nu}\nabla_\alpha u^\alpha$. So for first order theories with Bjorken flow we have $$\Pi=-\frac{\zeta}{\tau}\quad \rm{and} \quad\Phi=\frac{4\eta}{3\tau}.$$ The Navier-Stokes hydrodynamics is known to have instabilities and acausal behaviours[@lindblom; @BRW-06]; second order theories removes such unphysical artifacts. We use causal dissipative second order hydrodynamics of Isreal-Strewart[@Israel:1979wp] to study the expanding plasma in the fireball. In this theory we have evolution equations for $\Pi$ and $\Phi$ governed by their relaxation times $\tau_{\Pi}$ and $\tau_{\pi}$. We refer [@AM-DR:04; @ROM:09R] for more details on the recent developments in the theory and its application to relativistic heavy ion collisions. Under these assumptions, the set of equations (i.e., equation of motion (\[edot\]) and relaxation equations for viscous terms) dictating the longitudinal expansion of the medium are given by[@AM07; @BRW-06; @Heinz05] $$\begin{aligned} \frac{\partial\varepsilon}{\partial\tau} &=& - \frac{1}{\tau}(\varepsilon + P +\Pi - \Phi) \, , \label{evo1} \\ \frac{\partial\Phi}{\partial\tau} &=& - \frac{\Phi}{\tau_{\pi}}+\frac{2}{3}\frac{1}{\beta_2\tau} -\left[ \frac{4\tau_{\pi}}{3\tau}\Phi +\frac{\lambda_1}{2\eta^2}\Phi^2\right] \, , \label{evo2} \\ \frac{\partial\Pi}{\partial\tau} &=& - \frac{\Pi}{\tau_{\Pi}} - \frac{1}{\beta_0\tau} . \label{evo3}\end{aligned}$$ where $\Phi=\pi^{00}-\pi^{zz}$. The terms in the square bracket in Equation(\[evo2\]) are needed for the conformality of the theory[@Baier:2008:JHEP]. The coefficients $\beta_0$ and $\beta_2$ are related with the relaxation time by $$\tau_\Pi=\zeta\,\beta_0\,,\tau_\pi=2\eta\,\beta_2.$$ We use the $\mathcal N\,=\,4$ supersymmetric Yang-Mills theory expressions for $\tau_\pi$ and $\lambda_1$ [@Baier:2008:JHEP; @Okamura:N=4; @shiraz:N=4]: $$\label{tau_pi} \tau_{\pi} = \frac{2-\ln 2}{2\pi T}$$ and $$\lambda_1 = \frac{\eta}{2\pi T} \, .\label{lambda}$$ We set $\tau_\pi(T)=\tau_\Pi(T)$ as we don’t have any reliable prediction for $\tau_\Pi$[@fms08]. In order to close the hydrodynamical evolution equations (\[evo1\] - \[evo3\]) we need to supply the equation of state. Equation of state, $\zeta/s$ and $\eta/s$ ----------------------------------------- We are interested in the effect of bulk viscosity on the hydrodynamical evolution of the plasma and recent studies show that near the critical temperature $T_c$ effect of bulk viscosity becomes important[@karsch07; @Kharzeev:2007wb]. We use the recent lattice QCD result of A. Bazavov $\it{et ~al.}$[@Bazavov:2009] for equilibrium equation of state (EoS) (*non-ideal*: $\varepsilon-3P\neq 0$). Parametrised form of their result for trace anomaly is given by $$\frac{\varepsilon-3P}{T^4} = \left(1-\frac{1}{\left[1+\exp\left(\frac{T-c_1}{c_2}\right)\right]^2}\right) \left(\frac{d_2}{T^2}+\frac{d_4}{T^4}\right)\ , \label{e3pt4}$$ where values of the coefficients are $d_2= 0.24$ GeV$^2$, $d_4=0.0054$ GeV$^4$, $c_1=0.2073$ GeV, and $c_2=0.0172$ GeV[@kr:2010cv]. Their calculations predict a cross over from QGP to hadron gas around .200-.180 GeV. We take critical temperature as .190 GeV throughout the analysis. The functional form of the pressure is given by [@Bazavov:2009] $$\frac{P(T)}{T^4} - \frac{P(T_0)}{T_0^4} = \int_{T_0}^T dT' \,\frac{\varepsilon-3P}{T'^5}\ , \label{pt4}$$ with $T_0~$= 50 MeV and $P(T_0)$ = 0 [@kr:2010cv]. From Equations (\[e3pt4\]) and (\[pt4\]) we get $\varepsilon$ and $P$ in terms of $T$. ![$(\varepsilon-3P)/T^4\, ,\zeta/s$ (and $\eta/s=1/4\pi$) as functions of temperature T. One can see around critical temperature ($T_c=.190$ GeV) $\zeta\gg\eta$ and departure of equation of state from ideal case is large.[]{data-label="fig1"}](zeta-s.pdf){width="8.5cm" height="6cm"} We rely upon the lattice QCD calculation results for determining $\zeta/s$. We use the result of Meyer[@Meyer:2008], which indicate the existence a peak of $\zeta/s$ near $T_c$, however the height and width of this curve are not well understood. We follow parametrization of Meyer’s result from Ref.[@kr:2010cv], given by $$\frac{\zeta}{s} = a \exp\left( \frac{T_c - T}{\Delta T} \right) + b \left(\frac{T_c}{T}\right)^2\quad{\rm for}\ T>T_c, \label{zetabys}$$ where $b$ = 0.061. The parameter $a$ controls the height and $\Delta T$ controls the width of the $\zeta/s$ curve and are given by $$a=0.901,\, \Delta T=\frac{T_c}{14.5} .\label{zetacurve}$$ We will change these values to explore the various cases of $\zeta/s$ to account for the uncertainty of the height and width of the curve. In FIG.\[fig1a\] we show the change in bulk viscosity profile by varying the width of the $\zeta/s$ curve by keeping the height intact. ![Various bulk viscosity scenarios by changing the width of the curve through the parameter $\Delta T$.[]{data-label="fig1a"}](zeta-delt-a.pdf){width="8.5cm" height="6cm"} We use the lower bound of the shear viscosity to entropy density ratio known as KSS bound[@kss05] $$\eta/s=1/4\pi \label{KSS}$$ in our calculations. We note that the entropy density is obtained from the relation $$s=\frac{\varepsilon+P}{T}\label{s}.$$ In FIG.\[fig1\] we plot the trace anomaly $(\varepsilon-3P)/T^4$ and $\zeta/s$ for desired temperature range. We also plot the constant value of $\eta/s=1/4\pi$ for a comparison. It is clear that the *non-ideal* EoS deviates from the *ideal* case ($\varepsilon=3P$) significantly around the critical temperature. Around same temperature $\zeta/s$ starts to dominate over $\eta/s$ significantly. We would like to note that these results are qualitatively in agreement with Ref.[@fms08]. Thermal photons --------------- During QGP phase thermal photons are originated from various sources, like *Compton scattering* $q(\bar q)g\rightarrow q(\bar q)\gamma$ and annihilation processes $q\bar q\rightarrow g\gamma$. Recently Aurenche *et al.* showed that two loop level *bremsstrahlung* process contribution to photon production is as important as *Compton* or *annihilation* contributions evaluated up to one loop level[@aurenche:98]. They also discuss a new mechanism for hard photon production through the annihilation of an off-mass shell quark and an antiquark, with the off-mass shell quark coming from scattering with another quark or gluon. These processes in the context of hydrodynamics of heavy ion collisions were studied in Refs.[@dk99:EPJC; @thoma07]. Until recently only the processes of *Compton scattering* and *$q\bar q$-annihilation* were considered in studying the photon production rates. The production rate for hard ($E > T$) thermal photons from equilibrated QGP evaluated to the one loop order using perturbartive thermal QCD based on hard thermal loop (HTL) resummation to account medium effects. The *Compton scattering* and *$q\bar q$-annihilation* contribution is[@kapu91; @baier92; @traxler95] $$E\frac{dN}{d^4xd^3p}~=~\frac{1}{2\pi^2}\alpha\alpha_s \left(\sum_f e_f^2\right)~T^2~e^{-E/T}~\rm{ln} \left(\frac{cE}{\alpha_s T}\right),\label{Compt+Ann}$$ where the constant $c\approx$ 0.23 and $\alpha$ and $\alpha_s$ are the electromagnetic and strong coupling constants respectively. In summation $f$ denotes the flavours of the quarks and $e_f$ is the electric charge of the quark in units of the charge of the electron. The rate of photon production due to *Bremsstrahlung* processes is given by[@aurenche:98] $$\label{Brems} E\frac{dN}{d^4xd^3p}~=~\frac{8}{\pi^5}\alpha\alpha_s \left(\sum_f e_f^2\right)~\frac{T^4}{E^2}~e^{-E/T}~(J_T-J_L)~I(E,T)$$ where $J_T\approx1.11$ and $J_L\approx−1.06$ for two flavours and three colors of quarks[@thoma07]. The expression for $I(E,T)$ is given by $$I(E,T)=\left[ 3\zeta(3) +\frac{\pi^2}{6}\frac{E}{T}+\left(\frac{E}{T}\right)^2 \rm{ln}(2) + 4~\rm{Li}_3\left(-e^{-|E|/T}\right) + 2\left(\frac{E}{T}\right)\rm{Li}_2\left(-e^{-|E|/T}\right) - \left(\frac{E}{T}\right)^2~\rm{ln}\left(1+e^{-|E|/T}\right)\right]$$ and $\rm{Li}$ are the polylogarithmic functions given by $$\rm{Li_a}(z)=\sum_{n=1}^{+\infty}\frac{z^n}{n^a}.$$ Now the rate due to $q\bar q$*-annihilation with an additional scattering in the medium* is given by, $$\label{A+S} E\frac{dN}{d^4xd^3p}~=~\frac{8}{3\pi^5}\alpha\alpha_s \left(\sum_f e_f^2\right)~E~T~e^{-E/T}~(J_T-J_L).$$ We use the parametrization of $\alpha_s(T)$ by Karsch[@karsch88]: $$\alpha_s(T)=\frac{6\pi}{(33-2N_f)~\rm{ln(8T/T_c)}}$$ for our rate calculations. Here $N_f$ is the number of quark flavors in consideration.\ ![Hard thermal photon rates in QGP as a function of energy for a fixed temperature T=250 MeV. Photon rates are plotted for different relevant processes.[]{data-label="fig2"}](photonE-T.pdf){width="8.5cm" height="6cm"} In Fig.\[fig2\], we plot the different photon rates for a fixed temperature $T=250~MeV$. It shows the contributions from *Bremsstrahlung* (Brems), *annihilation with scattering* (A+S) and *Compton scattering* together with *$q\bar q$-annihilation* (C+A). *Bremsstrahlung* contributes to the photon production rate upto $E\sim 1~GeV$ only, afterwards A+S and C+A processes become dominant. This observation is in complete agreement with with Ref.[@thoma07].\ The total photon rate is obtained by adding different temperature depended photon rate expressions. Once the evolution of temperature is known from the hydrodynamical model, the *total photon spectrum* is obtained by integrating the total rate over the space time history of the collision[@thoma], $$\begin{aligned} \label{tot-yield} \left(\frac{dN}{d^2p_\bot dy}\right)_{y,p_\bot}&=&\int d^4x\left(E\frac{dN}{d^3pd^4x}\right)\\ &=&Q\int_{\tau_0}^{\tau_1} d\tau ~\tau \int_{-y_{nuc}}^{y_{nuc}}dy^{'} \left(E\frac{dN}{d^3pd^4x}\right)\nonumber\end{aligned}$$ where $\tau_0$ and $\tau_1$ are the initial and final values of time we are interested. $y_{nuc}$ is the rapidity of the nuclei whereas $Q$ is its transverse cross-section. For a $Au$ nucleus $Q \sim 180 fm^2$. $p_\bot$ is the photon momentum in direction perpendicular to the collision axis. The quantity $\left(E\frac{dN}{d^3pd^4x}\right)$ is Lorentz invariant and it is evaluated in the local rest frame in equation (\[tot-yield\]). Now the photon energy in this frame, i.e., in the frame comoving with the plasma, is given as $p_\bot cosh(y-y^\prime)$. So once the rapidity and $p_\bot$ are given we get the total photon spectrum.\ RESULTS AND DISCUSSION ====================== 0.1 in ----- ---------- --------- $~$ $(fm/c)$ $(GeV)$ 5.3 0.5 .310 ----- ---------- --------- : Initial conditions for RHIC In order to understand the temporal evolution of temperature $T(\tau)$, pressure $P(\tau)$ and viscous stresses - $\Phi(\tau)~\rm{and}~\Pi(\tau)$, we numerically solve the hydrodynamical equations describing the longitudinal expansion of the plasma: (\[evo1\]-\[evo3\]). We use the *non-ideal* EoS obtained from equations (\[e3pt4\]) and (\[pt4\]). Information about viscosity coefficients $\zeta$ and $\eta$ are obtained from equations (\[zetabys\]-\[KSS\]) using equation (\[s\]). We need to specify the initial conditions to solve the hydrodynamical equations, namely the initial time $\tau_0$ and $T_0$. We use the initial values relevant for RHIC experiment given in Table I, taken from Ref.[@dk99:EPJC]. We will take initial values of viscous contributions as $\Phi(\tau_0)=0$ and $\Pi(\tau_0)=0$. We would like to note that our hydrodynamical results are in complete agreement with that of Ref.[@kr:2010cv].\ Once we get the temperature profile we calculate the photon production rates. Total photon spectrum $E\frac{dN}{d^3pd^4x}$ (as a function of rapidity, $y$ and transverse momentum of photon, $p_\bot$) is obtained by adding different photon rates using equations (\[Compt+Ann\]),(\[Brems\]),(\[A+S\]) and convoluting with the space time evolution of the heavy-ion collision with equation (\[tot-yield\]). The final value of time $\tau_1$ is the time at which temperature evolves to critical value $\tau_f$, i.e.; $T(\tau_1)=T_c$. In all calculations we will consider the photon production in mid-rapidity region ($y=0$) only.\ We will be exploring various values of viscosity and its effect on the system. Since there is an ambiguity regarding the height and width of $\zeta/s$ curve, we will vary the parameters $a~\rm{and}~\Delta T$ from its base value given in equation (\[zetacurve\]). By this we will able to study the effect of variation of $\zeta$ on the system. The varied values of the parameters are represented by $a'~\rm{and}~\Delta T'$. We note that unless specified we will be using the base values of bulk viscosity parameters (\[zetacurve\]) in our calculations. Throughout the analysis we will keep the shear viscosity $\eta$ to its base value given by equation (\[KSS\]).\ In order to understand the effect of *non-ideal* EoS in hydrodynamical evolution and subsequent photon spectra we compare these results with that of an *ideal* EoS ($\varepsilon=3P$). We consider the EoS of a relativistic gas of massless quarks and gluons. The pressure of such a system is given by $$P=a~T^4\,;\,a=\left(16+\frac{21}{2}N_f\right)~\frac{\,\pi^2}{90}\label{idealP}$$ where $N_f=2$ in our calculations. Hydrodynamical evolution equations of such an EoS within ideal (without viscous effects) Bjorken flow can be solved analytically and the temperature dependence is given by[@bjor] $$T = T_0~\left(\frac{\tau_0}{\tau}\right)^{1/3} \label{idealT},$$ where $\tau_0~\rm{and}~T_0$ are the initial time and temperature. While considering the viscous effect of this *ideal* EoS, we will solve the set of hydrodynamical equations (\[evo1\] - \[evo2\]), since effect of bulk viscosity can be neglected in the relativistic limit when the equation of state $P=\varepsilon/3$ is obeyed [@wein].\ Hydrodynamics with *non-ideal* and *ideal* EoS {#hydrodynamics-with-non-ideal-and-ideal-eos .unnumbered} ----------------------------------------------- ![Temperature profile using massless (*ideal*) and *non-ideal* EoS in RHIC scenario. Viscous effects are neglected in both cases. System evolving with *non-ideal* EoS takes a significantly larger time to reach $T_c$ as compared to $ideal$ EoS scenario.[]{data-label="fig3"}](T-prof-ideal-vs-e3pt4.pdf){width="8.5cm" height="6cm"} FIG. (\[fig3\]) shows plots of temperature versus time for the *ideal* and *non-ideal* equation of states. The temperature profiles are obtained from the hydrodynamics without incorporating the effect of viscosity. The figure shows system with *non-ideal* EoS takes almost the double time than the system with *ideal* massless EoS to reach $T_c$. So even when the effect of viscosity is not considered, inclusion of the *non-ideal* EoS makes significant change in temperature profile of the system. This can affect the corresponding photon production rates (below).\ ![Figure shows time evolution of temperature with *non-ideal* EoS for different combinations of bulk ($\Pi$) and shear ($\Phi$) viscosities. Non zero value of bulk viscosity refers to equations (\[zetabys\]-\[zetacurve\]) and non zero shear viscosity is calculated from equation (\[KSS\]).[]{data-label="fig4"}](T-var-cases.pdf){width="8.5cm" height="6cm"} Now we analyse the viscous effects. Role of shear viscosity in the boost invariant hydrodynamics of heavy ion collisions, for a chemically nonequilibrated system, was already considered in Ref.[@skv-photon]. Next we consider possible combinations of $\Phi$ and $\Pi$ in *non-ideal* EoS case and study the corresponding temperature profiles as shown in FIG. (\[fig4\]). As expected viscous effects is slowing down temperature evolution. For the case of non zero bulk and shear viscosities ($\Pi\neq 0;~\Phi\neq 0$), temperature takes the longest time to reach $T_c$ as indicated by the top most curve. This is $\sim 1.5~ fm/c$ greater than the no viscosity case (the lowest curve). The remaining two curves show that the bulk viscosity dominates over the shear viscosity when the value of $T$ approaches $T_c$ and this makes the system to spend more time around $T_c$. However the intersection point of the two curves may vary with values of $a$ and $\Delta T$ as highlighted by FIG.\[fig1a\]. *Non-ideal* EoS and Cavitation {#non-ideal-eos-and-cavitation .unnumbered} ------------------------------ ![Longitudinal pressure $P_z$ for various viscosity cases shown in FIG.\[fig1a\].[]{data-label="fig5"}](pz-delT.pdf){width="8.5cm" height="6cm"} Let us note the fact that $\Pi<0$ [@kr:2010cv]. From the definition of longitudinal pressure $P_{z} = P + \Pi - \Phi$ it is clear that if either $\zeta$ ($\Pi$) or $\eta$ ($\Phi$) is large enough it can drive $P_z$ to negative values. $Pz=0$ defines the condition for the onset of *cavitation*. At this instant when of $P_z$ becoming zero the expanding fluid will break apart in to fragments and hydrodynamic treatment looses its validity (see for e.g. Ref.[@kr:2010cv]). Recent experiments at RHIC suggest $\eta/s$ to its smallest value $\sim 1/4\pi$. And such a small value of $\eta/s$ alone is inadequate to induce cavitation. Therefor we vary the bulk viscosity values by changing $a~\rm{and}~\Delta T$ to study the cavitation. In the discussion that follows we will use $\tau_c$ to denote the time when cavitation occurs.\ ![Temperature is plotted as a function of time. With peak value ($a$) of $\zeta/s$ remains same while width ($\Delta T$) varies. Solid line in the curve ends at the time of cavitation, while the dashed lines shows that how system would continue till $T_c$ if cavitation is ignored. Figure shows that larger the $\Delta T$ shorter the cavitation time.[]{data-label="fig7"}](cavitations.pdf){width="8.5cm" height="6cm"} In FIGs.\[fig5\] and \[fig7\] we plot $P_z$ and $T$ as functions of time for different values of $\Delta T$ while keeping $a$ (=0.901) fixed. It shows that higher value of $\Delta T$ is leading to a shorter cavitation time. For the values of $a~\rm{and}~\Delta T$ given by equation (\[zetacurve\]) we find that around $\tau_c=2.5~fm/c$, $P_z$ becomes zero as shown by the curve at the bottom of the FIG.\[fig5\]. In this case cavitation occurs when system temperature is larger than $T_c$. This can be seen from the top curve of the in FIG.\[fig7\]. End point of the solid line in the top curve occurs at $T\sim 210~MeV$ and $\tau_c=2.5~fm/c$. Had we ignored the cavitation, system would have taken a time $\tau_f=5.5~fm/c$ to reach $T_c$ which is significantly larger than $\tau_c$ as seen from FIG.\[fig7\]. This shows that cavitation occurs rather abruptly without giving any sign in the temperature profile of the system. The hydrodynamic evolution without calculating $P_z$ may end up in over estimating the evolution time and subsequent photon production.\ ![Cavitation time $\tau_c$ as a function of different values of height ($a'$) and width ($\Delta T'$) of $\zeta/s$ curve.[]{data-label="fig8"}](a-delT-tc.pdf){width="8.5cm" height="6cm"} A similar analysis can be carried out by keeping $\Delta T$ fixed ($=T_c/14.5$) and varying parameter $a$. We show the cavitation times corresponding to changes in $a$ and $\Delta T$ (denoted by $a'$ and $\Delta T'$) in FIG.\[fig8\]. The dashed curve in FIG.\[fig8\] shows $\tau_c$ as a function of $a$, while keeping $\Delta T$ fixed. The curve shows that $\tau_c$ decreases with with increasing $a$. Solid line shows how $\tau_c$ varies while keeping $a$ fixed and changing $\Delta T$. Thermal Photon Production {#thermal-photon-production .unnumbered} ------------------------- ![Photon flux as function of transverse momentum for different equation of states. No effect of viscosity included in the hydrodynamical equations.[]{data-label="fig9"}](phrate-ideal-vs-e3pt4.pdf){width="8.5cm" height="6cm"} We have already seen that the calculation of photon production rates require the initial time $\tau_0$, final time $\tau_1$ and $T(\tau)$. $\tau_1$ and $T(\tau)$ are determined from the hydrodynamics. Generally $\tau_1$ is taken as the time taken by the system to reach $T_c$, i.e.; $\tau_f$. But when there is cavitation, we must set $\tau_1=\tau_c$. Therefor photon productions will be influenced by cavitation, temperature profile and *non-ideal* EoS near $T_c$. ![Photon spectrum obtained by considering the effect of cavitation (dashed line). For a comparison we plot the spectrum without incorporating the effect of cavitation (solid line).[]{data-label="fig10"}](rate-over-estimation.pdf){width="8.5cm" height="6cm"} FIG. (\[fig9\]) shows the photon production rate calculated using *ideal* (massless) and *non-ideal* EoS. The figure shows that *non-ideal* EoS case can yield significantly larger photon flux as compared to the *ideal* EoS. At energy $E=1$ GeV, photon flux for the *non-ideal* EoS is 60$\%$ larger than that of *ideal* EoS case. This is because the calculation of the photon flux is done by performing time integral over the interval between the initial time $\tau_0$ and the final time $\tau_f$. $\tau_0$ is same for both the system while the $\tau_f$ for the case with *non-ideal* EoS is two times larger than the *ideal* EoS. Since the *non-ideal* EoS allows the system to have consistently higher temperature over a longer period as compared to the massless *ideal-gas* EoS, more photons are produced. Next we try to observe the effect of cavitation in photon production. We emphasis that rates should only be integrated up to $\tau_c$. In FIG.\[fig10\], photon rates are calculated for the two cases. In the dashed curve the effect of cavitation is taken into account and $\tau_1=\tau_c=2.5~fm/c$. The solid line represent the same case but with the effect of the cavitation is ignored and $\tau_1=\tau_f=5.5~fm/c$. We see from the solid curve that we end up over estimating the photon rates at $p_\bot=0.5$ GeV by $\sim$ 200 $\%$ and $p_\bot=2$ GeV over estimation is about 50$\%$. So it is clear that information about cavitation time is crucial for correctly estimating thermal photon production rate. In the FIG.\[fig11\] we plot photon production rates for various cavitation times obtained by varying $\Delta T$ (with $a=0.901$ is fixed). Here the enhancement in the photon production when $\Delta T$ is reduced to half of its base value is $\sim75\%$ at $p_\bot=0.5$ GeV and $\sim 55\%$ at $p_\bot=1$ GeV. A further reduction of the parameter value to $\Delta T/4$ is enhancing the photon production by $\sim 110\%$ at $p_\bot=0.5$ GeV and $\sim 80\%$ at $p_\bot=1$ GeV. Reduction in $\Delta T$ amounts to increase in the cavitation time, which in turn would increase the time interval over which photon production is calculated. Therefor this increases the photon flux. ![Photon production rates showing the effect of different cavitation time.[]{data-label="fig11"}](cav-ph-rates.pdf){width="8.5cm" height="6cm"} SUMMARY AND CONCLUSIONS ======================= Thus using the second order relativistic hydrodynamics we have analyzed the role of non-ideal effects near $T_c$ arising due to the equation of state, bulk-viscosity and cavitation on the thermal photon production. Since the experiments at RHIC imply extremely small values for $\eta/s$, the shear viscosity play a sub dominant role near $T_c$ in the photon production. We have shown using non-ideal EoS using the recent lattice results that the hydrodynamical expansion gets significantly slow down as compared to the case with the massless EoS. This in turn enhances the flux of hard thermal photons. 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--- abstract: | 0.5cm *Supertropical monoids* are a structure slightly more general than the supertropical semirings, which have been introduced and used by the first and the third authors for refinements of tropical geometry and matrix theory in [@IR1]–[@IzhakianRowen2009Resulatants], and then studied by us in a systematic way in [@IKR1]–[@IKR3] in connection with “*supervaluations*". In the present paper we establish a category $\STROP_m$ of supertropical monoids by choosing as morphisms the “*transmissions*", defined in the same way as done in [@IKR1] for supertropical semirings. The previously investigated category $\STROP$ of supertropical semirings is a full subcategory of $\STROP_m.$ Moreover, there is associated to every supertropical monoid $V$ a supertropical semiring $\htV$ in a canonical way. A central problem in [@IKR1]–[@IKR3] has been to find for a supertropical semiring $U$ the quotient $U / E $ by a “*TE-relation*", which is a certain kind of equivalence relation on the set $U$ compatible with multiplication (cf. [@IKR2 Definition 4.5]). It turns out that this quotient always exists in $\STROP_m$. In the good case, that $U / E$ is a supertropical semiring, this is also the right quotient in $\STROP.$ Otherwise, analyzing $(U / E)^\wedge,$ we obtain a mild modification of $E$ to a TE-relation $E'$ such that $U/ E' = (U/ E)^ \wedge$ in $\STROP.$ In this way we now can solve various problems left open in [@IKR1], [@IKR2] and gain further insight into the structure of transmissions and supervaluations. Via supertropical monoids we also obtain new results on totally ordered supervaluations and monotone transmissions studied in [@IKR3]. address: - 'Department of Mathematics, Bar-Ilan University, 52900 Ramat-Gan, Israel' - 'Department of Mathematics, NWF-I Mathematik, Universität Regensburg 93040 Regensburg, Germany' - 'Department of Mathematics, Bar-Ilan University, 52900 Ramat-Gan, Israel' author: - Zur Izhakian - Manfred Knebusch - Louis Rowen title: | Supertropical Monoids:\ Basics, Canonical Factorization,\ and Lifting Ghosts to Tangibles --- [^1] [^2] [^3] [^4] Introduction {#introduction .unnumbered} ============ To a large extent the algebra underpinning present day tropical geometry is based on the notion of a (commutative) **bipotent semiring**. Such a semiring $M$ is a totally ordered monoid under multiplication with smallest element $0$, the addition being given by $x+y = \max(x,y)$, cf. [@IKR1 §1] for details. In logarithmic notation, which most often is used in tropical geometry, bipotent semirings appear as totally ordered additive monoids with absorbing element $-\infty$. The primordial object here is the bipotent semifield $T(\Real) = \Real \cup \{-\infty \}, $ cf. e.g. [@IMS §1.5]. In [@I] the first author introduced a cover of $T(\Real)$, graded by the multiplicative monoid $(\Int_2, \cdot \; )$, which was dubbed the *extended tropical arithmetic*. Then, in [@IR1] and [@IR2], this structure has been amplified to the notion of a **supertropical semiring**. A supertropical semiring $U$ is equipped with a “ghost map" $\nu := \nu_U: U \to U$, which respects addition and multiplication and is idempotent, i.e., $\nu \circ \nu = \nu$. Moreover, in this semiring $a +a = \nu(a)$ for every $a \in U$ (cf. [@IKR3 §3]). This rule replaces the rule $a+a = a$ taking place in the usual max-plus (or min-plus) arithmetic. We call $\nu(a)$ the “**ghost**" of $a$ and we term the elements of $U$, which are not ghosts, “**tangible**". (The element $0$ is regarded both as tangible and ghost.) $U$ then carries a multiplicative idempotent $e = e^2$ such that $\nu(a) = ea$ for every $a \in U$. The image $eU$ of the ghost map, called the **ghost ideal** of $U$, is itself a bipotent semiring. Supertropical semirings allow a refinement of valuation theory to a theory of “supervaluations", the basics of which can be found in [@IKR1]-[@IKR3]. Supervaluations seem to be able to provide an enriched version of tropical geometry, cf. [@IKR1 §9. §11] and [@IR1]. We recall the initial definitions. An **-valuation** (= monoid valuation) on a semiring $R$ is a multiplicative map $v: R \to M $ to a bipotent semiring $M$ with $v(0) =0 $, $v(1) = 1$, and $$v(x+y) {\ {\leq} \ } v(x) + v(y) \quad [ = \max(v(x), v(y))],$$ cf. [@IKR1 §2]. We call $v$ a **valuation** if in addition the semiring $M$ is **cancellative**, by which we mean that $M \sm \00 $ is closed under multiplication and is a cancellative monoid in the usual sense. If $R$ happens to be a (commutative) ring, these valuations coincide with the valuations of rings defined by Bourbaki [@B] (except that we switched from additive notation there to multiplicative notation here). Given an -valuation $v: R \to M $ there exist multiplicative mappings $\vrp: R \to U$ into various supertropical semirings $U$ with $\vrp(0) = 0$, $\vrp(1) = 1$, such that $M$ is the ghost ideal of $U$ and $\nu_U \circ \vrp = v.$ These are the **supervaluations** covering $v$, cf. [@IKR1 §4]. The supervaluations lead us to the “right" class of maps between supertropical semirings $U$, $V$ which we have to admit as morphisms to obtain the category $\STROP$ of supertropical semirings (formally introduced in [@IKR2 §1]). These are the “transmissions". A **transmission** $\al: U \to V$ is a multiplicative map with $\al(0) = 0$, $\al(1) = 1$, $\al(e_U) = e_V$, whose restriction $\gm: eU \to eV$ to the ghost ideals is a semiring homomorphism. It turned out in [@IKR1 §5] that the transmissions from $U$ to $V$ are those maps $\al: U \to V$ such that for every supervaluation $\vrp: R \to U$ the map $\al \circ \vrp : R \to V$ is again a supervaluation. Every semiring homomorphism $\al: U \to V$ is a transmission, but there also exist transmissions which do not respect addition. This causes a major difficulty for working in $\STROP.$ A large part of the papers [@IKR1], [@IKR2] has been devoted to constructing various equivalence relations $E$ on a supertropical semiring $U$ such that the map $\pi_E: U \to U/E,$ which sends each $x\in U$ to its $E$-equivalence class $[x]_E$, induces the structure of a supertropical semiring on the set $U/E$ which makes $\pi_E$ a transmission. We call such an equivalence relation $E$ **transmissive**. It seems difficult to characterize transmissive equivalence relations in an axiomatic way. In [@IKR2 Proposition 4.4] three axioms TE1–TE3 have been provided, which obviously have to hold, but then, adding a fourth axiom, we only characterized those transmissive equivalence relations on $U$, where the ghost ideal of $U/E$ is cancellative [@IKR2 Theorem 4.5]. Dubbing the equivalence relations obeying axioms TE1–TE3 “TE-relations", we had to leave the following problem open in general: - When is a TE-relation $E$ on a supertropical semiring $U$ transmissive? The problem seems to be relevant since there exist natural classes of -valuations\ $v: R \to M,$ where the bipotent semiring $M$ has no reason to be cancellative, cf. [@IKR3 §1]. For $R$ a ring such -valuations already appeared in the work of Harrison–Vitulli [@HV] and D. Zhang [@Z]. The present paper gives a solution to the problem $(*)$ just described. We introduce a new category $\STROP_m$ containing the category $\STROP$ of supertropical semirings as a full subcategory. The objects of $\STROP_m$, called “**supertropical monoids**", are multiplicative monoids $U$ with an absorbing element $0$, an idempotent $e\in U,$ and a total ordering on the monoid $eU,$ which makes $eU$ a bipotent semiring. In a supertropical monoid it is natural to speak about tangibles and ghosts in the same way as in supertropical semirings. Every supertropical semiring can be regarded as a supertropical monoid, of course. Conversely, since addition in a supertropical semiring $U$ is determined by multiplication and the idempotent $e$ (cf. [@IKR3 Theorem 3.11]), we can turn a supertropical monoid $U$ into a supertropical semiring in at most one way, and then say that $U$ “is" a supertropical semiring. If $U$ and $V$ are supertropical monoids, the definition of a transmission $\al:U \to V$, as given above for $U$, $V$ supertropical semirings, still makes sense, and these “transmissions" between supertropical monoids (cf. Definition \[defn1.4\]) are taken as the morphisms in the category $\STROP_m$. The axioms TE1–TE3 mentioned above also make perfect sense for an equivalence relation $E$ on a supertropical monoid $U$ (cf. Definition \[defn1.6\] below). Thus, such a relation $E$ will again be called a **TE-relation**. But we have the important new fact that for a TE-relation $E$ on a supertropical monoid $U$ the quotient $U/E$ **always** exists in the category $\STROP_m$. More precisely, the set $U/E$ can be equipped with the structure of a supertropical monoid in a unique way, such that the map $\pi_E$ is a transmission (cf. Theorem \[thm1.7\] below). The solution of the problem $(*)$ from above now reads as follows (Scholium \[schol1.12\] below): If $U$ is a supertropical semiring and $E$ is a TE-relation on $U$, then $E$ is transmissive iff the quotient $U/E$ in $\STROP_m$ is a supertropical semiring. We will provide a necessary and sufficient condition that a given supertropical monoid $U$ is a supertropical semiring (Theorem \[thm1.2\] below). From this criterion it is immediate that $U$ is such a semiring if the bipotent semiring $U$ is cancellative, but there also are other cases, where this holds. A bipotent semiring $M$ may be viewed as a supertropical semiring, all of whose elements are ghosts. (This is the case $1 = e.$) Thus the category $\STROP_m /M$ of supertropical monoids over $M$ may be viewed as the category of supertropical monoids $U$ with a fixed ghost ideal $eU = M.$ Then the morphisms of $\STROP_m /M$ are the transmissions $\al: U \to V$ with $\al(x) = x$ for all $x \in M.$ We call the surjective transmissions over $M$ **fiber contractions** (over $M$), as we did for supertropical semirings [@IKR1 §6]. We note in passing that if $\al:U \onto V$ is a fiber contraction and $U$ is a supertropical semiring, then $V$ is again a supertropical semiring (cf. Theorem \[thm1.5\] below), and $\al$ is a semiring homomorphism [@IKR1 Propoistion 5.10.iii]. It turns out that for every supertropical monoid $U$ there exists a fiber contraction $\sig_U: U \to \htU$ with $\htU$ a supertropical semiring, called the **supertropical semiring associated** to $U$, such that every fiber contraction $\al:U \onto V$ factors through $\sig_U$ (in a unique way), $\al = \bt \circ \sig_U$ with $\bt:\htU \to V$ again a fiber contraction. In more elaborate terms, $\STROP /M$ is a full reflective subcategory of $\STROP_m /M$, cf. [@F p. 79], [@FS 1.813]. The reflections $\sig_U: U \to \htU$ turn out to be useful for solving problems of universal nature for supertropical semirings and supervaluations. The strategy is, first to solve such a problem in $\STROP_m$, which often is easy, and then to employ reflections to obtain a solution in $\STROP.$ Major instances for this are provided by Theorems \[thm4.6\] and \[thm6.9\] below. A large part of the paper is devoted to the factorization of transmissions into appropriately defined “basic" transmissions. Let $\al : U \onto V$ be a surjective transmission with $U,V$ supertropical monoids, and let $\al^\nu: = \gm : M \onto N$ denote the “ghost part" of $\al$, i.e., the semiring homomorphism obtained from $\al$ by restriction to the ghost ideals $M:= e_U U$, $N : =e_V V$. Then there exists an essentially unique factorization $$\label{eq:0.1} \xymatrix{ \al:U \ar[r]_{{\lambda}} & U_1 \ar[r]_{\bt} & V_1 \ar[r]_{\mu} & V, \\ }$$ for some supertropical monoids $U_1$ and $V_1$, with ${\lambda}$ and $\mu$ fiber contractions of certain types over $M$ and $N$ respectively and $\bt$ a so called “strict ghost-contraction", which means that $\bt$ restricts to a bijection from the set $\tT(U_1)$ of non-zero tangible elements of $U$ to $\tT(U_1)$, while $\bt^\nu = \gm,$ cf. Theorem \[thm2.9\]. (Notice that $\gm$ has convex fibers in $M$, since $\gm$ respects the orderings of $M$ and $N$. These convex sets are contracted by $\gm$ to one-point sets, hence the name “ghost-contraction". “Strict" alludes to the property that no element of $\tT(U)$ is sent to a ghost in $V$.) From we then obtain a factorization $$\label{eq:0.2} \xymatrix{ \al:U \ar[r]_{{\lambda}} & \brU \ar[r]_{\bt} & W \ar[r]_{\mu} & \brW \ar[r]_{\rho} & V \\ }$$ which is really unique. Here ${\lambda}, \bt, \mu$ are transmissions of the same types as before but are normalized to maps $\pi_E$ given by certain TE-relations $E$ on $U$, $\brU$, $W$ respectively, which are uniquely determined by $\al$, and $\rho$ is an isomorphism over $N = eV.$ This is the “**canonical factorization**" of $\al$, appearing in the title of the paper (Definition 2.12). The transmissions ${\lambda}, \bt, \mu, \rho$ are called the **canonical factors** of $\al.$ In §3 we make explicit how the canonical factors of a composition $\al_2 \circ \al_2$ of two transmissions $\al_1: U_1 \to U_2$, $\al_2: U_2 \to U_3$ can be obtained from the canonical factors of $\al_1$ and $\al_2.$ Our primary interest is not in supertropical monoids but in supertropical semirings. In this respect the following result is useful: If $U$ and $V$ are supertropical semirings, then in the canonical factorization all three supertropical monoids $\brU, W, \brW $ are again supertropical semirings, and thus all canonical factors are morphisms in $\STROP$ (Theorem \[thm2.9\]). Besides $\STROP$, the category $\STROPH$ deserves interest, whose objects are again the supertropical semirings but whose morphisms are only the semiring homomorphisms. (Thus $\STROPH$ is a subcategory of $\STROP$ and a full subcategory of the category of semirings.) In §5 we introduce a subcategory $\STROPH_m$ of $\STROP_m$ which turns out to be equally useful for working in $\STROPH$, as $\STROP_m$ has proved to be useful for $\STROP.$ The objects of $\STROPH_m$ are again the supertropical monoids, but the morphisms are suitably defined “**-transmissions**", which are designed in such a way that an -transmission $\al:U \to V$ between supertropical semirings is a semiring homomorphism, cf. Definition \[defn5.1\]. Thus $\STROPH$ is a full subcategory of $\STROPH_m.$ Again it turns out that for a given bipotent semiring $M$ the category $\STROPH / M$ is reflective in $\STROPH_m / M$ (Corollary \[cor5.10\]). If $\al: U \to V$ is a surjective -transmission, then the canonical factors of $\al$ are again -transmissions (Theorem \[thm5.11\]). It follows that, if $\al: U \to V$ is a surjective semiring homomorphism, the whole canonical factorization runs in $\STROPH$. In §6 we study supertropical monoids which have a total ordering defined to be compatible with the supertropical monoid structure in a rather obvious way (Definition \[defn61.1\]). We call them **ordered supertropical monoids** (= **OST-monoids**, for short). It turns out that the underlying supertropical monoids of an $\OST$-monoid is a supertropical semiring (Theorem  \[thm61.4\]). The “right" morphisms between $\OST$-monoids are the transmissions compatible with the given orderings, called **monotone transmissions**. It turns out that every monotone transmission is a semiring homomorphism (Theorem \[thm61.7\]). A major result now is that, given a monotone transmission $\al: U \to V$, all factors of the canonical factorization of $\al$ can be interpreted as monotone transmissions. More precisely, there exist unique total orderings on $\brU, W, \brW$, which make $\brU, W, \brW$ $\OST$-monoids and all factors ${\lambda}, \bt, \mu, \rho$ monotone transmissions (Theorem \[thm61.14\]). In the last sections §\[sec:6\]–§\[sec:8\] of the paper “*-supervaluations*" play the leading role. Given an -valuation $v: R \to M$ on a semiring $R$, an **-supervaluation $\vrp: R \to U$ covering** $v$ is defined in a completely analogous way as has been indicated above for a supervaluation. The only difference is that now $U$ is a supertropical monoid instead of a supertropical semiring (Definition \[defn6.1\]). The morphisms in $\STROP_m$ are adapted to the -supervaluations, as the morphisms in $\STROP$ were adapted to the supervaluations, to wit, a map $\al: U \to V$ between supertropical monoids is a transmission iff for every -supervaluation $\vrp:R \to U$ the map $\al \circ \vrp$ is again an -supervaluation. (This has not been detailed in the paper.) In order to avoid discussions about “equivalence" of -valuations we now tacitly assume, without essential loss of generality, that all occurring -supervaluations $\vrp:R \to U$ are “surjective", i.e., $U = \vrp (R) \cup e \vrp(R)$.[^5] Given such -supervaluations $\vrp : R \to U$, $\psi: R \to V$, w say that $\vrp$ **dominates** $\psi$, and write $\vrp \geq \psi$, of there exists a (unique) transmission $\al: U \to V$ with $\psi = \al \circ \vrp.$ In §\[sec:6\] we construct the **initial -supervaluation** $\vrp_v^0 : R \to U(v)^0$ covering a given -supervaluation $v: R \to M$, which means that $\vrp^0_v$ dominates all other -supervaluations covering $v$. It then is immediate that $$\vrp_v := \sig _{U(v)^0} \circ \vrp^0_v: R {\ {\to} \ } (U(v)^0)^\wedge$$ is the initial supervaluation covering $v$, i.e., dominates all supervaluations covering $v$ (Theorem \[thm6.9\]). Already in [@IKR1 §7] we could prove that an initial supervaluation $\vrp_v$ covering $v$ exists, but obtained an explicit description of $\vrp_v$ only in the case that $M$ is cancellative, while now we obtain an explicit description of $\vrp_v$ in general (Scholium \[schl6.11\]). (N.B. If $M$ is cancellative, then $\vrp_v = \vrp_v^0$.) More generally, if $\vrp: R \to U$ is an -supervaluation covering $v$, then $$\htvrp:= \sig_U \circ \vrp : R \to \htU$$ is a supervaluation covering $v$, and $\htvrp \geq \psi$ for any other supervaluation $\psi$ covering $v$ with $\vrp \geq \psi$. We call $\vrp$ **tangible**, if $\vrp(R) \subset \tT(U) \cup \00$. It turns out that $\vrp^0_v$ is always tangible, but if $U(v)^0$ is not a supertropical semiring, i.e., $\vrp^0_v \neq \vrp_v,$ then $\vrp_v$ is not tangible, and this implies that no supervaluation covering $v$ is tangible. The last two sections §\[sec:7\], §\[sec:8\] are motivated by our interest to put a supervaluation $\vrp: R \to U$ covering $v$ to use in tropical geometry. It will be relevant to apply $\vrp$ to the coefficients of a given polynomial $f({\lambda}) \in R[{\lambda}]$ in a set ${\lambda}$ of $n$ variables (or a Laurent polynomial), and to study the supertropical root sets and tangible components of polynomials $g({\lambda}) \in F[{\lambda}]$ in $F^n$ (cf. [@IR1 §5, §7]) obtained from $f^\vrp({\lambda}) \in U[{\lambda}]$ by passing from $U$ to various supertropical semifields $F$. For this purpose it will be important to have some control on the set $$\{ a \in R {\ {|} \ } \vrp(a) \in M \} {\ {=} \ } \{ a \in R {\ {|} \ } \vrp(a) = v(a) \}.$$ Given an -supervaluation $\vrp: R \to Y$ covering $v: R \to M$, we construct a tangible -supervaluation $\tlvrp: R \to \tlU,$ which is minimal with $\tlvrp \geq \vrp$ (Theorem \[thm7.10\]). In §\[sec:8\] we then classify the -supervaluations $\psi$ with $\vrp \leq \psi \leq \tlvrp,$ called the **partial tangible lifts** of $\vrp.$ They are uniquely determined by their **ghost value sets** $$G(\psi) := \psi (R) \cap M,$$ cf. Theorem \[thm8.4\]. These are ideals of the semiring $M,$ and all ideals $\mfa \subset G(\vrp)$ occur in this way (Theorem \[thm8.7\]). Unfortunately the ghost value set $G(\psi)$ does not control the set $\{ a \in R {\ {|} \ } \psi(a) \in M\} $ completely. We can only state that this set is contained in $\iv(G(\psi))$. If $\vrp$ is a supervaluation, then $\chvrp := (\tlvrp)^\wedge$ is the supervaluation, which is a partial tangible lift of $\vrp$ having smallest ghost value set. Given sets $X,Y$ we mean by $Y \subset X$ that $Y$ is a subset of $X$, with $Y = X$ allowed. If $E$ is an equivalence relation on $X$ then $X/E$ denotes the set of $E$-equivalence classes in $X$, and $\pi_E: X \to X/E$ is the map which sends an element $x$ of $X$ to its $E$-equivalence class, which we denote by $[x]_E$. If $Y \subset X$, we put $Y/E := \{[x]_E {\ {|} \ } x \in Y\}.$ If $U$ is a supertropical semiring, we denote the sum $1+1$ in $U$ by $e$, more precisely by $e_U$ if necessary. If $x \in U$ the **ghost companion** $ex$ is also denoted by $\nu(x)$, and the **ghost map** $U \to eU$, $x \mapsto \nu(x)$, is denoted by $\nu_U$. If $\al: U \to V$ is a transmission, then the semiring homomorphism $eU \to eV$ obtained from $\al$ by restriction is denoted by $\al^\nu$ and is called the **ghost part** of $\al$. Thus $\al^\nu \circ \nu_U = \nu_V \circ \al$. $\tT(U)$ and $\tG(U)$ denote the sets of tangible and ghost elements of $U$, respectively, cf. [@IKR1 Terminology 3.7]. We put $\tT(U)_0 := \tT(U) \cup \00.$ If $v : R \to M $ is an -valuation we call the ideal $v^{-1}(0)$ of $R$ the **support** of $v$, and denote it by ${\operatorname{supp}}(v)$. Supertropical monoids {#sec:1} ===================== \[defn1.1\] A **supertropical monoid** $U$ is a monoid $(U, \cdot \, )$ (multiplicative notation) which has an absorbing element $0 := 0_U$, i.e., $0 \cdot x = 0$ for every $x \in U$, and a distinguished central idempotent $e := e_U$ such that the following holds: $$\forall x \in U: \quad ex =0 \ \Rightarrow \ x = 0.$$ Further a total ordering, compatible with multiplication, is given on the submonoid $M := eU$ of $U$. We then regard $M$ as a bipotent semiring in the usual way [@IKR1 §1]. If $U$ is a supertropical monoid, we would like to enrich $U$ by a composition $ U \times U \overset{+}{\longrightarrow} U$ extending the addition on $M$, such that $U$ becomes a supertropical semiring with $$1_U + 1_U = e_U.$$ We are forced to define the addition on $U$ as follows ($x,y \in U$), cf. [@IKR1 Theorem 3.11]: $$x+y = \left\{ \begin{array}{cccc} y & & \text{if} & ex < ey, \\ x & & \text{if} & ex > ey , \\ ex & & \text{if} & ex= ey. \\ \end{array} \right.$$ If this addition obeys the associativity and distributivity laws, we say that the supertropical monoid $U$ “is” a semiring. In the commutative case we have the following criterion. \[thm1.2\] A supertropical commutative monoid is a semiring iff the following holds: $$\begin{aligned} \Dis: \quad & \forall x,y,z \in U: \text{ If } 0 < ex < ey, \text{ but}\\ & exz = eyz, \text{ then } yz = eyz \text{ (i.e., } yz \in eU).\end{aligned}$$ In this case the semiring $U$ is supertropical. Let $x,y,z \in U$ be given, Obviously, $x +y = y +x $ and $x +0 = x$, and it is easily checked that $(x+y) + z = x + (y+z)$. It remains to investigate when we have $$\renewcommand{\theequation}{$*$}\addtocounter{equation}{-1}\label{eq:str.1} (x+y)z = xz + yz.$$ We assume without loss of generality that $ex \leq ey$. If $ex=0$, then $x=0$ and $(*)$ is true. If $ex =e y$, then $exz = eyz$, hence $x +y = ey$ and $xz+ yz = eyz$. Thus $(*)$ is true again. We are left with the case that $0 < ex< ey$. Then $x+y =y$ and $exz \leq eyz$. If $exz < eyz$, then $xz + yz = yz$, and $(*)$ is true. But if $exz = eyz$, then $xz+yz = eyz$, while $(x+y)z =yz$. Thus $(*)$ holds iff $yz = eyz$. We conclude that $(*)$ holds for all triples $x,y,z$ iff condition $\Dis$ is fulfilled. \[rmk:1.13\] When $U$ is not commutative, we have an analogous result. We just need to add the same condition $\Dis$ for the monoid $U^{\opp}$ obtained from $U$ by changing the multiplication $(x,y) \mapsto xy$ to $(x,y) \mapsto yx$; i.e., $$\begin{aligned} \Dis': \quad & \forall x,y,z \in U: \text{ If } 0 < ex < ey, \text{ but}\\ & ezx = ezy, \text{ then } zy = ezy.\end{aligned}$$ Given an element $x$ of a supertropical monoid $U$, we call $ex$ the **ghost** of $x$, and we denote the **ghost map** $U \to eU$, $x \mapsto ex$, by $\nu_U$, as we did before for $U$ a supertropical semiring. By a (two sided) **ideal** $\ga$ of a supertropical monoid $U$ we mean a monoid ideal of $U$, i.e., a nonempty subset $\mfa$ of $U$ with $U \cdot \mfa \subset \mfa$ and $ \mfa \cdot U \subset \mfa$. Notice that in the case that $U$ is a supertropical semiring, such a set $\mfa$ is indeed an ideal of the semiring $U$ in the usual sense, cf. [@IKR2 Remark 6.21]. We call $eU$ the **ghost ideal** of $U$. Many more definitions in [@IKR1] and [@IKR2] retain their sense if we replace the supertropical semirings by supertropical monoids, in particular the following one. \[defn1.3\] Let $U$ and $V$ be supertropical monoids. We call a map $\al: U \to V$ a **transmission**, if the following holds (cf. [@IKR1 §5]): $$\begin{aligned} {2} &TM1: \quad&& \alpha(0)=0,\\ &TM2: \quad &&\alpha(1)=1,\\ &TM3: \quad &&\forall x,y\in U:\quad \alpha(xy)=\alpha(x)\alpha(y),\\ &TM4:\quad&&\alpha(e_U)=e_V,\\ &TM5:\quad &&\forall x,y\in eU: \quad x \leq y \Rightarrow \alpha(x) \leq \alpha(y). \end{aligned}$$ (N.B. $\al$ maps $eU$ to $eV$ due to TM3 and TM4.) Notice that this means that $\al$ is a monoid homomorphism sending $0$ to $0$ and $e$ to $e$, which restricts to a homomorphism $\gm: eU \to eV$ of bipotent semirings. We then call $\gm$ the **ghost part** of $\al$, and write $\gm = \al^\nu$. We also say that $\al$ **covers** $\gm$ (as we did in [@IKR1] for $U$, $V$ commutative supertropical semirings). Notice that $\al(U)$ is a supertropical submonoid of $V$ in the obvious sense. We introduce two sorts of “kernels” of transmissions. \[defn1.4\] Let $\al: U \to V$ be a transmission between supertropical monoids. 1. The **zero kernel** of $\al$ is the set $$\mfz_\al := \{ x \in U \ | \ \al(x) =0\}.$$ 2. The **ghost kernel** of $\al$ is the set $$\mfA_\al := \{ x \in U \ | \ \al(x) \in eV \}.$$ These sets are ideals of $U$, and $M \cup \mfz_\al \subset \mfA_\al$. If $U$ is a semiring, then $M \cup \mfz_\al = M + \mfz_\al$, (cf. [@IKR2 Remark 6.21]). If $ \mfA_\al = M$, we say that $\al$ has **trivial ghost kernel**, and if $ \mfz_\al = \{ 0 \}$, we say that $\al$ has **trivial zero kernel**. \[thm1.5\] Let $\al: U \to V$ be a transmission between supertropical monoids, which is injective on the set $(eU) \setminus \{ 0\}$. 1. If $\al$ has a trivial ghost kernel, and if $V$ is a semiring, then $U$ is a semiring. 2. If $\al$ is surjective, and if $U$ is a semiring, then $V$ is a semiring. Again we prove this for commutative monoids, leaving the obvious modifications in the noncommutative case to the interested reader. We use the criterion for a supertropical monoid to be a semiring given in Theorem \[thm1.2\]. (i): Let $x,y,z \in U$ with $0 < ex < ey$ and $exz = eyz$. Then $$e\al(x)= \al (ex) < \al(ey) = e \al(y),$$ since $\al$ is injective on $(eU) \setminus \{ 0 \}$, and $$e \al(x) \al(z) = e \al(y) \al(z).$$ Since $V$ is a semiring, we deduce that $\al(yz) = \al(y) \al(z) \in eV$. Since $\al $ has a trivial ghost kernel, it follows that $yz \in eU$, as desired. (ii): Let $x,y,z \in U$ with $$0 < e \al(x) < e \al(y) \quad \text{and} \quad e \al(x) \al(z) = e \al(y) \al(z).$$ We are done if we verify that $ \al(y) \al(z) \in e V$. We have $0 < \al (ex) < \al (ey)$ and $\al(exz) = \al(eyz)$.The inequalities imply $0 < ex < ey$. [Case I]{} : $\al(eyz) = 0$. Now $e \al(yz) = 0$, hence $\al(yz)= 0$, hence $\al(y) \al(z) =0.$ [Case II]{} : $\al(eyz) \neq 0$. Now $exz \neq 0$ and $eyz \neq 0 $. Since $\al$ is injective on $(eU) \setminus \{ 0 \} $, it follows that $exz = eyz$. Since $0 < ex < ey$ and $U$ is a semiring, we conclude that $yz \in eU$, hence $ \al(y)\al(z) = \al (yz) \in eV$. Thus $\al(y) \al(z) \in eV$ in both cases. \[defn1.6\] If $U$ is a supertropical monoid, we call an equivalence relation $E$ on the set $U$ a **TE-relation**, if the following holds (cf. [@IKR2 §4]): $$\begin{aligned} {2} &TE1: \quad&& \text{$E$ is multiplicative, i.e., } \forall x,y,z \in E:\\ & && x \sim_E y {\quad {\Rightarrow} \quad } xz \sim_E yz \ , zx \sim_E zy \\ &TE2: \quad && \text{The equivalence relation $E | M $ is order compatible, i.e.:} \\ & & & \text{If $x_1,x_2,x_3, x_4 \in M$ and $x_1 \leq x_2$, $x_3 \leq x_4$, $x_1 \sim_E x_4$, $x_2 \sim_E x_3$,} \\ & & & \text{then $x_1 \sim_E x_2$. } \text{(Hence all $x_i$ are $E$-equivalent.)} \\ &TE3: \quad&& \text{If $x \in U$ and $ex \sim_E 0$, then $x \sim_E 0$.} \end{aligned}$$ We have the following almost trivial but important fact. \[thm1.7\] Let $U $ be a supertropical monoid and $E$ a TE-relation on $U$. Then the set $U/E$ of equivalence classes carries a unique structure of a supertropical monoid such that the map $$\pi_E: U \to U/E, \qquad x \mapsto [x]_E,$$ is a transmission. This is just some universal algebra. We are forced to define the multiplication on the set $\brU := U/E$ by the rule ($x,y \in U$) $$[x]_E \cdot [y]_E = [xy]_E.$$ This makes sense since the equivalence relation $E$ is multiplicative. Now $\brU$ is a monoid with absorbing element $0_\brU := [0_U]_E$. We are further forced to take as distinguished idempotent on $\brU$ the element $e_\brU:= [e_U]_E$. Clearly $$e_\brU \brU = M/E := \{ [x]_E \ | \ x \in M\}.$$ Finally, we are forced to choose on the submonoid $M/E$ of $U/E$ the total ordering given by ($x,y \in M$) $$[x]_E \leq [y]_E {\quad {\Leftrightarrow} \quad } x \leq y.$$ This total ordering is well-defined since the restriction $E|M$ of $E$ to $M$ is order compatible (cf. [@IKR2 §2]). It is now evident that $\brU$ has become a supertropical monoid and $\pi_E$ a transmission. \[rem1.8\] Conversely, if $\al: U \to V$ is a transmission from $U$ to a supertropical monoid $V$, then the equivalence relation $E(\al)$ is TE, and the map $[x]_{E(\al)} \mapsto \al(x)$ is an isomorphism from the supertropical monoid $U/E(\al)$ onto the (supertropical) submonoid $\al(U)$ of $V$. Let $U$ be a supertropical monoid and $M:= eU$. As in the case of supertropical commutative semirings [@IKR1 §6] we define an **MFCE-relation** on $U$ as an equivalence relation $E$ on $U$, which is multiplicative, and is fiber conserving, i.e., $x \sim_E y \Rightarrow ex = ey$. Then we have an obvious identification $M/E = M$, and $E$ is a TE-relation. The functorial properties of transmissions between supertropical semirings stated in [@IKR1 Proposition 6.1] remain true if we admit instead supertropical monoids, and can be proved in exactly the same way. Thus we get: \[prop1.9\] Let $\al: U \to V$ and $\bt: V \to W$ be maps between supertropical monoids. 1. If $\al $ and $\bt $ are transmissions, then $\bt \al$ is a transmission. 2. If $\al$ and $\bt\al$ are transmissions and $\al$ is surjective, then $\bt$ is a transmission. Starting from now we assume that **all occurring supertropical monoids are commutative**. But we mention that all major results to follow can be established also for noncommutative monoids with obvious modifications of the proofs (in a similar way as indicated in Remark \[rmk:1.13\]). This will save space and hopefully help the reader not to get distracted from the central ideas of the paper. At the time being, the commutative case suffices for the applications we have in mind (cf. the Introduction). We define the **category of supertropical monoids** $\STROP_m$ as follows: the objects of $\STROP_m$ are the (commutative) supertropical monoids, and the morphisms are the transmissions between them. $\STROP_m$ contains the category $\STROP$ of supertropical semirings as a full subcategory. \[schol1.12\] Let $U$ be a supertropical semiring and $E$ a TE-relation on $U.$ Then the map $\pi_E: U \to U/E$ from $U$ to the supertropical monoid $U/E$ is a morphism in $\STROP_m.$ Since $\STROP$ is full in $\STROP_m$, it follows that $\pi_E$ is a morphism in $\STROP_m$ iff the supertropical monoid $U/E$ is a semiring. This means in terms of [@IKR2 §4], that the TE-relation $E$ is transmissive iff the supertropical monoid $U/E$ is a semiring. We define **initial transmissions** and **pushout transmissions** in $\STROP_m$ as we defined such transmissions in [@IKR2 §1] in the category $\STROP$. Just repeat [@IKR2 Definition 1.2] and [@IKR2 Definition 1.3], respectively, replacing everywhere the word “supertropical semiring” by “supertropical monoid”. The pleasant news now is that in $\STROP_m$ the pushout transmission exists for any supertropical monoid $U$ and surjective homomorphisms $\gm$ from $M := eU$ to a bipotent semiring $N$, and that it has the same explicit description as given in [@IKR2 Theorem 1.11] and [@IKR2 Example 4.9] (in the category $\STROP$) if $N$ is cancellative. More precisely the following holds and can be proved by the same arguments as used in [@IKR2 Example 4.9] and the proofs of [@IKR2 Theorems 1.11 and 4.14]. \[thm1.10\] Let $U$ be a supertropical monoid and $\gm$ a homomorphism from $M:=eU$ onto a (bipotent) semiring [^6] $N$. We obtain a TE-relation $F(U,\gm)$ on $U$ by decreeing for $x,y \in U$: $$x \sim_{F(U,\gm)}y \quad \Leftrightarrow \quad \left\{ \begin{array}{lll} \text{either} & & x=y, \\ \text{or} & & x=ey,\ y =ey, \ \gm(ex) = \gm(ey), \\ \text{or} & & \gm(ex) = \gm(ey) =0. \\ \end{array}\right.$$ The map $$\pi_{\FUgm} \twoheadrightarrow U/\FUgm$$ is a pushout transmission in $\STROP_m$ covering $\gm$. {Here we identify $M/\FUgm =N$ in the obvious way.} In particular every initial transmission in $\STROP_m$ is a pushout transmission in $\STROP_m$. \[schol1.11\] We consider the special case that $U$ is a supertropical semiring. If the supertropical monoid $U/\FUgm$ happens to be a semiring, then it is clear that $\pi_{\FUgm}$ is a pushout in the category $\STROP$. Thus, following [@IKR2 Notation 1.7], we now have $$\FUgm = \EUgm, \quad \pi_{\FUgm} = \al_{U,\gm}.$$ But if $U/\FUgm$ is not a semiring then the relation $\FUgm$ is different from $\EUgm$. If $U$ is a supertropical semiring and $\mfa$ is an ideal of $U$ we introduced in [@IKR2 §5] the saturum $\satUa$ and the equivalence relation $\Ea = \EUa$, and obtained there descriptions of these objects, which do not mention addition but only employ multiplication and the idempotent $e$ ([@IKR2 Corollary 5.5, Theorem 5.4]). We now use these descriptions to *define* $\satUa$ and $\EUa$ if $U$ is only a supertropical monoid. \[defn1.12\] Let $\mfa$ be an ideal of the supertropical monoid $U$. 1. The **saturum** $\satUa$ of $\mfa$ is the set of all $x \in U $ with $ex \leq ea$ for some $a \in \mfa$. We call $\mfa$ **saturated** if $\satUa = \mfa$. 2. The equivalence relation $E:= \Ea := \EUa$ is defined as follows: $$\begin{array}{lll} x \sim_E y & \Leftrightarrow & \text{either } x =y \\ && \text{or } x \in \satUa, \ y \in \satUa.\\ \end{array}$$ As in [@IKR2 §5] the following fact can be verified in an easy straightforward way. \[prop1.13\] $ $ 1. $\satUa$ is again a monoid ideal of $U$. 2. The saturated ideals $\mfa$ correspond uniquely with the ideals $\mfc$ of $M$ which are lower sets of $M$, via $$\mfc = \mfa \cap M = e \mfa \quad \text{and} \quad \mfa = \{ x \in U | ex \in \mfc \}.$$ 3. $\EUa$ is a TE-relation on $U$. 4. If $\mfb$ is a second ideal of $U$ then $$\EUa = \EUb \quad \text{iff} \quad \satUa = \satUb.$$ 5. $\piEa$ has the zero kernel $\satUa$. 6. If $\al : U \to V$ is a transmission then the zero kernel $\mfz_\al$ is a saturated ideal, and $\al$ factors through $\piEa$ iff $\mfa \subset \mfz_\al$. As in [@IKR2 §5] we will use the alleviated notation $x \sima y $ for $x \simEa y$, $\pia$ for $\piEa$, and $[x]_{\mfa}$ for the equivalence class $[x]_{E(\mfa)}$. It is immediate how to generalize the definition of the equivalence relation $\EUaPh$ given in [@IKR2 §6] to the case that $U$ is a supertropical monoid. We study only the case where these relations are TE-relations, and we encode (without loss of generality) the homomorphic equivalence relation $\Phi$ on $M := eU$ by a homomorphism from $M$ to another semiring. All the following can be verified in a straightforward way. \[thm1.14\] Let $U$ be a supertropical monoid and $\gm: M \to M'$ a surjective homomorphism from $M := eU$ to a (bipotent) semiring  $M'$. Further, let $\mfA$ be an ideal of $U$ containing $M$ and the saturated ideal $$\mfa_\gm := \{ x \in U \ | \ \gm(ex) = 0 \}.$$ 1. The equivalence relation $\EUAgm$ on $U$, given by ($x_1,x_2 \in U$) $$\begin{array}{lll} x_1 \sim_E x_2 & \Leftrightarrow & \text{either } x_1 =x_2 \\ && \text{or } x_1 \in \mfA, \ x_2 \in \mfA, \ \gm(ex_1) = \gm(ex_2),\\ \end{array}$$ is a TE-relation. 2. The transmission $\pi_E : U \to U/E$ has the ghost kernel $\mfA$. The ghost part $(\pi_E)^\nu$ is the map $\gm: M \to M'$. {Here we identify the ghost ideal $M/E$ of $U/E$ with $M'$ in the obvious way.} 3. Assume that a transmission $\bt: U \to V$ to a supertropical monoid $V$ is given with ghost kernel $\mfA_\bt \supset \mfA$, further a homomorphism ${\delta}: M' \to eV$ is given such that ${\delta}\gm = \bt^\nu$. Then there exists a unique transmission $\eta: U/ E \to V$ with $\eta^\nu = {\delta}$ and $\eta \al = \bt$. (cf. the diagram following [@IKR2 Problem 1.1].) \[rem1.15\] It can be readily verified that $$E(U, M\cup \mfa_\gm, \gm) = F(U,\gm).$$ Thus the present theorem is a generalization of Theorem \[thm1.10\]. For any ideal $\mfA \supset M$ of $U$ we define $$E(U,\mfA) : = E(U,\mfA, {\operatorname{id}}_M),$$ as we did in [@IKR2 §6] for $U$ a supertropical semiring. In this special case Theorem \[thm1.14\] reads as follows. \[cor1.16\] $E(U, \mfA)$ is a TE-relation on $U$. A transmission $\al: U \to V$ (with $V$ a supertropical monoid) factors through $\pi_{E(U,\mfA)}$ iff $\mfA \subset \mfA_\al$. Canonical factorization of a transmission {#sec:2} ========================================= Given a surjective transmission $\al : U \to V$ between supertropical semirings we start out to write $\al$ as a composition of transmissions of simple nature in a somewhat canonical way. More precisely, we will do this first in the category $\STROP_m$ of supertropical monoids. Afterward we will prove that, if $U$ and $V$ are semirings, this “canonical factorization” remains valid in the smaller category $\STROP$ of supertropical semirings, which has our primary interest. Let $U$ and $V$ be supertropical monoids, and $M:=eU$, $N := eV$ their ghost ideals. We first exhibit the “transmissions of simple nature” we have in mind. These are the *ideal compressions, tangible fiber contractions*, and *strict ghost contractions* to be defined now. \[defn2.1\] As in the case that $U$ and $V$ are supertropical semirings (cf. [@IKR1 §6]) we say that a surjective transmission $\al: U \to V$ is a **fiber contraction** if the ghost part $\al^\nu = \gm: M \to N$ is an isomorphism. We say that $\al$ is a **fiber contraction over** $M$, if $N = M$ and $\gm = {\operatorname{id}}_M$. Notice that $\al$ is a fiber contraction iff the equivalence relation $\Eal$ is an MFCE-relation, hence $\al = \rho \circ \pi_E$ with $E$ an MFCE-relation and $\rho$ an isomorphism. Then $\al$ is a fiber contraction over $M$ iff $M=N$ and $\rho$ is an isomorphism over $M$. \[defn2.2\] We call a surjective transmission $\al: U \to V$ an **ideal compression**, if $\al$ is a fiber contraction over $M$ which maps $U \setminus \mfA_\al$ bijectively on to $V \setminus N = \tT(M)$. {Recall that $\mfA_\al$ denotes the ghost kernel of $\al$.} This means that $\al = \rho \circ \pi_{E(U, \mfA)}$ with $\mfA$ an ideal of $U$ containing $M$ and $\rho$ an isomorphism from $\brU := U/ E(U,\mfA) $ onto $V$ over $M$. We have $\mfA = \mfA_\al$. \[defn2.3\] We call a transmission $\al$ **tangible** if $$\al(\tT(U)) {\ {\subset} \ } \tT(V) \cup \{ 0 \},$$ and **strictly tangible** if $$\al(\tT(U)) {\ {\subset} \ } \tT(V).$$ In other terms, $\al$ is tangible iff $\mfA_\al = M \cup \mfz_\al$, and $\al$ is strictly tangible iff $\mfA_\al = M$. What does this means in the case that $\al$ is a fiber contraction? Clearly, a tangible fiber contraction $\al: U \to V$ is strictly tangible. If $E$ is an MFCE-relation on $U$, then $\pi_E: U \to U/E$ is tangible iff $E$ is **ghost separating** (cf. [@IKR2 Definition 6.19]), in other terms, iff $E$ is finer than the equivalence relation $E_t := E_{t,U}$ on $U$ which has the equivalence classes $\{ a\in \tT(U) {\ {|} \ } ex = a \}$, $a \in M \sm \{ 0\}$, and the one-point equivalence classes $\{y\}$, $y \in M$ (cf. [@IKR1 Example 6.4.v]). \[defn2.4\] We call the MFCE-relations $E$ on $U$ with $E \subset E_t$ **tangible MFCE-relations.** In this terminology the tangible fiber contractions $\al: U \to V$ over $M := eU$ are the products $$\al = \rho \circ \pi_T$$ with $T$ a tangible MFCE-relation on $U$ and $\rho$ an isomorphism over $M$. \[defn2.5\] We call a transmission $\al: U \to V$ a **ghost contraction**, if $\al^\nu$ is a homomorphism from $M$ onto $N$, and if $\al$ maps $U \sm (M \cup \mfz_\al)$ bijectively onto $\tT(V) = V \sm N$. This means that $$\al = \rho \circ \pi_{F(U,\gm)}$$ with $\gm: M \to N$ a surjective homomorphism, namely $\gm = \al^\nu$, and $\rho$ an isomorphism over $N$ from $U / F(U,\gm)$ to $V$. Thus $\al$ is a ghost contraction iff $\al$ is a surjective pushout transmission in $\STROP_m$. {The equivalence relation $F(U,\gm)$ had been introduced in Theorem \[thm1.10\].} \[defn2.6.0\] In the situation of Definition \[defn2.2\] and Definition \[defn2.5\], respectively, we also say abusively that $V$ is an ideal compression (resp. a ghost contraction) of $U$. \[defn2.6\] We call a ghost contraction $\al : U \to V$ **strict**, if $\al^{-1}(0) \subset M$. This means that $\al$ is also a strict tangible transmission. Notice that every ghost contraction $\al: U \to V$ with $\gm^{-1}(0) = \{ 0 \}$, $\gm = \al^\nu$, is strict, and that $\gm^{-1}(0) = \{ 0 \}$ iff $\mfz_\al = \{ 0\}$. Of course, there exist other strict ghost contractions. The maps $\pi_{F(U,\gm)},$ where $\gm: M \to N$ is a homomorphism with $\igm(0) \neq \00,$ but where $U$ has no tangibles with ghost companion in $\igm(0),$ are main examples for this. \[defn2.7\] If $\gm: M \to N$ is a surjective homomorphism for $M = eU$ to a semiring $N$, we put $$\mfa_{U,\gm}:= \{ x \in U \ | \ \gm(ex) =0\},$$ an ideal already used in Theorem \[thm1.14\]. In this notation the ghost contraction $\pi_{F(U,\gm)}$ is strict iff $\mfa_{U,\gm} \subset M$. \[thm2.8\] Let $\al:U \to V $ be a surjective transmission between supertropical monoids and let $\gm: M \to N$ denote the homomorphism between the ghost ideals $M := eU$, $N:= eV$ obtained from $\al$ by restriction, $\gm = \al^\nu$. 1. There exists a factorization $$\al = \mu \circ \bt \circ {\lambda}$$ with ${\lambda}$ and ideal compression of $U$, $\bt$ a strict ghost contraction, and $\mu$ a tangible fiber contraction over $N := eV$. 2. The factorization is essentially unique. More precisely, if $ \al = \mu' \circ \bt' \circ {\lambda}'$ is a second such factorization of $\al$, then there exist isomorphisms $\rho$ over $M$ and $\sig$ over $N$ (of supertropical monoids) such that $${\lambda}' = \rho {\lambda}, \quad \mu' = \mu \sig^{-1}, \quad \bt' = \sig \bt \rho^{-1}.$$ 3. In particular we can choose $${\lambda}= \pi_{E(U,\mfA)}: U \longrightarrow \brU := U/E(U,\mfA)$$ with $\mfA := \mfA_\al$, the ghost kernel of $\al$, $$\bt = \pi_{\FbUgm}: \brU \longrightarrow W := \brU/\FbUgm$$ and $\mu: W \onto V$ the resulting tangible fiber contraction over $N$ such that $\al = \mu \bt {\lambda}$ (see proof below). a\) Let $\gm := \al^\nu : M \to N$, $\mfA:= \mfA_\al$, and $\brU:= U/E(U,\mfA)$. Then $\al$ factors through ${\lambda}:= \pi_{E(U,\mfA)}$ in a unique way, $$\xymatrix{ \al: U \ar @{>}[r]^{{\lambda}} & \brU \ar@{>}[r]^ {\bral} & V, \\ }$$ with $\bral$ a surjective transmission having trivial ghost kernel. This is clear from [@IKR2 Proposition 6.20], adapted to the category of supertropical monoids. b\) We have $(\bral)^\nu = \gm$. Let $\bt = \pi_{\FbUgm}$. By Theorem \[thm1.10\] we know that $\bt$ is an initial transmission in the category $\STROP_m$ (even a pushout). Thus we have a unique transmission $$\mu: W := \brU/ \FbUgm \ \to \ V$$ such that $\bral = \mu \circ \bt$, hence $\al = \mu \circ \bt \circ {\lambda}$. From $(\bral)^\nu = \gm = \mu ^\nu \circ \bt ^\nu$ and $\bt^\nu = \gm$ it follows that $\mu^\nu$ is the identity of $N$. Since $\bral$ has trivial ghost kernel and $\bt$ is surjective, both $\bt$ and $\mu$ have trivial ghost kernels. We conclude that $\bt$ is a strict ghost contraction and $\mu$ is a tangible fiber contraction over $N$. Parts i) and iii) of the theorem are proven. c\) Retaining the transmissions ${\lambda}, \bt, \mu$ which we have defined above, we turn to the claim of uniqueness in part ii) of the theorem. Let $ \al = \mu' \circ \bt' \circ {\lambda}'$ another factorization of $\al$ of the kind considered here. Both $\bt'$ and $\mu'$ have trivial ghost kernel. Thus the ideal compression  ${\lambda}'$ has the same ghost kernel $\mfA$ as $\al$. We conclude that $${\lambda}' = \rho \pi_{E(U, \mfA)} = \rho {\lambda}$$ with some isomorphism $\rho$ over $M$. From $ \al = (\mu' \bt' \rho) {\lambda}$ we then conclude that $\mu' \bt' \rho = \bral$. Now $\bt' \rho$ is a strict ghost contraction covering $\gm$, since $\bt'$ is such a ghost contraction and $\rho$ covers ${\operatorname{id}}_M$. It follows that $$\bt' \rho = \sig \pi_\FbUgm = \sig \bt$$ with some isomorphism $\sig$ over $N$, and hence $\bt' = \sig \bt \irho$. We finally obtain $$\al = \mu' \sig \bt {\lambda}= \mu \bt {\lambda},$$ and then $\mu' \sig = \mu$. \[thm2.9\] Let $\al:U \to V $ be a surjective transmission between supertropical semirings, and assume that $$\xymatrix{ \al: U \ar @{>}[r]^{{\lambda}} & U_1 \ar@{>}[r]^ {\bt} & V_1 \ar @{>}[r]^{\mu} & V, \\ }$$ is a factorization of $\al$ as described in Theorem \[thm2.8\].i (in the category $\STROP_m$). Then both $U_1$ and $V_1$ are supertropical semirings, hence all three factors ${\lambda},\bt,\mu$ are morphisms in $\STROP$. ${\lambda}$ and $\mu$ are surjective and ${\lambda}^\nu = {\operatorname{id}}_M$, $\mu^\nu = {\operatorname{id}}_N$. Moreover $\mu$ has trivial ghost kernel. Thus $V_1$ is a semiring by Theorem \[thm1.5\].i, and $U_1$ is a semiring by Theorem \[thm1.5\].ii. \[cor2.10\] Let $\al:U \to V $ be a surjective transmission between supertropical semirings covering $\bral = \gm : M \to N $. Then for the supertropical semiring $$\brU := U / E(U, \mfA_\al)$$ the transmission $$\pi_\FbUgm : \brU \to \brU / \FbUgm$$ is pushout in the category $\STROP$. In other terms (cf. [@IKR2 Notation 1.7]) $$\FbUgm = \EbUgm.$$ Theorem \[thm2.9\] tells us that $\brU / \FbUgm$ is a supertropical semiring. We know from §\[sec:1\] that $\pi_\FbUgm$ is pushout in $\STROP_m$. A fortiori this transmission is pushout in $\STROP$. \[defn2.11\] Let $\al:U \onto V $ be a surjective transmission covering $\alnu = \gm : M \onto N $. We know by Theorem \[thm2.8\] that there exists a unique factorization $$\renewcommand{\theequation}{$*$}\addtocounter{equation}{-1}\label{eq:str.1} \al = \rho \circ \pi_T \circ \pi_\FbUgm \circ \pi_{E(U,\mfA)}$$ with $\mfA$ an ideal of $U$ containing $M \cup \mfz_\al$, $\brU := U/ E(U,\mfA)$, $T$ a tangible MFCE-relation on $W:= \brU / \FbUgm$, and $\rho$ an isomorphism from $W/T$ to $V$ over $N$. Here $\mfA,$ $T$, and hence $\rho$ are uniquely determined by $\al$. We call $(*)$ the **canonical factorization** of $\al$, and $\pi_\FbUgm$, $\pi_{E(U,\mfA)}$, $\pi_{T}$, $\rho$ the **canonical factors** of $\al$. If one of these maps is the identity map, we feel justified to omit it in the list of the canonical factors of $\al$. We discuss some simple cases of canonical factorizations. \[schol2.12\] (The case of $M=N$.) Assume that $U$ and $V$ are supertropical monoids with $eU = eV = M$. Let $\al:U \to V$ be a fiber contraction over $M$ with ghost kernel $\mfA = \mfA_\al$. 1. $\al$ has the factorization $\al = \mu \circ {\lambda}$ with ${\lambda}= \pi_{E(U,\mfA)}$ and $$\mu: \brU := U/ E(U,\mfA) \to V$$ a (strict) tangible fiber contraction over $M$. This is clear from Theorem \[thm2.8\] or directly from the universal property of $\pi_{E(U,\mfA)}$, cf. Corollary \[cor1.16\]. Then $\mu = \rho \circ \pi_T$ with $T$ a tangible MFCE-relation on $\brU$. Thus $\al$ has the canonical factors $\pi_{E(U,\mfA)}$ , $\pi_T$, and $\rho$. 2. If $U$ is a semiring then both $\brU$ and $V$ are semirings, as follows directly from Theorem \[thm1.5\].ii. \[exmp2.13\] (Factorization of a ghost contraction.) Assume $\al:U \onto V $ is a ghost contraction covering $\alnu = \gm : M \onto N $. Let $\mfa$ denote the zero kernel of $\al$, $\mfa := \mfz_\al$. 1. $\al$ has the factorization $\al = \bt \circ {\lambda}$ with $ {\lambda}= \pi_{E(U, M\cup \mfa)}$ and $$\bt: \brU := U / E(U, M\cup \mfa) \ \to \ V$$ a strict ghost contraction. We have $$\bt = \rho \circ \pi_{\FbUgm}$$ with $\rho$ an isomorphism from $\brU / \FbUgm$ to $V$ over $N$. Thus $\al$ has the canonical factors $\pi_{E(U, M\cup \mfA) }$, $\pi_\FbUgm$, and $\rho$. 2. We further have the factorization $$\bt = \brbt \circ \pi_{E(\bmfa)}$$ with $\bmfa:= {\lambda}(\mfa)$ and $$\brbt : \brU/E(\bmfa) = U/E(\mfa) \ \to \ V,$$ which is a strict ghost contraction with zero kernel $\{0\}$. Notice that the ideal $\mfa =\mfz_\al$ is saturated in $U$ and $\bmfa$ is saturated in $\brU$. \[exmp2.14\] (The transmissions $\pi_{E(U,\mfA,\gm)}$.) Let $U$ be a supertropical monoid and let $\gm: M \to N$ be a surjective homomorphism from $M = eU$ to a semiring $N$. Further let $\mfA$ be an ideal of $U$ containing $M \cup \mfa_{U,\gm}$. 1. Then the transmission $$\pi_{E(U,\mfA,\gm)}: U \to V : = U/{E(U,\mfA,\gm)}$$ (cf. Theorem \[thm1.14\]) has the canonical factorization $$\pi_{E(U,\mfA,\gm)} = \pi_\FbUgm \circ \pi_{E(U,\mfA)}$$ with $\brU := U/E(U,\mfA) $. Indeed we have ${E(U,\mfA,\gm)}/ E(U,\mfA) = F(\brU, \gm)$, as has been stated in [@IKR2 Theorem 6.22]. 2. The ghost contractions $\al: U \to V$ covering $\gm$ are precisely the maps $$\al = \rho \circ \pi_{E(U,\mfA,\gm)}$$ with $\mfA = M \cup \mfa_{U,\gm}$ and $\rho$ an isomorphism over $N$, as is clear from the above and Example \[exmp2.13\]. The canonical factors of a product of two basic transmissions {#sec:3} ============================================================= \[defn3.1\] Let $\al : U \to V$ be a surjective transmission between supertropical monoids, and let $\gm:= \al^\nu : M \onto N$ denote the ghost part of $\al$. We call $\al$ a **basic transmission**, if  $\al$ is of one of the following 4 types. ***Type 1*** : $\al = \pi_{\EUmfA}$ with $\mfA$ an ideal of $U$ containing $M$. ***Type 2*** : $\al = \pi_{\FUgm}$ and $\al^{-1}(0) = \gm^{-1}(0)$. ***Type 3*** : $\al = \pi_T$ with $T$ a tangible MFCE-relation on $U$. ***Type 4*** : $\al = \rho$ with $\rho$ an isomorphism over $M$. Thus in all cases except the second we have $M=N$ and $\gm = {\operatorname{id}}_M$. In short, the basic transmissions are the factors occurring in the canonical factorizations of transmissions (cf. Definition \[defn2.11\]). \[prob3.2\] Given basic transmissions $\al: U \to V$ of type $i$ and $\bt: V \to W$ of type $j$ with $1 \leq j \leq i\leq 4$, find the canonical factorization of $\bt \al$ explicitly. It would be easy to find these canonical factorizations up to an undetermined isomorphism  $\rho$ as first factor (cf. Definition \[defn2.11\]) by running through parts a) and b) of the proof of Theorem \[thm2.9\]. But we want a completely explicit description of all factors. For this we will rely on realizations of the quotient monoids $U/\EUmfA$, $U/\FUgm$, $U/T$ arising up in Definition \[defn3.1\], such that the basic transmissions $\pi_{\EUmfA}$, $\pi_{\FUgm}$, $\pi_T$ have a particulary well amenable appearance, \[convs3.2\] Let $U$ be a supertropical monoid, $\mfA$ an ideal of $U$ containing $M := eU$, furthermore $\gm:M \to N$ a surjective homomorphism to a (bipotent) semiring $N$ with $\mfa_{U,\gm} \subset M$ (cf. Definition \[defn2.7\]), and $T$ a tangible MFCE-relation on $U$. 1. We write $\mfA := M \dcup S$ with $S$ a subset of $\tT(U)$ such that $S \cdot \tT(U) \subset S \cup M$. Justified by [@IKR2 Theorem 6.16], adapted to the monoid setting, we declare that $U/\EUmfA$ is the subset $$U \sm S = (\tT(U) \sm S) {\ {\dcup} \ } M$$ of $U$, and, for any $x\in M$ $$\pi_{\EUmfA} (x) = \left\{ \begin{array}{lll} x & \text{if} & x \in U \sm S, \\[1mm] ex & \text{if} & x \in S.\\ \end{array} \right.$$ For $x,y \in U \sm S$ the product $x \odot y$ in $U/ \EUmfA$ is given by $$x \odot y = \left\{ \begin{array}{rll} xy & \text{if} & xy \notin S, \\[1mm] exy & \text{if} & xy \in S.\\ \end{array} \right.$$ 2. We identify $\tT(U / \FUgm) $ with $\tT(U)$ such that $[x]_\FUgm = x$ for $x \in \tT(U)$. Now $$\tT(U / \FUgm) = \tT(U) {\ {\dcup} \ } N$$ and, for $x \in U$, $$\pi_{\FUgm} (x) = \left\{ \begin{array}{lll} x & \text{if} & x \in \tT(U), \\[1mm] \gm(x) & \text{if} & x \in M.\\ \end{array} \right.$$ If $x,y \in \tT(U) $, the product $x \odot y$ in $U/ \FUgm$ is given by $$x \odot y = \left\{ \begin{array}{lll} xy & \text{if} & xy \in \tT(U), \\[1mm] \gm(xy) & \text{if} & xy \in M.\\ \end{array} \right.$$ 3. For $x \in M $ we identify $x$ with $[x]_T$ (as we usually did for MFCE-relations before, but notice that now $[x]_T = \{ x\}$). We have $$U / T = \tT(U)/ T {\ {\dcup} \ } M,$$ and, for $x \in U$, $$\pi_{T} (x) = \left\{ \begin{array}{lll} [x]_T & \text{if} & x \in \tT(U), \\[1mm] x & \text{if} & x \in M.\\ \end{array} \right.$$ If $x,y \in \tT(U) $ the product of $[x]_T $ and $[y]_T$ in $U/T$ is given by $$[x]_T \odot [y]_T = \left\{ \begin{array}{lll} [xy]_T & \text{if} & xy \in \tT(U), \\[1mm] xy & \text{if} & xy \in M.\\ \end{array} \right.$$ We also need more terminology on equivalence relations. \[defn3.4\] $ $ 1. If $\eta : X \to Y$ is a map between sets and $F$ is an equivalence relation on $Y$, then  $\eta^{-1}(F)$ denotes the equivalence relation on $X$ given by $$x_1 \sim_{\eta^{-1}(F)} x_2 {\quad {\iff} \quad } \eta(x_1) \sim_F \eta(x_2).$$ Thus $\pi_{\eta^{-1}(F)} = \pi_F \circ \eta$. 2. We further have a unique map $$\xymatrix{ \breta: X / \eta^{-1}(F) \ar @{>}[r] & Y / F }$$ such that the diagram $$\xymatrix{ X \ar @{>}[d]^{\eta} \ar @{>}[rr]^{\pi_{\eta^{-1}(F)} } & & X / \eta^{-1}(F) \ar @{>}[d]^{\breta} \\ Y \ar @{>}[rr]^{\pi_F} & & Y / F }$$ commutes. We denote this map $\breta$ by $\eta^F$, and then have the formula $$\pi_F \circ \eta {\ {=} \ } \eta^F \circ \pi_{\eta^{-1}(F)}.$$ 3. If $E$ is an equivalence relation on the set $X$ and $F$ is an equivalence relation on $X / E$, then let $F \circ E$ denote the equivalence relation on $X$ given by $$x_1 \sim _{F \circ E} x_2 {\quad {\iff} \quad } x_1 \sim_E x_2 \ \text{ and } \ [x_1]_E \sim_F [x_2]_E.$$ We identify the sets $X / F \circ E$ and $(X / E) / F$ in the obvious way. Then $$\pi_{F \circ E} = \pi _ F \circ \pi_E.$$ The following fact is easily verified. \[lem3.5\] Let $\eta: V \to U$ be a transmission between supertropical monoids, and let $E$ be a TE-relation on $U$. Then $\ieta(E)$ is a TE-relation on $V$, and the induced map $$\eta^E : V/ \ieta(E) \to U/E$$ is again a transmission. We have the formula $$\pi_E \circ \eta = \eta^E \circ \pi_{\ieta(E)}.$$ This lemma already gives us the solution of Problem \[prob3.2\] for $\al$ basic of type 4. \[prop3.6\] Let $U$ and $V$ be supertropical monoids with $eU = eV = :M$, and let $\rho: V \to U$ be a tangible fiber contraction over $M$ (e.g. $\rho$ is an isomorphism over $M$). 1. If $T$ is a tangible MFCE-relation on $U$, then $\irho(T)$ is a tangible MFCE-relation on $V$ and $$\pi_T \circ \rho = \rho' \circ \pi_{\irho(T)}$$ with $\rho' : V/ \irho(T) \to U/T$ the obvious homomorphism over $M$ induced by $\rho$, namely $\rho' = \rho^T$. 2. If $\gm: M \onto N$ is a surjective homomorphism from $M$ to a semiring $N$ with $$\mfa_{V, \gm} := \{ x \in V {\ {|} \ } \gm(ex) = 0\} \subset M,$$ hence also $\mfa_{V, \gm} \subset M$, then $$\pi_{\FUgm} \circ \rho = \rho' \circ \pi_{\FUgm}$$ with $$\rho' := \rho^{\FUgm} : V/\FVgm {\ {\to} \ } U/ {\FUgm}.$$ $\rho'$ is a tangible fiber contraction over $N$ with the following explicit description:\ Writing $ V/\FUgm = \tT(V) \dcup N$ and $ U/\FUgm = \tT(U) \dcup N$ (cf. Convention \[convs3.2\].b), we have $\rho'(x) = \rho(x)$ if $x \in \tT(U)$ and $\rho'(x) = x $ if $x \in N$. 3. Let $\mfA$ be an ideal of $U$ containing $M$ and let $\mfB := \irho(\mfA)$, which is an ideal of $V$ containing $M$. Then $$\pi_{\EUmfA} \circ \rho = \rho' \circ \pi_{\EVmfB},$$ with $$\rho' := \rho^{\EVmfB} : V/\EVmfB {\ {\to} \ } U/ {\EUmfA}.$$ $\rho'$ is a tangible fiber contraction over $N$, which has the following explicit description: We write $ \mfA = M \dcup S$ with $S \subset \tT(U)$, and have $ \mfB = M \dcup \irho(S)$ with $\irho(S) \subset \tT(V)$. By Convention \[convs3.2\].a $$U/ {\EUmfA} = (\tT(U) \sm S) {\ {\dcup} \ } M,$$ $$V / {\EVmfB} = (\tT(V) \sm \irho(S)) {\ {\dcup} \ } M.$$ The map $\rho'$ is obtained from $\rho$ by restriction to these subsets of $U$ and $V$. {N.B. It is easy to check directly that $\rho'$ respects multiplication.} 4. If $\rho$ is an isomorphism, then in all three cases $\rho'$ is again an isomorphism. Thus, if $\bt:U \to W$ is a basic transmission of type $i = 1,2,3$ and $\rho: V \to U$ is basic of type 4, then $\bt \rho = \rho' \bt'$ with $\bt', \rho'$ again of type $i$ and 4 respectively. Straightforward by use of Lemma \[lem3.5\]. \[rem3.7\] If in Proposition \[prop3.6\].b we dismiss the assumption that $\mfa_{V, \gm} \subset M$, we have the same result, but with a slightly more complicated description of the tangible fiber contraction $\rho'$ as follows: We now have natural identifications $$V / \FVgm = ( \tTV \sm \mfa_{V, \gm}) {\ {\cup} \ } N,$$ $$U / \FUgm = ( \tTU \sm \mfa_{U, \gm}) {\ {\cup} \ } N,$$ and then $$\rho'(x) = \left\{ \begin{array}{lll} \rho(x) & \text{if} & x\in \tTV \sm \mfa_{V, \gm} \\[1mm] x & \text{if} & x\in N. \\ \end{array} \right.$$ The following three propositions contain the solution of Problem \[prob3.2\] in the remaining cases $i \leq j \leq 3$. The stated canonical factorizations can always quickly be verified by inserting an element $x$ of $\tTU$ and comparing both sides. {For $x \in eU$ equality is always evident.} Often more conceptional proofs are also possible. With one exception we do not give the details. \[prop3.8\] (The case $i=j$.) Let $U$ be a supertropical monoid and $M := eU$. 1. Assume that $\mfA = M \dcup S$ is an ideal of $U$ containing $M$ and $\bmfB = M \dcup S'$ is an ideal of $\brU := U / \EUmfA = U \sm S$ containing $M$. {Thus $S$ and $S'$ are disjoint subsets of $\tTU$.} Then $$\mfB := \pi^{-1}_{\EUmfA}(\bmfB) = M \dcup S \dcup S'$$ is an ideal of $U$ and $$\pi_{\bEUmfB} \circ \pi _{\EUmfA} = \pi_{\EUmfB}.$$ 2. Let $\gm: M \to N$ and ${\delta}: N \to L$ be surjective homomorphisms of bipotent semirings with $\mfa_{U,\gm} \subset M$ and $\mfa_{V,{\delta}} \subset N$, where $V := U / \FUgm$. Then $$\pi_{F(V,{\delta})} \circ \pi _{\FUgm} = \pi_{F(U, {\delta}\gm)}.$$ 3. If $T$ is a tangible MFCE-relation on $U$ and $T'$ is a tangible MFCE-relation on $U/ T,$ then $T' \circ T$ is again a tangible MFCE-relation and $$\pi_{T'} \circ \pi_T = \pi_{T' \circ T}.$$ \[prop3.9\] (The case $\al = \pi_T$.) Let $U$ be a supertropical monoid, $T$ a tangible MFCE-relation on $U$, and $$\brU := U/T = (\tTU/T) \dcup M.$$ 1. If $\gm$ is a surjective homomorphism from $M:= eU$ to a semiring $N$, and $\mfa_{U,\gm} \subset M$, then $$\pi _{F(\brU, \gm)} \circ \pi_T = \pi_{T'} \circ \pi_{\FUgm},$$ where $T'$ is the tangible MFCE-relation on $V:= U/ \FUgm = \tTU \cup N$ defined as follows. For any $x,y \in V$ $$x \sim_{T'}y \quad \Leftrightarrow \quad \left\{ \begin{array}{lll} \text{either} & & x,y \in \tTU \ \text{ and } \ x \sim_T y, \\ \text{or} & & x = y \in N. \\ \end{array}\right.$$ 2. Let $\mfB$ be an ideal of $\brU$ containing $M$, hence $\mfB = M \dcup \brS$ with $\brS$ a subset of $\tTU / T$. We put $S:= \ipi _T(\brS) \subset \tTU$. Then $$\pi_{E(\brU, \mfB)} \circ \pi_T = \pi_{T'} \circ \pi_{E(U, \mfA)},$$ with $\mfA := \ipi_T(\mfB) = M \cup S$, and $T'$ a tangible MFCE-relation on $$V:= U/ \EUmfA = (\tTU \sm S) \dcup M.$$ $T'$ is obtained from $T$ by restriction to the subset $U \sm S$ of $U$. {Notice that $\tT(U) \sm S$ is a union of $T$-equivalence classes.} \[prop3.10\] (The remaining case $j=1, i=2$.) Let $U$ be a supertropical monoid and $\gm$ a homomorphism from $M := eU$ onto a semiring $N$ with $\mfa_{U,\gm} \subset M$. Let $V:= U/ \FUgm = \tTU \dcup N$. 1. The ideals $\mfA \supset M $ of $U$ correspond uniquely with the ideals $\mfB \supset N $ of $V$ via $\mfA = \ibt(\mfB),$ $\mfB = \bt(\mfA)$, where $\bt:= \pi_\FUgm$. We then have $\mfA = M \dcup A$, $\mfB = N \dcup S$ with the same set $S \subset \tTU = \tTV$, and $\EUmfA = \ibt(\EVmfB)$. Finally 2. $$\pi_{\EVmfB} \circ \pi_{\FUgm} = \pi_{F(\brU,\gm)} \circ \pi_{\EUmfA}$$ with $\brU := U/ \EUmfA = (\tTU \sm S) \dcup M$. i): The point is that for $S$ a subset of $\tTU$ we have $S \cdot \tTU \subset M$ in $U$ iff $S \cdot \tTU \subset N$ in $V$, since $\ibt(N) = M$. (Recall that we identified $\tT(U) = \tT(V)$.) ii): Again just insert a given $x \in \tTU$ in both sides of the equation and compare. \[prop3.11\] If $\al:U \to V$ and $\bt: V \to W$ are basic transmissions, $\al$ of type $i$ and $\bt$ of type $j \leq i$, cf. Definition \[defn3.1\], then in case $i = j$ the transmission $\bt \al$ is again basic of type $i$, and otherwise $\bt \al = \al' \bt'$ with $\al'$ basic of type $i$ and $\bt'$ basic of type $j$, and the new basic transmissions can be determined from $\al$ and $\bt$ in an explicit way. If $\al:U \to V$ and $\bt: V \to W$ are any transmissions with known canonical factors, then the canonical factorization of $\bt \al$ can be determined explicitly in at most $4+3+2+1 = 10$ steps. The semiring associated to a supertropical monoid;\ initial transmissions {#sec:4} =================================================== Let $U$ be a supertropical monoid and $M:= eU$ its ghost ideal. We start out to convert $U$ into a supertropical semiring in a somewhat canonical way. If $S$ is any subset of $U$, then the set $(US) \cup M$ is the smallest ideal of $U$ containing both $S$ and $M$. For convenience we introduce the notation $$\EUS {\ {:=} \ } E(U, US \cup M).$$ Corollary \[cor1.16\] tells us the meaning of this equivalence relation. \[schol4.1\] A transmission $\al: U \to V$ factors through the ideal compression $\pi_{\EUS}$ (in a unique way), iff the ghost kernel $\mfA_\al$ contains the set $S$. We now define a subset $S(U)$ of $\tTU$, for which the relation $E(U, S(U))$ will play a central role for most of the rest of the paper. \[defn4.2\] $ $ 1. We call an element $x$ of $U$ an **NC-product** (in $U$), if there exist elements $y,z$ of $U$ and $y'$ of $M$ with $$x = yz, \quad y' < ey, \quad y' z = eyz.$$ Here the label “NC” alludes to the fact that we meet a non-cancellation situation in the monoid $M$: We have $y' \neq ey$, but $y' z = eyz$. 2. We denote the set of all NC-products in $U$ by $D_0(U)$ and the set $D_0(U) \cup M$ by $D(U)$. We finally put $$S(U):= D(U) \sm M = D_0(U) \cap \tTU.$$ This is the set of tangible NC-products in $U$. Clearly $D_0(U) \cdot U \subset D_0(U)$. Thus $D(U)$ is an ideal of $U$ containing $M$. We have $$E(U,S(U)) = E(U,D_0(U)) = E(U,D(U)).$$ Theorem \[thm1.2\] tell us that $U$ is a semiring iff $S(U) = \emptyset$, i.e., $D(U) = M$. We compare the set $S(U)$ with $S(V)$ for $V$ an ideal compression of $U$. \[lem4.3\] Let $\mfA$ be an ideal of $U$ containing $M$, $\mfA = M \dcup S$ with $S \subset \tTU$. We regard $V:= U/ \EUmfA$ as a subset of $U$, as explained in Convention \[convs3.2\].a. Then $$S(V) = S(U) \sm S.$$ It is obvious from the description of $V$ in Convention \[convs3.2\].a. that $S(V) = S(U) \cap \tTV$, and we have $\tTV = \tTU \sm S$. \[lem4.4\] If $T$ is a tangible MFCE-relation then $$S(U/T) = S(U)/T.$$ Look at the description of $U/T$ in Convention \[convs3.2\].c. \[thm4.5\] $ $ 1. The supertropical monoid $\htU:= U/ \EUSU $ is a semiring. 2. The ideal compression $$\sig_U := \pi_{\EUSU}: U {\ {\to} \ } \htU$$ is universal among all fiber contractions $\al: U {\ {\to} \ } V$ with $V$ a semiring. More precisely, given such a fiber contraction $\al$, we have a (unique) fiber contraction\ $\bt: \htU \to V$ with $\al = \bt \circ \sig_U$. {N.B. If $\al$ is a fiber contraction over $M$, the same holds for $\bt$.} (i): By Lemma \[lem4.3\], the set $S(\htU)$ is empty; hence $\htU$ is a semiring. (ii): We may assume that $\al: U \to V$ is a fiber contraction over $M$, and then that $\al = \pi_T \circ \pi_{\EUmfA}$ with an ideal $\mfA \supset M$ of $U$ and $T$ a tangible equivalence relation on $\brU := U / \EUmfA$. By Lemma \[lem4.4\] the set $S(V)$ is empty iff $S(\brU)$ is empty, and by Lemma \[lem4.3\] this happens iff $S(U) \subset \mfA$. Then $\pi_{\EUmfA}$, and hence $\al$, factors through $\pi_{\EUSU} = \sig_U$ (cf. Scholium \[schol4.1\]). Conversely, if $\al = \bt \circ \sig_U$ with $\bt: \htU \to V$ a fiber contraction, then we know already by Theorem \[thm1.5\].ii that $V$ is a semiring, since $\htU$ is a semiring. We call $\htU$ the **the semiring associated to the supertropical monoid** $U$. \[thm4.6\] Assume that $U$ is a supertropical semiring and $\gm$ is a surjective homomorphism from $M := e U$ to a (bipotent) semiring $N$. Let $V:= U / \FUgm$, which may be only a supertropical monoid. Then $$\al:= \sig_V \circ \pi _\FUgm : U \onto V \onto \htV$$ (with $\htV$ and $\sig_V$ as defined in the preceding theorem) is the initial transmission from $U$ to a supertropical semiring covering $\gm$ (cf. [@IKR2 Definition 1.3]). In the Notation 1.7 of [@IKR2] this reads $$\sig_V \circ \pi_{\FUgm} = \al_{U,\gm}.$$ Let $\bt: U \onto W$ be a transmission to a supertropical semiring $W$ covering $\gm$ (in particular, $eW = N$). Since $\pi_\FUgm$ is an initial transmission in the category $\STROP_m$ covering $\gm$, we have a (unique) transmission $\eta: V \to W$ over $N$, hence fiber contraction over $N$, with $\bt= \eta \circ \pi_\FUgm$. Theorem \[thm4.5\] gives us a factorization $\eta = \vrp \circ \sig_V$ with $\vrp : \htV \to W$ again a fiber contraction over $N$. Then $$\bt = \vrp \circ \sig_V \circ \eta = \vrp \circ \al$$ is the desired factorization of $\bt$ in the category $\STROP$. Of course, the factor $\vrp$ is unique, since $\al$ surjective. We want to find the canonical factorization of $\al_{U,\gm}$. More generally we look for the canonical factors of $$\al:= \pi_\EVmfB \circ \pi_{\FUgm}$$ with $V:= U/ \FUgm$ and $\mfB$ an ideal of $V$ containing $N = eV$. We allow $U$ to be any supertropical monoid. We write $\mfB = N \cup S$ with $S \subset \tT(V)$. Similarly to Convention \[convs3.2\].b (which treats a special case) we have a natural identification $$\tTV = \tTU \sm \mfa_{U,\gm}$$ in such a way that for every $x\in U$ $$\pi_\FUgm(x) = \left\{ \begin{array}{lll} x & \text{if } \ x\in \tTU \sm \mfa_{U, \gm}, \\[1mm] \gm(ex) & \text{otherwise} & .\\ \end{array} \right.$$ We then obtain the following generalization of Proposition \[prop3.10\], arguing essentially in the same way as in §\[sec:3\]. \[lem4.7\] Let $V:= U / \FUgm$ and $\bt:= \pi_{\FUgm}$. 1. The ideals $\mfB$ of $V$ containing $N = eV$ correspond uniquely with the ideals $\mfA$ of $U$ containing $M \cup \mfa_{U,\gm}$ via $\mfA = \ibt(\mfB)$, $\mfB = \bt (\mfA)$. Writing $\mfB = N \dcup S'$, with $$S' \subset \tTV = \tTU \sm \mfa_{U,\gm},$$ we have $\mfA = M \dcup S$ with $$S := \{ x\in \tTU {\ {|} \ } \gm(ex) = 0 \} \cup S' \subset \tTU.$$ 2. $\pi_\EVmfB \circ \pi_\FUgm = \pi_{F(\brU, \gm)} \circ \pi_\EUmfA $, with $\brU := U / \EUmfA = (\tT(U) \sm S ) \dot \cup M.$ In the case $\mfB = D(V)$ we have $S' = S(V)$. Thus the elements of $S'$ are the products $yz \in \tTV \subset \tTU$ with $\gm(y') < \gm (ey)$ and $\gm(y'z) = \gm (eyz)$ for some $y' \in M$. Notice that this forces $\gm(y') \neq 0$. \[defn4.8\] Let $U$ be any supertropical monoid. We call an element $x$ of $U$ a\ **$\gm$-NC-product** (in $U$), if there exist elements $y' \in M$, $y \in U$, $z \in U$ with $x = yz$ and $\gm(y') < \gm (ey)$, $\gm(y'z) = \gm (eyz)$. We denote the set of these elements $x$ by $D_0(U,\gm)$ and the set $D_0(U,\gm) \cap \tTU$ of tangible $\gm$-NC-products by $S(U,\gm)$. Notice that $D_0(U,\gm)$ is an ideal of $U$. We further define $$D(U,\gm) := M \cup D_0(U,\gm) = M \cup S(U,\gm),$$ which is an ideal of $U$ containing $M$. In this terminology we have $S' = S(U,\gm)$. If $U$ is a semiring, then we read off from Theorem \[thm4.6\] and Lemma \[lem4.7\] the following fact. \[thm4.9\] Let $U$ be a semiring and $\gm: eU =M \to N$ a surjective homomorphism from $M$ to a semiring $N$. Then $\al_{U,\gm}$ has the canonical factorization $$\al_{U,\gm} = \pi_{F(\brU, \gm)} \circ \pi_{\EUS}$$ with $$S= \{ x \in \tTU {\ {|} \ } \gm(ex) = 0\} \cup S(U,\gm)$$ and $\brU = U/\EUS$. It is now easy to write down the equivalence relation $E(\al_{U,\gm}) = E(U,\gm)$ (cf. Notation 1.7 in [@IKR2]). We obtain \[cor4.10\] For $U$ and $\gm$ as above, the equivalence relation $\EUgm$ reads as follows ($x_1,x_2 \in U$): $$x_1 \sim_{\EUgm} x_2 \quad \Leftrightarrow \quad \left\{ \begin{array}{lll} \text{either} & & x_1 = x_2,\\[1mm] \text{or} & & x_1, x_2 \in D(U,\gm), \ \gm(e x_1) = \gm (ex_2), \\[1mm] \text{or} & & \gm(e x_1) = \gm (ex_2) = 0. \\ \end{array}\right.$$ If $N$ is cancellative then $S(U,\gm) = \emptyset$, and we fall back to the description of $\EUgm$ in [@IKR2 Theorem 1.11]. Our arguments leading to Theorems \[thm4.6\] and \[thm4.9\] make sense if we only assume that $U$ is a supertropical monoid. To spell this out we introduce an extension of Notation 1.7 in [@IKR2]. \[defn4.11\] Let $U$ be a supertropical monoid with ghost ideal $M:= eU$, and let\ $\gm: M \to N$ be a surjective semiring homomorphism. 1. We **define** $U_\gm = \htV$ with $V:= U/ \FUgm$. Thus $U_\gm$ is a supertropical semiring. 2. We **define** $$\al_{U,\gm} := \sig_V \circ \pi_\FUgm : U \to U_\gm.$$ 3. We finally **define** $\EUgm := E(\al_{U,\gm})$ and then have $U_\gm = U/\EUgm$. The arguments leading to Theorems \[thm4.6\] and \[thm4.9\] give more generally the following \[thm4.12\] $ $ 1. Given a transmission $\bt: U \to W$ from a supertropical monoid $U$ to a supertropical semiring $W$ covering $\gm$ (in particular $eW= N$), there exists a unique semiring homomorphism $\eta: U_\gm \to W$ over $N$ such that $\bt = \eta \circ \al_{U,\gm}$. 2. $\al_{U,\gm}$ has the same canonical factorization as given in Theorem \[thm4.9\] for $U$ a semiring, and $\EUgm$ has the description written down in Corollary \[cor4.10\]. Given a further semiring homomorphism ${\delta}: N \to L$ we may ask whether there exists a transmission $\eta: U_\gm \to U_{{\delta}\gm}$ covering ${\delta}$. In other words, is $E(U, {\delta}\gm) \supset E(U,\gm)$? In general the answer will be negative. Assume for simplicity that $\mfa_{U,\gm} = \mfa_{U,{\delta}\gm} = \{0 \}$ (or even, that $\igm(0) = \{ 0\}$, $\idl(0) = \{ 0\}$). We have to study the commutative diagram $$\xymatrix{ & & U_\gm \ar@{-->}[rr]^?_{\eta} & & U_{{\delta}\gm}\\ U \ar[rr] \ar[rru]^{\al_{U,\gm}} & & U/\FUgm \ar[rr] \ar[u] && U/ F(U, {\delta}\gm) \ar[u] \\ M \ar[rr]_\gm \ar[u] && N \ar[rr]_{\delta}\ar[u] && L \ar[u] }$$ where the unadorned arrows are the obvious natural maps. Assume further that $L$ is cancellative. Using Convention \[convs3.2\].b we have $$\tTU = \tT(U/\FUgm) = \tT(U/ F(U, {\delta}\gm)) = \tT(U_{{\delta}\gm}),$$ but $\tT(U_\gm) = \tTU \sm S(U,\gm)$. If $\eta$ would exist then $\eta \circ \al_{U,\gm}$ would restrict to the identity on $\tTU$. But this cannot happen as soon as $S(U,\gm)$ is not empty. In particular we realize the following: \[rmk:4.13\] If $U$ is a semiring, $\igm(0) = \{0 \}$, but $S(U,\gm) \neq \emptyset$, and if there exist a homomorphism ${\delta}: N \onto L$ of semirings with $L$ cancellative and $\idl(0) = \{0 \}$, then the initial transmission $\al_{U,\gm}$ in $\STROP$ is **not** a pushout transmission. It is not difficult to find cases where the situation described here is met. \[exmp4.14\] $ $ 1. We choose a totally ordered abelian group $G$ and a convex subgroup $H$ of $G$ with $H \neq \{ 1 \}$, $H \neq G $. The group $G /H $ is again totally ordered in a unique way such that the map $G \to G /H$, $g \mapsto g H$, is order preserving. Thus we have bipotent semifields $\tlM := G \cup \00$ and $\tlL := (G/H) \cup \00$ at hand. Let $$A:= \{ a \in G {\ {|} \ } a < H \} {\quad {\text{and}} \quad } A/ H := \{ aH {\ {|} \ } a \in A\}.$$ Then $$M := A \cup H \cup \00, \qquad L:= (A/H) \cup \{ 1 \cdot H\} \cup \00$$ are subsemirings of $\tlM$ and $\tlL$, and hence are cancellative bipotent semidomains. 2. We construct a noncancellative bipotent semiring $N$ as follows. As an ordered set, we put $$N := (A/ H ) {\ {\dot \cup} \ } H {\ {\dot \cup} \ } \00$$ with $0 < A/H < H$, keeping the given orderings on $A/H$ and $H$. We decree that the multiplication on $N$ extends the given multiplication on $A/H$ and $H$, and, of course, $0 \cdot x = x \cdot 0 = 0 $ for all $x\in N$, while $(aH) \cdot h := a H$ for $a \in A$, $h \in H$. This multiplication clearly is associative and commutative, has the unit element $1 \in H$, and is compatible with the ordering on $N$. Thus $N$ can be interpreted as a supertropical semiring. 3. We define maps $\gm : M \to N$ and ${\delta}: N \to L$ by putting $\gm(0) : = 0 $ and ${\delta}(0) : = 0 $, $\gm(a) := a H$, $\gm(h) := h$, ${\delta}(aH) := a H$, ${\delta}(h) := 1 \cdot H = 1_L$ for $a \in A$, $h \in H$. Clearly $\gm$ and ${\delta}$ are order preserving surjective monoid homomorphisms, hence are surjective semiring homomorphisms. We have $\igm(0) = \00$, $\idl(0) = \00$. 4. We choose a homomorphism $\tlv: \tlT \to G$ from an abelian group $\tlT$ onto $G$. Then, by [@IKR1 Construction 3.16], we have a supertropical semifield $$\tlU := \STR(\tlT, G, \tlv)$$ at hand with $\tT(U) = \tlT$, $\tG(\tlU) = G$, $ex = \tlv(x)$ for $x \in \tlT$. Let $$T:= \tlv^{-1}(A \cup H),$$ and let $v: T\onto A\cup H$ denote the monoid homomorphism obtained from $\tlv$ by restriction. The subsemiring $$U := \STR(T, A\cup H, v)$$ of $\tlU$ is a supertropical domain with ghost ideal $M$ and $\tT(U) = T.$ 5. We take elements $h_1 < h_2$ in $H$ and $a \in A$. Then we take elements $x_1, x_2, y \in T$ with $v(x_1) = h_1$, $v(x_2) = h_2$, $v(y) = a$. Now we have $$\gm(ex_1) = h_1 < \gm(ex_2) = h_2 {\quad {\text{and}} \quad } \gm(ex_1y) = \gm(e x_2 y ) = aH.$$ Also $x_2 y \in \tT(U)$. Thus $x_2 y \in S(U,\gm).$ Since for every $h_2 \in H$ there exists some $h_1 \in H$ with $h_1 < h_2$, this shows that $$\iv(H) \cdot \iv(A) \subset S(U,\gm).$$ In particular, $S(U,\gm) \neq \emptyset.$ We conclude by Remark \[rmk:4.13\] that the initial transmission $\al_{U,\gm} : U \to U_\gm$ in $\STROP$ is not pushout in $\STROP$ (and all the more not pushout in $\STROP_m$). $\h$-transmissions {#sec:5} ================== In [@IKR2 §6] the equivalence homomorphic relations on a supertropical semiring $U$ have been studied in detail. These are the TE-relations on $U$ such that the supertropical monoid $U/ E$ is a semiring and $\pi_E: U \to U/E$ is a homomorphism of semirings. It turned out that these relations can be completely characterized in terms of $U$ as a supertropical monoid, cf. [@IKR2 Proposition 6.4], where the crucial compatibility of $E$ with addition is characterized in this way. Having this in mind we define “-transmissions" for supertropical monoids, \[defn5.1\] We call a map $\al: U \to V$ between supertropical monoids an **-transmission**, if $\al$ is a transmission and has also the following property $$\begin{aligned} & {\HT } \quad \forall x,y \in U: \ \text{ If } \ ex < ey \ \text{ and } \ \al(ex) = \al(ey), \text{ then } \al(y) \in eV.\end{aligned}$$ We can read off the following result from [@IKR2 Proposition 6.4]: \[prop5.2\] Assume that $U$ and $V$ are supertropical semirings. Then a map\ $\al: U \to~V$ is an -transmission iff $\al$ is a semiring homomorphism. \[rem5.3\] We note in passing that in Definition \[defn5.1\] the condition $\HT$ can be formally relaxed as follows. $$\begin{aligned} & {\HTp } \quad \forall x,y \in U: \ \text{ If } \ 0< ex < ey \ \text{ and } \ \al(ex) = \al(ey), \text{ then } \al(y) \in eV.\end{aligned}$$ Indeed if $\al$ is a transmission and $ex = 0$, $\al(ex) = \al(ey)$, we conclude right away that $0 = \al(ex) = \al(ey)$, hence $\al(y) = 0 \in eV$. We now study -transmissions between supertropical monoids with the primary goal to gain a more insight into the variety of homomorphisms between supertropical semirings. If nothing else is said, letters $U, V, W$ will denote supertropical monoids. \[exmp5.4\] Every transmission $\al: U \to V$, such that $\gm:= \al^\nu$ is injective on $(eU) \sm \{0\}$, is an -transmission. Indeed, now the condition $\HTp$ is empty. The functorial properties of transmissions stated in Proposition \[prop1.9\] have a counterpart for -transmissions. \[prop5.5\] Let $\al: U \to V $ and $\bt: V \to W$ be maps between supertropical monoids. 1. If $\al$ and $\bt$ are -transmissions, then $ \bt \al $ is an -transmission. 2. If $\al$ and $\bt \al $ are -transmissions and $\al$ is surjective, then $\bt$ is an -transmission. By Proposition \[prop1.9\] we may already assume that $\al$ and $\bt$ are transmissions. (i): Assume that $x,y \in U$ are given with $0 < ex < ey$ and $\bt \al(ex) = \bt \al (ey) $. We have to verify that $\bt \al (y) \in eW$. Case 1 : $\al (ex) = \al (ey)$. Now $\al(y) \in eV$, since $\al$ is an -transmission. This implies $\bt \al(y) \in eW.$ Case 2 : $\al (ex) < \al (ey)$. Since $\bt \al (ex) = \bt \al (ey)$ and $\bt$ is an -transmission, again $\bt \al (y) \in eW$. ii): Let $x,y \in U $ be given with $0 < \al(ex) < \al(ey)$ and $\bt \al (ex) = \bt \al (ey)$. Then $0 < ex < ey$. We conclude that $\bt\al(y) \in eW$. Since $\al$ is surjective and $\al(ex) = e\al(x)$, $\al(ey) = e\al(y)$, this proves that $\bt $ is an -transmission. \[note5.6\] $ $ 1. We introduce two new categories: 1. Let $\strop_m$ denote the category whose objects are the supertropical monoids and morphisms are the -transmissions. Notice that this makes sense by Proposition \[prop5.5\].i. 2. Let $\strop$ denote the category whose objects are the supertropical semirings and morphisms are the semiring homomorphisms between supertropical semirings. 2. We further denote by $\sring$ the category of all semirings and semiring homomorphisms. $\strop$ is a full subcategory of $\sring$ and, due to Proposition \[prop5.2\] also of $\strop_m$. Thus we have the following chart of categories, where “$\subset$” means “subcategory” and “$\subset_{\full}$” means “full subcategory” $$\begin{array}{ccccc} & & \STROP & \subset_{\full} & \STROP_m \\ & &\cup & & \cup \\ \sring & \supset_{\full} & \strop & \subset_{\full} & \strop_m .\\ \end{array}$$ Moreover, in slightly symbolic notation, $$\strop = \STROP \cap \sring = \STROP_m \cap \sring = \STROP \cap \strop_m.$$ Our main concern will be to understand relations between $\strop_m$ and $\STROP$ within the category $\STROP_m$, in order to get an insight into $\strop = \STROP \cap \strop_m$. \[thm5.7\] Assume that $\al : U \to V$ is a surjective -transmission and $U$ is a semiring. Then $V$ is a semiring. Let $M : = eU$, $N := eV$, and $\gm := \al^\nu$. We check the condition $\Dis$ in Theorem \[thm1.2\] for the supertropical monoid $V.$ Since $\al$ and hence $\gm$ is surjective, this means the following. Let $y,z \in U$ and $y' \in M$ be given with $0 < \gm(y') < \gm(ey)$ and $\gm(y'z) = \gm (eyz)$. Verify that $\al(yz) \in N$! We have $y' < ey$. If $y'z = eyz$ then $yz \in M$, since $U$ is a semiring, and we conclude that $\al(yz) \in N$. There remains the case that $y' z < eyz$. Since $\al$ is an -transmission, we conclude again that $\al(yz) \in N$. In the following we assume that $U$ is a supertropical monoid and $\gm$ is a homomorphism from $M := eU$ onto a (bipotent) semiring $N$. We look for -transmissions $\al : U \to V$ which cover $\gm$. We introduce the set $$\SigzUgm:= \{ x\in \tT(U) {\ {|} \ } \exists x_1 \in M : x_1 < ex, \gm(x_1) = \gm (ex) \neq 0 \}.$$ \[prop5.8\] A transmission $\al : U \to V$ covering $\gm$ is an -transmission iff the ghost kernel $\mfA_\al$ contains the set $\SigzUgm$. By Scholium \[schol4.1\] it is evident that $\SigzUgm \subset \mfA_\al$ iff $\al$ obeys the condition $\HTp$ from above. We further introduce the set $$\SigUgm:= \SigzUgm \cup \{ x\in \tT(U) {\ {|} \ } \gm (ex) = 0 \},$$ and the supertropical monoids $$\brU := U/ E(U, \SigUgm ),$$ $$U_\gm^h := \brU/ F( \brU, \gm ),$$ finally the transmission $\alhUgm:U \to U_\gm^h ,$ given by $$\renewcommand{\theequation}{$*$}\addtocounter{equation}{-1}\label{eq:str.1} \alhUgm:= \pi_{F(\brU,\gm)} \circ \pi_{ E(U, \SigUgm )}.$$ Notice that $\alhUgm$ covers $\gm$ and is the product of an ideal compression and a strict ghost contraction, so that $(*)$ gives already the canonical factorization of $\alhUgm$. {N.B. The ideal $U \cdot \SigUgm $ contains $\mfa_{U,\gm}$, hence $\mfa_{\brU,\gm} \subset M$.} The ghost kernel of $\alhUgm$ contains the set $\SigzUgm$, and thus we know by Proposition \[prop5.8\], that $\alhUgm$ is an -transmission. We call $\alhUgm$ a **pushout in the category** $\bf \strop_m$, since the following holds: \[thm5.9\] Assume that ${\delta}: N \to L$ is a surjective homomorphism from $N$ to a semiring $L$ and $\bt:U \to W$ is an -transmission covering ${\delta}\gm$ (in particular $eW =L$). Then there exists a (unique) -transmission $\eta: \Uhgm \to W$ covering ${\delta}$ such that $\bt = \eta \circ \alhUgm$. Let $\al := \alhUgm$ and ${\lambda}:= \pi_{E(U, \SigUgm)}$. We retain the notations from above, hence have $ \Uhgm = \brU/ F(\brU, \gm)$ with $\brU = U / E(U, \SigUgm)$. Now observe that $\SigUgm$ is contained in $\Sig(U, {\delta}\gm)$. Indeed, let $x \in \SigUgm$. If $\gm(ex)= 0$, then ${\delta}\gm(ex) =0$. If there exists some $x_1 \in M$ with $x_1 < ex$ and $\gm(x_1) = \gm(ex)$, then either ${\delta}\gm (ex) = 0$, or ${\delta}\gm (ex) \neq 0$, and then $x \in \Sig_0(U,{\delta}\gm)$. Thus $x \in \Sig(U, {\delta}\gm)$ in all cases. Since $\bt$ is an -transmission covering ${\delta}\gm$, the ghost kernel of $\bt$ contains $\Sig(U,{\delta}\gm)$ and hence $\SigUgm$. Thus we have a factorization $$\xymatrix{ \bt:U \ar[r]_{{\lambda}} & \brU \ar[r]_{\brBt} & W \\ }$$ with $\brBt$ a transmission again covering ${\delta}\gm$. We have a commuting diagram (solid arrows) $$\xymatrix{ \brU \ar[drr]_{\pi_{F(\brU,\gm)}} \ar[drrrr]^{\brBt} & & & & \\ U \ar[rr]_\al \ar[u]^{\lambda}&& U_\gm^h \ar@{-->}[rr]_\eta & & W\\ M \ar[rr]_\gm \ar[u] & & N \ar[u] \ar[rr]_{\delta}& & L \ar[u] &. \\ }$$ Since $\pi_{F(\brU, \gm)}$ is a pushout in the category $\STROP_m$ (Theorem \[thm1.10\]), we have a transmission $\eta: \Uhgm \to W$ covering ${\delta}$ such that $\eta \circ \pi_{F(\brU, \gm)} = \brBt$, hence $\eta \circ \al = \brBt \circ {\lambda}= \bt$. Since both $\al$ and $\bt$ are -transmissions, also $\eta$ is an -transmission (Proposition \[prop5.5\].i). \[cor5.10\] (Entering the category $\strop$.) Let $V : =\Uhgm$ and $\al := \alhUgm : U \to V$. Then $$\htal := \sig_V \circ \al : U {\ {\to} \ } \htV$$ is the **initial -transmission from $U$ to a semiring covering $\gm$**, i.e., given an -transmission $\bt: U \to W$ covering $\gm$ with $W$ a semiring, there exists a (unique) fiber contraction ${\zeta}: \htV \to W$ over $N$ with $\bt = \vrp \circ \htal$. By Theorem \[thm5.9\], applied with ${\delta}= {\operatorname{id}}_N$, we have fiber contraction $\eta:V \to W $ over $N$ such that $\bt = \eta \circ \al $. By Theorem \[thm4.5\] there exists a fiber contraction ${\zeta}: \htV \to W$ over $N$ such that $\eta = {\zeta}\circ \sig_V$. Thus $\bt = {\zeta}\circ \sig_V \circ \al = {\zeta}\circ \htal$. \[thm5.11\] Assume that $\al: U \to V$ is a surjective -transmission, and $$\xymatrix{ \al:U \ar[r]_{{\lambda}} & \brU \ar[r]_{\bt} & W \ar[r]_{\mu} & \brW \ar[r]_{\rho} & V \\ }$$ is the canonical factorization of $\al$ (cf. §\[sec:2\]). 1. The factors ${\lambda}, \bt, \mu, \rho$ are again -transmissions. 2. If $U$ is a semiring, then the supertropical monoids $U, W, \brW, V$ are semirings, and the maps ${\lambda}, \bt, \mu, \rho$ are semiring homomorphisms. i): We know already by Example \[exmp5.4\] that ${\lambda}, \mu, \rho$ are -transmissions since they cover the identities ${\operatorname{id}}_M$ and ${\operatorname{id}}_L$ respectively (and $\rho$ is even an isomorphism). We have $\brU = U/ E(U, \SigUgm)$. Now observe that, if $a'$ and $a$ are elements of $M$ with $a' < a$, $\gm(a') = \gm(a) \neq 0$, then the fiber $\brU_a = \igm_\brU(a) $ contains no tangible elements. {Recall the definition of the set $\SigzUgm \subset \SigUgm$.} Thus every transmission $\bt': U \to W' $ covering $\gm$ trivially obeys the condition $\HTp$ from above (Remark \[rem5.3\]), hence is an -transmission. In particular, $\bt$ is an -transmission. ii): If $U$ is a semiring, we conclude by Theorem \[thm5.7\] successively, that $\brU$, $W, \brW, V$ are semirings. Now invoke Proposition \[prop5.2\] to conclude that ${\lambda}, \bt, \mu, \rho$ are semiring homomorphisms. We strive for an explicit description of the initial -transmission $\alhUgm$ covering $\gm$. Let $\tH(U,\gm)$ denote the ideal of $U$ generated by $\SigUgm \cup M$, i.e., $$\tH(U,\gm) := (U \cdot \SigzUgm) {\ {\cup} \ } \mfa_{U,\gm} {\ {\cup} \ } M.$$ We have $$\alhUgm = \pi_{F(\brU, \gm)} \circ \pi_{E(U,\tH)}$$ with $\tH:= \tH(U,\gm)$ and $\brU = U/E(U,\tH)$. Invoking Example \[exmp2.14\] we learn that $$\alhUgm = \pi_{E(U,\tH(U,\gm),\gm)}.$$ We denote the equivalence relation $E(U,\tH(U,\gm),\gm)$ more briefly by $H(U,\gm)$. Our task is to describe this TE-relation explicitly. We will succeed if $U$ is a semiring. \[lem5.12\] If $U$ is a semiring, then $$\tH(U,\gm) = \SigzUgm \cup \mfa_{U,\gm}\cup M.$$ $\tH(U,\gm)$ contains the set on the right hand side. We are done, if we verify that a product $xy$ with $x\in \SigzUgm$, $y\in \tT(U)$, $xy \in \tT(U) \sm \mfa_{U,\gm}$ lies in $\SigzUgm$. We have $x\in\tT(U)$. By definition of $\SigzUgm$ there exists some $x' \in M$ with $x' < ex$ and $\gm(x') = \gm (ex) \neq 0$. Now $x'y \leq exy$, but equality here would imply that $xy \in M$, since $U$ is a semiring (cf. Theorem \[thm1.2\]). Thus $x'y < exy$. Further $\gm(x'y) = \gm (exy) \neq 0 $. This shows that indeed $xy \in \SigzUgm$. Starting from this lemma and the general description of the relations $E(U, \mfA, \gm)$ in\ Theorem \[thm1.14\] it is now easy to write out the TE-relation $H(U,\gm)$. We obtain a theorem which runs completely in the category $\strop$. \[thm5.13\] Assume that $U$ is a supertropical semiring. The initial semiring homomorphism $\alhUgm$ covering $\gm$ is the map $$\pi_{H(U,\gm)} : U {\ {\to} \ } U/ H(U,\gm)$$ corresponding to the following equivalence relation $H(U,\gm)$ on $U$: If $x_1,x_2 \in U$, then $x_1 \sim_{H(U,\gm)} x_2$ [ [iff]{}  ]{} $\gm(ex_1) = \gm(ex_2)$ and either $x_1 = x_2$, or $x_1, x_2 \in M \cup \SigzUgm$, or $\gm(ex_1)=0$. Ordered supertropical monoids ============================= In the paper [@IKR3] the present authors studied supervaluations with values in a “totally ordered supertropical semiring" [@IKR3 Definition 3.1] and obtained – as we believe – natural and useful examples of such supervaluations. This motivates us now to define “ordered supertropical monoids". \[defn61.1\] An **ordered supertropical monoid**, or **$\OST$-monoid** for short, is a supertropical monoid $U$ equipped with a total ordering $\leq$ of the set $U$, such that the following hold: $$\begin{aligned} {2} &(\OST1): \quad&& \text{The ordering $\leq$ is compatible with multiplication, i.e., }\\ & && \text{for $x,y,z \in U$, } \ x\leq y {\ {\Rightarrow} \ } xz \leq yz; \\ &(\OST2): \quad && \text{The ordering $\leq$ extends the natural total order of the bipotent}\\ & & & \text{semiring $M:= eU$, i.e., if $x,y \in M,$ then } x \leq y {\ {\Leftrightarrow} \ } x \leq_M y; \\ &(\OST3): \quad&& 0 \leq 1 \leq e. \end{aligned}$$ \[lem61.2\] Let $x \in U.$ 1. Then $ 0 \leq x \leq ex.$ 2. If $x \in \tT(U),$ then $x < ex.$ (a): This follows by multiplying the inequality $0 \leq 1 \leq e$ by $x.$ (b): We have $x \leq ex$ and $x \neq ex$; hence $x < ex.$ As common, we call a subset $C$ of a totally ordered set $X$ **convex** (in $X$) if for all $x,y\in C$, $z \in X$ with $x \leq z \leq y$ also $z \in C.$ (This definition still makes sense if $X$ is only partially ordered, but now we do not need this generality.) \[prop61.3\] $ $ 1. For every $c \in M \sm \00$ both the fiber $U_c := \inu_U(c)$ and the tangible fiber $\tT(U)_c := \tT(U) \cap U_c$ are convex in $U$. 2. If $c,d \in M$ and $c <d,$ then $$c < \tT(U)_d < d$$ (i.e., $c < x < d$ for every $x\in \tT(U)_d$). (a): Let $x,y \in U_c$, $z\in U,$ and $x \leq z \leq y.$ We can conclude from $c =ex \leq ez \leq ey = c$ that $ez =c;$ hence $z \in U_c.$ Now assume that in addition $x,y \in \tT(U).$ If $z$ were ghost, hence $ez =c$, it would follow by Lemma \[lem61.2\].b that $y < ey = z.$ Thus $z \in \tT(U).$ (b): Let $x \in \tT(U)_d.$ Then $x < ex = d$, by Lemma \[lem61.2\].b. Suppose $x \leq c$. Then it would follow that $ex = d \leq c,$ which is not true. Thus $c < x.$ \[thm61.4\] If $(U, \leq)$ is an $\OST$-monoid, then $U$ is a semiring. We verify condition $\Dis$ in Theorem \[thm1.2\]. Let $x,y \in U,$ $x' \in M,$ and assume that $x' < ex,$ but $x'y = exy.$ From $x' < ex$ we conclude by Proposition \[prop61.3\].b that $x' \leq x$. Furthermore $x \leq ex$ by Lemma \[lem61.2\].a. Multiplying by $y,$ we obtain $$x'y \leq xy \leq exy \leq x'y,$$ and we conclude that $xy = exy.$ \[thm61.5\] If $(U, \leq)$ is an $\OST$-monoid, then addition[^7] in the semiring $U$ is compatible with the ordering $\leq,$ i.e., $(x,y,z\in U)$ $$x \leq y {\quad {\Rightarrow} \quad } x+z \leq y + z.$$ We conclude from $x \leq y$ that $ex \leq ey.$ We distinguish the cases $ex < ey$ and $ex = ey$, and go through various subcases. Case 1 : $ex < ey.$ 1. If $ez \leq ex,$ then $e(x + z) = ex + ez = ex$ and $y + z = y.$ Since $ex < ey,$ we conclude that $x + z \leq e(x+z) < y + z$ (cf. Proposition \[prop61.3\].b). 2. If $ex < ez < ey,$ then $x + z = z$, $y + z = y$, and we conclude from $ ez < ey$ that $x + z < y + z$. 3. If $ex = ey,$ then $x + z = z$, $y + z = ey$. Since $ z \leq ez$ we obtain that $x + z \leq y + z$. 4. If $ey < ez,$ then $x + z = z$, $y + z = z$, hence $x + z = y + z$. Case 2 : $ex = ey.$ 1. If $z < ex,$ then $x + z = x,$ $y + z = y.$ 2. If $z = ex,$ then $x + z = ex,$ $y + z = ey.$ 3. If $ ex < z,$ then $x + z = y + z = z.$ Thus in all three cases $x + z \leq y + z. $ Starting from now we denote an $\OST$-monoid $(U, \leq)$ by the single letter $U$. From Theorems \[thm61.4\] and \[thm61.5\] it is obvious that the present $\OST$-monoids are the same objects as the totally ordered supertropical semirings defined in [@IKR3 Definition 3.1]. Examples of these structures can be found in [@IKR3 §3, §4, §6]. \[defn61.6\] Assume that $U$ and $V$ are $\OST$-monoids. We call a transmission $\al: U \to V$ (cf. Definition \[defn1.3\]) **monotone**, if $\al $ is compatible with the ordering on $U$ and $V$, i.e., $$\forall x,y \in U: \quad x \leq y {\quad {\Rightarrow} \quad } \al(x) \leq \al(y).$$ \[thm61.7\] Every monotone transmission $\al: U \to V$ is a semiring homomorphism. We verify condition $\HT$ in Definition \[defn5.1\], and then will be done by Proposition \[prop5.2\]. Let $x,y \in U$ with $ex < ey$ and $\al(ex) = \al(ey)$. By Proposition \[prop61.3\] we have $ex < y < ey.$ Applying $\al$, we obtain $$\al(ex) {\ {\leq} \ } \al(y) {\ {\leq} \ } \al(ey) {\ {=} \ } \al(ex),$$ hence $\al(ex) = \al(ey)$. But $\al(ey) = e\al(y)$ (cf. Definition \[defn3.1\]), and we conclude that $\al(y) \in eV,$ as desired. Another proof, which relies more on the semiring structure of $U $ and $V$, can be found in [@IKR3 §5]. \[defn61.8\] Every bipotent semiring can be regarded as an $\OST$-monoid. (This is the case $1 = e$.) If $U$ is an $\OST$-monoid, $M = eU$ (our present overall assumption), then $\nu_U: U \to M $ is a monotone transmission. We indicate a way how to obtain new $\OST$-monoids from given ones. First we quote a general fact about total orderings (cf. e.g. [@IKR2 Remark 4.1]). \[deflem61.9\] Let $X$ be a totally ordered set and $f: X \twoheadrightarrow Y$ a map from $X$ onto a set $Y.$ Then there exists a (unique) total ordering on $Y$, such that $f$ is order preserving, iff all fibers $f^{-1} (y),$ $y\in Y,$ are convex in $X.$ We call this total ordering the **ordering on $Y$ induced by $f$**. N.B. This ordering on $Y$ can be characterized as follows: For $x_1, x_2 \in X$ $$f(x_1) < f(x_2) {\quad {\Rightarrow} \quad } x_1 < x_2 {\quad {\Rightarrow} \quad } f(x_1) \leq f(x_2).$$ Alternatively, we can state: $$\begin{array}{ll} \text{If } x_1 < x_2, & \text{then } f(x_1) \leq f(x_2), \\ \text{If } x_1 > x_2, & \text{then } f(x_1) \geq f(x_2). \\ \end{array}$$ \[thm61.10\] Assume that $U$ is an $\OST$-monoid, $V$ is a supertropical monoid, and $\al: U \to V$ is a surjective transmission. Assume further that for every $p \in V$ the fiber $\ial(p)$ is convex in $U$. Then $V$, equipped with the total ordering induced by $\al$, is again an $\OST$-monoid. We verify the axioms $\OST1$-$\OST3$ in Definition \[defn61.1\] for the induced ordering $\leq_V$ on $V.$ $(\OST 1):$ Let $x,y,z \in U$ and $\al(x) < \al(y)$. Then $x < y,$ hence $xz \leq yz$, hence $$\al(x) \al(z) {\ {=} \ } \al(xz) {\ {\leq_V} \ } \al(yz) {\ {=} \ } \al(y) \al(z).$$ $(\OST 2):$ Let $M := eU,$ $N := eV. $ On $U$ and $V$ we have the given orderings $\leq_U,$ $\leq_V$, and on $M$ and $N$ we have the natural orderings $\leq_M,$ $\leq_N$ as bipotent semirings. The ordering $\leq_U$ restricts on $M$ to $\leq_M.$ We have to verify that $\leq_V$ restricts on $N$ to $\leq_N.$ The map $\al : U \to V$ restricts to a semiring homomorphism $\gm: M \to N$, which consequently is compatible with $\leq_M$ and $\leq_N$. Let $x,y \in M.$ If $\al(x) <_V \al(y)$ then $x < _U y,$ hence $x <_ M y$, hence $\gm(x) \leq _N \gm(y).$ Thus $$\al(x) \leq _V \al(y) {\quad {\Rightarrow} \quad } \gm(x) \leq _N \gm(y).$$ Conversely, if $\gm(x) < _N \gm(y)$, then $x < _M y$, hence $x <_U y$, hence $\al(x) \leq _V \al(y).$ Thus $$\gm(x) \leq _N \gm(y) {\quad {\Rightarrow} \quad } \al(x) \leq _V \al(y).$$ This proves that indeed the ordering $\leq_V$ restricts to $\leq_N$ on $N.$ $(\OST 3):$ Applying $\al$ to $0 \leq 1 \leq e$ in $U$, we obtain $0 \leq 1 \leq e$ in $V.$ We are ready for the main result of this section, which roughly states that, given a monotone transmission $\al : U \to V,$ the canonical factors of $\al$ may be viewed as monotone transmissions in a unique way. We will relay on three easy lemmas. \[lem61.11\] Assume that $U , V, W$ are $\OST$-monoids, and $\al: U \to V,$ $\bt: V \to W$ are transmissions. Assume further that $\al$ and $\bt \circ \al$ monotone and $\al$ surjective. Then $\bt$ is monotone. If $x,y \in U$ and $\al(x) < \al(y)$, then $x < y$, hence $\bt\al(x) \leq \bt\al(y).$ Thus $\al(x) \leq \al(y)$ implies $\bt(\al(x)) \leq \bt(\al(y))$. \[lem61.12\] Assume that $\al: U \to V$ is a monotone transmission (between $\OST$-monoids). Let $\mfA$ denote the ghost kernel of $\al,$ $\mfA = \mfA_\al$. Then for any $c \in eU$ the fiber $\mfA_c := \mfA \cap U_c$ is an upper set of the totally ordered set $U_c.$ Assume that $x \in \mfA_c,$ $y \in U_c,$ and $x < y.$ Then $x < y \leq c,$ hence $\al(x) \leq \al(y) \leq \al(c).$ Since $x$ lies in the ghost kernel $\mfA$ of $\al$ we have $$\al(x) {\ {=} \ } e \al(x) {\ {=} \ } \al(ex) {\ {=} \ } \al(c).$$ It follows that $\al(y) = \al(c) \in eV,$ hence $y \in \mfA,$ hence $y \in \mfA_c.$ \[lem61.13\] Assume that $\al: U \to V$ is a monotone transmission with trivial ghost kernel. Let $\gm = \al^\nu:M\to N$ denote the ghost part of $\al.$ Then $U_c = \{ c\}$ for any $c \in M$ such that there exists some $c_1 < c$ in $M$ with $\gm(c_1) = \gm(c).$ Precisely this has been verified in the proof of Theorem \[thm61.7\]. \[thm61.14\] Assume that $U , V$ are an $\OST$-monoids, and $\al: U \to V$ is a surjective monotone transmission. Assume further that $$\xymatrix{ \al:U \ar[r]_{{\lambda}} & \brU \ar[r]_{\bt} & W \ar[r]_{\mu} & \brW \ar[r]_{\rho} & V \\ }$$ is a canonical factorization (cf. §\[sec:2\]) of the transmission $\al$. Then the monoids $\brU, W, \brW$ can be equipped with total orderings (in a unique way), such that they become $\OST$-monoids and all factors ${\lambda}, \bt, \mu, \rho$ are monotone transmissions. a\) Let $\gm := \al^\nu: M \to N$ denote the ghost part of the transmission $\al$ and $\mfA$ denote the ghost kernel of $\al.$ Without loss of generality we may assume that $$\brU = U/ E(U,\mfA), \qquad {\lambda}= \pi_{E(U,\mfA)}, \qquad W= \brU/{F(\brU,\gm)},$$ $$\bt = \pi_{F(\brU,\gm)}, \qquad \brW = V, \qquad \rho = {\operatorname{id}}_V.$$ For any $c\in M$ we have $\ilm(c) = \mfA_c,$ which by Lemma \[lem61.12\] is an upper set $U_c,$ hence is convex in $U_c.$ Since $U_c$ is convex in $U$, it follows that $\ilm(c)$ is convex in $U.$ Invoking Lemma \[deflem61.9\], we equip the monoid $\brU$ with the total ordering induced by ${\lambda},$ and then know by Theorem \[thm61.10\] that $\brU$ has become an $\OST$-monoid and ${\lambda}$ has become a monotone transmission. By Lemma \[lem61.11\] also $ \mu \circ \bt: \brU \to V$ is monotone. b\) Replacing $U$ by $\brU,$ we are allowed to assume henceforth that $\al: U \to V$ has trivial ghost kernel, and may focus on the canonical factorization $\al = \mu \circ \bt$ with $ \bt = \pi_{F(U,\gm)}$ and $\mu : W \to V$ a tangible fiber contraction. We use the identifications in Convention \[convs3.2\].b to handle $W = U/{F(U,\gm)}$ and $\bt = \pi_{F(U,\gm)}$. For any $d\in N = eW$ the tangible fiber $\tT(W)_d$ is the union of all fibers $\tT(U)_c$ with $c \in \igm (d).$ Let $L(\gm)$ denote those $c \in M$ such that $c \neq 0$ and $c$ is the smallest element of $\igm(\gm(c))$. Lemma \[lem61.13\] tells us that $\tT(U)_c \neq \emptyset$ if $ c \in M\sm L(\gm).$ Thus we have the following picture: If $d\in N$ then $\tT(W)_d = \tT(U)_c$ if there exists $c\in L(\gm)$ with $\gm(c) =d,$ and this $c$ is then unique. Otherwise $\tT(W)_d = \emptyset.$ c\) Looking again at Convention \[convs3.2\].b we see that $\bt$ has the following fibers: If $p \in \tT(W)_d$, $d = \gm(c)$ with $c\in L(\gm)$, then $\ibt(p) = \{ p\}$ (using the identifications in Convention \[convs3.2\].b). If $d\in N$, then $\ial(d)= \igm(d).$ Recalling Proposition \[prop61.3\], we see that $\igm(d)$ is convex in $U.$ Thus all fibers of $\bt$ are convex in $U$. Invoking again Lemma \[deflem61.9\] and Theorem \[thm61.10\], we equip $W$ with that total ordering induced by $\bt$, which makes $W$ an $\OST$-monoid and $\bt$ a monotone transmission. By Lemma \[lem61.11\] we conclude that also $\mu$ is a monotone transmission. $\m$-supervaluations {#sec:6} ===================== \[defn6.1\] Let $R$ be a semiring. An **-supervaluation on $R$** is a map $\vrp: R \to U$ to a supertropical monoid which fulfills the axioms ${{\operatorname{SV1}}}$-${{\operatorname{SV4}}}$ required for a supervaluation in [@IKR1 Definition 4.1], there for $U$ a supertropical semiring. To repeat, $$\begin{aligned} {2} &{{\operatorname{SV1}}}:\ &&\varphi(0)=0,\\ &{{\operatorname{SV2}}}:\ &&\varphi(1)=1,\\ &{{\operatorname{SV3}}}:\ &&\forall a,b\in R: \varphi(ab)=\varphi(a)\varphi(b),\\ &{{\operatorname{SV4}}}:\ &&\forall a,b\in R: e\varphi(a+b)\le e(\varphi(a)+\varphi(b))\quad [=\max(e\varphi(a),e\varphi(b))].\end{aligned}$$ We then say that $\vrp$ **covers** the -valuation $$e \vrp : R {\ {\to} \ } eU, \qquad a \mapsto e\vrp(a).$$ Most notions developed for supervaluations in [@IKR1], [@IKR2 §2] make sense for -supervaluations in the obvious way and will be used here without further explanation, but we repeat the definition of dominance. \[defn6.2\] Assume that $\vrp: R \to U$ and $\psi: R \to V$ are -supervaluations. We say that $\vrp$ **dominates** $\psi$ and write $\vrp \geq \psi$, if for all $a, b \in R$ the following holds. $$\begin{aligned} {3} &{{\operatorname{D1}}}.\quad && \varphi(a)=\varphi(b)& &\Rightarrow \ \psi(a)=\psi(b),\\ &{{\operatorname{D2}}}.\quad && e\varphi(a)\le e\varphi(b)&&\Rightarrow \ e\psi(a)\le e\psi(b),\\ &{{\operatorname{D3}}}.\quad && \quad\varphi(a)\in eU && \Rightarrow \ \psi(a)\in eV.\end{aligned}$$ If $\vrp: R \to U$ is an -supervaluation and $\al: U \to V$ is a transmission, then clearly $\al \circ \vrp$ is an -supervaluation dominated by $\vrp$. Conversely, if $\vrp:R \to U$ is an -supervaluation which is surjective (i.e., $U = \vrp(R) \cup e \vrp(R)$, cf. [@IKR1 Definition 4.3]), and $\psi:R \to V$ is an -supervaluation dominated by $\vrp$, there exists a (unique) transmission $\al: U \to V$ with $\psi = \al \circ \vrp$. This can be proved in exactly the same way as done in [@IKR1 §5] for supervaluations. If $\vrp$ and $\psi$ cover the same -valuation $v: R \to M$, then $\al$ is a fiber contraction over $M$ (hence an -transmission). Let now $v: R \to M$ be a fixed -valuation. We call any -supervaluation $\vrp$ with $e \vrp = v$ an **-cover** of $v$. We call two -covers $\vrp: R \to U$ and $\psi: R \to V$ **equivalent**, if $\vrp \geq \psi$ and $\psi \geq \vrp$. If $\vrp$ and $\psi$ are surjective this means that $\psi = \al \circ \vrp$ with $\al : U \to V$ an isomorphism over $M$. We further denote the equivalence class of an -cover $\vrp$ of $v$ by $[\vrp]$, and the set of all these classes by ${\operatorname{Cov}}_m(\vrp)$. This set is partially ordered by declaring $$[\vrp] \geq [\psi] {\quad {\text{iff}} \quad } \vrp \geq \psi.$$ We now assume for simplicity and without loss of generality *that $v$ is surjective*. Then every class ${\zeta}\in {\operatorname{Cov}}_m(v)$ can be represented by a surjective -supervaluation. \[prop6.3\] If $\vrp: R \to U$ is an -cover of $v$, the subset $$C(\vrp) := \{ [\psi] \in {\operatorname{Cov}}_m(v) {\ {|} \ } \vrp \geq \psi \}$$ of the poset ${\operatorname{Cov}}_m(v)$ is a complete lattice. It has the top element $[\vrp]$ and the bottom element $[v]$. We may assume that the -supervaluation $\vrp: R \to U$ is surjective. Let $\MFC(U)$ denote the set of all $\MFC$-relations on $U$. This set is partially ordered by inclusion, $$E_1 \leq E_2 {\quad {\text{iff}} \quad } E_1 \subset E_2.$$ {We view the equivalence relations $E_i$ as subsets of $U \times U$ in the usual way.} We have a bijection $$\MFC(U) {\ {\tilde{\longrightarrow}} \ } C(\vrp), \qquad E \mapsto [\pi_E \circ \vrp],$$ since every fiber contraction $\al$ over $M$ is of the form $\rho \circ \pi_E$, with $E \in \MFC(U)$ uniquely determined by $\al$ and $\rho$ an isomorphism over $M$. Clearly the bijection reverses the partial orders on $\MFC(U)$ and $C(\vrp)$. Now it can be proved exactly as in [@IKR1 §7] for $U$ a supertropical semiring, that the poset $\MFC(U)$ is a complete lattice. Thus $C(\vrp)$ is a complete lattice. We construct a supertropical monoid $U$ which will be the target of an -cover\ $\vrp: R \to U$ dominating all other -covers. Let $\mfq := \iv(0) = {\operatorname{supp}}(v)$. As a set we define $U$ to be the disjoint union of $R \sm \mfq$ and $M$, $$U = (R \sm \mfq) {\ {\dot \cup} \ } M.$$ We introduce on $U$ the following multiplication: For $x,y \in U$ $$x \Udot y = y \Udot x = \left\{ \begin{array}{lll} x \Rdot y & \text{if} & x,y, xy \in R \sm \mfq, \\[1mm] 0 & \text{if} & x,y \in R \sm \mfq, \ xy \in \mfq, \\[1mm] v(x) \Mdot y & \text{if} & x \in R \sm \mfq, \ y \in M, \\[1mm] x \Mdot y & \text{if} & x,y \in M. \\ \end{array} \right.$$ It is readily checked that $U$ with this multiplication is a monoid with unit element $1_U = 1_R$ and absorbing idempotent $0_U = 0 _M$. Moreover $e:= 1_M$ is an idempotent of $U$ such that $M = e \cdot U $ and $M$ in its given multiplication is a submonoid of $U$. Finally $0_M$ is the only element $x$ of $U$ with $0_M \cdot x = 0_M$. Thus, if we choose the given total ordering on the submonoid $M$ of $U$, we have established on $U$ the structure of a supertropical monoid (cf.  Definition \[defn1.1\]). We denote this supertropical monoid now by $\Uz(v)$. \[thm6.4\] $ $ 1. The map $\vrpzv:R \to \Uz(v)$ with $$\vrpzv(a) = \left\{ \begin{array}{ll} a & \text{if } \ a \in R \sm \mfq, \\ 0_M & \text{if }\ a \in \mfq, \\ \end{array} \right.$$ is a surjective -valuation covering $v$. 2. $\vrpzv$ dominates every other -cover of $v$. (i): An easy verification. (ii): Let $\psi: R \to V $ be an -cover of $v$. We define a map $\al: \Uz(v) \to V$ by $\al(a) = \psi(a)$ for $a \in R\sm \mfq$ and $\al(x) = x$ for $x \in M$. It then can be verified in a straightforward way that $\al$ is a transmission and $\al \circ \vrpzv = \psi$. \[cor6.5\] The poset ${\operatorname{Cov}}_m(v)$ is a complete lattice with top element $[\vrpzv]$. By Theorem \[thm6.4\] we have ${\operatorname{Cov}}_m(v) = C(\vrpzv)$, and this is a complete lattice by Proposition \[prop6.3\]. $v: R \to M $ itself may be regarded as an -supervaluation (in fact a supervaluation) covering $v$, and thus $[v]$ is the bottom element of ${\operatorname{Cov}}_m(v).$ In [@IKR1] we had introduced the poset ${\operatorname{Cov}}(v)$ consisting of the equivalence classes $[\vrp]$ of supervaluations $\vrp : R \to U$ with $e \vrp = v,$ and we called these supervaluations $\vrp$ the **covers** of the -valuation $v$. Thus, a surjective -cover $\vrp: R \to U $ of $v$ is a cover of $v$ iff $U$ is a semiring. The set ${\operatorname{Cov}}(v)$ is a subposet of ${\operatorname{Cov}}_m(v)$. \[prop6.6\] If ${\zeta}\in {\operatorname{Cov}}(v)$, $\eta \in {\operatorname{Cov}}_m(v)$ and ${\zeta}\geq \eta$, then $\eta \in {\operatorname{Cov}}(v)$. We choose surjective -valuations $\vrp: R \to U$, $\psi: R \to V$ with $U$ a semiring and ${\zeta}= [\vrp]$, $\eta = [\psi]$. There exists a fiber contraction $\al: U \onto V$ over $M$ with $\al \circ \vrp = \psi$. Since $\vrp$ and $\psi$ are surjective, $U = \vrp(R) \cup e \vrp(R)$ and $V = \psi(R) \cup e \psi (R)$. We conclude that $$\al (U) = \al \vrp(R) \cup e \al \vrp(R) = \psi(R) \cup e \psi (R) = V.$$ Since $U$ is a semiring, it follows by Theorem \[thm5.7\], or already by Theorem \[thm1.5\].ii, that $V$ is a semiring, hence $\eta \in {\operatorname{Cov}}(v).$ Recall from §\[sec:4\] that every supertropical monoid $U$ gives us a supertropical semiring\ $\htU = U / E(U,S(U))$ together with an ideal compression $\sig_U := \pi_{E(U,S(U))}: U \onto \htU$. Here $S(U)$ is the set of tangible NC-elements in $U$ (cf. Definition \[defn4.2\]). \[defn6.7\] For every -supervaluation $\vrp: R \to U$ we define a supervaluation $$\htvrp := \sig_U \circ \vrp : R \to \htU.$$ \[prop6.8\] Let $\vrp$ be an -cover of $v$. 1. $\htvrp$ is a cover of $v$ and $\vrp \geq \htvrp$. 2. If $\psi$ is a cover of $v$ with $\vrp \geq \psi$, then $\htvrp \geq \psi.$ 3. If $\psi$ is an -cover of $v$ with $\vrp \geq \psi$, then $\htvrp \geq \htpsi$. i): This is obvious. ii): We may assume that $\vrp$ is a surjective -supervaluation. Then we have a fiber contraction $\al : U \to V$ over $M$ with $\psi = \al \circ \vrp$. By Theorem \[thm4.5\] we have a factorization $\al = \bt \circ \sig_U$ with $\bt$ another fiber contraction over $M$. We conclude that $\psi = \bt \sig_U \vrp = \bt \htvrp$. iii): $\vrp \geq \psi \geq \htpsi$ by i), hence $\htvrp \geq \htpsi$ by ii). \[thm6.9\] As before assume that $v: R \to M$ is a surjective -valuation. Let $$U(v) := (\Uz(v))^\wedge$$ and $$\vrp_v := (\vrpzv)^\wedge : R \to U(v).$$ The supervaluation $\vrp_v$ is an **initial cover of** $v$; i.e., given any supervaluation $\psi: R \to V$ covering $v$, there exists a (unique) semiring homomorphism $\al : U \to V$ over $M$ with $\psi = \al \circ \vrp_v.$ We may assume that $\psi$ is a surjective supervaluation covering $v$. We know by Theorem \[thm6.4\] that $\vrpzv \geq \psi$, and conclude by Proposition \[prop6.8\].ii that $\vrp_v = (\vrpzv)^\wedge \geq \psi.$ Thus, there exists a fiber contraction $\al: U(v) \to V$ over $M$ with $\psi = \al \circ \vrp$. Since $U(v)$ and $V$ are semirings, $\al$ is a semiring homomorphism over $M$ (cf. Proposition \[prop5.2\]). \[cor6.10\] The poset ${\operatorname{Cov}}(v)$ is a complete lattice with top element $[\vrp_v]$. We proved in [@IKR1 §7] that ${\operatorname{Cov}}(v)$ is a complete lattice, but – except in the case that $v$ is a valuation – there we have only proved that the a top element $[\vrp_v]$ exists (loc. cit, Proposition 7.5), without giving an explicit description of $\vrp_v$. Starting from the formula $\vrp_v = (\vrpzv)^\wedge$, this is now possible. Let $U := \Uz(v)$. Then $\tT(U) = R \sm \mfq$ and $eU= M$. We have $$\htU = U/ E (U, \tD(U)) = U / E(U, S(U))$$ with $S(U)$ the set of tangible NC-products in $U$ (cf. Definition \[defn4.2\]) and $\tD (U) = S(U) \cup M$, which is an ideal of $U$ (cf. §\[sec:4\]). We view $\htU$ as a subset of $U$, as indicated in Convention \[convs3.2\].a. Thus $e \htU = M$ and $\tT (\htU) = \tT(U) \sm S(U)$. Recalling the description of the supertropical monoid $\Uz(v)$ from above, we see that $S(U)$ is the following subset $Y(v)$ of $R \sm \mfq$: $$Y(v) := \{ ab {\ {|} \ } a,b \in R, \exists a' \in R \text{ with } v(a) < v(b) , v(a'b) = v(ab) \neq 0 \}.$$ Thus $\tT(\htU) = R \sm \mfq'$ with $\mfq' := \mfq \cup Y(v)$. Clearly $R \cdot Y(v) \subset \mfq \cup Y(v)$, and thus $\mfq'$ is an ideal of $R$. Looking again at Convention \[convs3.2\].a we obtain a completely explicit description of $U(v)$ and $\vrp_v$ as follows. \[schl6.11\] Assume that $v: R \to M$ is a surjective -valuation with support $\mfq = \iv(0)$. Let $\mfq' := \mfq \cup Y(v)$ with the set $Y(v) \subset R \sm \mfq$ given above. Then $ U(v)$ is the subset $(R \sm \mfq') \cup M$ of $\Uz(v) = (R \sm \mfq) \cup M$, with the following new product $\odot$: If $x,y \in U(v)$, then $$x \odot y = \left\{ \begin{array}{ll} xy & \text{if } x,y, xy \in R \sm \mfq',\\ exy & \text{otherwise},\\ \end{array} \right.$$ the products on the right hand side taken in $\Uz(v)$. {Here $e = e_{\Uz(v)} = e_{U(v)}$.} The initial covering $\vrp_v: R \to U(v)$ is given by $$\vrp_v(a) = \left\{ \begin{array}{ll} a & \text{if } a \in R \sm \mfq',\\ v(a) & \text{if } a\in \mfq'.\\ \end{array} \right.$$ \[schol6.11\] \[thm6.12\] As before assume that $v: R \to M$ is a surjective -valuation. 1. If $\vrp$ is any supervaluation covering $v$, then $\vrp(a)$ is ghost for every $a \in Y(v)$. 2. If $Y(v) = \emptyset$, then every surjective -supervaluation covering $v$ is a supervaluation. In short ${\operatorname{Cov}}_m (v) = {\operatorname{Cov}}(v)$. i): We read off from Scholium \[schl6.11\] that $\vrp_v(a) = v(a) \in M$ for any $a \in Y(v)$. Since $\vrp_v$ dominates $\vrp$, also $\vrp(a)= v(a) \in M$. ii): If $Y(v)= \emptyset$, it follows from Scholium \[schl6.11\] that $\vrpzv = \vrp_v$. Thus $[\vrpzv] \in {\operatorname{Cov}}(v)$. If $\vrp$ is any -supervaluation covering $v$, then $\vrpzv \geq \vrp$, hence $[\vrp] \in {\operatorname{Cov}}(v)$ by Proposition \[prop6.6\]. In particular $Y(v)$ is empty if $v$ is a valuation, since this means that the bipotent semiring $M$ is cancellative. Then we fall back on the explicit description of the initial cover $\vrp_v$ in [@IKR1] which in present notation says that $\vrp_v = \vrpzv$ (loc. cit, Example 4.5). In large parts of [@IKR1] and whole [@IKR2 §2], where we studied coverings of valuations, it was important that we have tangible supervaluations at our disposal. There remains the difficult task to develop a similar theory for coverings of -valuations, which are not valuations. We leave this to the future. But we mention that there exist many natural and beautiful -valuations which are not valuations, as is already clear from [@HV] and [@Z]. More on this can be found in a recent paper [@IKR3]. Lifting ghosts to tangibles {#sec:7} =========================== \[defn7.1\] We call a supertropical monoid $U$ **unfolded**, if the set $\tT(U)_0 := \tT(U) \cup \{0 \}$ is closed under multiplication. If $U$ is unfolded, then $N:= \tT(U)_0$ is a monoid under multiplication with absorbing element $0$. Further $M := eU$ is a totally ordered monoid with absorbing element $0$, and the restriction $$\rho := \nu_U |U : N {\ {\to} \ } M$$ is a monoid homomorphism with $\irho (0) = \{ 0 \}$. Observing also that $e_U = 1_M = \rho(1_N)$, we see that the supertropical monoid $U$ is completely determined by the triple $(N,M, \rho)$. This leads to a way to construct all unfolded supertropical monoids up to isomorphism. \[constr7.2\] Assume that we are given a totally ordered monoid $M$ with absorbing element $0_M \leq x$ for all $x \in M$, i.e., a bipotent semiring $M$, further an (always commutative) monoid $N$ with absorbing element $0_N$, and a multiplicative map $ \rho : N \to M$ with $\rho (1_N) = \rho(1_M)$, $\irho(0_M) = \{ 0 _N\} $. Then we define an unfolded supertropical monoid $U$ as follows: As a set $U$ is the disjoint union of $M \sm \{ 0_M\}$, $N \sm \{ 0_N\}$, and a new element $0$. We identify $0_M = 0_N = 0$, and then write $$U = M \cup N, \quad \text{with } M \cap N = \{ 0 \}.$$ The multiplication on $U$ is given by the rules, in obvious notation, $$x \cdot y = \left\{ \begin{array}{lll} x \Ndot y & \text{if} & x,y \in N, \\[1mm] \rho(x) \Mdot y & \text{if} & x \in N, y \in M, \\[1mm] x \Mdot \rho(y) & \text{if} & x \in M, y \in N, \\[1mm] x \Mdot y & \text{if} & x,y \in M. \\ \end{array} \right.$$ It is easy to verify that $(U, \cdot \; )$ is a (commutative) monoid with $1_U = 1_N$ and absorbing element $0$. Let $e := 1_M$. Then $eU = M$ and $\rho(x) = ex$ for $x \in M $, further $ex = 0$ iff $x=0$ for any $x \in U$, since $\irho(0) = \{ 0\}$. Thus $(U, \cdot \; , e)$ ,together with the given ordering on $M = eU$, is a supertropical monoid. It clearly is unfolded. We denote this supertropical monoid $U$ by $\STR(N,M, \rho)$. This construction generalizes the construction of supertropical domains [@IKR1] (loc. cit. Construction 3.16). There we assumed that $N \sm \{ 0 \}$ and $M \sm \{ 0 \}$ are closed under multiplication, and that the monoid $M \sm\{ 0 \}$ is cancellative, and we obtained all supertropical predomains up to isomorphism. Dropping here just the cancellation hypothesis would give us a class of supertropical monoids not broad enough for our work below. The present notation $\STR(N,M, \rho)$ differs slightly from the notation $\STR(\tT, \tG, v)$ in [@IKR1 Construction 3.16]. Regarding the ambient context this should not cause confusion. We add a description of the transmissions between two unfolded supertropical monoids. \[prop7.3\] Assume that $U' = \STR(N', M', \rho')$ and $U = \STR(N, M, \rho)$ are unfolded supertropical monoids. 1. If ${\lambda}: N' \to N$ is a monoid homomorphism with ${\lambda}(0) = 0$, and $\mu: M' \to M$ is a semiring homomorphism, and if $\rho' {\lambda}= \mu \rho$, then the well-defined map $$\STR({\lambda},\mu) : U' = N' \cup M' {\quad {\to} \quad } U = N \cup M,$$ which sends $x' \in N'$ to ${\lambda}(x')$ and $y' \in M'$ to $\mu(y')$, is a tangible transmission (cf. Definition \[defn2.3\]). 2. In this way we obtain all tangible transmissions from $U'$ to $U$. (i): A straightforward check. (ii): Obvious. We mention in passing, that given an -valuation $v: R \to M$ with support $\iv(0) = \mfq$, the supertropical semiring $U^0(v)$ occurring in Theorem \[thm6.4\] may be viewed as an instance of Construction \[constr7.2\], as follows. \[exm7.4\] Let $E$ denote the equivalence relation on $R$ with equivalence classes $[0]_E = \mfq$ and $[x]_E = \{ x \} $ for $x \in R\sm \mfq$. It is multiplicative, hence gives us a monoid $R / E$ with absorbing element $[0]_E = 0$, which we identify with the subset $(R \sm \mfq) \cup \{ 0 \}$ in the obvious way. The map $v: R\to M$ induces a monoid homomorphism $\brv: R/E \to M$ with values $\brv(x) = v(x)$ for $x \in R \sm \mfq$, $\brv(0) = 0 $. We have $\brv^{-1} (0) = \{ 0 \}$ and $$U^0(v) = \STR(R / E, M , \brv).$$ We now look for ways to “unfold” an arbitrary supertropical monoid $U$. By this we roughly mean a fiber contraction $\tau: \tlU \to U$ with $\tlU$ an unfolded supertropical monoid and fibers $\itau(x)$, $x\in U$ as small as possible. More precisely we decree \[defn7.4\] Let $M := eU$, and let $N$ be a submonoid of $(U, \cdot \;)$ which contains the set $\tT(U)_0$. An **unfolding of $U$ along $N$** is a fiber contraction $\tau: \tlU \to U$ over $M$ (in particular $e \tlU = M$), such that $$\itau(x) = \left\{ \begin{array}{lll} \{x, \tlx \} & \text{if} & x \in M \cap N, \\[1mm] \{ \tlx \} & \text{if} & x \in N \sm M, \\[1mm] \{x \} & \text{if} & x \in M \sm N, \\ \end{array} \right.$$ with $\tlx \in \tT(U)_0$. For any $x \in N$ we call $\tlx$ the **tangible lift** of $x$ (with respect to $N$). Notice that this forces $\tau(\tT(\tlU)_0)= N$, and that moreover for any $x \in N$ the tangible fiber $\tlx$ is the unique element of $\tT(\tlU)_0$ with $\tau(\tlx) = x$, hence $\tT(\tlU)_0 = \{ \tlx {\ {|} \ } x\in N \}$. Thus, if $\tau:\tlU \to U$ is an unfolding along $N$, then the map $\tlx \mapsto x$ from $\tT(\tlU)_0$ to $N$ obtained from $\tau$ by restriction is a monoid isomorphism, and $\tau$ itself is an ideal compression with ghost kernel ${(M \cap N)}^{\sim} \cup M$, where ${(M \cap N) }^{\sim}:= \{ \tlx {\ {|} \ } x \in M \cap N \}$. \[thm7.5\] $ $ 1. Given a pair $(U,N)$ consisting of a supertropical monoid $U$ and a multiplicative submonoid $N \supset \tT(U)_0$, there exists an unfolding $\tau: \tlU \to U$ of $U$ along $N$. 2. If $\tau': \tlU' \to U$ is a second unfolding of $U$ along $N$, then there exists a unique isomorphism of supertropical monoids $\al : \tlU {\ {\iso} \ } \tlU'$ with $\tau' \circ \al = \tau$. i\) *Existence:* Since $M$ is an ideal of $U$, the set $M \cap N$ is a monoid ideal of $N$. We have $U = N \cup M$, since $N \supset \tT(U)_0$. Let $\rho: N \to M$ denote the restriction of $\nu_U$ to $N$. It is a monoid homomorphism with $\irho(0) = \{ 0 \}$. Let $\tlN$ denote a copy of the monoid $N$ with copying isomorphism $x \mapsto \tlx$ ($x \in N$), and let $\tlrho: \tlN \to M$ denote the monoid homomorphism from $\tlN$ to $M$ corresponding to $\rho: N \to M$. Thus $\tlrho (\tlx) = \rho(x) = ex$ for $x\in N$. Now define the unfolded supertropical monoid $$\tlU := \STR(\tlN, M, \tlrho) = \tlN \cup M.$$ In $\tlU$ we have $\tl0_U = 0 $ and $\tlN \cap M = \{ 0 \}$. Further $\tT(\tlU)_0 = \tlN$ and $e \tlU = eU = M$. We obtain a well-defined surjective map $\tau: \tlU \to U$ by putting $\tau(\tlx) := x$ for $x\in N$, $\tau(y) := y$ for $y\in M$. This map $\tau$ is multiplicative, as checked easily, sends $0$ to $0$, $1 \in \tT(\tlU)$ to $1 \in N$, and restricts to the identity on $M$. Thus $\tau$ is a fiber contraction (cf. Definition \[defn2.1\]). The fibers of $\tau $ are as indicated in Definition \[defn7.4\]; hence $\tau$ is an unfolding of $U$ along $N$. ii\) *Uniqueness*: Let $\tltau: \tlU \to U$ and $\tltau': \tlU' \to U$ be unfoldings of $U $ along $N$ with tangible lifts $x \mapsto \tlx$ and $x \mapsto \tlx'$ respectively. Without loss of generality we assume that $\tlU = \STR(\tlN,M, \tlrho)$ and $\tlU' = \STR(\tlN',M, \tlrho')$ with tangible lifts $x \mapsto \tlx$ and $x \mapsto \tlx'$ ($x \in N$). Then $\tlrho(\tlx) = \tlrho'(\tlx') = ex$ for every $x \in N$. The map ${\lambda}: \tlN \to \tlN'$, given by ${\lambda}(\tlx) = \tlx'$ for $x \in N$, is a monoid isomorphism with $\tlrho' \circ {\lambda}= {\operatorname{id}}_M \circ \tlrho$. Thus we have a well defined transmission $$\al := \STR({\lambda}, {\operatorname{id}}_M): \tlU {\ {\to} \ } \tlU'$$ at hand (cf. Proposition \[prop7.3\]). $\al$ is an isomorphism over $U$, i.e., an isomorphism with $\tau' \circ \al = \tau$, clearly the only one. \[notation7.6\] We call the map $\tau: \tlU \to U$ constructed in part i) of the proof of Theorem \[thm7.5\] **“the” unfolding of $U$ along $N$** and write this map more precisely as $$\tau_{U,N}: \tlU(N) \to U$$ if necessary. But sometimes we abusively will denote any unfolding of $U$ along $N$ in this way, justified by part ii) of Theorem \[thm7.5\]. \[exam7.7\] We consider the very special case that $U = eU =M$. Then $N$ can be any submonoid of $M$ containing $0$. Now $\tlM(N) = \tlN \cup M$ with $\tlN \cap M = \{0\}$, and $$\tlM(N) \cong \STR(N,M, i)$$ with $i : N \into M$ the inclusion mapping. For every $x \in N$ there exists a unique tangible element $\tlx$ of $\tlM(N)$ with $e \tlx =x$, while for $x \in M \sm N$ there exists no such element. \[thm7.8\] Assume that $\al : U' \to U$ is a transmission between supertropical monoids, and that $N' \supset \tT(U')_0$, $N \supset \tT(U)_0$ are submonoids of $U'$ and $U$ with $\al(N') \subset N$. Then there exists a unique tangible transmission $$\tlal:= \tlal_{N',N} : \tlU' (N') {\ {\to} \ } \tlU(N),$$ called the **tangible unfolding of $\al$ along $N'$ and $N$**, such that the diagram $$\xymatrix{ \tlU'(N') \ar @{>}[d]^{\tau_{U',N'}} \ar @{>}[rr]^{\tlal } & & \tlU(N) \ar @{>}[d]^{\tau_{U,N}} \\ U' \ar @{>}[rr]_{\al} & & U }$$ commutes. Let $M' := eU'$, $M := eU$, and let $\rho': N' \to M$, $\rho: N \to M$ denote the monoid homomorphism obtained from $\nu_{U'}$ and $\nu_U$ by restriction to $N'$ and $N$. Then $$\tlU'(N') = \STR(N',M', \rho'), \qquad \tlU(N) = \STR(N,M,\rho).$$ The map $\al$ restricts to monoid homomorphisms ${\lambda}: N' {\ {\to} \ } N$ and $\gm: M' \to M$ with ${\lambda}(0) = 0$, $\gm(0) = 0 $, and $\gm$ order preserving. Now $\gm \circ \nu_{U'} = \nu_U \circ \al$, hence $\gm \rho' = \rho {\lambda}$. Thus we have the tangible transmission $$\tlal:= \STR({\lambda}, \gm): \tlU'(N') {\ {\to} \ } \tlU(N)$$ at hand. Clearly $\tau_{U,N} \circ \tlal = \al \circ \tau_{U',N'}$. Since any tangible transmission from $\tlU'(N')$ to $\tlU(N)$ maps $\tlN'$ to $\tlN$ and $M'$ to $M$, it is evident that $\tlal$ is the only such map. \[cor7.9\] Assume that $\al: U' \to U$ is a transmission between supertropical monoids which is tangibly surjective, i.e., $\tT(U) \subset \al(\tT(U'))$. Assume further that $U'$ is unfolded. Let $N:= \al (\tT(U')_0)$, which is a submonoid of $U$ containing $\tT(U)_0$. 1. There exists a unique tangible transmission $$\tlal: U' {\ {\to} \ } \tlU(N),$$ called the **tangible lift** of $\al$, such that $\tau_{U,N} \circ \tlal = \al.$ 2. If $x' \in U'$, then $$\tlal(x') = \left\{ \begin{array}{lll} \widetilde{\al(x')} & \text{if} & x' \in \tT(U')_0, \\[1mm] \al(x') & \text{if} & x' \in eU'. \\ \end{array} \right.$$ (i): applying Theorem \[thm7.8\] with $N' := \tT(U')_0$, and observe that $\tlU'(N') = U'$, since  $U'$ is unfolded. (ii): Now obvious, since $\tau_{U,N} (\tlal(x')) = \al (x')$ and $\tlal(x') \in \tT(\tlU)_0$ iff $x' \in \tT(U')_0$. We are ready to construct “tangible lifts” of -supervaluations. \[thm7.10\] Assume that $\vrp: R \to U$ is an -supervaluation which is tangibly surjective, i.e., $\tT(U) \subset \vrp(R)$ {e.g. $\vrp$ is surjective; $U = \vrp(R) \cup e \vrp(R)$}. Let $N := \vrp(R)$, which is a submonoid of $U$ containing $\tT(U)$. 1. The map $$\tlvrp : R \to \tlU(N), \qquad a \mapsto \widetilde{\vrp(a)},$$ with $\widetilde{\vrp(a)}$ denoting the tangible lift of $\vrp(a)$ w.r.t. $N$, is a tangible -supervaluation of $\vrp$, called the **tangible lift** of $\vrp$. 2. If $\vrp' : R \to U'$ is a tangible -supervaluation dominating $\vrp$, then $\vrp'$ dominates $\tlvrp$. (i): $\tlvrp$ is multiplicative, $\tlvrp(0) = 0$, $\tlvrp(1) = 1$, and $e \tlvrp = e \vrp$ is an -valuation. Thus $\tlvrp$ is an -supervaluation. By construction $\tlvrp$ is tangible. (ii): We may assume that the -supervaluation $\vrp': R \to U'$ is surjective, and hence $\vrp'(R) \supset \tT'(U)_0$. Since $\vrp'$ is tangible, this forces $\vrp'(R) =\tT'(U)_0$. Thus $\tT'(U)_0$ is a submonoid of $U'$, i.e., $U'$ is unfolded. Since $\vrp'$ dominates $\vrp$, there exists a transmission $\al: U' \to U$ with $\vrp = \al \circ \vrp'$. We have $$\al(\tT'(U)_0) = \al (\vrp'(R)) = \vrp (R) = N.$$ Thus we have the tangible lift $$\tlal: U' {\ {\to} \ } \tlU(N)$$ of $\al$ at hand. For any $a\in R,$ $$\tlal(\vrp'(a)) = [\al (\vrp'(a))]^\sim = \widetilde{\vrp(a)} = \tlvrp(a).$$ Thus $\tlvrp = \tlal \circ \vrp',$ which proves that $\vrp'$ dominates $\tlvrp.$ \[adden7.11\] As the proof has shown, if the -valuation $\vrp'$ is surjective, then $U'$ is unfolded, and the transmission $$\al_{\tlvrp,\vrp'}: U' {\ {\to} \ } \tlU(N)$$ (cf. [@IKR1 Definition 5.3]) is the tangible lift of $\al_{\vrp, \vrp'}: U' \to U$. \[cor7.12\] If $\vrp$, $\psi$ are -supervaluations covering $v$ and $\vrp \leq \psi$, then $\tlvrp \leq \tlpsi$. We have $\vrp \leq \psi \leq \tlpsi$. It follows by Theorem \[thm7.10\].ii that $\tlvrp \leq \tlpsi$. The partial tangible lifts of an $\m$-supervaluation {#sec:8} ==================================================== In all the following $v: R \to M$ is a fixed -valuation and $\vrp: R \to U$ is a tangible surjective -supervaluation covering $v$. (Most often $v$ and $\vrp$ will both be surjective.) In §\[sec:7\] we introduced the tangible lift $\tlvrp: R \to \tlU$ (cf. Theorem 7.10). We now strive for an explicit description of the -supervaluations $\psi$ covering $v$ with $\vrp \leq \psi \leq \tlvrp$. We warm up with two general observations. \[defn8.1\] If $\psi: R \to V$ is an -supervaluation covering $v: R \to M$, we call $$G(\psi) := \psi(R) \cap M = \{ \psi(a) {\ {|} \ } a\in R, \ \psi(a)= v(a) \}$$ the **ghost value set** of $\psi$. {Notice that $eV = M$.} \[lem8.2\] Let $\psi_1, \psi_2$ be -supervaluations covering $v$. If $\psi_1 \geq \psi_2$, then $G(\psi_1) \subset G(\psi_2)$. If $\psi_1 \sim \psi_2$, then $G(\psi_1) = G( \psi_2)$. Let $a\in R$. If $\psi_1 \geq \psi_2$, then $\psi_1(a) \in M$ implies that $\psi_2(a) \in M$, due to condition ${{\operatorname{D3}}}$ in the definition of dominance (cf. Definition \[defn6.2\]). Thus, for $\psi_1 \sim \psi_2$ we have $\psi_1(a) \in M$ iff $\psi_2(a) \in M$. \[lem8.3\] Assume that the -valuation $v: R \to M$ is surjective. Then the ghost value set $G(\psi)$ of any -supervaluation $\psi$ covering $v$ is an ideal of the semiring $M$. If $x \in G(\psi)$ and $y \in M$, there exist $a,b \in R$ with $\psi(a) =x$, $e \psi(b) = y$. It follows that $$xy = e \psi(a) \psi(b) = \psi(a) \psi(b) = \psi(a b).$$ Thus $xy \in \psi(R) \cap M = G(\psi)$. This proves that $G(\psi) \cdot M \subset G(\psi)$. Since $M$ is bipotent, $G(\psi)$ is also closed under addition. \[thm8.4\] Assume that $\vrp: R \to U$ is an -supervaluation covering $v: R \to M$, and that $\psi_1, \psi_2$ are -supervaluations covering $v$ with $$\vrp \leq \psi_1 \leq \tlvrp, \quad \vrp \leq \psi_2 \leq \tlvrp.$$ 1. $\psi_1 \geq \psi_2 {\quad {\Leftrightarrow} \quad } G(\psi_1) \subset G(\psi_2)$. 2. $\psi_1 \sim \psi_2 {\quad {\Leftrightarrow} \quad } G(\psi_1) = G(\psi_2)$. We assume without loss of generality that $\vrp$ is surjective. Then also the -super-valuations $\psi_1,$ $\psi_2,$ $ \tlvrp$ are surjective. By Corollary \[cor7.12\] the tangible lifts $\tlpsi_1$ and $\tlpsi_2$ are both equivalent to $\tlvrp$. Again without loss of generality we moreover assume that $\vrp = \tlvrp / E := \pi_E \circ \tlvrp$ with $E$ an -relation on $\tlU$, and also $\psi_i = \tlvrp / E_i$ with an -relation $E_i$ ($i =1,2$). Let us describe these relations $E, E_1, E_2$ explicitly. We have $\tlU = \tlN \cup M$, with $$\tlN := \tT(\tlU)_0 = \tlvrp(R), \qquad \tlN \cap M = \{ 0\},$$ further $U = N \cup M$ with $$N:= \vrp (R), \qquad N\cap M = G(\vrp),$$ and we have a copying isomorphism $$s: N {\ {\iso} \ } \tlN$$ of monoids (new notation!), which sends each $x \in N$ to its tangible lift $\tlx$, as explained in §\[sec:7\] (Definition \[defn7.4\], Proof of Theorem \[thm7.5\].i). Notice that $es(x)=x $ for $x \in N \cap M = G(\vrp).$ The relation $E$ has the 2-point equivalence classes $\{x, s(x) \}$ with $x$ running through $G(\vrp)$, while all other $E$-equivalence classes are one-point sets. Analogously, $E_i$ has the 2-point set equivalence classes $\{x, s(x) \}$ with $x$ running through $G(\psi_i) \subset G(\vrp)$, while again all other $E$-equivalence classes are one-point sets. Thus it is obvious that[^8] $E_1 \subset E_2$ iff $G(\psi_1) \subset G(\psi_2)$. But $E_1 \subset E_2$ means that $\psi_1 \geq \psi_2$. This gives claim (i), and claim (ii) follows. \[defn8.5\] We call the monoid isomorphism $$s: \vrp(R) {\ {\to} \ } \tT(\tlU)_0 = \tlvrp(R),$$ i.e., the copying isomorphism $s: N \iso \tlN$ occurring in the proof of Theorem \[thm8.4\], the **tangible lifting map for** $\vrp$. Notice that for $x\in \vrp(R)$, $y\in \tT(\tlU)_0$ we have $s(x)y = s(xy)$. *We assume henceforth that the -valuation $v: R \to M$ is surjective, and that $\vrp: R \to U$ is a surjective -supervaluation with $e \vrp = v$.* The question arises whether every ideal $\mfa$ of $M$ with $\mfa \subset G(\vrp)$ occurs as the ghost value set $G(\psi)$ of some -supervaluation $\psi$ covering $v$ with $\vrp \leq \psi \leq \tlvrp$. This is indeed true. \[const8.6\] We employ the tangible lifting map $ s: \vrp(R) {\ {\to} \ } \tlvrp(R) = \tT(\tlU)_0$ defined above. Assume that $\mfa$ is an ideal of $M$ contained in $G(\vrp)$. We have $$s(\mfa) \cdot \tlU {\ {\subset} \ } s(\mfa) \cup M,$$ since $s(x)y = s(xy) \in s(\mfa)$ for $x \in \mfa$ and $y \in \tT(U)$. We conclude that $s(\mfa) \cup M$ is an ideal of $U$. Let $$E_\mfa : = E(\tlU, s(\mfa)) = E(\tlU, s(\mfa)\cup M)$$ and $\tlU_\mfa := \tlU/ E_\mfa$. We regard $\tlU_\mfa$ as a subset of $\tlU$, as indicated in Convention \[convs3.2\].a. The map $\pi_{E_\mfa} : \tlU \onto \tlU_\mfa$ is the ideal compression with ghost kernel $s(\mfa) \cup M$, and $$\tlvrp_\mfa := \tlvrp/ E_\mfa = \pi_{E_\mfa} \circ \tlvrp : R {\quad {\to} \quad } \tlU_\mfa$$ is an -supervaluation. For any $a\in R$ $$\tlvrp_\mfa (a) = \left\{ \begin{array}{lll} \vrp(a) = v(a) & \text{if} \ \vrp(a) \in \mfa , &\\[1mm] \tlvrp(a) & \text{else} \ . & \\ \end{array} \right.$$ Clearly $\vrp \leq \tlvrp_\mfa \leq \tlvrp$ and $G(\tlvrp_\mfa) = \mfa$. We call $\tlvrp_\mfa$ **the tangible lift of $\vrp$ outside $\mfa$**, and we call any such map $\tlvrp_\mfa$ a **partial tangible lift of $\vrp$.** Let $[\vrp, \tlvrp]$ denote the “interval” of the poset $ {\operatorname{Cov}}_m(v)$ containing all classes $[\psi]$ with $\vrp \leq \psi \leq \tlvrp$, and let $[0,G(\vrp)]$ be the set of ideals $\mfa$ of $M$ with $\mfa \subset G(\vrp)$, ordered by inclusion. By the Lemmas \[lem8.2\] and \[lem8.3\] we have a well defined order preserving map $$[\vrp, \tlvrp] {\ {\to} \ } [0, G(\vrp)],$$ which sends each class $[\psi] \in [\vrp, \tlvrp]$ to the ideal $G(\psi)$. By Theorem \[thm8.4\] this map is injective, and by Construction \[const8.6\] we know that it is also surjective. Thus we we have proved \[thm8.7\] The map $$[\vrp, \tlvrp] {\ {\to} \ } [0, G(\vrp)], \qquad [\psi] \mapsto G(\psi),$$ is a well defined order preserving bijection. The inverse of this map sends an ideal $\mfa \subset G(\vrp)$ to the class $[\tlvrp_\mfa]$ of the tangible lift of $\vrp$ outside $\mfa$. The poset ${\operatorname{Cov}}_m(v)$ is a complete lattice (cf. Corollary \[cor6.5\]). The poset $I(M)$ consisting of the ideals of $M$ and ordered by inclusion, is a complete lattice as well. Indeed, the infimum of a family $(\mfa_i {\ {|} \ } i\in I)$ in $I(M)$ is the ideal $\bigcap_i \mfa_i$, while the supremum is the ideal $\sum_i \mfa_i = \bigcup_i \mfa_i.$ {Recall once more that every subset of $M$ is closed under addition.} The intervals $[\vrp, \tlvrp]$ and $[0,G(\vrp)]$ are again complete lattices, and thus the map $ [\vrp, \tlvrp] {\ {\to} \ } [0, G(\vrp)]$ in Theorem \[thm8.7\] is an anti-isomorphism of complete lattices. This implies the following \[cor8.8\] Assume that $(\psi_i {\ {|} \ } i \in I)$ is a family of supervaluations covering $v$ with $\vrp \leq \psi_i \leq \tlvrp$ for each $i \in I$. Let $\bigvee_i \psi_i$ and $\bigwedge_i \psi_i$ denote respectively representatives of the classes $\bigvee_i [\psi_i]$ and $\bigwedge_i [\psi_i]$ (as described in [@IKR1 §7]). Then $$G\bigg(\bigvee_i \psi_i \bigg) = \bigcap_i G(\psi_i), \qquad G\bigg(\bigwedge_i \psi_i\bigg) = \bigcup_iG(\psi_i).$$ We switch to the case where $\vrp: R \to U$ is a supervaluation, i.e., the supertropical monoid  $U$ is a semiring. We want to characterize the partial tangible lifts $\psi$ of $\vrp$ which are again supervaluations; in other terms, we want to determine the subset $[\vrp, \tlvrp] \cap {\operatorname{Cov}}(v)$ of the interval $[\vrp, \tlvrp]$ of ${\operatorname{Cov}}_m(v)$. The set $Y(v)$ introduced near the end of §\[sec:6\] will play a decisive role. It consists of the products $a b \in R$ of elements $a,b \in R$ for which there exists some $a' \in R$ with $$v(a') < v(a), \quad v(a'b) = v(ab) \neq 0.$$ Henceforth we call these products $ab$ the **$v$-NC-products** (in $R$). Let further $$\mfq' := \mfq \cup Y(v)$$ with $\mfq$ the support of $v$, $\mfq = v^{-1}(0)$. As observed in §\[sec:6\], $\mfq'$ is an ideal of the monoid $(R, \cdot \;)$, while $\mfq$ is an ideal of the semiring $R$. \[exmp8.9\] Let $R$ be a supertropical semiring and $\gm: eR \to M$ a semiring homomorphism to a bipotent semiring $M$. Then $$v:= \gm \circ \nu_R: R {\ {\to} \ } M$$ is a strict -valuation. The $v$-NC-products are the products $yz$ with $y,z \in U$ such that there exists some $y' \in R$ with $$\gm(ey') < \gm (ey), \quad \gm(ey'z) = \gm (eyz).$$ Thus $Y(v)$ is the ideal $D_0(R,\gm)$ of the supertropical semiring $R$ introduced in Definition \[defn4.8\]. \[prop8.10\] If $\vrp$ is a supervaluation then $\vrp(\mfq')$ is contained in the ghost value set  $G(\vrp)$. We have seen in §\[sec:6\] that $\vrp(Y(v)) \subset M$. Since $\vrp(\mfq) = \{ 0\}$, this implies that $\vrp(\mfq') \subset M \cap \vrp(R) = G(\vrp)$. \[rem8.11\] Here is a more direct argument that $\vrp(Y(v)) \subset M$, than given in the proof of Theorem \[thm6.12\].i. If $x \in Y(v)$, then we have $a',a,b \in R$ with $x =ab$, $v(a') < v(a)$, $v(a'b) = v(ab) \neq 0$. Clearly $\vrp(x) = \vrp (a) \vrp (b)$ is an NC-product in the supertropical semiring  $U$ (recall Definition \[defn4.2\]), and thus $\vrp(x)$ is ghost, as observed already in Theorem \[thm1.2\]. \[lem8.12\] Assume that $\vrp: R \to U$ is a surjective tangible -supervaluation covering $v$. Then $\vrp(R \sm \mfq) = \tT(U)$, $v(R) = M$, and $\vrp(Y(v)) = S(U)$.[^9] a\) We have $U = \vrp(R) \cup v(R)$, $\vrp(R \sm \mfq) \subset \tT(U)$, and $v(R) \subset M$. Since $U = \tT(U) \dot \cup M$, this forces $\vrp(R \sm \mfq) = \tT(U)$ and $v(R) = M.$ b\) Let $c \in Y(v)$. There exist $a,b,a' \in R$ with $c =ab$, $v(a') < v(a)$, $v(a'b) = v(ab) \neq 0.$ It follows that $\vrp(c) = xy \neq 0$ with $x:= \vrp(a)$, $y:= \vrp(b)$, $v(a') < ex$, $v(a') y = exy.$ Thus $\vrp(c)$ is an NC-product in $U$. Moreover $\vrp(c)$ is tangible, hence $\vrp(c) \in S(U).$ Thus $\vrp(Y(v)) \subset S(U).$ c\) Let $x \in S(U)$ be given. Then $x = yz \in \tT(U)$ with $y,z \in U$ and $y' < ey$, $y'z = eyz \neq 0$ for some $y' \in M$. Clearly $y,z\in \tT(U)$. We choose $a, b, a' \in R$ with $\vrp(a) = y$, $\vrp(b) = z$, $v(a') = y'.$ Then $ey = v(a)$, $ez = v(b)$, and it follows that $v(a') < v(a)$, $v(a'b) = v(ab) \neq 0$. Thus $ab \in Y(v)$ and $x = \vrp(ab)$. This proves that $S(U) \subset \vrp (Y(v))$. \[thm8.13\] Assume that $\vrp: R \to U$ is a supervaluation, i.e., $U$ is a semiring. Let $\chvrp$ denote the tangible lift of $\vrp$ outside the ideal $v(\mfq') = \{ 0 \} \cup v(Y(v))$ of $M$, $$\chvrp := (\tlvrp)_{v(\mfq')} : R {\ {\to} \ } \chU := \tlU/E_{v(\mfq')}$$ (cf. Construction \[const8.6\]). 1. $\chvrp$ is again a supervaluation. More precisely, $\chvrp$ coincides with the supervaluation $(\tlvrp)^\wedge$ associated to the tangible lift $\tlvrp: R \to \tlU$ of $\vrp$ (cf. Definition \[defn6.7\]). 2. If $\psi$ is an -supervaluation covering $v$ with $\vrp \leq \psi \leq \tlvrp$, then $\psi$ is a supervaluation iff $\psi \leq \chvrp$. Thus $$[\vrp, \tlvrp] \cap {\operatorname{Cov}}(v) = [\vrp, \chvrp].$$ (i): $(\tlvrp)^\wedge$ is the map $\tlvrp/ E(\tlU, S(\tlU)) = \pi_{E(\tlU, S(\tlU)} \circ \tlvrp$ from $R$ to $\chU:= \tlU/ E(\tlU, S(\tlU).$ By Lemma \[lem8.12\], applied to $\vrp$, we have $$S(\tlU) \cup \{ 0 \} = \tlvrp (\mfq') = s\vrp(\mfq')$$ with $s: \vrp(R) \to \tT(\tlU)_0$ denoting the tangible lifting map for $\vrp$. Moreover $\vrp(\mfq') = v(\mfq')$ by Proposition \[prop8.10\]. Thus $ \chU = \tlU/ E_{v(\mfq')}$ and $(\tlvrp)^\wedge = \tlvrp/ E_{v(\mfq')} = \chvrp.$ (ii): If $\psi$ is a supervaluation, then we know by Proposition \[prop8.10\] that $G(\psi) \supset v(\mfq') = G(\chvrp)$, and hence by Theorem \[thm8.4\] that $\psi \leq \chvrp$. Conversely, if $\psi \leq \chvrp$, then $\psi$ is a supervaluation since $\chvrp$ is a supervaluation (cf. Proposition \[prop6.6\]). \[defn8.14\]$ $ 1. Given a supervaluation $\vrp:= R \to U$ covering $v$ we call $$\chvrp := (\tlvrp)^\wedge: R {\ {\to} \ } \chU = (\tlU)^\wedge$$ the **almost tangible lift of $\vrp$** (to a supervaluation) and we call $[\chvrp] \in {\operatorname{Cov}}(v)$ the almost tangible lift (in ${\operatorname{Cov}}(v)$) of the class $[\vrp] \in Cov(v)$. 2. If $\chvrp = \vrp$, we say that $\vrp$ itself is **almost tangible.** \[rem8.15\] $ $ 1. Clearly $\vrp$ is almost tangible iff $G(\vrp) = v(\mfq')$. A subtle point here is that then there may nevertheless exist elements $a \in R \sm \mfq'$ with $\vrp(a)$ ghost. 2. If $\vrp$ is any supervaluation, then $\chvrp$ is almost tangible. 3. If $v$ happens to be a valuation, i.e., $M$ is cancellative, then $\chvrp = \tlvrp$. \[prop8.16\] If $\psi$ is any almost tangible supervaluation dominating the supervaluation $\vrp$ (but not necessarily covering $v$), then $\psi$ dominates $\chvrp$. $\tlpsi \geq \tlvrp$, and hence $\psi= (\tlpsi)^\wedge \geq (\tlvrp)^\wedge = \chvrp.$ [IMS]{} N. Bourbaki, *Alg. Comm. VI*, §3, No.1. P. Freyd, *Abelian Categories, an Introduction to the Theory of Functors*, Harper & Row, New York, 1964. P. Freyd and A. Scedrov, *Categories, Allegories*. North-Holland, 1979. R. Huber and M. Knebusch, *On valuation spectra*, Contemp. Math. **155** (1994), 167–206. D.K. Harrison and M.A. Vitulli, *$V$-valuations of a commutative ring I*, J. Algebra **126** (1989), 264-292. I. Itenberg, G. Mikhalkin, and E. Shustin, *Tropical Algebraic Geometry*, Oberwolfach Seminars, 35, Birkhäuser Verlag, Basel, 2007. Z. Izhakian, *Tropical arithmetic and matrix algebra*, Commun. Algebra **37**:4 (2009), [1445–1468]{}. (Preprint at arXiv: math/0505458v2.) Z. Izhakian, M. Knebusch, and L. Rowen, *Supertropical semirings and supervaluations*, J. Pure and Applied Alg., [**215**]{}:10 (2011), 2431-2463. (Preprint at arXiv:1003.1101.) Z. Izhakian, M. Knebusch, and L. Rowen, *Dominance and transmissions in supertropical valuation theory*, preprint at arXiv:1102.1520, 2011. Z. Izhakian, M. Knebusch, and L. Rowen, *Monoid valuations and value ordered supervaluations*, preprint at arXiv:1104.2753, 2011. Z. Izhakian and L. Rowen. *Supertropical algebra*, Advances in Math., **225**:4 (2010), 2222––2286. (Preprint at arXiv:0806.1175.) Z. Izhakian and L. Rowen, *Supertropical matrix algebra*. Israel J. Math., **182**:1 (2011), 383–424. (Preprint at arXiv:0806.1178.) Z. Izhakian and L. Rowen, *Supertropical matrix algebra II: Solving tropical equations*, Israel J. Math., to appear. (Preprint at arXiv:0902.2159, 2009.) Z. Izhakian and L. Rowen, *Supertropical matrix algebra III: Powers of matrices and generalized eigenspaces.*, [J. of Algebra]{}, to appear (2011). Preprint at arXiv:1008.0023. Z. Izhakian and L. Rowen, , **324**:8 (2010), 1860–1886. (Preprint at arXiv:0902.2155.) M. Knebusch and D. Zhang, *Manis Valuations and Prüfer Extensions. I. A New Chapter in Commutative Algebra*, Lecture Notes in Mathematics, 1791, Springer-Verlag, Berlin, 2002. M. Knebusch and D. Zhang, *Convexity, valuations, and Prüfer extensions in real algebra*, Doc. Math. **10** (2005), 1–109. S. Maclane, *Categories for the working mathemtician*, 4th ed., Springer-Verlag, 1998. J. Rhodes and B. Steinbergy. . Springer-Verlag, 2008. D. Zhang, *The $M$-valuation spectrum of a commutative ring*, Commun. Algebra **30** (2002), 2883–2896. [^1]: The research of the first author was supported by the Oberwolfach Leibniz Fellows Programme (OWLF), Mathematisches Forschungsinstitut Oberwolfach, Germany. [^2]: The research of the first and third authors have been supported by the Israel Science Foundation (grant No. 448/09). [^3]: The research of the second author was supported in part by the Gelbart Institute at Bar-Ilan University, the Minerva Foundation at Tel-Aviv University, the Department of Mathematics of Bar-Ilan University, and the Emmy Noether Institute at Bar-Ilan University. [^4]: \ File name: [^5]: Although this does not mean surjectivity in the usual sense, there is no danger of confusion since a supervaluation $\vrp:R \to V$ can hardly ever be surjective as a map except in the degenerate case $V=M$. [^6]: Notice that a homomorphic image of a bipotent semiring is again bipotent. [^7]: Recall the formulas for $x + y$ in §\[sec:1\], preceding Theorem \[thm1.2\]. [^8]: As in [@IKR1] we view an equivalence relation on a set $X$ as a subset of $X \times X$ in the usual way. [^9]: Recall that $S(U)$ denotes the set of tangible NC-products in $U$ (Definition \[defn4.2\]).

--- abstract: | From high-resolution spectra a non-LTE analysis of the [Mg[ii]{}]{} 4481.2 Å feature is implemented for 52 early and medium local B stars on the main sequence (MS). The influence of the neighbouring line [Al[iii]{}]{} 4479.9 Å is considered. The magnesium abundance is determined; it is found that $\log \varepsilon({\rm Mg})=7.67\pm0.21$ on average. It is shown that uncertainties in the microturbulent parameter [$V_t$]{} are the main source of errors in $\log \varepsilon({\rm Mg})$. When using 36 stars with the most reliable [$V_t$]{} values derived from [O[ii]{}]{} and [N[ii]{}]{} lines, we obtain the mean abundance $\log \varepsilon({\rm Mg})=7.59\pm0.15$. The latter value is precisely confirmed for several hot B stars from an analysis of the [Mg[ii]{}]{} 7877 Å weak line. The derived abundance $\log \varepsilon({\rm Mg})=7.59\pm0.15$ is in excellent agreement with the solar magnesium abundance $\log \varepsilon_{\sun}({\rm Mg})=7.55\pm0.02$, as well as with the proto-Sun abundance $\log \varepsilon_{ps}({\rm Mg})=7.62\pm0.02$. Thus, it is confirmed that the Sun and the B-type MS stars in our neighbourhood have the same metallicity. date: 'Accepted 2004 May 15. Received 2004 May 15; in original form 2004 May 20' title: 'Surface abundances of light elements for a large sample of early B-type stars - IV. The magnesium abundance in 52 stars - a test of metallicity' --- \[firstpage\] stars: atmospheres – stars: early-type – stars: magnesium abundance. [^1] INTRODUCTION ============ In the 1990s a number of works were published, in which the carbon, nitrogen and oxygen abundances were derived for local early B-type stars on the main sequence (MS) (Kilian 1992; Gies & Lambert 1992; Cunha & Lambert 1994; Daflon et al. 1999). It was shown that these B stars were deficient in C, N and O typically by 0.2–0.4 dex with respect to the published abundances for the Sun. One might ask whether this deficiency reflects a real difference in the metal abundances between the B-type MS stars and the Sun. (It should be remembered that it is accepted in astrophysics to denote by metals all chemical elements heavier than hydrogen and helium). If so, then, following Nissen (1993), it should be recognized that the metallicity of the B stars is lower by about a factor of two than the metallicity of the Sun, i.e., the metal mean mass fraction must be Z = 0.01 for them instead of the usually accepted solar value Z = 0.02. An assumption arose that the Sun formed out of material enriched by metals (see, e.g., Daflon et al. 1999). Moreover, a hypothesis appeared that other F–G main sequence stars with planet systems are also more metal rich than similar stars without planets (Gonzalez 1997, 2003). There was a different viewpoint, namely that the initial C, N and O abundances in the Sun and in the nearby B stars were really the same, but the observed deficiency in the B stars results from either the effect of gravitational-radiative separation of elements in the stellar atmosphere or systematic errors in the abundance analysis of the B stars. A partial solution of the problem came unexpectedly from the recent C, N and O abundance determinations for the Sun on the basis of a three-dimensional time-dependent hydrodynamical model solar atmosphere (Allende Prieto et al. 2001, 2002; Asplund 2002; Asplund et al. 2004). It was shown that in this more realistic 3D case the solar C, N and O abundances decrease markedly as compared with the standard 1D case (see, e.g., Holweger 2001). So, the discrepancy between the B stars and the Sun decreases, too. Carbon, nitrogen and oxygen participate in the H-burning CNO-cycle that is a main source of energy of B-type MS stars. Their abundances in stellar interiors change significantly during the MS phase. There are empirical data on mixing between the interiors and surface layers of early B stars (see, e.g., Lyubimkov 1996, 1998; Maeder & Meynet 2000). Mixing will reduce the surface C abundance and increase the surface N abundance as material from layers exposed to the N-cycle is brought to the surface. Mixing to deeper depths will bring products of the ON-cycles to the surface resulting in a reduction of the surface O abundance and further increase of the N abundance. In fact, our analysis of the helium abundance in early B stars led to the conclusion that the observed He enrichment during the MS phase can be explained by rotation with velocities $v=250-400$ [kms$^{-1}$]{}(Lyubimkov et al. 2004). When adopting, for instance, $v=400$ [kms$^{-1}$]{}, one may find from Heger & Langer’s (2000) computations that for a 15 Mo star the surface C, N and O abundances change by -1.2, +0.9 and -0.4 dex, respectively. Therefore, use of C, N and O as indicators of metallicity of early B stars is not strictly correct. In the case of cool stars, iron is usually used as an indicator of metallicity. On the one hand, this element does not participate in thermonuclear reactions during early evolutionary phases, and, therefore, its abundance corresponds to the initial metallicity of a star. On the other hand, it gives a lot of spectral lines, which allow its abundance to be derived with high accuracy. The metallicity \[$Fe/H$\] is determined then as the difference between the Fe abundances of the star and the Sun, i.e. $$[Fe/H] = \log \varepsilon_*({\rm Fe}) - \log \varepsilon_{\sun}({\rm Fe}),$$ where the abundance $\log \varepsilon$ corresponds to the standard logarithmic scale with the hydrogen abundance $\log \varepsilon({\rm H})=12.00$. Unfortunately, in spectra of early B stars the iron lines are weak. One of the strongest lines in spectra of early and medium B stars is the [Mg[ii]{}]{} 4481.2 Å line. Magnesium is quite suitable as an indicator of metallicity, because it does not change its abundance during the MS evolutionary phase. Even its participation in the MgAl-cycle leads to a negligible decrease of the abundance, by about 0.02–0.03 dex (Daflon et al. 2003). Moreover, for magnesium, unlike C, N and O, the precise solar abundance is known. In fact, according to Holweger (2001), its abundance derived from photospheric lines is $\log \varepsilon({\rm Mg})=7.54\pm0.06$. This spectroscopic value is very close to the meteoritic abundance $7.56\pm0.02$. Averaging these two abundances, Lodders (2003) recommends the value $\log \varepsilon_{\sun}({\rm Mg})= 7.55\pm0.02$ as the solar magnesium abundance (’meteoritic’ refers to the CI-type chondrites). We shall use the latter value. The metallicity of stars will be determined from formula $$[Mg/H] = \log \varepsilon_*({\rm Mg}) - \log \varepsilon_{\sun}({\rm Mg}).$$ It should be noted that Lodders presents also the proto-Sun magnesium abundance $\log \varepsilon_{ps}({\rm Mg})=7.62\pm0.02$. This value differs slightly from the above-mentioned photospheric value because heavy-element fractionation in the Sun has altered original abundances. At present we continue our study of large sample of early and medium B stars. Our general goal was to provide new comprehensive observations of the stars and to determine and analyse the abundances of CNO-cycle elements, namely helium, carbon, nitrogen and oxygen. A series of papers is being published. First, we obtained high-resolution spectra of more than 100 stars at two observatories, namely the McDonald Observatory of the University of Texas and the Crimean Astrophysical Observatory (Ukraine); equivalent widths of 2 hydrogen lines and 11 helium lines were measured (Lyubimkov et al. 2000, hereinafter Paper I). Second, we determined a number of fundamental parameters of the stars including the effective temperature [$T_{\rm eff}$]{}, surface gravity [$\log\,g$]{}, interstellar extinction $A_v$, distance $d$, mass $M$, radius $R$, luminosity $L$, age $t$, and relative age [$t/t_{\rm MS}$]{}, where $t_{\rm MS}$ is the MS lifetime (Lyubimkov et al. 2002, Paper II). Third, from non-LTE analysis of [He[i]{}]{} lines we derived the helium abundance $He/H$, microturbulent parameter [$V_t$]{} and projected rotational velocity [$v$sin$i$]{} (Lyubimkov et al. 2004, Paper III). Presently, we are analysing [C[ii]{}]{}, [N[ii]{}]{} and [O[ii]{}]{} lines; in particular, the microturbulent parameter [$V_t$]{} is determined from these lines, too. We present in this paper a new determination of the Mg abundances in B stars that is based on our spectra and our atmospheric parameters. We use the [Mg[ii]{}]{} 4481.2 Å line and for a sample of hot stars a weaker [Mg[ii]{}]{} line at 7877 Å. SELECTION OF STARS ================== Influence of the AlIII 4479.9 Å line ------------------------------------ An analysis of the [Mg[ii]{}]{} 4481.2 Å line can be complicated by presence of the neighbouring [Al[iii]{}]{} 4479.9 Å line. If a star has a rather high rotational velocity [$v$sin$i$]{}, the two lines form a common blend. In such cases the computation of synthetic spectra is necessary. The [Al[iii]{}]{} 4479.9 Å line intensity depends strongly on [$T_{\rm eff}$]{}. In Fig.1 we show as examples the observed spectra in the [Mg[ii]{}]{} 4481.2 Å vicinity for three B stars with the different effective temperatures [$T_{\rm eff}$]{} and low rotational velocities [$v$sin$i$]{}(i.e. with sharp spectral lines). One sees that the [Al[iii]{}]{} 4479.9 Å line is relatively strong at [$T_{\rm eff}$]{} = 22500 K, but weak both at the high temperature, [$T_{\rm eff}$]{} = 30700 K, and at the low one, [$T_{\rm eff}$]{} = 15800 K. ![The lines [Mg[ii]{}]{} 4481.2 Å and [Al[iii]{}]{} 4479.9 Å in observed spectra of three B stars with the different effective temperatures [$T_{\rm eff}$]{} and low rotational velocities [$v$sin$i$]{}.](fig1.eps){width="84mm"} In order to study the dependence on [$T_{\rm eff}$]{} in more detail, we measured the equivalent width $W$ of this line for 28 B stars from our list, which have the little rotational velocities (5 [kms$^{-1}$]{}$\leqslant$ [$v$sin$i$]{} $\leqslant$ 33 [kms$^{-1}$]{}) and the effective temperatures [$T_{\rm eff}$]{} $\geqslant$ 15800 K. In Fig.2a we present the $W$ value as a function of [$T_{\rm eff}$]{}. To guide the eye, we approximated the relation by the fourth-order polynomial (solid curve in Fig.2a). One sees that there is a maximum of the $W$ values in the [$T_{\rm eff}$]{} range between 21000 and 24000 K. ![Dependence on the effective temperature of (a) the [Al[iii]{}]{}4479.9 Å line equivalent widths and (b) the ratio of the [Al[iii]{}]{}and [Mg[ii]{}]{} line equivalent widths. Solid curves correspond to approximations by the fourth-order polynomials.](fig2.eps){width="84mm"} In Fig.2b we show the dependence on [$T_{\rm eff}$]{} for the ratio of equivalent widths of the [Al[iii]{}]{} 4479.9 Å and [Mg[ii]{}]{} 4481.2 Ålines. One sees that in the [$T_{\rm eff}$]{} range from 19000 to 27000 K this ratio is equal to about 20–30 per cent; so, if the lines form a common blend, a contribution of the Al line must be taken into account. We believe that the influence of the Al line can be ignored only for B stars with temperatures [$T_{\rm eff}$]{} $<$ 17000 K. Two stars, namely HR1923 and HR2633, are ’outliers’ in Fig.2a (they are marked in Fig.2a). We may not explain this discrepancy by errors in the $W$([Al[iii]{}]{} 4479.9) measurements, because the profiles of the Al line for these stars are sharp and symmetric and, therefore, measured with confidence. The explanation cannot be connected with errors in [$T_{\rm eff}$]{} too, because they are small, $\pm500$ and $\pm400$ K respectively. The star HR2633 is the only B giant in Fig.2, it has [$\log\,g$]{} = 3.38, whereas for all other stars [$\log\,g$]{} $>$ 3.8, i.e., they are dwarfs. If the [Al[iii]{}]{} 4479.9 Å line is stronger for B giants than for B dwarfs with the same temperatures [$T_{\rm eff}$]{}, then the elevated $W$([Al[iii]{}]{} 4479.9) value for HR2633 becomes understandable. As far as HR1933 is concerned, it is possible that this star has a somewhat lowered Al abundance. It is interesting that in Fig.2b, unlike Fig.2a, neither of these stars is an ‘outlier’. Selection of stars ------------------ Although the empirical relations in Fig.2 would in principle allow one to correct for the [Al[iii]{}]{} contribution to the [Al[iii]{}]{} - [Mg[ii]{}]{} blend in the spectrum of a rapidly-rotating star, we chose in the interest of controlling the uncertainties affecting the the Mg abundance to restrict the selection of stars to those where the [Al[iii]{}]{} and [Mg[ii]{}]{} lines are clearly resolved. In Paper III, we determined accurate rotational velocities [$v$sin$i$]{}for 100 normal B stars. It was shown that the [$v$sin$i$]{} values of the stars span the range from 5 to 280 [kms$^{-1}$]{}. In order to separate clearly the [Mg[ii]{}]{} and [Al[iii]{}]{} lines in question, we should exclude the stars with high [$v$sin$i$]{}. We see from our spectra that for the stars with [$v$sin$i$]{} $\gtrsim$ 100 [kms$^{-1}$]{} the [Mg[ii]{}]{} line is substantially blended by the [Al[iii]{}]{} line, especially in the above-mentioned [$T_{\rm eff}$]{}range between 19000 and 27000 K. So, we reject stars with such large [$v$sin$i$]{}. Eventually, we singled out the stars with [$v$sin$i$]{} $\leqslant$ 54 [kms$^{-1}$]{} when [$T_{\rm eff}$]{} $>$ 17000 K; one relatively cool star ([$T_{\rm eff}$]{} = 16600 K) with [$v$sin$i$]{} = 70 [kms$^{-1}$]{}was added. It is important to note that in spectra of all these stars the [Al[iii]{}]{} 4479.9 Å line is distinctly separated from the [Mg[ii]{}]{} 4481.2 Å line, so equivalent widths of the latter are measured with confidence. A list of selected 52 stars is presented in Table 1. We give here for each star its HR number, as well as the effective temperature [$T_{\rm eff}$]{} and surface gravity [$\log\,g$]{} from Paper II. Next the microturbulent parameter [$V_t$]{} is given, which was inferred by us for the most of the stars from [N[ii]{}]{} and [O[ii]{}]{} lines. However, for 16 cool stars these lines are too weak and, therefore, unfit for the [$V_t$]{} determination by the standard method, when an agreement in abundances derived from relatively strong and weak lines should take place. We present for these stars the [$V_t$]{} values obtained in Paper III from [He[i]{}]{} lines (marked by asterisks in Table 1). The projected rotational velocities [$v$sin$i$]{} from Paper III are given next in Table 1. We remark below on the equivalent widths $W$(4481) and magnesium abundances $\log \varepsilon({\rm Mg})$ presented in two last columns. ------ ------------------- --------------- ---------------- ---------------- ----------- ------------------------------ HR [$T_{\rm eff}$]{} [$\log\,g$]{} [$V_t$]{} [$v$sin$i$]{} $W$(4481) $\log \varepsilon({\rm Mg})$ K [kms$^{-1}$]{} [kms$^{-1}$]{} mÅ 38 18400 3.82 2.7 11 194 7.51 561 13400 3.75 0.0$^*$ 24 307 8.06 1072 22300 3.81 6.5 39 184 7.57 1363 13700 3.67 1.8$^*$ 36 304 7.99 1595 22500 4.17 1.0 6 157 7.69 1617 18300 3.96 0.5$^*$ 46 233 8.02 1640 19400 4.11 0.0 54 201 7.82 1731 17900 3.98 2.2 41 214 7.67 1753 15500 4.10 0.0$^*$ 27 274 7.97 1756 27900 4.22 5.1 33 135 7.67 1781 22500 4.08 1.5 5 150 7.60 1783 22300 4.00 1.4 14 152 7.64 1810 20000 4.01 0.0 24 191 7.83 1820 18600 4.08 1.0 14 205 7.77 1840 21300 4.26 1.2 11 171 7.67 1848 18900 4.23 0.0 25 196 7.70 1855 30700 4.42 5.0 20 93 7.41 1861 25300 4.11 1.7 10 128 7.58 1886 23300 4.11 1.5 13 134 7.49 1887 27500 4.13 4.8 30 130 7.64 1923 20900 3.84 1.2 17 146 7.44 1950 23100 4.13 1.8 34 147 7.60 2058 21000 4.20 1.0 21 179 7.75 2205 18800 3.74 2.0 9 184 7.54 2222 25200 4.22 0.0 17 116 7.51 2494 17500 4.09 1.6$^*$ 34 175 7.32 2517 16600 3.31 9.5 70 304 7.43 2618 22900 3.39 12.0 47 227 7.68 2621 15800 4.11 0.0$^*$ 11 258 7.89 2633 18000 3.38 7.0 18 229 7.37 2688 19200 3.86 0.8 17 139 7.24 2739 29900 4.10 10.0 47 118 7.48 2756 16200 3.96 0.0$^*$ 26 248 7.92 2824 19400 3.89 1.2 42 188 7.70 2928 22800 3.90 4.4 26 136 7.31 3023 21200 4.02 1.3 42 175 7.73 6588 17000 3.77 1.3$^*$ 45 205 7.62 6787 20000 3.54 4.2 44 174 7.39 6941 17700 3.82 2.0$^*$ 18 208 7.64 6946 19100 3.76 0.6 45 206 7.90 7426 16100 3.62 0.0$^*$ 29 230 7.89 7862 17100 4.00 2.0$^*$ 34 223 7.66 7929 16700 3.64 1.0$^*$ 4 227 7.88 7996 15600 3.65 0.1$^*$ 35 251 8.01 8385 15000 3.50 1.2$^*$ 19 238 7.80 8403 14100 3.70 0.0$^*$ 17 282 8.03 8439 17400 3.31 5.1 19 255 7.62 8549 19700 3.95 2.6 8 165 7.40 8554 14000 4.03 0.0$^*$ 18 300 7.96 8768 17500 3.81 0.0 8 200 7.70 8797 27200 3.98 10.7 47 161 7.65 9005 21900 4.03 2.9 15 177 7.68 ------ ------------------- --------------- ---------------- ---------------- ----------- ------------------------------ : Parameters of 52 B stars, equivalent width of the [Mg[ii]{}]{} 4481 Å line and the magnesium abundance $^*$ [$V_t$]{} is derived from [He[i]{}]{} lines It should be noted that the distances of programme stars, according to Paper II, are less than 800 pc, so the stars are really in the solar neighbourhood. OBSERVATIONAL MATERIAL AND EQUIVALENT WIDTHS ============================================ High-resolution spectra of programme stars have been acquired in 1996–1998 at two observatories: the McDonald Observatory (McDO) of the University of Texas and the Crimean Astrophysical Observatory (CrAO). At the McDO the 2.7 m telescope and coudé echelle spectrometer (Tull et al. 1995) was used. A resolving power was $R$ = 60000 and the typical signal-to-noise ratio was between 100 and 300. At the CrAO we observed on the 2.6 m telescope with coudé spectrograph. In this case we had $R$ = 30000 and a signal-to-noise ratio between 50 and 200. A more detailed description of the observations and reductions of spectra can be found in Paper I. The equivalent width $W$ of the [Mg[ii]{}]{} 4481.2 Å line was measured by direct integration. The measurements were effected independently by two of us (TMR and partially by DBP). A comparison of these two sets of measurements showed good agreement and an absence of a systematic difference. The averaged $W$ values were adopted. It should be noted that a similar approach has been used by us in Paper I in measurements of hydrogen and helium lines. The $W$ values of the [Mg[ii]{}]{} 4481.2 Å line are given in Table 1. According to our estimates, the error in the W determination is $\pm5$ per cent on average. In our list there are 12 common stars, which have been observed both at the McDO and at the CrAO. A comparison of the $W$ values for two observatories showed good agreement: the mean difference is $\pm4$ per cent and the maximum discrepancy is $\pm9$ per cent. Note that the very good agreement (a scatter within $\pm5$ per cent) has been obtained as well in Paper I for [He[i]{}]{} lines. ![Comparison of our measurements of the [Mg[ii]{}]{} 4481.2 Åline equivalent widths with Kilian & Nissen’s (1989) data for 10 common stars.](fig3.eps){width="84mm"} It would be interesting to compare our $W$ values for the [Mg[ii]{}]{}4481.2 Å line with other published data. Such a comparison with measurements of Kilian & Nissen (1989) for 10 common stars is shown in Fig.3. One sees that for all the stars excepting one a difference in $W$ lies within $\pm4$ per cent. The only exception is the hot B star HR 2739, for which our equivalent width is greater by 15 per cent than Kilian & Nissen’s value. It should be noted in this connection that use of Kilian & Nissen’s equivalent width for this star, according to our calculations, leads to the magnesium abundance $\log \varepsilon({\rm Mg})=7.37$, whereas our $W$ value gives $\log \varepsilon({\rm Mg})=7.48$ that is substantially closer to the solar abundance $\log \varepsilon_{\sun}({\rm Mg})=7.55$ (Lodders 2003). On the whole, Fig.3 confirms the stated accuracy of our $W$ values. NON-LTE COMPUTATIONS OF THE MgII 4481 LINE ========================================== More than 30 years ago Mihalas (1972) first showed that a non-LTE approach is necessary in the [Mg[ii]{}]{} 4481.2 Å line analysis for early B stars and especially O stars. According to his calculations, the non-LTE equivalent widths $W$ of the line are greater than the LTE ones, and the discrepancy increases with increasing [$T_{\rm eff}$]{}. Our computations confirm this conclusion; moreover, the $W$ increment seems to be greater for B giants than for B dwarfs. In particular, when [$\log\,g$]{} = 3.5, a value corresponding approximately to giants, the difference in $W$ between the non-LTE and LTE cases is about 5–6 per cent at [$T_{\rm eff}$]{} = 20000–25000 K, i.e. comparable with errors in the measured W values, whereas it exceeds 50 per cent at [$T_{\rm eff}$]{} = 30000 K. Since our sample includes both dwarfs and giants, we apply the non-LTE approach to all stars. The [Mg[ii]{}]{} 4481.2 Å line was treated in our computations as a triplet, and for its three components the same oscillator strengths were adopted as in Daflon et al.’s (2003) work. In the non-LTE calculations of the line we used the codes [detail]{} (Giddings 1981; Butler 1994) and [surface]{} (Butler 1984). The former code determines the atomic level populations by jointly solving the radiative transfer and statistical equilibrium equations. The latter one computes the emergent flux and, therefore, the line profile. We employed Przybilla et al.’s (2001) magnesium model atom. It includes all Mg[i]{} atomic levels with principal quantum numbers n $\leqslant$ 9, all [Mg[ii]{}]{} levels with n $\leqslant$ 10 and the Mg[iii]{} ground state. This model atom is likely the most complete one for magnesium today. The computations were implemented on the basis of the LTE model atmospheres calculated in Paper III with the code [atlas9]{} of Kurucz (1993). In order to analyse the effect of the microturbulent parameter [$V_t$]{} on the [Mg[ii]{}]{} 4481.2 Å line equivalent width, we made trial calculations of the line for some model atmospheres from Kurucz’s (1993) grid. We confirmed that the line strengthens and its sensivity to [$V_t$]{} increases as the effective temperature [$T_{\rm eff}$]{}decreases from 30000 to 14000 K. Therefore, the effect of errors in [$V_t$]{} on the Mg abundance determination is largest for coolest B stars from our list. THE MAGNESIUM ABUNDANCES ======================== Errors in the derived abundances -------------------------------- The derived magnesium abundances $\log \varepsilon({\rm Mg})$ are presented in Table 1. When averaging these values for all 52 stars, we obtained the mean abundance $\log \varepsilon({\rm Mg}) = 7.67\pm0.21$. The difference with the solar value of 7.55 is 0.12 dex, i.e., substantially less than the uncertainty in the mean value. We estimated typical errors in the abundances, which originate from uncertainties in the equivalent width $W$, effective temperature [$T_{\rm eff}$]{}, surface gravity [$\log\,g$]{}, and microturbulent velocity [$V_t$]{}. As mentioned above, the typical uncertainty in $W$ is $\pm5$ per cent. Contributions of three other parameters can depend on [$T_{\rm eff}$]{}, so we divided the stars into four groups as follows: [$T_{\rm eff}$]{}$\leqslant$ 15000 K, [$T_{\rm eff}$]{} between 15000 and 19000 K, [$T_{\rm eff}$]{} between 19000 and 24000 K, [$T_{\rm eff}$]{} $>$ 24000 K. In each group the mean errors in [$T_{\rm eff}$]{} and [$\log\,g$]{} are evaluated from the individual errors (see Paper II). As far as uncertainties in [$V_t$]{} are concerned, we adopted the values $\pm2.0$ [kms$^{-1}$]{} and $\pm2.5$ [kms$^{-1}$]{} for stars with [$V_t$]{} $<$ 5 [kms$^{-1}$]{} and [$V_t$]{} $\sim$ 10 [kms$^{-1}$]{}, respectively. We found by this method that the full typical error in $\log \varepsilon({\rm Mg})$ is about 0.14 dex for stars with [$T_{\rm eff}$]{}between 19000 and 30000 K, about 0.20 dex for stars with [$T_{\rm eff}$]{}between 15000 and 19000 K and can attain 0.30 dex for coolest programme stars with [$T_{\rm eff}$]{} around 14000 K. It is important to note that the uncertainty in the microturbulent parameter [$V_t$]{} gives the dominant contribution to the full error, namely from 40 to 65 per cent when [$T_{\rm eff}$]{} decreases from 30000 to 19000 K and 70–80 per cent for cooler stars. The latter conclusion will play an important role in further discussion. Comparison with $T_{\rm eff}$, $\log g$, $V_t$ and $v\,sin\,i$ -------------------------------------------------------------- ![The Mg abundance as a function of (a) the effective temperature [$T_{\rm eff}$]{} and (b) the surface gravity [$\log\,g$]{}. Filled circles correspond to stars with [$V_t$]{} from [N[ii]{}]{} and [O[ii]{}]{} lines, open circles - stars with [$V_t$]{} from [He[i]{}]{} lines. Dashed line corresponds to the solar abundance $\log \varepsilon_{\sun}({\rm Mg}) = 7.55$.](fig4.eps){width="84mm"} We show in Fig.4 the derived Mg abundances as a function of the effective temperature [$T_{\rm eff}$]{} and surface gravity [$\log\,g$]{}. In Fig.5 the Mg abundances are displayed as a function of the microturbulent parameter [$V_t$]{} and projected rotational velocity [$v$sin$i$]{}. Filled circles correspond to 36 stars with the [$V_t$]{} values derived from [N[ii]{}]{} and [O[ii]{}]{} lines; open circles correspond to 16 stars, where we had to adopt the [$V_t$]{} values determined from [He[i]{}]{} lines (see above). Dashed lines correspond to the solar abundance $\log \varepsilon_{\sun}({\rm Mg})=7.55$ (Lodders 2003). ![The Mg abundance as a function of (a) the microturbulent parameter [$V_t$]{} and (b) the projected rotational velocity [$v$sin$i$]{}.](fig5.eps){width="84mm"} A few points are of special interest in Fig.4 and 5, namely: 1. filled circles group well around the solar Mg abundance; 2. open circles tend to show elevated $\log \varepsilon({\rm Mg})$ values; 3. open circles are collected in the region of lowest [$T_{\rm eff}$]{} (Fig.4a); 4. open circles correspond to the [$V_t$]{} values close to zero (Fig.5a). One may see from Fig.4a that for 11 of 16 open circles the stellar Mg abundance exceeds the solar one by greater than 0.30 dex, i.e., greater than the largest error found for the coolest programme stars (see above). These 11 stars have effective temperatures [$T_{\rm eff}$]{}$<$ 17000 K excepting one star and the microturbulent velocities 0 $\leqslant$ [$V_t$]{} $<$ 2 [kms$^{-1}$]{} derived from [He[i]{}]{} lines. We can not explain the Mg overabundance for 11 stars by uncertainties in the $W$ measurements. In fact, it is necessary to lower $W$ by 15–20 per cent to eliminate the overabundance, whereas the typical error in $W$ is 5 per cent. We considered a systematic overestimation of $W$ due to blending of the [Mg[ii]{}]{}4481.2 Å line by some other lines. According to the VALD-2 database (Kupka et al. 1999), there is only one potential line, Fe[ii]{} 4480.7 Å, in the vicinity of [Mg[ii]{}]{} 4481.2 Å. This very weak line, as is the [Al[iii]{}]{} 4479.9 Å line, is clearly separated from [Mg[ii]{}]{} 4481.2 Å on our spectra of cool B stars with low [$v$sin$i$]{}. As mentioned above, the coolest programme B stars are the most sensitive to uncertainties in [$V_t$]{}; moreover, errors in [$V_t$]{} give for them a dominating contribution, up to 80 per cent, to the errors in $\log \varepsilon({\rm Mg})$. So, errors in [$V_t$]{} are very likely to be a main cause of the Mg overabundance for such stars. We present in Table 2 the thermal velocity $V_{\rm therm}$ of the He, N, O and Mg atoms for three temperatures T of interest. One sees that the $V_{\rm therm}$ values for the He atoms are by about a factor of two higher than for the N and O atoms. Even at $T$ = 15000 K, $V_{\rm therm}$ for helium is rather high, $V_{\rm therm}$ $\approx$ 8 [kms$^{-1}$]{}. Therefore, when the microturbulent parameter [$V_t$]{} is small but not zero, a broadening and strengthening of [He[i]{}]{} lines by microturbulence, unlike [N[ii]{}]{} and [O[ii]{}]{} lines, is small compared with the effects the thermal broadening. So, such a small [$V_t$]{} is derived from [He[i]{}]{} lines with appreciable uncertainty. One may suppose that the values [$V_t$]{}$\sim$ 0 [kms$^{-1}$]{} adopted for coolest stars are underestimated. Our analysis showed that it is enough to increase [$V_t$]{} for such stars from 0 to 3 or 4 [kms$^{-1}$]{} in order to arrive at agreement in $\log \varepsilon({\rm Mg})$ with other stars. $T$, K He N O Mg -------- ------ ----- ----- ----- 15000 7.8 4.2 3.9 3.2 22000 9.6 5.1 4.8 3.9 30000 11.2 6.0 5.6 4.6 : Thermal velocities of the He, N, O, and Mg atoms (in [kms$^{-1}$]{}) Thus, we arrive at the conclusion that the abundances $\log \varepsilon({\rm Mg})$ represented in Fig.4 and 5 by open circles are less reliable, because they are likely based on the underestimated [$V_t$]{} values. When excluding these abundances, we found the mean magnesium abundance $\log \varepsilon({\rm Mg}) = 7.59\pm0.15$ for the 36 remaining stars. This value differs from the solar abundance only by 0.04 dex. It should be noted that it is also very close to the proto-Sun Mg abundance $\log \varepsilon_{ps}({\rm Mg}) = 7.62\pm0.02$ (Lodders 2003). Abundances from the MgII 7877 Å line for hot B stars ---------------------------------------------------- As mentioned above, uncertainties in the microturbulent parameter [$V_t$]{} are the main sources of errors in the Mg abundances. On our spectra, there is a detectable [Mg[ii]{}]{} line at 7877.05 Å, which is rather weak in spectra of hot B stars and, therefore, insensitive to [$V_t$]{}. This line is used by us as a check on the $\log \varepsilon({\rm Mg})$ determination from the 4481 Å line. Computations of the 7877 Å line with the same codes [detail]{} and [surface]{} showed that for B stars with the effective temperatures [$T_{\rm eff}$]{} between 25000 and 30000 K its equivalent widths $W$ vary between 15 and 30 mÅ and, in fact, are almost independent of [$V_t$]{}. At lower [$T_{\rm eff}$]{}, the 7877 Å line is stronger and sensitive to the adopted [$V_t$]{}. We selected from Table 1 five stars with [$T_{\rm eff}$]{} $>$ 25000 K, namely HR1756, 1855, 1861, 2739, and 8797, for which we could measure the line from the McDO spectra. The measurements were complicated by blending due to telluric H$_2$O lines. To correct for the latter we used the spectrum of the hot B star HR7446 with rapid rotation ([$T_{\rm eff}$]{} = 26800 K, [$v$sin$i$]{} = 270 [kms$^{-1}$]{}, see Paper III). Weak lines, like 7877 Å, are not seen in its spectrum because of the rotation, so the telluric spectrum can be clearly observed. When comparing this spectrum with individual spectra of 5 stars, we allowed for the differences in airmasses. Dividing out the telluric lines in the vicinity of the 7877 Å line, we obtained its profile and measured the equivalent width $W$. The observed $W$ values for 5 stars range from 21 to 29 mÅ. The corresponding Mg abundances are very close to the values derived from the 4481 Å line. The mean Mg abundance from 7877 Å for these stars is $\log \varepsilon({\rm Mg}) = 7.59\pm0.05$, i.e., it coincides exactly with the above-mentioned mean abundance found from the 4481 Å line for the 36 stars with most reliable [$V_t$]{}. Note that the mean Mg abundance from 4481 Å for the same 5 stars is $7.56\pm0.11$. Therefore, very good agreement is found between the Mg abundances determined from the [Mg[ii]{}]{} lines at 7877 Å and 4481 Å. Thus, we confirmed the reliability of our results derived from the [Mg[ii]{}]{} 4481 Å line. In our further discussion, the value $\log \varepsilon({\rm Mg}) = 7.59\pm0.15$ is adopted as the final mean magnesium abundance in B stars. Therefore, according to formula (2), the mean index of metallicity is \[$Mg/H$\] = $0.04\pm0.15$. Thus, the mean metallicity of the B-type MS stars in the solar neighborhood is very close or equal to the metallicity of the Sun. Distribution of the stars on $\log \varepsilon({\rm\bf Mg})$ ------------------------------------------------------------ The derived magnesium abundances $\log \varepsilon({\rm Mg})$ range from 7.24 to 8.06 in the case of all 52 stars and from 7.24 to 7.90 in the case of the 36 stars with the most reliable microturbulent parameters [$V_t$]{}. What are causes of the spread in the Mg abundances? Is it possible to explain the spread only by errors in the $\log \varepsilon({\rm Mg})$ values, or are there some real star-to-star variations? ![Distribution on the Mg abundance of (a) all 52 stars and (b) the 36 stars with most reliable [$V_t$]{}.](fig6.eps){width="84mm"} In order to answer these questions, we plotted distributions of the abundances $\log \varepsilon({\rm Mg})$ separately for the 52 stars (Fig.6a) and the 36 stars (Fig.6b). The pronounced feature in both histograms is the sharp maximum at $\log \varepsilon({\rm Mg}) = 7.65$ and 7.64, respectively. It is interesting to note that the maximum position is very close to the proto-Sun magnesium abundance $\log \varepsilon_{ps}({\rm Mg})=7.62\pm0.02$ (Lodders 2003). The histogram in Fig.6a shows one more feature: a second and weak maximum between $\log \varepsilon({\rm Mg}) = 7.88$ and 8.06. This maximum is due completely to the relatively cool B stars with small [$V_t$]{} derived from [He[i]{}]{} lines. These stars create a substantial asymmetry in the distribution; this fact can be considered as one more confirmation that the corresponding abundances $\log \varepsilon({\rm Mg})$ were overestimated. The second maximum is absent in Fig.6b, where the stars with less reliable Vt are excluded. In Fig.6b 27 of 36 stars have effective temperatures [$T_{\rm eff}$]{} $>$ 19000 K; so, the typical error $\sigma$ in $\log \varepsilon({\rm Mg})$ for them is 0.14 dex (see above). For the remaining 9 stars with 16600 K $\leqslant$ [$T_{\rm eff}$]{} $<$ 19000 K the typical error is 0.20 dex. Adopting for each star the error $\sigma$ in accordance with its [$T_{\rm eff}$]{}, we found that for 34 of 36 stars a scatter of $\log \varepsilon({\rm Mg})$ around the maximum of the distribution (i.e. $\log \varepsilon({\rm Mg}) = 7.64$) is less than 2$\sigma$. Only two stars, HR2688 and 2928, have a Mg underabundance that corresponds to 2.9$\sigma$ and 2.4$\sigma$. One may conclude that the scatter in Fig.6b is well accounted for by the errors in $\log \varepsilon({\rm Mg})$. Nevertheless, some asymmetry of the star’s distribution in Fig.6b (the left wing is more extended than the right one) may indicate the existence of local B stars with a small Mg deficiency. DISCUSSION ========== Daflon and her colleagues have analysed the [Mg[ii]{}]{} 4481 Å line in several samples of B stars with their ultimate goal being to determine the Galactic abundance gradient of Mg from about 5 kpc to 13 kpc. Our and their analysis techniques have much in common including the use of high-resolution high S/N spectra, use of the codes DETAIL and SURFACE, the same Mg model atom, and the same oscillator strengths for the components of the 4481 line. Our methods for determining the atmospheric parameters and the adopted values for those parameters differ. A comparison of Mg abundances and a discussion of the reasons for their slight difference is given in this section. Daflon et al. (2003) determined Mg abundances for 23 stars from six OB associations within 1 kpc of the Sun: Galactocentric distances ranged from 6.9 kpc to 8.2 kpc on the assumption that the Sun is at 7.9 kpc. The mean Mg abundance for 20 stars from the five associations with Galactocentric distances of 7.6 kpc to 8.2 kpc is 7.37$\pm$0.14 or 0.22 dex less than our final mean abundance. In other papers, Daflon and colleagues (Daflon, Cunha, & Butler 2004a,b) obtain Mg abundances for stars inside and outside the solar circle. Daflon & Cunha (2004) collate the abundances (and those of other elements) to determine and discuss the Galactic abundance gradients. Linear fits to the Mg abundances from 5 kpc to 13 kpc imply a solar abundance of around 7.4, i.e., a subsolar abundance but with a scatter from association to association that brushes the solar abundance. Systematic differences exist between our atmospheric parameters and those used by Daflon and colleagues. There are clear differences in [$T_{\rm eff}$]{} and [$V_t$]{}. Our [$T_{\rm eff}$]{}’s are about 1500–1700 K cooler than Daflon’s. What differences in the magnesium abundances $\log \varepsilon({\rm Mg})$ can arise because of differences in the [$T_{\rm eff}$]{} scales? We considered as an example the star HR1861 with parameters [$T_{\rm eff}$]{} = 25300 K and [$\log\,g$]{} = 4.11 (Table 1) and increased its [$T_{\rm eff}$]{} value by 1700 K. As mentioned in Paper II, an overestimation of effective temperatures [$T_{\rm eff}$]{} leads automatically to an overestimation of [$\log\,g$]{}. (Note that traces of such overestimation are likely seen in the [$\log\,g$]{} values of Daflon et al. (2003); unrealistically large [$\log\,g$]{} are found for some stars: [$\log\,g$]{} = 4.38–4.57 whereas on the ZAMS [$\log\,g$]{} $\approx$ 4.25). In other words, if we raise [$T_{\rm eff}$]{}, we should raise simultaneously [$\log\,g$]{}. Using the [$T_{\rm eff}$]{}-[$\log\,g$]{} diagram for HR 1861, we found that [$\log\,g$]{} must be increased by 0.17 dex. A redetermination of $\log \varepsilon({\rm Mg})$ with the revised model atmosphere ([$T_{\rm eff}$]{} = 27000 K, [$\log\,g$]{} = 4.28) showed that the magnesium abundance increases by 0.12 dex. (Note that the computed equivalent width $W$ of the line decreases with increasing [$T_{\rm eff}$]{}, but increases with increasing [$\log\,g$]{}, so a partial compensation takes place). We arrive at a conclusion that the difference in the [$T_{\rm eff}$]{} scales results in a higher Mg abundance for Daflon et al.’s [$T_{\rm eff}$]{} scale, but their published abundances are lower than ours. So, the difference with Daflon et al.’s scale cannot account for the difference in magnesium abundance between us. As mentioned above, uncertainties in the microturbulent parameter [$V_t$]{} give the dominating contribution to errors in $\log \varepsilon({\rm Mg})$. There is a significant discrepancy in the [$V_t$]{} values between us and Daflon et al. Just this discrepancy can be the main cause of systematic differences in $\log \varepsilon({\rm Mg})$. The microturbulent parameter [$V_t$]{} was derived by us for 100 B stars from [He[i]{}]{} lines in Paper III. The number of stars studied was large, so we had an opportunity to divide the stars into three groups according to their masses M. It was shown that in the group of relatively low-massive stars (4.1 $\leqslant M/M_{\sun} \leqslant$ 6.9) the [$V_t$]{} values are mostly within the 0–5 [kms$^{-1}$]{}range. The same result is obtained for stars of intermediate masses (7.0 $\leqslant M/M_{\sun} \leqslant$ 11.2) with the relative ages [$t/t_{\rm MS}$]{} $<$ 0.8; however, for evolved giants with 0.80 $<$ [$t/t_{\rm MS}$]{} $\leqslant$ 1.02 there is a large scatter in [$V_t$]{}up to [$V_t$]{} = 11 [kms$^{-1}$]{}. For the most massive stars of the sample (12.4 $\leqslant M/M_{\sun} \leqslant$ 18.8) a strong correlation between [$V_t$]{} and [$t/t_{\rm MS}$]{} was found; for unevolved stars of the group ([$t/t_{\rm MS}$]{} $\leqslant$ 0.24) the velocities [$V_t$]{} vary from 4 to 7 [kms$^{-1}$]{}. Very similar results were found by us recently from [N[ii]{}]{} and [O[ii]{}]{}lines. Since Daflon and colleagues derived [$V_t$]{} from [O[ii]{}]{}lines, we consider at first our data inferred only from these lines. In particular, when considering all stars of the first group and the stars of the second group with [$t/t_{\rm MS}$]{} $<$ 0.7, [O[ii]{}]{}lines give [$V_t$]{} between 0 and 5 [kms$^{-1}$]{} and a mean value [$V_t$]{} = 2.5 [kms$^{-1}$]{}; note that these stars have effective temperatures [$T_{\rm eff}$]{} $<$ 25500 K and surface gravities [$\log\,g$]{} $\geqslant$ 3.9. For the unevolved stars of the third group ([$T_{\rm eff}$]{} $>$ 27000 K and [$\log\,g$]{} $>$ 4.0) we found from [O[ii]{}]{} lines, as from [He[i]{}]{} lines, substantially greater [$V_t$]{} values, 5.8 [kms$^{-1}$]{} on average. In accordance with this finding, we selected from the lists of Daflon et al. (2003, 2004a,b) two groups of stars, namely the stars with [$T_{\rm eff}$]{} $<$ 25500 K and the stars with [$T_{\rm eff}$]{} $>$ 27000 K; in both cases [$\log\,g$]{} $>$ 4.0. The corresponding mean [$V_t$]{} values are compared with our data in Table 3. Source [$T_{\rm eff}$]{} $<$ 25500 K [$T_{\rm eff}$]{} $>$ 27000 K ------------------------- ------------------------------- ------------------------------- -- -- Daflon et al. (2003) 7.0 (4) 8.4 (10) Daflon et al. (2004a,b) 4.4 (7) 6.5 (11) our data 2.5 (21) 5.8 (3) : The mean [$V_t$]{} values (in [kms$^{-1}$]{}) derived from [O[ii]{}]{} lines for two groups of unevolved B stars by Daflon et al. (2003, 2004a,b) in comparison with our data. The number of stars used is indicated in bracket One may see from Table 3 that there is a significant difference in [$V_t$]{} for both groups between our and Daflon et al.’s (2003) results, while the difference with Daflon et al. (2004a,b) is markedly smaller. It is necessary to remember that we have used the [$V_t$]{} values averaged on [O[ii]{}]{} and [N[ii]{}]{} lines, so our real mean [$V_t$]{} values are somewhat smaller than in Table 3. It is important to note once again that we determined [$V_t$]{}independently from lines of three chemical elements, namely [He[i]{}]{}, [N[ii]{}]{} and [O[ii]{}]{}. Moreover, the Vt determination from [N[ii]{}]{} and [O[ii]{}]{}lines was implemented by the standard method, but that from [He[i]{}]{}lines was effected by a quite different method (Paper III). Nevertheless, all three sets of [$V_t$]{} are in rather good agreement. In particular, for the B stars with [$T_{\rm eff}$]{} $<$ 25500 K we found that (i) individual [$V_t$]{} values range, as a rule, between 0 and 5 [kms$^{-1}$]{}; (ii) mean [$V_t$]{} values are 1.0, 0.5 and 2.5 [kms$^{-1}$]{} for [He[i]{}]{}, [N[ii]{}]{} and [O[ii]{}]{}, respectively. (Note that we have not found reasons to prefer [$V_t$]{}([N[ii]{}]{}) over [$V_t$]{}([O[ii]{}]{}), so we used an averaged value [$V_t$]{}([N[ii]{}]{},[O[ii]{}]{})). Thus, our results from [He[i]{}]{}, [N[ii]{}]{} and [O[ii]{}]{} lines confirm that our lower [$V_t$]{} scale is preferable to the higher [$V_t$]{}scale of Daflon et al. (2003, 2004a,b). It is important to remember as well that our mean Mg abundance 7.59 obtained with the [$V_t$]{}([N[ii]{}]{},[O[ii]{}]{}) values from the [Mg[ii]{}]{} 4481 Å line is precisely confirmed from an analysis of the weak [Mg[ii]{}]{} 7877 Å line that is insensitive to [$V_t$]{}. We implemented trial determinations of $\log \varepsilon({\rm Mg})$ with various [$V_t$]{} for some stars from first and second groups. The changes in $\log \varepsilon({\rm Mg})$ depend substantially on the effective temperature [$T_{\rm eff}$]{}; nevertheless, the final conclusion is clear: the differences in the [$V_t$]{} values explain completely the above-mentioned discrepancies between our and Daflon et al.’s (2003) mean magnesium abundance. It is interesting to note that, according to Daflon (2004), the lower [$V_t$]{} values can be a result of the cooler effective temperatures [$T_{\rm eff}$]{}. The difference between our and Daflon et al.’s [$T_{\rm eff}$]{} scales cannot be sufficient to cause a marked change in thermal velocities and, hence, a change in [$V_t$]{}. We believe that the difference in [$V_t$]{} discussed above may be connected with a rough correlation between the observed equivalent widths $W$ and excitation potentials $\chi_e$: lines with higher $\chi_e$ tend to show lower $W$ on average. These lines are more sensitive to [$T_{\rm eff}$]{}than weaker lines, so their calculated $W$ values change more significantly when [$T_{\rm eff}$]{} increases. In this case one should increase [$V_t$]{} to eliminate a discrepancy in the derived abundances between relatively weak and strong lines. CONCLUSIONS =========== For more than ten years the C, N, and O abundances have been considered as indicators of the metallicity of early B stars in reference to the Sun. However, it is gradually becoming clear that this choice is not the best one. On the one hand, there are empirical data that mixing exists in the early B-type MS stars between their interiors and surface layers, so the observed abundances of the CNO-cycle elements may be affected by evolutionary alterations. On the other hand, the solar C, N and O abundances were continuously revised and tended to decrease during this period. Unfortunately, it was impossible in these cases to use the accurate meteoritic abundances, because C, N and O are incompletely condensed in meteorites, so their meteoritic abundances are significantly lower than the solar ones. Magnesium, unlike C, N and O, does not have such demerits. First, this chemical element should not alter markedly its abundance in B stars during the MS phase. Second, its solar abundance is known now very precisely from spectroscopic and meteoritic data. Displaying the rather strong [Mg[ii]{}]{} 4481.2 Å line in spectra of early and medium B stars, this element is appropriate as a reliable indicator of their metallicity. Using the high-resolution spectra of 52 B stars we effected a non-LTE analysis of the [Mg[ii]{}]{} 4481.2 Å line and determined the magnesium abundance. We studied the role of the neighbouring [Al[iii]{}]{}4479.9 Å line and selected stars with rather low rotational velocities [$v$sin$i$]{} and, therefore, with the [Mg[ii]{}]{} 4481.2 Å line well resolved from the [Al[iii]{}]{} 4479.9 Å line. We found for 52 stars the mean abundance $\log \varepsilon({\rm Mg}) = 7.67\pm0.21$. The effect of uncertainties in the microturbulent parameter [$V_t$]{} on $\log \varepsilon({\rm Mg})$ is important, especially for coolest programme stars. For 16 such stars the [$V_t$]{}values were derived from [He[i]{}]{} lines, but not from [N[ii]{}]{} and [O[ii]{}]{}lines as for other stars. Being close to zero these [$V_t$]{}([He[i]{}]{}) values are less accurate than the [$V_t$]{}([N[ii]{}]{},[O[ii]{}]{}) values. When excluding these 16 stars, we obtained the mean abundance $\log \varepsilon({\rm Mg}) = 7.59\pm0.15$ for the remaining 36 stars. This abundance is precisely confirmed from an analysis of the weak [Mg[ii]{}]{} 7877 Å line for several hot B stars. This is our recommended Mg abundance for the B-type MS stars in the solar neighbourhood (with $d <$ 800 pc). Comparing the latter value with the solar magnesium abundance $\log \varepsilon_{\sun}({\rm Mg}) = 7.55\pm0.02$, one may conclude that the metallicity of the stars is very close to the solar one. Our mean Mg abundance in B stars, as well as the position of the maximum in Fig.6, $\log \varepsilon({\rm Mg}) = 7.64$, is also very close to the proto-Sun magnesium abundance $\log \varepsilon_{ps}({\rm Mg})=7.62\pm0.02$. We discussed the preceding determinations of the Mg abundance in B stars by Daflon et al. (2003). Their $\log \varepsilon({\rm Mg})$ values are somewhat lower than ours. We showed that this difference in $\log \varepsilon({\rm Mg})$ is explained by differences in [$V_t$]{}. Thus, our results show that the Sun is not measurably enriched in metals as compared with the neighbouring young stars. ACKNOWLEDGEMENTS ================ Two of us, LSL and SIR, are grateful to the staff of the Astronomy Department and McDonald Observatory of the University of Texas for hospitality during the visit in spring 2004. DLL acknowledges the support of the Robert A. Welch Foundation of Houston, Texas. [99]{} Asplund M., 2002, in Charbonnel C., Schaerer D. and Meynet G., eds, CNO in the Universe, ASP Conf. Ser., 304, 275 Asplund M., Grevesse N., Sauval A.J., Allende Prieto C., Kiselman D., 2004, A&A, 417, 751 Allende Prieto C., Lambert D.L., Asplund M., 2001, ApJ, 556, L63 Allende Prieto C., Lambert D.L., Asplund M., 2002, ApJ, 573, L137 Butler K., 1984, PhD thesis, Univ. London Butler K., 1994, http://ccp7.dur.ac.uk./Docs/detail.ps Cunha K., Lambert D.L., 1994, ApJ, 426, 170 Daflon S., 2004, private communication Daflon, S., Cunha, K. 2004, ApJ, in press (astro-ph/0409804) Daflon S., Cunha K., Becker S., 1999, ApJ, 522, 950 Daflon S., Cunha K., Butler K., 2004a, ApJ, 604, 362 Daflon S., Cunha K., Butler K., 2004b, ApJ, 606, 514 Daflon S., Cunha K., Smith V.V., Butler K., 2003, A&A, 399, 525 Giddings J.R., 1981, PhD thesis, Univ. London Gies D.R., Lambert D.L., Astrophys. J. 1992. V.387. P.673 Gonzalez G., 1997, MNRAS, 285, 403 Gonzalez G., 2003, Rev. Mod. Phys., 75, 101 Heger A., Langer N., 2000, ApJ, 544, 1016 Holweger H., 2001, in Wimmer-Schweingruber R.F. ed, Solar and Galactic Composition, American Institute of Physics Conf. Proc., 598, 23 Kilian J., 1992, A&A, 262, 171 Kilian J., Nissen P.E., 1989, A&AS, 80, 255 Kupka F., Piskunov N.E., Ryabchikova T.A., Stempels H.C., Weiss W.W., 1999, A&AS, 138, 119 Kurucz R., 1993, ATLAS9 Stellar Atmosphere Programs and 2 [kms$^{-1}$]{}Grid, CD-ROM No.13, Cambridge, Mass., Smithsonian Astrophys. Obs. Lodders K., 2003, ApJ, 591, 1220 Lyubimkov L.S., 1996, Ap&SS, 243, 329 Lyubimkov L.S., 1998, AZh, 75, 61 (also Astron. Rep., 42, 52) Lyubimkov L.S., Lambert D.L., Rachkovskaya T.M., Rostopchin S.I., Tarasov A.E., Poklad D.B. Larionov V.M., Larionova L.V., 2000, MNRAS, 316, 19 (Paper I) Lyubimkov L.S., Rachkovskaya T.M., Rostopchin S.I., Lambert D.L., 2002, MNRAS, 333, 9 (Paper II) Lyubimkov L.S., Rostopchin S.I., Lambert D.L., 2004, MNRAS, 351, 745 (Paper III) Maeder A., Meynet G., 2000, Ann. Rev. Astron. Astrophys., 38, 143 Mihalas D., 1972, ApJ, 177, 115 Napiwotzki R., Sch[ö]{}nberner D., Wenske V., 1993, A&A, 268, 653 Nissen P.E., 1993, in Weiss W.W., Baglin A., eds, Inside the stars, ASP Conf. Ser., 40, 108 Przybilla N., Butler K., Becker S.R., Kudritzki R.P., 2001, A&A, 369, 1009 Tull R. G., MacQueen P. J., Sneden S., Lambert D. L., 1995, PASP, 107, 251 \[lastpage\] [^1]: $^\star$ E-mail: [email protected] (LSL); [email protected] (DLL)

--- abstract: 'In this pedagogical text aimed at those wanting to start thinking about or brush up on probabilistic inference, I review the rules by which probability distribution functions can (and cannot) be combined. I connect these rules to the operations performed in probabilistic data analysis. Dimensional analysis is emphasized as a valuable tool for helping to construct non-wrong probabilistic statements. The applications of probability calculus in constructing likelihoods, marginalized likelihoods, posterior probabilities, and posterior predictions are all discussed.' --- Data analysis recipes:\ Probability calculus for inference {#data-analysis-recipes-probability-calculus-for-inference .unnumbered} ================================== David W. Hogg\ [[*Center for Cosmology and Particle Physics, Department of Physics, New York University*]{}]{}\  [[*Max-Planck-Institut für Astronomie, Heidelberg*]{}]{} When constructing the plan or basis for a probabilistic inference—a data analysis making use of likelihoods and also prior probabilities and posterior probabilities and marginalization of nuisance parameters—the options are *incredibly strongly constrained* by the simple rules of probability calculus. That is, there are only certain ways that probability distribution functions (“pdfs” in what follows) *can* be combined to make new pdfs, compute expectation values, or make other kinds of non-wrong statements. For this reason, it behooves the data analyst to have good familiarity and facility with these rules. Formally, probability calculus is extremely simple. However, it is not always part of a physicist’s (or biologist’s or chemist’s or economist’s) education. That motivates this [document]{}. For space and specificity—and because it is useful for most problems I encounter—I will focus on continuous variables (think “model parameters” for the purposes of inference) rather than binary, integer, or discrete parameters, although I might say one or two things here or there. I also won’t always be specific about the domain of variables (for example whether a variable $a$ is defined on $0<a<1$ or $0<a<\infty$ or $-\infty<a<\infty$); the limits of integrals—all of which are definite—will usually be implicit. Along the way, a key idea I want to convey is—and this shows that I come from physics—that it is very helpful to think about the units or dimensions of probability quantities. I learned the material here by using it; probability calculus is so simple and constraining, you can truly learn it as you go. However if you want more information or discussion about any of the subjects below, [@sivia] provide a book-length, practical introduction and the first few chapters of [@jaynes] make a great and exceedingly idiosyncratic read. Generalities ============ A probability distribution function has units or dimensions. Don’t ignore them. For example, if you have a continuous parameter $a$, and a pdf $p(a)$ for $a$, it must obey the normalization condition $$\begin{aligned} \displaystyle 1 &=& \int p(a)\,{\mathrm{d}}a \quad ,\end{aligned}$$ where the limits of the integral should be thought of as going over the entire domain of $a$. This (along with, perhaps, $p(a)\geq 0$ everywhere) is almost the *definition* of a pdf, from my (pragmatic, informal) point of view. This normalization condition shows that $p(a)$ has units of $a^{-1}$. Nothing else would integrate properly to a dimensionless result. Even if $a$ is a multi-dimensional vector or list or tensor or field or even point in function space, the pdf must have units of $a^{-1}$. In the multi-dimensional case, the units of $a^{-1}$ are found by taking the product of all the units of all the dimensions. So, for example, if $a$ is a six-dimensional phase-space position in three-dimensional space (three cartesian position components measured in m and three cartesian momentum components measured in kgms$^{-1}$), the units of $p(a)$ would be kg$^{-3}$m$^{-6}$s$^3$. Most problems we will encounter will have multiple parameters; even if we *condition* $p(a)$ on some particular value of another parameter $b$, that is, ask for the pdf for $a$ *given* that $b$ has a particular, known value to make $p(a {\,|\,}b)$ (read “the pdf for $a$ given $b$”), it must obey the same normalization $$\begin{aligned} \displaystyle 1 &=& \int p(a {\,|\,}b)\,{\mathrm{d}}a \quad ,\end{aligned}$$ but you can *absolutely never do* the integral $$\begin{aligned} \displaystyle\label{eq:wrong1} \mbox{\textbf{wrong:}} & & \int p(a {\,|\,}b)\,{\mathrm{d}}b\end{aligned}$$ because that integral would have units of $a^{-1}\,b$, which is (for our purposes) absurd. If you have a probability distribution for two things (“the pdf for $a$ and $b$”), you can always factorize it into two distributions, one for $a$, and one for $b$ given $a$ or the other way around: $$\begin{aligned} \displaystyle p(a, b) &=& p(a)\,p(b {\,|\,}a) \\ p(a, b) &=& p(a {\,|\,}b)\,p(b) \quad ,\end{aligned}$$ where the units of both sides of both equations are $a^{-1}\,b^{-1}$. These two factorizations taken together lead to what is sometimes called “Bayes’s theorem”, or $$\begin{aligned} \displaystyle p(a {\,|\,}b) &=& \frac{p(b {\,|\,}a)\,p(a)}{p(b)} \quad ,\end{aligned}$$ where the units are just $a^{-1}$ (the $b^{-1}$ units cancel out top and bottom), and the “divide by $p(b)$” aspect of that gives many philosophers and mathematicians the chills (though certainly not me). Conditional probabilities factor just the same as unconditional ones (and many will tell you that there is no such thing as an unconditional probability); they factor like this: $$\begin{aligned} \displaystyle p(a, b {\,|\,}c) &=& p(a {\,|\,}c)\,p(b {\,|\,}a, c) \\ p(a, b {\,|\,}c) &=& p(a {\,|\,}b, c)\,p(b {\,|\,}c) \\ p(a {\,|\,}b, c) &=& \frac{p(b {\,|\,}a, c)\,p(a {\,|\,}c)}{p(b {\,|\,}c)} \quad ,\end{aligned}$$ where the condition $c$ must be carried through all the terms; the whole right-hand side must be conditioned on $c$ if the left-hand side is. Again, there was Bayes’s theorem, and you can see its role in conversions of one kind of conditional probability into another. For technical reasons, I usually write Bayes’s theorem like this: $$\begin{aligned} \displaystyle\label{eq:bayes} p(a {\,|\,}b, c) &=& \frac{1}{Z}\,p(b {\,|\,}a, c)\,p(a {\,|\,}c) \\ Z &\equiv& \int p(b {\,|\,}a, c)\,p(a {\,|\,}c)\,{\mathrm{d}}a \quad .\end{aligned}$$ Here are things you *can’t* do: $$\begin{aligned} \displaystyle \mbox{\textbf{wrong:}} & & p(a {\,|\,}b, c)\,p(b {\,|\,}a, c) \\ \mbox{\textbf{wrong:}} & & p(a {\,|\,}b, c)\,p(a {\,|\,}c) \quad;\end{aligned}$$ the first over-conditions (it is not a factorization of anything possible) and the second has units of $a^{-2}$, which is absurd (for our purposes). Know these and *don’t do them*. One important and confusing point about all this: The terminology used throughout this [document]{} *enormously overloads* the symbol $p(\cdot)$. That is, we are using, in each line of this discussion, the function $p(\cdot)$ to mean something different; it’s meaning is set by the letters used in its arguments. That is a nomenclatural abomination. I apologize, and encourage my readers to do things that aren’t so ambiguous (like maybe add informative subscripts), but it is so standard in our business that I won’t change (for now). The theory of continuous pdfs is measure theory; measure theory (for me, anyway) is the theory of things in which you can do integrals. You can *integrate out* or *marginalize away* variables you want to get rid of (or, in what follows, *not* infer) by integrals that look like $$\begin{aligned} \displaystyle p(a {\,|\,}c) &=& \int p(a, b {\,|\,}c)\,{\mathrm{d}}b \\\label{eq:marginalize} p(a {\,|\,}c) &=& \int p(a {\,|\,}b, c)\,p(b {\,|\,}c)\,{\mathrm{d}}b \quad ,\end{aligned}$$ where the second is a factorized version of the first. Once again the integrals go over the entire domain of $b$ in each case, and again if the left-hand side is conditioned on $c$, then everything on the right-hand side must be also. This equation is a natural consequence of the things written above and dimensional analysis. Recall that because $b$ is some kind of arbitrary, possibly very high-dimensional mathematical object, these integrals can be extremely daunting in practice (see below). Sometimes equations like (\[eq:marginalize\]) can be written $$\begin{aligned} \displaystyle p(a {\,|\,}c) &=& \int p(a {\,|\,}b)\,p(b {\,|\,}c)\,{\mathrm{d}}b \quad ,\end{aligned}$$ where the dependence of $p(a {\,|\,}b)$ on $c$ has been dropped. This is only permitted if it *happens to be the case* that $p(a {\,|\,}b, c)$ doesn’t, in practice, depend on $c$. The dependence on $c$ is really there (in some sense), it just might be trivial or null. In rare cases, you can get factorizations that look like this: $$\begin{aligned} \displaystyle p(a, b {\,|\,}c) &=& p(a {\,|\,}c)\,p(b {\,|\,}c) \quad ;\end{aligned}$$ this factorization doesn’t have the pdf for $a$ depend on $b$ or vice versa. When this happens—and it is rare—it says that $a$ and $b$ are “independent” (at least conditional on $c$). In many cases of data analysis, in which we want probabilistic models of data sets, we often have some number $N$ of data $a_n$ (indexed by $n$) each of which is independent in this sense. If for each datum $a_n$ you can write down a probability distribution $p(a_n {\,|\,}c)$, then the probability of the full data set—when the data are independent—is simply the product of the individual data-point probabilities: $$\begin{aligned} \displaystyle p({\{{a_n}\}_{{n=1}}^{{N}}} {\,|\,}c) &=& \prod_{n=1}^N p(a_n {\,|\,}c) \quad .\end{aligned}$$ This is the definition of “independent data”. If all the functions $p(a_n {\,|\,}c)$ are identical—that is, if the functions don’t depend on $n$—we say the data are “iid” or “independent and identically distributed”. However, this will not be true in general in data analysis. Real data are heteroscedastic at the very least. I am writing here mainly about continuous variables, but one thing that comes up frequently in data analysis is the idea of a “mixture model” in which data are produced by two (or more) qualitatively different processes (some data are good, and some are bad, for example) that have different relative probabilities. When a variable ($b$, say) is discrete, the marginalization integral corresponding to [equation]{} (\[eq:marginalize\]) becomes a sum $$\begin{aligned} \displaystyle p(a {\,|\,}c) &=& \sum_b p(a {\,|\,}b, c)\,p(b {\,|\,}c) \quad ,\end{aligned}$$ and the normalization of $p(b {\,|\,}c)$ becomes $$\begin{aligned} \displaystyle 1 &=& \sum_b p(b {\,|\,}c) \quad ;\end{aligned}$$ in both sums, the sum is implicitly over the (countable number of) possible states of $b$. If you have a conditional pdf for $a$, for example $p(a {\,|\,}c)$, and you want to know the expectation value $E(a {\,|\,}c)$ of $a$ under this pdf (which would be, for example, something like the mean value of $a$ you would get if you drew many draws from the conditional pdf), you just integrate $$\begin{aligned} \displaystyle E(a {\,|\,}c) &=& \int a\,p(a {\,|\,}c)\,{\mathrm{d}}a \quad .\end{aligned}$$ This generalizes to any function $f(a)$ of $a$: $$\begin{aligned} \displaystyle E(f {\,|\,}c) &=& \int f(a)\,p(a {\,|\,}c)\,{\mathrm{d}}a \quad .\end{aligned}$$ You can see the marginalization integral (\[eq:marginalize\]) that converts $p(a {\,|\,}b, c)$ into $p(a {\,|\,}c)$ as providing the *expectation value* of $p(a {\,|\,}b, c)$ under the conditional pdf $p(b {\,|\,}c)$. That’s deep and relevant for what follows. You have conditional pdfs $p(a {\,|\,}d)$, $p(b {\,|\,}a, d)$, and $p(c {\,|\,}a, b, d)$. Write expressions for $p(a, b {\,|\,}d)$, $p(b {\,|\,}d)$, and $p(a {\,|\,}c, d)$. You have conditional pdfs $p(a {\,|\,}b, c)$ and $p(a {\,|\,}c)$ expressed or computable for any values of $a$, $b$, and $c$. You are not permitted to multiply these together, of course. But can you use them to construct the conditional pdf $p(b {\,|\,}a, c)$ or $p(b {\,|\,}c)$? Did you have to make any assumptions? You have conditional pdfs $p(a {\,|\,}c)$ and $p(b {\,|\,}c)$ expressed or computable for any values of $a$, $b$, and $c$. Can you use them to construct the conditional pdf $p(a {\,|\,}b, c)$? You have a function $g(b)$ that is a function only of $b$. You have conditional pdfs $p(a {\,|\,}c)$ and $p(b {\,|\,}a, c)$. What is the expectation value $E(g {\,|\,}c)$ for $g$ conditional on $c$ but *not* conditional on $a$? Take the integral on the right-hand side of [equation]{} (\[eq:marginalize\]) and replace the “${\mathrm{d}}b$” with a “${\mathrm{d}}a$”. Is it permissible to do this integral? Why or why not? If it *is* permissible, what do you get? Likelihoods =========== Imagine you have $N$ data points or measurements $D_n$ of some kind, possibly times or temperatures or brightnesses. I will say that you have a “generative model” of data point $n$ if you can write down or calculate a pdf $p(D_n {\,|\,}\theta, I)$ for the measurement $D_n$, conditional on a vector or list $\theta$ of parameters and a (possibly large) number of other things $I$ (“prior information”) on which the $D_n$ pdf depends, such as assumptions, or approximations, or knowledge about the noise process, or so on. If all the data points are independently drawn (that would be one of the assumptions in $I$), then the pdf for the full data set ${\{{D_n}\}_{{n=1}}^{{N}}}$ is just the product $$\begin{aligned} \displaystyle\label{eq:likelihood} p({\{{D_n}\}_{{n=1}}^{{N}}} {\,|\,}\theta, I) &=& \prod_{n=1}^N p(D_n {\,|\,}\theta, I) \quad .\end{aligned}$$ (This requires the data to be independent, but not necessarily iid.) When the pdf of the data is thought of as being a *function* of the parameters at *fixed* data, the pdf of the data given the parameters is called the “likelihood” (for historical reasons I don’t care about). In general, in contexts in which the data are thought of as being fixed and the parameters are thought of as variable, any kind of conditional pdf for the data—conditional on parameters—is called a likelihood “for the parameters” even though it is a pdf “for the data”. Now imagine that the parameters divide into two groups. One group $\theta$ are parameters of great interest, and another group $\alpha$ are of no interest. The $\alpha$ parameters are nuisance parameters. In this situation, the likelihood can be written $$\begin{aligned} \displaystyle p({\{{D_n}\}_{{n=1}}^{{N}}} {\,|\,}\theta, \alpha, I) &=& \prod_{n=1}^N p(D_n {\,|\,}\theta, \alpha, I) \quad .\end{aligned}$$ If you want to make likelihood statements about the important parameters $\theta$ without committing to anything regarding the nuisance parameters $\alpha$, you can marginalize rather than infer them. You might be tempted to do $$\begin{aligned} \displaystyle\label{eq:wrongmlikelihood} \mbox{\textbf{wrong:}} & & \int p({\{{D_n}\}_{{n=1}}^{{N}}} {\,|\,}\theta, \alpha, I)\,{\mathrm{d}}\alpha \quad ,\end{aligned}$$ but that’s not allowed for dimensional arguments given in the previous [Section]{}. In order to integrate over the nuisances $\alpha$, something with units of $\alpha^{-1}$ needs to be multiplied in—a pdf for the $\alpha$ of course: $$\begin{aligned} \displaystyle\label{eq:mlikelihood} p({\{{D_n}\}_{{n=1}}^{{N}}} {\,|\,}\theta, I) &=& \int p({\{{D_n}\}_{{n=1}}^{{N}}} {\,|\,}\theta, \alpha, I)\,p(\alpha {\,|\,}\theta, I)\,{\mathrm{d}}\alpha \quad ,\end{aligned}$$ where $p(\alpha {\,|\,}\theta, I)$ is called the “prior pdf” for the $\alpha$ and it *can* depend (but doesn’t *need to* depend) on the other parameters $\theta$ and the more general prior information $I$. This marginalization is incredibly useful but note that it comes at a substantial cost: It required specifying a prior pdf over parameters that, by assertion, you don’t care about! [Equation]{} (\[eq:mlikelihood\]) could be called a “partially marginalized likelihood” because it is a likelihood (a pdf for the data) but it is conditional on fewer parameters than the original, rawest, likelihood. Sometimes I have heard concern that when you perform the marginalization (\[eq:mlikelihood\]), you are allowing the nuisance parameters to “have any values they like, whatsoever” as if you are somehow not constraining them. It is true that you are not *inferring* the nuisance parameters, but you certainly are using the data to limit their range, in the sense that the integral in (\[eq:mlikelihood\]) only gets significant weight where the likelihood is large. That is, if the data strongly constrain the $\alpha$ to a narrow range of good values, then only those good values are making any (significant) contribution to the marginalization integral. That’s important! Because the data were independent by assumption (the full-data likelihood is a product of individual-datum likelihoods), you might be tempted to do things like $$\begin{aligned} \displaystyle\label{eq:wrong2} \mbox{\textbf{wrong:}} & & \prod_{n=1}^N \left[\int p(D_n {\,|\,}\theta, \alpha, I)\,p(\alpha {\,|\,}\theta, I)\,{\mathrm{d}}\alpha\right] \quad .\end{aligned}$$ This is wrong because if you do the integral *inside* the product, you end up doing the integral $N$ times over. Or another way to put it, although you don’t want to infer the $\alpha$ parameters, you want the support in the marginalization integral to be consistently set by *all* the data taken together, not inconsistently set by each individual datum separately. One thing that is often done with likelihoods (and one thing that many audience members think, instinctively, when you mention the word “likelihood”) is “maximum likelihood”. If all you want to write down is your likelihood, and you want the “best” parameters $\theta$ given your data, you can find the parameters that maximize the full-data likelihood. The only really probabilistically responsible use of the maximum-likelihood parameter value is—when coupled with a likelihood width estimate—to make an approximate description of a likelihood function. Of course a maximum-likelihood value could in principle be useful on its own when the data are *so decisive* that there is (for the investigator’s purposes) no significant remaining uncertainty in the parameters. I have never seen that happen. The key idea is that it is the *likelihood function* that is useful for inference, not some parameter value suggested by that function. To make all of the above concrete, we can consider a simple example, where measurements $D_n$ are made at a set of “horizontal” postions $x_n$. Each measurement is of a “vertical” position $y_n$ but the measurement is noisy, so the generative model looks like: $$\begin{aligned} \displaystyle\label{eq:example0} y_n &=& a\,x_n + b \\ D_n &=& y_n + e_n \\ p(e_n) &=& N(e_n {\,|\,}0, \sigma_n^2) \\ p(D_n {\,|\,}\theta, I) &=& N(D_n {\,|\,}a\,x_n + b, \sigma_n^2) \\\label{eq:examplelike} p({\{{D_n}\}_{{n=1}}^{{N}}} {\,|\,}\theta, I) &=& \prod_{n=1}^N p(D_n {\,|\,}\theta, I) \\ \theta &\equiv& [a, b] \\\label{eq:example1} I &\equiv& [{\{{x_n, \sigma_n^2}\}_{{n=1}}^{{N}}}, \mbox{and so on\ldots} ] \quad,\end{aligned}$$ where the $y_n$ are the “true values” for the heights, which lie on a straight line of slope $a$ and intercept $b$, the $e_n$ are noise contributions, which are drawn independently from Gaussians $N(\cdot{\,|\,}\cdot)$ with zero means and variances $\sigma_n^2$, the likelihood is just a Gaussian for each data point, there are two parameters, and the $x_n$ and $\sigma_n^2$ values are considered prior information. Show that the likelihood for the model given in [[equation]{}s]{} (\[eq:example0\]) through (\[eq:example1\]) can be written in the form $Q\,\exp(-\chi^2/2)$, where $\chi^2$ is the standard statistic for weighted least-squares problems. On what does $Q$ depend, and what are its dimensions? The likelihood in [equation]{} (\[eq:examplelike\]) is a product of Gaussians in $D_n$. At fixed data and $b$, what shape will it have in the $a$ direction? That is, what functional form will it have when thought of as being a function of $a$? You will have to use the properties of Gaussians (and products of Gaussians). Posterior probabilities ======================= A large fraction of the inference that is done in the quantitative sciences can be left in the form of likelihoods and marginalized likelihoods, and probably should be. However, there are many scientific questions the answers to which require going beyond the likelihood—which is a pdf for the data, conditional on the parameters—to a pdf for the parameters, conditional on the data. To illustrate, imagine that you want to make a probabilistic prediction, given your data analysis. For one example, you might want to predict what $\theta$ values you will find in a subsequent experiment. Or, for another, say some quantity $t$ of great interest (like, say, the age of the Universe) is a function $t(\theta)$ of the parameters and you want to predict the outcome you expect in some future independent measurement of that same parameter (some other experiment that measures—differently—the age of the Universe). The distributions or expectation values for these predictions, conditioned on your data, will require a pdf for the parameters or functions of the parameters; that is, if you want the expectation $E(t {\,|\,}D)$, where for notational convenience I have defined $$\begin{aligned} \displaystyle D &\equiv& {\{{D_n}\}_{{n=1}}^{{N}}} \quad ,\end{aligned}$$ you need to do the integral $$\begin{aligned} \displaystyle\label{eq:posteriorexpectation} E(t {\,|\,}D) &=& \int t(\theta)\,p(\theta {\,|\,}D, I)\,{\mathrm{d}}\theta \quad ,\end{aligned}$$ this in turn requires the pdf $p(\theta {\,|\,}D, I)$ for the parameters $\theta$ given the data. This is called the “posterior pdf” because it is the pdf you get *after* digesting the data. The posterior pdf is obtained by Bayes rule (\[eq:bayes\]) $$\begin{aligned} \displaystyle\label{eq:posterior} p(\theta {\,|\,}D, I) &=& \frac{1}{Z}\,p(D {\,|\,}\theta, I)\,p(\theta {\,|\,}I) \\ Z &\equiv& \int p(D {\,|\,}\theta, I)\,p(\theta {\,|\,}I)\,{\mathrm{d}}\theta \quad ,\end{aligned}$$ where we had to introduce the prior pdf $p(\theta {\,|\,}I)$ for the parameters. The prior can be thought of as the pdf for the parameters *before* you took the data $D$. The prior pdf brings in new assumptions but also new capabilities, because posterior expectation values as in (\[eq:posteriorexpectation\]) and other kinds of probabilistic predictions become possible with its use. Some notes about all this: *(a)* Computation of expectation values is not the only—or even the primary—use of posterior pdfs; I was just using the expectation value as an example of *why* you might want the posterior. You can use posterior pdfs to make predictions that are themselves pdfs; this is in general the only way to propagate the full posterior uncertainty remaining after your experiment. *(b)* The normalization $Z$ in [equation]{} (\[eq:posterior\]) is a marginalization of a likelihood; in fact it is a fully marginalized likelihood, and could be written $p(D {\,|\,}I)$. It has many uses in model evaluation and model averaging, to be discussed in subsequent [[document]{}s]{}; it is sometimes called “the evidence” though that is a very unspecific term I don’t like so much. *(c)* You can think of the posterior expression (\[eq:posterior\]) as being a “belief updating” operation, in which you start with the prior pdf, multiply in the likelihood (which probably makes it narrower, at least if your data are useful) and re-normalize to make a posterior pdf. There is a “subjective” attitude to take towards all this that makes the prior and the posterior pdfs specific to the individual inferrer, while the likelihood is (at least slightly) more objective. *(d)* Many committed Bayesians take the view that you *always* want a posterior pdf—that is, you are never satisfied with a likelihood—and that you therefore *must* always have a prior, even if you don’t think you do. That view is false, but it contains an element of truth: If you eschew prior pdfs, then you are relegated to only ever asking and answering questions about the probability of the data. You can answer questions like “what regions of parameter space are consistent with the data?” but within the set of consistent models, you can’t answer questions like “is this parameter neighborhood more plausible than that one?” You also can’t marginalize out nuisance parameters. Although you might use the posterior pdf to report some kind of mean prediction as discussed above, it almost never makes sense to just *optimize* the posterior pdf. The posterior-optimal parameters are called the “maximum [*a posteriori*]{}” (MAP) parameters. Like the maximum likelihood parameters, these only make sense to compute if they are being used to provide an approximation to the posterior pdf (in the form, say, of a mode and a width). The key idea is that the results of responsible data analysis is not *an answer* but a *distribution over answers*. Data are inherently noisy and incomplete; they never answer your question precisely. So no single number—no maximum-likelihood or MAP value—will adequately represent the result of a data analysis. Results are always pdfs (or full likelihood functions); we must embrace that. Continuing our example of the model given in [[equation]{}s]{} (\[eq:example0\]) through (\[eq:example1\]), if we want to learn the posterior pdf over the parameters $\theta \equiv [a, b]$, we need to choose a prior pdf $p(\theta {\,|\,}I)$. In principle this should represent our actual prior or exterior knowledge. In practice, investigators often want to “assume nothing” and put a very or infinitely broad prior on the parameters; of course putting a broad prior is *not* equivalent to assuming nothing, it is just as severe an assumption as any other prior. For example, even if you go with a very broad prior on the parameter $a$, that is a *different* assumption than the same form of very broad prior on $a^2$ or on $\arctan(a)$. The prior doesn’t just set the ranges of parameters, it places a *measure on parameter space.* That’s why it is so important. If you choose an infinitely broad prior pdf, it can become *improper*, in the sense that it can become impossible to satisfy the normalization condition $$\begin{aligned} \displaystyle 1 &=& \int p(\theta {\,|\,}I)\,{\mathrm{d}}\theta \quad .\end{aligned}$$ The crazy thing is that—although it is not advised—even with improper priors you can still often do inference, because an infinitely broad Gaussian (for example) is a well defined limit of a wide but finite Gaussian, and the posterior pdf can be well behaved in the limit. That is, posterior pdfs can be proper even when prior pdfs are not. Sometimes your prior knowledge can be very odd. For example, you might be willing to take any slope $a$ over a wide range, but require that the line $y=a\,x+b$ pass through a specific point $(x,y)=(X_0,Y_0)$. Then your prior might look like $$\begin{aligned} \displaystyle p(\theta {\,|\,}I) &=& p(a {\,|\,}I)\,p(b {\,|\,}a, I) \\\label{eq:dirac} p(b {\,|\,}a, I) &=& \delta(b + a\,X_0 - Y_0) \quad ,\end{aligned}$$ where $p(a {\,|\,}I)$ is some broad function but $\delta(\cdot)$ is the Dirac delta function, and, implicitly, $X_0$ and $Y_0$ are part of the prior information $I$. These examples all go to show that you have an enormous range of options when you start to write prior pdfs. In general, you should include all the things you know to be true when you write your priors. With great power comes great responsibility. In other [[document]{}s]{} in this series (for example @straightline), more advanced topics will be discussed. One example is the (surprisingly common) situation in which you have far more parameters than data. This sounds impossible, but the rules of probability calculus don’t prohibit it, and once you marginalize out most of them you can be left with extremely strong constraints on the parameters you care about. Another example is that in many cases you care about the *prior* on your parameters, not the parameters themselves. Imagine, for example, that you don’t know what prior pdf to put on your parameters $\theta$ but you want to take it from a family that itself has parameters $\beta$. Then you can marginalize out the $\theta$—yes, marginalize out all the parameters we once thought we cared about—using the prior $p(\theta {\,|\,}\beta, I)$ and be left with a likelihood for the parameters $\beta$ of the prior, like so: $$\begin{aligned} \displaystyle p(D {\,|\,}\beta, I) &=& \int p(D {\,|\,}\theta, I)\,p(\theta {\,|\,}\beta, I)\,{\mathrm{d}}\theta \quad .\end{aligned}$$ The parameters $\beta$ of the prior pdf are usually called “hyperparameters”; this kind of inference is called “hierarchical”. \[prob:gaussianprior\] Show that if you take the model in [[equation]{}s]{} (\[eq:example0\]) through (\[eq:example1\]) and put a Gaussian prior pdf on $a$ and an independent Gaussian prior pdf on $b$ that your posterior pdf for $a$ and $b$ will be a two-dimensional Gaussian. Feel free to use informal or even hand-waving arguments; there are no mathematicians present. \[prob:improper\] Take the limit of your answer to [Exercise]{} \[prob:gaussianprior\] as the width of the $a$ and $b$ prior pdfs go to infinity and show that you still get a well-defined Gaussian posterior pdf. Feel free to use informal or even hand-waving arguments. Show that the posterior pdf you get in [Exercise]{} \[prob:improper\] is just a rescaling of the likelihood function by a scalar. Two questions: Why *must* there be a re-scaling, from a dimensional point of view? What is the scaling, specifically? You might have to do some linear algebra, for which I won’t apologize; it’s awesome. [Equation]{} (\[eq:dirac\]) implies that the delta function $\delta(q)$ has what dimensions? Hogg, D. W., 2009, “Is cosmology just a plausibility argument?”, arXiv:0910.3374 Hogg, D. W., Bovy, J., & Lang, D., 2010a, “Data analysis recipes: Fitting a model to data”, arXiv:1008.4686 Hogg, D. W., Myers, A. D., & Bovy, J., 2010b, “Inferring the eccentricity distribution”, arXiv:1008.4146 Jaynes, E. T., 2003, *Probability theory: The logic of science* (Cambridge University Press) Sivia, D. S. & Skilling, J., 2006, *Data analysis: A Bayesian tutorial*, 2ed (Oxford University Press)

--- abstract: 'We construct an analogue of the normaliser decomposition for $p$–local finite groups $(S,\mathcal{F},\mathcal{L})$ with respect to collections of $\mathcal{F}$–centric subgroups and collections of elementary abelian subgroups of $S$. This enables us to describe the classifying space of a $p$–local finite group, before $p$–completion, as the homotopy colimit of a diagram of classifying spaces of finite groups whose shape is a poset and all maps are induced by group monomorphisms.' address: | Department of Mathematical Sciences\ King’s College\ University of Aberdeen\ Aberdeen\ AB24 3UE\ Scotland\ United Kingdom author: - Assaf Libman bibliography: - 'link.bib' title: 'The normaliser decomposition for $p$–local finite groups' --- We construct an analogue of the normaliser decomposition for p-local finite groups (S,F,L) with respect to collections of F-centric subgroups and collections of elementary abelian subgroups of S. This enables us to describe the classifying space of a p-local finite group, before p-completion, as the homotopy colimit of a diagram of classifying spaces of finite groups whose shape is a poset and all maps are induced by group monomorphisms. We construct an analogue of the normaliser decomposition for p&ndash;local finite groups (S,F,L) with respect to collections of F&ndash;centric subgroups and collections of elementary abelian subgroups of S. This enables us to describe the classifying space of a p&ndash;local finite group, before p&ndash;completion, as the homotopy colimit of a diagram of classifying spaces of finite groups whose shape is a poset and all maps are induced by group monomorphisms. The main results ================ For finite groups Dwyer [@Dwyer-decomp] defined three types of homology decompositions of classifying spaces of finite groups known as the “subgroup”, “centraliser” and “normaliser” decompositions. These decompositions are functors $F\co D \to \spaces$, where $D$ is a small category which is constructed using collections $\H$ of carefully chosen subgroups of $G$. The essential property of these functors is, that given a finite group $G$, the spaces $F(d)$ have the homotopy type of classifying spaces of subgroups of $G$. Moreover the category $D$ is constructed using information about the conjugation in $G$ of the subgroups in $\H$. We say that $D$ depends on the fusion of the collection $\H$ of $G$. The purpose of this note is to construct an analogue of the normaliser decomposition for $p$–local finite groups in certain important cases. Throughout this note we will freely use the terminology and notation that by now has become standard in the theory for $p$–local finite groups. The reader who is not familiar with the jargon is advised to read prior to this section, and is also referred to [@BLO2] where $p$–local finite groups were initially defined. It should be noted that the analogues of the “subgroup” and the “centraliser” decompositions for $p$–local finite groups was already known to Broto, Levi and Oliver [@BLO2 Section 2]. The normaliser decomposition which is introduced in this note enabled the author together with Antonio Viruel to analyze the nerve $|\L|$ of $p$–local finite groups $(S,\F,\L)$ with small Sylow subgroups $S$. We prove that these are classifying spaces of, generally infinite, discrete groups [@LV]. The author also used normaliser decompositions to give an analysis of the spectra associated with the nerve, $|\L|$, of the linking systems due to Ruiz and Viruel in [@RV-extraspecial] and other “exotic” examples, see [@Li]. These results will appear separately as they involve techniques that have little to do with the actual construction of the normaliser decomposition. We now describe the main results of this paper. Throughout we work simplicially, thus a space means a simplicial set. The category of simplicial sets is denoted by $\spaces$. The nerve of a small category $\mathbf{D}$ is denoted $\Nr(\mathbf{D})$ or $|\mathbf{D}|$. We obtain a functor $|-|\co\cat\to\spaces$ where $\cat$ is the category of small categories. A more detailed discussion can be found in Let $(S,\F,\L)$ be a $p$–local finite group. A collection is a set ${\mathcal{C}}$ of subgroups of $S$ which is closed under conjugacy in $\F$. That is if $P\leq S$ belongs to ${\mathcal{C}}$ then so do all the $\F$–conjugates of $P$. A collection ${\mathcal{C}}$ is called $\F$–centric if it consists of $\F$–centric subgroups of $S$. \[def k-simplices\] A $k$–simplex in a collection ${\mathcal{C}}$ is a sequence $\PPP$ of proper inclusions $P_0<P_1<\cdots <P_k$ of elements of ${\mathcal{C}}$. Two $k$–simplices $\PPP$ and $\PPP'$ are called conjugate if there exists an isomorphism $f\in \Iso_\F(P_k,P_k')$ such that $f(P_i)=P_i'$ for all $i=0,\ldots,k$. The conjugacy class of $\PPP$ is denoted $[\PPP]$. \[def bsdc\] The category $\bsd{\mathcal{C}}$ is a poset whose objects are the conjugacy classes $[\PPP]$ of all the $k$–simplices in ${\mathcal{C}}$ where $k=0,1,2,\ldots$. A morphism $[\PPP] \to [\PPP']$ in $\bsd{\mathcal{C}}$ exists if $\PPP'$ is conjugate to a subsimplex of $\PPP$. Recall from that in every $p$–local finite group it is possible to choose morphisms $\iota_P^Q$ in the linking system $\L$ which are lifts of inclusions $P\leq Q$ of $\F$–centric subgroups. The choice can be made in such a way that $\iota_Q^R\circ\iota_P^Q=\iota_P^R$ for inclusions $P\leq Q\leq R$. Let ${\mathcal{C}}$ be an $\F$–centric collection in $(S,\F,\L)$ and let $\PPP$ be a $k$–simplex in ${\mathcal{C}}$. Define $\Aut_\L(\PPP)$ as the subgroup of $\prod_{i=0}^k\Aut_\L(P_i)$ whose elements are the $(k{+}1)$–tuples $(\vp_i)_{i=0}^k$ which render the following ladder commutative in $\L$ $$\begin{CD} P_0 @>{\iota_{P_0}^{P_1}}>> P_1 @>{\iota_{P_1}^{P_2}}>> \cdots @>{\iota_{P_{k-1}}^{P_k}}>> P_k \\ @V{\vp_0}VV @V{\vp_1}VV @. @VV{\vp_k}V \\ P_0 @>>{\iota_{P_0}^{P_1}}> P_1 @>>{\iota_{P_1}^{P_2}}> \cdots @>>{\iota_{P_{k-1}}^{P_k}}> P_k \end{CD}$$ \[prop resL\] The assignment $(\vp_i)_{i=0}^k \mapsto \vp_0$ gives rise to a canonical isomorphism of $\Aut_\L(\PPP)$ with a subgroup of $\Aut_\L(P_0)$. More generally, if $\PPP'$ is a subsimplex of $\PPP$ in ${\mathcal{C}}$ then restriction induces a monomorphism of groups $\Aut_\L(\PPP) \to \Aut_\L(\PPP')$. The second assertion follows immediately from . The first follows from the second by letting $\PPP'$ be the $1$–simplex $P_0$. $\B\Aut_\L(\PPP)$ denotes the subcategory of $\L$ whose only object is $P_0$ and whose morphism set is $\Aut_\L(\PPP)$. Given an $\F$–centric collection ${\mathcal{C}}$ in a $p$–local finite group $(S,\F,\L)$, let $\L^{\mathcal{C}}$ denote the full subcategory of $\L$ generated by the objects set ${\mathcal{C}}$. Frequently, the inclusion $\L^{\mathcal{C}} \subseteq \L$ induces a weak homotopy equivalence on nerves. For example, this happens when ${\mathcal{C}}$ contains all the $\F$–centric $\F$–radical subgroups of $S$. This fact is proved by Broto, Castellana, Grodal, Levi and Oliver [@BCGLO1 Theorem 3.5]. The following theorem applies to all $\F$–centric collections. The decomposition approximates $\L$ if the inclusion $\L^{\mathcal{C}} \subseteq \L$ induces an equivalence as explained above. \[thmA\] Fix an $\F$–centric collection ${\mathcal{C}}$ in a $p$–local finite group $(S,\F,\L)$. Then there exists a functor $\de_{\mathcal{C}}\co\bsd{\mathcal{C}}\to\spaces$ such that 1. \[thmA:target\] There is a natural weak homotopy equivalence $${\hocolim_{\bsd{\mathcal{C}}}}\, \de_{\mathcal{C}} {\xrightarrow{~~\simeq~~}} |\L^{\mathcal{C}}|.$$ 2. There is a natural weak homotopy equivalence $B\Aut_\L(\PPP) {\xrightarrow{\simeq}} \de_{\mathcal{C}}([\PPP])$ for every $k$–simplex $\PPP$. \[thmA:terms\] 3. The natural maps $\de_{\mathcal{C}}([\PPP]) \to |\L^{\mathcal{C}}|$ are induced by the inclusion of categories $\B\Aut_\L(\PPP) \subseteq \L^{\mathcal{C}}$. \[thmA:augmentation\] 4. If $\PPP'$ is a subsimplex of $\PPP$ then the equivalence renders the following square commutative $$\xymatrix{ B\Aut_\L(\PPP) \ar[r]^\simeq \ar[d]_{B\res^\PPP_{\PPP'}} & \de_{\mathcal{C}}([\PPP]) \ar[d] \\ B\Aut_\L(\PPP') \ar[r]^\simeq & \de_{\mathcal{C}}([\PPP']) }$$ Moreover if $\PPP$ and $\PPP'$ are conjugate $k$–simplices and $\psi\in\Iso_\L(P_0,P_0')$ maps $\Aut_\L(\PPP')$ onto $\Aut_\L(\PPP)$ by conjugation then the following square commutes $$\xymatrix{ B\Aut_\L(\PPP') \ar[r]^{\simeq} \ar[d]_{Bc_\psi} & \de_{\mathcal{C}}([\PPP']) \ar@{=}[d] \\ B\Aut_\L(\PPP) \ar[r]_{\simeq} & \de_{\mathcal{C}}([\PPP]) }$$ \[thmA:maps\] When $(S,\F,\L)$ is associated with a finite group $G$ one may consider the $G$–collection $\H$ consisting of all the subgroups of $G$ which are conjugate to elements of the $\F$–collection ${\mathcal{C}}$. Dwyer [@Dwyer-decomp Section 3] constructs a poset $\bsd\H$ and a functor $\de^{\text{Dwyer}}_\H\co\bsd\H \to \spaces$ which he calls the normaliser decomposition. We will show in that $\bsd\H=\bsd{\mathcal{C}}$ and that $\de_{\mathcal{C}}$ and $\de^{\text{Dwyer}}_\H$ can be connected by a natural zigzag of mod–$p$ equivalences. That is, a zigzag of natural transformations which at every object of $\bsd{\mathcal{C}}$ give rise to an $H_*(-;\ZZ/p)$–isomorphism. We now describe the second type of normaliser decomposition that we shall construct in this note. It is based on collections $\E$ of elementary abelian subgroups of $S$. \[def autF\] For a $k$–simplex $\EEE$ in $\E$ define $\Aut_\F(\EEE)$ as the subgroup of $\Aut_\F(E_k)$ consisting of the automorphisms $f$ such that $f(E_i)=E_i$ for all $i=0,\ldots,k$. Consider an $\F$–centric collection ${\mathcal{C}}$ in $(S,\F,\L)$. \[def barcl\] Fix an elementary abelian subgroup $E$ of $S$. The objects of the category $\bar{C}_{\L}({\mathcal{C}};E)$ are pairs $(P,f)$ where $P\in {\mathcal{C}}$ and $f\co E \to Z(P)$ is a morphism in $\F$. Morphisms $(P,f) \to (Q,g)$ in $\bar{C}_{\L}({\mathcal{C}};E)$ are morphisms $\psi \in \L(P,Q)$ such that $g=\pi(\psi)\circ f$ where $\pi\co\L \to \F$ is the projection functor. Observe that $\Aut_\F(E)$ acts on $\bar{C}_{\L^{\mathcal{C}}}(E)$ by pre-composition. That is, every $h\in\Aut_\F(E)$ indices the assignment $(P,f)\mapsto (P,f\circ h)$. \[def NoL\] For a $k$–simplex $\EEE$ in $\E$ let $\No_\L({\mathcal{C}};\EEE)$ denote the subcategory of $\L$ whose objects are $P\in{\mathcal{C}}$ for which $E_k \leq Z(P)$. A morphism $\vp\in\L(P,Q)$ belongs to $\No_\L({\mathcal{C}};\EEE)$ if $\pi(\vp)|_{E_k}$ is an element of $\Aut_\F(\EEE)$. Recall that the homotopy orbit space of a $G$–space $X$, ie the Borel construction $EG\times_G X$, is denoted by $X_{hG}$. \[No equiv orbit\] Let $\EEE$ be a $k$–simplex in $\E$. There is a map $$\epsilon_\EEE \co | \No_\L({\mathcal{C}};\EEE)| \to |\bar{C}_\L({\mathcal{C}};E_k)|_{h\Aut_\F(\EEE)}$$ which is a homotopy equivalence if $E_k$ is fully $\F$–centralised. The map is natural with respect to inclusion of simplices. This is immediate from . A comment on the categories $\bar{C}_{\L}({\mathcal{C}};E)$ is in place. If ${\mathcal{C}}$ is the collection of all the $\F$–centric subgroups of $S$ and $E$ is fully $\F$–centralised, then it is shown by Broto, Levi and Oliver [@BLO2 Theorem 2.6] that $|\bar{C}_\L({\mathcal{C}};E)|$ has the homotopy type of the nerve of the centraliser linking system $|C_\L(E)|$. The categories $\No_\L({\mathcal{C}};\EEE)$ are more mysterious. Even when $\EEE$ is a $1$–simplex $E$, the category $\No_\L({\mathcal{C}};E)$ is in general only a subcategory of the normaliser linking system $N_\L(E)$ because the largest subgroup which appears as an object of $\No_\L({\mathcal{C}};E)$ is $C_S(E)$ which in general is smaller than $N_S(E)$. When $C_S(E)=N_S(E)$ these categories are equal. The next decomposition result, , depends on a collection of elementary abelian groups $\E$ and a collection ${\mathcal{C}}$ of $\F$–centric subgroups of $S$. It approximates $\L$ if ${\mathcal{C}}$ contains, for example, all the $\F$–centric $\F$–radical subgroups of $S$. The collection $\E$ must be large enough as explicitly stated in the theorem. For example the collection of all the non-trivial elementary abelian subgroups will always be a valid choice. \[def omega\_p\] Given a group $H$ and a prime $p$ let $\Omega_p(H)$ denote the subgroup of $H$ generated by all the elements of order $p$ in $H$. \[thmB\] Consider a $p$–local finite group $(S,\F,\L)$, an $\F$–centric collection ${\mathcal{C}}$ and a collection $\E$ of elementary abelian subgroup of $S$ which contains the subgroups $\Omega_pZ(P)$ for all $P\in {\mathcal{C}}$. Then there exists a functor $\de_\E\co\bsd\E \to \spaces$ with the following properties. 1. There is a natural weak homotopy equivalence $ {\hocolim_{\bsd\E}} \, \de_\E {\xrightarrow{\ \ \simeq \ \ }} |\L^{\mathcal{C}}|. $ \[thmB:target\] 2. For a $k$–simplex $\EEE$ in $\E$ there is a weak homotopy equivalence \[thmB:terms\] $$|\bar{C}_{\L}({\mathcal{C}};E_k)|_{h\Aut_\F(\EEE)} {\xrightarrow{ \ \ \simeq \ \ }} \de_\E([\EEE]).$$ 3. Fix a $k$–simplex $\EEE$ where $E_k$ is fully $\F$–centralised. The equivalences and give a natural map $\de_\E([\EEE]) \to |\L^{\mathcal{C}}|$ whose precomposition with $\epsilon_\EEE$ of is induced by the realization of the inclusion of $\No_\L({\mathcal{C}};\EEE)$ in $\L^{\mathcal{C}}$. \[thmB:augment\] 4. If $\EEE'$ is a $k$–subsimplex of an $n$–simplex $\EEE$ then the following square commutes up to homotopy $$\xymatrix{ |\bar{C}_{\L}({\mathcal{C}};E_n)|_{h\Aut_\F(\EEE)} \ar[d] \ar[rr]^\simeq & & \de_\E([\EEE]) \ar[d] \\ |\bar{C}_{\L}({\mathcal{C}};E'_k)|_{h\Aut_\F(\EEE')} \ar[rr]^\simeq & & \de_\E([\EEE']) }$$ The homotopy is natural with respect to inclusion of simplices. In addition, the square commutes on the nose if $E_k'=E_n$. \[thmB:maps\] ### Acknowledgments {#acknowledgments .unnumbered} The author was supported by grant NAL/00735/G from the Nuffield Foundation. Part of this work was supported by Institute Mittag-Leffler (Djursholm, Sweden). On $p$–local finite groups {#sec plfg} ========================== The term $p$–local finite group was coined by Broto, Levi and Oliver [@BLO2]. It cropped up naturally in their attempt [@BLO1] to describe the space of self equivalences of a $p$–completed classifying space of a finite group $G$. They discovered that the relevant information needed to solve this problem lies in the fusion system of the $p$–subgroups of $G$ and certain categories which they later on called “linking systems”. Historically, fusion systems were first introduced by Lluis Puig [@Puig]. Fix a prime $p$ and let $S$ be a finite $p$–group. A *fusion system* over $S$ is a sub-category $\F$ of the category of groups whose objects are the subgroups of $S$ and whose morphisms are group monomorphisms such that - All the monomorphisms that are induced by conjugation by elements of $S$ are in $\F$. - Every morphism in $\F$ factors as an isomorphism in $\F$ followed by an inclusion of subgroups. We say that two subgroups $P,Q$ of $S$ are $\F$–*conjugate* if they are isomorphic as objects of $\F$. When $g$ is an element of $S$ and $P,Q$ are subgroups of $S$ such that $g P g^{-1}\leq Q$, we let $c_g$ denote the morphism $P \to Q$ defined by conjugation, namely $c_g(x)=gxg^{-1}$ for every $x\in P$. We let $\Hom_S(P,Q)$ denote the set of all the morphisms $P \to Q$ in $\F$ that are induced by conjugation in $S$. Also notice that the factorization axiom (2) implies that all the $\F$–endomorphisms of a subgroup $P$ are in fact automorphisms in $\F$. Thus we write $\Aut_\F(P)$ for the set of morphisms $\F(P,P)$. A subgroup $P$ of $S$ is called *fully $\F$–centralised* (resp. *fully $\F$–normalised*) if its $S$–centraliser $C_S(P)$ (resp. $S$–normaliser $N_S(P)$) has the maximal possible order in the $\F$–conjugacy class of $P$. That is, $|C_S(P)|\geq |C_S(P')|$ (resp. $|N_S(P)|\geq |N_S(P')|$) for every $P'$ which is $\F$–conjugate to $P$. \[def sat fus\] A fusion system $\F$ over a finite $p$–group $S$ is called *saturated* if - Every fully $\F$–normalised subgroup $P$ of $S$ is fully $\F$–centralised and moreover $\Aut_S(P)=N_S(P)/C_S(P)$ is a Sylow $p$–subgroup of $\Aut_\F(P)$. - Every morphism ${\ensuremath{\vp\co P\rightarrowS}}$ in $\F$ whose image $\vp(P)$ is fully $\F$–centralised can be extended to a morphism ${\ensuremath{\psi\co N_\vp\rightarrowS}}$ in $\F$ where $$N_{\vp}=\{ g\in N_S(P) : \vp c_g\vp^{-1} \in\Aut_S(P)\}.$$ A subgroup $P$ of $S$ is called $\F$–*centric* if $P$ and all of its $\F$–conjugates contain their $S$–centralisers, that is $C_S(P')=Z(P')$ for every subgroup $P'$ of $S$ which is $\F$–conjugate to $P$. A *centric linking system* associated to a saturated fusion system $\F$ over $S$ consists of 1. A small category $\L$ whose objects are the $\F$–centric subgroups of $S$, 2. a functor $\pi\co\L \to \F$ and 3. group monomorphisms $\de_P\co P \to \Aut_\L(P)$ for every $\F$–centric subgroup $P$ of $S$, Such that the following axioms hold - The functor $\pi$ acts as the inclusion on object sets, that is $\pi(P)=P$ for every $\F$–centric subgroup $P$ of $S$. For any two objects $P,Q$ of $\L$, the group $Z(P)$ acts freely on the morphism set $\L(P,Q)$ via the restriction of $\de_P\co P \to \Aut_\L(P)$ to $Z(P)$. The induced map on morphisms sets $$\pi\co\L(P,Q) \to \F(P,Q)$$ identifies $\F(P,Q)$ with the quotient of $\L(P,Q)$ by the free action of $Z(P)$. - For every $\F$–centric subgroup $P$ of $S$ the map $\pi\co\Aut_\L(P) \to \Aut_\F(P)$ sends $\de_P(g)$, where $g\in P$, to $c_g$. - For every $f\in\L(P,Q)$ and every $g\in P$ there is a commutative square in $\L$ $$\begin{CD} P @>{f}>> Q \\ @V{\de_P(g)}VV @VV{\de_Q(\pi(f)(g))}V \\ P @>>{f}> Q. \end{CD}$$ A morphism $f\in\L(P,Q)$ is called a *lift* of a morphism $\vp\in\F(P,Q)$ if $\vp=\pi(f)$. A $p$–*local finite group* is a triple $(S,\F,\L)$ where $\F$ is a saturated fusion system over the finite $p$–group $S$ and $\L$ is a centric linking system associated to $\F$. The *classifying space* of $(S,\F,\L)$ is the space ${{|\L|}^\wedge_p}$, that is the $p$–completion in the sense of Bousfield and Kan [@BK], of the realization of the small category $\L$. \[non exotic examples\] When $S$ is a Sylow $p$–subgroup of a finite group $G$, there is an associated $p$–local finite group denoted $(S,\F_S(G),\L_S(G))$. See [@BLO2 Proposition 1.3, remarks after Definition 1.8]. We shall write $\F$ for $\F_S(G)$ and $\L$ for $\L_S(G)$. Morphism sets between $P,Q\leq S$ are $$\F(P,Q)=\Hom_G(P,Q)=N_G(P,Q)/C_G(P)$$ where $N_G(P,Q)=\{g\in G : gPg^{-1}\leq Q\}$ and $C_G(P)$ acts on $N_G(P,Q)$ by right translation. A subgroup $P$ of $S$ is, by [@BLO2 Proposition 1.3], $\F$–centric precisely when it is $p$–centric in the sense of [@Dwyer-decomp Section 1.19], that is, $Z(P)$ is a Sylow $p$–subgroup of $C_G(P)$. In this case $C_G(P)=Z(P)\times C'_G(P)$ where $C_G'(P)$ is the maximal subgroup of $C_G(P)$ of order prime to $p$. Morphism sets of $\L=\L_S(G)$ have, by definition, the form $$\L(P,Q) = N(P,Q)/C_G'(P).$$ The functor $\pi\co\L_S(G)\to\F_S(G)$ is the obvious projection functor. The monomorphism $\de_P\co P\to \Aut_\L(P)$ is induced by the inclusion of $P$ in $N_G(P)$. It is shown by Broto, Levi and Oliver [@BLO2 after Definition 1.8] that $(S,\F_S(G),\L_S(G))$ is a $p$–local finite group and that ${{|\L_S(G)|}^\wedge_p}\simeq {{BG}^\wedge_p}$. It should also be remarked that there are examples of $p$–local finite groups that cannot be associated with any finite group. These are usually referred to as “exotic examples”. \[iota morphisms\] In every $p$–local finite group $(S,\F,\L)$ one can choose morphisms $\iota_P^Q\in\L(P,Q)$ for every inclusion of $\F$–centric subgroups $P\leq Q$, in such a way that 1. $\pi(\iota_P^Q)$ is the inclusion $P\leq Q$, 2. $\iota_Q^R\circ \iota_P^Q=\iota_P^R$ for every $\F$–centric subgroups $P\leq Q\leq R$ of $S$, and 3. $\iota_P^P=\id$ for every $\F$–centric subgroup $P$ of $S$. This follows from [@BLO2 Proposition 1.11]. Using the notation there, one chooses $\iota^Q_P=\delta_{P,Q}(e)$ where $e$ is the identity element in $S$. Whenever possible, in order to avoid cumbersome notation, we shall write $\iota$ for $\iota_P^Q$. \[unique factor in L\] From [@BLO2 Lemma 1.10(a)] it also follows that every morphism $\vp\co P \to Q$ in $\L$ factors uniquely as an isomorphism $\vp'\co P \to P'$ in $\L$ followed by the morphism $\iota\co P' \to Q$. In fact $P'=\pi(\vp)(P)$ \[morphisms in L are mono and epi\] It was observed by Broto, Levi and Oliver [@BLO2 remarks after Lemma 1.10] that every morphism in $\L$ is a monomorphism in the categorical sense. It was later observed by Broto, Castellana, Grodal, Levi and Oliver [@BCGLO1 Corollary 3.10] and independently by others, that every morphism in $\L$ is also an epimorphism. As an easy consequence we record for further use: \[restn in L\] Consider a $p$–local finite group $(S,\F,\L)$ and a commutative square in $\F$ on the left of the display below $$\begin{CD} P @>{f}>> P' \\ @V{\incl}VV @VV{\incl}V \\ Q @>>{g}> Q' \end{CD} \qquad \qquad \qquad \qquad \qquad \begin{CD} P @>{\tilde{f}}>> P' \\ @V{\iota_P^Q}VV @VV{\iota_{P'}^{Q'}}V \\ Q @>>{\tilde{g}}> Q' \end{CD}$$ where $P,P',Q$ and $Q'$ are $\F$–centric subgroups of $S$. Then for every lift $\tilde{g}$ of $g$ in $\L$ there exists a unique lift $\tilde{f}$ of $f$ in $\L$ which render the square on the right commutative in $\L$. We denote $\tilde{f}$ by $\tilde{g}|_P$. Given a lift $\tilde{f}$ for $f$, if there exists a lift $\tilde{g}$ for $g$ rendering the square on the right commutative, then it is unique. The first assertion follows immediately from [@BLO2 Lemma 1.10(a)] by setting $\psi=\incl_{P'}^{Q'}, \tilde{\psi}=\iota_{P'}^{Q'}$ and $\tilde{\psi\vp}=\tilde{g}\iota_P^Q$. The second assertion follows immediately from the fact that $\iota_P^Q$ is an epimorphism. \[centraliser and normaliser systems\] Fix a $p$–local finite group $(S,\F,\L)$. Given a subgroup $P$ of $S$, there are two important $p$–local finite groups associated with it: the centraliser of $P$ when $P$ is fully $\F$–centralised and the normaliser of $P$ when $P$ is fully $\F$–normalised. Both were defined by Broto Levi and Oliver in [@BLO2]. The centraliser fusion system $C_\F(P)$, where $P$ is fully $\F$–centralised, is a subcategory of $\F$. As a fusion system it is defined over the $S$–centraliser of $P$ denoted $C_S(P)$. Morphisms $Q \to Q'$ in $C_\F(P)$ are those morphisms $\vp\co Q \to Q'$ in $\F$ that can be extended to a morphism $\bar{\vp}\co PQ \to PQ'$ in $\F$ which induces the identity on $P$. The objects of the centric linking system $C_\L(P)$ associated to $C_\F(P)$ are the $C_\F(P)$–centric subgroups of $C_S(P)$. The set of morphisms $Q \to Q'$ in $C_\L(P)$ is a subset of $\L(PQ,PQ')$ consists of those morphisms $f\co PQ \to PQ'$ such that $\pi(f)$ induces the identity on $P$ and carries $Q$ to $Q'$. It is shown in [@BLO2] that $(C_S(P),C_\F(P),C_\L(P))$ is a $p$–local finite group. Now fix a subgroup $K \leq \Aut_\F(P)$ where $P$ is fully normalised in $\F$. The $K$–normaliser fusion system $N^K_\F(P)$ is a subcategory of $\F$ defined over $N_S(P)$. The objects of $N_\F^K(P)$ are the subgroups of $N_S(P)$. A morphisms $\vp\in\F(Q,Q')$ belongs to $N^K_\F(P)$ if it can be extended to a morphism $\bar{\vp}\co PQ \to PQ'$ in $\F$ which induces an automorphism from $K$ on $P$. The fusion system $N_\F^K(P)$ is saturated. When $K=\Aut_\F(P)$ we denote this category by $N_\F(P)$ and call it the normaliser fusion system of $P$. The centric linking system $N_\L(P)$ associated to $N_\F(P)$ has the $N_\F(P)$–centric subgroups of $N_S(P)$ as its object set. The set of morphisms $Q \to Q'$ is the subset of $\L(PQ,PQ')$ consisting of those $f\co PQ \to PQ'$ such that $\pi(f)$ carries $Q$ to $Q'$ and induces an automorphism on $P$. The Grothendieck construction {#sec homotopy colimits} ============================= Throughout this paper we work simplicially, namely a “space” means a simplicial set. For further details, the reader is referred to Bousfeld and Kan [@BK], May [@May-ss], Goerss and Jardine [@Goerss-sht] and many other sources. In this section we collect several results from general simplicial homotopy theory that we shall use repeatedly in the rest of this note. **Homotopy colimits** Fix a small category $\KKK$ and a functor $U\co\KKK\to\spaces$. The simplicial replacement of $U$ is the simplicial space $\coprod_*U$ which has in simplicial dimension $n$ the disjoint union of the spaces $U(K_0)$ for every chain $$K_0 \to K_1 \to \cdots \to K_n$$ of $n$ composable arrows in $\KKK$. The homotopy colimit of $U$ denoted ${\hocolim_{\KKK}}U$ is the diagonal of $\coprod_*U$ regarded as a bisimplicial set. See Bousfield and Kan [@BK Section XII.5]. Consider a functor $F\co\KKK \to \LLL$ between small categories. For a functor $U\co\LLL\to \spaces$ there is an obvious natural map, cf [@BK Section XI.9]. $${\hocolim_{\KKK}} F^*U \to {\hocolim_{\LLL}} U.$$ For an object $L\in \LLL$, the comma category $(L \darrow F)$ has the pairs $\smash{(K, L {\xrightarrow{k\in\KKK}} FK)}$ as its objects. Morphisms $(K,L \smash{\stackrel{\raisebox{-1pt}{\scriptsize$k$}}{\longrightarrow}} FK) \to (K',L \smash{\stackrel{\raisebox{-1pt}{\scriptsize$k'$}}{\longrightarrow}} FK')$ are the morphisms $x\co K \to K'$ such that $Fx \circ k = k'$. Similarly one defines the category $(F \darrow L)$ whose object set consists of the pairs $(K,k\co FK\to L)$. Compare MacLane [@MacLane-working]. \[def right cofinal\] The functor $F\co\KKK \to \LLL$ is called right-cofinal if for every object $L\in \LLL$ the category $(L \darrow F)$ has a contractible nerve. The following theorem was probably first proved by Quillen [@Quillen Theorem A]. See also Hollender and Vogt [@Ho-Vo Section 4.4] and Bousfield and Kan [@BK Section XI.9]. \[cofinal thm\] Let $F\co\KKK \to \LLL$ be a right cofinal functor between small categories. Then for every functor $U\co\LLL\to\spaces$ the natural map $${\hocolim_{\KKK}}F^*U\to{\hocolim_{\LLL}}U$$ is a weak homotopy equivalence. Associated with a functor $U\co\KKK\to\spaces$ there is a functor $F_*U\co\LLL\to\spaces$ called the homotopy left Kan extension of $U$ along $F$. It is defined on every object $L\in\LLL$ by $$F_*U(L) = \hocolim \Big( (F\darrow L) {\xrightarrow{\text{proj}}} \KKK {\xrightarrow{F}} \spaces \Big).$$ See [@Ho-Vo Section 5], [@Dw-Ka Section 6]. The following theorem is originally due to Segal. See eg [@Ho-Vo Theorem 5.5]. \[pushdown thm\] Fix a functor $F\co\KKK\to\LLL$ of small categories. Then for every functor $U\co\KKK\to\spaces$ there is a natural weak homotopy equivalence $${\hocolim_{\LLL}} F_*U {\xrightarrow{ \ \ \simeq \ \ }} {\hocolim_{\KKK}}U.$$ **The Grothendieck construction**Recall that a small category $\KKK$ gives rise to a simplicial set $\Nr(\KKK)$ called the *nerve* of $\KKK$. Its $n$–simplices are the chains of $n$ composable arrows $K_0\to K_1\to\cdots\to K_n$ in $\KKK$. See, for example, Goerss and Jardine [@Goerss-sht Example 1.4] or Bousfield and Kan [@BK Section XI.2]. We shall also use the notation $|\KKK|$ for the nerve of $\KKK$. Given a functor $U\co\KKK\to\Cat$ Thomason [@Thomason] defined the translation category $\KKK\int U$ associated to $U$ as follows. The object set consists of pairs $(K,X)$ where $K$ is an object of $\KKK$ and $X$ is an object of $U(K)$. Morphisms $(K_0,X_0)\to(K_1,X_1)$ are pairs $(k,x)$ where $k\co K_0\to K_1$ is a morphism in $\KKK$ and $x\co U(k)(X_0) \to X_1$ is a morphism in $U(K_1)$. Composition of $(K_0,X_0) \smash{\raisebox{-2pt}{${\xrightarrow{(k_0,x_0)}}$}} (K_1,X_1)$ and $(K_1,X_1) \smash{\raisebox{-2pt}{${\xrightarrow{(k_1,x_1)}}$}} (K_2,X_2)$ is given by $$(k_1,x_1)\circ(x_0,k_0) = (k_1\circ k_0, x_1\circ U(k_1)(x_0)).$$ This category is also called the Grothendieck construction of $U$ and the notation $\Tr_\KKK U$ is also used. Thomason [@Thomason] shows that there is a natural weak homotopy equivalence $$\label{Thomason map} \eta\co {\hocolim_{\KKK}}\, |U| {\xrightarrow{\ \ \simeq \ \ }} |\Tr_{\KKK}U|$$ A natural transformation $U\Rightarrow U'$ gives rise to a canonical functor $\Tr_\KKK U \to \Tr_{\KKK}U'$. The induced map $|\Tr_\bfK(U)|\to|\Tr_\bfK(U')|$ corresponds via $\eta$ to the induced map ${\hocolim_{\bfK}}\, |U| \to {\hocolim_{\bfK}}\, |U'|$. Furthermore, for every object $K$ in $\KKK$ the natural map $$|U(K)| \to {\hocolim_{\bfK}}\, |U|$$ corresponds under to the inclusion of categories $$\label{translation cone} U(K) \to \Tr_{\KKK}\, U, \qquad \text{ where } X \mapsto (K,X) \text{ and } x \mapsto (1_K,x).$$ Consider now a functor $F\co\KKK\to\LLL$ of small categories. Given $U\co \LLL\to\cat$ there is a naturally defined functor $$\label{def F shriek} F_! \co \Tr_\KKK F^*U \to \Tr_\LLL U, \qquad \text{where} \qquad \left\{ \begin{array}{l} F_!(K,X\in F^*U(K)) = (FK,X) \\ F_!(k,x) = (Fk,x). \end{array} \right.$$ The functor $F_!$ is a model for the map $\hocolim\,F^*|U| \to \hocolim\,|U|$ in the sense that the following square commutes $$\begin{CD} |\Tr_\KKK F^*U| @>{\eta}>> {\hocolim_{\KKK}} F^*|U| \\ @V{|F_!|}VV @VVV \\ |\Tr_\LLL U| @>{\eta}>> {\hocolim_{\LLL}} |U| \end{CD}$$ \[def catF star\] For a functor $U\co \KKK\to\cat$ define $F_*U\co \LLL\to\cat$ by $$F_*U(L) = \Tr \bigl( (F\darrow L) {\xrightarrow{\text{proj}}} \KKK {\xrightarrow{U}} \spaces \bigr)$$ The maps $\eta$ provide a natural weak homotopy equivalence $|F_*U| {\xrightarrow{\simeq}} F_*|U|$. The equivalence in the pushdown theorem can be realized as the nerve of a functor between the transporter categories as follows. \[def F sharp\] The functor $F_\# \co \Tr_\LLL F_*U \to \Tr_\KKK U$ defined by $$\begin{array}{l} F_\# \co \big( L,(K,FK \to L),X\in UK \big) \mapsto (K,X) \\ F_\# \co \big( L {\xrightarrow{\ell}} L', K {\xrightarrow{k}} K', U(k)(X) {\xrightarrow{x}} X' \big) \mapsto (k,x). \end{array}$$ renders the following diagram commutative where the arrow at the top of the square is an equivalence by the pushdown theorem. $$\xymatrix{ {{\hocolim_{\LLL}}}\, |F_*U| \ar[dr]^\eta_\simeq & {\hocolim_{\LLL}}\, F_*|U| \ar[r]^{\hbox{\footnotesize$\sim$}} \ar[d]_{\hbox{\footnotesize$\sim$}} \ar[l]_{\hbox{\footnotesize$\sim$}} & {\hocolim_{\KKK}}\, |U| \ar[d]_{\hbox{\footnotesize$\sim$}}^\eta \\ & |\Tr_{\LLL}(F_*U)| \ar[r]_{|F_\#|} & |\Tr_{\KKK}(U)| }$$ It is useful to point out that if $\star\co \KKK \to \cat$ is the constant functor on the trivial category with one object and an identity morphism, then $\Tr_\KKK(\star)=\Nr(\KKK)$. EI categories {#sec EI} ============= Fix an EI category $\A$, namely a category all of whose endomorphisms are isomorphisms. We shall assume that the category $\A$ is finite. We shall also assume that $\A$ is equipped with a height function, namely a function $h\co \Obj(\A) \to \NN$ such that $h(A) \leq h(A')$ if there exists a morphism $A\to A'$ in $\A$ and equality holds if and only if $A\to A'$ is an isomorphism. Clearly, if $\A$ is an EI-category then so is $\A^\op$. The finiteness condition also implies that if $\A$ is heighted then so is $\A^\op$. We can always choose a full subcategory $\A_{\sk}$ of $\A$ which contains one representative from each isomorphism class of objects in $\A$. We say that $\A_{\sk}$ is skeletal in $\A$. Clearly the inclusion $\A_{\sk}\subseteq \A$ is an equivalence of categories. In the language of Słomińska [@Slominska] $\A_{\sk}$ is an EIA category. Throughout we let $\underk$ denote the poset $\{0\to 1\to\cdots \to k\}$ considered as a small category. \[def sA\] The subdivision category $s(\A)$ is the category whose objects are height increasing functors $\AAA\co \underk\to \A$, namely $h(\AAA(i))<h(\AAA(i+1))$ for all $i<k$. Morphisms $\AAA \to \AAA'$ in $s(\A)$ are pairs $(\epsilon,\vp)$ where $\epsilon\co \underk'\to\underk$ is a strictly increasing function and $\vp\co \epsilon^*(\AAA) \to \AAA'$ is a natural isomorphism of functors $\underk' \to \A$. Composition of $(\epsilon,\vp)\co \AAA\to \AAA'$ and $(\epsilon',\vp')\co \AAA' \to \AAA''$ is given by $(\epsilon\circ\epsilon',{\epsilon'}^*(\vp)\circ \vp')$. Note that $\epsilon$ is determined by the heights of the values of $\AAA$ namely $\epsilon(i)=j$ if and only if $h(\AAA'(i))=h(\AAA(j))$. We shall further assume that $\A$ contains a subcategory $\I$ which is a poset with the property that every morphism $\vp\co A \to A'$ in $A$ can be factored uniquely as $\vp = \iota \vp'$ where $\vp'$ is an isomorphism in $\A$ and $\iota$ is a morphism in $\I$. The ladder $$\xymatrix{ {\cdots} \ar[r] & \AAA(n-2) \ar[r]^{\vp_{n-1}} \ar[d]_{(\vp_n'\circ\vp_{n-1})'}^\cong & \AAA(n-1) \ar[r]^{\vp_n} \ar[d]_{\vp_n'}^\cong & \AAA(n) \ar@{=}[d] \\ {\cdots} \ar[r] & \AAA'(n-2) \ar[r]_{\iota} & \AAA'(n-1) \ar[r]_\iota & \AAA'(n) }$$ shows that the the full subcategory $s_\I(\A)$ of $s(\A)$ consisting of the objects $\AAA$ in which all the arrows $\AAA(i) \to \AAA(i+1)$ belong to $\I$ is a skeletal subcategory of $s(\A)$. We obtain two skeletal subcategories of $s(\A)$ $$\label{skel sA} s(\A_{\sk}) \subseteq s(\A) \qquad \text{ and } \qquad s_\I(\A) \subseteq s(\A).$$ We observe that $\Hom_{s(\A)}(\AAA,\AAA')$ has a free action of $\Aut_{s(\A)}(\AAA')$ with a single orbit. Also every $(\epsilon,\vp)\co \AAA\to \AAA'$ in $s(\A)$ gives rise to a natural group homomorphism upon restriction and conjugation with the isomorphism $\vp\co \epsilon^*\AAA\approx\AAA'$ $$\label{sA auto maps} \vp_*\co \Aut_{s(\A)}(\AAA) \to \Aut_{s(\A)}(\AAA').$$ \[p cofinal\] There is a right cofinal functor $p\co s(\A) \to \A$ defined by $$p(\AAA) = \AAA(0), \qquad \qquad (\AAA\co \underk \to \A).$$ S[ł]{}omińska [@Slominska Proposition 1.5] shows that the functor $p\co s(\A_{\sk}) \to \A_{\sk}$ is right cofinal () hence so is $p\co s(\A) \to \A$. \[def barsA\] The category $\bar{s}(\A)$ has the isomorphism classes $[\AAA]$ of the objects of $s(\A)$ as its object set. There is a unique morphism $[\AAA] \to [\AAA']$ if there exists a morphism $\AAA \to \AAA'$ in $s(\A)$. There is an obvious projection functor $$\pi\co s(\A) \to \bar{s}(\A), \qquad \AAA \mapsto [\AAA].$$ When $\D$ is a full subcategory of $s(\A)$ one obtains a sub-poset $\bar{\D}$ of $\bar{s}(\A)$ whose objects are the isomorphism classes of the objects of $\D$. Clearly $\bar{s}(\A)$ is a poset and it should be compared with Słomińska’s construction of $s_0(\A)$ in [@Slominska Section 1]. Also note that $\bar{s}_\I(\A)=\bar{s}(\A)$ because $s_\I(\A)$ is skeletal in $s(\A)$. Similarly $\bar{s}(\A_{\sk})=\bar{s}(\A)$. \[adj cof lem\] Let $J\co {\mathcal{C}} \to \D$ be a functor of small categories with a left adjoint $L\co \D \to {\mathcal{C}}$ such that $L \circ J=\Id$. Then $J$ is right cofinal. Fix an object $d\in \D$. We have to prove that the category $(d \darrow J)$ has a contractible nerve. Let $(d \darrow \D)$ denote the category $(d \darrow 1_\D)$. It clearly has a contractible nerve because it has an initial object. The functors $J$ and $L$ induce obvious functors $$\begin{aligned} && J_* \co (d\darrow J) \to (d\darrow \D), \qquad (c, d{\xrightarrow{f}} Jc) \mapsto (Jc, d {\xrightarrow{f}} Jc) \\ && L_*\co (d\darrow\D) \to (d\darrow J), \qquad (d', d {\xrightarrow{f}} d') \mapsto (Ld', d {\xrightarrow{f}} d' {\xrightarrow{\eta}} JLd').\end{aligned}$$ It is obvious that $L_*\circ J_*=\Id$. Furthermore the unit $\eta\co \Id \to LJ$ gives rise to a natural transformation $\Id \to J_*\circ L_*$. Therefore $J_*$ induces a homotopy equivalence on nerves so $|(d\darrow J)| \simeq |d\darrow \D)|\simeq *$. \[commapi\] For every functor $F\co s(\A) \to\cat$ and every $\AAA\in s(\A)$ there is a functor $$\Tr_{\B\Aut(\AAA)} F(\AAA) \to (\pi_* F)([\AAA])$$ which induces a weak homotopy equivalence on nerves. It is natural in the sense that every morphism $(\epsilon,\vp)\co \AAA \to \AAA'$ in $s(\A)$, gives rise to a square $$\xymatrix{ {\Tr_{\Aut(\AAA)} F(\AAA)} \ar[r] \ar[d]_{\Tr_{\vp_*} F(\vp)} & (\pi_*F)([\AAA]) \ar[d]^{\pi_*([\vp])} \ar@{}|-{\overset{\tau}{\Leftarrow}}[dl] \\ {\Tr_{\Aut(\AAA')} F(\AAA')} \ar[r] & (\pi_*F)([\AAA']) }$$ which commutes up to a natural transformation $\tau$ which is functorial in $(\epsilon,\vp)$. Here $\vp_*\co \Aut(\AAA) \to \Aut(\AAA')$ is the homomorphism induced by restriction and conjugation by $\vp\co \epsilon^*\AAA \approx \AAA'$ as we described in . The square commutes on the nose if $F(\vp)\co F(\AAA) \longrightarrow F(\AAA')$ is the identity. Fix an object $\AAA\co \underk \to \A$ in $s(\A)$. Let $\Pi_\AAA$ be the full subcategory of $(\pi\darrow [\AAA])$ consisting of the objects $(\AAA',[\AAA']{\xrightarrow{=}} [\AAA])$. It is isomorphic to the full subcategory of $s(\A)$ consisting of the isomorphism class of $\AAA$. The inclusion $J\co \Pi_\AAA \to (\pi\darrow [\AAA])$ has a left adjoint $L$ where $$L \co (\BBB,[\BBB] \to [\AAA]) \mapsto \epsilon^*\BBB, \qquad \text{ where } [\epsilon^*\BBB]=[\AAA] \text{ for } \epsilon\co \underk' \hookrightarrow \underk.$$ Clearly $\epsilon$ is unique if it exists. There is a natural map $\BBB \to \epsilon^*\BBB$ induced by the identity on $\epsilon^*\BBB$ under the bijection $s(\A)(\BBB,\epsilon^*\BBB) \approx s(\A)(\epsilon^*\BBB,\epsilon^*\BBB)$. We obtain a natural transformation $\Id \to JL$ which gives rise to bijections for every object $(\AAA',[\AAA']\to [\AAA])$ in $(\pi \darrow [\AAA])$ $$\Hom_{(\pi \darrow [\AAA])}(\BBB,J\AAA') = \Hom_{s(\A)}(\BBB,\AAA') \approx \Hom_{s(\A)}(\epsilon^*\BBB,\AAA') = \Hom_{\Pi_\AAA}(L\BBB,\AAA').$$ Thus $L$ is left adjoint to $J$ and we apply . By definition $\Pi_\AAA$ is a connected groupoid with automorphism group $\B\Aut_{s(\A)}(\AAA)$. Therefore upon realization, the functor $$\Tr\bigl( \B\Aut_{s(\A)}(\AAA){\to}s(\A) {\xrightarrow{F}} \Cat \bigr) {\xrightarrow{\text{restriction}}} \Tr\bigl((\pi{\darrow}[\AAA]){\to}s(\A) {\xrightarrow{F}} \Cat\bigr) = (\pi_*F)([\AAA])$$ induces a weak homotopy equivalence. Also, for a morphism $\vp\co \AAA\to \AAA'$ we get an obvious $\vp_*\co \Pi_\AAA \to \Pi_\AAA'$ by restriction and conjugation by the isomorphism $\vp\co \epsilon^*\AAA \to \AAA'$. It gives rise to the following diagram $$\xymatrix{ {\B\Aut_{s(\A)}(\AAA)} \ar[r]^{\incl} \ar[d]^{\vp_*} & \Pi_\AAA \ar[d]_{\vp_*} \ar[r]^{J} & (\pi\darrow [\AAA]) \ar[d]^{[\vp]_*} \\ {\B\Aut_{s(\A)}(\AAA')} \ar[r]^{\incl} & \Pi_{\AAA'} \ar[r]^J & (\pi\darrow [\AAA']) }$$ The morphism $\vp$ provides a canonical natural transformation $[\vp]_*\circ J \circ \incl \to J \circ \incl \circ \vp_*$. This provides the natural transformation $\tau$ in the statement of the proposition and its naturality with $\vp$. If $F(\vp)\co F(\AAA) \longrightarrow F(\AAA')$ is the identity, then $F(\tau)$ becomes the identity and the square in the statement of the proposition commutes. Proof of the main results ========================= Fix a $p$–local finite group $(S,\F,\L)$ and an $\F$–centric collection ${\mathcal{C}}$. Choose a subcategory $\I \subseteq \L^{\mathcal{C}}$ of distinguished inclusions, cf . Note that $\L^{\mathcal{C}}$ possesses a height function, see , by assigning to a subgroup $P$ in ${\mathcal{C}}$ its order. Also every morphism in $\L^{\mathcal{C}}$ factors uniquely as an isomorphism followed by a morphism in $\I$. We claim that (Definitions \[def bsdc\] and \[def barsA\]) $$\bsd{\mathcal{C}} = \bar{s}(\L^{\mathcal{C}}).$$ To see this recall that $s_\I(\L^{\mathcal{C}})$ is a skeletal subcategory of $s(\L^{\mathcal{C}})$, see , hence $\bar{s}(\L^{\mathcal{C}}) = \bar{s}_\I(\L^{\mathcal{C}})$. The functor $\pi\co \L \to \F$ gives a functor $\bar{s}_\I(\L^{\mathcal{C}}) \to \bsd{\mathcal{C}}$ because it maps the morphisms $\iota\in \I$ to inclusion of subgroups of $S$. It is an isomorphism of categories because conjugation of two $k$–simplices $P_0<\cdots<P_k$ and $P'_0<\cdots< P'_k$ induced by an isomorphism $\vp_k\in\Iso_\F(P_k,P_k')$ can be lifted to an isomorphism of the corresponding objects in $s_\I(\L^{\mathcal{C}})$ by lifting the isomorphism $\vp_k\co P_k \to P_k'$ to $\L$ and using to obtain the commutative ladder in $\L^{\mathcal{C}}$: $$\xymatrix{ P_0 \ar[r]^{\iota} \ar[d]_\cong & P_1 \ar[r]^\iota \ar[d]_\cong & \cdots \ar[r]^{\iota} & P_k \ar[d]^{\tilde{\vp_k}} \\ P_0' \ar[r]_\iota & P_1' \ar[r]^\iota & \cdots \ar[r]^{\iota} & P_k' }$$ We remark that a $k$–simplex $P_0<\cdots<P_k$ in ${\mathcal{C}}$ can be identified with the object $P_0 {\xrightarrow{\iota}} \cdots {\xrightarrow{\iota}} P_k$ of $s(\L^{\mathcal{C}})$. Under this identification we clearly have $$\Aut_\L(\PPP) = \Aut_{s(\L^{\mathcal{C}})}(\PPP).$$ When $G$ is a discrete group we let $\B G$ denote the category with one object and $G$ as its set of morphisms. For every $k$–simplex $\PPP$ in ${\mathcal{C}}$ we identify $\B\Aut_\L(\PPP)$ with the obvious subcategory of $\B\Aut_\L(P_0)$. \[thmC\] Let ${\mathcal{C}}$ be an $\F$–centric collection in a $p$–local finite group $(S,\F,\L)$. Then there exists a functor $\tilde{\de}_{\mathcal{C}}\co \bsd{\mathcal{C}} \to \Cat$ with the following properties 1. There is a naturally defined functor $\Tr_{\bsd{\mathcal{C}}}(\tilde{\de}_{\mathcal{C}}) \to \L^{\mathcal{C}}$ which induces a weak homotopy equivalence on nerves. \[thmC:target\] 2. For every $k$–simplex $\PPP$ there is a canonical functor $ \B\Aut_\L(\PPP) \to \tilde{\de}_{\mathcal{C}}([\PPP]) $ which induces a weak homotopy equivalence on nerves. If $\PPP'$ is a subsimplex of $\PPP$ then the following square commutes $$\xymatrix{ {\B\Aut_\L(\PPP)} \ar[r] \ar[d]_{\res^\PPP_{\PPP'}} & \tilde{\de}_{\mathcal{C}}([\PPP]) \ar[d] \\ {\B\Aut_\L(\PPP')} \ar[r] & \tilde{\de}_{\mathcal{C}}([\PPP']) }$$ \[thmC:BPtype\] 3. The natural inclusion $\B\Aut_\L(\PPP) \subseteq \L^{\mathcal{C}}$ is equal to the composition $$\B\Aut_\L(\PPP) \to \tilde{\de}_{\mathcal{C}}([\PPP]) \subseteq \Tr_{\bsd{\mathcal{C}}}(\tilde{\de}_{\mathcal{C}}) \to \L^{\mathcal{C}}$$ \[thmC:augment\] 4. An isomorphism of $k$–simplices $\psi\co \PPP'{\xrightarrow{\approx}} \PPP$ in $s(\L^{\mathcal{C}})$ induces a commutative square $$\xymatrix{ {\B\Aut_\L(\PPP)} \ar[r] \ar[d]_{c_\psi} & \tilde{\de}_{\mathcal{C}}([\PPP]) \ar[d] \\ {\B\Aut_\L(\PPP')} \ar[r] & \tilde{\de}_{\mathcal{C}}([\PPP']) }$$ \[thmC:morphisms\] We have seen that $\bsd{\mathcal{C}} = \bar{s}(\L^{\mathcal{C}})$. Let $\star\co \bar{s}(\L^{\mathcal{C}}) \to \cat$ denote the constant functor on the trivial small category with one object and identity morphism. Use the projection functor $\pi\co s(\L^{\mathcal{C}}) \to \bar{s}(\L^{\mathcal{C}})$ to define $$\tilde{\de}_{\mathcal{C}} = \pi_*(\star)$$ According to we have a canonical functor $$\B\Aut_\L(\PPP) = \Tr_{\B\Aut_\L(\PPP)}(\star) \to \pi_*(\star)([\PPP]) = \tilde{\de}_{\mathcal{C}}([\PPP])$$ which induces a weak homotopy equivalence. Since $\star$ is constant, the square in the statement of commutes and we obtain the naturality assertions in point and . The natural functor of is defined using and by $$\Tr_{\bsd{\mathcal{C}}} (\tilde{\de}_{\mathcal{C}}) = \Tr_{\bar{s}(\L^{\mathcal{C}})} (\pi_*(\star)) {\xrightarrow{ \ \pi_\# \ }} \Tr_{s(\L^{\mathcal{C}})}(\star) = s(\L^{\mathcal{C}}) {\xrightarrow{p}} \L^{\mathcal{C}}.$$ It induces a weak homotopy equivalence by , and . Whence point . Inspection of the functor $\pi_\#$, the inclusion $\B\Aut_\L(\PPP) \subseteq (\pi\darrow [\PPP]) = \pi_*(\star)([\PPP])$ and yield point Apply above and define $\de_{\mathcal{C}} = |\tilde{\de}_{\mathcal{C}}|$. \[compare with Dwyer\] We now relate the construction in to Dwyer’s normaliser decomposition [@Dwyer-decomp Section 3]. We will show that the two functors are related by a zigzag of natural transformations which induce a mod–$p$ equivalence. Fix a finite group $G$ and the $p$–local finite group $(S,\F,\L)$ associated with it. A collection ${\mathcal{C}}$ of $\F$–centric subgroups of $S$ gives rise to a $G$–collection $\H$ of $p$–centric subgroups of $G$ (cf [@Dwyer-decomp Section 1.19], ) by taking all the $G$–conjugates of the elements of ${\mathcal{C}}$. We let $\T^\H$ denote the transporter category of $\H$. That is, the object set of $\T^\H$ is $\H$ and the morphism set $\T^\H(H,K)$ is the set $N_G(H,K) = \{ g\in G : g^{-1}Hg\leq K\}$. We also let $\T^{\mathcal{C}}$ denote the full subcategory of $\T^\H$ having ${\mathcal{C}}$ as its object set. Almost by definition $\T^{\mathcal{C}}$ is skeletal in $\T^\H$. We also obtain a zigzag of functors (see ) $$\T^\H \leftarrow \T^{\mathcal{C}} \to \L^{\mathcal{C}}.$$ Dwyer [@Dwyer-decomp Section 3] defines a category $\bsd\H$ whose objects are the $G$–conjugacy classes $[\HHH]$ of the $k$–simplices $H_0<\cdots<H_k$ in $\H$. There is a unique morphism $[\HHH] \to [\HHH']$ in $\bsd\H$ if and only if $\HHH'$ is conjugate in $G$ to a subsimplex of $\HHH$. It follows directly from the definition of $\H$ as the smallest $G$–collection containing ${\mathcal{C}}$ and from the definition of $\F=\F_S(G)$ that $\bsd\H=\bsd{\mathcal{C}}$. We obtain a commutative diagram (see ) $$\xymatrix{ s(\T^\H) \ar[d]_{\pi_2} & s(\T^{\mathcal{C}}) \ar[r] \ar[d]_{\pi_1} \ar[l]_{\supseteq} & s(\L^{\mathcal{C}}) \ar[d]^\pi \\ {\bar{s}}(\T^\H) & \bar{s}(\T^{\mathcal{C}}) \ar@{=}[l] \ar@{=}[r] & \bar{s}(\L^{\mathcal{C}}). }$$ Fix a $k$–simplex $\PPP=P_0<\cdots<P_k$ in ${\mathcal{C}}$. Note that $(\pi_2\darrow [\PPP])$ is isomorphic to the subcategory of $\T^\H$ of the objects $\PPP'$ which admit a morphism to $\PPP$. It contains a full subcategory $\Pi_\PPP$ of the objects of $s(\T^\H)$ that are isomorphic to $\PPP$; cf the proof of . By inspection $\Pi_\PPP$ is the translation category of the action of $G$ on the orbit of $\PPP$, that is it is the transported category of the $G$–set $[\PPP]$ in $\H$ thought of as a functor $\B\Aut_\L(\PPP)\to \sets$, cf [@Dwyer-decomp Section 3.3]. Thus $$\de^{\text{Dwyer}}_\H([\PPP]) = EG \times_G [\PPP] = \Nr(\Pi_\PPP).$$ The inclusion $J\co \Pi_\PPP\hookrightarrow (\pi_2 \darrow [\PPP])$ has a left adjoint $L\co (\PPP', [\PPP'] \to [\PPP]) \mapsto \epsilon^*\PPP'$ where $(\epsilon,\vp)\co \PPP' \to \PPP$ is a morphism in $s(\L^{{\mathcal{C}}})$, see . Compare the proof of . implies that $J$ is right cofinal. We obtain a zigzag of functors $$\de^{\text{Dwyer}}_\H \xrightarrow[\simeq]{\phantom{-----}} |(\pi_2)_*(\star)| \xleftarrow[\simeq]{\ \incl \ } |(\pi_1)_*(\star)| {\xrightarrow{\text{ mod-}p \ }} |\pi_*(\star)| = |\tilde{\de}_{\mathcal{C}}| = \de_{\mathcal{C}}.$$ The third map induces a mod–$p$ equivalence by the following argument. For any object $[\PPP]$ we obtain a map $(\pi_1)_*(\star)([\PPP]) \to \pi_*(\star)([\PPP])$ which by is equivalent to the map $$B\Aut_G(\PPP) \to B\Aut_\L(\PPP).$$ Since $P_0$ is $p$–centric then $C_G(P_0)=Z(P_0)\times C_G'(P_0)$ where $C_G'(P_0)$ is a characteristic $p'$–subgroup of $C_G(P_0)$ and $\Aut_\L(P_0) = N_G(P_0)/C_G'(P_0)$. Therefore $\Aut_G(\PPP) \to \Aut_G(\PPP)/C_G'(P_0)$ induces a mod–$p$ equivalence as needed. We shall now prove . Fix an $\F$–centric collection ${\mathcal{C}}$ and a collection $\E$ of elementary abelian subgroups in $(S,\F,\L)$. Recall from and the definitions of $\bar{C}_\L({\mathcal{C}};E_k)$ and $\No_\L({\mathcal{C}};\EEE)$ where $\EEE$ is a $k$–simplex in $\E$. \[NoTrCbar\] Fix a $k$–simplex $\EEE$ in $\E$, namely a functor $\EEE\co \underk \to \F^\E$. There is a functor $$\epsilon\co \No_\L({\mathcal{C}};\EEE) \to \Tr_{\B\Aut_\F(\EEE)} \Big(\bar{C}_\L({\mathcal{C}};\EEE) \Big)$$ which is fully faithful and natural with respect to inclusion of simplices. If $E_k$ is fully $\F$–centralised, its image is also skeletal and in particular induces homotopy equivalence $ |\No_\L({\mathcal{C}};\EEE)| {\xrightarrow{ \ \ \simeq \ \ }} |\bar{C}_\L({\mathcal{C}};E_k)|_{h\Aut_\F(\EEE)}. $ The objects of $\H:=\B\Aut_\F(\EEE)\int \bar{C}_\L({\mathcal{C}};\EEE)$ are pairs $(P,f)$ where $P\in {\mathcal{C}}$ and $f \in \F(E,Z(P))$. Morphisms are pairs $(\vp,g)$ where $\vp\in \L(P,P')$ and $g\in\Aut_\F(\EEE)$ such that $f'=\pi(\vp)\circ f \circ g$ (see ). Since $f,f'$ are monomorphisms, $g$ is determined by $\vp$. Define $\epsilon\co \No_\L({\mathcal{C}};\EEE) \to \H$ by $$\begin{aligned} &&\epsilon(P) = (P,E {\xrightarrow{\incl}} Z(P) \leq P) \\ \nonumber &&\epsilon( P{\xrightarrow{\vp}} P') = (P{\xrightarrow{\vp}} P', \pi(\vp)|_{E_k}^{-1}).\end{aligned}$$ It is well defined and fully faithful by the definition of $\No_\L({\mathcal{C}};\EEE)$. Naturality with respect to inclusion of simplices is readily verified. Consider an object $(P,f) \in \H$. Note that $f(E_k)\leq Z(P)$, hence for $g:=f^{-1}\in \Iso_\F(f(E_k),E_k)$ we must have (see ) $N_g \supseteq C_S(f(E_k)) \supseteq P$. By axiom II of we can extend $g$ to an isomorphism $h\co P \to P'$ in $\F$. Clearly $P'$ is in ${\mathcal{C}}$ because the latter is a collection. Fix a lift $\tilde{h}\in\L(P,P')$ for $h$. We have $\pi(\tilde{h})\circ f = h \circ \smash{\incl^P_{f(E_k)}} \circ g^{-1} = \smash{\incl_{E_k}^{P'}}$. Therefore $(\id_{E_k},\tilde{h})$ is an isomorphism $(P,f) \cong \bigl(P',\smash{\incl_{E_k}^{P'}}\bigr)$ in $\H$. This shows that $\epsilon$ embeds $\No_\L({\mathcal{C}};\EEE)$ into a skeletal subcategory of $\H$ and the result follows. For every $P\in{\mathcal{C}}$ define $\zeta(P)=\Omega_pZ(P)$, see . Note that if $f\co P\to P'$ is an isomorphism in $\F$ then $f^{-1}\co \zeta(P') \to \zeta(P)$ is an isomorphism in $\F$. Also if $P \leq P'$ in ${\mathcal{C}}$ then $Z(P) \leq Z(P')$ because $P$ and $P'$ are $\F$–centric so their centres are equal to their $S$–centralisers. It easily follows that this assignment forms a functor $$\zeta\co \L^{\mathcal{C}} \to {\F^\E}^\op.$$ Fix $E\in\E$. Since every homomorphism $f\co E \to Z(P)$ factors through $\zeta(P)$ we see that $\bar{C}_\L({\mathcal{C}};E) =(\zeta \darrow E)$. In particular $$\label{kan zeta} \zeta_*(\star) \co E \mapsto \bar{C}_\L({\mathcal{C}};E).$$ We now observe that $\F^\E$ is an EI-category. The assignment $E \mapsto |E|$ gives rise to a height function in the sense of . Furthermore the set $\I$ of inclusions in $\F^\E$ forms a poset where every morphism in $\F^\E$ factors uniquely as an isomorphism followed by an element in $\I$. We conclude that $$\bar{s}\E = \bar{s}_\I(\F^\E) \approx \bar{s}(\F^\E).$$ There is an isomorphism $\tau\co s({\F^\E}^\op) \to s(\F^\E)$ which is the identity on objects. On the morphism set between $\EEE\co \underk\to\F^\E$ and $\EEE'\co \undern\to\F^\E$ such that $\epsilon^*\EEE\approx \EEE'$ for some injective $\epsilon\co \undern \to \underk$ (see ), $\tau$ has the effect $$\begin{gathered} \Hom_{s({\F^\E}^\op)}(\EEE,\EEE') = \Iso_{{\F^\E}^\op}(\epsilon^*\EEE,\EEE') = \\ \Iso_{\F^\E}(\EEE',\epsilon^*\EEE) \xrightarrow[\approx]{\ \ \ f \mapsto f^{-1} \ \ } \Iso_{\F^\E}(\epsilon^*\EEE,\EEE') = \Hom_{s(\F^\E)}(\EEE,\EEE').\end{gathered}$$ The functor $p\co s({\F^\E}^\op) \to {\F^\E}^\op$ of fits into a commutative diagram $$\xymatrix{ s({\F^\E}^\op) \ar[r]^p \ar[d]_\tau^\approx & {\F^\E}^\op \ar@{=}[d] \\ s(\F^\E) \ar[r]^\mu & {\F^\E}^\op }$$ where $\mu(\EEE)=E_k$. Since $\tau$ is an isomorphism, $\mu$ is right cofinal. We obtain a zigzag of functors $$\bsd\E \xleftarrow{\ \ \pi \ \ } s(\F^\E) {\xrightarrow{\ \ \mu \ \ }} {\F^\E}^\op \xleftarrow{ \ \ \zeta \ \ } \L^{\mathcal{C}} {\xrightarrow{\ \ \star \ \ }} \Cat.$$ Define $$\tilde{\de}_\E= \pi_*\circ \mu^* \circ \zeta_*(\star), \qquad \text{ and } \qquad \de_\E=|\tilde{\de}_\E|.$$ Since $\mu$ is right cofinal, the and imply a weak homotopy equivalence $${\hocolim_{\bsd\E}} \, \de_\E {\xrightarrow{\simeq}} |\L^{\mathcal{C}}|.$$ This is point of the theorem. Inspection of , and shows that this equivalence is given as the realization of a functor $\Tr_{\bsd\E}\tilde{\de}_\E \to \L^{\mathcal{C}}$ where $$\begin{aligned} (\EEE, E_k {\xrightarrow{f}} P) &\mapsto& P \\ (\EEE \to \EEE', g\in\Aut_\F(\EEE'), P {\xrightarrow{\vp\in\L}} P') &\mapsto& \vp.\end{aligned}$$ Point follows from and . Point follows from , inspection of $\epsilon$ in , of and the functor $\Tr_{\bsd\E}(\tilde{\de}_\E) \to \L^{\mathcal{C}}$. Point is a consequence of .

--- abstract: 'We investigate the stability and coherence properties of one-dimensional exciton-polariton condensates under nonresonant pumping. We model the condensate dynamics using the open-dissipative Gross-Pitaevskii equation. In the case of spatially homogeneous pumping, we find that the instability of the steady state leads to significant reduction of the coherence length. We consider two effects that can lead to the stabilization of the steady state, i.e. the polariton energy relaxation and the influence of an inhomogeneous pumping profile. We find that, while the former has little effect on the stability, the latter is very effective in stabilizing the condensate which results in a large coherence length.' author: - 'Nataliya Bobrovska$^1$, Elena A. Ostrovskaya$^2$ and Micha[ł]{} Matuszewski$^1$' bibliography: - 'references.bib' title: 'Stability and spatial coherence of nonresonantly pumped exciton-polariton condensates' --- Introduction ============ Exciton-polaritons are light-matter bosonic quasiparticles created due to strong coupling between excitons and photons [@kavokin2007-book; @Weisbuch_Polaritons; @Kasprzak_BEC]. Their extremely light effective mass combined with strong exciton-mediated interparticle interactions makes them an ideal system for investigation of fundamental phenomena such as room temperature Bose-Einstein condensation [@Grandjean_RoomTempLasing; @Kena_Organic; @Mahrt_RTCondensatePolymer] as well as applications [@Bramati_SpinSwitches; @Savvidis_TransistorSwitch; @Sanvitto_Transistor; @Hofling_ElectricallyPumped; @Battacharya_RoomTemperatureElectrically]. A wide range of phenomena observed in exciton-polariton systems, such as superfluidity [@amo2009], or spontaneous creation of self-localized structures, including solitons and vortices, has attracted great interest in recent years [@Carusotto_QuantumFluids; @Yamamoto_RMP]. In contrast to condensates of ultracold atoms, polariton superfluids are nonequilibrium systems in which continuous pumping is required to maintain the condensate population [@kavokin2007-book; @Kasprzak_BEC; @Yamamoto_RMP; @wouters2007]. In a number of experiments, spatial coherence extending over the whole polariton cloud was demonstrated [@Kasprzak_BEC; @Yamamoto_SpatialCoherence; @Bloch_ExtendedCondensates; @Hofling_1DcoherenceInteractions; @Yamamoto_RMP]. Recently, stability limits for polariton condensates under nonresonant pumping were determined within the open-dissipative Gross-Pitaevskii model [@ostrovskaya2014]. It was predicted that [*spatially homogeneous*]{} steady states are stable in regions of parameter space determined by the ratio $\gamma_R g_C / \gamma_C g_R$, where $\gamma_{C,R}^{-1}$ are the lifetimes of the polariton condensate and the exciton reservoir and $g_{C,R}$ are the coefficients of interaction within the condensate and between the condensate and the reservoir, respectively. Importantly, for comparable interaction constants, stability close to the threshold was predicted only for relatively small values of $\gamma_C / \gamma_R$. It is important to note that values of $\gamma_R$ varying by orders of magnitude are used throughout the literature [@Bloch_1DAmplification; @Deveaud_VortexDynamics; @Yamamoto_VortexPair]. According to independent measurements [@Szczytko_ExcitonFormation; @Bloch_DynamicsElectronGas], physical values of $\gamma_R$ correspond to the unstable regime, where, as we show below, a significant reduction of the coherence length can be expected. In this work, we investigate whether large coherence length can emerge in condensates even in the unstable region of parameter space predicted by the homogeneous theory. We analyse the stability and coherence properties of polariton condesates in more detail, taking into account the effects of polariton relaxation and the inhomogeneous pumping profile. We use the Bogoliubov-de Gennes theory as well as direct numerical integration of the open-dissipative Gross-Pitaeskii equations. We demonstrate that full coherence can be achieved even for large values of $\gamma_C / \gamma_R$ once the finite size of the pumping spot is taken into account. We believe that further experiments, in particular with homogeneous or ring-shaped pumping profiles, are necessary to determine stability limits in parameter space and fix the values of phenomenological parameters of the model. In particular, the observation of coherence reduction or its absence in the case of almost-homogeneous pumping would verify whether the ratio $\gamma_C / \gamma_R$ corresponds to the unstable regime. The model ========= In one dimension (e.g. in a microwire [@Bloch_ExtendedCondensates]) the exciton-polariton condensate with the wavefunction $\psi(x,t)$ can be modelled with the generalized open-dissipative Gross-Pitaevskii equation coupled to the rate equation for the polariton reservoir density, $n_R(x,t)$ [@wouters2007; @Xue_DarkSoliton] $$\begin{split} \label{GP1} i \hbar\frac{\partial \psi}{\partial t} &=-\frac{\hbar^2 }{2 m^*} \frac{\partial^2 \psi}{\partial x^2} + g_{\rm C}^{\rm 1D} |\psi|^2 \psi + g_{\rm R}^{\rm 1D} n_{\rm R} \psi \\ &+i\frac{\hbar}{2}\left(R^{\rm 1D} n_{\rm R} - \gamma_{\rm C} \right) \psi, \\ \frac{\partial n_{\rm R}}{\partial t} &= P(x) - (\gamma_{\rm R}+ R^{\rm 1D} |\psi|^2) n_{\rm R} \end{split}$$ where $P(x)$ is the exciton creation rate determined by the pumping profile, $m^*$ is the effective mass of lower polaritons, $\gamma_{\rm C}$ and $\gamma_{\rm R}$ are the polariton and exciton loss rates, and $(R^{\rm 1D},g_i^{\rm 1D})=(R^{\rm 2D},g_i^{\rm 2D})/\sqrt{2\pi d^2}$ are the rates of stimulated scattering into the condensate and the interaction coefficients, rescaled in the one-dimensional case. Here, we assumed a Gaussian transverse profile of $|\psi|^2$ and $n_{\rm R}$ of width $d$. In the case of a one-dimensional microwire [@Bloch_ExtendedCondensates], the profile width $d$ is of the order of the microwire thickness. We note that the exciton field correspond to the “active” exciton population rather than the reservoir at high energy levels [@Deveaud_VortexDynamics]. While the latter may have much longer lifetime $\gamma^{-1}$, it is not subject to a considerable back-action from polaritons, such as stimulated scattering, which is relevant for the stability properties of the system. By rescaling time, space, wavefunction amplitude and material coefficients as $t=\tau \widetilde{t}$, $x=\xi\widetilde{x}$, $\psi=(\xi\beta)^{-1/2}\widetilde{\psi}$, $n_R=(\xi\beta)^{-1}\widetilde{n}_R$ $R^{\rm 1D}=(\xi\beta/\tau)\widetilde{R}$, $(g^{\rm 1D},g_R^{\rm 1D})=(\hbar\xi\beta/\tau)(\widetilde{g},\widetilde{g}_R)$, $(\gamma,\gamma_R)=\tau^{-1}(\widetilde{\gamma},\widetilde{\gamma}_R)$, $P(x)=(1/\xi\beta\tau)\widetilde{P}(x)$, where $\xi=\sqrt{\hbar \tau /2m^*}$, while $\tau$ and $\beta$ are arbitrary scaling parameters, we can rewrite the above equation in the dimensionless form (from now on we omit the tildas for convenience) $$\begin{aligned} \label{psi} &i\frac{\partial \psi}{\partial t}=\left[ -\frac{\partial^2}{{\partial x}^2}+\frac{i}{2}\left(Rn_R-\gamma\right)+g|\psi|^2+g_Rn_R\right]\psi,\\ \label{nr} &\frac{\partial n_R}{\partial t}=P(x)-\left(\gamma_R+R|\psi|^2\right)n_R,\end{aligned}$$ In particular, we may choose the time scaling $\tau$ in such a way that $\gamma_R=1$ without loss of generality. The norms of both fields $N_\psi=\int|\psi|^2 dx$ and $N_R=\int n_R dx$ are scaled by the factor of $\beta$. Homogeneous pumping =================== We investigate the stability of a uniform condensate solution with the assumption of homogeneous pumping (${P=const}$) $$\begin{split} &\psi(x,t)=\psi_0 e^{-i\mu_0 t},\\ &n_R(x,t) = n_R^0, \label{steady-state} \end{split}$$ where $\mu_0$ is the chemical potential of the condensate. Substitution of the above equations into Eqs. (\[psi\]) gives the steady state solutions. For $P<P_{th}=\gamma\gamma_R/R$ only the noncondensed solution $\psi_0=0$ exists with $n_R ^0=P/\gamma_R$. Above threshold, the condensate density is ${|\psi_0|^2=(P/\gamma)-(\gamma_R/R)}$ with the chemical potential $\mu_0=g|\psi_0|^2+g_Rn_R ^0$ and the reservoir density $n_R ^0=\gamma/R$ [@wouters2007; @ostrovskaya2014]. ![(Colour online).  Evolution of the stationary state ${\psi_0=\sqrt{(P-P_{th})/\gamma}}$ with additional noise in the case of homogeneous pumping. The evolution of density in (a) the stable case and (b), (c) unstable case near and far from the critical point, respectively. Parameters are $R=1$, $g=1$, $g_R=2g$, $P/P_{th}=1.2$ and $\gamma/\gamma_R=0.1, 0.66, 4.5$ for (a), (b), (c), respectively. Bottom panel shows the first-order correlation functions $g^{(1)}(d)=\langle \psi^*(x)\psi(x+d) \rangle/\langle|\psi(x)|^2\rangle$ at ${t=t_{max}}$, calculated from about 500 samples. Solid lines show $g^{(1)}$ calculated with Eqs. (\[GP1\]) and dashed lines depict the corresponding results with quantum fluctuations included (see text). Corresponding parameters in physical units are: time unit $\tau=\gamma_R^{-1}=10\,$ps, length unit $\xi=3.4\,\mu$m, $g=3.4\,\mu$eV$\mu$m$^2$, $R=5.1\times 10^{-3}\,\mu$m$^2$ ps$^{-1}$ for $d=2\,\mu$m, $m^*=5\times 10^{-5} m_{\rm e}$, and $\beta=0.003$. TWA correlation lengths are (a) $\xi=70 \mu$m, (b) $\xi=17 \mu$m, and (c) $\xi=3 \mu$m. []{data-label="fig:stab-noise"}](evolution1.eps){width="1.\columnwidth"} ![(Colour online). (a) Stability limits of a static condensate created with homogeneous pumping (solid line) and for a uniform condensate flowing with momentum $p=1$ (dashed line). Parameters are ${R=1}$, ${g=1}$, ${g_R=2g}$, ${\gamma_R=1}$. The dash-dotted line corresponds to the crtical velocity condition $p=c_s=\sqrt{2 g |\psi|^2}$. Black solid line corresponds to the analytical condition (\[condition\]) for stability of the homogeneous solution. Blue points correspond to particular cases from Fig. \[fig:stab-noise\]. (b) Evolution of $|\psi(x,t)|^2$ for a uniform flow with homogeneous pumping and momentum $p=1$ in the motionally unstable regime, $P/P_{th}=1.5$, $\gamma/\gamma_R=0.17$.[]{data-label="fig:with-p"}](stability-1.eps){width="1.0\columnwidth"} The stability of the condensed state depends on the system parameters. In Fig. \[fig:stab-noise\] we show typical examples of dynamics of the condensate density in stable, weakly unstable, and strongly unstable cases. Periodic boundary conditions are imposed which corresponds to a ring-shaped geometry of the microwire. The initial state is perturbed by a white noise, which mimics classical (eg. thermal) density fluctuations in the initial polariton field. We checked that stability is independent of the amplitude of the noise provided that it is much smaller than the amplitude of the steady state. It is important to note that while in all three cases the population of low-momentum states is much larger than that below threshold, there is no long-range range order in the final states in Figs. \[fig:stab-noise\](b) and \[fig:stab-noise\](c), even in the mean field limit. As shown in Fig. \[fig:stab-noise\](d), the coherence length is of the order of the typical size of the structures visible on the figures. The reason for the reduced coherence is different than in the case of 1D quasicondensates of ultracold atoms [@Petrov2000; @Bagnato1991; @Bouchoule2007] since here we work in the mean-field limit, and the same instability is present also in the higher-dimensional versions of the model. The above modulational instability of the condensate was first reported in [@wouters2007], where it was named the “hole-burning effect”. The analytical condition for stability was derived in [@ostrovskaya2014] $$\label{condition} \frac{P}{P_{th}}>\frac{g_R}{g}\frac{\gamma}{\gamma_R}.$$ Stability limit for $g_R=2g$ is marked in Fig. \[fig:with-p\](a) with a thick black line. The above result has important practical consequences. As shown in several experiments, the lifetime of polaritons is typically much shorter than the exciton lifetime. This can be easilly understood since the cavity photon lifetime even in very high-Q cavities [@Snoke_BallisticMotion] is still shorter than the natural lifetime of excitons, which are of the order of hundreds of picoseconds. The reported exciton lifetimes were as high as $\gamma_R^{-1}=700$ ps [@Szczytko_ExcitonFormation; @Bloch_DynamicsElectronGas] or $\gamma_R^{-1}=300$ ps [@Bloch_1DAmplification]. Taking into account that typical polariton lifetimes $\gamma^{-1}$ are of the order of a few or a few tens of picoseconds, these values correspond to values of $\gamma/\gamma_R$ in the phase diagram Fig. \[fig:with-p\](a) where stability can be achieved only at a very high pumping powers. Here, we suggest how this fact can be reconciled with the emergence of large coherence length in experiments. It was pointed out that the lifetime of active excitons, that is excitons that can scatter directly to the condensate, can be much shorter than the average lifetime of excitons in the system [@Deveaud_VortexDynamics]. Since these excitons become dressed with light field at low momenta, their lifetime can be reduced due to a nonzero photonic Hopfield coefficient. However, there is no fundamental reason why the lifetime should become shorter than the photon lifetime, which is necessary for stability at [*any power*]{} $P>P_{th}$ in Eq. (\[condition\]). On the other hand, we show that this condition can be relaxed in the case of a finite pumping spot. We show that in the case of inhomogeneous pumping profiles large coherence lengths can emerge for relatively large values of $\gamma/\gamma_R$. We note that while the above calculations take into account only the classical fluctuations in the initial polariton field, it is possible to include the effect of quantum fluctuations on the level of classical fields approximation [@Ciuti2005; @WoutersSavona2009; @Matuszewski2014]. In the truncated Wigner approximation (TWA), the quantum field is simulated by an ensemble of realizations of the Gross-Pitaevskii equations in the form similar to (\[psi\])-(\[nr\]), with the addition of a stochastic term. The extended version of Eq. (\[psi\]) reads $d\psi= (\dots) dt + dW$, where $dW$ is a complex stochatic variable with [@Ciuti2005; @WoutersSavona2009; @Matuszewski2014] $$\begin{aligned} \label{dW} \langle dW(x) dW(x')\rangle &= 0\,, \\ \nonumber \langle dW(x) dW^*(x')\rangle &= \beta\frac{dt}{2 \Delta x} (R n_R + \gamma_C) \delta_{x,x'}\,,\end{aligned}$$ reflecting the quantum noise due to the particles entering and leaving the condensate. In Fig. \[fig:stab-noise\] (bottom panel, dashed lines) we show the effect of quantum fluctuations on the correlation functions. While quantum fluctuations lead to decay of correlation functions over long distance [@Chiocchetta2013; @Gladilin2013; @Bloch_ExtendedCondensates], this effect is overwhelmed by the instability in the unstable case (c) (black lines), which marks a dramatic reduction of the correlation length. Stability of a uniform flow --------------------------- Another important issue in the case of uniform pumping is the stability of a condensate with a finite momentum. In this case the stationary solution is $$\psi(x,t)=\psi_0 e^{ipx-i\mu_0 t}, \label{steady-state-momentum}$$ where $p$ is the condensate momentum and ${\mu_0=g\psi_0^2+g_Rn_R^0+p^2}$ the chemical potential. Such polariton flow can occur naturally eg. in the presence of a spatially varying exciton-photon detuning which generates a potential gradient, or due to repulsive interactions with reservoir excitons which are generated by the pump [@Bloch_ExtendedCondensates]. Similarly to the case of a static condensate [@wouters2007; @ostrovskaya2014], the modulational stability of the steady-state solution (\[steady-state-momentum\]) can be investigated within the Bogoliubov-de Gennes approximation [@Wouters_CriticalVelocity; @Ostrovskaya_PersistenCurrents]. Small fluctuations around the steady state have the form \[bogo-uv\] &=\_0 e\^[ipx-i\_0 t]{},\ n\_R(x,t)&=n\_R \^0, where $\omega_k$ is the frequency of the mode with the wavenumber $k$, and $a_k,b_k,c_k$ are small fluctuations. Substituting Eqs. (\[bogo-uv\]) into the system of Eqs. (\[psi\]) and keeping linear terms only, we get the standard eigenvalue problem $\mathcal{L}_k \mathcal{U}_k=\omega_k \mathcal{U}_k$, where ${\mathcal{U}_k=(a_k, b_k, c_k)^T}$ and $$\begin{aligned} &\mathcal{L}_k =\\ \nonumber &= \begin{pmatrix} g\psi_0^2+2kp+k^2 & g{\psi_0}^2 & \frac{i}{2}R+g_R\\ -g\psi_0^2 & -g\psi_0^2+2kp-k^2 & \frac{i}{2}R-g_R \\ % \vdots & \vdots & \ddots & \vdots \\ -i\gamma \psi_0^2 & -i\gamma \psi_0^2 & -i\left(\gamma_R+R\psi_0^2\right) \end{pmatrix} \label{matrix1}\end{aligned}$$ The existence of an eigenvalue $\omega_k$ with a positive imaginary part marks the dynamical instability of the flow. We solved the above eigenvalue problem numerically to obtain the stability limits in parameter space. The results are presented in Fig. \[fig:with-p\], where the stability of the flow with a finite momentum are marked with a dashed line. Similarly to [@Ostrovskaya_PersistenCurrents], the region of stability of solutions is further decreased with respect to the static case. There is a certain area for small $P/P_{th}$ and $\gamma/\gamma_R$ in which the condensate becomes unstable. In this region, the dynamics of the instability is qualitatively different to the one presented in Fig. \[fig:stab-noise\]. Instead of chaotic dynamics, regular waves are created by the unstable flow, as shown in Fig. \[fig:with-p\](b). Here, instead of periodic boundary conditions, we used boundary conditions that provide a flow of particles with a prescribed momentum through the borders of the computational window: $\partial \psi/\partial x=ip\psi$. The origin of the above modulational instability of the flow is clearly due to the movement of polaritons with respect to the practically immobile reservoir excitons. This instability is absent in the standard version of the polariton model which does not include a separate equation for the reservoir [@Berloff_VortexLattices; @Wouters_CriticalVelocity], since in this case the system is Galilean invariant. However, it can be recovered by inclusion of frequency dependent pumping [@Wouters_CriticalVelocity], which resembles momentum dependence of scattering from the immobile reservoir. Notably, similar to [@Wouters_CriticalVelocity], the modulational instability does not correspond to $p$ being higher than the critical velocity, see dash-dotted line in Fig. \[fig:with-p\](a). The latter conditon defines the limit on stability of the flow with respect to scattering against defects. Effects of polariton energy relaxation ====================================== ![(Colour online). Comparison of stability regions with and without the energy relaxation term included. In both cases $p=0$. Red dash-dot lines mark out the stable region for $A=0.5$, which partially overlaps with the stable region for $A=0$. Parameters are ${R=1}$, ${g=1}$, ${g_R=2g}$, ${\gamma_R=1}$, $\gamma\in[0,5]$.[]{data-label="fig:with-A"}](stability-2.eps){width="0.89\columnwidth"} One of the drawbacks of the Gross-Pitaevskii model based on Eqs. (\[psi\]) is that the relaxation of polariton energy is not taken into account. Several experiments [@Bloch_ExtendedCondensates; @Bloch_1DAmplification; @Bloch_GapStates] have shown that energy relaxation (or thermalization) may play an important role in the dynamics. It is important to investigate whether it can have a stabilizing effect on the condensate. To estimate the effect of relaxation we follow Refs. [@Bloch_1DAmplification; @Bloch_GapStates] by adding a phenomenological relaxation term to the condensate evolution equation (\[psi\]) $$\begin{aligned} \label{eq:bloch} \frac{\partial \psi}{\partial t}&=&\\ \nonumber &=&\left[ \left(i+A\right)\frac{\partial^2}{{\partial x}^2}+\frac{1}{2}\left(Rn_R-\gamma\right)-ig|\psi|^2-ig_Rn_R\right]\psi,\end{aligned}$$ where the real coefficient $A$ corresponds to the energy relaxation in the condensate. Following the steps of the previous section, we obtain the matrix which describes the Bogoliubov-de Gennes excitations in the static ($p=0$) case $$\begin{aligned} &\mathcal{L}_k = \\ \nonumber &= \begin{pmatrix} g\psi_0^2+i\alpha k^2 & g{\psi_0}^2 & \frac{i}{2}R+g_R\\ -g\psi_0^2 & -g\psi_0^2+i \alpha k^2 & \frac{i}{2}R-g_R \\ % \vdots & \vdots & \ddots & \vdots \\ -i\gamma \psi_0^2 & -i\gamma \psi_0^2 & -i\left(\gamma_R+R\psi_0^2\right) \end{pmatrix} \label{matrix1}\end{aligned}$$ where $\alpha=i+A$. The correction to the stability diagram with the relaxation term included is shown in Fig. \[fig:with-A\], where we compare the stability limits obtained numerically by solving (\[matrix1\]) with the analytical condition for the relaxation-free case (\[condition\]). It is clear that even the large relaxation rate ($A=0.5$) cannot lead to the stabilization of the condensate in a large parameter range, although the region of stability is increased with respect to the $A=0$ case. We note that excitation spectrum of a formally similar hybrid Boltzmann-Gross Pitaevskii model was calculated recently [@Malpuech_Hybrid]. In this model, steady states were found to be dynamically (modulationally) stable in the whole parameter range. This stability is a consequence of the assumption of an ideally thermalized reservoir, which translates into immediate response of the reservoir to the change of the condensate density. This includes the response of the reservoir density distribution $n_R(x,t)$. To describe the instability shown in Fig. \[fig:stab-noise\], it is important to take into account that the relatively heavy excitons remain practically immobile on a short time scale, so that their density distribution can become out of equilibrium (although the reservoir may be thermalized locally). Inhomogeneous pumping ===================== ![(Colour online). Evolution of the polariton density for Gaussian pumping ${P(x)=P_{\rm max} \exp(-{x^2}/{W^2})}$ ($A=0$). Frames (a) and (b) show condensate densities in the coherent and incoherent cases, respectively. Parameters are $R=1$, $\gamma/\gamma_R=1.5, 5 $ for (a) and (b), respectively, $g=0.38$, $P_{max}/P_{th}=1.7$, $W=44.7$. Frames below show examples of coherent nonstationary states: (c), (d) oscillating wavepackets with parameters $R=2.5$, $\gamma/\gamma_R=0.15$, $g=3$ and $R=1$, $\gamma/\gamma_R=0.7$, $g=0.38$, $P_{max}/P_{th}=1.8$, $W=15.8$, respectively, and (e) a breather with parameters $R=1$, $\gamma/\gamma_R=0.7$, $g=0.38$, $P_{max}/P_{th}=1.6$, $W=15.8$.[]{data-label="fig:gauss-cos"}](soli-gauss.eps){width="1.\columnwidth"} ![(Colour online). Stability regions for Gaussian pumping profiles. The lines mark out the coherent regions for narrower pumping of width $W=15.8$ (dash-dotted lines) and for wider pumping of width $W=44.7$ (dashed lines). Black solid line corresponds to the analytical stability condition of a homogeneous condensate [@ostrovskaya2014]. Blue points correspond to particular cases from Fig. \[fig:gauss-cos\]. Parameters are ${R=1}$, ${g=0.38}$, ${g_R=2g}$, and ${\gamma_R=1}$. For the choice of scaling parameters as in Fig. \[fig:stab-noise\], the width of the pumping is $W=54\,\mu$m (dash-dotted lines) and $W=152\,\mu$m (dashed lines).[]{data-label="fig:with-gauss"}](stability-Pmax.eps){width="0.89\columnwidth"} Experiments with nonresonantly pumped polariton condensates are performed using inhomogeneous, typically Gaussian-shaped, pumping beams. In this section we investigate how the shape of the pump can influence the stability of the condensate. In Fig. \[fig:gauss-cos\] we show examples of dynamics for various parameters of the system with a Gaussian pumping profile, ${P(x)=P_{\rm max} \exp(-{x^2}/{W^2})}$. The initial state is a small white noise in the polariton field $\psi(x)$. The frames (a) and (b) show examples of stable and unstable dynamics, corresponding to Fig. \[fig:stab-noise\](a) and (c). There is also a number of other possible nonstationary states, as shown in Fig. \[fig:gauss-cos\](c)-(e), including the oscillating wavepackets and breathers. These states, despite the complex dynamics, display correlation length that is comparable with the size of the Gaussian pump. For this reason, we classify these states as “coherent” in contrast to the “incoherent” unstable state of Fig. \[fig:gauss-cos\](b). We note that nonstationary condensate states have been previously shown to emerge in two-dimensional models with inhomogeneous pumping [@Berloff_VortexLattices; @Berloff_PatternFormation]. In Fig. \[fig:with-gauss\] we show the regions of parameter space where coherent (stationary or nonstationary) states exist. The limits of the regions were determined numerically by solving the Gross-Pitaevskii equation (\[psi\]). Clearly, the region of coherence in the case of inhmogeneous pumping is much larger than in the homogeneous case of Fig. \[fig:with-p\]. Moreover, it expands significantly as the pump size is decreased. The reduction of the pumping spot appears to be a very effective way to stabilize condensates even at high ratios of $\gamma/\gamma_R$. Condensates with $\gamma/\gamma_R = 3.5$ are fully coherent at all pumping powers already at a relatively wide pumping spot with $W=15.8$. We note that the threshold value of $P_{max}$ for the appearence of the condensate in the case of inhomogeneous pumping is higher than $P_{th}$ [@Ostrovskaya_DissipativeSolitons]. However, since our pumping profiles are relatively wide, this difference cannot be seen in the area covered by Fig. \[fig:with-gauss\] which begins at $P/P_{th}=1.2$. conclusions =========== In conclusion, we investigated the dynamical stability and its relation to spatial coherence properties of one-dimensional exciton-polariton condensates under nonresonant pumping. In the case of spatially homogeneous pumping, we found that the instability of the steady state leads to a significant reduction of the condensate coherence. This instability is predicted to occur in the physically relevant case when the loss rate of the exciton reservoir is lower than the loss rate of the exciton-polaritons. Since the experiments with inhomogeneous optical pumping reported large coherence lengths [@Kasprzak_BEC; @Yamamoto_SpatialCoherence; @Bloch_ExtendedCondensates; @Hofling_1DcoherenceInteractions; @Yamamoto_RMP], we considered two effects that can potentially lead to the stabilization of condensates. These are the effect of polariton energy relaxation and the inhomogeneous pumping profiles. We found that while the former has little effect on the stability, the latter is very effectve in stabilizing the condensate which results in a large coherence length. It is noteworthy that in the two-dimensional experiment with a top-hat shaped pumping profile, which is the closest to the homogeneous profile, strong internal dynamics and vortex creation were observed [@Yamamoto_VortexPair], which evidences the existence of a nonstationary state. However, we believe that further experiments, especially with almost-homogeneous or ring-shaped pumps, are necessary to determine the stability limits in the parameter space. Such experiments would also provide verification of the open-dissipative Gross-Pitaevskii model (\[GP1\]) and its modifications widely used in the literature, and allow for the determination of their phenomenological parameters. N.B. and M. M. acknowledge support from the National Science Center grant DEC-2011/01/D/ST3/00482. E.A.O. acknowledges support by the Australian Research Council (ARC).

--- abstract: | We study the long-time behavior of solutions to a measure-valued selection-mutation model that we formulated in [@CLEVACK]. We establish permanence results for the full model, and we study the limiting behavior even when there is more than one strategy of a given fitness; a case that arises in applications. We show that for the pure selection case the solution of the dynamical system converges to a Dirac measure centered at the fittest strategy class provided that the support of the initial measure contains a fittest strategy; thus we term this Dirac measure an Asymptotically Stable Strategy (ASS). We also show that when the strategy space is discrete, the selection-mutation model with small mutation has a locally asymptotically stable equilibrium that attracts all initial conditions that are positive at the fittest strategy.\ [**Key Words:**]{} Evolutionary game theory, selection-mutation models, cone of nonnegative measures, long time behavior, [persistence theory]{}, permanence, survival of the fittest, asymptotically stable strategy, [Lyapunov functions]{}.\ [**AMS Subject Classification:**]{} 91A22, 34G20, 37C25, 92D25. author: - | Azmy S. Ackleh$^{\ddag}$, John Cleveland$^{\dag}$ and Horst R. Thieme$^*$\ $^\ddag$Department of Mathematics\ University of Louisiana at Lafayette\ Lafayette, Louisiana 70504-1010\ $^\dag$Department of Mathematics\ University of Wisconsin\ Richland Center, WI 53581\ $^*$School of Mathematical and Statistical Sciences\ Arizona State University\ Tempe, Arizona 85787-1804 title: | **[Selection-Mutation Differential Equations:\ Long-Time Behavior of Measure-Valued Solutions]{}** --- Introduction {#sec:intro} ============= A significant part of evolutionary game theory (EGT) focuses on the creation and study of mathematical models that describe how the strategy profiles in games change over time due to mutation and selection (replication) [@Nowak; @BrownVincent]. In [@CLEVACK] we defined an evolutionary game theory (EGT) model as an ordered triple $(Q,\mu(t),F(\mu(t)))$ subject to: $$\label{mconstraint} \frac{d}{dt}\mu(t)(E)=F(\mu(t))(E), \text{ for every} ~~E \in \mathcal{B}(Q).$$ Here $Q$ is the strategy (metric) space, $\mathcal{B}(Q)$ is the $\sigma$-algebra of Borel subsets of $Q$, $\mu(t)$ is a time dependent family of nonnegative finite Borel measures on $Q$, and $F$ is a density dependent vector field such that $\mu$ and $F$ satisfy equation . For a Borel subset $E$ of $Q$, $\mu(t) (E)$ denotes the measure $\mu(t)$ applied to $E$ [and can be interpreted as the number of individuals at time $t$ that carry a strategy from the set $E$.]{} We also formulated the following selection-mutation EGT model as a dynamical system $\phi(t,u, \gamma)$ on the state space of finite nonnegative Borel measures under the weak$^*$ topology with $\mu(t) = \phi(t,u,\gamma)$ and $\bar \mu(t) = \mu(t)(Q)$: $$\left\{\begin{array}{rl} \label{M1} \displaystyle \frac{d}{dt}{\mu}(t)(E) = & \displaystyle \int_Q {B}\big(\bar \mu(t), q\big) \gamma( q)(E)\mu(t)(d q) \; -\; \int_E {D}\big(\bar \mu(t), q\big) \mu(t)(d q) , \\ = & {F} (\mu(t), \gamma)(E), \\ \mu(0)=& u. \end{array}\right.$$ Here $B(\bar \mu(t),q)$ and $D(\bar \mu(t),q)$ represent the reproduction [(replication)]{} and mortality rates of individuals carrying strategy $q$ when the total population size is $\bar \mu(t)= \mu(t)(Q)$. The probability kernel $\gamma(q)(E)$ represents the probability that an individual carrying strategy $q$ produces [offspring]{} carrying strategies in the Borel set $E$. We call $\gamma$ a [*mutation kernel*]{}. The purpose of this paper is to complement the well-posedness theory established in [@CLEVACK] with a study of the long-time behavior of solutions to the model . It is well known that the solutions of many such models constructed on the state space of continuous or integrable functions converge to a Dirac measure [concentrated]{} at the fittest [strategy or trait]{} [@AFT; @AMFH; @calsina; @CALCAD; @GVA; @P; @GR1; @GR2; @ThiYang]. In [@P ch.2], these measure-valued limits are illustrated in a biological and adaptive dynamics environment. This convergence is in the weak$^* $ topology [@AMFH]. Thus, the asymptotic limit of the solution is not in such state spaces; it is a measure. Some models (e.g. [@AFT], [[@ThiYang]]{}) have addressed this problem. In [@AFT], the authors formulated a [*pure*]{} selection model [ on the space of finite signed measures with density dependent birth and mortality functions and a 2-dimensional [strategy]{} space]{}. They discussed existence-uniqueness of solutions and studied the long term behavior of the model. Here, we substantially generalize the results in that paper in several directions: - We model selection and mutation, and we allow for the selection-mutation kernel to be a family of measures (thus simultaneously treating discrete and continuous strategy spaces). - We consider more general nonlinearities in the rates, and thus the results apply to a wider class of models. - We allow for more than one fittest strategy. To motivate our attempt of allowing more than one fittest strategy, recall that in [@AMFH] the authors considered the following logistic growth with pure selection (i.e., strategies replicate themselves exactly and no mutation occurs) model: $$\frac{d}{dt} x(t,q) = x(t,q) ( q_1 -q_2 \bar x(t)), \label{logiseq}$$ [where $\bar x(t) = \int_Q x(t,q) dq$]{} is the total population, $Q \subset \text{int}(\mathbb{R}_+^2)$ is a rectangle and the state space is the set of continuous real valued functions $C(Q)$. Each $ q=(q_1, q_2) \in Q$ is a pair where $q_1$ is an intrinsic replication rate and $q_2$ is an intrinsic mortality rate. The fittest strategy was defined as the one with the highest replication to mortality ratio, $\max_Q\{q_1/q_2\}$, which is unique for this choice of $Q$. Utilizing the uniqueness of the fittest strategy, the authors show that the solution converges to the Dirac mass centered at [the strategy]{} with this ratio. In Figure 1, we present two examples of strategy spaces $Q\subset \text{int}(\mathbb{R}_+^2)$: [one that is similar to that considered in [@AMFH] and has a unique fittest strategy (left) and another that has a continuum of fittest strategies (right).]{} \[Fig1\] ![ Two examples of strategy spaces.[]{data-label="strategyspace"}](strat_space00.eps){width="10cm" height="10cm"} This paper is organized as follows. In section \[sec:ass\], we provide some background material and make assumptions on the model parameters. In section \[sec:asymp-gen\], we provide some permanence and persistence results. In section \[sec:pure-sel\], we study the asymptotic behavior for the pure selection kernel, and in section \[sec:dir-mut\], we study the asymptotic behavior for directed mutation kernels. In section \[sec:disc\], we consider discrete strategy spaces and establish asymptotic behavior results for this case. In section \[sec:conclud\], we provide concluding remarks. Assumptions and background material {#sec:ass} =================================== In this section, we state assumptions and define notation that we will use throughout the paper, and we recall the main well-posedness result established in [@CLEVACK]. Here $ \mathcal{M}= \mathcal{M}(Q)$ are the finite signed Borel measures on $Q$, a compact metric space. $\mathcal{M}_{V,+}$ represents the positive cone under the total variation topology, and $\mathcal{M}_{w,+}$ represents the positive cone under the weak\* topology. Let $\mathcal P_w= \mathcal P_w(Q) $ denote the probability measures under the weak\* topology and $ C(Q,\mathcal{P}_w)$ denote the continuous $\mathcal{P}_w$ valued functions on $Q$ with the topology of uniform convergence. The mutation kernel $\gamma$ is assumed to be an element of $C(Q,\mathcal{P}_w)$, in other words, $\gamma $ has the Feller property [(see [@LaTh] and the references therein)]{}. Also, for any time dependent mapping, $f(t)$, we let $f^\prime (t)=\frac{d}{dt} f(t)$. \[re:measures-weak-top\] The following hold: - ${{\cal M}}_{w+}$ is a closed convex subset (actually cone) of the locally convex topological vector space ${{\cal M}}_w$ in which all bounded closed subsets are compact. - There exists a norm $p$ on ${{\cal M}}$ such that, for each bounded closed subset of ${{\cal M}}_w$, the induced metric space is complete and topologically equivalent to the topological space induced by the weak$^*$ topology. The norm $p$ makes ${{\cal M}}_+$ into a complete metric space which is topologically equivalent to ${{\cal M}}_{w+}$. - $C(Q, {{\cal P}}_w)$ is a complete convex subset of the normed vector space $C(Q, {{\cal M}}_p)$ where ${{\cal M}}_p$ is ${{\cal M}}$ equipped with the norm $p$. Since $\mathcal{M}_{V}$ is the norm dual of $C(Q)$, the first statement is a consequence of the Alaoglu-Bourbaki theorem. Since $Q$ is a compact metric space, its topology has a countable base. So $C(Q)$ with the supremum norm is separable (Theorem 7.6.3 in [@Bau]), and so is its unit ball. Choose a dense subset $S= \{f_k; k \in \N\}$ of the unit ball of $C(Q)$. For each $k\in \N$, define the seminorm $p_k$ on ${{\cal M}}(Q)$ by $$\label{eq:seminorm} p_k ( \nu) = \Big | \int_Q f_k(q) \nu(dq) \Big |.$$ The following standard construction defines a norm $p$ on ${{\cal M}}(Q)$, $$\label{eq:norm-weak} p(\nu) = |\nu(Q)| + \sum_{k=1}^\infty 2^{-k} p_k (\nu).$$ [This norm induces the weak$^*$ topology on every bounded subset of ${{\cal M}}_w$ [[@ALI Thm.6.30]]{} and on ${{\cal M}}_{w+}$. Further, any sequence that is a Cauchy sequence with respect to $p$ and bounded in ${{\cal M}}_w$ is a weak$^*$ Cauchy sequence and so converges in the weak$^*$ topology and thus also with respect to $p$. Any sequence in ${{\cal M}}_+$ that is a Cauchy sequence with respect to $p$ is automatically bounded in total variation and converges with respect to $p$ by the same arguments.]{} The topology on $C(Q,\mathcal{P}_w)$ is induced by the norm $$\label{eq:metric-mutation} \|\gamma\| = \sup_{q \in Q} p ( \gamma(q)), \qquad { \gamma \in C(Q, {{\cal M}}_p).}$$ Standard considerations show that $C(Q,\mathcal{P}_w)$ is a closed convex subset of the [normed vector space]{} $C(Q, {{\cal M}}_p)$. \[re:semicont2\] If $f: Q \to \R$ is bounded below (above) and lower (upper) semicontinuous, then the mapping $$\begin{split} \nu & \mapsto \int f d\nu, \\ \cal{M_+} & \to [-\infty, \infty], \end{split}$$ is lower (upper) semicontinuous. We will demonstrate the result for lower semicontinuous functions that are bounded below; the other case follows from this one by considering $-f$. Let $\nu_m \rightarrow \nu$ in the weak\* topology. Then there exists a sequence $ ( f_n)$ with each $f_n $ being Lipschitz continuous such that $f_n(x) \uparrow f(x)$ for $x \in X $ [@ALI Thm.3.13]. From $$\int f_n d \nu_m \le \int f d\nu_m \mbox { and } \int f_n d\nu_m \stackrel{m \to \infty}{\longrightarrow} \int f_n d\nu,$$ we see that $ \int f_n d\nu \le \liminf_{m} \int f d\nu_m $ for each $n.$ We use the Monotone Convergence Theorem and obtain $$\int f d\nu = \lim_{n \rightarrow \infty} \int f_n d\nu \le \liminf_{m} \int f d\nu_m .$$ By Lemma \[re:measures-weak-top\], this sequential characterization of lower semicontinuity is equivalent to the topological one (inverse images of intervals $(b, \infty]$, $b \in \R$, are open sets), and the function $ \nu \mapsto \int f d\nu $ is lower semicontinuous. \[re:semicont\] Let $E$ be an open (closed) subset of the compact metric space $Q$. - Then the function $\rho: {{\cal M}}_{w,+} \to \R_+$, $\rho (\nu) = \nu(E)$, is lower (upper) semicontinuous. - Also the function $\psi:C(Q,\mathcal{P}_w)\times Q \to \R_+$ defined by $\psi(\gamma, q) = \gamma(q)(E)$ is lower (upper) semicontinuous. We first notice that the characteristic function of an open set is lower semicontinuous and the characteristic function of a closed set is upper semicontinuous. If a set is both open and closed, then its characteristic function is continuous. Hence (a) is immediate from Lemma [\[re:semicont2\]]{}. Likewise (b) follows once we notice $$C( Q, {{\cal P}}_w) \times Q \rightarrow \cal{M}_{w+} \rightarrow \mathbb{R_+}$$ given by $$( \gamma, q) \mapsto \gamma(q) \mapsto \int f d \gamma(q)$$ is a composition of a continuous and a lower (upper) semicontinuous function. Birth and Mortality Rates ------------------------- Concerning the birth rate, $B(s,q)$, and the mortality rate, $ D(s,q)$, [where $s \in [0,\infty)$ is the total population size and $q \in Q$ a strategy (trait)]{}, we make assumptions similar to those used in [@AFT; @CLEVACK]: - [$B: [0,\infty) \times Q \rightarrow [0,\infty)$ is continuous, and $B(s,q)$ is locally Lipschitz continuous in $s\ge 0$, uniformly with respect to $q \in Q$, and nonincreasing in $s \ge 0$.]{} - [$D: [0,\infty) \times Q \rightarrow [0,\infty)$ is continuous, and $D(s,q)$ is locally Lipschitz continuous in $s \ge 0$, uniformly with respect to $q \in Q$, and nondecreasing in $s \ge 0$ ]{} and $ \inf_{q \in Q} {D(0, q)} =\varpi >0 $. (This means that there is some inherent, [density unrelated, mortality.]{}) The [*reproduction number*]{} of strategy $q\in Q$ at population size $s$ is defined by $$\label{eq:rep-num} \cR( s, q) = \frac{B(s,q)}{D(s,q)}.$$ The [*basic reproduction number*]{} of strategy $q$ is defined by $$\label{eq:rep-num-basic} \cR_0(q) = \cR(0,q), \qquad q \in Q.$$ The following additional assumption is made. - For each $q \in Q$ with $\cR_0(q) \ge 1$, there exists a unique $K(q) \ge 0$ such that $\cR(K(q), q)$ $ =1$. - [If $\cR_0(q) < 1$, we define $K(q) =0$.]{} [The number $K(q)$ in (A3) is the [*carrying capacity*]{} of the environment if everyone in the population were subject to strategy $q$.]{} Since (A1)-(A3) imply that the function $K(\cdot)$ is continuous, it has a maximum and a minimum on the compact set $Q$. We define $$\label{Max} K^\diamond = \max_{ q \in Q} K(q)$$ and $$\label{Min} k_{\diamond}= \min_{ q \in Q} K(q).$$ Let $Q^\di$ be the subset of $Q$ where the maximal carrying capacity is taken, $$\label{eq:Q-max} Q^\di = \{q \in Q; K(q) = K^\di\}.$$ Then $Q^\di$ is a nonempty compact subset of $Q$ and $K(q) = K^\di$ for all $q \in Q^\di$. Further, if $K^\di >0$, $$\label{eq:carry-cap-more} \left . \begin{array}{rl}\cR(K^\di, q) =1 & \\ \cR(x,q) >1, & 0 \le x< K^\di \\ \cR(x, q) < 1, & x > K^\di \end{array} \right \} \quad q \in Q^\di.$$ Main Theorem from [@CLEVACK] ---------------------------- The following is the main well-posedness theorem taken from [@CLEVACK]. \[main\] Assume that (A1)-(A2) hold. There exists a continuous dynamical system $({\cal M}_{w,+ }, C(Q, {\cal P}_w),\phi)$ where $\phi: {\mathbb {R}}_+ \times{\cal M} _{w,+} \times C(Q, {\cal P}_w) \to {\cal M}_{w,+} $ satisfies the following: 1. The mapping $(t,u,\gamma) \mapsto \phi(t,u, \gamma)$ is continuous. 2. For fixed $ u, \gamma $, the mapping $t \mapsto \phi(t, u,\gamma)$ is continuously differentiable in total variation, i.e., $\phi(\cdot, u, \gamma): {\mathbb {R}}_+ \to {\cal M}_{V,+}$ is continuously differentiable. 3. For fixed $ u, \gamma $, the mapping $t \mapsto \phi(t, u,\gamma)$ is the unique *solution* $\mu$ to $$\left\{\begin{array}{rl} \label{M} \displaystyle {\mu}'(t)(E) = & \int_Q {B}(\bar \mu(t), q) \,\gamma( q)(E)\,\mu(t)(d q) \; - \; \displaystyle \int_E {D}(\bar \mu(t), q) \,\mu(t)(d q) \\ = & {F} (\mu(t), \gamma)(E), \\ \mu(0)= &u, \end{array}\right.$$ where $\bar \mu (t) = \mu(t) (Q)$. Asymptotic Results with an Arbitrary [Mutation Kernel]{} $\gamma$ {#sec:asymp-gen} =================================================================== In this section we begin studying the long time behavior with an arbitrary [mutation]{} kernel $\gamma \in C(Q, {\cal P}_w)$. In particular, we provide sufficiency for permanence and uniform persistence.\ \[re:pos-pres\] Let (A1)-(A2) hold and let $\mu$ be a solution of (\[M\]). Then the following holds. - If $E$ is a Borel subset of $Q$ and $\mu(0)(E) > 0$, then $\mu(t) (E) > 0$ for all $t \ge 0$. Assume in addition that $B(x,q) > 0$ for all $x \ge 0$ and $q \in Q$. Then the following holds. - If $E$ and $\tilde E$ are Borel subsets of $Q$ with $\inf_{q\in {\tilde E}} \gamma(q)(E) >0$ and $\mu(t)({\tilde E}) >0$ for all $t >0$, then $\mu(t)(E) > 0$ for all $t >0$. [For a solution $\mu$ of (\[M\]) and $\bar \mu(t) = \mu(t)(Q)$, set]{} $$\label{eq:extreme-vitals} b(t) = \min_{q\in Q} B(\bar \mu(t),q), \qquad \theta(t) = \max_{q \in Q} D(\bar \mu(t), q), \qquad t \ge 0.$$ Since $B$ and $D$ are continuous and $Q$ compact, $b$ and ${\theta}$ are continuous. We have the differential inequalities, $$\mu'(t) (E) \ge b(t) \int_Q \gamma(q)(E) \mu(t)(dq) - \theta(t) \mu(t) (E) \ge - \theta(t) \mu(t)(E).$$ The second inequality can be integrated as $$\mu(t) (E) \ge \mu(0)(E) \exp \Big ( - \int_0^t \theta(s) ds \Big),$$ which provides the first statement. Let [$\tilde E$]{} also be a Borel subset of $Q$. Then $$\mu'(t) (E) \ge b(t) \inf_{q \in {\tilde E}} \gamma(q)(E) \mu(t)({\tilde E}) - \theta(t) \mu(t) (E).$$ This can be integrated as $$\mu(t)(E) \ge \inf_{q \in {\tilde E}} \gamma(q)(E) \int_0^t b(r) \mu(r)({\tilde E}) \exp \Big (- \int_r^t \theta(s)ds\Big ) dr.$$ This implies the second statement. Uniform eventual boundedness ---------------------------- A system $ \frac{dx}{dt} =F(x)$ is called dissipative and its solution uniformly eventually bounded, if all solutions exist for all forward times and if there exists some $c>0$ such that $$\limsup_{t \rightarrow \infty}||x(t)||< c$$ for all solutions $x$ [@Thi03 pg. 153]. Next, we will show that the solutions of (\[M\]) are uniformly eventually bounded. Recall that $\bar \mu(t)= \mu(t)(Q)$ denotes the total population size at time $t$. \[BS\](Bounds for Solution) Assume that (A1)-(A3) hold. Then, for any solution $\mu$ of and $\bar \mu = \mu( \cdot)(Q)$, we have the following: $$\label{limbound} \min\{ k_{\diamond}, \bar \mu(0)\} \le \bar \mu(t) \le \max \{\bar \mu(0), K^{\diamond} \}, \qquad \text{for all }~ t ~\geq 0,$$ and $$\label{limsupbound} k_{\diamond} \leq \liminf_{t \to \infty} \bar \mu (t) \leq \limsup_{t \to \infty} \bar \mu(t) \le K^{\diamond} .$$ Hence, if $k_{\diamond} >0$ then the population is permanent. It seems at first glance that $k_{\diamond} >0$ is too restrictive of an assumption for proving persistence, i.e., $ \liminf_{t \rightarrow \infty} \mu(t)(Q) >0 .$ However, if $k_{\diamond} =0$ and $ \gamma(\hat q) = \delta_{\mathfrak{q}}$ (i.e., individuals with any strategy only reproduce individuals with strategy $\mathfrak{q}$), then it is an exercise to show that the model converges to the zero measure even in setwise convergence. We first prove the rightmost inequalities, i.e., those for $\max$ and $\limsup$. Let $\bar \mu (t)=\mu(t)(Q)$. First notice that $$\label{TOTPOPBOUND} \begin{array}{lll} \bar \mu'(t) &=& \displaystyle \int_{Q}\Bigl[B(\bar \mu(t),q)-D(\bar \mu(t),q)\Bigr]\mu(t)(d q) \\[3mm] &=& \displaystyle \int_{Q}\Bigl[\cR(\bar \mu(t),q )-1 \Bigr]D(\bar \mu (t),q)\mu(t)(d q). \end{array}$$ For the second inequality in we have cases. First assume $K^{\diamond} =0$. Using and the fact that for every $ q \in Q $, $ \cR(\cdot,q )$ is nonincreasing we see that $\bar \mu'(t) \leq 0$, for all $t \geq 0 $. Now assume $0<K^{\diamond} < \infty $. Starting from , we see that if $\bar \mu(t)> K^{\diamond} $ then $ \bar \mu' \leq 0$; therefore it follows from basic analysis or [@Thi03 Lemma A.6] that $$\label{limbound1} \bar \mu(t)\leqslant \max\{ K^{\diamond}, \bar \mu(0)\}, \text { for all } t \ge~ 0 .$$ In particular, $\bar \mu$ is bounded and we can define $$\bar \mu^\infty := \limsup_{t \to \infty} \bar \mu(t).$$ By the fluctuation lemma [@HHG][@Thi03 Prop.A.22], there exists a sequence $(t_j)$ such that $t_j \to \infty$, $\bar \mu(t_j) \to \bar \mu^\infty$ and $\bar\mu'(t_j) \to 0$ as $j \to \infty$. By (\[TOTPOPBOUND\]) and the continuity of $\cR$ and $D$ $$0 = \lim_{j \to \infty} \int_Q [ \cR(\bar \mu^\infty,q) -1 ] D(\bar \mu^\infty, q ) \mu(t_j) (dq).$$ Suppose that $\bar \mu^\infty > K^\di$. Then there exists some $\xi \in [0,1)$ and $\delta >0$ such that $\cR(\bar \mu^\infty, q) \le \xi$ and $D(\bar \mu^\infty, q ) \ge \delta$ for all $q \in Q$. So $$0 \le (\xi-1) \delta \bar \mu^\infty <0,$$ a contradiction. Exchanging $\liminf$ arguments for $\limsup$ arguments, the lefthand inequalities are proved similarly. Recall $\phi$ from Theorem \[main\]. Let $d (\nu, A) = \inf \{ p(\nu - u); u \in A\}$ be the distance from the point $\nu \in {{\cal M}}_{w+}$ to the set $A \subseteq {{\cal M}}_{w+}$, where $p$ is the norm defined in (\[eq:norm-weak\]). \[re:compact-glob-attr\] Assume that (A1)-(A3) hold. Then, for any $\gamma \in C(Q, {{\cal P}}_w)$, there exists a compact attractor of bounded sets, i.e., a compact invariant subset $A_\gamma $ of ${{\cal M}}_{w+}$ such that, for all bounded subsets $B $ of ${{\cal M}}_{w+}$, $ d( \phi(t, u, \gamma), A_\gamma) \to 0$ as $t \to \infty$ uniformly for $u \in B$. Moreover, $\kappa (Q) \le K^\di$ for all $\kappa \in A_\gamma$, and there is a compact set $ C$ such that $A_\gamma \subseteq C$ for all $\gamma \in C(Q, {{\cal P}}_w)$. Finally, the attractors $A_\gamma$ are upper semicontinuous, i.e., for any $ \eta \in C(Q, {{\cal P}}_w)$, $$\sup_{\nu \in A_\gamma} d (\nu, A_{\eta}) \to 0 \hbox{ as }\gamma \to \eta.$$ Consider the semiflow $\phi(\cdot, \gamma)$ defined in Theorem \[main\]. By Theorems \[main\] and \[BS\] and Lemma \[re:measures-weak-top\], in the language of [@SmTh Def.2.25], this semiflow is point-dissipative, asymptotically smooth, and eventually bounded on bounded sets. Existence of the attractors $A_\gamma$ now follows from [@SmTh Thm.2.33]. We define a function $ V : {{\cal M}}_{w+}(Q)\rightarrow \mathbb{R_+}$ by $V(\nu) = [\nu(Q) - K^\di]_+^2$, where $r_+ = \max\{r,0\}$ is the positive part of a real number $r$. $V$ is continuous by Lemma \[re:semicont\]. Furthermore, $r_+^2$ is differentiable and $\frac{d}{dr} r_+^2 = 2 r_+$. [Then the orbital derivative of $V$ along (\[M\]) [@Hal p.313] is $$\dot V(\nu) := \limsup_{t \to 0+} \frac{1}{t} ( V(\phi(t, \nu, \gamma)) - V(\nu)) = 2 [\nu(Q) - K^\di]_+ \int_Q [\cR(\nu(Q),q) -1 ] D(\nu(Q),q ) \nu (dq).$$ This implies that $\dot V \le 0$ and $(d/dt) V (\phi(t,\nu, \gamma)) = \dot V (\phi(t,\nu,\gamma)) \le 0$. So $V$ is a Lyapunov function in the sense of Definition 2.49 [@SmTh pg.52]. Finally, $\{ \dot V = 0 \} = \{\nu ; \nu(Q) \le K^\di \} = \{V(\nu) =0\}$. ]{}[ Let $\kappa \in A_\gamma$. Since $A_\gamma$ is invariant, by [@SmTh Thm.1.40], there exists a total solution $\mu : \R \to A_\gamma$ of (\[M\]) such that $\mu (0) = \kappa$. Then $(d/dt) V(\mu(t)) = \dot V (\mu(t)) \le 0$ and $V \circ \mu$ is decreasing on $\R$. Since $A_\gamma$ is compact, $\mu$ has an $\alpha$-limit set $\alpha$ on which $V$ is constant. So $\dot V=0$ on $\alpha$ which implies $\alpha \subseteq \{\nu; \nu(Q) \le K^\di\}= \{V(\nu) =0\}$. Since $\alpha $ attracts $\mu (t)$ as $t \to - \infty$, $\lim_{t\to -\infty} V(\mu(t)) =0$. Since $V(\mu(t))$ decreases, $0 = V(\mu(0)) = V(\kappa)$. This implies $\kappa (Q) \le K^\di$. This holds for any $\kappa \in A_\gamma$ and so $A_\gamma \subseteq \{\kappa \in {{\cal M}}_{w,+}; \kappa (Q) \le K^\di\}$ which is a compact subset in the $w^*$-topology by the Alaoglu-Bourbaki theorem.]{} The last statement follows from [@Mag09 Prop.2.7]. We consider the state space $ X = {{\cal M}}_{w,+} \times C(Q,{{\cal P}}_w)$ with the product topology and the semiflow $\Phi$ on $X$ given by $\Phi(t, (u, \gamma)) = (\phi(t, u, \gamma), \gamma)$. $X$ can be made a metric space with the metric $d_X ((u,\gamma), (v, \eta)) = p(u-v) + \|\gamma- \eta\|$ where $u,v \in {{\cal M}}_{w\,+}$ and $\gamma, \eta \in C(Q, {{\cal P}}_w)$. $\Phi$ is a continuous from $\R_+ \times X$ to $X$ by Theorem \[main\]. \[re:compact-attr-Cart\] If $\Gamma$ is a compact subset of $C(Q, {{\cal P}}_w)$, the restriction of the semiflow $\Phi$ to $X={{\cal M}}_{w,+} \times \Gamma$ has a compact attractor of bounded sets, $\cA_\Gamma$, which satisfies $\cA_\Gamma = \bigcup_{\gamma \in \Gamma} ( A_\gamma \times \{\gamma\} )$, where $A_\gamma \subseteq {{\cal M}}_{w+}$ is the attractor of $\phi (\cdot, \gamma)$ in Corollary \[re:compact-glob-attr\]. In particular, all elements in $\cA_\Gamma$ are or the form $(\kappa, \gamma)$ with $\kappa (Q) \le K^\di$. We apply [@SmTh Thm.2.33]. By Corollary \[re:compact-glob-attr\], the compact set $\{u \in M_{w,+}; u(Q)\le K^\di\} \times \Gamma$ attracts all points in $X$. In particular, $\Phi$ is point-dissipative. To check that $\Phi$ is asymptotically smooth, let $ B $ be a bounded subset of $X$ that is forward invariant under $\Phi$. Then $\Phi(\R_+ \times B) \subseteq B$. Since $X = {{\cal M}}_{w,+} \times \Gamma$, $\Phi(\R_+ \times B) \subseteq K \times \Gamma$ with $K$ being a bounded subset of ${{\cal M}}_{w,+}$. [By the Alaoglu-Bourbaki theorem, $\Phi(\R_+ \times B)$ is contained in a compact subset of $X$. This implies that $\Phi$ is asymptotically compact on $B$.]{} $\Phi$ is bounded on every bounded subset $B$ of $X$ by Theorem \[BS\]. By [@SmTh Thm.2.33], the restriction of $\Phi$ to $X$ has a compact attractor of bounded sets, $\cA_\Gamma$. Now let $\gamma \in \Gamma$ and $\tilde A_\gamma = \{\kappa \in {{\cal M}}_{w+}; (\kappa, \gamma) \in \cA_\Gamma \}$. Then $\tilde A_\gamma$ is a compact subset of ${{\cal M}}_{w+}$ that is invariant under $\phi(\cdot, \gamma)$. Since $A_\gamma$ is the compact attractor of bounded subsets for $\phi(\cdot, \gamma)$, $\tilde A_\gamma \subseteq A_\gamma$. By Corollary \[re:compact-glob-attr\], $\tilde \cA_\Gamma= \bigcup_{\gamma \in \Gamma} ( A_\gamma \times \{\gamma\})$ is a compact subset of $X$. Since each $A_\gamma$ is invariant under $\phi(\cdot, \gamma)$, $\tilde \cA_\Gamma$ is invariant under $\Phi$. Since $\cA_\Gamma$ is the compact attractor of all bounded subsets of $X$ for $\Phi$, $\tilde \cA_\Gamma\subseteq \cA_\Gamma$. Persistence ----------- Next, we prove a persistence result for the case where $k_\diamond$ is not necessarily positive and (A3) does not necessarily hold. In order to obtain population persistence, we impose a balancing inequality on some sets with strong strategies. A strong strategy, $q$, is one that has $\mathcal{R}(0,q)> 1.$ So if [$E \subseteq Q$]{} consists entirely of strong strategies, and if a member of $E$ contributes on average more than one of its offspring to $E$, then the population will persist. Mostly, the strong strategies need to play the balancing act if the population is to survive, and on average only extremely strong traits can afford to have large proportions of their offspring be weak. \[def:irred\] Let $E$ be a Borel subset of $Q$. A kernel $\gamma \in C(Q, {{\cal P}}_w )$ is called $E$-irreducible if for every solution $\mu$ of (\[M\]) with $\mu(0)(Q) > 0$ there exists some $r \ge 0$ such that $\mu(r)(E) >0$. The kernel $\gamma$ is called uniformly $E$-irreducible if $r$ does not depend on $\mu$. \[persistence\] Assume that (A1)-(A2) hold and let $\epsilon > 0$ and $ E \subseteq Q $ be a Borel set such that $$\label{balance} \inf_{q \in E} \mathcal{R}(\epsilon,q) \gamma(q)(E) > 1 .$$ - Then the population is uniformly weakly persistent in the sense that $\limsup_{t\to \infty} \mu(t)(Q) \ge \epsilon$ for all solutions with $\mu(0)(E) > 0$. - Assume in addition that [the]{} kernel $\gamma$ is $E$-irreducible. Then $\limsup_{t\to \infty} \mu(t)(Q) \ge \epsilon$ for all solutions with $\mu(0)(Q) > 0$. \(a) Assume that $\limsup_{t\to \infty} \mu(t)(Q)< \epsilon$. Then for sufficiently large $t$, since $B$ is nonincreasing and $D$ is nondecreasing in the first variable and $\mu(t)(Q)$ is nonnegative, $$\begin{array}{ll} \mu' (t)(E) & \displaystyle \ge \int_E B(\epsilon,q) \gamma(q) (E) \mu(t)(d q) - \int_E D(\epsilon,q) \mu(t)( dq)\\ & \displaystyle = \int_E [{\cal R} (\epsilon,q) \gamma(q)(E) -1] D(\epsilon,q) \mu(t)(dq) . \end{array}$$ Thus, $$\mu'(t)(E) \ge [\inf_{q\in E} {\cal R}(\epsilon,q) \gamma(q)(E) -1] \int_E D(\epsilon,q) \mu(t)(dq).$$ Then, $$\mu'(t)(E) \ge [\inf_{q\in E} {\cal R}(\epsilon,q)\gamma(q)(E)-1] \inf_{q\in E} D(\epsilon,q) \mu(t)(E).$$ So $\mu(t) (E) \to \infty$ because $\mu(0)(E) > 0$ and $\inf_{q\in E} D(\epsilon,q)> 0$ by (A2). This contradicts $\limsup_{t\to \infty} \mu(t) (Q) < \epsilon$. \(b) Now assume that $\gamma$ is $E$-irreducible and $\mu$ is a solution with $\mu(0)(Q) > 0$. Then there exists some $r >0$ such that $\mu(r) (E) >0$ and $\limsup_{t \to \infty} \mu(t) (Q) > \epsilon$ by our previous result. \[re:persistence-strong\] Assume that (A1)-(A2) hold and let $ E \subseteq Q $ be a Borel set such that $\gamma$ is $E$-irreducible and $$\label{balance2} \inf_{q\in E} \mathcal{R}(\epsilon,q) \gamma(q)(E) > 1, \text { for some } \epsilon > 0 .$$ Then the following hold: - The population is uniformly persistent in the following sense: There exist some $\epsilon_0 > 0$ such that $\liminf_{t\to \infty} \mu(t)(Q) \ge \epsilon_0$ for all solutions with $\mu(0)(Q) > 0$. - If (A3) holds as well and $\gamma$ is uniformly $E$-irreducible, then for every $f \in C_+(Q)$ with $\inf f(E) >0$ there exists some $\delta_f >0$ such that $\liminf_{t \to \infty} \int_Q f(q) \mu(t)(dq) \ge \delta_f$ for all solutions with $\mu(0)(Q) > 0$. - If (A3) holds and $E$ is open and $\gamma$ is uniformly $E$-irreducible, then there exists some $\delta_E >0$ such that $\liminf_{t \to \infty} \mu(t) (E) \ge \delta_E$ for all solutions with $\mu(0)(Q) > 0$. <!-- --> - By Theorem \[persistence\], in the language of [@Thi03 A.5], the semiflow induced by (\[M\]) on ${{\cal M}}_{w,+}$ is uniformly weakly $\rho$-persistent for $\rho: {{\cal M}}_{w+} \to \R_+$, $\rho(\mu) = \mu(Q)$. Further the sets $\{\rho \le c\}$ are compact in ${{\cal M}}_{w,+}$ for each $c >0$. By [@Thi03 Thm.A.32], the semiflow induced by (\[M\]) on ${{\cal M}}_{w,+}$ is uniformly $\rho$-persistent. - Now assume (A3) in addition and that $\gamma$ is uniformly $E$-irreducible. [For $f \in C_+(Q)$]{}, we apply [@SmTh Thm.4.21] with $$\tilde \rho (\mu) = \int_Q f(q) \mu(dq).$$ By Theorem \[BS\], there exists a compact subset $C$ of $M_{w_+}$ such that $\mu(t) \in C$ for sufficiently large $t > 0$. Let $\mu(\cdot): \R \to C_\epsilon $ be a total trajectory where $C_\epsilon = C \cap \{{\tilde \rho }\ge \epsilon\}$. In our situation, a total trajectory is a solution $\mu$ that is defined for all $t \in \R$. Choose $r >0$ from the uniform $E$-irreducibility definition. Since $\mu(-r)(Q) > 0$, we have $\mu(0)(E) >0$. Since $\inf f(E)>0$, $\tilde \rho(\mu(0))> 0$.   Since $\tilde \rho$ is continuous, the semiflow is uniformly $\tilde \rho$-persistent by [@SmTh Thm.4.21].   - Now assume (A3) and that $E$ is open. This time, we apply [@SmTh Thm4.21] with $\tilde \rho(\nu) = \nu (E)$. By Lemma \[re:semicont\], $\tilde \rho$ is lower semicontinuous. \[re:gammas\] If $E \subseteq Q$ is open, the following hold: - Assumption can be replaced by $\cR(0,q)\gamma(q)(E) >1$ for all $q \in \bar E$. - If $\Gamma$ is a compact subset of $C(Q, {{\cal P}}_w )$ and $\cR(0,q)\gamma(q)(E) >1$ for all $q \in \bar E$ and all $\gamma \in \Gamma$, then there exists some $\epsilon > 0$ such that $\inf_{q \in Q} \cR(\epsilon,q)\gamma(q)(E) >1$ for all $\gamma \in \Gamma$. \(a) Suppose there is no $\epsilon >0$ such that $$\inf_{q \in E} \cR(\epsilon, q) \gamma(q,E) > 1.$$ Let $(\epsilon_n)$ be a sequence of positive numbers such that $\epsilon_n \to 0$. Then there exists a sequence $(q_n)$ in $E$ such that $\liminf_{n \to \infty} \cR(\epsilon_n, q_n) \gamma(q_n,E) \le 1$. After choosing subsequences, we can assume that $q_n \to q$ for some $q \in \bar E$. By continuity of $\cR$ and lower semi-continuity of $\gamma(\cdot) ( E)$ (recall Lemma \[re:semicont\] (a)), $$\liminf_{n \to \infty} \cR(\epsilon_n, q_n) \gamma(q_n)(E) \ge \liminf_{n \to \infty} \cR(\epsilon_n, q_n) \liminf_{n \to \infty} \gamma(q_n)(E) \ge \cR(0,q) \gamma(q)(E) >1,$$ a contradiction. Part (b) is shown similarly using Lemma \[re:semicont\] (b). \[re:persistence-unif\] Assume that (A1)-(A3) hold and let $ E \subseteq Q $ be an open set and $\Gamma$ be a compact subset of $C(Q, {{\cal P}}_w)$ with the following properties: 1. $\displaystyle \mathcal{R}(0,q) \gamma(q)(E) > 1$ for all $q \in \bar E$ and all $\gamma \in \Gamma$. 2. All $\gamma \in \Gamma$ are $E$-irreducible. Then the population is uniformly persistent in the following sense: There exist some $\epsilon_0 > 0$ such that $\liminf_{t\to \infty} \mu(t)(E) \ge \epsilon_0$ for all solutions $\mu$ of (\[M\]) with $\mu(0)(Q) > 0$ and $\gamma \in \Gamma$. We consider the state space $X = {{\cal M}}_{w+} \times \Gamma$ with the product topology and the semiflow $\Phi$ on $X$ given by $\Phi(t, (u, \gamma)) = (\phi(t, u, \gamma), \gamma)$. $X$ can be made a metric space and $\Phi$ is continuous on $\R_+ \times X$. We first choose the persistence function $\rho(u, \gamma) = u(Q)$. By [Theorem \[persistence\]]{} and Remark \[re:gammas\] (b), $\Phi$ is uniformly weakly $\rho$-persistent. By Theorem \[BS\] and Lemma \[re:measures-weak-top\], there exists a compact subset $C$ of $X$ such that, for all $(u,\gamma) \in X$, $\Phi(t,u, \gamma) \in C$ for all sufficiently large $t > 0$. By [@SmTh Thm.4.13], $\Phi$ is uniformly $\rho$-persistent. Now use a second persistence functions $\tilde \rho(u, \gamma)= u(E)$. By Lemma \[re:semicont\], $\tilde \rho$ is lower semicontinuous. The statement now follows from [@SmTh Thm.4.21] similarly as in the proof of Theorem \[re:persistence-strong\]. Robust persistence for optimum preserving mutation kernels ---------------------------------------------------------- We now consider $E = Q^\di$ and try to drop the irreducibility assumption. As trade-off we assume that $Q^\di$ is open (which implies that $Q$ is disconnected) and that there are no mutation losses for strategies in $Q^\di$. \[def:kernel-opt-pres\] A mutation kernel $\gamma^\di \in C(Q, {{\cal P}}_w)$ is called [*optimum preserving*]{} if $$\gamma^\di (q) (Q^\di) =1 \hbox{ for all } q \in Q^\di.$$ Assume that $Q^\di$ is an open subset of $Q$. Recall that $K^\di = \sup K(Q)$ and $Q^\di = \{q \in Q; K(q) = K^\di\}$. Since $\tilde Q = Q \setminus Q^\di$ is compact, $\tilde K := \sup K(\tilde Q) < K^\di$. Choose some $\epsilon_0 > 0$ such that $$\label{eq:pers-rob1} \sup K(\tilde Q ) + 3 {\epsilon_0} < K^\di.$$ Let $ E = Q^{\diamond} $ and $\psi( \gamma, q)=\gamma(q)(E)$ be as in Lemma \[re:semicont\]. Then $${ \cal F}(\gamma, q) =\cR( K^{\diamond} - \epsilon_0, q)\psi(\gamma,q)$$ is continuous, because $Q^{\diamond}$ is both open and closed. Hence, there exists some $\delta_0 > 0$ such that $$\label{eq:pers-rob2} \inf_{q \in Q^\di} \cR(K^\di - {\epsilon_0}, q) \gamma(q)(Q^\di) > 1, \qquad \|\gamma -\gamma^\di\| < \delta_0.$$ Here, $\|\cdot \|$ is the norm on $C(Q, {{\cal M}}_p)$ defined in (\[eq:metric-mutation\]).\ \[re:pers-robust\] Assume (A1), (A2) and (A3), $K^\di > 0$, and that $Q^\di$ is an open subset of $Q$. If $\gamma^\di \in C(Q, {{\cal P}}_w)$ is an optimum preserving mutation kernel, the following hold: - Then there exists some $\tilde \delta \in (0, \delta_0) $ such that $\displaystyle \liminf_{t \to \infty} \phi(t, u, \gamma)(Q^\di) \ge \tilde \delta $ for all $u \in {{\cal M}}_{w+}$ with $u(Q^\di) > 0$ and all $\gamma \in C(Q, {{\cal P}}_w)$ with $\|\gamma- \gamma^\di\| < \tilde \delta$. - For all $\gamma \in C(Q, {{\cal P}}_w)$ with $\|\gamma- \gamma^\di\| < \tilde \delta$ with $\tilde \delta > 0$ from (A), there exists a persistence attractor $\tilde A_\gamma$, i.e., a compact invariant stable subset $\tilde A_\gamma$ with the following properties: - $\nu (Q^\di) > 0 $ for all $\nu \in \tilde A_\gamma $. - For all compact subsets $V$ of ${{\cal M}}_{w+}$ with $\inf_{u \in V} u(Q^\di) > 0$ there exists a neighborhood $U$ such that $d( \phi(t, u, \gamma), \tilde A_\gamma) \to 0$ as $t \to \infty$ uniformly for $u \in U$. - The attractors $\tilde A_\gamma$, $\|\gamma - \gamma^\di\|< \tilde \delta$, in part (B) are upper semicontinuous at $\gamma^\di$: For all open subsets $W $ with $\tilde A_{\gamma^\di} \subseteq W \subseteq {{\cal M}}_{w+}$, there exists some $\delta_W \in (0,\tilde \delta)$ such that $\tilde A_\gamma \subseteq W$ for all $\gamma \in C(Q, {{\cal P}}_w)$ with $\|\gamma- \gamma^\di\| < \delta_W$. Here $d (\nu, A) = \inf \{ p(\nu - u); u \in A\}$ is the distance from the point $\nu \in {{\cal M}}_{w+}$ to the set $A \subseteq {{\cal M}}_{w+}$ where $p$ is the norm defined in (\[eq:norm-weak\]). Property (A) makes the semiflow $\Phi$ robustly persistent at an optimum preserving mutation kernel $\gamma^\di$ as $\breve \epsilon >0$ can be chosen uniformly for all $\gamma$ in a neighborhood of $\gamma^\di$ [(see [@HoSc; @Sal] and the references therein).]{} Suppose that (A) is false. Then there exist sequences $(\gamma_n)$ in $C(Q, {{\cal P}}_w)$ and $(u_n) $ in ${{\cal M}}_{w+}$ such that $\delta_0 \ge \|\gamma_n -\gamma^\di\| \to 0$, $u_n(Q^\di) > 0$ and $$\liminf_{t \to \infty} \phi(t, u_n, \gamma_n)(Q^\di) \to 0, \qquad n \to \infty.$$ Here $\delta_0 >0$ is from (\[eq:pers-rob2\]). The set $\Gamma = \{\gamma_n; n \in \N\} \cup \{\gamma^\di\}$ is compact in [$C(Q, {{\cal P}}_w)$]{}. Let $X = {{\cal M}}_{w+} \times \Gamma$ with the metric $\tilde D( (u, \gamma), (v, \eta)) = p (u-v) + \|\gamma- \eta\|$ for $u,v \in {{\cal M}}_{w+}$ and $\gamma,\eta \in \Gamma$. We apply [@SmTh Thm.8.20] to the semiflow $\Phi$ on $X$ given by $\Phi(t,(u,\gamma)) = (\phi(t,u,\gamma), \gamma)$ and the persistence function $\rho(u, \gamma) = u (Q^\di)$. By Theorem \[re:compact-attr-Cart\], $\Phi$ has a compact attractor $\cA$ of bounded sets on ${{\cal M}}_{w+} \times \Gamma$. Let $$\label{eq:Xzero} \begin{split} X_0 := & \{(u, \gamma) \in X; \forall t \ge 0: \rho (\Phi(t, (u, \gamma) ) =0 \} \\= & \{ (u, \gamma)\in X; \forall t \ge 0: \phi(t, u, \gamma)(Q^\di ) =0 \}. \end{split}$$ By [@SmTh Thm.5.21], $\cA_0 = X_0 \cap \cA$ is a compact attractor of compact sets in $X_0$ and attracts all subsets of $X_0$ that are attracted by $\cA$. So $\cA_0$ attracts all bounded subsets of $X_0$ and is isolated in $X_0$ by [@SmTh Thm.2.19]. By [@SmTh Thm.1.40], $\cA_0$ is acyclic in $X_0$ and, by [@SmTh Thm.2.17], contains the $\omega$-limit sets of all points in $X_0$ under $\Phi$. Next we show that $\cA_0$ is uniformly weakly $\rho$-repelling for $\rho(u,\gamma) = u(Q^\di)$. Let $(u,\gamma) \in X_0$ and $\mu (t) = \phi(t, u, \gamma)$. [Set $\tilde Q = Q \setminus Q^\di$.]{} Then $$\mu(t) (E) = \int_{\tilde Q} B\big ( \mu(t) (\tilde Q) , q \big ) \gamma(q)(E) \mu(t)(dq) - \int_E D \big (\mu(t)(\tilde Q),q \big ) \mu(t)(dq), \qquad E \subseteq \tilde Q.$$ So the restriction of $\Phi$ to $X_0$ corresponds to solutions of (\[M\]) with $Q$ being replaced by its compact subset $\tilde Q = Q \setminus Q^\di$. By Theorem \[re:compact-attr-Cart\], $$\label{eq:Azero} \cA_0 \subseteq \{(u, \gamma); \gamma \in \Gamma, u (\tilde Q) \le \tilde K\}, \qquad \tilde K := \sup K(\tilde Q).$$ Recall (\[eq:pers-rob1\]) and (\[eq:pers-rob2\]). Suppose that $\cA_0$ is not uniformly weakly $\rho$-repelling. If [ $\epsilon_0>0$ is as in (\[eq:pers-rob1\])]{}, then there exists some $u \in {{\cal M}}_{w+}$ and $\gamma \in \Gamma$ such that for $\mu = \phi(\cdot, u, \gamma)$ we have $\mu(0)(Q^\di) > 0$ and $\limsup_{t \to \infty} d( \mu(t), \cA_0) < \epsilon_0.$ [By (\[eq:Xzero\]) and (\[eq:Azero\]),]{} $$\limsup_{t \to \infty} \mu(t)(Q^\di) < \epsilon_0 \quad \hbox{ and } \quad \limsup_{t \to \infty} \mu(t)(\tilde Q) < \tilde K + \epsilon_0.$$ By (\[eq:pers-rob1\]) and these results for $\mu$, for sufficiently large $t > 0$, $\mu(t) (Q) < K^\di - \epsilon$. Also $\mu(t) (Q^\di) > 0$ for all $t \ge 0$. [By (\[eq:pers-rob2\])]{}, there is some $r > 0$ such that for all $t \ge r >0$, $$\begin{split} \mu'(t) (Q^\di ) \ge & \int_{Q^\di} B(K^\di - \epsilon, q) \gamma (q) (Q^\di) \mu(t)(dq) - \int_{Q^\di} D (K^\di - \epsilon, q) \mu(t) (dq) \\ = & \int_{Q^\di} \big [\cR(K^\di - \epsilon, q)\gamma (q) (Q^\di)- 1\big] D(K^\di - \epsilon, q) \mu(t) (dq) \ge \breve \delta \mu(t) (Q^\di) \end{split}$$ with some $\breve \delta > 0$. So $\mu(t) (Q^\di ) \to \infty$, a contradiction. This proves that $\cA_0$ is uniformly weakly $\rho$-repelling. By [@SmTh Thm.8.20], [with $\Omega \subseteq M_1:=\cA_0 $]{}, $\Phi$ is uniformly weakly $\rho$-persistent on $X$. [[@SmTh Thm.4.13] implies that $\Phi$ is uniformly $\rho$-persistent on $X$. This contradicts $\liminf_{t \to \infty} \phi(t, u_n, \gamma_n)(Q^\di) \to 0$ as $n \to \infty$ because $(u_n,\gamma_n) \in X$ and $\rho(u_n,\gamma_n) > 0$.\ ]{} \(B) follows from part (A) and [@SmTh Thm.5.6] applied to each semiflow $\phi(\cdot, \gamma)$ with $\|\gamma - \gamma^\di\| < \tilde \delta$. \(C) follows from [@Mag09 Thm.1.1]. Pure Selection Dynamics {#sec:pure-sel} ======================= When attempting to analyze the asymptotic behavior of an EGT model, one usually first forms the appropriate notion of an *Evolutionary Stable Strategy* or *ESS.* The concept of an *ESS* was introduced into biology from the field of game theory by Maynard Smith and Price to study the behavior of animal conflicts [@Maynard2]. Intuitively an *ESS* is a strategy such that if all members of a population adopt it, no differing behavior could invade the population under the force of natural selection. So at the “equilibrium" of an *ESS* all other strategies, if present in small quantities, should have negative fitness and die out. We use the above discussion to define ESS and ASS as follows. We define for two strategies $q$ and $\hat q $ a relative fitness. Then using this definition we define an *ESS*. To this end define the relative fitness between two strategies as $$\lambda_R(q, \hat q) =\mathcal{R}(K(q), \hat q) -1.$$ This is clearly well-defined and is a measure of the long term fitness of $\hat q$ when the subpopulation with trait $q$ is at its carrying capacity $K(q)$. A strategy $q$ is a **(local) global *ESS*** if $ q $ satisfies $$\lambda_{R}(q,\hat q) < \lambda_{R}(q,q), \text{ for all } \hat q, \hat q \neq q, ~(\text{in a neighborhood of } q).$$ In this work an ESS is global unless explicitly mentioned as local. Notice that $\lambda_{R}(q,q) =0$ for all $ q $ and hence $q$ is an (local) *ESS* if and only if $\lambda_{R}(q, \hat q) <0$ for all $\hat q \neq q$ (in a neighborhood of $q)$. All other strategies (in a neighborhood) have negative fitness when the subpopulation with trait $q$ is at its carrying capacity $K(q)$ and hence die out. Assumption (A3) implies that $R(K(q),\hat q) \ne 1$ for any $\hat q \ne q$. Recall, that $I(q,\hat q)=R(K(q),\hat q)$ is the invasion reproductive number of strategy $\hat q$ with respect to strategy $q$, i.e., it is a measure of the ability of strategy $\hat q$ to invade strategy $q$ when the subpopulation with strategy $q$ is at its carrying capacity $K(q)$ [@MM]. From the definition of an ESS it is evident that the following are equivalent: 1) finding an ESS; 2) finding a strategy $q$ such that the **relative fitness** $\lambda_R(q,\hat q) <0$ for all $\hat q \ne q$; 3) finding a strategy $q$ such that the **invasion reproductive number** $I(q,\hat q) < 1$ for all $\hat q \ne q$; 4) finding a strategy $q$ that has the **largest carrying capacity** $K(q)$. The concept of an ESS is insufficient to determine the outcome of the evolutionary game, since a strategy that is an *ESS* need not be an evolutionary attractor [@BrownVincent ch.6]. It is not the case that all members of the population will end up playing that strategy. An *ESS* simply implies that if a population adopts a certain strategy (phenotype, language or cultural norm etc.), then no mutant small in quantity can invade or replace this strategy. Suppose a population is evolving according to . If $c_{q}\delta_{q}$ for some finite number $c_{q}$ attracts any solution $\mu(t)$ of satisfying $q \in supp(\mu(0))$, then we call the strategy $q$ an **Asymptotically Stable Strategy** or ***ASS***. This strategy, if it exists, is the endgame of the evolutionary process. As it is attractive and once adopted, it cannot be invaded or replaced. Let $\gamma( q) =\delta_{q} \in C(Q, {\cal P}_w)$ for all $q \in Q$ and $u\in \mathcal{M}_+.$ Substituting these parameters in and setting $\mu(t) = \phi(t, u, \gamma)$ and $\bar \mu(t) = \mu(t)(Q)$, one obtains the pure selection model $$\left\{\begin{array}{ll} \label{selection} \displaystyle {\mu}^\prime (t)(E) = \int_E \left [B(\bar \mu(t), q)-D(\bar \mu(t),q) \right ]\mu(t)(d q) \\ \mu(0)=u. \end{array}\right.$$ [Using the uniqueness of solutions in Theorem \[main\], one observes that the solution to satisfies the following integral representation : $$\label{pureintrep} \mu(t)(E)= \int_{E}\exp \Big (\int_{0}^{t}B(\bar \mu(\tau),q)-D(\bar \mu(\tau),q)d\tau \Big )\mu(0)(dq) .$$]{} The following easy consequence will be used without further mentioning. \[re:posit-pres\] If $\mu(0)(E) =0$, then $\mu(t)(E) =0 $ for all $t \ge 0$. If $\mu(0)(E) >0$, then $\mu(t)(E) >0 $ for all $t \ge 0$. \[re\_equilibria-select\] Every $u \in {{\cal M}}_{w+}$ with $u(Q^\di) = K^\di$ and $u (Q \setminus Q^\di) =0$ is an equilibrium of (\[selection\]). In particular, the collection ${{\cal M}}^\di$ of all such measures is a compact invariant set. Let $u$ be as described above. Then $u(D) =0$ for all Borel subsets of $Q \setminus Q^\di$ and, for all Borel subsets $E$ of $Q$, $$\begin{split} & \int_E [B(u(Q),q) - D(u(Q),q)] u(dq) \\ = & \int_{E \cap Q^\di} [B(K^\di,q) - D(K^\di,q)] u(dq) + \int_{E \setminus Q^\di} [B(u(Q),q) - D(u(Q),q)] u(dq) \\ = & 0 +0 =0. \end{split}$$ Any set of equilibria is invariant. To show that ${{\cal M}}^\di $ is closed, let $(u_k)$ be a sequence in ${{\cal M}}^\di$ and $u \in {{\cal M}}_{w+}$ such that $u_k \to u$ in the weak$^*$ topology. Since $\chi_Q$ is continuous, $u(Q)= \lim_{j\to \infty} u_j(Q) = K^\di$. Since $Q^\di$ is compact, $Q \setminus Q^\di$ is open and there exists an increasing sequence $(f_j)$ of continuous functions such that $0 \le f_j \le \chi_{Q \setminus Q^\di}$ and $f_j \to \chi_{Q \setminus Q^\di}$ pointwise [@ALI Thm.3.13]. Then, for each $ j, k$, $\int_Q f_j(q) u_k(dq)=0$ and so $ \int_Q f_j(q) u(dq) =0$. Thus $ \int_Q \chi_{Q \setminus Q^{\diamond}}(q)u(dq) = u(Q \setminus Q^\di)=0$ by the monotone convergence theorem. \[re:select1\] Assume (A1)-(A3) and $K^\di > 0$. For every $\epsilon \in (0, K^\di)$, there exist an open set $U_\epsilon$ with $Q^\di \subseteq U_\epsilon \subseteq Q$ and some $\xi_\epsilon > 1$ such that $\cR( K^\di- \epsilon, q) > \xi_\epsilon $ for all $ q \in U_\epsilon$. Further, if $\mu$ is a solution of (\[selection\]) such that $\mu(0)(U_\epsilon )>0$, then $$\limsup_{t \to \infty} \mu(t)( Q) \in ( K^\di - \epsilon, K^\di].$$ We know from Theorem \[BS\] that $$\bar\mu^\infty: = \limsup_{t\to \infty} \mu(t)(Q) \le K^\di$$ for all solutions $\mu $ of (\[selection\]). Let $\epsilon \in (0, K^\di)$. For all $q \in Q^\di$, $$1 = \cR( K^\di, q) < \cR(K^\di -\epsilon, q ).$$ Since $\cR( K^\di -\epsilon, \cdot)$ is continuous and $Q^\di$ is compact, there exists some $\xi_\epsilon > 1$ such that $\cR( K^\di -\epsilon, q) > \xi_\epsilon$ for all $q \in Q^\di$. We claim that there exists some open set $U_\epsilon$ with $Q^\di \subseteq U_\epsilon \subseteq Q $ such that $$\cR( K^\di - \epsilon, q) > \xi_\epsilon , \qquad q \in U_\epsilon.$$ Recall that $ U_\delta (Q^\di)=\{ \hat q\in Q; d(\hat q, Q^\di)< \delta \}$ is an open subset of $Q$ that contains $Q^\di$ for any $\delta > 0$. So, if our claim does not hold, there exists a sequence $(\hat q_n)$ in $Q$ such that $d(\hat q_n, Q^\di) \to 0$ and $\cR( K^\di -\epsilon, \hat q_n) \le \xi_\epsilon$. By definition of the distance function, there exists a sequence $(q_n)$ in $Q^\di$ such that $d(q_n, \hat q_n) \to 0$. After choosing subsequences, $q_n \to q$ for some $q \in Q^\di$ and also $\hat q_n \to q$. By continuity, $\cR(K^\di - \epsilon, q) \le \xi_\epsilon$, a contradiction. Now consider a solution $\mu$ of (\[selection\]) with $\mu(0)(U_\epsilon) > 0$. So ${\mu(t)( U_\epsilon) }> 0$ for all $t \ge 0$. Suppose ${\bar\mu^\infty}< K^\di -\epsilon$. Then there exists some $r \ge 0$ such that $\bar \mu(t) < K^\di - \epsilon$ for all $t \ge r$. For $t \ge r$, $$\begin{split} \mu'(t)(U_\epsilon) = & \int_{U_\epsilon} [\cR(\bar \mu(t), q) -1] D(\bar \mu(t),q) \mu (t)(dq) \\ \ge & \int_{U_\epsilon} [\cR( K^\di -\epsilon, q) -1] D(\bar \mu(t),q) \mu (t)(dq) \\ \ge & \int_{U_\epsilon} [\xi -1] D(\bar \mu(t),q) \mu (t)(dq) \ge [\xi -1] \inf_{q \in U_\epsilon} D(0,q) \mu (t)(U_\epsilon). \end{split}$$ Since $[\xi -1] \inf_{q \in U_\epsilon} D(0,q) >0$, ${\mu(t)(U_\epsilon)}$ increases exponentially and $\bar \mu(t) \ge \mu(t)(U_\epsilon)$ grows unbounded, a contradiction. We add another assumption which states that the strategies that maximize the carrying capacity are also [ superior at all other relevant population densities.]{} - Let $Q^\di$ be defined by (\[eq:Q-max\]). $$\hbox{ For each } q^\di \in Q^\di \hbox{ and } q \in Q \setminus Q^\di, \quad \cR(X, q^\di) > \cR(X, q) \hbox{ for all } X \in [k_\di, K^\di].$$ Let $K^\di >0$. Assume that $\cR(X, q) = L(X) M(q)$ for all $X \ge 0$ and $q \in Q$ with continuous functions $L: \R_+ \to (0,\infty)$ and $M: Q \to \R_+$. Define $M^\circ = \max_{q \in Q} M(q)$ and $Q^\circ = \{q \in Q; M(q) = M^\circ\}$. Then $Q^\circ = Q^\di$ and (A4) follows. This result is very similar to one for chemostats namely that maximizing the basic reproduction number amounts to the same as minimizing the break-even concentration if the reproduction number factorizes as above [@SmTh13]. Since $K^\di > 0$, $1 < \cR(0, q)= L(0) M(q) $ for some $q \in Q$. This implies that $1 < L(0) M(q) $ for all $q \in Q^\circ$ and so $K(q) > 0$ for all $q \in Q^\circ$. > Step 1: $K(\cdot)$ is constant on $Q^\circ$ Let $q_1, q_2 \in Q^\circ$. Then $M(q_1) = M^\circ = M(q_2)$. Recall that $$1= \cR(K(q_1), q_1) = L(K(q_1)) M(q_1)= L(K(q_1)) M(q_2)= \cR(K(q_1),q_2).$$ Since, by assumption, $K(q_2)$ is uniquely determined by $\cR(K(q_2), q_2)=1$, $K(q_1) = K(q_2)$. > Step 2: If $q^\circ \in Q^\circ$ and $q \in Q \setminus Q^\circ$, then $K(q^\circ) > K(q)$. Suppose that $q^\circ \in Q^\circ$ and $q \in Q \setminus Q^\circ$. Then $K(q^\circ) >0$ and we can assume that $K(q) > 0$. By definition of $K$, $$1= \cR(K(q^\circ), q^\circ) = L(K(q^\circ)) M(q^\circ) = L(K(q^\circ)) M^\circ$$ and $$1 = \cR(K(q), q) = L(K(q)) M(q) < L(K(q))M^\circ.$$ Then $L(K(q))> L(K(q^\circ))$. Since $L$ is decreasing, $K(q)< K(q^\circ)$. Step 1 and Step 2 imply that $Q^\circ= Q^\di$. The following example in which the birth rate is of Ricker type and the death rate is constant shows that (A4) is very restrictive and that without (A4) maximizing the carrying capacity may be different from maximizing the basic reproduction number. We will learn in the next section that, without (A4), it is the carrying capacity that is maximized. $$B (x,q) = \kappa_q e^{- \eta_q x} , \qquad D(x,q) = e^{\theta x}.$$ Then $$\cR(x,q) = \kappa_q e^{ -(\eta_q + \theta) x}.$$ Further $$\cR_0(q) = \kappa_q, \qquad K(q) = \frac{\ln \kappa_q}{\eta_q + \theta}.$$ Now let $Q = \{q_1, q_2, q_3\}.$ We choose $\kappa_{q_1} > \kappa_{q_2} >1\ge \kappa_{q_3}$, but $\eta_{q_1} $ much larger than $\eta_{q_2} $. Then strategy $q_1$ has a larger basic reproduction number but a smaller carrying capacity than strategy $q_2$, $K^\di = K(q_2)$ and $k_\di = K(q_3)=0$. So $$\cR(k_\di, q_2) = \cR(0, q_2) = \kappa_{q_2} < \kappa_{q_1} = \cR(k_\di, q_1),$$ falsifying (A4). It is not clear whether this counterexample works if $k_\di > 0$. In this subsection we will sometimes assume the following: - There is a unique strategy with largest carrying capacity, i.e., there is a unique $q^\di$ such that $ K (q^\di) = K^{\diamond}.$ Under these assumptions, we show that if a population is evolving according to the pure selection dynamics , then a multiple of $\delta_{q^\di} $ attracts all solutions $\mu(t)$ that embrace $q^\di$ as a possible strategy. \[CSS1\] Assume that (A1)-(A5) hold, then $q^\di$ is an $ASS$. That is, if the population $ \mu(t) $ is evolving according to the pure selection dynamics and $ q^\di \in supp (\mu(0))$, then $$\mu{(t)} \to K^\di \delta_{q^\di}, \quad t \to \infty.$$ in the weak$^*$ topology. Two technical propositions are required. We first show that strategies different from the optimal strategies are not adopted in the long run. \[extinction\] Assume (A1)-(A4). Let $U_0$ be an open set such that $Q^\di \subseteq U_0 \subseteq Q$. Then there exists some open set $U$ with $Q^\di \subseteq U \subseteq Q$ such that $\mu (t, Q \setminus U_0) \to 0$ as $t \to \infty$ for all solutions $ \mu$ with $\mu(0)(U) > 0$. The following statement will provide the assertion of the proposition. > Claim: > > : If $\check q \in Q \setminus Q^\di$, then there exists some $\delta = \delta(\check q) >0$ such that ${\mu(t)( U_\delta(\check q))} \to 0$ as $ t \to \infty$ for all solutions $\mu$ with $\mu(0)(U_\delta (Q^\di))>0$. > > Here $U_\delta(\check q)$ denotes the $\delta$-neighborhood of $\check q$. We first show that this claim implies the assertion of the proposition, indeed. Set $Q_0 = Q \setminus U_0$. Then $Q_0$ is a compact subset of $Q$ and $Q_0 \cap Q^\di = \emptyset$. There is a finite subset $\check Q$ of $Q_0$ such that $Q_0$ is contained in the union of finitely many open sets $V_q = U_{\delta}( q) $, $ q \in Q_0$, where $\delta=\delta_q$ has been chosen according to the claim. Let $\epsilon = \min_{q \in Q_0} \delta_q$ and $U = U_\epsilon (Q^\di)$. Then, for all solutions $\mu$ with $\mu(0)(U) > 0$, $$\mu(t)( Q_0) \le \sum_{q \in \check Q} \mu(t)( V_q ) \to 0, \qquad t \to \infty,$$ because $\check Q$ is finite. We now turn to proving the claim. Let $\breve q \in Q \setminus Q^\di $ and $\delta > 0$. Let $$U_\delta = U_\delta (\breve q), \qquad V_\delta = U_\delta (Q^\di),$$ where $U_\delta (Q^\di)= \{q \in Q; d(x, Q^\di)< \delta\}$ is the $\delta$-neighborhood of $Q^\di$ and $d$ the distance function extending the metric $d$. We consider $$x(t)= \mu(t)( V_\delta), \qquad y(t) = \mu(t)( U_\delta), \qquad t \ge 0.$$ Assume that $x(0) > 0$, and so $x(t) > 0$ for all $t > 0$. So we can also consider the function $$z = y^\xi x^{-1},$$ where the number $\xi>0$ will be suitably determined. (A similar function has been considered in [@AA1; @AMFH].) The strategy of our proof is to show that, for sufficiently small $\delta >0$, $z(t) \to 0$ and so $y(t) \to 0$ because $x$ is bounded. We can assume that $y(0) > 0$, otherwise $y$ is identically 0. Thus $y(t) >0$ for all $t \ge 0$. Notice that $$\label{eq:quotient-function} {\frac{z'}{z} = \xi \frac{y'}{y} - \frac{x'}{x}.}$$ Further $$y' = \int_{U_\delta} (B(\bar \mu,q) - D(\bar \mu,q)) \mu(dq), \qquad x' = \int_{V_\delta} (B(\bar \mu,q) - D(\bar \mu,q)) \mu(dq).$$ By continuity, there exist $ q^\di \in \bar V_\delta$ and $\hat q \in \bar U_\delta$ such that $B(\bar \mu,q) - D(\bar \mu,q) \ge B(\bar \mu,q^\di ) - D(\bar \mu,q^\di)$ for all $q \in U_\delta$ and $B(\bar \mu,q) - D(\bar \mu,q) \le B(\bar \mu,\hat q ) - D(\bar \mu,\hat q)$ for all $q \in V_\delta$. [ (Note $q^\di$ is not necessarily a point in $Q^{\di})$]{}. Hence $$\begin{split} y' \le & (B(\bar \mu,\hat q) - D(\bar \mu,\hat q )) y = D(\bar \mu, \hat q) (\cR(\bar \mu, \hat q) -1)y, \\ x' \ge & (B(\bar \mu, q^\di) - D(\bar \mu, q^\di )) x = D(\bar \mu, q^\di) (\cR(\bar \mu, q^\di ) -1)x. \end{split}$$ We substitute these formulas into (\[eq:quotient-function\]), $$\frac{z'}{z} \le \xi D(\bar \mu, \hat q) (\cR(\bar \mu, \hat q) -1)- D(\bar \mu, q^\di) (\cR(\bar \mu, q^\di) -1).$$ This can be rewritten as $$\frac{z'}{z} \le [\xi D(\bar \mu, \hat q)- D(\bar \mu, q^\di)] (\cR(\bar \mu, q^\di) -1) + D(\bar \mu, \hat q) [\cR(\bar \mu, \hat q) - \cR(\bar \mu, q^\di) ].$$ By continuity, compactness and (A4), one can find some $\eta > 0$ and some $\bar K > K^\di$ and, if $k_\di > 0$, some $ \bar k \in [0, k_\di)$ such that, for all sufficiently small $\delta > 0$, $$\cR(\bar \mu, \hat q) - \cR(\bar \mu,q^\di) < - \eta$$ whenever $\bar \mu \in [\bar k, \bar K]$ and $q^\di \in \bar V_\delta$ and $\hat q \in \bar U_\delta $. By Theorem \[BS\], for sufficiently large $t > 0$, $\bar \mu \in [\bar k, \bar K]$ and $$\frac{z'}{z} \le [\xi D(\bar \mu, \hat q)- D(\bar \mu, q^\di)] (\cR(\bar \mu, q^\di) -1) - \eta D(0, \hat q) .$$ Recall that $D$ is bounded on $[\bar k, \bar K] \times Q$ and bounded away from zero on $\R_+ \times Q$. So, by choosing $\xi >0 $ small enough, we obtain that $\xi D(\bar \mu, \hat q)- D(\bar \mu, q^\di) \le 0$ for all sufficiently large $t$. Since $\cR(\bar \mu, q) \ge 1$ for all $\bar \mu \in [0, K^\di]$ and $q \in Q^\di$, for arbitrary $\epsilon > 0$ we can arrange by choosing $\delta >0$ small enough and $\bar K$ close enough to $K^\di$ that $$\cR(\bar \mu, q^\di) \ge 1 - \epsilon, \qquad \bar \mu \in [0, \bar K], q^\di \in \bar V_\delta.$$ So $$\frac{z'}{z} \le [\xi D(\bar \mu, \hat q)- D(\bar \mu, q^\di)] \epsilon - \eta D(0, \hat q) .$$ By choosing $\delta >0$ small enough, we can have $\epsilon >0$ small enough to have the right hand side being smaller than a negative constant. This implies that $z(t)$ decreases exponentially as we wanted to show. \[re:converge-total\] Assume (A1)-(A4). Let $K^\di >0$. Then, for any $\epsilon \in (0,K^\di)$, there exists an open set $W_\epsilon$ such that $Q^\di \subseteq W_\epsilon \subseteq Q$ and $\liminf_{t \to \infty} \mu(t)( Q) \ge K^\di -\epsilon$ for all solutions of (\[selection\]) with $\mu(0)(W_\epsilon) > 0$. Let $\epsilon \in (0,K^\di)$. Set $K_\epsilon = K^\di -\epsilon$. By Proposition \[re:select1\], there exist a neighborhood $\tilde U_\epsilon$ of $Q^\di$ and some $\xi_\epsilon > 1$ such that $\cR (s, q) \ge \xi_\epsilon$ for all $s\in [0,K_\epsilon]$ and $q \in \tilde U_\epsilon$. By Proposition \[extinction\], there exists an open set $V_\epsilon$ such that $Q^\di \subseteq V_\epsilon \subseteq Q$ and $\mu(t)( Q \setminus \tilde U_\epsilon) \to 0$ as $t \to \infty$ for all solutions $\mu$ with $\mu(0) (V_\epsilon) > 0$. By Proposition \[re:select1\], $\limsup_{t\to \infty} \mu(t)( \tilde U_\epsilon) > K_\epsilon$ if $\mu(0) (\tilde U_\epsilon) > 0$. Set $W_\epsilon = \tilde U_\epsilon \cap {V_\epsilon}$. Suppose $$\bar \mu_\infty:= \liminf_{t \to \infty} \mu(t)(Q) < K_\epsilon \quad \hbox{ and }\mu(0)(W_\epsilon ) >0.$$ Since ${\mu(t)(\tilde U_\epsilon)}$ does not converge as $t \to \infty$, a version of the fluctuation lemma [@HHG][@Thi03 Prop.A.20] provides a sequence $(t_n)$ with $t_n \to \infty$ such that $$\mu(t_n)( \tilde U_\epsilon) \to \bar \mu_\infty < K_\epsilon$$ and $(d/dt) \mu(t_n)( \tilde U_\epsilon) =0$. Then $$0= \mu'(t_n)( \tilde U_\epsilon) = \int_{\tilde U_\epsilon} [\cR(\bar \mu (t_n), q) -1] D(\bar \mu(t_n) ,q) \mu (t_n)( dq).$$ For sufficiently large $n$, $\bar \mu(t_n) \le K^\di -\epsilon$ and $$0 \ge (\xi -1) \inf_{q \in Q} D(0,q) \mu(t_n)( \tilde U_\epsilon) >0.$$ This contradiction finishes the proof. The next result now follows in combination with Proposition \[re:select1\]. \[corollary\] Assume (A1)-(A4). If $\mu$ is a solution of (\[selection\]) such that $\mu(0)(U) >0$ for all open sets $U$ with $Q^\di \subseteq U \subseteq Q$, then $\mu(t)( Q) \to K^\di$ as $t \to \infty$. The following result extends Theorem \[CSS1\], since Theorem \[CSS1\] is an obvious corollary with $Q^\di =\{q^\di \}$. \[re:converge-prep\] Assume (A1)-(A4). Let $\mu$ be a solution of (\[selection\]) such that $\mu(0)(U) > 0$ for all open sets $U$ with $Q^\di \subseteq U \subseteq Q$. Then, for all $f \in C(Q)$, $$\liminf_{t\to \infty } \int_Q f(q) \mu(t)(dq) \ge K^\di \inf_{Q^\di} f$$ and $$\limsup_{t\to \infty } \int_Q f(q) \mu(t)(dq) \le K^\di \sup_{Q^\di} f.$$ Define $$f^\di = \sup_{Q^\di} f, \qquad f_\di = \inf_{Q^\di} f.$$ Let $\epsilon > 0$ and $U = \{q \in Q; f(q) < f^\di + \epsilon\}$. Since $f$ is continuous, $U$ is an open subset of $Q$ and $Q^\di \subseteq U \subseteq Q$. By Proposition \[extinction\], ${\mu(t)( Q \setminus U)} \to 0$ as $t \to \infty$. Now $$\begin{split} & \limsup_{t \to \infty } \int_Q f(q) \mu(t)(dq) \le \limsup_{t \to \infty } \Big (\int_U f(q) \mu(t)(dq) + \sup_Q f \; \mu(t)(Q \setminus U) \Big ) \\ \le & (f^\di + \epsilon) \limsup_{t \to \infty } \mu(t)( U) = (f^\di + \epsilon) \limsup_{t \to \infty } \mu(t)( Q) = (f^\di + \epsilon) K^\di. \end{split}$$ Since this holds for any $\epsilon >0$, the statement for the limit superior follows. The proof for the limit inferior is similar. Recall the set of equilibrium measures $${{\cal M}}^\di = \{ \nu \in {{\cal M}}_{w+}; \nu(Q) = \nu(Q^\di) = K^\di\} \textcolor{red}{.}$$ \[re:converge-prep2\] Assume (A1)-(A4). Let $\mu$ be a solution of (\[selection\]) such that $\mu(0)(U) > 0$ for all open sets $U$ with $Q^\di \subseteq U \subseteq Q$. For every sequence $(t_n)$ with $t_n \to \infty$, there exists a subsequence $(t_{n_j})$ and some $\nu \in {{\cal M}}^\di$ such that, for all $f \in {C(Q)}$, $$\int_Q f(q) \mu(t_{n_j})( dq) \to \int_{Q^\di} f(q) \nu(dq), \qquad j \to \infty.$$ Let $(t_n)$ be a sequence with $t_n \to \infty$. By Lemma \[re:measures-weak-top\], there exists some $\nu \in {{\cal M}}_+(Q)$ and a subsequence $(t_{n_j})$ such that $$\lim_{j \to \infty} \int_Q f(q) \mu(t_{n_j})(dq) \to \int_Q f(q) \nu (dq), \qquad f \in {C(Q)}.$$ For $f \equiv 1$, $\nu(Q) = \lim_{j \to \infty} \mu(t_{n_j}) (Q) =K^\di$ by Corollary \[corollary\]. Let $U$ be open such that $Q^\di \subseteq U \subseteq Q$. Then there exists some $\delta > 0$ such that $U \supseteq \bar V_\delta$ where $\bar V_\delta$ is the closure of the open set $V_\delta:= \{q \in Q ; d(q, Q^\di) < \delta\}$. Now $Q \setminus U$ is compact and $Q \setminus \bar V_\delta$ is open and $Q \setminus U \subseteq Q \setminus \bar V_\delta$. So there exists some $f \in {C_{+}(Q)}$ with values between 0 and 1 such that $f \equiv 1 $ on $Q \setminus U$ and $f \equiv 0$ on $\bar V_\delta$. By Proposition \[extinction\], $$\nu (Q \setminus U) \le \int_Q f(q) \nu(dq) = \lim_{j\to \infty} \int_Q f(q) \mu(t_{n_j})( dq) \le \limsup_{j\to \infty} { \mu(t_{n_j})(Q \setminus V_\delta)} =0.$$ Now $Q^\di = \bigcap_{m\in \N} V_{1/m} $ and $Q \setminus Q^\di = \bigcup_{m\in \N} (Q \setminus V_{1/m} )$. Since $\nu$ is countably additive, $ \nu (Q \setminus Q^\di) = \lim_{m \to \infty } \nu (Q \setminus V_{1/m} )=0. $ The following result also extends Theorem \[CSS1\]. \[re:meas-select\] Assume (A1)-(A4). Let $\mu$ be a solution of (\[selection\]) such that $\mu(0)(U) > 0$ for all open sets $U$ with $Q^\di \subseteq U \subseteq Q$. Then there exists a function $\mu^\di: \R_+ \to {{\cal M}}^\di$ with the following properties. - For all $B \in \cB$, $\mu^\di (\cdot)( B)$ is measurable on $\R_+$; - $ \displaystyle \int_Q f(q) \mu(t)(dq) - \int_{Q^\di} f(q) \mu^\di (t)(dq) \to 0, \qquad t \to \infty, $ for all $f \in C(Q)$. ${{\cal M}}^\di$ is compact and sequentially compact with respect to the weak$^*$ topology and the weak$^*$ topology on ${{\cal M}}^\di$ can be induced by the norm $p$ in Lemma \[re:measures-weak-top\]. Now let $\mu(\cdot) $ be a solution of the pure selection equation. Define $g: \R_+ \times {{\cal M}}^\di\to \R_+$ by $$g(t, \nu) = p(\mu(t) - \nu), \qquad t \ge 0, \nu \in {{\cal M}}^\di.$$ Then $g$ is continuous on $\R_+ \times {{\cal M}}^\di$. Since ${{\cal M}}^\di$ is a compact metric space, it is complete and separable. By a measurable selection theorem (see [@BrPu], e.g.), there exists a Borel measurable function $\mu^\di: \R_+ \to {{\cal M}}^\di$ such that $$\label{eq:meas-select} g(t, \mu^\di(t))= \inf_{\nu \in {{\cal M}}^\di} g(t, \nu).$$ We claim that $g(t, \mu^\di(t)) \to 0$ as $t \to \infty$. Suppose not. Then there exists some $\epsilon > 0$ and a sequence $(t_n)$ with $t_n \to \infty$ such such $g(t_n, \mu^\di(t_n)) \ge \epsilon$ for all $n \in \N$. By Proposition \[re:converge-prep2\], there exists some $\nu \in {{\cal M}}^\di $ and a subsequence $(t_{n_j})$ such that $g(t_{n_j}, \nu)= p(\mu (t_{n_j})- \nu) \to 0$. By (\[eq:meas-select\]), $g(t_{n_j}, \nu (t_{n_j})) \to 0$, a contradiction. By construction, $p(\mu(t)- \mu^\di(t)) \to 0$ as $t \to \infty$ and so $$\int_Q f(q) \mu(t)(dq) - \int_{Q^\di} f(q) \mu^\di(t)(dq) \to 0, \qquad t \to \infty, f \in C(Q).$$ For all $f \in C(Q)$, $\int_Q f(q) \mu^\di(t)(dq)$ is a Borel measurable function of $t$. Standard arguments imply that $\mu^\di (t)( B)$ is a Borel measurable function of $t$ first for all open subsets of $Q$ and then for all Borel subsets of $Q$. Directed mutation kernels {#sec:dir-mut} ========================= Alternatively to (A4), we assume that $Q^\di$ is an open subset of $Q$. We make this assumption because it makes $\chi_{Q^\di}$ continuous. It has the unfortunate consequence that $Q^\di$ is separated from the rest of $Q$ and that $Q$ is not connected. \[def:directed\] Let $q^\di \in Q^\di$. A mutation kernel $\gamma: Q \to {{\cal P}}_{w}$ is called [*directed to $q^\di $*]{} if $q^\di$ is an isolated point of $Q$ and the following hold: - For all $q \in Q^\di $, $ \gamma(q)( \{q^\di\}) > 0$ and $\gamma (q) ( Q^\di) =1$. - $\gamma (q^\di) (\{q^\di\}) =1$. It is easy to see that there is at most one $q^\di \in Q^\di$ a mutation kernel can be directed to. Notice that every directed mutation kernel is optimum preserving (Definition \[def:kernel-opt-pres\]). In turn, if $Q^\di =\{q^\di\}$ and $\gamma$ is optimum preserving, then $\gamma$ is directed to $q^\di$. If $\gamma$ is a mutation kernel directed to $q^\di$, then $K^\di \delta_{q^\di}$ is an equilibrium of (\[M1\]). If $\gamma$ is the no-mutation kernel $\gamma(q) = \delta_q$ and $Q^\di =\{q^\di\}$ is a singleton set and $q^\di$ is an isolated point of $Q$, then the no-mutation kernel is trivially directed to $q^\di$. \[re:directed\] Assume (A1)-(A3). Let $q^\di \in Q^\di $ and $\gamma$ be a mutation kernel directed to $q^\di $. Then, for all compact subsets $C$ of ${{\cal M}}_{w+}$ with $u (Q^\di) > 0$ for all $u \in C$, $\phi (t, u, \gamma) \to K^\di \delta_{q^\di}$ as $t \to \infty$ uniformly for $u \in C$. By Theorem \[re:pers-robust\] (c), in the language of [@SmTh], the semiflow $\phi (\cdot, \gamma)$ induced by (\[M1\]) is uniformly $\rho$-persistent for $\rho (\nu) = \nu (Q^\di)$. By Corollary \[re:compact-glob-attr\], this semiflow has a compact attractor $A_{\gamma}$ of bounded sets in ${{\cal M}}_{w+}$ with $\nu (Q) \le K^\di$ for all $\nu \in A_{\gamma}$. By Theorem \[re:pers-robust\] (B), the semiflow has a $\rho$-persistence attractor $A_1\subseteq A$, i.e., a compact invariant set $A_1\subseteq A$ which attracts all compact sets $V$ in ${{\cal M}}_{w_+}$ for which $\nu(Q^\di) >0$ for all $\nu \in V$. We claim that $A_1 = \{K^\di \delta_{q^\di}\}$. We apply [@SmTh Thm.2.53] with $A= A_1$ and $\tilde A = \{K^\di \delta_{q^\di}\}$. Let $\mu: \R \to A_1$ be a solution of (\[M1\]) on $\R$. Since $\mu$ is defined on the whole real line, it corresponds to a total trajectory. Since $\mu$ takes its values in the $\rho$-persistence attractor, we have $\inf_{t \in \R} \mu(t)(Q^\di) > 0$. By a similar proof as the one of Lemma \[re:pos-pres\] (b), $\mu(t)(\{q^\di\})>0$ for all $t \in \R$ because of Definition \[def:directed\] (a). For $\nu \in A_1$ with $\nu (\{q^\di\}) > 0$, we define the Volterra type Lyapunov-function-to-be $$L(\nu) = \nu(\{q^\di\}) + K^\di (\ln K^\di - \ln \nu(\{q^\di\}) + c \nu (Q \setminus Q^\di).$$ It follows from our assumptions that $\chi_{\{q^\di\}}$ and $\chi_{Q \setminus Q^\di}$ are continuous. So $L$ depends continuously on $\nu$ in the $w^*$ topology. $L(\mu(t))$ is differentiable in $t$ and $(d/dt) L(\mu(t)) = \dot L (\mu(t))$ where $\dot L$ is the orbital derivative of $L$ along (\[M\]), $$\begin{split} \dot L(\nu) = & \int_Q B(\nu (Q), q) \gamma(q)( \{q^\di\}) \nu(dq) \Big (1 - \frac{K^\di}{\nu(\{q^\di\})}\Big ) \\ - & D(\nu(Q), q^\di) (\nu(\{q^\di\}) - K^\di ) \\ + & c \int_{Q } B(\nu(Q),q ) \gamma(q)( Q\setminus Q^\di) \nu(dq) \\ - & c \int_{Q\setminus Q^\di} D(\nu(Q),q ) \nu(dq). \end{split}$$ Set $G (x,q) = B(x,q) - D(x,q)$ for $x \ge 0$. Since $ \gamma(q) (Q \setminus Q^\di) =0 $ for all $q \in Q^\di$ and $ \gamma (q^\di) ( \{q^\di\}) =1 $ by assumption, $$\begin{split} \dot L(\nu) \le & \int_{Q \setminus \{q^\di\}} B(\nu (Q), q) \gamma(q)( \{q^\di\}) \nu(dq) \Big (1 - \frac{K^\di}{\nu(\{q^\di\})}\Big ) \\ + & G(\nu(Q), q^\di) (\nu(\{q^\di\}) - K^\di ) \\ + & c \int_{Q \setminus Q^\di} G(\nu(Q),q ) \nu(dq). \end{split}$$ Since $\nu$ is an element in the global attractor, $\nu(\{q^\di\}) \le K^\di$ and the first term on the right hand side is nonpositive, $$\begin{split} \dot L(\nu) \le & \int_{Q^\di \setminus \{q^\di\}} B(\nu (Q), q) \gamma(q) ( \{q^\di\}) \nu(dq) \Big (1 - \frac{K^\di}{\nu(\{q^\di\})}\Big ) \\ & + G(\nu(Q), q^\di) (\nu(\{q^\di\}) - K^\di ) + c \int_{Q\setminus Q^\di} G(\nu(Q),q ) \nu(dq). \\ \end{split}$$ Define $$b = \inf \{ B(x,q); 0 \le x \le K^\di, q \in Q^\di\}, \qquad \gamma_\di =\inf_{q \in Q^\di\setminus \{q^\di\}} \gamma (q)( \{q^\di\}).$$ It follows from (A3) that $b > 0$. Since $\gamma^\di$ is directed towards $q^\di$, $\gamma_\di > 0$. After rearrangement, $$\begin{split} \dot L(\nu) \le & \; b \gamma_\di \nu(Q^\di \setminus \{q^\di\}) \Big (1 - \frac{K^\di}{\nu (\{q^\di\})} \Big ) \\ & + G(\nu(Q), q^\di) (\nu(Q) - K^\di ) + G(\nu(Q), q^\di) (\nu(\{q^\di\}) - \nu(Q) ) \\ & + c \int_{Q \setminus Q^\di} G(\nu(Q),q ) \nu(dq). \end{split}$$ Since $\nu(Q) \le K^\di$, $G(\nu(Q), q^\di) \ge 0$ and so $$\begin{split} \dot L(\nu) \le & \; b \gamma_\di \nu(Q^\di \setminus \{q^\di\}) \Big (1 - \frac{K^\di}{\nu (\{q^\di\})} \Big ) + G(\nu(Q), q^\di) (\nu(Q) - K^\di ) \\&+ G(\nu(Q), q^\di) (\nu({Q^\di}) - \nu(Q) ) + c \int_{Q \setminus Q^\di} G(\nu(Q),q ) \nu(dq). \end{split}$$ We rearrange, $$\begin{split} \dot L(\nu) \le & \;b \gamma_\di \nu(Q^\di \setminus \{q^\di\}) \Big (1 - \frac{K^\di}{\nu (\{q^\di\})} \Big ) + G(\nu(Q), q^\di) (\nu(Q) - K^\di ) \\& + \int_{Q \setminus Q^\di} [cG(\nu(Q),q ) - G(\nu(Q), q^\di)] \nu(dq). \end{split}$$ Let $\tilde K = \sup_{q \in Q \setminus Q^\di} K(q)$. Since $Q \setminus Q^\di$ is compact, $\tilde K < K^\di$. Let $q \in Q \setminus Q^\di$. Notice that $G(X, q^\di) > 0$ for all $X \in [0,\tilde K]$. So, by choosing $c>0$ small enough, we can achieve that $$cG(x, q) - G(x, q^\di) <0 , \qquad x \in [0,\tilde K], q \in Q \setminus Q^\di.$$ For $x \in (\tilde K, K^\di]$, we have $G(x,q) <0$ for $q \in Q \setminus Q^\di$, and for $x \in [\tilde K, K^\di)$ we have $G(x, q^\di)>0$. So for all $q \in Q \setminus Q^\di$ and $x \in [\tilde K, K^\di]$, $$cG(x, q) - G(x, q^\di) <0 .$$ In combination, this inequality holds for all $q \in Q \setminus Q^\di$ and $x \in [0,K^\di]$. So, if $c>0$ is chosen small enough, $\dot L(\nu) \le 0$ and $\dot L(\nu) =0$ only if $\nu(Q) =K^\di $ and $\nu(Q\setminus Q^\di)=0 $ and so $\nu(Q^\di)=K$ as well. Moreover $\dot L(\nu) =0$ only if $\nu(\{q^\di\} ) = K^\di$ or $\nu (Q \setminus \{q^\di\}) =0$. Combined with the other information, $\dot L(\nu) =0$ only if $\nu(Q) =K^\di = \nu(\{q^\di\})$, i.e., $\nu $ is the point measure concentrated at $q^\di$ taking the value $K^\di$. Recall that we apply [@SmTh Thm.2.53] with $A= A_1$ and $\tilde A = \{K^\di \delta_{q^\di}\}$. All we need to show is that every solution $\mu : \R \to A_1 $ of (\[M\]) with $\dot L (\mu(t)) = 0 $ satisfies $\mu(t) = K^\di \delta_{q^\di}$ which we just did. Assume (A1)-(A3) and let $Q^\di$ be open in $Q$ and $q^\di \in Q^\di$ be an isolated point of $Q$ and $\gamma^\di $ be a mutation kernel directed towards $q^\di$. Assume that $K^\di > 0$. Then, for any $\epsilon > 0$, there exists some $\delta_\epsilon > 0$ such that $$\limsup_{t\to \infty} \phi(t, u, \gamma) (Q \setminus \{q^\di\}) < \epsilon, \quad \limsup_{t\to \infty} |\phi(t,u, \gamma ) (Q) - K^\di| < \epsilon$$ for all $\gamma \in C(Q, {{\cal P}}_w)$ with $\|\gamma- \gamma^\di\| < \delta_\epsilon$ and all $u \in {{\cal M}}_{w+}$ with $u(Q^\di)>0$. Let $\epsilon > 0$ and let $W$ be the set of measures $\nu$ with $\nu(Q\setminus \{q^\di\}) < \epsilon $ and $|\nu (Q) - K|< \epsilon$. $W$ is an open neighborhood of $ K^\di \delta_{q^\di}$ in the weak$^*$-topology. By Theorem \[re:pers-robust\], there exists some $\delta_\epsilon > 0$ such that $\tilde A_\gamma \subseteq W$ for all $\gamma \in C(Q, {{\cal P}}_w) $ with $\|\gamma- \gamma^\di\| < \delta_\epsilon $. Since $\tilde A_\gamma$ is the persistence attractor for $\phi(\cdot, \gamma)$, for any $u \in {{\cal P}}_{w+}$ with $u(Q^\di) > 0$ there exists some $t_u > 0$ such that $\phi (t,u,\gamma) \in W$ for $t \ge t_u$. This implies the assertion. Discrete strategies and small mutations {#sec:disc} ======================================= The results of this section are for finitely many strategies. This means that $Q$ is a finite set. Any finite set becomes a metric space if equipped with the discrete metric to which any other metric on it is equivalent. Notice that all points of $Q$ are isolated and all subsets of $Q$ are both compact and open. The main result of this section is to demonstrate that, if there is a unique strategy $q^\di \in Q$ under which the carrying capacity is maximal, $K(q^\di) = K^\di$, then there is a neighborhood around the pure selection kernel where unique equilibria are obtained which attract all solutions which adopt this strategy at least partially. We first show that in the above case the model reduces to a system of ordinary differential equations. To this end, let $Q=\{q_{i}\}_{i=1}^{N}$. Then $\mathcal{M}= \text{span}_{\mathbb{R}}\{\delta_{e_{i}}\}$ can be identified with $\R^N$ and $C(Q,{\cal P}_w )$ with the set $\Gamma$ of nonnegative $ N\times N$ matrices whose rows sum to one. Let $ x_j(t) =\mu(t)(\{q_j\})$, $B_{j}(\bar x) = B(\bar x, q_j)$ and $D_{j}(\bar x) = D(\bar x, q_j)$ where $ \bar x=\sum_{j=1}^{N} x_j$, then the system reduces to the following differential equations system: $$\left\{\begin{array}{ll} \displaystyle \frac{d}{dt}{x_j}(t;u, \gamma) = \sum_{i=1}^N B_i (\bar x(t))x_i(t)\gamma_{ij} - D_j(\bar x(t))x_j(t), \qquad j = 1, . . . , N , \\ x_j(0;u,\gamma) = u_j, \end{array}\right. \label{discrete}$$ where $\gamma=\{ \gamma_{ij} \}$ is a row stochastic matrix. The pure selection kernel $ \gamma( \hat q) = \delta_{\hat q} \in C(Q,{\cal P}_w )$ is represented as the $\I = I_{N\times N}$ identity matrix. Note that in $\gamma_{ij}$ represents the proportion of strategy $i$ offspring that belong to strategy $j$. Since the sum of the proportions of offspring of strategy $i$ must be one, $\sum_{j=1}^N \gamma_{ij}= 1$ for all $i=1, \ldots, N$. The norm on $C(Q, {{\cal P}}_w)$ given by (\[eq:metric-mutation\]) is equivalent to any of the matrix norms on the set $\Gamma$ of row stochastic matrices. Notice that $\Gamma$ is compact. Furthermore, let $K_j$ denote $K(q_j)$, the carrying capacity of strategy $j$. The goal in this section is to study the dynamics of when $\gamma$ is a small perturbation of the identity. To this end, we assume for the rest of this section that the fittest strategy is unique and that it occurs at $q_1$, without loss of generality. Thus, in this case $K^\diamond = K_j = K_1$. We assume $K_1 > 0$ and $K_1 > K_j$ for $j=2, \ldots, N$ and - $B_j$ and $D_j$ are continuously differentiable [on $[0,\infty)$]{} and $ \displaystyle B_1'(K_1) - D_1'(K_1) < 0 $. To establish our asymptotic behavior result we recall the following theorem: [Let $x:[0,\infty) \to \R^N$ and]{} $$\label{finite} {x}^\prime = f(x, \lambda),$$ where $ f: U \times \Lambda \rightarrow \mathbb{R}^N $ is continuous, $ U \subseteq \mathbb{R}^{N}$, $\Lambda \subseteq \mathbb{R}^{k}$ and $ \frac{\partial f}{\partial x} (x,\lambda) $ is continuous on $ U \times \Lambda $. We write $x(t, z, \lambda) $ for the solution of satisfying $x(0)=z.$ \[pert1\] (Smith and Waltman [@SmithWalt]) Assume that $(x_0, \lambda_{0}) \in U\times\Lambda $, $x_0 \in int(U)$, $f(x_0, \lambda_0) =0$, all eigenvalues of $\frac{\partial f}{\partial x}(x_0, \lambda_0)$ have negative real part, and $x_0$ is globally attracting in $U$ for solutions of with $\lambda =\lambda_0 $. If (H1) there exists a compact set $ \mathcal{D} \subseteq U$ such that for each $ \lambda \in \Lambda$ and each $ z \in U$, $x(t,z, \lambda) \in \mathcal{D}$ for all large t, then there exists $\epsilon >0$ and a unique point $\widehat{x}( \lambda) \in U$ for $\lambda \in B_{\Lambda}( \lambda_0, \epsilon)$ (the ball in $\Lambda$ of radius $\epsilon$ around $\lambda_0$) such that - $\hat x(\lambda)$ depends continuously on $\lambda \in B_{\Lambda}( \lambda_0, \epsilon)$ and $\hat x (\lambda_0) = x_0$, - $f(\widehat{x}( \lambda),\lambda)=0$ and - $x(t,z, \lambda) \rightarrow \widehat{x}(\lambda)$, as $t\rightarrow \infty$ for all $z \in U$ and $\lambda \in B_{\Lambda}( \lambda_0, \epsilon)$. In order to apply Theorem \[pert1\] to our model we let $U=\{ x \in \mathbb{R}^N_+ ; x_1 > 0\}$, $\Lambda$ be an appropriate subset of $\Gamma$ still to be determined, and $ f: U \times \Lambda \rightarrow \mathbb{R}^N $, where $f= (f_j)_{j=1}^{N}$, $x= (x_j)_{j=1}^{N}$, $\lambda= \gamma= (\gamma _{ij})_{i,j=1}^N$ and $$f_j(x,\lambda)= \sum_{i=1}^N B_i (\bar x) x_i \gamma_{ij} - D_j(\bar x)x_j, \qquad \bar x = \sum_{k=1}^N x_k.$$ We also let $$(x_0,\lambda_0)=( K_1 e^1, \I),$$ with $e^1$ denoting the first of the canonical basis vectors of $\R^N$. One may notice that $(K_1 e^1)$ mentioned above is not an interior point of $U$. However, in Theorem \[pert1\] the assumption that $x_0$ is an interior point of $U$ is unnecessarily restrictive. One can use one-sided derivatives with respect to some cone or wedge [@SmithWalt]. Thus, for the model one can use one-sided derivatives of $f$ with respect to $\mathbb{R}^N_+$. We now have the following theorem describing the dynamics of the model when mutation is small: \[main-finite\] Assume that (A1)-(A3) and (A5), (A6) hold. Then there exists some $\delta > 0$ such that, for each matrix $\gamma \in \Gamma$ with $\|\gamma - \I \| < \delta$, there exists a stable equilibrium $x^*_\gamma $ of the ordinary differential equation system with $x^*_\gamma $ converging to $x_\I^*= K_1 e^1$ as $ \| \gamma - \I \| \rightarrow 0$. Furthermore, if $\|\gamma - \I \| < \delta $, $x^*_\gamma$ attracts all solutions $x$ of (\[discrete\]) with $x_1(0) > 0$. [Here $\|\cdot\|$ is any matrix norm for $N\times N$ matrices.]{} First note that $ f$ is continuous. Moreover, $ f( x^*_\I ,\I)= 0$ and, by Theorem \[re:directed\], $ x^*_\I$ is globally attractive for initial measures in $U$. Also, observe that the Jacobian matrix $\frac{\p f}{\p x} (x, \gamma)$ is continuous on $U \times \Gamma$, and evaluating it at $ ( x^*_\I ,\I)$ we obtain an upper triangular matrix with elements: $$\frac{\p f_j}{ \p x_i} (x^*_\I, \I) = \left \{ \begin{array}{ll} \displaystyle ( B_1'(K_1) - D_1'(K_1)) K_1, \quad& j=1, \; i=1,\dots, N, \\ B_j (K_1) - D_j (K_1), & i=j=2, \dots, N, \\ 0, & \hbox{ otherwise.} \end{array} \right.$$ Thus, the eigenvalues of the Jacobian matrix $\frac{\p f}{\p x} ( x^*_\I ,I)$ are given by the diagonal elements of this matrix, namely, $$\Big ( \frac{\partial B_1}{\partial x_1}(K_1)-\frac{\partial D_1}{\partial x_1}(K_1)\Big ) K_1, \qquad B_j (K_1) - D_j (K_1), \quad j =2, \ldots, N.$$ From assumptions (A1)-(A6) it is clear that these eigenvalues are real and negative. As for (H1), we use Theorem \[re:pers-robust\] with $Q = \{1, \ldots, N\}$ and $Q^\di = \{1\}$. Then there exists some $\tilde \delta > 0$ and some $\check \epsilon >0$ such $\liminf_{t\to \infty} x_1(t) \ge \check \epsilon $ for all solutions of (\[discrete\]) with $x_1(0) > 0$. To satisfy (H1) of Theorem \[pert1\], set $\Lambda = \{\gamma \in \Gamma: \|\gamma - \I\|< \tilde \delta \}$ and $D = \{y \in \R^N_+; y_1 \ge \check \epsilon, \sum_{j=1}^N y_j \le K_1 +1 \} \subseteq U $. All assertions follow from Theorem \[pert1\] except local stability of the equilibria $x^*_\gamma$ which follows from the fact that the Jacobian matrices $\frac{\p f}{\p x} (x^*_\gamma, \gamma)$ continuously depend on $\gamma$ and the spectral bound of a matrix continuously depends on the matrix. So all eigenvalues of the Jacobian matrices $\frac{\p f}{\p x} (x^*_\gamma, \gamma)$ are negative if $\|\gamma - \I \|$ is sufficiently small. We turn to the case that there are several strategies for which the carrying capacity is maximal. We assume that the mutation kernel is directed to one of those strategies. Without loss of generality, we assume that the carrying capacity is maximal for the first $m$ strategies and the mutation matrix is directed to the first strategy. More precisely, let $N \ge 2$ and $m \in \{2,\ldots, N\}$ and $K_1 = \cdots =K_m = K^\di$ and $K_j < K^\di$ for $j =m+1, \ldots, N$. In case that $m =N$, the last inequality is omitted. Then $\gamma^\di \in \Gamma$ is directed to the first strategy if $$\begin{array}{cl} \gamma_{11}^\di = 1, \qquad \gamma_{1j}^\di= 0 , \quad & j=2, \ldots, N, \\ \gamma_{i1}^\di > 0, \qquad & i =1 , \ldots, N, \\ \gamma_{ij}^\di =0, \qquad & i =1, \ldots, m, \quad j = m+1, \ldots, N. \end{array}$$ \[main-finite2\] Assume that (A1)-(A3) and (A6) hold and that $\gamma^\di$ is a mutation matrix that is directed to the first strategy as just explained. Then there exists some $\delta > 0$ such that, for each matrix $\gamma \in \Gamma$ with $\|\gamma - \gamma^\di \| < \delta$, there exists a stable equilibrium $x^*_\gamma $ of the ordinary differential equation system with $x^*_\gamma $ converging to $x^*_{\gamma^\di}= K_1 e^1$ as $ \| \gamma - \gamma^\di \| \rightarrow 0$. Furthermore, if $\|\gamma - \gamma^\di \| < \delta $, $x^*_\gamma$ attracts all solutions $x$ of (\[discrete\]) with $x_1(0) > 0$. The proof is the same as for Theorem \[main-finite\] except for proving that the Jacobian of $f$ evaluated at $(K^\di e^1, \gamma^\di)$ has a negative spectral bound. Notice that $$\begin{split} f_1( x, \gamma^\di) = & \sum_{i=1}^N B_i(\bar x) x_i \gamma^\di_{i1} - D_1(\bar x) x_1, \\ f_j(x, \gamma^\di) = & \sum_{i=2}^N B_i(\bar x) x_i \gamma^\di_{i1} - D_j(\bar x) x_j, \qquad j =2, \ldots, N, \\ \bar x = & \sum_{k=1}^N x_j. \end{split}$$ It is easy to see that $x^\di = x^*_{\gamma^\di} = K^\di e^1$ is an equilibrium of $f(\cdot, \gamma^\di)$. We do not need to compute all entries of the Jacobian matrix in order to determine the sign of its spectral bound, $$\begin{split} \frac{\p f_1}{\p x_1} (x^\di, \gamma^\di)= & (B_1'(K^\di) - D_1'(K^\di)) K^\di, \\ \frac{\p f_j}{\p x_1} (x^\di, \gamma^\di)= & 0, \qquad j =2, \ldots , N, \\ \frac{\p f_j}{\p x_j} (x^\di, \gamma^\di)= & B_j(K^\di) \gamma^\di_{jj} - D_j(K^\di), \qquad j =2, \ldots , N, \\ \frac{\p f_j}{\p x_i} (x^\di, \gamma^\di)= & B_i(K^\di) \gamma^\di_{ij} , \qquad i, j =2 , \ldots, N,\; i \ne j. \end{split}$$ From these equations, it is apparent that the spectral bound of $\frac{\p f}{\p x} (x^\di,\gamma^\di)$ is the larger of $ (B_1'(K^\di) - D_1'(K^\di)) K^\di < 0$ and the spectral bound of the matrix $\big (\frac{\p f_j}{\p x_i} (x^\di,\gamma^\di) \big )_{2 \le i,j\le N} $. The row sums of this latter matrix are $$\sum_{j=2}^N \frac{\p f_j}{\p x_i} (x^\di,\gamma^\di) = \sum_{j=2}^N B_i(K^\di) \gamma_{ij}^\di - D_i(K^\di) = B_i(K^\di) - D_i(K^\di) - B_i(K^\di) \gamma^\di_{i1}, \quad i =2, \ldots, N.$$ The last expression is negative: for $i=2, \ldots, m$ because $B_i(K^\di) = D_i(K^\di) >0$ and $\gamma^\di_{i1} > 0$; for $i =m+1, \ldots, N$ because $B_i(K^\di) < D_i(K^\di)$. By [@Thi03 Rem.A.48], the spectral bound of $(\frac{\p f_j}{\p x_i} (x^\di,\gamma^\di))_{2 \le i,j\le N} $ is negative and so is the spectral bound of $\frac{\p f}{\p x} (x^\di, \gamma^\di)$. Concluding Remarks {#sec:conclud} ================== We studied the long-time behavior of measure-valued solutions of a differential equation [that can be viewed as the model for an evolutionary game or as a selection-mutation population model]{}. We first assume that there is a unique fittest [strategy or trait]{}. This unique fittest trait is characterized by maximizing the carrying capacity $K(q)$. We [provided conditions under which]{}, in the pure replication case with a unique fittest [trait]{}, the model converges to a Dirac measure [concentrated]{} at the fittest trait [provided that the fittest trait is contained in the support of the initial measure.]{} [One sufficient condition is that this trait (strategy) does not only maximize the carrying capacity but also the reproduction numbers at all relevant population densities. Another sufficient condition is that the fittest strategy is isolated in the strategy space. If there are several fittest traits, the solutions converges to a set of measures the support of which is contained in the set of fittest traits.]{} [If mutations are allowed, we only could get results on the long-time behavior of solutions if we assumed that the set of fittest traits was topologically separated from the less fit traits. Convergence to the Dirac measure concentrated at the fittest trait holds if the mutations are directed in the sense that there are no mutation losses from the set of fittest traits and that among the fittest traits there is one particular trait such that mutations within this set are directed towards this particular trait. ]{} We also showed that, [for a discrete strategy space and mutations that are small or directed]{}, there is a locally asymptotically stable equilibrium that attracts all solutions with initial conditions that are positive at the fittest strategy. The abstract theory presented in this paper finds practical application in epidemic models. Epidemic models which consider the dynamics of multi-strain pathogens have been studied in the literature (e.g., [[@AA1; @BT; @SmTh13]]{}). These models have been formulated as systems of ordinary differential equations where infected individuals are distributed over a set of $n$ classes each carrying a particular strain of a finite and discrete [trait]{} (strategy) space. However, often a continuous (strategy) space is needed. For example, think of a particular disease that has transmission rate $\beta$ with possible values in the interval $[\underline \beta,\overline \beta]$. Then infected individuals are distributed over a [trait]{} space with transmission taking values in this interval. The theory presented here will have potential applications in treating such distributed rate epidemic models with possibly more than one fittest strain; however, at the present state of the theory, selection of the fittest strains could only be concluded if mutations are excluded.\ [**Acknowledgements:**]{} We thank Paul Salceanu and Ping Ng for helpful discussions. 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--- abstract: 'In this paper we study the following set$$A=\{p(n)+2^nd \mod 1: n\geq 1\}\subset [0.1],$$ where $p$ is a polynomial with at least one irrational coefficient on non constant terms, $d$ is any real number and for $a\in [0,\infty)$, $a \mod 1$ is the fractional part of $a$. By a Bernoulli decomposition method, we show that $A$ must have box dimension equal to one.' address: | Han Yu\ School of Mathematics & Statistics\ University of St Andrews\ St Andrews\ KY16 9SS\ UK\ author: - Han Yu title: Bernoulli decomposition and arithmetical independence between sequences --- [^1] Introduction ============ This paper is a self-contained sequel of [@Y19]. We use the idea of Bernoulli decomposition to consider a number theoretic problem. Given two sequences $x=\{x_n\}_{n\geq 1}, y=\{y_n\}_{n\geq 1}$ in $[0,1],$ it is often of great interest to study their independence. In terms of sequences with dynamical background, this can be also understood as the disjointness between dynamical systems, see [@F67] for more details. Intuitively, we want to say that two sequences $x,y$ are independent if $\{(x_n,y_n)\}_{n\geq 1}$ is in some sense close to the product set $X\times Y,$ where $X,Y$ are the sets of numbers in the sequence $x,y$ respectively. We give two natural ways of expressing this idea. Let $x=\{x_n\}_{n\geq 1}, y=\{y_n\}_{n\geq 1}$ be two sequences in $[0,1].$ We write $X,Y$ to be the sets of numbers in the sequence $x,y$ respectively. Then we say that $x$ and $y$ are: - [topologically independent]{} if $X\times Y$ is equal to the closure of the set of elements in the sequence $\{(x_n,y_n)\}_{n\geq 1}.$ - [arithmetically independent]{} if the set $H(x,y)$ of numbers in the sequence $\{x_n+y_n\}_{n\geq 1}$ attains the maximal box dimension, namely, $${\underline{\dim_{\mathrm{B}}}}H(x,y)=\min\{1, {\underline{\dim_{\mathrm{B}}}}X+{\underline{\dim_{\mathrm{B}}}}Y\}.$$ As an easy example we see that $\{n\alpha\}_{n\geq 1}$ and $\{n\beta\}_{n\geq 1}$ are topologically and arithmetically independent if $1,\alpha,\beta$ are linearly independent over the field $\mathbb{Q}.$ It is also possible to study the independence between $\{n\alpha\}_{n\geq 1}$ and $\{n^2\beta\}_{n\geq 1}$ based on Weyl’s equidistribution theorem. Naturally, a next question is to ask about the independence between $\{n\alpha\}_{n\geq 1}$ and $\{2^n d\}_{n\geq 1}$, where $d$ is any real number. For a polynomial $p$ with degree $k$ with real coefficients, we write $p(n)=\sum_{i=0}^k a_i n^i.$ We say that $p$ is irrational if at least one of the numbers $a_1,\dots,a_k$ is an irrational number. In this paper, we show the following result. See Section \[Ab\] for a clarification of the notation that appear below. \[main\] Let $p$ be an irrational polynomial and let $d$ be any real number. Then the sequences $\{p(n) \mod 1\}_{n\geq 1}$ and $\{2^n d\mod 1\}_{n\geq 1}$ are arithmetically independent. We note that there is a curious connection between sequences of form $\{p(n)+2^n d\mod 1\}_{n\geq 1}$ and $\alpha\beta$-sequences. Let $\alpha,\beta$ be two real numbers, an $\alpha\beta$-sequence $\{x_n\}_{n\geq 1}$ is such that $x_1=0$ and for each $i\geq 1$ we can choose $x_{i+1}=x_i+\alpha\mod 1$ or $x_{i+1}=x_i+\beta\mod 1$ freely. We have the following interesting problem. \[feng\] Let $\alpha,\beta$ be such that $1,\alpha,\beta$ are independent over the field of rational numbers. Then any $\alpha\beta$-sequence has full box dimension. This conjecture is related to affine embeddings between Cantor sets, symbolic dynamics and Diophantine approximation, see [@K79], [@FX18] and [@Y18b]. A lot of ideas for proving Theorem \[main\] appeared in [@Y18b] for $\alpha\beta$-sets. For this reason, we can consider Theorem \[main\] as a cousin of Conjecture \[feng\]. Although the Bernoulli decomposition method we introduce in this paper cannot be used directly for $\alpha\beta$-sequences, it still sheds some lights on Conjecture \[feng\]. We also consider here a number theoretic result which is closely related to the independence of sequences. Let $m$ be an odd number and we consider the ring $R[m]$ of residues modulo $m.$ This is the finite set $\{0,\dots,m-1\}$ together with the integer multiplication and addition modulo $m$. In this setting we can also consider the sequence $\{2^n+cn\mod m\}_{n\geq 0}$ where $c$ is an integer such that $gcd(c,m)=1.$ On one hand the $+c \mod m$ action on $R[m]$ can be seen as uniquely ergodic, which is analogous to $+\alpha\mod 1$ action on the unit interval with an irrational number $\alpha.$ On the other hand, $\{2^n \mod m\}_{n\geq 0}$ is an orbit under the $\times 2\mod m$ action. A analogy of Theorem \[main\] would be that $\{2^n+cn\mod m\}_{n\geq 0}$ is large in $R[m].$ We prove the following result which confirm this intuition. We remark that the method for proving the following result shares some strategy for proving Theorem \[main\]. This is a special case of Problem 6 in the third round of the 27-th Brazilian Mathematical Olympiad, [@MO]. For this reason, we will not provide the solution for the fully generalized case. \[num\] Let $m\geq 3$ be an odd number and $c$ be such that $gcd(c,m)=1$. Let $D(m)$ be the number of residue classes visited by $\{2^n+cn\mod m\}_{n\geq 0}.$ Then $D(m)=m.$ In other words, for each $r\in R[m],$ there is an integer $n_r$ such that $2^{n_r}+cn_r\equiv r\mod m.$ Preliminaries ============= Dimensions ---------- We list here some basic definitions of dimensions mentioned in the introduction. For more details, see [@Fa Chapters 2,3] and [@Ma1 Chapters 4,5]. We shall use $N(F,r)$ for the minimal covering number of a set $F$ in $\mathbb{R}^n$ with balls of side length $r>0$. ### Hausdorff dimension Let $g: [0,1)\to [0,\infty)$ be a continuous function such that $g(0)=0$. Then for all $\delta>0$ we define the following quantity $$\mathcal{H}^g_\delta(F)=\inf\left\{\sum_{i=1}^{\infty}g(\mathrm{diam} (U_i)): \bigcup_i U_i\supset F, \mathrm{diam}(U_i)<\delta\right\}.$$ The $g$-Hausdorff measure of $F$ is $$\mathcal{H}^g(F)=\lim_{\delta\to 0} \mathcal{H}^g_{\delta}(F).$$ When $g(x)=x^s$ then $\mathcal{H}^g=\mathcal{H}^s$ is the $s$-Hausdorff measure and Hausdorff dimension of $F$ is $${\dim_{\mathrm{H}}}F=\inf\{s\geq 0:\mathcal{H}^s(F)=0\}=\sup\{s\geq 0: \mathcal{H}^s(F)=\infty \}.$$ ### Box dimensions The upper box dimension of a bounded set $F$ is $$\overline{{\dim_{\mathrm{B}}}} F=\limsup_{r\to 0} \left(-\frac{\log N(F,r)}{\log r}\right).$$ Similarly the lower box dimension of $F$ is $${\underline{\dim_{\mathrm{B}}}}F=\liminf_{r\to 0} \left(-\frac{\log N(F,r)}{\log r}\right).$$ If the limsup and liminf are equal, we call this value the box dimension of $F$ and we denote it as ${\dim_{\mathrm{B}}}F.$ The unconventional fractional part symbol {#Ab} ----------------------------------------- For a real number $\alpha$, it is conventional to use $\{\alpha\}$ for its fractional part. It is unfortunate that $\{.\}$ is also used to denote a set or a sequence as well. For this reason we will use $\mod 1$ for the fractional part. More precisely, for a real number $x$ we write $x\mod 1$ to denote the unique number $a$ in $[0,1)$ such that $a-x$ is an integer. Sets and sequences ------------------ We write $\{x_n\}_{n\geq 1}$ for the sequence $x_1x_2x_3\dots.$ Sometimes it is convenient to use $\{x_n\}_{n\geq 1}$ to denote the following set $$\{x: \exists n\in\mathbb{N}, x=x_n\}.$$ Thus $\overline{\{x_n\}_{n\geq 1}}$ and ${\underline{\dim_{\mathrm{B}}}}\{x_n\}_{n\geq 1}$ should be understood in this way. Filtrations, atoms and entropy ------------------------------ Let $X$ be a set with $\sigma$-algebra $\mathcal{X}.$ A filtration of $\sigma$-algebras is a sequence $\mathcal{F}_n\subset\mathcal{X},n\geq 1$ such that $$\mathcal{F}_1\subset \mathcal{F}_2\subset\dots \subset \mathcal{X}.$$ Given a measurable map $S:X\to X$ and a finite measurable partition $\mathcal{A}$ of $X$, we denote $S^{-n} \mathcal{A}$ to be the following finite collection of sets (notice that $S$ might not be invertible) $$\{S^{-n}(A) :A\in \mathcal{A}\}.$$ Then we use $\vee_{i=0}^{n-1} S^{-i}\mathcal{A}$ to be the $\sigma$-algebra generated by $S^{-i} \mathcal{A},i\in [0,n-1].$ An atom in $\vee_{i=0}^{n-1} S^{-i}\mathcal{A}$ is a set $A$ that can be written as $$A=\bigcap_{i} C_i$$ where for each $i\in\{0,\dots,n-1\}$, $C_i\in S^{-i}\mathcal{A}$. In this sense $\vee_{i=0}^{n-1} S^{-i}\mathcal{A}$ is generated by a finite partition $\mathcal{A}_{n-1}$ of $X$ which is finer than $\mathcal{A}.$ Let $\mu$ be a probability measure, then we define the entropy of $\mu$ with respect to a finite partition $\mathcal{A}$ as follows $$H(\mu,\mathcal{A})=-\sum_{A\in\mathcal{A}} \mu(A)\log \mu(A).$$ We define the entropy of $S$ as follows $$h(S,\mu)=\lim_{n\to\infty} \frac{1}{n} H(\mu, \mathcal{A}_{n-1}),$$ where $\mathcal{A}$ is a partition such that $\vee_{i=1}^\infty S^{-i}\mathcal{A}=\mathcal{X}.$ Here we implicitly used Sinai’s entropy theorem, see [@PY Lemma 8.8] Dynamical systems and factors ----------------------------- A measurable dynamical system is in general denoted as $(X,\mathcal{X},S,\mu)$ where $X$ is a set with $\sigma$-algebra $\mathcal{X}$ and measure $\mu$ and a measurable map $S:X\to X.$ In case when $\mathcal{X}$ is clear in context (for example Borel $\sigma$-algebra in Borel spaces) then we do not explicitly write it down. Given two dynamical systems $(X,\mathcal{X},S,\mu),(X_1,\mathcal{X}_1,S_1,\mu_1)$, a measurable map $f:X\to X_1$ is called a factorization map and $(X_1,\mathcal{X}_1,S_1,\mu_1)$ is called a factor of $(X,\mathcal{X},S,\mu)$ if $\mu_1=f\mu$ and $f\circ S=S_1\circ f.$ Bernoulli system ---------------- Let $\Lambda$ be a finite set of symbols and let $\Omega=\Lambda^{\mathbb{N}}$ be the space of one sided infinite sequences over $\Lambda.$ We define $S$ to be the shift operator, namely, for $\omega=\omega_1\omega_2\dots\in\Omega,$ $$S(\omega)=\omega_2\omega_3\dots.$$ Then we take a $\sigma$-algebra on $\Omega$ generated by cylinder subsets. A cylinder subset $Z\subset\Omega$ is such that $ Z=\prod_{i\in\mathbb{N}}Z_i $ and $Z_i=\Lambda$ for all but finitely many integers $i\in\mathbb{N}.$ We construct a probability measure $\mu$ on $\Omega$ by giving a probability measure $\mu_{\Lambda}=\{p_\lambda\}_{\lambda\in \Lambda}$ on $\Lambda$ and set $\mu=\mu^{\mathbb{N}}_{\Lambda}.$ We require here that $p_\lambda\neq 0$ for all $\lambda\in \Lambda$. Then this system is weak-mixing and has entropy $h(S,\mu)=\sum_{\lambda\in\Lambda} -p_\lambda\log p_\lambda.$ We call this system a Bernoulli system. We can also introduce a metric topology on $\Omega$ by defining $d(\omega,\omega')=\#\Lambda^{-\min\{i\in\mathbb{N}: \omega_i\neq\omega'_i\}}.$ This turns $\Omega$ into a compact and totally disconnected space. For $\omega\in\Omega$ and $r\in (0,1)$, we use $B(\omega,r)$ to denote the $r$-ball around $\omega$ with radius $r$ with respect to the metric $d$ constructed above. A mathematical Olympiad problem =============================== We first illustrate a short proof of Theorem \[num\] which provides us some motivation. Let $l=ord(2,m)$ be the order of $2$ in the multiplication group $(\mathbb{Z}/m\mathbb{Z})^*.$ This can be done because $gcd(2,m)=1.$ For convenience we consider $c=1$ and note that other cases can be shown with the same method. Since $l=ord(2,m)$ we consider the following sequence $$\{2^{nl}+nl\mod m\}_{n\geq 0}.$$ We see that $2^{nl}\equiv 1\mod m$ for all $n\geq 0.$ However $H=\{nl \mod m\}_{n\geq 0}$ is a subgroup of $\mathbb{Z}/m\mathbb{Z}$ of order $m/gcd(l,m).$ For convenience we write $\Delta=gcd(l,m).$ This $\Delta$ plays the same role of the entropy in the proof of Theorem \[main\]. If $\Delta=1$ then $D(m)=m$ follows automatically. We consider the case when $\Delta>1.$ Now for each integer $r$ we consider the following sequence $$\{2^{r+nl}+r+nl\mod m\}.$$ This sequence forms a coset of $H.$ More precisely it is $2^r+r+H.$ Now if $\{2^r+r\mod \Delta\}_{r\geq 0}$ would visit all residue classes modulo $\Delta,$ then $2^r+r+H,r\geq 0$ would visit all cosets of $H$ in $\mathbb{Z}/m\mathbb{Z}$ and $\{2^n+n\}_{n\geq 1}$ would visit all residue classes modulo $m.$ Since $\Delta$ is an odd number as well we see that we have reduced the problem for $m$ to the problem for $\Delta$ which is strictly smaller than $m.$ We can iterate this reduction procedure. Since we are considering positive integer set, either we eventually obtain $\Delta=1$ or else we can consider further $gcd(\Delta,ord(2,\Delta))<\Delta.$ The later can not happen infinitely often. This concludes the proof. Equidistributed sequences {#ES} ========================= In this section, we prove some results for equidistributed sequences. We are particularly interested in taking subsequences. \[DEF1\] Let $H=\{x_n\}_{n\geq 1}$ be an equidistributed sequence in $[0,1]$ with respect to the Lebesgue measure. Let $K\subset\mathbb{N}=\{k_1,k_2,\dots\}$ be an integer sequence. We introduce the following sequence $$A_K(H)=\{x_{k_i}\}_{i\geq 1}.$$ We will be interested mostly in the closure of the set of numbers in $A_K(H).$ For convenience we denote $C_K(H)$ to be the closure of the set of numbers in $A_K(H).$ Observe that $C_{\mathbb{N}}(H)=[0,1].$ $C_K(H)$ does not need to contain any intervals. Indeed, let $\{y_n\}_{n\geq 1}$ be an enumeration of rational numbers in $(0,1).$ For each $n\geq 1,$ let $I_n\subset [0,1]$ be the open interval centred at $y_n$ with length $\epsilon_n>0.$ Assume further that $\sum_n \epsilon_n=0.5.$ One thing to observe is that $I=\cup_{n}I_n$ has Lebesgue measure at most $0.5.$ Since $H$ equidistributes, for each $n\geq 1,$ $$\{k: x_k\in I_n\}$$ has natural density equal to $\epsilon_n.$ Then we see that $$\{k: x_k\in \cup_{n\leq N}I_n\}$$ has natural density at most $0.5$ for each integer $N\geq 1.$ We construct a sequence $K\subset\mathbb{N}$ in the following way. First, for $N=1,$ we choose an integer $k_1$ and a subset $K_1\subset\{1,\dots,k_1\}$ such that $$\{x_n\}_{n\in K_1}\cap I_1=\emptyset.$$ We can achieve further that $\#K_1\geq 0.49k_1.$ Now let $N=2$, similar as the above step, we can choose an integer $k_2>1$ and a subset $K_2\subset\{k_1+1,k_1+k_2\}$ such that $$\{x_n\}_{n\in K_2}\cap (I_1\cup I_2)=\emptyset.$$ Continuing this manner, for each $N\geq 1$, we obtain a non-empty set $K_N\subset \mathbb{N}$ such that $$\{x_n\}_{n\in K_N}\cap (I_1\cup\dots\cup I_N)=\emptyset.$$ Now let $K=\cup_N K_N.$ Then it is possible to see that for each integer $n\geq 1,$ $x_k\in I_n$ for at most finitely many $k\in K.$ Thus $C_K(H)$ either does not contain $y_n$ or contains $y_n$ as a isolated point. Since $\{y_n\}_{n\geq 1}$ is dense in $[0,1],$ $C_K(H)$ can not contain any intervals. We can even achieve that $K$ has upper natural density at least $0.49.$ The above argument shows that we can not say much about topological structures of $C_K(H).$ In turn, we focus on the Lebesgue measure. \[lma1\] Let $H$ be as in Definition \[DEF1\]. Suppose that $K$ has upper natural density $\rho>0.$ Then $C_K(H)$ has Lebesgue measure at least $\rho.$ We first examine a simple case. Let $I\subset [0,1]$ be an interval with length larger than $1-\rho.$ Then since $H$ equidistributes we see that $$K'=\{k: x_k\in I\}$$ has natural density larger than $1-\rho.$ Therefore $K\cap K'$ cannot be empty. This implies that $C_K(H)\cap I\neq\emptyset.$ Now let $I$ be any open set with Lebesgue measure larger than $1-\rho.$ By [@SS Theorem 1.3], $I$ can be written as a countable union of closed intervals with disjoint interiors, say $I=\cup_{n\geq 1} I_n.$ Then for a large enough integer $N$ the Lebesgue measure of $\cup_{n\leq N}I_n$ is greater than $1-\rho.$ Now we want to study the following set, $$K''=\{k: x_k\in \cup_{n\leq N}I_n\}.$$ It is tempting to claim that $K''$ natural density greater than $1-\rho.$ The difficulty here is that the intervals $I_n,n\geq 1$ are not really disjoint. In fact for $I_n\cap I_{n'}$ can contain at most one point. For this reason we shrink each interval $I_n$ to $I'_n$ by fixing their centres and reducing their lengths to $(1-\epsilon)I_n.$ By choosing $\epsilon$ small enough, $\cup_{n\leq N}I'_n$ has Lebesgue measure greater than $1-\rho.$ Now we see that $$K'''=\{k: x_k\in \cup_{n\leq N}I'_n\}.$$ has natural density greater than $1-\rho.$ Then $K\cap K'''\neq\emptyset$ and $C_K(H)\cap (\cup_{n\leq N}I'_n)\neq\emptyset.$ This implies that $[0,1]\setminus C_K(H)$ is an open set with Lebesgue measure at most $1-\rho.$ Thus $C_K(H)$ has Lebesgue measure at least $\rho.$ The above lemma is essentially sharp. We can not hope to bound the Lebesgue measure of $C_K(H)$ to be larger than the upper natural density of $K.$ Indeed, let $I=[0,0.5]$ and we take $$K=\{k: x_k\in I\}.$$ Then $C_K(H)=I$ and it has Lebesgue measure $0.5.$ Because of the equidistribution property of $H$, $K$ has natural density $0.5.$ However, it is possible to obtain better bounds for specially chosen sequences $K.$ \[lma2\] Let $H,K$ as in Definition \[DEF1\]. Let $(\Omega,S,\nu)$ be a Bernoulli system and let $\Omega'\subset \Omega$ be a measurable set with positive $\nu$ measure. For each $\omega\in\Omega$, we define $$K(\omega,\Omega')=\{k\in\mathbb{N}: S^k(\omega)\in\Omega'\}.$$ Then $A_{K(\omega,\Omega')}(H)$ equidistributes for $\nu$ almost all $\omega\in\Omega$. Let $I\subset [0,1]$ be a closed interval. For each $\omega$ we are interested in the following limit, $$\lim_{n\to\infty} \frac{1}{\sum_{i=1}^n \mathbbm{1}_{\Omega'}(S^i(\omega))} \sum_{i=1}^n \mathbbm{1}_{\Omega'}(S^n(\omega)) \mathbbm{1}_{I}(x_n).$$ Of course we need to require that $\sum_{i=1}^n\mathbbm{1}_{\Omega'}(S^n(\omega))$ is not zero for large enough $n.$ This happens for almost all $\omega.$ We see that almost surely $$\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^n\mathbbm{1}_{\Omega'}(S^n(\omega))=\nu(\Omega')=p>0.$$ This can be seen by using the ergodicity of Bernoulli systems. Now we want to estimate the following sum $$\sum_{i=1}^n \mathbbm{1}_{\Omega'}(S^n(\omega)) \mathbbm{1}_{I}(x_n).$$ For a fixed interval $I$ with length $|I|$ we know that the following sequence $$K_I=\{k\in\mathbb{N}: x_k\in I \}$$ has natural density $|I|.$ If we are able to show that almost surely $$\lim_{n\to\infty} \frac{1}{n} \sum_{i=1}^n \mathbbm{1}_{\Omega'}(S^n(\omega)) \mathbbm{1}_{I}(x_n)=p|I|,\tag{*}$$ then we will see that for almost all $\omega,$ $$\lim_{n\to\infty} \frac{1}{\sum_{i=1}^n \mathbbm{1}_{\Omega'}(S^i(\omega))} \sum_{i=1}^n \mathbbm{1}_{\Omega'}(S^n(\omega)) \mathbbm{1}_{I}(x_n)=|I|.$$ For convenience we write $X_n(\omega)=\mathbbm{1}_{\Omega'}(S^n(\omega))$. If $\Omega'$ is a ball of radius $2^{-r}$, we decompose the integer set with modulo $r.$ For each $j\in\{0,\dots,r-1\}$ we denote $$N_j=\{kr+j: k\geq 0\}$$ and $K_j=K_I\cap N_j.$ For each $j\in\{0,\dots,r-1\}$, if $\#K_j=\infty$, we enumerate $K_j=\{k_j(1),k_j(2),\dots\}$ in increasing order. By the law of large numbers we see that for $\nu$ almost all $\omega,$ $$\lim_{n\to\infty} \frac{1}{n}\sum_{i=1}^n X_{k_j(i)}(\omega)=p.$$ We can apply this to each $j$ with $\#K_j=\infty.$ Now we see that, $$\frac{1}{n}\sum_{i=1}^n X_n(\omega)\mathbbm{1}_I(x_n)= \frac{1}{n}\sum_{j=0}^{r-1}\sum_{i:k_j(i)\leq n} X_{k_j(i)}(\omega).$$ If $\#K_j=\infty,$ we saw that for $\nu$ almost all $\omega,$ $$\sum_{i:k_j(i)\leq n} X_{k_j(i)}(\omega)=\#(K_j\cap [1,n])p+o(\#(K_j\cap [1,n])).$$ If $\#K_j< \infty$ then we see that for all $\omega,$ $$\sum_{i:k_j(i)\leq n} X_{k_j(i)}(\omega)\leq \#K_j=O(1).$$ We want to sum up the above bounds over $j\in\{0,\dots,r-1\}.$ As there are finitely many terms to sum we see that the little-$o$ parts sum up to $$o(\#K_I\cap [1,n])$$ and we have $$\sum_{j=0}^{r-1}\sum_{i:k_j(i)\leq n} X_{k_j(i)}(\omega)=\#(K_I\cap [1,n])p-\sum_{j:\#K_j<\infty} \#(K_j\cap [1,n])p+o(\#K_I\cap [1,n])+O(1).$$ Dividing $n$ in the above expression we see that for $\nu$ almost all $\omega$, $$\frac{1}{n}\sum_{i=1}^n X_n(\omega)\mathbbm{1}_I(x_n)=p\frac{\#(K_I\cap [1,n])}{n}+o\left(\frac{\#(K_I\cap [1,n])}{n}\right)+O(n^{-1})$$ Taking $n\to\infty$ we see that $$\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^n X_n(\omega)\mathbbm{1}_I(x_n)=p|I|.\tag{**}$$ This proves $(*)$ for $\Omega'$ being a ball of radius $2^{-r}$ for each $r\geq 1.$ Suppose that $\Omega'$ satisfies $(*)$ then ${\Omega'}^c$ satisfies $(*)$ as well because $$\mathbbm{1}_{\Omega'}(S^n(\omega))+\mathbbm{1}_{{\Omega'}^c}(S^n(\omega))=1$$ for all $\omega$ and all $n\geq 1.$ A cylinder set in $\Omega$ can be written as a finite disjoint union of balls in $\Omega.$ Let $\Omega'_1,\dots,\Omega'_k$ be $k\geq 2$ pairwise disjoint balls in $\Omega.$ Then we know that $(*)$ holds for each $\Omega'_i,i\in\{1,\dots,k\},$ for almost all $\omega\in\Omega.$ Thus we can deduce that for almost all $\omega\in\Omega,$ $(*)$ holds simultaneously for all $\Omega'_1,\dots,\Omega'_k.$ This implies that $(*)$ holds for $\cup_{i=1}^k \Omega'_i$ as well. Thus we have shown that $(*)$ holds for all cylinder subsets of $\Omega.$ We want to upgrade this result for the $\sigma$-algebra generated by cylinder sets, which will be sufficient since this is the $\sigma$-algebra we are considering. Let $\Omega'$ be an element in this $\sigma$-algebra with $\nu$-measure $p$. Then for each $\epsilon>0$ we can find a cylinder set $\Omega''$ such that $$\nu(\Omega'\Delta\Omega'')\leq \epsilon.$$ Then we see that $$\sum_{i=1}^n \mathbbm{1}_{\Omega'}(S^n(\omega)) \mathbbm{1}_{I}(x_n)\leq\sum_{i=1}^n \mathbbm{1}_{\Omega''}(S^n(\omega)) \mathbbm{1}_{I}(x_n)+\sum_{i=1}^n \mathbbm{1}_{\Omega'\setminus\Omega''}(S^n(\omega)) \mathbbm{1}_{I}(x_n).$$ For $\nu$ almost all $\omega$, the first summation is $\nu(\Omega'')|I|n+o_N(n)$ (by what we have shown in above) and the second can be bounded by $$\sum_{i=1}^n \mathbbm{1}_{\Omega'\setminus\Omega''}(S^n(\omega)) \mathbbm{1}_{I}(x_n)\leq\sum_{i=1}^n\mathbbm{1}_{\Omega'\setminus\Omega''}(S^n(\omega))=\nu(\Omega'\setminus\Omega'')n+o_N(n),$$ this is because the Bernoulli system in consideration is ergodic. This implies that for almost all $\omega$ (in a manner that depends on $\epsilon$), $$\limsup_{n\to\infty} \frac{1}{n} \sum_{i=1}^n \mathbbm{1}_{\Omega'}(S^n(\omega)) \mathbbm{1}_{I}(x_n)\leq \nu(\Omega'')|I|+\epsilon\leq (p+\epsilon)|I|+\epsilon.$$ We apply the above with a sequence $\epsilon_i\downarrow 0$, as a result for almost all $\omega,$ $$\limsup_{n\to\infty} \frac{1}{n} \sum_{i=1}^n \mathbbm{1}_{\Omega'}(S^n(\omega)) \mathbbm{1}_{I}(x_n)\leq p|I|.\tag{1}$$ For the other direction, observe that $$\sum_{i=1}^n \mathbbm{1}_{\Omega'}(S^n(\omega)) \mathbbm{1}_{I}(x_n)\geq\sum_{i=1}^n \mathbbm{1}_{\Omega''}(S^n(\omega)) \mathbbm{1}_{I}(x_n)-\sum_{i=1}^n \mathbbm{1}_{\Omega''\setminus \Omega'}(S^n(\omega)) \mathbbm{1}_{I}(x_n).$$ With similar argument as above we see that, $$\liminf_{n\to\infty} \frac{1}{n} \sum_{i=1}^n \mathbbm{1}_{\Omega'}(S^n(\omega)) \mathbbm{1}_{I}(x_n)\geq (p-\epsilon)|I|-\epsilon.$$ We apply the above with $\epsilon\downarrow 0$ again and we see that for almost all $\omega,$ $$\liminf_{n\to\infty} \frac{1}{n} \sum_{i=1}^n \mathbbm{1}_{\Omega'}(S^n(\omega)) \mathbbm{1}_{I}(x_n)\geq p|I|.\tag{2}$$ Combining $(1),(2)$ we see that $(*)$ holds for the $\sigma$-algebra generated by cylinder sets. Now we apply the above argument to intervals with rational end points, as there are countably many of them, we see that almost surely $(*)$ holds for all $I$ with rational end points. This concludes the proof. In particular, the above lemma says that if we choose a subset $K$ of integers by choosing each integer with probability $p,$ then almost surely $C_K(H)=[0,1].$ This happens no matter how small $p$ is, as long as $p>0.$ We see that almost surely, $K$ has natural density $p$ and if we would apply Lemma \[lma1\] we only obtain that $C_K(H)$ has Lebesgue measure at least $p$, which is much weaker than the result of Lemma \[lma2\]. Now we finish this section by combining the above lemmas and obtain the following theorem. \[thm1\] Let $H$ be a equidistributed sequence in $[0,1].$ Let $K$ be an integer sequence with lower natural density $\rho>0.$ Let $(\Omega,S,\mu)$ be a Bernoulli system. Let $\Omega_i,i\in\mathcal{I}$ be a finite collection of pairwise disjoint events (measurable subsets of $\Omega$). Suppose that $\sum_{i\in\mathcal{I}}\mu(\Omega_i)=1-\epsilon,$ where $\epsilon<\rho.$ Then there is a full measure set $\Omega'\subset\Omega$( not depending on $K$) such that for each $\omega\in \Omega',$ there is an $i=i(\omega,K)$ such that $$C_{K\cap K_{\Omega_i}} (H)$$ has Lebesgue measure at least $\rho-\epsilon.$ Here $K_{\Omega_i}$ is defined as follows $$K_{\Omega_i}=\{k\in\mathbb{N}: S^k(\omega)\in \Omega_i \}.$$ We see that almost surely, $K_{\Omega_i}$ has natural density $\mu(\Omega_i)$ for each $i\in\mathcal{I}.$ By Lemma \[lma2\], we see that almost surely, $A_{K_{\Omega_i}}(H)$ equidistributes in $[0,1]$ for each $i\in\mathcal{I}.$ Let $\Omega'\subset\Omega$ be a full measure set in which the above two properties hold for each $i\in\mathcal{I}$. Then for each $\omega\in\Omega'$ we see that $$d(\cup_{i\in\mathcal{I}} K_{\Omega_i})=1-\epsilon.$$ Since $K$ has lower natural density $\rho>\epsilon$, we see that $K\cap \cup_{i\in\mathcal{I}} K_{\Omega_i}$ has lower natural density at least $\rho-\epsilon>0.$ Notice that $K_{\Omega_i},i\in\mathcal{I}$ are pairwisely disjoint. For each $i\in\mathcal{I}$ we write $K_i=K\cap K_{\Omega_i}$ and $\rho_i=\overline{d}(K_i).$ Then we see that $$\sum_{i\in\mathcal{I}} \rho_i\geq \rho-\epsilon.$$ We remark here that the above inequality may not hold if $\rho_i,i\in\mathcal{I}$ are replaced by lower natural densities. We see that there is an $i\in\mathcal{I}$ such that $$\rho_i\geq (\rho-\epsilon)\mu(\Omega_i).$$ Recall that $A_{K_{\Omega_i}}(H)$ equidistributes. We enumerate $K_{\Omega_i}$ as $k_1< k_2<\dots.$ Then $K_i\subset K$, viewed in this enumeration, has upper natural density at least $\rho_i/\mu(\Omega_{i})$ which is at least $\rho-\epsilon.$ Now let $H'=A_{K_{\Omega_i}}(H)$ and let $K'_i$ be the following sequence $$K'_i=\{j\in\mathbb{N}: k_j\in K_i \}.$$ Then by Lemma \[lma1\] we see that $C_{K'_i}(H')$ has Lebesgue measure at least $\rho-\epsilon.$ This concludes the proof. Bernoulli factors and a combinatorial Sinai factor result ========================================================= In this section we closely follow [@Y19 Section 9]. In order to increase the readability and self-containing, we present here full details. \[factor\] Let $(X,S,\mu)$ be an ergodic dynamical system then any Bernoulli system $(\Omega,S_B,\nu)$ with $h(S_B,\nu)\leq h(S,\mu)$ is a factor of $(X,S,\mu).$ When $(\Omega,S_B,\nu)$ is a factor of $(X,S,\mu)$ with the same entropy, then intuitively all the complicities are carried by $(\Omega,S_B,\nu)$ and therefore the fibres of $f$ should not be too complicated with respect to the map $S$. The following result expresses this intuition in a clear way. The following result is known as Rohlin’s disintegration theorem, and we adopt the version in [@S12]. Let $f:X\to Y$ be a measurable map between two measurable spaces and let $\mu$ be a measure on $X$ with projection $\mu_Y=f\mu$ on $Y$. We call a collection of measures $\{\mu_y\}_{y\in Y}$ a system of conditional measures if the following properties hold, - : For all $y\in Y$, $\mu_y$ is a measure supported on $f^{-1}(y)$ and for $\mu_Y$ almost all $y\in Y$, $\mu_y$ is a probability measure. - : We have the law of measure disintegration. For all Borel set $B\subset X$, we have $$\mu(B)=\int \mu_{y}(B)d\mu_Y(y).$$ If $X,Y$ are also metric spaces ($f$ need not to be continuous) we require further that the following holds for $\mu_Y$ almost all $y\in Y$. - : $\mu_y=\lim_{r\to 0} \mu_{f^{-1}(B(y,r))},$ where the limit is in the weak\* sense and $\mu_{f^{-1}(B(y,r))}$ is the conditional measure of $\mu$ on $f^{-1}(B(y,r))$, namely, for any Borel set $B\subset X$ with positive $\mu$ measure, $$\mu_{f^{-1}(B(y,r))}(B)=\frac{\mu(B\cap f^{-1}(B(y,r)))}{\mu(B)}.$$ Let $f:X\to Y$ be a measurable map between two metric spaces with corresponding Borel $\sigma$-algebra. Then there exists a system of conditional measures. Then we have the following result due to [@Wu Lemma 6.4] which is a direct consequence of conditional Shannon-McMillan-Breiman theorem, Egorov’s theorem and the Portmanteau theorem. \[Wu\] Let $(X,S,\mu)$ be an ergodic dynamical system with $X$ being a Borel space. Let $\mathcal{A}$ be a finite partition of $X$ such that $\vee_{i=0}^\infty S^{-i}\mathcal{A}$ generates the $\sigma$-algebra of $X.$ For each $x\in X$ not on the boundaries of sets in $\vee_{i=1}^{n} S^{-i}\mathcal{A}$, for each $n\in\mathbb{N}$ we denote $A_n(x)$ the unique atom $A$ of $\vee_{i=0}^n S^{-i}\mathcal{A}$ such that $x\in A.$ If $\mu$ does not positive measures to boundaries of $S^{-i}\mathcal{A}$ for all $i\in\mathbb{N}$ and $h(S,\mu)>0$ then there exist a Bernoulli factor $(\Omega,S_B,\nu)$ with measurable factorization map $f:X\to \Omega$ and for each $\delta>0$ there exist a $X_\delta\subset X$ and a constant $C_\delta$ with the following properties, - :$\mu(X_\delta)>1-\delta.$ - :For all $x\in X_\delta$ and $n\geq 1$, $\mu_{f(x)}(A_n(x))\geq C_\delta2^{-n\delta}$ and $\mu_{f(x)}$ is a probability measure. - :For all integers $n\geq 1$, there exists a measurable set $B^n_\delta\subset\Omega$ with $\nu(B^n_\delta)\geq 1-\delta$ and a $r=r(\delta,n)>0$ such that for all $\omega\in B^{n}_\delta$ and all atoms $A_n$ we have $$\frac{\mu(f^{-1}(B(\omega,r))\cap A_n)}{\mu(f^{-1}(B(\omega,r)))}\geq (1-\delta)\mu_{\omega}(A_n).$$ From the above results we can obtain the following combinatorial version of Sinai’s factor theorem. \[def1\] Let $(X,S,\mu)$ be an ergodic system. Let $\mathcal{A}$ be a finite partition generating the Borel $\sigma$-algebra of $X.$ For $\delta>0,$ we say that $(X,S,\mu)$ is $\delta$-Bernoulli if the following statements hold: There is a number $c_\delta>0$ and for each integer $n\geq 1,$ there is an integer $N(n)$ and a measurable decomposition $\mathcal{D}_n=\{D_n(1),\dots,D_n(N(n))\}$ of $X$ such that $(\mathcal{D}^{\mathbb{N}}_n,\pi S,\pi\mu)$ is a Bernoulli system where $\pi:X\to\mathcal{D}$ is defined by taking $\pi(x)$ to be the sequences of sets $D_n\in\mathcal{D}_n$ such that $S^n(x)\in D_n.$ For each $i\in\{1,\dots,N(n)\},$ we write $\tilde{\mathcal{M}}_n(i)$ to be the collection of atoms of $\mathcal{A}_n$ intersecting $D_n(i).$ For each $i$, there is a subcollection $\mathcal{M}_n(i)\subset \tilde{\mathcal{M}}_n(i)$ consisting at most $c_\delta 2^{n\delta}$ many elements such that the union of atoms in $\cup_{i} \mathcal{M}_n(i)$ has $\mu$ measure at least $1-2\delta.$ We say that $(X,T,\mu)$ satisfies the weak Bernoulli property if it is $\delta$-Bernoulli for each $\delta>0.$ The following result is a variant of [@Wu Theorem 6.1] and it is closely related to Austin’s theorem on the weak Pinsker property, see [@A18]. \[thmAustin\] We adopt the conditions in Theorem \[Wu\]. Then $(X,S,\mu)$ satisfies the weak Bernoulli property. Moreover, let $\epsilon>0$ be arbitrarily chosen in $(0,1)$ and $H=\{h_k\}_{k\geq 1}$ be an equidistributed sequence in $[0,1].$ For each $\delta\in (0,1)$, there is a constant $c_\delta>0$ and $X'_\delta$ with full $\mu$ measure such that for all $n\geq 1,$ all $x\in X'_\delta$ and all $K\subset\mathbb{N}$ with lower natural density at least $\rho>2\delta+\epsilon,$ there is a collection $\mathcal{M}_n=\mathcal{M}_n(x,K)$ of at most $c_\delta 2^{n\delta}$ atoms of $\mathcal{A}_n$ with the following property: - Denote the union of elements in $\mathcal{M}_n$ as $M_n$. We construct the following sequence $$K'(x)=\{k\in\mathbb{N}: S^k(x)\in M_n \}.$$ Then the following set has Lebesgue measure at least $\epsilon$ $$C_{K\cap K'(x)}(H).$$ We use Theorem \[Wu\] to find a set $X_\delta$ with $\mu(X_\delta)>1-\delta$. Then for each integer $n\geq 1$ we can find $B^n_\delta$ with $\nu(B^n_\delta)\geq 1-\delta$ and $r=r(\delta,n)>0.$ Without loss of generality we shall assume that $r=d^{-k}$ where $d$ is the number of digits of the Bernoulli system and $k$ is an integer. For each $\omega\in B^n_\delta$ we have $$\frac{\mu(f^{-1}(B(\omega,r))\cap A_n)}{\mu(f^{-1}(B(\omega,r)))}\geq (1-\delta)\mu_{\omega}(A_n).$$ Now because of the topology we chose for $\Omega$, we see that $B(\omega,r)$ consists all sequences in $\Omega$ with the same first $k$ digits as $\omega.$ In particular if $\omega'\in B(\omega,r)$ then $B(\omega',r)=B(\omega,r).$ This property reflects the fact that $\Omega$ is an ultrametric space. Notice that for any Bernoulli system $(\Omega,S_B,\nu)$, any ball of positive radius has positive $\nu$ measure. In particular $\mu(f^{-1}(B(\omega,r)))>0$ and by properties $(2)(3)$ in Theorem \[Wu\], for each $\omega'\in B^n_\delta\cap B(\omega,r)$ we have $$\frac{\mu(f^{-1}(B(\omega,r))\cap A_n)}{\mu(f^{-1}(B(\omega,r)))}=\frac{\mu(f^{-1}(B(\omega',r))\cap A_n)}{\mu(f^{-1}(B(\omega',r)))}\geq (1-\delta)\mu_{\omega'}(A_n)\geq (1-\delta)C_{\delta} 2^{-n\delta}$$ whenever $A_n=A_n(x)$ for some $x\in X_\delta\cap f^{-1}(\omega')$. Now it is possible to see that for all $x$ in the set $X_\delta\cap f^{-1}(B(\omega,r)\cap B^n_\delta)$ we have the following result $$\frac{\mu(f^{-1}(B(\omega,r))\cap A_n(x))}{\mu(f^{-1}(B(\omega,r)))}\geq (1-\delta)C_{\delta} 2^{-n\delta}.$$ On the other hand we clearly have $$\sum_{\text{atoms }A_n}\frac{\mu(f^{-1}(B(\omega,r))\cap A_n)}{\mu(f^{-1}(B(\omega,r)))}=1,$$ therefore $X_\delta\cap f^{-1}(B(\omega,r)\cap B^n_\delta)$ can be covered by at most $$\frac{2^{n\delta}}{(1-\delta)C_\delta}$$ many atoms of $\vee_{i=0}^n S^{-i}\mathcal{A}.$ Now let $Y(\omega)=X_\delta\cap f^{-1}(B(\omega,r)\cap B^n_\delta)$. Since there are only finitely many $r$ balls in $\Omega$ we see that as $\omega$ varies in $B^n_\delta$ there are finitely many different sets of form $Y(\omega).$ Denote the collection of these sets as $\{Y_1,\dots, Y_{N'(n)}\}$ where $N'(n)$ is an integer. For each $i\in\mathcal{I}=\{1,\dots,N'(n)\}$, let $\Omega(i)\subset B^n_\delta$ be the set of form $B(\omega,r)\cap B^n_\delta$ such that $Y_i=X_\delta\cap f^{-1}(\Omega(i)).$ We notice here that the union of all $Y_i$ is a rather large subset of $X$, more precisely we have the following result, $$\mu\left(\bigcup_{i\in\mathcal{I}}Y_i\right)= \mu\left(X_\delta\cap f^{-1}(B^n_\delta)\right)\geq 1-2\delta.$$ For each $i\in \mathcal{I}$ we write the collection of atoms intersecting $Y_i$ as $\mathcal{M}_n(i)$ and write their union as $M_n(i).$ Then we saw that $$\#\mathcal{M}_n(i)\leq \frac{2^{n\delta}}{(1-\delta)C_\delta}$$ and the following collection of atoms in $$\cup_{i}\mathcal{M}_n(i)$$ covers $\cup_i Y_i.$ In particular, we have $$\mu(\cup_{A\in\cup_{i}\mathcal{M}_n(i)} A)\geq \mu(\cup_i Y_i)\geq 1-2\delta.$$ In order to finish the proof for the weak Bernoulli property, we need to specify $\mathcal{D}_n.$ Denote the number of $r$-balls of $\Omega$ as $N(n)$, which is in general larger than $N'(n).$ We enumerate them as $B_1,\dots,B_{N(n)}.$ Then precisely $N'(n)$ of them intersect $B_\delta^n.$ We assume that these are the first $N'(n)$ elements. We take $D_n(i)$ to be $f^{-1}(B_i)$ for each $i.$ For each $i\in\{1,\dots,N(n)\},$ we take $\tilde{\mathcal{M}}_n(i)$ to be the collection of atoms in $\mathcal{A}_n$ intersecting $D_n(i).$ For each $i\in\{1,\dots, N'(n)\}$ we already constructed $\mathcal{M}_n(i)$ in above. For $i\in\{N'(n)+1,N(n)\}$ we can choose $\mathcal{M}_n(i)$ to be empty and concludes the proof of the first part. Now we consider the following sequence for $x\in X,$ $$K(x)=\{k\in\mathbb{N}: S^k(x)\in X_\delta \}$$ by the ergodic theorem we see that for $\mu$ almost all $x\in X,$ $ K(x) $ has natural density at least $1-\delta.$ For each $i\in\mathcal{I}$ and $x\in X$ we construct the following set $$K'_i(x)=\{k\in\mathbb{N}: S^k(x)\in M_n(i) \}$$ and we see that $$\begin{aligned} K(x)\cap K'_i(x)&=&\{k\in\mathbb{N}: S^k(x)\in M_n(i)\cap X_\delta\}\\ &\supset& \{k\in\mathbb{N}: S^k(x)\in Y_i\}\\ &=&\{k\in\mathbb{N}: S^k(x)\in f^{-1}(\Omega(i))\cap X_\delta\}\\ &=&K(x)\cap K(f(x),\Omega(i))).\end{aligned}$$ Here $K(f(x),\Omega(i))=\{k\in\mathbb{N}: S^k\circ f(x)\in \Omega(i) \}.$ By Lemma \[lma2\], we claim that for $\mu$ almost all $x\in X$ and any sequence $K$ with lower natural density at least $2\delta+\epsilon$ there exists an $i\in\mathcal{I}$ such that $$C_{K\cap K(x)\cap K(f(x),\Omega(i))}(H)$$ has Lebesgue measure at least $\epsilon.$ Indeed, we know that $K\cap K(x)$ has lower natural density at least $\delta+\epsilon$ for $\mu$ almost all $x\in X$ and $\sum_{i\in\mathcal{I}}\nu(\Omega(i))=\nu(B^n_\delta)\geq 1-\delta$. By Lemma \[lma2\], there is a full $\nu$ measure set $\Omega'$ with the property that for all sequence $K_0$ with lower natural density at least $\delta+\epsilon,$ for each $\omega'\in\Omega'$ there is a $i\in\mathcal{I}$ such that $$C_{K_0\cap K(\omega',\Omega(i))}$$ has Lebesgue measure at least $\epsilon.$ As $\Omega'$ has full $\nu$ measure, there is at least one (in fact, $\mu$-almost all) $x\in X$ such that $f(x)\in \Omega'.$ Now if $x\in X$ is such that $K(x)\cap K$ has lower density at least $\delta+\epsilon$ and at the same time $f(x)\in\Omega'$, then there is $i\in\mathcal{I}$ such that $$C_{K\cap K(x)\cap K(f(x),\Omega(i))}(H)$$ has Lebesgue measure at least $\epsilon.$ However, such choices of $x\in X$ ranges over a full $\mu$-measure set, which proves the claim. The second part of the theorem follows since the above argument holds for all $n\geq 1$ and we can find a full measure set $X'_\delta\subset X$ which satisfies all our requirements. On sequences $\{p(n)+2^n d \mod 1\}_{n\geq 1}$ ============================================== Given any irrational number $c$ and any number $d$ we consider the sequence $\{nc+2^n d \mod 1\}_{n\geq 1}.$ If $d$ is a rational number then $2^n d$ takes only finitely many values. From here we know that in this case $\{nc+2^n d \mod 1\}_{n\geq 1}$ have box dimension $1$, in fact it must be dense in $[0,1].$ If $d$ is randomly picked according to the Lebesgue measure then intuitively $nc$ and $2^n d$ are somehow independent and $\{nc+2^n d \mod 1\}_{n\geq 1}$ can be viewed as an equidistributed sequence with a random pertubation and it is plausible that $\{nc+2^n d \mod 1\}_{n\geq 1}$ should have box dimension $1.$ \[Thm1\] Let $c$ be an irrational number and $d$ be any real number. We have $${\underline{\dim_{\mathrm{B}}}}\{nc+2^n d\mod 1\}_{n\geq 1}=1.$$ Let $Or_2(d)$ be the following orbit closure $$\overline{\{2^n d\}_{n\geq 1}}.$$ $Or_2(d)$ is then a closed $\times 2\mod 1$ invariant subset of $[0,1]$ and therefore there exist $\times 2 \mod 1$ invariant probability measures whose supports are contained in $Or_2(d).$ By taking ergodic components we can find an invariant measure $\mu$ whose support is contained in $Or_2(2)$ and the $\times 2\mod 1$ action is ergodic with respect to $\mu.$ Our idea is to take a $\mu$-typical point $d'$ and study the sequence $\{nc+2^n d'\mod 1\}_{n\geq 1}.$ In order to set up the link between $\{nc+2^n d\mod 1\}_{n\geq 1}$ and $\{nc+2^n d'\mod 1\}_{n\geq 1}$ we first observe that $d'\in Or_2(d)$ and therefore there exists a sequence $\{i_k\}_{k\geq 1}$ such that $2^{i_k} d \mod 1\to d'$ as $k\to\infty.$ Consider the points $i_k c\mod 1$, without loss of generality we shall assume that $i_kc\mod 1\to c^*$ as $k\to\infty,$ where $c^*\in [0,1]$. Consider the points $(i_k+1)c+2^{i_k+1} d$ for integers $k.$ We see that this sequence of points converge to $c^*+c+2d' \mod 1.$ Similarly for each integer $M$ we see that $(i_k+M)c+2^{i_k+M} d$ converges to $c^*+Mc+2^M d' \mod 1.$ As this holds for any integer $M$ we see that $$c^*+ \overline{\{nc+2^n d'\}_{n\geq 1}}\subset \overline{\{nc+2^n d\}_{n\geq 1}}.$$ Therefore we see that it is enough to show that $ \overline{\{nc+2^n d'\}_{n\geq 1}}$ has full lower box dimension. Now consider the ergodic dynamical system $([0,1],T,\mu)$ where $T$ is the $\times 2\mod 1$ action. Consider the partition $[0,0.5]\cup [0.5,1]$ which we denote as $\mathcal{A}.$ It is easy to show that $T^{-n}\mathcal{A}$ generates the Borel $\sigma$-algebra as $n\to\infty.$ Then we have $h(T,\mu)=h(T,\mu,\mathcal{A}).$ If $\mu$ supports on boundaries of $T^{-n}\mathcal{A}$ for integers $n$ then as those boundaries are rational numbers we see that we can pick $d'\in\mathbb{Q}$ and in this case $\overline{\{nc+2^n d'\}_{n\geq 1}}=[0,1].$ Now assume that $\mu$ does not have any boundary supports. Our further argument can be split into two cases based on whether or not $h(T,\mu)$ is $0$. Zero entropy ------------ When $h(T,\mu)$ is equal to zero, then for each integer $n$ we have $2^n$ many disjoint dyadic intervals of length $2^{-n}$ covering $[0,1].$ Denote this collection of intervals as $\mathcal{D}_n.$ We see that $$-\sum_{I\in\mathcal{D}_n} \mu(I)\log \mu(I)=o(n).$$ Suppose that the above sum is bounded from above by $\epsilon'n$ for each large enough $n$. Then for each $\delta'>0$, we define the following subset $$V_{\delta'}=\{v\in \mathcal{D}_n: \mu(v)\geq 2^{-n\delta'}\}.$$ Then it is easy to see that $$\sum_{v\in V}\mu(v)(-\log \mu(v))=\sum_{v\in V_{\delta'}}\mu(v)(-\log \mu(v))+\sum_{v\notin V_{\delta'}}\mu(v)(-\log \mu(v))<n\epsilon'.$$ If $v\notin V_{\delta'}$ then $-\log \mu(v)>n\delta'\log 2$ and therefore $$n\epsilon'>\sum_{v\notin V_{\delta'}}\mu(v)(-\log \mu(v))> \sum_{v\notin V_{\delta'}} n\delta' \mu(v)\log 2.$$ Then we see that $$\sum_{v\in V_{\delta'}}\mu(v)=1-\sum_{v\notin V_{\delta'}}\mu(v)>1-\frac{1}{\log 2}\frac{\epsilon'}{\delta'}.$$ On the other hand because $\sum_{v\in V}\mu(v)=1$ then we see that $$\#V_{\delta'}\leq 2^{n\delta'}.$$ Let $\delta'=\sqrt{\epsilon'}$ we see that for all large enough $n$, at least $1-10\delta'$ portion of $\mu$ measure is supported in a collection of at most $2^{n\delta'}$ many dyadic intervals in $\mathcal{D}_n.$ Let $d'$ be a $\mu$-typical number which is generic with respect to $\mu.$ Let $\delta>0$ be a small number. Let $M$ be a large integer and there is a collection of at most $2^{M\delta}$ many dyadic intervals in $\mathcal{D}_{M}$ which support at least $1-10\delta$ portion of $\mu$ measure. Denote this collection of dyadic intervals as $\mathcal{M}_M$ and write their union as $S_M.$ As $d'$ is generic with respect to $\mu$ we see that for large enough integers $n$, $\{2^n d'\mod 1\}$ spends at most $1-20\delta$ portion of time in $S_M.$ More precisely, $A_M=\{n\in\mathbb{N}: 2^n d'\mod 1\in S_M\}$ is an integer sequence with natural density at least $1-20\delta.$ By Lemma \[lma1\], the closure $E$ of $\{nc\mod 1\}_{n\in A_M}$ has Lebesgue measure at least $1-20\delta.$ We will need at least $(1-20\delta)2^{M}$ many dyadic intervals in $\mathcal{D}_M$ to cover $E$ for otherwise the Lebesgue measure of $E$ is smaller than $1-20\delta.$ Denote this collection of dyadic intervals as $\mathcal{F}_M.$ We say that the pair $(s_M,f_M)\in \mathcal{M}_M\times \mathcal{F}_M$ related (or $s_M$ is related to $f_M$) if there is an integer $n\in A_M$ such that $2^n d' \mod 1\in s_M$ and $nc\mod 1 \in f_M.$ Now each $s_M\in S_M$ is related to at least one of $f_M$ in $\mathcal{F}_M.$ Then by the pigeonhole principle we see that there exists at least one $f_M$ which is related to at least $(1-20\delta)2^{M}/2^{M\delta}$ many elements in $\mathcal{M}_M.$ This implies that in order to cover $\{nc+2^n d'\mod 1\}_{n\in A_M}$ with dyadic intervals in $\mathcal{D}_M$, we need at least $0.01(1-20\delta)2^{M}/2^{M\delta}$ many of them. As the above argument holds for all large enough integer $M$ we see that ${\underline{\dim_{\mathrm{B}}}}\{nc+2^n d'\mod 1\}_{n\geq 1}\geq 1-\delta$. As $\delta$ can be arbitrarily small, this concludes the case when $h(T,\mu)=0.$ Positive entropy ---------------- In this case we use the second part of Theorem \[thmAustin\]. In the statement we use $K=\mathbb{N}$, $\delta>0$ be an arbitrarily chosen small number and $X_\delta$ is full $\mu$ subset of $[0,1].$ For each integer $M$, we can find a collection $\mathcal{M}_M$ of at most $c_\delta2^{M\delta}$ many dyadic intervals in $\mathcal{D}_M$, where $c_\delta>0$ is a constant depending on $\delta.$ Denote the union of those dyadic intervals as $S_M.$ Let $A_M=\{n\in\mathbb{N}: 2^n d' \in S_M \}$. Then the closure of $\{nc\}_{n\in A_M}$ has Lebesgue measure at least $1-2\delta.$ The rest of the proof is exactly the same as the zero entropy case and this concludes the positive entropy case. Having established the result for $\{nc+2^n d \mod 1\}_{n\geq 1}$ it is natural to ask what is the situation for $\{p(n)+2^n d\mod 1 \}_{n\geq 1}$ where $p:\mathbb{Z}\to \mathbb{R}$ is a polynomial with at least one irrational coefficient. In fact we can treat the polynomial case with a similar method. If we know that $d$ is a generic point with respect to an ergodic measure of the doubling map, then the argument in the above proof can be applied to show that $\{p(n)+2^n d \mod 1\}_{n\geq 1}$ has full box dimension. The only thing we need here is that $\{p(n)\mod 1\}_{n\geq 1}$ equidistributes in $[0,1].$ If $d$ is not such a generic point we want to choose another number $d'$ in $Or_2(d)$. This step is very crucial. In the case when $p(n)=nc$ for a irrational number $c$, we can use some sort of translational invariance of the sequence $\{nc \mod 1\}$ to set up the link between $\{nc+2^n d\mod 1 \}$ and $\{nc+2^n d'\mod 1 \}.$ We can not directly use this argument for the general polynomial situation. Some modifications are needed. As an illustration, we first deal with the sequence $\{n^2 c+2^n d\mod 1 \}.$ Again we have to re-select a number $d'$ in $Or_2(d)$ and we want to set up a link between $\{n^2 c+2^n d\mod 1 \}$ and $\{n^2 c+2^n d'\mod 1 \}.$ Let $i_k\to\infty$ be a sequence of integers such that $2^{i_k}d\to d'$ as $i_k\to\infty.$ Then we consider points $i^2_k c, (i_k+1)^2 c, (i_k+2)^2 c.$ It is of no loss of generality to assume that there exist three numbers $c_1,c_2,c_3\in [0,1]$ such that the fraction parts of $i^2_k c, (i_k+1)^2 c, (i_k+2)^2 c$ converge to $c_1,c_2,c_3$ respectively. Indeed, we only need to take a subsequence out of a subsequence three times to ensure all numbers converge as required. Now, observe the following relation which holds for all integer $k\geq 1,$ $$((i_k+2)^2 c-(i_k+1)^2 c)-((i_k+1)^2 c-i^2_k c)=2c.$$ This implies that (by treating the integer part and fractional part in the above expression separately) we have $$c_3-c_2\mod 1=c_2-c_1+2c \mod 1.$$ Now for each integer $M$ we consider the $M+1$ points $i^2_k c,\dots, (i_k+M)^2 c$. We can still assume that they converge ($\mod 1$) respectively to $c_0,c_1,\dots,c_M.$ Arguing as above we see that for each integer $m\in \{1,\dots, M\}$, $$c_{m+1}-c_{m}\mod 1= c_{m}-c_{m-1}+2c\mod 1.$$ This implies that $\{c_0,c_1,\dots c_M\}$ is actually $\{c_0+n(c_1-c_0)+n(n-1)c\}_{n=1}^{n=M}.$ It is evident that along the sequence $i_k\to\infty$ we have $2^{i_k+m}d\mod 1\to 2^md' \mod 1$ for each $m\in\{0,\dots,M\}.$ This implies that $\{c_0+n(c_1-c_0)+n(n-1)c+2^n d'\}_{n=1}^{n=M}\subset \overline{\{n^2 c+2^n d\}_{n\geq 1}}.$ Now for each integer $M$ we obtained a number $c_0(M)$ such that $$\{c_0(M)+n(c_1(M)-c_0(M))+n(n-1)c+2^n d'\}_{n=1}^{n=M}\subset \overline{\{n^2 c+2^n d\}_{n\geq 1}}.$$ We can assume that for a subsequence $M_k\to\infty$ both $c_0(M_k), c_1(M_k)$ converges. Therefore there exist numbers $c_0^*, c_1^*\in [0,1]$ such that, $$\{c_0^*+nc_1^*+n(n-1) c+2^n d'\}_{n=1}^{n=\infty}\subset \overline{\{n^2 c+2^n d\}_{n\geq 1}}.$$ This means that $\overline{\{n^2 c+2^n d\}_{n\geq 1}}$ contains a translational copy of $\overline{\{nc_1^*+n(n-1) c+2^n d'\}_{n\geq 1}}$ and this is exactly the result we want. Notice that $\{nc_1^*+n(n-1) c\mod 1\}$ equidistributes in $[0,1].$ Let $c$ be an irrational number and $d$ be any real number. We have $${\underline{\dim_{\mathrm{B}}}}\{n^2c+2^n d\mod 1\}_{n\geq 1}=1.$$ The above method can be iterated to obtain the result for general polynomials. We omit here the full detail and only state the result. Let $d$ be any real number and $p$ be a polynomial with at least one irrational coefficient of non constant term. We have $${\underline{\dim_{\mathrm{B}}}}\{p(n)+2^n d\mod 1\}_{n\geq 1}=1.$$ Our task is to show that if we choose a number $d'$ inside the orbit closure $Or_2(d)$ then there is an equidistributed sequence $\{x_n\}_{n\geq 1}$ such that $\{x_n+2^n d'\}\subset \overline{\{p(n)+2^n d\mod 1\}_{n\geq 1}}.$ As we can choose $d'$ to be generic with respect to an ergodic measure of the doubling map, the argument in the proof of Theorem \[Thm1\] can be applied to show the result of this theorem. If $c$ is a rational number, it is easy to see that for each $k\geq 1$, the fractional part of $n^k c$ for integers $n$ can attain at most finitely many values. Therefore we can assume that $p$ has degree at least two and the coefficient of the highest term is a irrational number. Given $m\geq 2$ many real numbers $x_1,\dots,x_m.$ We define the $m$-th order difference inductively as follows. For $m=2$ we write $$\Delta_2(x_1,x_2)=x_1-x_2.$$ Having defined the $(k-1)$-th order difference $\Delta(x_1,\dots,x_{k-1})$ we can define the $k$-th order difference as $$\Delta_k(x_1,\dots,x_k)=\Delta_{k-1}(\Delta_2(x_1,x_2),\Delta_2(x_2,x_3),\dots,\Delta_2(x_{k-1},x_{k})).$$ For example we see that $$\Delta_3(x_1,x_2,x_3)=(x_1-x_2)-(x_2-x_3)=-2x_2+x_1+x_3$$ and $$\Delta_4(x_1,x_2,x_3,x_4)=\Delta_3(x_1-x_2,x_2-x_3,x_3-x_4)=-2(x_2-x_3)+(x_1-x_2)+(x_3-x_4).$$ If $p$ has degree $m\geq 2$, then $$\Delta_m(p(n),p(n+1),\dots,p(n+m-1)),$$ viewed as a function of $n$, is a polynomial of degree $0$, namely, a constant $c'$. On the other hand, for any given numbers $x_1,\dots,x_{m-1},$ there is a unique way to extend these numbers to an infinite sequence $x_1,\dots,x_n,\dots$ such that $$\Delta_{m}(x_{n},\dots,x_{n+m-1})=c'$$ for all $n\geq 1.$ It is also possible to see that $x_n=p_1(n)\mod 1$ for a polynomial $p_1$ with degree $m$ and the coefficient of the highest order term is the same as that of $p$. Then the same argument as we have shown in the case when $p(n)=n^2 c$ can be applied here. As a result we see that there is a irrational polynomial $p_0$, which is in general different than $p$ such that $$\{p_0(n)+2^n d'\}_{n\geq 1}\subset \overline{\{p(n)+2^n d\mod 1\}_{n\geq 1}}.$$ From here we can use the entropy argument in the proof of Theorem \[Thm1\] and finish the proof of this Theorem. It is interesting to see whether we can replace $\{p(n)\mod 1\}$ with an arbitrary equidistributed sequence $x_n$ in $[0,1].$ The difficulty is that in this case we cannot easily shift $d$ in $\{x_n+2^n d\}$ into $d'.$ The above results seem to suggest the plausibility the following statement. Let $H=\{x_n\}_{n\geq 1}$ be an equidistributed sequence in $[0,1]$. Then we have for all $d\in [0,1],$ $${\underline{\dim_{\mathrm{B}}}}\{x_n+2^n d\mod 1\}_{n\geq 1}=1.$$ However, this is not the case. There is an equidistributed sequence $H=\{x_n\}_{n\geq 1}$ and a real number $d$ such that $$\{x_n+2^n d \mod 1\}_{n\geq 1}=\{0\}.$$ Let $d'$ be a generic point with respect to the doubling map and the Lebesgue measure. Then $\{2^n d'\}_{n\geq 1}$ equidistributes in $[0,1].$ Take $d=1-d'$ and we see that $\{2^n d+2^n d'\mod 1\}_{n\geq 1}=\{0\}.$ Despite the above result, it leaves open the following problem. \[23\] Let $d',d''$ be two real numbers. Is it true that $${\underline{\dim_{\mathrm{B}}}}\{3^n d'+2^n d'' \mod 1\}_{n\geq 1}\geq \max\{{\underline{\dim_{\mathrm{B}}}}\{3^n d' \mod 1\}_{n\geq 1},{\underline{\dim_{\mathrm{B}}}}\{2^n d'' \mod 1\}_{n\geq 1}\}$$ or even $${\underline{\dim_{\mathrm{B}}}}\{3^n d'+2^n d'' \mod 1\}_{n\geq 1}\geq \min\{1,{\underline{\dim_{\mathrm{B}}}}\{3^n d' \mod 1\}_{n\geq 1}+{\underline{\dim_{\mathrm{B}}}}\{2^n d'' \mod 1\}_{n\geq 1}\}?$$ We note that the above problem is closely related to the results in [@HS12]. In fact $Or_{3}(d')=\overline{\{3^n d'\mod 1\}_{n\geq 1}}$ and $Or_2(d'')=\overline{\{2^n d''\mod 1\}_{n\geq 1}}$ are $\times 3,\times 2$ invariant sets respectively. Then by [@HS12 Theorem 1.3], for each projection $\pi: (x,y)\in\mathbb{R}^2\to ux+vy\in\mathbb{R}$ with $u,v\in\mathbb{R}$ and $uv\neq 0$, the projected image $\pi(Or_3(d')\times Or_2(d''))$ always attains the maximal dimension, namely, $${\dim_{\mathrm{H}}}\pi(Or_3(d')\times Or_2(d''))=\min\{1,{\dim_{\mathrm{H}}}Or_2(d'')+{\dim_{\mathrm{H}}}Or_3(d')\}.$$ In particular if we choose $u=v=1$ in above we see that the sum set $Or_3(d')+Or_2(d'')$ attains the maximal dimension. This argument almost gives an answer to Question \[23\]. The difficulty of Question \[23\] is that we are not taking the sum set of $Or_2(d'')$ and $Or_3(d'')$ but sums between individual elements in the sequences $\{2^n d''\mod 1\}_{n\geq 1}$ and $\{3^n d'\mod 1\}_{n\geq 1}.$ Further questions ================= Hausdorff dimension ------------------- So far we considered the box dimension sequences of form $\{p(n)+2^n d\mod 1\}_{n\geq 1}.$ A natural question is what about the Hausdorff dimension of its closure? One observation is that if $\overline{\{2^n d\mod 1\}}$ has full Hausdorff dimension, then because it is a closed $\times 2\mod 1$ invariant set, it must be $[0,1]$. In particular we have $0\in \overline{\{2^n d\mod 1\}}.$ By the argument in the proof of Theorem \[Thm1\] we see that there is another polynomial with at least one irrational coefficient such that $\{p'(n)+2^n 0\}$ is contained in the closure of $\{p(n)+2^n d\mod 1\}_{n\geq 1}.$ This implies that $\{p(n)+2^n d\mod 1\}_{n\geq 1}$ itself must be dense in $[0,1].$ On the other hand, if $\overline{\{2^n d\mod 1\}}$ has Hausdorff dimension strictly smaller than $1$, then observe that the following difference set $$\{p(n)+2^n d\mod 1\}_{n\geq 1}-\{2^n d\mod 1\}_{n\geq 1}$$ is dense in $[0,1].$ Since $\overline{\{2^n d\mod 1\}}$ is $\times 2\mod 1$ invariant we see that the box dimension of $\overline{\{2^n d\mod 1\}}$ is equal to its Hausdorff dimension which is strictly smaller than $1$. This implies that $\overline{\{p(n)+2^n d\mod 1\}_{n\geq 1}}$ must have positive Hausdorff dimension. Thus we proved the following theorem. Let $d$ be any real number and $p$ be a polynomial with at least one irrational coefficient of non-constant term. We have $${\dim_{\mathrm{H}}}\overline{\{p(n)+2^n d\mod 1\}_{n\geq 1}}>0.$$ The above result and argument is from D-J. Feng and we thank him for permission to include it here. It is interesting to see whether stronger results can hold. We pose here the following problem. \[Q1\] Let $d$ be any real number and $p$ be a polynomial with at least one irrational coefficient of non constant term. Is it true that there is a absolute constant $a>0$ such that $${\dim_{\mathrm{H}}}\overline{\{p(n)+2^n d\mod 1\}_{n\geq 1}}\geq a.$$ Can $a$ be chosen to be $1$? In the beginning of Section \[ES\], we see that a subsequence of an equidistributed sequence may not be anywhere dense. Motivated by this result we ask here the following question. \[Q2\] Let $d$ be any real number and $p$ be a polynomial with at least one irrational coefficient of non constant term. Can $\{p(n)+2^n d\mod 1\}_{n\geq 1}$ be nowhere dense in $[0,1]?$ A negative answer to the above question would affirmatively answer Question \[Q1\]. Acknowledgement =============== HY was financially supported by the University of St Andrews. [dABCFS11]{} 27-th Brazilian Mathematical Olympiad, third round, problem 6, 2005. T. Austin. *Measure concentration and the weak Pinsker property*, Publications mathématiques de l’IHÉS, (2018), 1-119. K. Falconer. *Fractal geometry: Mathematical foundations and applications, second edition*, John Wiley and Sons, Ltd, 2005. H. Furstenberg. *Disjointness in Ergodic Theory, Minimal Sets, and a Problem in Diophantine Approximation*, Mathematical systems theory, **1**(1),(1967), 1-49. D-J. Feng and Y. Xiong. *Affine embeddings of Cantor sets and dimension of $\alpha\beta$-sets*, Israel J. Math., 226(2):805–826, 2018. M. Hochman and P. Shmerkin. *Local entropy averages and projections of fractal measures*,Annals of Mathematics, **175**,(2012), 1001-1059. Y. Katznelson. *On $\alpha\beta$-sets*, Israel J. Math., 33(1):1–4, 1979. P. Mattila. *Geometry of sets and measures in Euclidean spaces: Fractals and rectifiability*, Cambridge Studies in Advanced Mathematics, Cambridge University Press, 1999. M. Pollicott and M. Yuri. *Dynamical systems and ergodic theory*, London Mathematical Society Student Texts, Cambridge University Press, (1998). D. Simmons. *Conditional measures and conditional expectation; Rohlin’s disintegration theorem*, Discrete and continuous dynamical systems, **32**(7), (2012), 2565-2582. E. Stein and R. Shakarchi. *Real analysis: Measure theory, integration and Hilbert spaces*,Princeton University Press, (2005). M. Wu. *A proof of Furstenberg’s conjecture on the intersections of $\times p$ and $\times q$-invariant sets*, preprint,arxiv:1609.08053, (2016). H. Yu. *Rotations of tori and number theory I: Discrepancies of irrational rotations, binary expansions of powers of 3 and an improvement on Furstenberg’s slicing problem*, preprint, arxiv:1811.11073, (2018). H. Yu. *Multi-rotations on the unit circle*, preprint, (2018), arXiv:1808.09911 [^1]:

--- abstract: 'We examine the Butcher-Oemler effect and its cluster richness dependence in the largest sample studied to date: 295 Abell clusters. We find a strong correlation between cluster richness and the fraction of blue galaxies, [[$f_B$]{}]{}, at every redshift. The slope of the $f_B(z)$ relation is similar for all richnesses, but at a given redshift, [[$f_B$]{}]{}is systematically higher for poor clusters. This is the chief cause of scatter in the [[$f_B$]{}]{} vs. $z$ diagram: the spread caused by the richness dependence is comparable to the trend in [[$f_B$]{}]{} over a typical redshift baseline, so that conclusions drawn from smaller samples have varied widely. The two parameters, $z$, and a consistently defined projected galaxy number density, $N$, together account for all of the observed variation in [[$f_B$]{}]{} within the measurement errors. The redshift evolution of [[$f_B$]{}]{} is real, and occurs at approximately the same rate for clusters of all richness classes.' author: - 'V. E. Margoniner' - 'R. R. de Carvalho' - 'R. R. Gal and S. G. Djorgovski' title: 'The Butcher-Oemler Effect in 295 Clusters: Strong Redshift Evolution and Cluster Richness Dependence' --- Introduction ============ The Butcher-Oemler (BO) effect provided some of the first evidence for the evolution of galaxies and clusters. [Butcher & Oemler ]{}(1978, 1984) found an excess of blue galaxies in high redshift clusters in comparison with the typical early-type population observed in the central region of local clusters (Dressler 1980). The [BO effect ]{}has also been observed in more recent studies, which indicate an even stronger evolution of the fraction of blue galaxies in clusters (Rakos & Schombert 1995, Margoniner & de Carvalho 2000). The BO effect is an indicator of evolution in the cluster galaxy population, which may result from changes in the morphology and star-formation rates of member galaxies with redshift. The fact that blue galaxies are more commonly found at higher redshifts, and the observation of an apparent excess of S0 galaxies in low redshift clusters, lead to the suggestion by Larson [[*et al. *]{}]{}(1980) of an evolutionary connection between S0 and spiral galaxies. This idea can explain the population of blue galaxies observed by Butcher & Oemler as spiral galaxies seen just before running out of gas, and the disappearance of this population in more evolved, low redshift rich clusters (Dressler [[*et al. *]{}]{}1997, Couch [[*et al. *]{}]{}1998). In the last decade, high-resolution [[*Hubble Space Telescope *]{}]{}images have allowed the determination of the morphology of these high redshift blue galaxies. Dressler [[*et al. *]{}]{}(1994), Couch [[*et al. *]{}]{}(1994, 1998), and Oemler [[*et al. *]{}]{}(1997) found that the [Butcher & Oemler ]{}galaxies are predominantly normal late-type spirals, and that dynamical interactions and mergers between galaxies may be a secondary process responsible for the enhanced star formation. A recent study by Fasano [[*et al. *]{}]{}(2000) indicates that as the redshift decreases, the S0 cluster population increases while the number of spiral galaxies decreases, supporting Larson’s original idea of spirals evolving into S0s. A second factor affecting the number of blue galaxies may be their environment. The fraction of blue galaxies seems to be correlated with local galaxy density, and with the degree of substructure in the cluster. Many studies have shown that the blue galaxies lie preferentially in the outer, lower density cluster regions (Butcher & Oemler 1984, Rakos [[*et al. *]{}]{}1997, Margoniner & de Carvalho 2000). Also, Smail [[*et al. *]{}]{}(1998) studied 10 massive clusters and found that the fraction of blue galaxies in these clusters is smaller than observed in [Butcher & Oemler ]{}(1984) clusters at the same redshift range. However, an opposite trend is found in non-relaxed clusters with a high degree of substructure, where the fraction of blue galaxies is higher than that observed in regular clusters (Caldwell & Rose 1997, Metevier [[*et al. *]{}]{}2000). The evolution of these galaxies is probably correlated with its environment in the sense that spirals might consume and/or lose their gas, evolving to SOs, while falling into the higher density cluster regions. Although the exact mechanism responsible for the [BO effect ]{}is not completely understood (Kodama & Bower 2000), most authors seem to agree that the effect is real. A different idea is presented by Andreon & Ettori (1999) who show that the mean X-ray luminosity and richness of [Butcher & Oemler ]{}(1984) clusters increases with redshift, and argue that the increase in the fraction of blue galaxies with redshift may not represent the evolution of a single class of objects. We present observations of the [BO effect ]{}in a large sample of 295 Abell (1958, 1989) clusters of all richnesses, with no further selection on the basis of morphology or X-ray luminosities. This is important because all previous studies were based on samples biased toward richer clusters, with some samples being further selected according to morphology and/or X-ray luminosities, so that a combination of different selection effects could be mimicking the observed $f_B(z)$ relation. Although our sample inherits biases existent in the Abell catalog, it should contain a more representative variety of clusters (in terms of degree of substructure, richness, and mass) at each redshift than any previous sample, and because of its large size should allow a better determination of the relation. Any results driven by selection effects present in previous samples might become apparent when compared with this one. We describe the data in Section 2, the [BO effect ]{}analysis in Section 3, and our conclusions in Section 4. Sample Selection and Input Data =============================== The data presented in this paper were obtained to calibrate the DPOSS-II (the Digitized Second Palomar Observatory Sky Survey, Djorgovski [[*et al. *]{}]{}1999). It comprises 44 Abell clusters imaged at the 0.9m telescope at the Cerro Tololo Interamerican Observatory (CTIO) (Margoniner & de Carvalho 2000, hereafter MdC00), and 431 clusters observed at the Palomar Observatory 1.5m telescope (Gal [[*et al. *]{}]{}2000, hereafter G00, and in preparation). The CCD images were taken in the [[*g*]{}]{}, [[*r*]{}]{}  and [[*i*]{}]{}  filters of the Thuan & Gunn (1976) photometric system, with typical $1$-$\sigma$ magnitude errors of 0.12 in [[*g*]{}]{}, 0.10 in [[*r*]{}]{}, and 0.16 in [[*i*]{}]{}  at $r=20.0^m$. We have also observed 22 control fields in order to assess the background contribution. More details concerning the data reduction, photometry, and catalog construction can be found in MdC00 and G00. From this original sample we excluded 120 clusters observed with a small field of view CCD, and 31 very low redshift ($z<0.05$) clusters, for which only the core region can be observed with our $13^{\prime} \times 13^{\prime}$ images. Also, 26 cluster fields which exhibited galaxy counts comparable to the mean background, and one cluster with a bright star occupying $\sim 25\%$ of the CCD region, were excluded from the analysis. Since 2 Abell clusters had repeated observations from G00 and MdC00, our final sample comprises 295 clusters. It is important to note that at variance with previous studies, the sample presented in this work is representative of all richness class clusters (21% $R=0$, 50% $R=1$, 21% $R=2$, and 8% $R\ge3$). No further selection regarding richness, morphology or degree of sub-clustering was applied when determining our sample. Spectroscopic redshifts for 77 clusters were obtained from the literature, and for the remainder photometric redshifts were estimated with an rms accuracy of approximately 0.04 using the methodology described in MdC00. Analysis of the Butcher-Oemler Effect ===================================== The BO effect can be measured by comparing the fraction of blue galaxies in clusters at different redshifts. The $(g-r)~vs.~r$ [[color-magnitude ]{}]{}(CM) relation was determined by fitting a linear relation to the red galaxy envelope using the same prescription as in MdC00. Red envelopes were subjectively classified as well-defined or not by visual inspection. Galaxies were defined as [[*blue *]{}]{}if they had $(g-r)$ colors $0.2^m$ below the linear locus in the CM relation. We used the spectral energy distribution of a typical elliptical galaxy (Coleman [[*et al. *]{}]{}1980) to derive $k(z)$ corrections for our sample, and also corrected the data for extinction using the maps of Schlegel [[*et al. *]{}]{}(1998). Because the main purpose of this work is to study the evolution of galaxy populations, fainter galaxies play a crucial role. Whereas [Butcher & Oemler ]{}were limited to brighter galaxies by their photographic data, our CCD sample allows us to probe significantly deeper, selecting galaxies with [[*r*]{}]{}  magnitude between $M^*-1$ and $M^*+2$, inside a region of radius $0.7$ Mpc around the cluster. This fixed linear size was chosen because our CCD images cover a field of radius $\sim 0.5-4.0$ Mpc at $z=0.03-0.38$, and we want to study the same physical region and the same galaxy luminosity range for the entire sample. We assume a cosmology with $H_{\circ} = 67$ Km s$^{-1}$ Mpc$^{-1}$, and $q_{\circ}=0.1$, in which $M_r^*=-20.16$ (Lin [[*et al. *]{}]{}1996). Most ($61\%$) of the clusters in our sample are at $z=0.1-0.2$, so that the $0.7$ Mpc central region, and the entire luminosity range between $M^*-1$ and $M^*+2$ can be observed. Clusters at higher redshifts are limited at brighter absolute magnitudes and a correction was applied in order to compare the fraction of blue galaxies in these clusters with the rest of our sample. The lower redshift clusters suffer from the opposite problem, since the brighter galaxies will be saturated, and we are also limited by the field of view. Details about these corrections can be found in MdC00, the only difference being that in the present work we adopt a more conservative brighter limiting magnitude of $M^*+2$ instead of $M^*+3$. The blue and total counts contain a mix of cluster members and background galaxies. Background variations place a fundamental limit on the accuracy of [[$f_B$]{}]{} measurements: no matter how well one determines the [*mean*]{} background from a number of control fields, the background estimate for any particular cluster is no more accurate than the [*scatter*]{} in the control fields. For each cluster, we measured the effect of this scatter by computing [[$f_B$]{}]{} using background corrections from the 22 individual control fields scaled to the area of the CCD and the CM relation appropriate to that cluster. The final [[$f_B$]{}]{} is given by the median and the rms is based on the central two quartiles of the distribution. Simple propagation of errors through the equation $f_B \equiv {{n_{blue,cluster} - n_{blue,background}}\over{n_{total,cluster} - n_{total,background}}}$ would give an overestimate because $n_{blue,background}$ and $n_{total,background}$ are correlated. The usual assumption that the error in [[$f_B$]{}]{} is mainly due to Poisson statistics in the background-corrected cluster counts does not properly take into account the background variations and errors thus derived are about three times smaller than ours when applied to the same data. The final fractions of blue galaxies for the entire sample are shown in the upper panel of Figure 1. For those clusters with multiple observations, we have used the observation with smaller $\sigma_{f_B}$. Clusters with spectroscopic and photometric redshift measurements are indicated by solid and open circles respectively. The individual error bars are not presented to avoid confusion in the plot, but the median $\sigma_{f_B}$ is 0.071. The lower panel presents only clusters that have: (1) $\sigma_{f_B}<0.071$, (2) spectroscopic redshift measurements, and (3) a well-defined CM relation. The $1$-$\sigma_{f_B}$ errors are indicated for these clusters. In both panels, the solid lines indicates a linear fit derived from the clusters with $z\leq0.25$ ($f_B=(1.24 \pm 0.07)z - 0.01$, with an rms scatter of $0.1$ for the entire sample shown in the upper panel, and $f_B=(1.34 \pm 0.11)z - 0.03$, with an rms of $0.07$ for the sample shown in the lower panel). The fits were done with the GaussFit program (Jefferys [[*et al. *]{}]{}1988) taking into account the measurement errors in [[$f_B$]{}]{}. The dashed line indicates the rms scatter around the fit, and while the upper panel shows a larger scatter, the two derived relations are the same within the errors. A clear trend of strong evolution with redshift is seen. A $\chi_{\nu}$ test applied to the fit presented in the upper panel results in 1.36 if only [[$f_B$]{}]{}  errors are considered, and 1.13 when the uncertainties in the photometric redshifts are also taken into account. Such small $\chi_{\nu}$ is due however to large measurement errors in this sample. A higher $\chi_{\nu}$ of 1.72 is obtained for the subsample of clusters with smaller $\sigma_{f_B}$ and spectroscopic redshift (lower panel of Figure 1). Richness is a natural second parameter which might cause the range in [[$f_B$]{}]{} at given redshift. The left upper panel of Figure 2 shows the $f_B(z)$ diagram with symbol sizes scaled by $N$, the number of galaxies between $M^*-1$ and $M^*+2$, inside $0.7$ Mpc, after background correction. Only clusters from the lower panel in Figure 1 and at redshifts between 0.1 and 0.2, where no corrections needed to be applied to compute [[$f_B$]{}]{}, are presented in this figure. The solid line is the best-fit linear $f_B(z)$ relation for the 26 clusters shown, and the dashed lines indicate the relations obtained when the sample is subdivided in two 13 cluster samples according to richness. The slopes of the three relations are the same within the errors ($0.86\pm0.23$ for the entire sample, $0.96\pm0.26$ for the subsample of richest clusters, and $0.86\pm0.56$ for the poorest ones). Although the uncertainties are large, the rate of evolution is approximately constant for all richnesses, and there is clearly an effect, with richer clusters tending to lower [[$f_B$]{}]{}. The suggestion by Andreon & Ettori (1999) that the observed increase of [[$f_B$]{}]{}  with redshift is caused by missing poor clusters at higher redshift is therefore no longer tenable, because any such selection effects would serve to [*decrease*]{} the redshift evolution of [[$f_B$]{}]{}. The evolution in [[$f_B$]{}]{}  is real, and takes place in all richness classes. To ensure that richness and redshift are not correlated in this subsample, we plot $N~vs.~z$ in the right upper panel of Figure 2. The best-linear fit ($N = -(38.2\pm106.2)z + (89.5\pm14.5)$) is indicated by the solid line, and the dashed lines represent adding/subtracting $1$-$\sigma$ uncertainties to its coefficients. Richness and redshift are uncorrelated in this sample. To gauge the richness importance, we investigate $f_B(z,N)$ relations with various richness dependences ($N^{-1}$, $N^{-3/2}$, $N^{-2}$, and $N^{-3}$), and found that $N^{-2}$ correlates slightly better with [[$f_B$]{}]{}. The final best-fit relation for the data, when both redshift and richness are considered, and taking into account the errors in [[$f_B$]{}]{}and $N$, is $f_B = (1.03\pm0.25)z + (388.3\pm111.4)N^{-2} - 0.04$, with $\chi_{\nu}=0.99$. The $F$ test indicates with $>99\%$ confidence ($F_{\chi_{\nu}}=23.8$, for 25 to 24 degrees of freedom) that richness is responsible for the improvement observed in $\chi_{\nu}$. Richness is therefore extremely important in determining [[$f_B$]{}]{} and it is in fact enough to account for the scatter observed in a simple $f_B(z)$ relation. In the lower left panel of Figure 2 we plot [[$f_B$]{}]{} [*vs.*]{} $z + 378.6 N^{-2}$, which represent an edge-on view of the best-fit plane $f_B(z,N)$, and in the lower right panel we present the redshift dependence of a richness-corrected-[[$f_B$]{}]{}: the redshift evolution is clear. Although all 26 clusters shown in Figure 2 were used to compute the relations indicated by solid lines in the figure, the two clusters represented by open circles (Abell 520, and Abell 1081) seem to deviate from the trend. These clusters also have the largest $\sigma_N$ (derived from the scatter in the 22 control regions), and if excluded from the fitting, the relation changes to $f_B = 1.13 z + 480.56 N^{-2} - 0.06$ ($\chi_{\nu}=0.95$), indicating a slightly stronger redshift dependence. Finally, we used 11 clusters with ROSAT X-ray luminosities to check for correlations with [[$f_B$]{}]{}, but this small sample did not allow any conclusions. There is a possible trend of increasing [[$L_X$]{}]{} with redshift in the sense found by Andreon & Ettori (1999) in BO’s original sample. However, there is no clear trend of [[$f_B$]{}]{} with [[$L_X$]{}]{}, arguing against any contamination of the $f_B(z)$ relation by selection effects. A comparison of the X-ray luminosities of these clusters with their [[$f_B$]{}]{} could provide new insight to the BO effect, and we are in the process of obtaining X-ray fluxes and upper limits from the RASS (Rosat All Sky Survey) for all of these clusters, as well as a larger sample of new clusters generated from DPOSS-II. We stress that these results are not to be directly compared with fractions of blue galaxies as originally defined in BO84. The most important reason for differences is the magnitude range used to calculate [[$f_B$]{}]{}. The number of blue galaxies increases at fainter magnitudes and it is therefore natural that our [[$f_B$]{}]{} measurements are in general higher. Also, the usage of a rigid physical scale for all clusters, instead of regions determined individually for each cluster according to its density profile, should yield different [[$f_B$]{}]{}estimates. When [[$f_B$]{}]{} is computed using the same absolute magnitude range used by BO84, we find signs of evolution that are stronger than originally suggested in their work, and which are consistent with recent works by Rakos & Schombert (1995) and MdC00. Limiting our sample at brighter absolute magnitudes however increases the $\sigma_{f_B}$ (and $\sigma_{N}$) because of the smaller galaxy count statistics, and probes a smaller fraction of the cluster population. Summary and Conclusions ======================= We compute [[$f_B$]{}]{} for 295 randomly selected Abell (1989) clusters, including galaxies as faint as $M^* + 2$ to sample a larger range of the luminosity function and provide better statistics in each cluster. The resulting [[$f_B$]{}]{} shows a stronger trend with redshift than did [[$f_B$]{}]{}as originally defined by BO84, consistent with the idea that the [BO effect ]{}is stronger among the late-type spirals and irregulars which dominate the galaxy populations at intermediate and lower luminosities. This is the first sample large enough to allow the study of the [[$f_B$]{}]{}variations at a given redshift. A $\chi_{\nu}$ test applied to a simple linear $f_B(z)$ relation shows that the scatter in [[$f_B$]{}]{} at a given redshift is consistently larger than the measurement error, indicating a real cluster-to-cluster variation. We investigate the richness dependence of the Butcher-Oemler effect, and find a strong correlation between [[$f_B$]{}]{} and galaxy counts in the sense that poorer clusters tend to have larger [[$f_B$]{}]{} than richer clusters at the same redshift. The inclusion of poor clusters tends therefore to increase the slope of the $f_B(z)$ relation, and is another factor (together with the inclusion of fainter galaxies) responsible for the stronger evolution observed in this sample when compared to previous works based mostly on rich clusters. We show that the fraction of blue galaxies in a cluster can be completely determined by its redshift and richness (within the measurement errors). The evolution in [[$f_B$]{}]{} with redshift is real, and occurs at approximately the same rate for clusters of all richnesses. We thank D. Wittman, J.A. Tyson, A. Dressler, S. Andreon, and the anonymous referee for very helpful comments and suggestions which helped to improve the paper. We also thank the Palomar TAC and Directors for generous time allocations for the DPOSS calibration effort, and numerous past and present Caltech undergraduates who assisted in the taking of the data utilized in this paper. RRG was supported in part by an NSF Fellowship and NASA GSRP NGT5-50215. The DPOSS cataloging and calibration effort was supported by a grant from the Norris Foundation. Abell G.O., 1958, , 3, 211. Abell G.O., Corwin H.G., Olowin R.P., 1989, , 70, 1. Andreon S., Ettori S., 1999, , 516, 647. Butcher H. & Oemler A.Jr., 1978, , 219, 18. Butcher H. & Oemler A.Jr., 1984, , 285, 426. Caldwell N. & Rose J.A., 1997, , 113, 492. Coleman G.D., Wu C-C., Weedman D.W. 1980, , 43, 393. Connolly A.J., Csabi I., Szalay A.S., 1995, , 110, 6. Couch W.J., Ellis R.S., Sharples R.M., Smail I., 1994, , 430, 121. Couch W.J., Barger A.J., Smail I., Ellis R.S., Sharples R.M., 1998, , 497, 188. Djorgovski S.G., Gal R.R., Odewahn S.C., de Carvalho R.R., Brunner R., Longo G., and Scaramella R., 1999, in Wide Field Surveys in Cosmology, eds. S. Colombi, Y. Mellier, and B. Raban, Gif sur Yvette: Eds. Frontières p. 89. Dressler A., 1980, , 236, 351. Dressler A., Oemler A.Jr., Butcher H.R., Gunn J.E., 1994, , 430, 107. Dressler A., Oemler A.Jr., Couch W.J., Smail I., Ellis R.S., Barger A., Butcher H.R., Poggianti B.M., Sharples R.M, 1997, , 490, 577. Fasano G., Poggianti B.M., Couch W.J., Bettoni D., Kjærgaard P., Moles M., 2000, , in press. Gal R.R., de Carvalho R.R., Brunner R., Odewahn S.C., Djorgovski S.G., 2000, , 120, 540. Jefferys, W.H., Fitzpatric M.J., McArthur B.E., 1988 Celest. Mech., 41, 39 Kodama T. & Bower R.G., 2000, submitted to . Larson R.B., Tinsley B.M., Caldwell N., 1980, , 237, 692. Lin H., Kirshner R.P., Shectman S.A., Landy S.D., Oemler A., Tucker D.L., Shechter P.L., 1996, , 464, 60. Margoniner V.E. & de Carvalho R.R., 2000, , 119, 1562. Metevier A.J., Romer A.K., Ulmer M.P. , 2000 , 119, 1090. Oemler A.Jr., Dressler A., Butcher H.R., 1997, , 474, 561. Rakos K.D., Schombert J.M., 1995, , 439, 47. 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--- abstract: | We have constructed a preonic model starting from a coloured fermionic preon and by postulating a new symmetry, MUSY. This new symmetry is defined via the MU number involving colour, charge and spin properties of the preons. We show that all the known fields of the Standard Model (SM) can be constructed using the fermionic preon and 6 preonic scalars, its MUSY partners. As an example, we present well known $\beta$-decay process at MUSY level. MUSY also forbids some processes such as proton decay (preserving the baryon number) and hence, it is compatible with current experimental results. In this model the number of SM generations arises to be three naturally. The MUSY generalization of the SUSY algebra is constructed and the MUSY invariant Lagrangian is also explicitly written. Similar to other preonic and supersymmetric models, a number of new particles are predicted. These particles do not interact with any of the SM fermions but only with the gauge bosons. These particles could be dark matter candidates. author: - Oktay Doğangün - Saleh Sultansoy - 'N. Gökhan Ünel' title: 'A Preonic Model with Colour - Spin Symmetry' --- Introduction ============ Although the Standard Model (SM) of particle physics is able to explain the experimental results obtained so far [@PDG-1], its problems and open questions shed some doubt on its prospects as being the ultimate theory of fundamental particles and their interactions. The large number of free parameters in the model, the nature of the elementary particles, the reason for their observed mass hierarchy, the electroweak symmetry breaking mechanism are some examples of these open issues. A number of alternative theories have been proposed as the cure to these problems. For example, Grand Unified Models [@GUT-1], Technicolour [@Technicolour-1], Compositeness [@Compositeness-1] and Supersymmetry [@SUSY-1] can be cited. The composite models are particularly interesting for the continued simplification they offer. Similar situations have been seen at least twice in the the past: The big and complex periodic table of elements has been understood in terms of three constituents and their combinations: the proton, the neutron and the electron. Similarly, to solve the hadron inflation of the 1960ies, the quark model had to be introduced. The hope of the authors is to propose a solution to some of the aforementioned problems with an effective model at the preonic level. Considering elementary particle family replication and especially the fermion mixings as hints for preonic structure of some of SM particles, a number of preonic models, with different levels of compositeness, have been studied by various groups. Below we list a few examples. H. Harari [@harari] and A. M. Shupe [@shupe] proposed two coloured fermionic preons (T,V) with which the first generation of SM quarks and leptons can be formulated as three preon bound states. In this model, the other fermion generations are thought to be excitations of the first generation and the SM bosons are assumed to be fundamental particles. Considering Harari-Shupe model, Buchmann and Schmid [@BuSch-1] later proposed some quark-like structures with hypercolours so that the model become compatible with ’t Hooft anomaly. H. Fritzsch and G. Mandelbaum considered two fermionic and two bosonic preons ($\alpha,\beta,x,y$) to construct all SM fundamental particles except the photon and gluons [@FriztschM-1]. In this model, the first generation of SM quarks and leptons are composed of a fermionic and a bosonic preon, while $W$, $Z$ and Higgs bosons are assumed to consist of a fermionic preon and its anti-particle. The other generations are constructed by adding one or more “gluons” of the preon binding force (haplodynamics) to the appropriate state. Therefore, this model is not able to give any constraint on the number of the generations in the SM. J. Dugne, S. Fredriksson, J. Hansson and E. Predazzi, in their Preon Trinity Model [@Trinity-1], proposed three coloured fermionic preons ($\alpha,\beta,\delta$) with which they construct three scalar bound states ($x,y,z$). With the bound state of a fermionic and a scalar preon (or its anti-particle) they construct all SM fermions and more: one additional charged lepton, two neutral leptons and three additional quarks emerge naturally from the model. With the various combinations of the fermionic preons, they construct the electroweak gauge bosons and more: the model predicts two additional charged bosons and four additional neutral ones. A. Celikel, M. Kantar and S. Sultansoy [@Celikel] proposed two scalar and two fermionic preons, all coloured. In this model leptons are bound states of one fermionic preon and one scalar antipreon whereas antiquarks consist of one fermionic and one scalar preon. As a result each SM lepton has its colour octet partner and each SM antiquark has its colour sextet partner. S. Ishida and Sekiguchi [@ishida] have discussed a left-right symmetric preon model assuming the quarks, leptons and weak bosons as composite states and the weak interaction is a secondary effective one without the need of breaking a fundamental symmetry. The fermions are formed by a fermionic and a bosonic preons where the fermionic preons are colour singlet but isospin doublet and bosonic preons are $\bar{\bf 3} \oplus {\bf 1}$ colour. The conservation of baryon and lepton number is included by hand. The goal of this work is to use the key concepts from compositeness and supersymmetry ideas for constructing a new model with charge-colour and spin symmetry at preonic level. Using this preonic model, we attempt to reduce the number of free parameters, answer the questions on the number of elementary particle families, to propose a candidate for the Dark Matter (DM) and to solve the hierarchy problem. It will be shown that all three fermion generations of the SM can be obtained from a single chiral coloured preon using appropriate symmetries. Using current-current interactions, the known bosons of the SM naturally emerge in the right form, e.g. gluon masses vanishingly small. The baryon and lepton number conservations are emergent from the model itself, prohibiting the anomalies like the proton decay. The rest of this manuscript is organized as follows: in the next section, the reader will be motivated for a colour and spin symmetric preonic model, MUSY model, which defines all known SM fundamental particles as preonic bound states. The section \[sec:Dynamics-of-MUSY\] will go into the dynamics of the MUSY model to define the MUSY algebra and to construct the effective MUSY Lagrangian. In Section \[sec:SMparticles\], the preonic bound states forming SM particles aer discussed. Section \[sec:fenomenoloji\] contains a MUSY description of the well known processes of proton and $\beta$-decay followed by an assesment of the CKM and MNS mixings in MUSY model. In section \[sec:Predictions-of-MUSY\], a number of testable MUSY predictions are considered: excited fermions and dark matter candidates. The last section contains some discussions on the phenomenology of the MUSY model. Building the MUSY model\[sec:Building-the-MUSY\] ================================================ The construction of the model starts with a single fermionic chiral preon colour triplet of charge $-e/6$. Firstly, we define a conserved “MU”[^1] number for each preon as follows: $$MU=QC+S+H\,,\label{eq:MUnumber}$$ where Q is electrical charge, C is number of colours (3 for colour, -3 for anticolour triplet, 1 for colour singlet), S is spin and H is helicity ( $+1/2$ for right, $-1/2$ for left components). The conserved quantity MU will arise from a transformation which makes the action invariant, i.e., a MU symmetry or MUSY in short. Q C S H MU ------------------ -------- ------- ------- -------- -------- $\psi_{a}$ $-1/6$ **3** $1/2$ $+1/2$ $+1/2$ $\bar{\psi}^{a}$ $-1/6$ **3** $1/2$ $-1/2$ $-1/2$ $\phi_{i}$ $+1/2$ **1** $0$ $0$ $+1/2$ $\hat{\phi}_{i}$ $-1/2$ **1** $0$ $0$ $-1/2$ : Preons of the MUSY model: musions. The subscript $i$ runs from 1 to 3 and $a$ runs for three colour charges (Red, Green and Blue). \[Tab:MUSY-preons\] A symmetry transformation that changes spin-1/2 to spin-0, colour triplet to singlet (or vice versa) leads us to generate six scalar preons starting from the single initial fermionic preon with left and right states. Therefore, each helicity state of the fermionic preon will give three complex scalars since it is a colour triplet as well as a complex spinor field. According to these constraints, one may write the MU transformation mappings as follows using the notation of [@Martin]: $$\begin{aligned} \label{eq:MUSY-transformation} \begin{array}{rcl} \psi_{\alpha, a} & \rightarrow & \phi_{i}\label{eq:aksi2-1}\\ \bar{\psi}^{\dot{\alpha}, a} & \rightarrow & \hat{\phi}_{i}^{*}\\ \phi_{i} & \rightarrow & \psi_{\alpha, a}\\ \hat{\phi}_{i} & \rightarrow & \bar{\psi}^{\dot\alpha, a} \end{array}\end{aligned}$$ where $\psi_{\alpha, a}$ and $\bar{\psi}^{\dot\alpha, a}$ are respectively left and Hermitian conjugate of the right handed fermionic preons with colour index $a=R,G,B$, and the fields $\phi_{i}$ and $\hat{\phi}_{i}$ denote the scalar preons with family index $i=1,2,3$. The quantum numbers of the MUSY preons are presented in Table \[Tab:MUSY-preons\]. One may notice that the transformation maps colours to families in addition to mapping fermions to bosons as in supersymmetry [@Martin]. This property makes the model look like a “colour-family extension of supersymmetry”, that is, MUSY will reduce to a SUSY if one equates the colour indices to the fermion family indices, which means setting $a=i$ [@WessZumino]. In supersymmetry, a fermionic parameter, denoted by $\epsilon$, was introduced to suppress the spinor indices in one side of the supersymmetric transformations [@SUSY-1]. In MU symmetry, as one may notice, there has to be a similar variable that would suppress both colour and family indices. Hence, a MU symmetric parameter, denoted by $\zeta_{i}^{a}$, is to be introduced. It will lead to CKM mixing of quarks and electroweak symmetry breaking as it will be discussed later in Section \[sec:fenomenoloji\]. Notice that the reduction of MUSY to SUSY is then nothing but setting the parameter $\zeta_{ia}$ proportional to Kronecker delta $\delta_{ia}$. The complete transformation will be discussed in Section \[sub:MUSY-transformation\]. Dynamics of the MUSY model\[sec:Dynamics-of-MUSY\] ================================================== Generally, most of the preonic models suffer from the lack of a convincing dynamical explanation for the problem of the bound states consisting of both fermion and scalar preons. The crucial problem in those models arises from the absence of an explicit interaction between preons. Some models constist of a Yukawa interaction in which the preons have a common gauge symmetry. ![Vertex of fermionic and scalar preons (left). Bound state of a fermionic preon and a scalar preon forming a quark (right). \[fig:boundstates\]](boundstates) In MUSY model, there does not exist a Yukawa interaction between scalar preons and fermionic preons because of the chirality, i.e., a Yukawa term is not invariant under $SU(3)_L \otimes SU(3)_R$ or $SU(2)_L \otimes SU(2)_R$ global symmetry for which the scalar sector and the fermionic sector remains invariant, respectively. However, the MUSY model has an advantage of having Fermi-like interactions which lead one to have an approximate gauge symmetry as will be discussed in the next sections. These approximate gauge interaction terms, here denoted by ${\cal L}_{int}$, will effectively generate supersymmetric Noether currents [@Iorio:2000xv] of preons in low energies (i.e., SM scale) as follows: $$\begin{aligned} {\cal L}_{\mbox{eff}} &=& - \frac{i}{2} \int d {\bf x} \, T \, \lbrace {\cal L}_{int} ({\bf x}), {\cal L}_{int} ({\bf x}^\prime) \rbrace \\ &=& \frac{G}{\sqrt2} \, \bar\zeta J_N^\mu \bar\zeta J_{N\mu} + \frac{G}{\sqrt2} {\bar J}_N^\mu \zeta {\bar J}_{N\mu} \zeta + \mbox{Higher dimensional terms}\end{aligned}$$ where it is used the Wilson short distance method [@wilson], $$\begin{aligned} \label{eq:noether_current} J_{N}^{\mu} & = & \gamma^{\nu}\gamma^{\mu}\Psi\partial_{\nu}\Phi^{\dagger}\end{aligned}$$ is a fermionic current and have both colour and family indices, $\zeta$ is the MUSY parameter, $\gamma^{\mu}$ are Dirac matrices, $\Psi$ is the four-component fermionic preon, $\Phi$ is the scalar preon doublet. The $\left( \Phi \partial \Psi \right)^2$ vertex occured in the effective Lagrangian of the MUSY model is shown on the left of Figure \[fig:boundstates\]. On the right of the figure, it is described the low energy transition of the bound state schematically. In this Section it will be described the dynamics of the MUSY model in the following picture: $$\begin{aligned} {\cal L}_{\mbox{preon}} \left( \Psi, \Phi \right) \rightarrow {\cal L}_{\mbox{app.}} \left( \Psi, \Phi, {\bf G} , {\bf W} , {B} \right) \rightarrow {\cal L}_{\mbox{eff}} \left( q, \ell, {\bf G} , {\bf W} , { B}\right) + {\cal L}_{\mbox{DM}}\end{aligned}$$ where $\Psi, \Phi$ are preons, $q, \ell$ are quarks and leptons, ${\bf G} , {\bf W} , {B}$ are vector bosons with approximate guage symmetry and $ {\cal L}_{\mbox{DM}}$ is the Lagrangian of the composite fields other than SM fields. MUSY transformation\[sub:MUSY-transformation\] ---------------------------------------------- MUSY transformation maps a colour-triplet Weyl spinor to three scalar fields, and vice versa, since bosonic number of degrees of freedom ($n_{B}$) and fermionic number of degrees of freedom ($n_{F}$) should be equal in order to handle the naturalness problem of scalar preons, that is, when $n_{F}=n_{B}$, the huge mass correction to a scalar field vanishes. Indeed, a (complex) colour-triplet Weyl spinor has 12 degrees of freedom and each (complex) scalar have 2 degrees of freedom. The MU transformation also conserves the number of degrees of freedom between bosonic preons and fermionic preons as shown in Table \[tab:ndf\_musy\]. ----- -- -- ------------------- ------------------- ------------------- ---------------------- -- -- -- -- $\phi_{i}$ $\phi_{i}^{*}$ $\hat{\phi}_{i}$ $\hat{\phi}_{i}^{*}$ ndf $1\times3\times1$ $1\times3\times1$ $1\times3\times1$ $1\times3\times1$ ----- -- -- ------------------- ------------------- ------------------- ---------------------- -- -- -- -- : Fermionic and bosonic numbers of degrees of freedom (ndf) in MUSY model. The order of the product in ndf row is colour $\times$ family $\times$ spinor. \[tab:ndf\_musy\] Naturally, the infinitisimal change in a scalar under a spin transformation will require its right-hand side to be a scalar such that the spinor indices are suppressed. Also, it can be seen immediately that MUSY transformation \[eq:MUSY-transformation\] should have a new parameter discussed previously to satisfy the family index in both sides of the equation. Therefore, the change in scalar takes the following form: $$\begin{aligned} \label{eq:scalar-transformation} \begin{array}{rcl} \delta\phi_{i} & = & \zeta_{i}^{a}\epsilon^{\alpha}\psi_{\alpha a}\\ \delta\hat{\phi}_{i} & = & \zeta_{ia}\bar{\epsilon}_{\dot{\alpha}}\bar{\psi}^{\dot{\alpha}a} \end{array}\end{aligned}$$ where $\epsilon_{\alpha}$ is the so-called SUSY parameter, and $\zeta_{ia}$ are complex constants in a global MUSY transformation, i.e. $\partial_{\mu}\zeta_{ia}=0$ [@Martin]. One can note that $$\begin{aligned} \begin{array}{rcl} \delta\phi^{*} & = & \delta\hat{\phi}\\ \delta\hat{\phi}^{*} & = & \delta\phi \end{array}\end{aligned}$$ although $\phi$ and $\hat{\phi}^{*}$ themselves are not equal (as it should not), due to the condition of $n_{F}=n_{B}$. Now, we may deduce the transformation of the fermionic preon for the free case by writing the MUSY Lagrangian density, which will be discussed in the next section, as follows: $$\begin{aligned} \mathcal{L}_{free} & = & -\partial^{\mu}\phi_{i}^{*}\partial_{\mu}\phi_{i} -\partial^{\mu}\hat{\phi}_{i}^{*}\partial_{\mu}\hat{\phi}_{i} +i\bar{\psi}^{a}\bar{\sigma}^{\mu}\partial_{\mu}\psi_{a}\end{aligned}$$ where summation convention is used over repeated indices $i$, $a$ and $\mu$. Therefore, the change in fermion field, which makes the action invariant under MUSY transformations up to total derivatives, is written as follows: $$\begin{aligned} \delta\psi_{\alpha,a} & = & -i\zeta_{ia}\sigma_{\alpha\dot{\beta}}^{\mu}\bar{\epsilon}^{\dot{\beta}} \partial_{\mu}\left(\phi_{i}+\hat{\phi}_{i}^{*}\right) \label{eq:psi-degis}\end{aligned}$$ where $\sigma_{\mu}$ are the Pauli matrices. In summary, the MUSY transformations take the following form, immediately after taking the Hermitian conjugate of Eq. (\[eq:psi-degis\]): $$\begin{aligned} \begin{array}{rcl} \delta\phi_{i} & = & \bar{\zeta}_{i}^{a}\epsilon\psi_{a}\\ \delta\hat{\phi}_{i} & = & \zeta_{ia}\bar{\epsilon}\bar{\psi}^{a}\\ \delta\psi_{a} & = & -i\zeta_{ia}\sigma^{\mu}\bar{\epsilon}\partial_{\mu}\left(\phi_{i}+\hat{\phi}_{i }^{*}\right) \\ \delta\bar{\psi}^{a} & = & i\bar{\zeta}_{i}^{a}\epsilon\sigma^{\mu}\partial_{\mu}\left(\phi_{i}^{*}+\hat{ \phi}_{i}\right) \end{array}\end{aligned}$$ where the spinor indices are suppressed. This transformation indeed looks like SUSY transformation of Wess-Zumino model, only with an additional colour transformation. Effective MUSY Lagrangian\[sec:effective-lagrangian\] ----------------------------------------------------- The indices of fermionic preons $\psi_{a}$, and scalars $\phi_{i}$ and $\hat{\phi_{i}}$ do not indicate colour and family *a priori*. Instead, when the gauge bosons emerges with an approximate gauge symmetry from the current-current interactions as discussed in [@suzuki], preons will have colour or flavour *post-priori*. The simplest effective Lagrangian containing free fields and current-current interactions at preonic level can be written as follows (again suppressing the family and colour indices): $$\begin{aligned} \mathcal{L}_{preon} & = & -\partial^{\mu}\Phi^{\dagger}\partial_{\mu}\Phi +i\psi^{\dagger}\bar{\sigma}^{\mu}\partial_{\mu}\psi +i\bar{\psi}^{\dagger}\sigma^{\mu}\partial_{\mu}\bar{\psi}+f_{s}^{2}j_{m}^{\mu} j_{m\mu} +f_{W}^{2}J_{k}^{\mu}J_{k\mu}+f^{\prime2}J^{\mu}J_{\mu} + f_0^2 j^{\mu} j_{\mu} \label{eq:preonic-lagrangian}\end{aligned}$$ where $\Phi$ is a doublet consist of the scalar preons, and $f_{s}$, $f_0$, $f_{W}$ and $f^{\prime}$ are constants of dimension mass$^{-1}$, $j_{m}^{\mu}$, $j^\mu$, $J_{k}^{\mu}$ and $J^{\mu}$ are the currents of the preons which can be explicitly written as: $$\begin{aligned} \begin{array}{rcl} j_{m}^{\mu} & = & \psi^{\dagger}\lambda_{m}\bar{\sigma}^{\mu}\psi+\bar{\psi}^{\dagger}\lambda_{m} \sigma^{\mu}\bar{\psi} ,\\ j^{\mu} & = & \psi^{\dagger}\lambda_{m}\bar{\sigma}^{\mu}\psi+\bar{\psi}^{\dagger} \sigma^{\mu}\bar{\psi} ,\\ J_{k}^{\mu} & = & \Phi^{\dagger}\tau_{k}\partial^{\mu}\Phi+(\partial^{\mu}\Phi^{\dagger})\tau_{k} \Phi , \\ J^{\mu} & = & \Phi^{\dagger}\partial^{\mu}\Phi+(\partial^{\mu}\Phi^{\dagger})\Phi . \end{array}\end{aligned}$$ Here $m=1,\cdots,8$ and $k=1,2,3$ indicates the indices of the generators $\lambda_{m}$ and $\tau_{k}$ of groups $SU(3)$ and $SU(2)$, respectively. One could easily see that Eq. (\[eq:preonic-lagrangian\]) is a non-renormalizable Lagrangian including a current-current interaction and it is not invariant under MU transformations anymore. This non-invariance occurs due to the current-current interaction, and it could be cured by adding an auxiliary Lagrangian to $\mathcal{L}_{preon}$. The auxiliary Lagrangian could be written as: $$\begin{aligned} \mathcal{L}_{aux} & = & \left(m^{\prime}B^{\mu}-f^{\prime}J^{\mu}\right)\left(m^{\prime} B^{\mu}-f^{\prime}J^{\mu} \right)+\left(m_{W}W_{k}^{\mu}-f_{W}J_{k}^{\mu}\right)\left(m_{W}W_{k\mu}-f_{W} J_{k\mu}\right)\\ & + & \left(m_{s}G_{m}^{\mu}-f_{s}j_{m}^{\mu}\right)\left(m_{s}G_{m}^{\mu}-f_{s}j_{m}^ {\mu}\right) + \left(m_{0} G_0^{\mu}-f_{0} j^{\mu}\right) \left(m_{0}G_{0}^{\mu} - f_{0}j^ {\mu}\right) \nonumber \end{aligned}$$ where the new fields, $G_{m}^{\mu}$, $G_0^\mu$, $W_{k}^{\mu}$ and $B^{\mu}$ are the, so called, auxiliary fields. Then the two Lagrangians $\mathcal{L_{\mbox{preon}}}$ and $\mathcal{L}=\mathcal{L_{\mbox{preon}}}+\mathcal{L_{\mbox{aux}}}$ are equivalent since their partition functions are equal. A general approach to preonic models where currents act as approximate gauge fields, were investigated elsewhere [@suzuki]. When off-shell, the new fields give an extra degree of freedom and they should satisfy the MU invariance as these also transform under MU transformation. This requirement implies an additional term in the MU transformations of the preons as follows: $$\begin{aligned} \begin{array}{rcl} \delta\phi_{i} & = & \bar{\zeta}_{i}^{a}\epsilon\psi_{a}\\ \delta\hat{\phi}_{i} & = & \zeta_{ia}\bar{\epsilon}\bar{\psi}^{a}\\ \delta\psi_{a} & = & -i\zeta_{ia}\sigma^{\mu}\bar{\epsilon} D_{\mu}\left(\phi_{i}+\hat{\phi}_{i }^{*}\right)\label{eq:psi-degis-1}\\ \delta\bar{\psi}^{a} & = & i\bar{\zeta}_{i}^{a}\epsilon\sigma^{\mu} \left[ D_{\mu}\left(\phi_{i}+\hat{\phi}_{i }^{*}\right) \right]^{\dag} \end{array}\end{aligned}$$ where $D_\mu \Phi = \partial_\mu \Phi - i g^\prime Y^\prime B_\mu \Phi - i g_W T_k W_\mu^k \Phi $ is the covariant derivatives are presented via the occurance of the approximate gauge symmetry as motivated. Since auxiliary fields do not have kinetic terms, the variation of the total Lagrangian will give equations of motion of these fields as follows: $$\begin{aligned} \begin{array}{rcl} G_{m}^{\mu} & = & \left(f_{s}/m_{s}\right)\left(\psi^{\dagger}\lambda_{m}\bar{\sigma}^{\mu} \psi+\bar{\psi}^{\dagger}\lambda_{m}\sigma^{\mu}\bar{\psi}\right)\\ G_{0}^{\mu} & = & \left(f_{s}/m_{s}\right)\left(\psi^{\dagger}\bar{\sigma}^{\mu} \psi+\bar{\psi}^{\dagger}\sigma^{\mu}\bar{\psi}\right)\\ W_{k}^{\mu} & = & \left(f_{W}/m_{W}\right)\left(\Phi^{\dagger}\tau_{k}\partial^{\mu} \Phi+(\partial^{\mu}\Phi^{\dagger})\tau_{k}\Phi\right)\\ B^{\mu} & = & \left(f^{\prime}/m^{\prime}\right)\left(\Phi^{\dagger}\partial^{\mu} \Phi+(\partial^{\mu}\Phi^{\dagger})\Phi\right) \end{array}\end{aligned}$$ The coupling constants and masses of bosons obtained from loop diagrams will give rise to an approximate $SU(3)\times SU(2)\times U(1)$ gauge symmetry, with an extra singlet, where all the gauge non-invariant parts are embedded into the mass term of the vector fields: $$\begin{aligned} \mathcal{L} & = & -\partial^{\mu}\Phi^{\dagger}\partial_{\mu}\Phi+i\psi^{\dagger}\bar{\sigma}^{\mu }\partial_{\mu}\psi+i\bar{\psi}^{\dagger}\sigma^{\mu}\partial_{\mu}\bar{\psi}+m^ {\prime2}B^{\mu}B_{\mu}+m_0^2 G_0^{\mu}G_{0\mu}\\ & + & m_{W}^{2}W_{k}^{\mu}W_{k\mu}+m_{s}^{2}G_{m}^{\mu}G_{m\mu}+g_{s}j_{m}^{\mu}G_{ m\mu}+g_{W}j_{k}^{\mu}W_{k\mu}+g_0 j_{0}^{\mu}G_{0\mu}\nonumber\end{aligned}$$ where $g_s$, $g_0$, $g^{\prime}$ and $g_W$ are dimensionless coupling constants satisfying $$\begin{aligned} \label{eq:coupling-constants} \begin{array}{rcl} g_{s} &=& f_{s}m_{s} \\ g_{0} &=& f_{0}m_{0} \\ g_{W} &=& f_{W}m_{W} \\ g^{\prime} &=& f^{\prime}m^{\prime} \end{array}\end{aligned}$$ The mass of the vector bosons $G^{\mu}$ and $G_0^\mu$ converges to zero as the corresponding constant $f$ diverges and the masses of weak bosons arise from the spontaneous symmetry breaking as will be discussed in the next section. The procedure for obtaining mass to composite vector bosons from loop diagrams has already been investigated in detail [@suzuki]. The corresponding representations for the composite vector bosons are presented in Table \[Tab:reps\]. Vector boson Constituent Representation -------------- ------------------------------ ---------------- -- -- -- $G_m^\mu$ $\bar\Psi \gamma \Psi$ **8** $G_0^\mu$ $\bar\Psi \gamma \Psi$ **1** $W_k^\mu$ $\Phi^\dagger \partial \Phi$ **3** $B^\mu$ $\Phi^\dagger \partial \Phi$ **1** : Composite SM vector bosons \[Tab:reps\] Standard Model Particles in MUSY Model\[sec:SMparticles\] ========================================================= In this section quarks, leptons and gauge bosons will be defined in the flavour basis. The corresponding states in the mass basis, resulting in CKM mixings, are discussed in the next section. The SM fermions are proposed as doublets made from fermionic and bosonic preons. Similarly, SM gauge bosons are assumed to be matrices acting on those doublets. Standard Model fermions in MUSY model ------------------------------------- The SM fermions are in the form: $$\begin{aligned} Q_{i} & = & \left(\begin{array}{c} u\\ d\end{array}\right),\ \left(\begin{array}{c} c\\ s\end{array}\right),\ \left(\begin{array}{c} t\\ b\end{array}\right),\\ \bar{u}_{i} & = & \bar{u},\ \bar{c},\ \bar{t},\\ \bar{d}_{i} & = & \bar{d},\ \bar{s},\ \bar{b},\\ L_{i} & = & \left(\begin{array}{c} \nu_{e}\\ e\end{array}\right),\ \left(\begin{array}{c} \nu_{\mu}\\ \mu\end{array}\right),\ \left(\begin{array}{c} \nu_{\tau}\\ \tau\end{array}\right),\\ \bar{e}_{i} & = & \bar{e},\ \bar{\mu},\ \bar{\tau},\\ \bar{\nu}_{i} & = & \bar{\nu}_{e},\ \bar{\nu}_{\mu},\ \bar{\nu}_{\tau},\end{aligned}$$ where $Q_i$ and $L_i$ are weak isospin doublets and the remaining ones are singlets. In order to construct SM fermions in terms of MUSY, one can write the SM particles as composite states of preons. A quark is then considered as a composite state made of a fermionic preon and a scalar preon. Thus, it has colour and fractional electric charge as it should. The preonic scalar is to be postulated as $\phi_{i}$ for up-type and $\hat{\phi}_{i}$ for down-type quarks where $i=1,2,3$ denotes the quark family index. Similarly, up and down type antiquarks are in the form $\bar{\psi}\phi_{i}^{*}$ and $\bar{\psi}\hat{\phi}_{i}^{*}$, respectively. The full preon content of all left and right handed quark states can be found in Table \[Tab:quarks-musy\]. Note that we have written the MU numbers of composite particles by summing MU numbers of the preons. Quark (left) Q C MU$_{L}$ Quark (right) MU$_{R}$ ------------------------ -------- ------- ---------- ---------------------------------------- ---------- $u=\psi\phi_{1}$ $2/3$ **3** 1 $\bar{u}=\bar{\psi}\phi_{1}^{*}$ 0 $d=\psi\hat{\phi}_{1}$ $-1/3$ **3** 0 $\bar{d}=\bar{\psi}\hat{\phi}_{1}^{*}$ -1 $c=\psi\phi_{2}$ $2/3$ **3** 1 $\bar{c}=\bar{\psi}\phi_{2}^{*}$ 0 $s=\psi\hat{\phi}_{2}$ $-1/3$ **3** 0 $\bar{s}=\bar{\psi}\hat{\phi}_{2}^{*}$ -1 $t=\psi\phi_{3}$ $2/3$ **3** 1 $\bar{t}=\bar{\psi}\phi_{3}^{*}$ 0 $b=\psi\hat{\phi}_{3}$ $-1/3$ **3** 0 $\bar{b}=\bar{\psi}\hat{\phi}_{3}^{*}$ -1 : SM quarks according to the MUSY model in flavour basis \[Tab:quarks-musy\] Leptons are thought to be composed of three fermionic preons and one bosonic preon as listed in Table \[tab:leptons\_musy\]. To keep compatibility with the SM spin 1/2 particles, one fermionic preon will be opposite handed with respect to the other two. In terms of the colour structure, the state counting gives $\mathbf{3}\otimes\mathbf{3}\otimes\mathbf{3}=\mathbf{1}\oplus\mathbf{8} \oplus\mathbf{8}\oplus\mathbf{10}$ states. The colour singlet state is taken to yield the SM leptons. The remaining states will be discussed later in Section \[sec:Oghuz\]. The masses of the SM fermions could arise from the quantum mass corrections from the four-point interaction of the fermionic preons, similar to the mass term in $\phi^4$-model [@ryder]. Fermionic component of the bound state resulting in SM particles could also help acquiring mass through (approximate) guage interactions even if the fermionic preon does not interact directly with the scalar one [@Martin]. In general, the correction to the squared mass of the scalar preon has two leading terms one from self-interaction and another one from an indirect interaction with the fermionic preon. However, since the self-interaction term does not exist due to the absence of the Yukawa coupling as MU symmetry requires, the correction to the squared mass of the scalar preon would be as follows: $$\begin{aligned} \Delta M_{\phi}^{2} & = & -\frac{g_s^{2}+g_0^{2}}{16\pi^{2}}\left[\Lambda^{2}-24 M_{\psi}^{ 2}\ln\left(\Lambda/M_{\psi}\right)+\cdots\right]\end{aligned}$$ where $\Lambda$ is the cut-off scale and it gives contribution even if the bare mass $M_{\psi}$ is set to zero. The mass terms will be finite since $f_s$ and $f_0$ in Eq. (\[eq:coupling-constants\]) are expected to be in the same order of magnitude. At higher energies, the contribution from the fermionic preons is suppressed since there aren’t any direct interactions between fermionic and scalar preons. However, the masses are still finite since $f^{2}$ and $1/\Lambda^{2}$ terms are expected to be at the same order of magnitude. This can be understood by considering the preonic four-fermion interaction constant $f^2$ as an analogy to the Fermi constant $G_F = g^2_W / M_W^2$ which is in the same order of magnitude with $1/\Lambda_{\mbox{weak}}^2$ where $\Lambda_{\mbox{weak}} \approx M_W$ is the cut-off scale of the Fermi theory. According to the discussion above, the masses of the composite states could be obtained without having an hierarchy problem and furthermore the composite states would acquire their masses with respect to the chiral symmetry breaking of the effective action similar to the baryons in chiral perturbation theory [@Ecker:1994gg]. Left handed states Q C MU$_{L}$ Right handed states MU$_{R}$ ----------------------------------------------------- ---- ------- ---------- --------------------------------------------------------------------- ---------- $\nu_{e}=\psi_{R}\bar{\psi}_{G}\psi_{B}\phi_{1}$ 0 **1** 1 $\bar{\nu}_{e}=\bar{\psi}_{R}\psi_{G}\bar{\psi}_{B}\phi_{1}^{*}$ 0 $e=\psi_{R}\bar{\psi}_{G}\psi_{B}\hat{\phi}_{1}$ -1 **1** 0 $\bar{e}=\bar{\psi}_{R}\psi_{G}\bar{\psi}_{B}\hat{\phi}_{1}^{*}$ -1 $\nu_{\mu}=\psi_{R}\bar{\psi}_{G}\psi_{B}\phi_{2}$ 0 **1** 1 $\bar{\nu}_{\mu}=\bar{\psi}_{R}\psi_{G}\bar{\psi}_{B}\phi_{2}^{*}$ 0 $\mu=\psi_{R}\bar{\psi}_{G}\psi_{B}\hat{\phi}_{2}$ -1 **1** 0 $\bar{\mu}=\bar{\psi}_{R}\psi_{G}\bar{\psi}_{B}\hat{\phi}_{2}^{*}$ -1 $\nu_{\tau}=\psi_{R}\bar{\psi}_{G}\psi_{B}\phi_{3}$ 0 **1** 1 $\bar{\nu}_{\tau}=\bar{\psi}_{R}\psi_{G}\bar{\psi}_{B}\phi_{3}^{*}$ 0 $\tau=\psi_{R}\bar{\psi}_{G}\psi_{B}\hat{\phi}_{3}$ -1 **1** 0 $\bar{\tau}=\bar{\psi}_{R}\psi_{G}\bar{\psi}_{B}\hat{\phi}_{3}^{*}$ -1 : SM leptons according to the MUSY model in Flavour Basis \[tab:leptons\_musy\] Standard Model gauge bosons in MUSY model ----------------------------------------- At the preonic scale, MUSY model does not have fundamental gauge interactions at all. This means that a preon does not decay into another preon. Since SM consists of the gauge group $SU(3)\times SU(2)\times U(1)$, MUSY model should produce those gauge bosons, out of the preons in the model. One way of forming the gauge bosons, similar to the construction of the scalar bosons in Technicolour models, is to form a fermionic bound state. However, one could easily see that fermionic composite gauge bosons would have to be flavour singlets since the flavour is originated only from bosonic preons in MUSY model. Another way is current-current interactions becoming current-gauge interactions of SM fermions and thus all terms that are not invariant under global gauge transformations transferred into boson mass [@suzuki]. Fermionic (left and right) and bosonic current definitions are repeated here for the reader’s convenience: $$\begin{aligned} \begin{array}{rcl} G_{m}^{\mu} & = & g_{s}\left(\psi^{\dagger}\lambda_{m}\bar{\sigma}^{\mu}\psi+\bar{\psi}^{\dagger} \lambda_{m}\sigma^{\mu}\bar{\psi}\right)\\ G_{m}^{\mu} & = & g_{0}\left(\psi^{\dagger} \bar{\sigma}^{\mu}\psi+\bar{\psi}^{\dagger} \lambda_{m}\sigma^{\mu}\bar{\psi}\right)\\ W_{k}^{\mu} & = & g_{W}\left(\Phi^{\dagger}\tau_{k}\partial^{\mu}\Phi+(\partial^{\mu}\Phi^{\dagger })\tau_{k}\Phi\right)\\ B^{\mu} & = & g^{\prime}\left(\Phi^{\dagger}\partial^{\mu}\Phi+(\partial^{\mu}\Phi^{\dagger} )\Phi\right) \end{array}\end{aligned}$$ where the fields $\psi$, $\bar{\psi}$ and $\Phi$ are multiplets of the groups generated by $\lambda_{m}$ and $\tau_{n}$, respectively. ### Gluons The fermionic preon current gives a colour-changing vector gauge boson which is nothing but the gluon. Since the multiplets in the current are colour triplets, the gauge group is nothing but the $SU(3)_{\text{colour}}$. Therefore, gluons are defined proportional to the colour-changing current which is unique in the model: $$G_{m}^{\mu}\sim\bar{\Psi}\gamma^{\mu}\Psi$$ where $\lambda_{m}$ are the Gell-Man matrices for $m=1,\ \cdots,\ 8$ and $\bar{\Psi}$ refers to a four-component fermion taking $\psi$ and $\bar{\psi}$ as left and right handed colour triplet fermions, respectively. Table \[tab:gluons\_in\_musy\] contains a summary of the preonic contents of the gluons according to the MUSY model. The mass of the gluons emerges as approximately zero since the four-point interaction of the fermionic preons is infinitely strong, i.e., $m_{s}\rightarrow0$ as $f_{s}\rightarrow\infty$ since $g_{s}$ is known and finite from the experiments. It is crucial to have a sufficiently large $f_s^2$ because the preonic four-fermion interaction is expected to have a very large cut-off scale with respect to the scalar current-current interactions. This cut-off scale is the cut-off scale of the effective Lagrangian discussed in Section \[sec:effective-lagrangian\]. The same approach is applied to the field $G_0^\mu$ which is a colour singlet and interacts only with fermionic preons. --------------------------------------------------------------------------------------------- Gluon Preonic Contents --------------- ----------------------------------------------------------------------------- $G_{1}^{\mu}$ $\frac{1}{\sqrt{2}}\left(\bar{\Psi}^{R}\gamma^{\mu}\Psi_{B}+\bar{\Psi}^{B} \gamma^{\mu}\Psi_{R}\right)$ $G_{2}^{\mu}$ $\frac{-i}{\sqrt{2}}\left(\bar{\Psi}^{R}\gamma^{\mu}\Psi_{B}-\bar{\Psi}^{B} \gamma^{\mu}\Psi_{R}\right)$ $G_{3}^{\mu}$ $\frac{1}{\sqrt{2}}\left(\bar{\Psi}^{R}\gamma^{\mu}\Psi_{R}-\bar{\Psi}^{B} \gamma^{\mu}\Psi_{B}\right)$ $G_{4}^{\mu}$ $\frac{1}{\sqrt{2}}\left(\bar{\Psi}^{R}\gamma^{\mu}\Psi_{G}+\bar{\Psi}^{G} \gamma^{\mu}\Psi_{R}\right)$ $G_{5}^{\mu}$ $\frac{-i}{\sqrt{2}}\left(\bar{\Psi}^{R}\gamma^{\mu}\Psi_{G}-\bar{\Psi}^{G} \gamma^{\mu}\Psi_{R}\right)$ $G_{6}^{\mu}$ $\frac{1}{\sqrt{2}}\left(\bar{\Psi}^{B}\gamma^{\mu}\Psi_{G}+\bar{\Psi}^{G} \gamma^{\mu}\Psi_{B}\right)$ $G_{7}^{\mu}$ $\frac{i}{\sqrt{2}}\left(\bar{\Psi}^{B}\gamma^{\mu}\Psi_{G}+\bar{\Psi}^{G} \gamma^{\mu}\Psi_{B}\right)$ $G_{8}^{\mu}$ $\frac{1}{\sqrt{6}}\left(\bar{\Psi}^{R}\gamma^{\mu}\Psi_{R}+\bar{\Psi}^{G} \gamma^{\mu}\Psi_{G}-2\bar{\Psi}^{B}\gamma^{\mu}\Psi_{B}\right)$ $G_{0}^{\mu}$ $\left(\bar{\Psi}^{R}\gamma^{\mu}\Psi_{R}+\bar{\Psi}^{G} \gamma^{\mu}\Psi_{G}+\bar{\Psi}^{B}\gamma^{\mu}\Psi_{B}\right)$ --------------------------------------------------------------------------------------------- : SM Gluons according to the MUSY model. Here the indices of fermionic preons refers R for red, B for blue and G for green. \[tab:gluons\_in\_musy\] ### Electroweak bosons and weak mixing angle The definition of the weak isotriplet and isosinglet fields emerge from the MUSY model as bosonic currents with $SU(2)\times U(1)$ generators since the only available preons are $\phi$ and $\hat{\phi}$, corresponding to up and down isospin states. The explicit form of the electroweak gauge bosons are shown in Table \[tab:EWbosons\]. -------------------- ---------------------------------------------------------------------------------- Fields Contents \[\] $W_{1}^{\mu}$ $\hat{\phi}^{\dagger}\partial^{\mu}\phi+\phi^{\dagger}\partial^{\mu}\hat{\phi} $ \[\] $W_{2}^{\mu}$ $i\left(\hat{\phi}^{\dagger}\partial^{\mu}\phi-\phi^{\dagger}\partial^{\mu}\hat{ \phi}\right)$ \[\] $W_{3}^{\mu}$ $\phi^{\dagger}\partial^{\mu}\phi-\hat{\phi}^{\dagger}\partial^{\mu}\hat{\phi} $ \[\] $B^{\mu}$ $\phi^{\dagger}\partial^{\mu}\phi+\hat{\phi}^{\dagger}\partial^{\mu}\hat{\phi} $ \[\] -------------------- ---------------------------------------------------------------------------------- : Electroweak Gauge Fields according to the MUSY model \[tab:EWbosons\] The explicit preon content of the weak and electromagnetic gauge bosons are given in Table \[tab:WZA\]. The masses of the weak bosons are obtained via the mixing of the weak currents. With the well-known mixing mechanism as introduced in [@FriztschM-1], it is possible to obtain a massless photon and masses of $W$ and $Z$ bosons compatible with the experimental results. Following the electroweak model, the physical states of charged gauge bosons ($W^{+}$ and $W^{-}$) can be defined as: $$\begin{aligned} W^{\pm\mu} & = & (W_{1}^{\mu}\mp iW_{2}^{\mu})/\sqrt{2}\end{aligned}$$ and the neutral bosons ($Z^{0}$ and photon) are defined through the weak mixing angle: $$\begin{aligned} \label{eq:mixSM} \begin{array}{rcl} Z^{\mu} & = & \cos\theta_{W}W_{3}^{\mu}-\sin\theta_{W}B^{\mu}\,, \\ A^{\mu} & = & \sin\theta_{W}B^{\mu}+\cos\theta_{W}W_{3}^{\mu}\,. \end{array}\end{aligned}$$ Before the electroweak mixing, the masses of $W_1$, $W_2$ and $W_{3}$ are equal as discussed in Ref. [@FriztschM-1]. However, when $W^{3}$ and $B$ are mixed to form the $Z$ boson and the physical photon as in Eq. (\[eq:mixSM\]), the contribution to the mass of $Z$ boson will be related to the decay constant $F_{3}$ to the constituents in the vector boson, which reads as follows: $$\begin{aligned} \langle0|\ J^{3\mu}\ |W^{3}\rangle & = & \langle0|\ \phi^{\dagger}\partial^{\mu}\phi-\hat{\phi}^{\dagger}\partial^{\mu}\hat{\phi}\ |W^{3}\rangle\\ & = & iM_{w} F_{3} p^{\mu}\end{aligned}$$ where $M_{W}$ is the mass of the $W^{3}$ boson (before the contribution), $p^{\mu}$ is the momentum. Therefore, the mixing parameter (as in [@FriztschM-1]) becomes $\lambda=\sqrt{g/e}\ \sin\theta_{W}=gM_{W}/F_{3}$. Therefore the relation between the masses of the $Z$ and $W$ bosons can be written as: $$\begin{aligned} M_{Z}^{2} & = & M_{W}^{2}/(1-\lambda^{2}) \, .\end{aligned}$$ This mechanism is an anology of the mixing of the vector mesons of QCD and obtaining their masses. -------------------- ---------------------------------------------------------------------------------- Fields Contents \[\] $W_{\mu}^{+}$ $\hat{\phi}_{j}^{\dagger}\partial_{\mu}\phi_{i}$ $W_{\mu}^{-}$ $\phi_{i}^{\dagger}\partial_{\mu}\hat{\phi}_{j}$ $Z_{\mu}$ $\frac{1}{\sqrt{2}}\left(g_{1}\phi_{i}^{\dagger}\partial_{\mu}\phi_{i}-g_{2}\hat {\phi}_{j}^{\dagger}\partial_{\mu}\hat{\phi}_{j}\right)$ $A_{\mu}$ $\frac{1}{\sqrt{2}}\left(g_{2}\phi_{i}^{\dagger}\partial_{\mu}\phi_{i}+g_{1}\hat {\phi}_{j}^{\dagger}\partial_{\mu}\hat{\phi}_{j}\right)$ -------------------- ---------------------------------------------------------------------------------- : Weak and Electromagnetic Gauge Fields according to the MUSY model where $g_1=\cos \theta_W + \sin \theta_W$ and $g_2 = \cos \theta_W - \sin \theta_W$. \[tab:WZA\] SM Phenomenology within MUSY \[sec:fenomenoloji\] ================================================= Baryon and Lepton number conservation ------------------------------------- Let $\Delta_{\psi}$ indicate the difference between the number of fermionic preons and fermionic anti-preons, and similarly $\Delta_{\phi}$ the difference between the number of scalars and anti-scalars. Since a lepton includes three fermionic preons and one scalar preon, lepton number is defined as: $$\begin{aligned} L & = & N_{\ell}-N_{\bar{\ell}}=\frac{\Delta\psi-\Delta\phi}{2}\,.\end{aligned}$$ As for quarks, there is one fermionic preon for each scalar. Thus, one can define baryon number as follows: $$\begin{aligned} B & = & \frac{N_{q}-N_{\bar{q}}}{3}=\frac{3\Delta_{\phi}-\Delta_{\psi}}{6}\,.\end{aligned}$$ Since, there is no decay between fermionic preons and scalars in MUSY model, these quantum numbers are always conserved. Hence, baryon number violations such as proton decays is not allowed in MUSY model. Beta decay ---------- Remembering the MUSY content of the quarks as $u=\psi\phi_{1}$ and $d=\psi\hat{\phi}_{1}$ where the fermions are of the same colour, the decay can be written as $$\begin{aligned} \psi\hat{\phi}_{1} & \rightarrow & \psi\phi_{1}+W^{-}\,.\end{aligned}$$ It is already introduced that the charged weak boson is originating from the current of scalar preons leading to an interaction of current and vector boson just as in SM. The decay would consist of four fermion and four scalar vertices at the preonic level as shown in the Figure \[fig:SMbeta\]. ![Beta Decay: on the left according to the SM and on the right according to MUSY model \[fig:SMbeta\]](beta2 "fig:") $\quad$![Beta Decay: on the left according to the SM and on the right according to MUSY model \[fig:SMbeta\]](beta_decay "fig:") CKM and MNS mixing ------------------ From some decay processes, it is known that quarks are observed in mixed flavour while leptons remain in flavour basis. One could suggest that this is because of the nature of quarks but not leptons. However, MUSY model gives a different mixing mechanism which is originated from the MUSY parameter $\zeta$ and the mixing is generated by the weak bosons. This can be seen immediately by writing the low-energy approximation of the effective MUSY Lagrangian since the operators obtained would include $\zeta_{ia}$ parameter as Noether current in Eq. (\[eq:noether\_current\]) arises. In order to have the mixing correct which the values would be obtained from experiments, the leptonic vertex would have $\zeta^3$ where the deviation from the unitarity is in the order of MNS mixing. Predictions of the MUSY model\[sec:Predictions-of-MUSY\] ======================================================== Excited fermions ---------------- As previously discussed, SM fermions are constructed as bound states of preons: one scalar and one fermionic preon for quarks and one scalar and three fermionic preons for leptons. Therefore, the MUSY model contains a rich phenomenology due to the orbital or spin excitations of the SM fermions. For example, for each quark and lepton one expects to have a multitude of orbital excitations, denoted as $q^{(n)}$ and $\ell^{(n)}$. In the lepton sector, an additional possibility are the spin excitations of the from $\ell_{3/2}\equiv\psi\psi\psi\phi$, giving spin 3/2 particles and naturally their orbital excitations as well. So, each SM lepton has two colour-octet and one colour-decouplet partners. New matter particles: The Oghuz \[sec:Oghuz\] --------------------------------------------- As discussed in the construction of the SM leptons, three fermionic preons yield a singlet, two octet and one decouplet in terms of group representations of colours. It has been stated that the singlet state corresponds to SM leptons, we therefore propose the remaining objects as dark matter candidates with an appropriate helicity state to give a spin 1/2 particle because there occurs chargeless and colour-singlet fermions (like a spin-3/2 neutrino). Left handed states Q C S ---------------------------- ------ ------------------------------------------------------------------- ----- $\bar{\psi}\psi\phi$ 1/3 **$\mathbf{\bar{3}\oplus}\mathbf{6}$** 0 $\bar{\psi}\psi\hat{\phi}$ -2/3 **$\mathbf{\bar{3}\oplus}\mathbf{6}$** 0 $\psi\psi\phi$ 1/3 **$\mathbf{\bar{3}\oplus}\mathbf{6}$** 1 $\psi\psi\hat{\phi}$ -2/3 **$\mathbf{\bar{3}\oplus}\mathbf{6}$** 1 $\psi\psi\psi\phi$ 0 **$\mathbf{1}\oplus\mathbf{8}\oplus\mathbf{8}\oplus\mathbf{10}$** 3/2 $\psi\psi\psi\hat{\phi}$ -1 **$\mathbf{1}\oplus\mathbf{8}\oplus\mathbf{8}\oplus\mathbf{10}$** 3/2 : BSM particle content of the MUSY model \[tab:oghuz\] Possible bound states other than SM particles are in the form given in Table \[tab:oghuz\] with their charge, colour and spin properties. These bound states are split into two subgroups which do not interact with each other: a group of bicoloured bosonic (vector or scalar) particles and another group of fermionic particles with 3/2 spin. A suitable name these new particles could be “Oghuz”[^2], and the two sub-groups could be called as “Bozoklar” and “Üçoklar” respectively. The Bozoklar interact within the group via four point interactions consist of either fermionic preons and bosonic anti-preons or fermionic anti-preons and bosonic preons. Therefore these “particles” can not interact with any of the SM bosons which consist of either all preons or all anti-preons and can be though as the Dark Matter (DM) candidates of the MUSY model. Additionally, these DM candidates form a group of particles with two fermionic preons and another group with either one or three fermionic preons. Discussion and conclusion ========================= A new symmetry, MU symmetry, including spin, colour and charge was introduced. It was shown that the MU symmetry is a generalization of the well known supersymmetry concept and in fact, reduces to SUSY in a specific scenario. Using MU symmetry, and starting from a single chiral colour triplet preon, the electroweak and QCD was built. The MUSY model proposes SM fermions and gauge bosons as the bound states of scalar and fermionic preons. It also predicts the number of families as three, relating it to the number of colours in strong interactions. Other naturally occurring features of the model are the inclusion of the lepton and baryon number conservation, the existence of the quark mixings and new fermions which do not interact with the SM fermions. These new fermions, named the Oghuz, could be the Dark Matter candidates. The higher excitations of the SM bound states, are also expected to be present in Nature and are to be sought at the present and future colliders. With an effective Lagrangian the MUSY model proposes the gauge bosons of the SM as current-current interactions. It is also contains vanishingly small gluon and photon masses due to infinitely strong four-point interaction of the fermionic preons. The electroweak sector remains very similar to the SM, including the weak mixing angle, arising from four-point scalar preon interactions. Although the effective Lagrangian approach does provide a working model, it would have been more elegant to embed all preonic states in a large enough gauge group similar to GUT models, i.e., a gauge interaction between preons an extension of MUSY model. The applicability of this idea is yet to be investigated. Acknowledgements {#acknowledgements .unnumbered} ---------------- The authors would like to thank D. A. Demir, A. Havare and V. E. Ozcan for useful discussions. [15]{} K. Nakamura et al. (Particle Data Group), J. Phys. G.37, 075021 (2010). G. Ross, Grand Unified Theories, Westview Press 1984. C.T. Hill and E. H. Simmons, Phys. Rep. 381, 235 (2003). I. A. D’Souza and C. S. Kalman, Preons, World Scientific, 1992. J. Wess and J. Bagger, Supersymmetry and Supergravity, Princeton Univ. Press, 1991. Harari, H, Phys. Lett. 86B, 83 (1979). Shupe, M. Phys. Lett. 86B, 87 (1979). A. J. Buchmann and M. L. Schmid, Physical Review D 71, 055002, 2005. H. Fritzsch and G. Mandelbaum, Phys. Lett. 102B, 319, 1981. J.-J. Dugne, S. Fredriksson, J. Hansson, and E. Predazzi, hep-ph/9802339, 1998. A. Celikel, M. Kantar, S. Sultansoy, Phys. Lett. 443B, 35 (1998). S. Ishida and M. Sekiguchi, Progress of Theoretical Physics, [**86**]{}, No. 2, 1991. S. P. Martin, A Supersymmetry Primer, arXiv:hep-ph/9709306. J. Wess and B. Zumino, Nucl. Phys. B70, 39 (1974). A. Iorio, L. O’Raifeartaigh and S. Wolf, Annals Phys.  [**290**]{}, 156 (2001) \[hep-th/0008196\]. K. G. Wilson, Phys. Rev.  [**179**]{}, 1499 (1969). L. H. Ryder, Cambridge, Uk: Univ. Pr. ( 1985) 443p G. Ecker, Prog. Part. Nucl. Phys.  [**35**]{}, 1 (1995) \[hep-ph/9501357\]; A. E. C. Hernandez, arXiv:1108.0115 \[hep-ph\]. M. Suzuki, Phys. Rev. D82, 045026 (2010). H. Fritzsch and G. Mandelbaum, Phys. Lett. 109B, 224, 1982. [^1]: The word “mu” means “mystery, yet unknown, hidden” in Old Turkic (and in some Altaic languages). Today the word is living in Modern Turkish as interrogative. [^2]: The name Oghuz is derived from the Turkish word “ok”, which means arrow or tribe. The reason for calling these particles as Oghuz is simple: in Turkish history, the Oghuz tribes were split into two fractions “Üçoklar” (Three Arrows) and “Bozoklar” (Grey Arrows). These two fractions would not interact one with another, just like the two sub-groups in the MUSY model.

--- abstract: 'We report the development of a magnetically driven Persistent Current Switch operated in a dilution refrigerator. We show that it can be safely used to charge a $60$ mH coil with $0.5$ A at $11$ mK, which heats up the dilution refrigerator to $60.5$ mK. Measurements at $4$ K on a $440$ coil reveal a residual resistance of $R \leqslant 3.3$ p${\ensuremath{\Omega}}$.' author: - Bob van Waarde - Olaf Benningshof - Tjerk Oosterkamp title: A Magnetic Persistent Current Switch at milliKelvin Temperatures --- Introduction ============ There are many applications in which a stable, low noise magnetic field is desired, such as in MRI, NMR or qubit studies. An elegant way of establishing such a magnetic field is by the use of a Persistent Current Switch (PCS). In a PCS a superconducting coil is shunted by a superconducting shortcut such that together they form a closed resistanceless circuit. The coil can be charged by briefly switching the shortcut to the resistive state. The circuit then becomes an $RL$-circuit, and a power source can be connected to inject a current into the coil. Back in the superconducting state, the flowing current is in principle stable and low-noise, and, hence, so too is the magnetic field induced by the current in the coil. The switching of the shortcut can be accomplished in a number of ways: one can heat the shortcut to above its critical temperature, as is most often done for large magnets, or create a magnetic field higher than the shortcut’s critical field[@AmeenWiederhold1964; @HagedornDullenkopf1974august; @NotoEtAl1995; @NotoEtAl1996; @GotoEtAl1999], or even mechanically interrupt it[@TsudaEtAl2000; @TomitaEtAl2002]. We do SQUID-based experiments on the mixing chamber stage of a dilution refrigerator cryostat, which means that aside from being sensitive to magnetic noise, we are also concerned about heat input. It is our wish to place a PCS close to our experiment on the mixing chamber stage such that the wiring between PCS and experiment can be kept short — shorter wires are less susceptible to noise and also mechanically less vulnerable — while keeping the heat input manageable. The magnetic PCS presented in this paper provides us with a solution that suits our needs. Design and Fabrication ====================== We set out to design a PCS that: 1. Can be operated in a dilution refrigerator, i.e. an environment with very little cooling power on the order of 1 . 2. Introduces only a low (ideally zero) resistance such that it may be used with small coils of and still yield a long lifetime $\tau$. 3. Can be charged with a user-adjustable current in the range of $0.01$ A $-$ $1$ A. 4. Has reasonable dimensions, preferably smaller than $5$ cm. To our knowledge, a PCS that combines all of these features has not yet been constructed. In a magnetic PCS we distinguish two coils. The experiment coil, which is to be charged with a current $I_{exp}$ and used to perform the experiment of choice, and the switch coil with current $I_{sw}$, which is used to bring a superconducting shortcut to the normal state. Figure \[electrical\_scheme\] gives a schematic overview. ![\[electrical\_scheme\] In black the basic electrical circuit for making a magnetic field with a coil L[$_{\text{exp}}$]{} and a current source. Added are a that is brought to its resistive state by a . The response is measured by a . The contains the parts that are at low temperature inside a cryostat.](mPCS.pdf) We fabricate experiment coils from $100$ diameter NbTi wire with a $13$ Formvar insulating layer. We know the inductances of our experiment coils by integrating the voltage response upon charging them: $$\label{Vcharge} \int V(t) dt = L_{exp} \Delta I_{exp}.$$ The experiment coils discussed in this paper have inductances $L_{exp} = 60$ mH and $L_{exp} = 440$ . For convenience, we charge the experiment coil from empty $I_{exp} = 0$ A to $I_{exp} = I_0$ in a single step and we power up the switch coil as fast as our setup allows for. The energy dissipated in the shortcut while charging then totals $\frac{1}{2} L_{exp} I_0^2$ as we will derive. Dissipation can also come from other sources. There is Ohmic dissipation in the current lines towards the coils, which we find to be negligible if care is taken to use superconducting wiring from $4$ K to the mixing chamber stage of the dilution refrigerator and if care is taken to thermalize the wiring well. Further, the rapid change in magnetic field in both the experiment coil and the switch coil may cause dissipation due to eddy currents in normal metals in their vicinity. The superconducting shortcut is made of an insulated Niobium wire, $50$ in diameter. We twist the wire around itself, such that the mutual inductance to the switch coil is minimized, thereby minimizing the noise input through this channel. We have chosen to make the shortcut out of Niobium because of its high critical current density and critical temperature and its relatively low critical field. This means that at the currents we intend to use, quasiparticle dynamics are of no concern, and that tests in liquid Helium are possible. All superconducting connections are made by spot welding[@PhillipEtAl1995] the wires to Niobium sheets of $100$ thickness. In order to get a good superconducting connection we strip the wires of their Formvar insulation with a knife and clean them with IPA. We clean the Niobium sheets by sanding them lightly with sand paper and then wiping them with IPA. We spot weld the NbTi wires to the Nb sheets with $20-25$ Watt-seconds and the thinner Nb shortcut wire with $8-10$ Watt-seconds. We have also tried to laser weld the connections, but this resulted in higher contact resistances as well as less mechanical stability. The switch coil should be able to deliver at least $B_{c2} = 400$ mT, the upper critical field of Niobium at $0$ K[@FinnemoreEtAl966]. We construct it from $100$ diameter Copper clad single core NbTi wire, $62$ diameter NbTi core, with a $13$ insulating Formvar layer. The spindle on which the switch coil is wound is made from PEI and allows for a coil with an inner diameter of $9$ mm, an outer diameter of $22$ mm and a length of $18$ mm. In the center of the spindle we leave a hole of $3$ mm diameter through which the shortcut is put. For sturdiness, we give the spindle $4$ mm thick walls on either side of the coil and apply a layer of Stycast 2850FT to the whole after winding. We were able to put $N=8672$ windings on the spindle. The inductance of the switch coil is about $L_{sw} = 0.5$ H. The NbTi wire leads of the switch coil are twisted and led from the mixing chamber stage of the cryostat, where the switch coil is mounted, uninterrupted to the $4$ K stage, thermalized at the intermediary stages on Copper bobbins. From $4$ K to room temperature, the wires are from Copper. We should avoid sending too high currents, because these could cause dissipation in the non-superconducting parts of the wiring. We aimed for a switching current $I_{sw}$ on the order of $1$ A, at which the switch coil makes a field of $$B_{sw} = \frac{\mu_0 N I_{sw}}{l} = 600 \text{mT} ~ > ~ B_{c2} .$$ Results and Discussion ====================== We placed the PCS with a $60$ mH experiment coil on the mixing chamber stage of a dilution refrigerator which reached a minimum temperature of $10.5$ mK. After some tweaking, we found that we could charge and discharge the experiment coil without dangerously warming up the cryostat by using a switch current of $I_{sw} = 2$ A during $2$ seconds. Figure \[fits\] shows a typical voltage response for the case that we charge the experiment coil from $0$ A to $0.5$ A. If we left the switch activated for longer than $\sim 10$ s, we saw a sudden, vigorous increase in temperature. To avoid having to recondense into the dilution refrigerator every time we switch, we therefore limit ourselves to quick switches. Alternatively, one can choose to charge the experiment coil at a higher temperature when there is more cooling power available and cool further down afterwards. As we will see, the superconducting contacts have such a low resistance that there is ample time before the stored current is appreciably diminished, making this a viable possibility. However, we wanted to verify that it is possible to charge a coil even at the lowest temperature attainable in our cryostat without introducing a worryingly high heat input. ![\[fits\] The voltage response of a $60$ mH coil at $T = 11$ mK when charged with $0.5$ A using $I_{sw} = 2$ A. The shape of $V(t)$ is determined by the time-dependent normal state resistance $R_{pcs}(t)$ of the shortcut; assuming it be exponential, eq. , we find $R_{ns} = 1.4$ ${\ensuremath{\Omega}}$ and $\tau_{ns} = 73$ ms.](10mK_pcspeak_fit_exp.pdf) The shape of $V(t)$ can be understood by assuming that the normal state resistance of the superconducting shortcut grows exponentially to its final value $R_{ns}$ with a time constant $\tau_{ns}$ upon switching[@AmeenWiederhold1964] $$\label{Rpcs_t} R_{pcs}(t) = R_{ns} ( 1 - e^{-t/\tau_{ns}} ) .$$ The values of $R_{ns}$ and $\tau_{ns}$ depend on a multitude of factors including the switching field $B_{sw}$ ($\propto I_{sw}$), temperature and cooling power. The general solution for the PCS’s voltage response with time-varying $R_{pcs}(t)$ when charged $I_{exp} = 0\text{ A} \to I_0$ is given by $$\label{PCS_V_t} V(t) = I_0 R_{pcs}(t) ~ e^{- \int_0^t \frac{R_{pcs}(t')}{L_{exp}} dt'}.$$ Fitting this to the measurement, using the exponential $R_{pcs}(t)$ of eq. , we find that $R_{ns} = 1.4$ ${\ensuremath{\Omega}}$ and $\tau_{ns} = 73$ ms. The energy that is dissipated in the shortcut due to (dis)charging of the experiment coil is $$E_{diss}(t) = \int^t_0 P_{diss}(t') dt' = \int^t_0 \frac{V^2(t')}{R_{pcs}(t')} dt'.$$ For large $t$, it is easy to show that $E_{diss}(t)$ converges to $\frac{1}{2} L_{exp} I_0^2$: the energy that needs to be dissipated in the shortcut is exactly the energy that is stored in the experiment coil. This contribution to the dissipation can be made smaller by charging the experiment coil in a number of steps rather than in a single step as is done now. However, in our experiment it is not the dominating factor. During (dis)charging of the coils, we monitor the temperature of the mixing chamber stage onto which the PCS is mounted. Figure \[temp\_peak\_dilfridge\] shows the temperature of the mixing chamber stage as a function of time during the switch of figure \[fits\]. At the moment of switching, the temperature quickly increases to $60.5$ mK and gradually decreases again afterwards. Our calibration of the cooling power versus temperature $P_{cool}(T)$ allows us to convert the temperature to power dissipation, and, consequently, integration yields the dissipated energy in the temperature peak. We estimate $E_{diss} \approx 100$ mJ, see the inset in figure \[temp\_peak\_dilfridge\]. ![\[temp\_peak\_dilfridge\] The temperature of the mixing chamber stage versus time, $\Delta t = 1$ minute. When charging the experiment coil with $0.5$ A, the temperature increases to $60.5$ mK. The inset shows the estimated dissipated energy in time, which levels off to $103$ mJ.](thermalpeakInset.pdf) Note that this is much more than the energy dissipated in the shortcut, $\frac{1}{2} L_{exp} I_{0}^2 = 7.5$ mJ. We attribute this to eddy currents: for the sake of thermalization our coils are securely fastened to the mixing chamber stage of the dilution refrigerator, which is a gold-coated copper disk of $1$ cm thickness. Being a normal metal, induced eddy currents are dissipated here and cause a heating that scales with $(dB/dt)^2$ [@Pobell]. A straightforward way of reducing this heat input would be to ramp the switch current up and down more gradually and to divide the charging of the experiment coil into several small steps. The eddy currents could be further reduced by keeping the switch and experiment coils away from normal metals or encasing them in a superconducting shield and use normal metal only to provide cooling to the coils. We measured the quality of the spot welded superconducting joints by evaluating the residual resistance $R$ in a long-lasting measurement. Because of the exceptionally low resistance of the joints, there is no measurable decrease in $I_{exp}$ even after a few days. We therefore placed a PCS with a $440$ experiment coil in a vacuum dipstick inserted in a liquid Helium dewar, charged it from $0$ A to $350$ mA, and left it untouched for a little under $17$ days before discharging. Figure \[long\_4K\_meas\] shows the two voltage responses. Fitting to equation gives $R_{ns} = 375$ m${\ensuremath{\Omega}}$ and $\tau_{ns} = 1.1$ ms. ![\[long\_4K\_meas\] The voltage responses at $4$ K when charging to $350$ mA, waiting for a time $t_W = 16.8$ days and then fully discharging. For the discharge voltage response we plot $-V(t)$ to emphasize how little current has been lost. We measure the current still flowing in the experiment coil to be $I_W = 346$ mA, therefore $\tau \geqslant 4.2$ years and $R \leqslant 3.3$ p${\ensuremath{\Omega}}$. ](chargedischarge4K.pdf) The integration of $V(t)$, eq. , when discharging $I_{exp} = I_W \to 0$ A yields the current still flowing in the experiment coil $I_W$ after waiting a time $t_W$. It is related to the injected current $I_0$ as $$\frac{I_W}{I_0} = e^{-t_W /\tau} , \quad \tau = \frac{L_{exp}}{R} .$$ The amount of current still flowing thus allows us to measure the residual resistance in the superconducting circuit $R$. We measure $I_W = 346$ mA, which translates to $\tau \geqslant 4.2$ years and $R \leqslant 3.3$ p${\ensuremath{\Omega}}$. Conclusions =========== We have shown that it is possible to put a $60$ mH coil in the persistent mode carrying a current of $0.5$ A in a dilution refrigerator at $11$ mK using a magnetic Persistent Current Switch. We find that spot welding all wires to each other via Nb sheets ensures a resistance of less than $3.3$ p${\ensuremath{\Omega}}$. Acknowledgements ================ The authors wish to thank J.J.T. Wagenaar and M. de Wit for helpful discussions and G. Koning and F. Schenkel for technical support. This research is part of the Single Phonon Nanomechanics project of the Dutch Foundation for Fundamental Research on Matter (FOM). [10]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{} , ****, (). , ****, (). , , , , , , ****, (). , , , , , , , , , , , ****, (). , , , , ****, (). , , , , , , , , ****, (). , , , , ****, (). , , , ****, (). , , , ****, (). , ** (, ), ed., .